LMFDB - Embedded newform 1875.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1875,2,Mod(1,1875)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1875, 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("1875.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1875 = 3 \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1875.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:	$14.9719503790$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 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			1.1
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	1875.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.618034 q^{2} +1.00000 q^{3} -1.61803 q^{4} -0.618034 q^{6} -2.00000 q^{7} +2.23607 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.618034 q^{2} +1.00000 q^{3} -1.61803 q^{4} -0.618034 q^{6} -2.00000 q^{7} +2.23607 q^{8} +1.00000 q^{9} -3.00000 q^{11} -1.61803 q^{12} -1.00000 q^{13} +1.23607 q^{14} +1.85410 q^{16} +0.236068 q^{17} -0.618034 q^{18} +6.70820 q^{19} -2.00000 q^{21} +1.85410 q^{22} +7.61803 q^{23} +2.23607 q^{24} +0.618034 q^{26} +1.00000 q^{27} +3.23607 q^{28} -1.38197 q^{29} -4.70820 q^{31} -5.61803 q^{32} -3.00000 q^{33} -0.145898 q^{34} -1.61803 q^{36} -2.00000 q^{37} -4.14590 q^{38} -1.00000 q^{39} -11.6180 q^{41} +1.23607 q^{42} -9.61803 q^{43} +4.85410 q^{44} -4.70820 q^{46} -9.23607 q^{47} +1.85410 q^{48} -3.00000 q^{49} +0.236068 q^{51} +1.61803 q^{52} +6.76393 q^{53} -0.618034 q^{54} -4.47214 q^{56} +6.70820 q^{57} +0.854102 q^{58} -13.9443 q^{59} -4.70820 q^{61} +2.90983 q^{62} -2.00000 q^{63} -0.236068 q^{64} +1.85410 q^{66} +9.18034 q^{67} -0.381966 q^{68} +7.61803 q^{69} -1.09017 q^{71} +2.23607 q^{72} +2.29180 q^{73} +1.23607 q^{74} -10.8541 q^{76} +6.00000 q^{77} +0.618034 q^{78} -15.8541 q^{79} +1.00000 q^{81} +7.18034 q^{82} +9.00000 q^{83} +3.23607 q^{84} +5.94427 q^{86} -1.38197 q^{87} -6.70820 q^{88} +11.1803 q^{89} +2.00000 q^{91} -12.3262 q^{92} -4.70820 q^{93} +5.70820 q^{94} -5.61803 q^{96} -2.85410 q^{97} +1.85410 q^{98} -3.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} + 2 q^{3} - q^{4} + q^{6} - 4 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + q^2 + 2 * q^3 - q^4 + q^6 - 4 * q^7 + 2 * q^9	$ 2 q + q^{2} + 2 q^{3} - q^{4} + q^{6} - 4 q^{7} + 2 q^{9} - 6 q^{11} - q^{12} - 2 q^{13} - 2 q^{14} - 3 q^{16} - 4 q^{17} + q^{18} - 4 q^{21} - 3 q^{22} + 13 q^{23} - q^{26} + 2 q^{27} + 2 q^{28} - 5 q^{29} + 4 q^{31} - 9 q^{32} - 6 q^{33} - 7 q^{34} - q^{36} - 4 q^{37} - 15 q^{38} - 2 q^{39} - 21 q^{41} - