LMFDB - Embedded newform 1075.2.a.o.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1075,2,Mod(1,1075)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1075, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1075.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1075 = 5^{2} \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1075.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$8.58391821729$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.24217.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 5x^{3} - x^{2} + 3x + 1 $ x^5 - 5x^3 - x^2 + 3x + 1
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			1.2
Root			$-0.722813$ of defining polynomial
Character	$\chi$	$=$	1075.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.722813 q^{2} -1.13013 q^{3} -1.47754 q^{4} +0.816870 q^{6} -0.754729 q^{7} +2.51361 q^{8} -1.72281 q^{9} +O(q^{10})$	\(q-0.722813 q^{2} -1.13013 q^{3} -1.47754 q^{4} +0.816870 q^{6} -0.754729 q^{7} +2.51361 q^{8} -1.72281 q^{9} -1.36316 q^{11} +1.66981 q^{12} +7.11818 q^{13} +0.545528 q^{14} +1.13821 q^{16} -2.27409 q^{17} +1.24527 q^{18} +3.30131 q^{19} +0.852940 q^{21} +0.985308 q^{22} +3.01498 q^{23} -2.84070 q^{24} -5.14511 q^{26} +5.33738 q^{27} +1.11514 q^{28} -9.17534 q^{29} -7.20959 q^{31} -5.84994 q^{32} +1.54054 q^{33} +1.64374 q^{34} +2.54553 q^{36} -0.480643 q^{37} -2.38623 q^{38} -8.04445 q^{39} +1.96029 q^{41} -0.616516 q^{42} +1.00000 q^{43} +2.01412 q^{44} -2.17927 q^{46} +5.74014 q^{47} -1.28633 q^{48} -6.43038 q^{49} +2.57001 q^{51} -10.5174 q^{52} -12.3818 q^{53} -3.85793 q^{54} -1.89710 q^{56} -3.73090 q^{57} +6.63205 q^{58} +6.62955 q^{59} +9.53738 q^{61} +5.21118 q^{62} +1.30026 q^{63} +1.95198 q^{64} -1.11352 q^{66} -8.25480 q^{67} +3.36006 q^{68} -3.40731 q^{69} -8.70502 q^{71} -4.33048 q^{72} -6.45527 q^{73} +0.347415 q^{74} -4.87782 q^{76} +1.02882 q^{77} +5.81463 q^{78} -16.0502 q^{79} -0.863478 q^{81} -1.41692 q^{82} -3.58875 q^{83} -1.26025 q^{84} -0.722813 q^{86} +10.3693 q^{87} -3.42645 q^{88} -12.4291 q^{89} -5.37230 q^{91} -4.45476 q^{92} +8.14776 q^{93} -4.14904 q^{94} +6.61117 q^{96} -3.08745 q^{97} +4.64796 q^{98} +2.34847 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 3 q^{6} + 3 q^{8} - 5 q^{9}+O(q^{10}) $ 5 * q - 3 * q^6 + 3 * q^8 - 5 * q^9	$ 5 q - 3 q^{6} + 3 q^{8} - 5 q^{9} - 9 q^{11} - 8 q^{12} + q^{13} - 7 q^{14} - 2 q^{16} - 3 q^{17} + 10 q^{18} - 11 q^{19} - 5 q^{21} - 13 q^{22} - 9 q^{24} - 5 q^{26} - 3 q^{27} + 15 q^{28} - 22 q^{29} - 5 q^{31} - 4 q^{32} + 12 q^{33} - 7 q^{34} + 3 q^{36} - 7 q^{37} + 3 q^{38} - 3 q^{39} - 21 q^{41} - 7 q^{42} + 5 q^{43} - 20 q^{44} + 13 q^{46} + 3 q^{47} + 6 q^{48} - 13 q^{49} - 5 q^{51} - 18 q^{52} + 5 q^{53} + 4 q^{54} + 4 q^{56} - 13 q^{57} + 16 q^{58} - 8 q^{59} - 16 q^{61} + 2 q^{62} - 7 q^{63} - 17 q^{64} + 19 q^{66} + 8 q^{67} + 7 q^{68} - 15 q^{69} - 10 q^{71} - 5 q^{72} - q^{73} - 3 q^{76} - 7 q^{77} + 25 q^{78} + 12 q^{79} - 15 q^{81} - 41 q^{82} - 20 q^{83} + 5 q^{84} + 22 q^{87} + 2 q^{88} - 36 q^{89} - 13 q^{91} + q^{92} + 13 q^{93} + 28 q^{96} + 8 q^{97} - 25 q^{98} - 4 q^{99}+O(q^{100}) $ 5 * q - 3 * q^6 + 3 * q^8 - 5 * q^9 - 9 * q^11 - 8 * q^12 + q^13 - 7 * q^14 - 2 * q^16 - 3 * q^17 + 10 * q^18 - 11 * q^19 - 5 * q^21 - 13 * q^22 - 9 * q^24 - 5 * q^26 - 3 * q^27 + 15 * q^28 - 22 * q^29 - 5 * q^31 - 4 * q^32 + 12 * q^33 - 7 * q^34 + 3 * q^36 - 7 * q^37 + 3 * q^38 - 3 * q^39 - 21 * q^41 - 7 * q^42 + 5 * q^43 - 20 * q^44 + 13 * q^46 + 3 * q^47 + 6 * q^48 - 13 * q^49 - 5 * q^51 - 18 * q^52 + 5 * q^53 + 4 * q^54 + 4 * q^56 - 13 * q^57 + 16 * q^58 - 8 * q^59 - 16 * q^61 + 2 * q^62 - 7 * q^63 - 17 * q^64 + 19 * q^66 + 8 * q^67 + 7 * q^68 - 15 * q^69 - 10 * q^71 - 5 * q^72 - q^73 - 3 * q^76 - 7 * q^77 + 25 * q^78 + 12 * q^79 - 15 * q^81 - 41 * q^82 - 20 * q^83 + 5 * q^84 + 22 * q^87 + 2 * q^88 - 36 * q^89 - 13 * q^91 + q^92 + 13 * q^93 + 28 * q^96 + 8 * q^97 - 25 * q^98 - 4 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−0.722813	−0.511106	−0.255553	−	0.966795i	$-0.582257\pi$
$2$	−0.722813	−0.511106	−0.255553	+	0.966795i	$0.582257\pi$
$3$	−1.13013	−0.652479	−0.326240	−	0.945287i	$-0.605782\pi$
$3$	−1.13013	−0.652479	−0.326240	+	0.945287i	$0.605782\pi$
$4$	−1.47754	−0.738771
$5$	0	0
$5$	0	0
$6$	0.816870	0.333486
$7$	−0.754729	−0.285261	−0.142630	−	0.989776i	$-0.545556\pi$
$7$	−0.754729	−0.285261	−0.142630	+	0.989776i	$0.545556\pi$
$8$	2.51361	0.888696
$9$	−1.72281	−0.574271
$10$	0	0
$11$	−1.36316	−0.411008	−0.205504	−	0.978656i	$-0.565883\pi$
$11$	−1.36316	−0.411008	−0.205504	+	0.978656i	$0.565883\pi$
$12$	1.66981	0.482033
$13$	7.11818	1.97423	0.987114	−	0.160019i	$-0.0511557\pi$
$13$	7.11818	1.97423	0.987114	+	0.160019i	$0.0511557\pi$
$14$	0.545528	0.145798
$15$	0	0
$16$	1.13821	0.284553
$17$	−2.27409	−0.551547	−0.275773	−	0.961223i	$-0.588934\pi$
$17$	−2.27409	−0.551547	−0.275773	+	0.961223i	$0.588934\pi$
$18$	1.24527	0.293513
$19$	3.30131	0.757372	0.378686	−	0.925525i	$-0.376376\pi$
$19$	3.30131	0.757372	0.378686	+	0.925525i	$0.376376\pi$
$20$	0	0
$21$	0.852940	0.186127
$22$	0.985308	0.210068
$23$	3.01498	0.628667	0.314334	−	0.949313i	$-0.398219\pi$
$23$	3.01498	0.628667	0.314334	+	0.949313i	$0.398219\pi$
$24$	−2.84070	−0.579856
$25$	0	0
$26$	−5.14511	−1.00904
$27$	5.33738	1.02718
$28$	1.11514	0.210742
$29$	−9.17534	−1.70382	−0.851908	−	0.523691i	$-0.824555\pi$
$29$	−9.17534	−1.70382	−0.851908	+	0.523691i	$0.824555\pi$
$30$	0	0
$31$	−7.20959	−1.29488	−0.647440	−	0.762116i	$-0.724161\pi$
$31$	−7.20959	−1.29488	−0.647440	+	0.762116i	$0.724161\pi$
$32$	−5.84994	−1.03413
$33$	1.54054	0.268174
$34$	1.64374	0.281899
$35$	0	0
$36$	2.54553	0.424255
$37$	−0.480643	−0.0790172	−0.0395086	−	0.999219i	$-0.512579\pi$
$37$	−0.480643	−0.0790172	−0.0395086	+	0.999219i	$0.512579\pi$
$38$	−2.38623	−0.387097
$39$	−8.04445	−1.28814
$40$	0	0
$41$	1.96029	0.306146	0.153073	−	0.988215i	$-0.451083\pi$
$41$	1.96029	0.306146	0.153073	+	0.988215i	$0.451083\pi$
$42$	−0.616516	−0.0951304
$43$	1.00000	0.152499
$43$	1.00000	0.152499
$44$	2.01412	0.303641
$45$	0	0
$46$	−2.17927	−0.321316
$47$	5.74014	0.837285	0.418642	−	0.908151i	$-0.362506\pi$
$47$	5.74014	0.837285	0.418642	+	0.908151i	$0.362506\pi$
$48$	−1.28633	−0.185665
$49$	−6.43038	−0.918626
$50$	0	0
$51$	2.57001	0.359873
$52$	−10.5174	−1.45850
$53$	−12.3818	−1.70077	−0.850387	−	0.526157i	$-0.823632\pi$
$53$	−12.3818	−1.70077	−0.850387	+	0.526157i	$0.823632\pi$
$54$	−3.85793	−0.524997
$55$	0	0
$56$	−1.89710	−0.253510
$57$	−3.73090	−0.494170
$58$	6.63205	0.870831
$59$	6.62955	0.863094	0.431547	−	0.902090i	$-0.357968\pi$
$59$	6.62955	0.863094	0.431547	+	0.902090i	$0.357968\pi$
$60$	0	0
$61$	9.53738	1.22114	0.610568	−	0.791964i	$-0.290941\pi$
$61$	9.53738	1.22114	0.610568	+	0.791964i	$0.290941\pi$
$62$	5.21118	0.661821
$63$	1.30026	0.163817
$64$	1.95198	0.243998
$65$	0	0
$66$	−1.11352	−0.137065
$67$	−8.25480	−1.00848	−0.504242	−	0.863562i	$-0.668228\pi$
$67$	−8.25480	−1.00848	−0.504242	+	0.863562i	$0.668228\pi$
$68$	3.36006	0.407467
$69$	−3.40731	−0.410192
$70$	0	0
$71$	−8.70502	−1.03310	−0.516548	−	0.856258i	$-0.672783\pi$
$71$	−8.70502	−1.03310	−0.516548	+	0.856258i	$0.672783\pi$
$72$	−4.33048	−0.510352
$73$	−6.45527	−0.755532	−0.377766	−	0.925901i	$-0.623308\pi$
$73$	−6.45527	−0.755532	−0.377766	+	0.925901i	$0.623308\pi$
$74$	0.347415	0.0403861
$75$	0	0
$76$	−4.87782	−0.559525
$77$	1.02882	0.117244
$78$	5.81463	0.658377
$79$	−16.0502	−1.80579	−0.902896	−	0.429859i	$-0.858563\pi$
$79$	−16.0502	−1.80579	−0.902896	+	0.429859i	$0.858563\pi$
$80$	0	0
$81$	−0.863478	−0.0959420
$82$	−1.41692	−0.156473
$83$	−3.58875	−0.393917	−0.196958	−	0.980412i	$-0.563106\pi$
$83$	−3.58875	−0.393917	−0.196958	+	0.980412i	$0.563106\pi$
$84$	−1.26025	−0.137505
$85$	0	0
$86$	−0.722813	−0.0779429
$87$	10.3693	1.11171
$88$	−3.42645	−0.365261
$89$	−12.4291	−1.31748	−0.658741	−	0.752370i	$-0.728911\pi$
$89$	−12.4291	−1.31748	−0.658741	+	0.752370i	$0.728911\pi$
$90$	0	0
$91$	−5.37230	−0.563170
$92$	−4.45476	−0.464441
$93$	8.14776	0.844883
$94$	−4.14904	−0.427941
$95$	0	0
$96$	6.61117	0.674750
$97$	−3.08745	−0.313483	−0.156742	−	0.987640i	$-0.550099\pi$
$97$	−3.08745	−0.313483	−0.156742	+	0.987640i	$0.550099\pi$
$98$	4.64796	0.469515
$99$	2.34847	0.236030
$100$	0	0
$101$	−8.69949	−0.865631	−0.432816	−	0.901482i	$-0.642480\pi$
$101$	−8.69949	−0.865631	−0.432816	+	0.901482i	$0.642480\pi$
$102$	−1.85763	−0.183933
$103$	9.64093	0.949949	0.474974	−	0.880000i	$-0.342457\pi$
$103$	9.64093	0.949949	0.474974	+	0.880000i	$0.342457\pi$
$104$	17.8923	1.75449
$105$	0	0
$106$	8.94974	0.869276
$107$	9.44527	0.913109	0.456554	−	0.889696i	$-0.349083\pi$
$107$	9.44527	0.913109	0.456554	+	0.889696i	$0.349083\pi$
$108$	−7.88620	−0.758850
$109$	−3.04096	−0.291271	−0.145635	−	0.989338i	$-0.546523\pi$
$109$	−3.04096	−0.291271	−0.145635	+	0.989338i	$0.546523\pi$
$110$	0	0
$111$	0.543187	0.0515571
$112$	−0.859043	−0.0811719
$113$	−17.1573	−1.61402	−0.807011	−	0.590537i	$-0.798916\pi$
$113$	−17.1573	−1.61402	−0.807011	+	0.590537i	$0.798916\pi$
$114$	2.69674	0.252573
$115$	0	0
$116$	13.5569	1.25873
$117$	−12.2633	−1.13374
$118$	−4.79192	−0.441132
$119$	1.71632	0.157335
$120$	0	0
$121$	−9.14180	−0.831073
$122$	−6.89374	−0.624130
$123$	−2.21538	−0.199754
$124$	10.6525	0.956620
$125$	0	0
$126$	−0.939842	−0.0837278
$127$	−13.8232	−1.22661	−0.613305	−	0.789846i	$-0.710160\pi$
$127$	−13.8232	−1.22661	−0.613305	+	0.789846i	$0.710160\pi$
$128$	10.2890	0.909424
$129$	−1.13013	−0.0995021
$130$	0	0
$131$	2.39345	0.209117	0.104559	−	0.994519i	$-0.466657\pi$
$131$	2.39345	0.209117	0.104559	+	0.994519i	$0.466657\pi$
$132$	−2.27622	−0.198119
$133$	−2.49159	−0.216049
$134$	5.96667	0.515442
$135$	0	0
$136$	−5.71617	−0.490157
$137$	−7.22243	−0.617054	−0.308527	−	0.951216i	$-0.599836\pi$
$137$	−7.22243	−0.617054	−0.308527	+	0.951216i	$0.599836\pi$
$138$	2.46285	0.209652
$139$	12.2069	1.03538	0.517688	−	0.855570i	$-0.326793\pi$
$139$	12.2069	1.03538	0.517688	+	0.855570i	$0.326793\pi$
$140$	0	0
$141$	−6.48708	−0.546311
$142$	6.29210	0.528021
$143$	−9.70321	−0.811423
$144$	−1.96093	−0.163411
$145$	0	0
$146$	4.66595	0.386157
$147$	7.26715	0.599385
$148$	0.710170	0.0583756
$149$	−1.17814	−0.0965174	−0.0482587	−	0.998835i	$-0.515367\pi$
$149$	−1.17814	−0.0965174	−0.0482587	+	0.998835i	$0.515367\pi$
$150$	0	0
$151$	1.64702	0.134033	0.0670164	−	0.997752i	$-0.478652\pi$
$151$	1.64702	0.134033	0.0670164	+	0.997752i	$0.478652\pi$
$152$	8.29821	0.673074
$153$	3.91782	0.316737
$154$	−0.743641	−0.0599243
$155$	0	0
$156$	11.8860	0.951642
$157$	10.2466	0.817769	0.408884	−	0.912586i	$-0.365918\pi$
$157$	10.2466	0.817769	0.408884	+	0.912586i	$0.365918\pi$
$158$	11.6013	0.922951
$159$	13.9930	1.10972
$160$	0	0
$161$	−2.27550	−0.179334
$162$	0.624133	0.0490365
$163$	5.73234	0.448992	0.224496	−	0.974475i	$-0.427926\pi$
$163$	5.73234	0.448992	0.224496	+	0.974475i	$0.427926\pi$
$164$	−2.89641	−0.226172
$165$	0	0
$166$	2.59400	0.201333
$167$	12.8464	0.994083	0.497041	−	0.867727i	$-0.334420\pi$
$167$	12.8464	0.994083	0.497041	+	0.867727i	$0.334420\pi$
$168$	2.14396	0.165410
$169$	37.6685	2.89758
$170$	0	0
$171$	−5.68754	−0.434937
$172$	−1.47754	−0.112662
$173$	−10.2649	−0.780426	−0.390213	−	0.920725i	$-0.627599\pi$
$173$	−10.2649	−0.780426	−0.390213	+	0.920725i	$0.627599\pi$
$174$	−7.49506	−0.568199
$175$	0	0
$176$	−1.55157	−0.116954
$177$	−7.49223	−0.563151
$178$	8.98391	0.673372
$179$	−8.54837	−0.638935	−0.319467	−	0.947597i	$-0.603504\pi$
$179$	−8.54837	−0.638935	−0.319467	+	0.947597i	$0.603504\pi$
$180$	0	0
$181$	−6.87941	−0.511343	−0.255671	−	0.966764i	$-0.582297\pi$
$181$	−6.87941	−0.511343	−0.255671	+	0.966764i	$0.582297\pi$
$182$	3.88316	0.287839
$183$	−10.7784	−0.796766
$184$	7.57850	0.558694
$185$	0	0
$186$	−5.88930	−0.431825
$187$	3.09994	0.226690
$188$	−8.48129	−0.618562
$189$	−4.02828	−0.293014
$190$	0	0
$191$	21.1212	1.52828	0.764139	−	0.645052i	$-0.223164\pi$
$191$	21.1212	1.52828	0.764139	+	0.645052i	$0.223164\pi$
$192$	−2.20599	−0.159204
$193$	5.46190	0.393156	0.196578	−	0.980488i	$-0.437017\pi$
$193$	5.46190	0.393156	0.196578	+	0.980488i	$0.437017\pi$
$194$	2.23165	0.160223
$195$	0	0
$196$	9.50116	0.678654
$197$	10.5839	0.754072	0.377036	−	0.926199i	$-0.376943\pi$
$197$	10.5839	0.754072	0.377036	+	0.926199i	$0.376943\pi$
$198$	−1.69750	−0.120636
$199$	25.5885	1.81392	0.906962	−	0.421213i	$-0.138396\pi$
$199$	25.5885	1.81392	0.906962	+	0.421213i	$0.138396\pi$
$200$	0	0
$201$	9.32897	0.658015
$202$	6.28810	0.442429
$203$	6.92489	0.486032
$204$	−3.79729	−0.265864
$205$	0	0
$206$	−6.96859	−0.485524
$207$	−5.19425	−0.361025
$208$	8.10201	0.561773
$209$	−4.50021	−0.311286
$210$	0	0
$211$	−6.06412	−0.417472	−0.208736	−	0.977972i	$-0.566935\pi$
$211$	−6.06412	−0.417472	−0.208736	+	0.977972i	$0.566935\pi$
$212$	18.2947	1.25648
$213$	9.83778	0.674074
$214$	−6.82716	−0.466695
$215$	0	0
$216$	13.4161	0.912850
$217$	5.44129	0.369379
$218$	2.19804	0.148870
$219$	7.29527	0.492969
$220$	0	0
$221$	−16.1874	−1.08888
$222$	−0.392623	−0.0263511
$223$	−23.9705	−1.60519	−0.802593	−	0.596527i	$-0.796547\pi$
$223$	−23.9705	−1.60519	−0.802593	+	0.596527i	$0.796547\pi$
$224$	4.41512	0.294998
$225$	0	0
$226$	12.4015	0.824936
$227$	−1.69021	−0.112183	−0.0560916	−	0.998426i	$-0.517864\pi$
$227$	−1.69021	−0.112183	−0.0560916	+	0.998426i	$0.517864\pi$
$228$	5.51256	0.365078
$229$	−11.7533	−0.776678	−0.388339	−	0.921517i	$-0.626951\pi$
$229$	−11.7533	−0.776678	−0.388339	+	0.921517i	$0.626951\pi$
$230$	0	0
$231$	−1.16269	−0.0764995
$232$	−23.0632	−1.51418
$233$	−28.1758	−1.84586	−0.922928	−	0.384973i	$-0.874211\pi$
$233$	−28.1758	−1.84586	−0.922928	+	0.384973i	$0.874211\pi$
$234$	8.86406	0.579462
$235$	0	0
$236$	−9.79544	−0.637629
$237$	18.1388	1.17824
$238$	−1.24058	−0.0804147
$239$	−23.6151	−1.52753	−0.763766	−	0.645493i	$-0.776652\pi$
$239$	−23.6151	−1.52753	−0.763766	+	0.645493i	$0.776652\pi$
$240$	0	0
$241$	9.15352	0.589630	0.294815	−	0.955554i	$-0.404742\pi$
$241$	9.15352	0.589630	0.294815	+	0.955554i	$0.404742\pi$
$242$	6.60781	0.424766
$243$	−15.0363	−0.964579
$244$	−14.0919	−0.902140
$245$	0	0
$246$	1.60130	0.102095
$247$	23.4993	1.49523
$248$	−18.1221	−1.15076
$249$	4.05575	0.257022
$250$	0	0
$251$	−15.0824	−0.951994	−0.475997	−	0.879447i	$-0.657913\pi$
$251$	−15.0824	−0.951994	−0.475997	+	0.879447i	$0.657913\pi$
$252$	−1.92118	−0.121023
$253$	−4.10990	−0.258387
$254$	9.99158	0.626928
$255$	0	0
$256$	−11.3410	−0.708810
$257$	−3.48315	−0.217273	−0.108636	−	0.994082i	$-0.534648\pi$
$257$	−3.48315	−0.217273	−0.108636	+	0.994082i	$0.534648\pi$
$258$	0.816870	0.0508561
$259$	0.362755	0.0225405
$260$	0	0
$261$	15.8074	0.978453
$262$	−1.73002	−0.106881
$263$	−10.1942	−0.628602	−0.314301	−	0.949323i	$-0.601770\pi$
$263$	−10.1942	−0.628602	−0.314301	+	0.949323i	$0.601770\pi$
$264$	3.87233	0.238325
$265$	0	0
$266$	1.80096	0.110424
$267$	14.0465	0.859629
$268$	12.1968	0.745039
$269$	3.01134	0.183605	0.0918024	−	0.995777i	$-0.470737\pi$
$269$	3.01134	0.183605	0.0918024	+	0.995777i	$0.470737\pi$
$270$	0	0
$271$	18.2703	1.10984	0.554922	−	0.831903i	$-0.312748\pi$
$271$	18.2703	1.10984	0.554922	+	0.831903i	$0.312748\pi$
$272$	−2.58840	−0.156945
$273$	6.07138	0.367457
$274$	5.22046	0.315380
$275$	0	0
$276$	5.03445	0.303038
$277$	16.7310	1.00527	0.502636	−	0.864498i	$-0.332364\pi$
$277$	16.7310	1.00527	0.502636	+	0.864498i	$0.332364\pi$
$278$	−8.82330	−0.529186
$279$	12.4208	0.743612
$280$	0	0
$281$	−32.0823	−1.91387	−0.956935	−	0.290301i	$-0.906245\pi$
$281$	−32.0823	−1.91387	−0.956935	+	0.290301i	$0.906245\pi$
$282$	4.68895	0.279223
$283$	6.28426	0.373561	0.186780	−	0.982402i	$-0.440195\pi$
$283$	6.28426	0.373561	0.186780	+	0.982402i	$0.440195\pi$
$284$	12.8620	0.763221
$285$	0	0
$286$	7.01360	0.414723
$287$	−1.47949	−0.0873314
$288$	10.0783	0.593872
$289$	−11.8285	−0.695796
$290$	0	0
$291$	3.48921	0.204541
$292$	9.53793	0.558165
$293$	2.19299	0.128116	0.0640579	−	0.997946i	$-0.479596\pi$
$293$	2.19299	0.128116	0.0640579	+	0.997946i	$0.479596\pi$
$294$	−5.25279	−0.306349
$295$	0	0
$296$	−1.20815	−0.0702222
$297$	−7.27569	−0.422179
$298$	0.851578	0.0493306
$299$	21.4612	1.24113
$300$	0	0
$301$	−0.754729	−0.0435019
$302$	−1.19049	−0.0685049
$303$	9.83152	0.564806
$304$	3.75759	0.215513
$305$	0	0
$306$	−2.83185	−0.161886
$307$	−3.85368	−0.219941	−0.109971	−	0.993935i	$-0.535076\pi$
$307$	−3.85368	−0.219941	−0.109971	+	0.993935i	$0.535076\pi$
$308$	−1.52012	−0.0866168
$309$	−10.8955	−0.619822
$310$	0	0
$311$	−4.99766	−0.283391	−0.141696	−	0.989910i	$-0.545255\pi$
$311$	−4.99766	−0.283391	−0.141696	+	0.989910i	$0.545255\pi$
$312$	−20.2206	−1.14477
$313$	9.53286	0.538829	0.269414	−	0.963024i	$-0.413170\pi$
$313$	9.53286	0.538829	0.269414	+	0.963024i	$0.413170\pi$
$314$	−7.40638	−0.417966
$315$	0	0
$316$	23.7149	1.33407
$317$	13.9190	0.781768	0.390884	−	0.920440i	$-0.372169\pi$
$317$	13.9190	0.781768	0.390884	+	0.920440i	$0.372169\pi$
$318$	−10.1143	−0.567184
$319$	12.5074	0.700282
$320$	0	0
$321$	−10.6744	−0.595784
$322$	1.64476	0.0916587
$323$	−7.50746	−0.417726
$324$	1.27583	0.0708792
$325$	0	0
$326$	−4.14341	−0.229482
$327$	3.43667	0.190048
$328$	4.92740	0.272070
$329$	−4.33225	−0.238845
$330$	0	0
$331$	−24.3757	−1.33981	−0.669904	−	0.742447i	$-0.733665\pi$
$331$	−24.3757	−1.33981	−0.669904	+	0.742447i	$0.733665\pi$
$332$	5.30253	0.291014
$333$	0.828057	0.0453773
$334$	−9.28553	−0.508081
$335$	0	0
$336$	0.970827	0.0529630
$337$	2.71275	0.147773	0.0738864	−	0.997267i	$-0.476460\pi$
$337$	2.71275	0.147773	0.0738864	+	0.997267i	$0.476460\pi$
$338$	−27.2273	−1.48097
$339$	19.3899	1.05312
$340$	0	0
$341$	9.82782	0.532206
$342$	4.11102	0.222299
$343$	10.1363	0.547309
$344$	2.51361	0.135525
$345$	0	0
$346$	7.41961	0.398880
$347$	11.1283	0.597397	0.298699	−	0.954347i	$-0.403447\pi$
$347$	11.1283	0.597397	0.298699	+	0.954347i	$0.403447\pi$
$348$	−15.3211	−0.821295
$349$	−14.5059	−0.776484	−0.388242	−	0.921557i	$-0.626918\pi$
$349$	−14.5059	−0.776484	−0.388242	+	0.921557i	$0.626918\pi$
$350$	0	0
$351$	37.9924	2.02789
$352$	7.97439	0.425037
$353$	24.7317	1.31634	0.658168	−	0.752871i	$-0.271332\pi$
$353$	24.7317	1.31634	0.658168	+	0.752871i	$0.271332\pi$
$354$	5.41548	0.287830
$355$	0	0
$356$	18.3645	0.973317
$357$	−1.93966	−0.102658
$358$	6.17887	0.326563
$359$	−30.9025	−1.63097	−0.815485	−	0.578778i	$-0.803530\pi$
$359$	−30.9025	−1.63097	−0.815485	+	0.578778i	$0.803530\pi$
$360$	0	0
$361$	−8.10136	−0.426387
$362$	4.97253	0.261350
$363$	10.3314	0.542258
$364$	7.93779	0.416053
$365$	0	0
$366$	7.79080	0.407232
$367$	6.83336	0.356699	0.178349	−	0.983967i	$-0.442924\pi$
$367$	6.83336	0.356699	0.178349	+	0.983967i	$0.442924\pi$
$368$	3.43169	0.178889
$369$	−3.37721	−0.175811
$370$	0	0
$371$	9.34492	0.485164
$372$	−12.0386	−0.624175
$373$	8.93469	0.462621	0.231310	−	0.972880i	$-0.425699\pi$
$373$	8.93469	0.462621	0.231310	+	0.972880i	$0.425699\pi$
$374$	−2.24068	−0.115863
$375$	0	0
$376$	14.4285	0.744092
$377$	−65.3117	−3.36372
$378$	2.91169	0.149761
$379$	10.4408	0.536307	0.268154	−	0.963376i	$-0.413587\pi$
$379$	10.4408	0.536307	0.268154	+	0.963376i	$0.413587\pi$
$380$	0	0
$381$	15.6220	0.800338
$382$	−15.2667	−0.781112
$383$	−3.20266	−0.163648	−0.0818242	−	0.996647i	$-0.526075\pi$
$383$	−3.20266	−0.163648	−0.0818242	+	0.996647i	$0.526075\pi$
$384$	−11.6278	−0.593380
$385$	0	0
$386$	−3.94793	−0.200944
$387$	−1.72281	−0.0875755
$388$	4.56184	0.231592
$389$	2.48235	0.125860	0.0629301	−	0.998018i	$-0.479955\pi$
$389$	2.48235	0.125860	0.0629301	+	0.998018i	$0.479955\pi$
$390$	0	0
$391$	−6.85633	−0.346740
$392$	−16.1635	−0.816379
$393$	−2.70491	−0.136445
$394$	−7.65018	−0.385411
$395$	0	0
$396$	−3.46996	−0.174372
$397$	9.67082	0.485364	0.242682	−	0.970106i	$-0.421973\pi$
$397$	9.67082	0.485364	0.242682	+	0.970106i	$0.421973\pi$
$398$	−18.4957	−0.927107
$399$	2.81582	0.140967
$400$	0	0
$401$	−6.40054	−0.319628	−0.159814	−	0.987147i	$-0.551089\pi$
$401$	−6.40054	−0.319628	−0.159814	+	0.987147i	$0.551089\pi$
$402$	−6.74310	−0.336315
$403$	−51.3192	−2.55639
$404$	12.8539	0.639503
$405$	0	0
$406$	−5.00540	−0.248414
$407$	0.655192	0.0324767
$408$	6.46000	0.319818
$409$	6.29968	0.311499	0.155750	−	0.987797i	$-0.450221\pi$
$409$	6.29968	0.311499	0.155750	+	0.987797i	$0.450221\pi$
$410$	0	0
$411$	8.16226	0.402615
$412$	−14.2449	−0.701795
$413$	−5.00351	−0.246207
$414$	3.75447	0.184522
$415$	0	0
$416$	−41.6409	−2.04161
$417$	−13.7953	−0.675561
$418$	3.25281	0.159100
$419$	−32.5423	−1.58980	−0.794899	−	0.606742i	$-0.792476\pi$
$419$	−32.5423	−1.58980	−0.794899	+	0.606742i	$0.792476\pi$
$420$	0	0
$421$	24.5612	1.19704	0.598520	−	0.801108i	$-0.295756\pi$
$421$	24.5612	1.19704	0.598520	+	0.801108i	$0.295756\pi$
$422$	4.38323	0.213372
$423$	−9.88918	−0.480828
$424$	−31.1231	−1.51147
$425$	0	0
$426$	−7.11087	−0.344523
$427$	−7.19814	−0.348342
$428$	−13.9558	−0.674578
$429$	10.9659	0.529437
$430$	0	0
$431$	−16.8372	−0.811019	−0.405509	−	0.914091i	$-0.632906\pi$
$431$	−16.8372	−0.811019	−0.405509	+	0.914091i	$0.632906\pi$
$432$	6.07508	0.292287
$433$	24.4883	1.17683	0.588417	−	0.808558i	$-0.299751\pi$
$433$	24.4883	1.17683	0.588417	+	0.808558i	$0.299751\pi$
$434$	−3.93303	−0.188792
$435$	0	0
$436$	4.49314	0.215182
$437$	9.95339	0.476135
$438$	−5.27311	−0.251959
$439$	18.7571	0.895228	0.447614	−	0.894227i	$-0.352274\pi$
$439$	18.7571	0.895228	0.447614	+	0.894227i	$0.352274\pi$
$440$	0	0
$441$	11.0783	0.527540
$442$	11.7004	0.556532
$443$	−4.69401	−0.223019	−0.111509	−	0.993763i	$-0.535569\pi$
$443$	−4.69401	−0.223019	−0.111509	+	0.993763i	$0.535569\pi$
$444$	−0.802582	−0.0380889
$445$	0	0
$446$	17.3262	0.820419
$447$	1.33145	0.0629756
$448$	−1.47322	−0.0696030
$449$	−11.6165	−0.548217	−0.274108	−	0.961699i	$-0.588383\pi$
$449$	−11.6165	−0.548217	−0.274108	+	0.961699i	$0.588383\pi$
$450$	0	0
$451$	−2.67218	−0.125828
$452$	25.3506	1.19239
$453$	−1.86135	−0.0874536
$454$	1.22170	0.0573375
$455$	0	0
$456$	−9.37803	−0.439166
$457$	−19.1207	−0.894428	−0.447214	−	0.894427i	$-0.647584\pi$
$457$	−19.1207	−0.894428	−0.447214	+	0.894427i	$0.647584\pi$
$458$	8.49541	0.396965
$459$	−12.1377	−0.566537
$460$	0	0
$461$	−37.0695	−1.72650	−0.863250	−	0.504776i	$-0.831575\pi$
$461$	−37.0695	−1.72650	−0.863250	+	0.504776i	$0.831575\pi$
$462$	0.840409	0.0390994
$463$	18.9470	0.880544	0.440272	−	0.897865i	$-0.354882\pi$
$463$	18.9470	0.880544	0.440272	+	0.897865i	$0.354882\pi$
$464$	−10.4435	−0.484827
$465$	0	0
$466$	20.3658	0.943427
$467$	−22.5354	−1.04281	−0.521407	−	0.853308i	$-0.674593\pi$
$467$	−22.5354	−1.04281	−0.521407	+	0.853308i	$0.674593\pi$
$468$	18.1195	0.837575
$469$	6.23014	0.287681
$470$	0	0
$471$	−11.5800	−0.533577
$472$	16.6641	0.767028
$473$	−1.36316	−0.0626781
$474$	−13.1110	−0.602206
$475$	0	0
$476$	−2.53593	−0.116234
$477$	21.3316	0.976705
$478$	17.0693	0.780731
$479$	20.8173	0.951169	0.475584	−	0.879670i	$-0.342237\pi$
$479$	20.8173	0.951169	0.475584	+	0.879670i	$0.342237\pi$
$480$	0	0
$481$	−3.42130	−0.155998
$482$	−6.61628	−0.301363
$483$	2.57160	0.117012
$484$	13.5074	0.613972
$485$	0	0
$486$	10.8684	0.493002
$487$	−27.4719	−1.24487	−0.622435	−	0.782671i	$-0.713857\pi$
$487$	−27.4719	−1.24487	−0.622435	+	0.782671i	$0.713857\pi$
$488$	23.9733	1.08522
$489$	−6.47827	−0.292958
$490$	0	0
$491$	−19.5158	−0.880737	−0.440369	−	0.897817i	$-0.645152\pi$
$491$	−19.5158	−0.880737	−0.440369	+	0.897817i	$0.645152\pi$
$492$	3.27331	0.147572
$493$	20.8655	0.939735
$494$	−16.9856	−0.764218
$495$	0	0
$496$	−8.20605	−0.368463
$497$	6.56993	0.294702
$498$	−2.93155	−0.131366
$499$	11.7368	0.525409	0.262705	−	0.964876i	$-0.415386\pi$
$499$	11.7368	0.525409	0.262705	+	0.964876i	$0.415386\pi$
$500$	0	0
$501$	−14.5180	−0.648618
$502$	10.9018	0.486570
$503$	23.2118	1.03496	0.517482	−	0.855694i	$-0.326869\pi$
$503$	23.2118	1.03496	0.517482	+	0.855694i	$0.326869\pi$
$504$	3.26834	0.145583
$505$	0	0
$506$	2.97069	0.132063
$507$	−42.5702	−1.89061
$508$	20.4244	0.906184
$509$	2.32207	0.102924	0.0514620	−	0.998675i	$-0.483612\pi$
$509$	2.32207	0.102924	0.0514620	+	0.998675i	$0.483612\pi$
$510$	0	0
$511$	4.87198	0.215524
$512$	−12.3805	−0.547147
$513$	17.6203	0.777957
$514$	2.51767	0.111049
$515$	0	0
$516$	1.66981	0.0735093
$517$	−7.82472	−0.344131
$518$	−0.262204	−0.0115206
$519$	11.6007	0.509212
$520$	0	0
$521$	−15.6692	−0.686481	−0.343241	−	0.939248i	$-0.611525\pi$
$521$	−15.6692	−0.686481	−0.343241	+	0.939248i	$0.611525\pi$
$522$	−11.4258	−0.500093
$523$	−14.5045	−0.634237	−0.317118	−	0.948386i	$-0.602715\pi$
$523$	−14.5045	−0.634237	−0.317118	+	0.948386i	$0.602715\pi$
$524$	−3.53643	−0.154490
$525$	0	0
$526$	7.36851	0.321282
$527$	16.3952	0.714188
$528$	1.75347	0.0763098
$529$	−13.9099	−0.604777
$530$	0	0
$531$	−11.4215	−0.495650
$532$	3.68143	0.159610
$533$	13.9537	0.604401
$534$	−10.1530	−0.439361
$535$	0	0
$536$	−20.7494	−0.896236
$537$	9.66074	0.416892
$538$	−2.17664	−0.0938415
$539$	8.76563	0.377563
$540$	0	0
$541$	−32.1733	−1.38324	−0.691619	−	0.722263i	$-0.743102\pi$
$541$	−32.1733	−1.38324	−0.691619	+	0.722263i	$0.743102\pi$
$542$	−13.2060	−0.567247
$543$	7.77461	0.333641
$544$	13.3033	0.570373
$545$	0	0
$546$	−4.38847	−0.187809
$547$	−0.314657	−0.0134538	−0.00672688	−	0.999977i	$-0.502141\pi$
$547$	−0.314657	−0.0134538	−0.00672688	+	0.999977i	$0.502141\pi$
$548$	10.6714	0.455861
$549$	−16.4311	−0.701263
$550$	0	0
$551$	−30.2906	−1.29042
$552$	−8.56466	−0.364536
$553$	12.1136	0.515122
$554$	−12.0934	−0.513800
$555$	0	0
$556$	−18.0362	−0.764905
$557$	−19.3532	−0.820020	−0.410010	−	0.912081i	$-0.634475\pi$
$557$	−19.3532	−0.820020	−0.410010	+	0.912081i	$0.634475\pi$
$558$	−8.97790	−0.380065
$559$	7.11818	0.301067
$560$	0	0
$561$	−3.50333	−0.147911
$562$	23.1895	0.978190
$563$	32.6648	1.37666	0.688328	−	0.725399i	$-0.258345\pi$
$563$	32.6648	1.37666	0.688328	+	0.725399i	$0.258345\pi$
$564$	9.58494	0.403599
$565$	0	0
$566$	−4.54235	−0.190929
$567$	0.651692	0.0273685
$568$	−21.8810	−0.918108
$569$	−33.8596	−1.41947	−0.709735	−	0.704469i	$-0.751185\pi$
$569$	−33.8596	−1.41947	−0.709735	+	0.704469i	$0.751185\pi$
$570$	0	0
$571$	−21.3458	−0.893295	−0.446647	−	0.894710i	$-0.647382\pi$
$571$	−21.3458	−0.893295	−0.446647	+	0.894710i	$0.647382\pi$
$572$	14.3369	0.599456
$573$	−23.8697	−0.997170
$574$	1.06939	0.0446356
$575$	0	0
$576$	−3.36290	−0.140121
$577$	−28.6602	−1.19314	−0.596569	−	0.802562i	$-0.703470\pi$
$577$	−28.6602	−1.19314	−0.596569	+	0.802562i	$0.703470\pi$
$578$	8.54981	0.355625
$579$	−6.17264	−0.256526
$580$	0	0
$581$	2.70854	0.112369
$582$	−2.52205	−0.104542
$583$	16.8784	0.699032
$584$	−16.2260	−0.671438
$585$	0	0
$586$	−1.58512	−0.0654807
$587$	48.1469	1.98723	0.993617	−	0.112805i	$-0.0359835\pi$
$587$	48.1469	1.98723	0.993617	+	0.112805i	$0.0359835\pi$
$588$	−10.7375	−0.442808
$589$	−23.8011	−0.980707
$590$	0	0
$591$	−11.9612	−0.492017
$592$	−0.547074	−0.0224846
$593$	−17.6142	−0.723330	−0.361665	−	0.932308i	$-0.617792\pi$
$593$	−17.6142	−0.723330	−0.361665	+	0.932308i	$0.617792\pi$
$594$	5.25896	0.215778
$595$	0	0
$596$	1.74076	0.0713042
$597$	−28.9183	−1.18355
$598$	−15.5124	−0.634350
$599$	−5.38379	−0.219976	−0.109988	−	0.993933i	$-0.535081\pi$
$599$	−5.38379	−0.219976	−0.109988	+	0.993933i	$0.535081\pi$
$600$	0	0
$601$	38.1849	1.55759	0.778797	−	0.627276i	$-0.215830\pi$
$601$	38.1849	1.55759	0.778797	+	0.627276i	$0.215830\pi$
$602$	0.545528	0.0222341
$603$	14.2215	0.579143
$604$	−2.43355	−0.0990195
$605$	0	0
$606$	−7.10635	−0.288676
$607$	−38.9139	−1.57947	−0.789734	−	0.613450i	$-0.789781\pi$
$607$	−38.9139	−1.57947	−0.789734	+	0.613450i	$0.789781\pi$
$608$	−19.3125	−0.783223
$609$	−7.82601	−0.317126
$610$	0	0
$611$	40.8593	1.65299
$612$	−5.78875	−0.233996
$613$	−43.6651	−1.76362	−0.881808	−	0.471608i	$-0.843674\pi$
$613$	−43.6651	−1.76362	−0.881808	+	0.471608i	$0.843674\pi$
$614$	2.78549	0.112413
$615$	0	0
$616$	2.58604	0.104195
$617$	41.8500	1.68482	0.842409	−	0.538839i	$-0.181137\pi$
$617$	41.8500	1.68482	0.842409	+	0.538839i	$0.181137\pi$
$618$	7.87539	0.316795
$619$	−4.71748	−0.189611	−0.0948057	−	0.995496i	$-0.530223\pi$
$619$	−4.71748	−0.189611	−0.0948057	+	0.995496i	$0.530223\pi$
$620$	0	0
$621$	16.0921	0.645754
$622$	3.61237	0.144843
$623$	9.38060	0.375826
$624$	−9.15630	−0.366545
$625$	0	0
$626$	−6.89047	−0.275399
$627$	5.08581	0.203108
$628$	−15.1398	−0.604144
$629$	1.09302	0.0435817
$630$	0	0
$631$	39.8839	1.58775	0.793876	−	0.608080i	$-0.208060\pi$
$631$	39.8839	1.58775	0.793876	+	0.608080i	$0.208060\pi$
$632$	−40.3440	−1.60480
$633$	6.85323	0.272391
$634$	−10.0608	−0.399566
$635$	0	0
$636$	−20.6753	−0.819829
$637$	−45.7726	−1.81358
$638$	−9.04053	−0.357918
$639$	14.9971	0.593277
$640$	0	0
$641$	21.7445	0.858856	0.429428	−	0.903101i	$-0.358715\pi$
$641$	21.7445	0.858856	0.429428	+	0.903101i	$0.358715\pi$
$642$	7.71556	0.304509
$643$	5.60270	0.220949	0.110474	−	0.993879i	$-0.464763\pi$
$643$	5.60270	0.220949	0.110474	+	0.993879i	$0.464763\pi$
$644$	3.36214	0.132487
$645$	0	0
$646$	5.42649	0.213502
$647$	21.5558	0.847446	0.423723	−	0.905792i	$-0.360723\pi$
$647$	21.5558	0.847446	0.423723	+	0.905792i	$0.360723\pi$
$648$	−2.17045	−0.0852633
$649$	−9.03713	−0.354738
$650$	0	0
$651$	−6.14935	−0.241012
$652$	−8.46977	−0.331702
$653$	23.1930	0.907611	0.453806	−	0.891101i	$-0.350066\pi$
$653$	23.1930	0.907611	0.453806	+	0.891101i	$0.350066\pi$
$654$	−2.48407	−0.0971347
$655$	0	0
$656$	2.23123	0.0871148
$657$	11.1212	0.433880
$658$	3.13140	0.122075
$659$	−16.7203	−0.651332	−0.325666	−	0.945485i	$-0.605588\pi$
$659$	−16.7203	−0.651332	−0.325666	+	0.945485i	$0.605588\pi$
$660$	0	0
$661$	−19.5757	−0.761408	−0.380704	−	0.924697i	$-0.624318\pi$
$661$	−19.5757	−0.761408	−0.380704	+	0.924697i	$0.624318\pi$
$662$	17.6191	0.684784
$663$	18.2938	0.710471
$664$	−9.02073	−0.350072
$665$	0	0
$666$	−0.598530	−0.0231926
$667$	−27.6635	−1.07113
$668$	−18.9811	−0.734399
$669$	27.0897	1.04735
$670$	0	0
$671$	−13.0010	−0.501896
$672$	−4.98964	−0.192480
$673$	9.29531	0.358308	0.179154	−	0.983821i	$-0.442664\pi$
$673$	9.29531	0.358308	0.179154	+	0.983821i	$0.442664\pi$
$674$	−1.96081	−0.0755276
$675$	0	0
$676$	−55.6567	−2.14064
$677$	−6.72027	−0.258281	−0.129141	−	0.991626i	$-0.541222\pi$
$677$	−6.72027	−0.258281	−0.129141	+	0.991626i	$0.541222\pi$
$678$	−14.0153	−0.538253
$679$	2.33019	0.0894245
$680$	0	0
$681$	1.91015	0.0731972
$682$	−7.10367	−0.272014
$683$	4.20342	0.160840	0.0804198	−	0.996761i	$-0.474374\pi$
$683$	4.20342	0.160840	0.0804198	+	0.996761i	$0.474374\pi$
$684$	8.40357	0.321319
$685$	0	0
$686$	−7.32665	−0.279733
$687$	13.2827	0.506766
$688$	1.13821	0.0433940
$689$	−88.1361	−3.35772
$690$	0	0
$691$	52.1283	1.98306	0.991528	−	0.129896i	$-0.0414645\pi$
$691$	52.1283	1.98306	0.991528	+	0.129896i	$0.0414645\pi$
$692$	15.1668	0.576556
$693$	−1.77246	−0.0673300
$694$	−8.04366	−0.305333
$695$	0	0
$696$	26.0644	0.987968
$697$	−4.45787	−0.168854
$698$	10.4851	0.396866
$699$	31.8422	1.20438
$700$	0	0
$701$	−4.56564	−0.172442	−0.0862210	−	0.996276i	$-0.527479\pi$
$701$	−4.56564	−0.172442	−0.0862210	+	0.996276i	$0.527479\pi$
$702$	−27.4614	−1.03646
$703$	−1.58675	−0.0598454
$704$	−2.66086	−0.100285
$705$	0	0
$706$	−17.8764	−0.672787
$707$	6.56575	0.246931
$708$	11.0701	0.416039
$709$	−37.3798	−1.40383	−0.701914	−	0.712261i	$-0.747671\pi$
$709$	−37.3798	−1.40383	−0.701914	+	0.712261i	$0.747671\pi$
$710$	0	0
$711$	27.6515	1.03701
$712$	−31.2419	−1.17084
$713$	−21.7368	−0.814050
$714$	1.40201	0.0524689
$715$	0	0
$716$	12.6306	0.472026
$717$	26.6880	0.996683
$718$	22.3367	0.833598
$719$	39.7114	1.48099	0.740493	−	0.672064i	$-0.234592\pi$
$719$	39.7114	1.48099	0.740493	+	0.672064i	$0.234592\pi$
$720$	0	0
$721$	−7.27629	−0.270983
$722$	5.85576	0.217929
$723$	−10.3446	−0.384721
$724$	10.1646	0.377765
$725$	0	0
$726$	−7.46766	−0.277151
$727$	39.8444	1.47775	0.738874	−	0.673844i	$-0.235358\pi$
$727$	39.8444	1.47775	0.738874	+	0.673844i	$0.235358\pi$
$728$	−13.5039	−0.500487
$729$	19.5834	0.725310
$730$	0	0
$731$	−2.27409	−0.0841101
$732$	15.9256	0.588627
$733$	−47.1495	−1.74150	−0.870752	−	0.491722i	$-0.836368\pi$
$733$	−47.1495	−1.74150	−0.870752	+	0.491722i	$0.836368\pi$
$734$	−4.93924	−0.182311
$735$	0	0
$736$	−17.6375	−0.650126
$737$	11.2526	0.414495
$738$	2.44109	0.0898578
$739$	15.7982	0.581147	0.290574	−	0.956853i	$-0.406154\pi$
$739$	15.7982	0.581147	0.290574	+	0.956853i	$0.406154\pi$
$740$	0	0
$741$	−26.5572	−0.975603
$742$	−6.75463	−0.247970
$743$	33.2757	1.22077	0.610384	−	0.792106i	$-0.291015\pi$
$743$	33.2757	1.22077	0.610384	+	0.792106i	$0.291015\pi$
$744$	20.4803	0.750844
$745$	0	0
$746$	−6.45811	−0.236448
$747$	6.18275	0.226215
$748$	−4.58029	−0.167472
$749$	−7.12862	−0.260474
$750$	0	0
$751$	44.8868	1.63794	0.818972	−	0.573834i	$-0.194544\pi$
$751$	44.8868	1.63794	0.818972	+	0.573834i	$0.194544\pi$
$752$	6.53350	0.238252
$753$	17.0451	0.621157
$754$	47.2081	1.71922
$755$	0	0
$756$	5.95194	0.216470
$757$	−0.201966	−0.00734058	−0.00367029	−	0.999993i	$-0.501168\pi$
$757$	−0.201966	−0.00734058	−0.00367029	+	0.999993i	$0.501168\pi$
$758$	−7.54673	−0.274110
$759$	4.64471	0.168592
$760$	0	0
$761$	−5.35609	−0.194158	−0.0970790	−	0.995277i	$-0.530950\pi$
$761$	−5.35609	−0.194158	−0.0970790	+	0.995277i	$0.530950\pi$
$762$	−11.2918	−0.409057
$763$	2.29510	0.0830881
$764$	−31.2075	−1.12905
$765$	0	0
$766$	2.31492	0.0836416
$767$	47.1903	1.70394
$768$	12.8167	0.462484
$769$	14.2818	0.515015	0.257508	−	0.966276i	$-0.417099\pi$
$769$	14.2818	0.515015	0.257508	+	0.966276i	$0.417099\pi$
$770$	0	0
$771$	3.93640	0.141766
$772$	−8.07018	−0.290452
$773$	50.9848	1.83380	0.916899	−	0.399120i	$-0.130684\pi$
$773$	50.9848	1.83380	0.916899	+	0.399120i	$0.130684\pi$
$774$	1.24527	0.0447603
$775$	0	0
$776$	−7.76065	−0.278591
$777$	−0.409959	−0.0147072
$778$	−1.79428	−0.0643279
$779$	6.47152	0.231866
$780$	0	0
$781$	11.8663	0.424610
$782$	4.95584	0.177221
$783$	−48.9722	−1.75013
$784$	−7.31915	−0.261398
$785$	0	0
$786$	1.95514	0.0697376
$787$	43.6102	1.55453	0.777267	−	0.629171i	$-0.216605\pi$
$787$	43.6102	1.55453	0.777267	+	0.629171i	$0.216605\pi$
$788$	−15.6382	−0.557087
$789$	11.5208	0.410150
$790$	0	0
$791$	12.9491	0.460417
$792$	5.90313	0.209759
$793$	67.8888	2.41080
$794$	−6.99019	−0.248072
$795$	0	0
$796$	−37.8081	−1.34007
$797$	2.95371	0.104626	0.0523129	−	0.998631i	$-0.483341\pi$
$797$	2.95371	0.104626	0.0523129	+	0.998631i	$0.483341\pi$
$798$	−2.03531	−0.0720492
$799$	−13.0536	−0.461802
$800$	0	0
$801$	21.4130	0.756591
$802$	4.62639	0.163364
$803$	8.79955	0.310529
$804$	−13.7839	−0.486122
$805$	0	0
$806$	37.0941	1.30659
$807$	−3.40320	−0.119798
$808$	−21.8671	−0.769283
$809$	−29.3341	−1.03133	−0.515665	−	0.856790i	$-0.672455\pi$
$809$	−29.3341	−1.03133	−0.515665	+	0.856790i	$0.672455\pi$
$810$	0	0
$811$	24.0217	0.843516	0.421758	−	0.906708i	$-0.361413\pi$
$811$	24.0217	0.843516	0.421758	+	0.906708i	$0.361413\pi$
$812$	−10.2318	−0.359066
$813$	−20.6478	−0.724150
$814$	−0.473581	−0.0165990
$815$	0	0
$816$	2.92522	0.102403
$817$	3.30131	0.115498
$818$	−4.55349	−0.159209
$819$	9.25546	0.323412
$820$	0	0
$821$	36.9129	1.28827	0.644134	−	0.764913i	$-0.277218\pi$
$821$	36.9129	1.28827	0.644134	+	0.764913i	$0.277218\pi$
$822$	−5.89979	−0.205779
$823$	−51.6280	−1.79964	−0.899819	−	0.436263i	$-0.856302\pi$
$823$	−51.6280	−1.79964	−0.899819	+	0.436263i	$0.856302\pi$
$824$	24.2335	0.844216
$825$	0	0
$826$	3.61660	0.125838
$827$	−20.5580	−0.714872	−0.357436	−	0.933938i	$-0.616349\pi$
$827$	−20.5580	−0.714872	−0.357436	+	0.933938i	$0.616349\pi$
$828$	7.67472	0.266715
$829$	−21.8000	−0.757145	−0.378573	−	0.925572i	$-0.623585\pi$
$829$	−21.8000	−0.757145	−0.378573	+	0.925572i	$0.623585\pi$
$830$	0	0
$831$	−18.9082	−0.655918
$832$	13.8946	0.481707
$833$	14.6232	0.506666
$834$	9.97145	0.345283
$835$	0	0
$836$	6.64925	0.229969
$837$	−38.4803	−1.33007
$838$	23.5220	0.812555
$839$	−44.1508	−1.52426	−0.762128	−	0.647426i	$-0.775845\pi$
$839$	−44.1508	−1.52426	−0.762128	+	0.647426i	$0.775845\pi$
$840$	0	0
$841$	55.1868	1.90299
$842$	−17.7532	−0.611814
$843$	36.2571	1.24876
$844$	8.96000	0.308416
$845$	0	0
$846$	7.14802	0.245754
$847$	6.89958	0.237072
$848$	−14.0932	−0.483961
$849$	−7.10202	−0.243741
$850$	0	0
$851$	−1.44913	−0.0496755
$852$	−14.5357	−0.497986
$853$	−31.8487	−1.09048	−0.545240	−	0.838280i	$-0.683562\pi$
$853$	−31.8487	−1.09048	−0.545240	+	0.838280i	$0.683562\pi$
$854$	5.20290	0.178040
$855$	0	0
$856$	23.7417	0.811476
$857$	−25.5131	−0.871512	−0.435756	−	0.900065i	$-0.643519\pi$
$857$	−25.5131	−0.871512	−0.435756	+	0.900065i	$0.643519\pi$
$858$	−7.92626	−0.270598
$859$	0.980066	0.0334394	0.0167197	−	0.999860i	$-0.494678\pi$
$859$	0.980066	0.0334394	0.0167197	+	0.999860i	$0.494678\pi$
$860$	0	0
$861$	1.67201	0.0569819
$862$	12.1701	0.414516
$863$	−28.7874	−0.979934	−0.489967	−	0.871741i	$-0.662991\pi$
$863$	−28.7874	−0.979934	−0.489967	+	0.871741i	$0.662991\pi$
$864$	−31.2233	−1.06224
$865$	0	0
$866$	−17.7005	−0.601486
$867$	13.3677	0.453992
$868$	−8.03973	−0.272886
$869$	21.8790	0.742194
$870$	0	0
$871$	−58.7591	−1.99098
$872$	−7.64378	−0.258851
$873$	5.31910	0.180024
$874$	−7.19444	−0.243355
$875$	0	0
$876$	−10.7791	−0.364191
$877$	−5.36800	−0.181264	−0.0906322	−	0.995884i	$-0.528889\pi$
$877$	−5.36800	−0.181264	−0.0906322	+	0.995884i	$0.528889\pi$
$878$	−13.5579	−0.457556
$879$	−2.47836	−0.0835929
$880$	0	0
$881$	24.8137	0.835995	0.417998	−	0.908448i	$-0.362732\pi$
$881$	24.8137	0.835995	0.417998	+	0.908448i	$0.362732\pi$
$882$	−8.00757	−0.269629
$883$	−15.4394	−0.519577	−0.259788	−	0.965666i	$-0.583653\pi$
$883$	−15.4394	−0.519577	−0.259788	+	0.965666i	$0.583653\pi$
$884$	23.9175	0.804432
$885$	0	0
$886$	3.39289	0.113986
$887$	0.335606	0.0112686	0.00563428	−	0.999984i	$-0.498207\pi$
$887$	0.335606	0.0112686	0.00563428	+	0.999984i	$0.498207\pi$
$888$	1.36536	0.0458185
$889$	10.4328	0.349904
$890$	0	0
$891$	1.17706	0.0394329
$892$	35.4175	1.18586
$893$	18.9500	0.634136
$894$	−0.962391	−0.0321872
$895$	0	0
$896$	−7.76538	−0.259423
$897$	−24.2539	−0.809813
$898$	8.39656	0.280197
$899$	66.1504	2.20624
$900$	0	0
$901$	28.1573	0.938057
$902$	1.93149	0.0643116
$903$	0.852940	0.0283841
$904$	−43.1267	−1.43437
$905$	0	0
$906$	1.34540	0.0446981
$907$	19.9085	0.661051	0.330525	−	0.943797i	$-0.392774\pi$
$907$	19.9085	0.661051	0.330525	+	0.943797i	$0.392774\pi$
$908$	2.49736	0.0828776
$909$	14.9876	0.497107
$910$	0	0
$911$	50.1160	1.66042	0.830209	−	0.557453i	$-0.188221\pi$
$911$	50.1160	1.66042	0.830209	+	0.557453i	$0.188221\pi$
$912$	−4.24656	−0.140618
$913$	4.89204	0.161903
$914$	13.8207	0.457147
$915$	0	0
$916$	17.3659	0.573787
$917$	−1.80641	−0.0596529
$918$	8.77325	0.289561
$919$	36.9785	1.21981	0.609905	−	0.792475i	$-0.291208\pi$
$919$	36.9785	1.21981	0.609905	+	0.792475i	$0.291208\pi$
$920$	0	0
$921$	4.35515	0.143507
$922$	26.7943	0.882425
$923$	−61.9639	−2.03957
$924$	1.71793	0.0565156
$925$	0	0
$926$	−13.6952	−0.450051
$927$	−16.6095	−0.545528
$928$	53.6751	1.76197
$929$	57.1602	1.87537	0.937683	−	0.347492i	$-0.112967\pi$
$929$	57.1602	1.87537	0.937683	+	0.347492i	$0.112967\pi$
$930$	0	0
$931$	−21.2287	−0.695742
$932$	41.6309	1.36366
$933$	5.64799	0.184907
$934$	16.2889	0.532988
$935$	0	0
$936$	−30.8251	−1.00755
$937$	−9.06198	−0.296042	−0.148021	−	0.988984i	$-0.547290\pi$
$937$	−9.06198	−0.296042	−0.148021	+	0.988984i	$0.547290\pi$
$938$	−4.50322	−0.147035
$939$	−10.7733	−0.351575
$940$	0	0
$941$	−53.7491	−1.75217	−0.876086	−	0.482155i	$-0.839854\pi$
$941$	−53.7491	−1.75217	−0.876086	+	0.482155i	$0.839854\pi$
$942$	8.37015	0.272714
$943$	5.91024	0.192464
$944$	7.54584	0.245596
$945$	0	0
$946$	0.985308	0.0320351
$947$	15.0395	0.488717	0.244358	−	0.969685i	$-0.421423\pi$
$947$	15.0395	0.488717	0.244358	+	0.969685i	$0.421423\pi$
$948$	−26.8008	−0.870451
$949$	−45.9497	−1.49159
$950$	0	0
$951$	−15.7302	−0.510087
$952$	4.31416	0.139823
$953$	29.4427	0.953742	0.476871	−	0.878973i	$-0.341771\pi$
$953$	29.4427	0.953742	0.476871	+	0.878973i	$0.341771\pi$
$954$	−15.4187	−0.499200
$955$	0	0
$956$	34.8923	1.12850
$957$	−14.1350	−0.456919
$958$	−15.0470	−0.486148
$959$	5.45098	0.176021
$960$	0	0
$961$	20.9782	0.676717
$962$	2.47296	0.0797314
$963$	−16.2724	−0.524372
$964$	−13.5247	−0.435601
$965$	0	0
$966$	−1.85878	−0.0598054
$967$	21.8407	0.702349	0.351175	−	0.936310i	$-0.385782\pi$
$967$	21.8407	0.702349	0.351175	+	0.936310i	$0.385782\pi$
$968$	−22.9789	−0.738571
$969$	8.48439	0.272558
$970$	0	0
$971$	−44.8646	−1.43977	−0.719887	−	0.694091i	$-0.755806\pi$
$971$	−44.8646	−1.43977	−0.719887	+	0.694091i	$0.755806\pi$
$972$	22.2168	0.712603
$973$	−9.21290	−0.295352
$974$	19.8570	0.636261
$975$	0	0
$976$	10.8556	0.347478
$977$	−18.9916	−0.607597	−0.303798	−	0.952736i	$-0.598255\pi$
$977$	−18.9916	−0.607597	−0.303798	+	0.952736i	$0.598255\pi$
$978$	4.68258	0.149732
$979$	16.9428	0.541495
$980$	0	0
$981$	5.23900	0.167268
$982$	14.1063	0.450150
$983$	12.3803	0.394872	0.197436	−	0.980316i	$-0.436739\pi$
$983$	12.3803	0.394872	0.197436	+	0.980316i	$0.436739\pi$
$984$	−5.56859	−0.177520
$985$	0	0
$986$	−15.0819	−0.480304
$987$	4.89599	0.155841
$988$	−34.7212	−1.10463
$989$	3.01498	0.0958709
$990$	0	0
$991$	26.7648	0.850213	0.425106	−	0.905143i	$-0.360237\pi$
$991$	26.7648	0.850213	0.425106	+	0.905143i	$0.360237\pi$
$992$	42.1757	1.33908
$993$	27.5476	0.874197
$994$	−4.74883	−0.150624
$995$	0	0
$996$	−5.99254	−0.189881
$997$	36.0725	1.14243	0.571214	−	0.820801i	$-0.306473\pi$
$997$	36.0725	1.14243	0.571214	+	0.820801i	$0.306473\pi$
$998$	−8.48347	−0.268540
$999$	−2.56537	−0.0811648

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1075.2.a.o.1.2	yes	5
3.2	odd	2		9675.2.a.cb.1.4		5
5.2	odd	4		1075.2.b.i.474.4		10
5.3	odd	4		1075.2.b.i.474.7		10
5.4	even	2		1075.2.a.n.1.4	✓	5
15.14	odd	2		9675.2.a.cc.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1075.2.a.n.1.4	✓	5	5.4	even	2
1075.2.a.o.1.2	yes	5	1.1	even	1	trivial
1075.2.b.i.474.4		10	5.2	odd	4
1075.2.b.i.474.7		10	5.3	odd	4
9675.2.a.cb.1.4		5	3.2	odd	2
9675.2.a.cc.1.2		5	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1075 = 5^{2} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1075.a (trivial)

\(f(q)\)	\(=\)	\(q-0.722813 q^{2} -1.13013 q^{3} -1.47754 q^{4} +0.816870 q^{6} -0.754729 q^{7} +2.51361 q^{8} -1.72281 q^{9} +O(q^{10})\)	\(q-0.722813 q^{2} -1.13013 q^{3} -1.47754 q^{4} +0.816870 q^{6} -0.754729 q^{7} +2.51361 q^{8} -1.72281 q^{9} -1.36316 q^{11} +1.66981 q^{12} +7.11818 q^{13} +0.545528 q^{14} +1.13821 q^{16} -2.27409 q^{17} +1.24527 q^{18} +3.30131 q^{19} +0.852940 q^{21} +0.985308 q^{22} +3.01498 q^{23} -2.84070 q^{24} -5.14511 q^{26} +5.33738 q^{27} +1.11514 q^{28} -9.17534 q^{29} -7.20959 q^{31} -5.84994 q^{32} +1.54054 q^{33} +1.64374 q^{34} +2.54553 q^{36} -0.480643 q^{37} -2.38623 q^{38} -8.04445 q^{39} +1.96029 q^{41} -0.616516 q^{42} +1.00000 q^{43} +2.01412 q^{44} -2.17927 q^{46} +5.74014 q^{47} -1.28633 q^{48} -6.43038 q^{49} +2.57001 q^{51} -10.5174 q^{52} -12.3818 q^{53} -3.85793 q^{54} -1.89710 q^{56} -3.73090 q^{57} +6.63205 q^{58} +6.62955 q^{59} +9.53738 q^{61} +5.21118 q^{62} +1.30026 q^{63} +1.95198 q^{64} -1.11352 q^{66} -8.25480 q^{67} +3.36006 q^{68} -3.40731 q^{69} -8.70502 q^{71} -4.33048 q^{72} -6.45527 q^{73} +0.347415 q^{74} -4.87782 q^{76} +1.02882 q^{77} +5.81463 q^{78} -16.0502 q^{79} -0.863478 q^{81} -1.41692 q^{82} -3.58875 q^{83} -1.26025 q^{84} -0.722813 q^{86} +10.3693 q^{87} -3.42645 q^{88} -12.4291 q^{89} -5.37230 q^{91} -4.45476 q^{92} +8.14776 q^{93} -4.14904 q^{94} +6.61117 q^{96} -3.08745 q^{97} +4.64796 q^{98} +2.34847 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 3 q^{6} + 3 q^{8} - 5 q^{9}+O(q^{10}) \) 5 * q - 3 * q^6 + 3 * q^8 - 5 * q^9	\( 5 q - 3 q^{6} + 3 q^{8} - 5 q^{9} - 9 q^{11} - 8 q^{12} + q^{13} - 7 q^{14} - 2 q^{16} - 3 q^{17} + 10 q^{18} - 11 q^{19} - 5 q^{21} - 13 q^{22} - 9 q^{24} - 5 q^{26} - 3 q^{27} + 15 q^{28} - 22 q^{29} - 5 q^{31} - 4 q^{32} + 12 q^{33} - 7 q^{34} + 3 q^{36} - 7 q^{37} + 3 q^{38} - 3 q^{39} - 21 q^{41} - 7 q^{42} + 5 q^{43} - 20 q^{44} + 13 q^{46} + 3 q^{47} + 6 q^{48} - 13 q^{49} - 5 q^{51} - 18 q^{52} + 5 q^{53} + 4 q^{54} + 4 q^{56} - 13 q^{57} + 16 q^{58} - 8 q^{59} - 16 q^{61} + 2 q^{62} - 7 q^{63} - 17 q^{64} + 19 q^{66} + 8 q^{67} + 7 q^{68} - 15 q^{69} - 10 q^{71} - 5 q^{72} - q^{73} - 3 q^{76} - 7 q^{77} + 25 q^{78} + 12 q^{79} - 15 q^{81} - 41 q^{82} - 20 q^{83} + 5 q^{84} + 22 q^{87} + 2 q^{88} - 36 q^{89} - 13 q^{91} + q^{92} + 13 q^{93} + 28 q^{96} + 8 q^{97} - 25 q^{98} - 4 q^{99}+O(q^{100}) \) 5 * q - 3 * q^6 + 3 * q^8 - 5 * q^9 - 9 * q^11 - 8 * q^12 + q^13 - 7 * q^14 - 2 * q^16 - 3 * q^17 + 10 * q^18 - 11 * q^19 - 5 * q^21 - 13 * q^22 - 9 * q^24 - 5 * q^26 - 3 * q^27 + 15 * q^28 - 22 * q^29 - 5 * q^31 - 4 * q^32 + 12 * q^33 - 7 * q^34 + 3 * q^36 - 7 * q^37 + 3 * q^38 - 3 * q^39 - 21 * q^41 - 7 * q^42 + 5 * q^43 - 20 * q^44 + 13 * q^46 + 3 * q^47 + 6 * q^48 - 13 * q^49 - 5 * q^51 - 18 * q^52 + 5 * q^53 + 4 * q^54 + 4 * q^56 - 13 * q^57 + 16 * q^58 - 8 * q^59 - 16 * q^61 + 2 * q^62 - 7 * q^63 - 17 * q^64 + 19 * q^66 + 8 * q^67 + 7 * q^68 - 15 * q^69 - 10 * q^71 - 5 * q^72 - q^73 - 3 * q^76 - 7 * q^77 + 25 * q^78 + 12 * q^79 - 15 * q^81 - 41 * q^82 - 20 * q^83 + 5 * q^84 + 22 * q^87 + 2 * q^88 - 36 * q^89 - 13 * q^91 + q^92 + 13 * q^93 + 28 * q^96 + 8 * q^97 - 25 * q^98 - 4 * q^99

Introduction

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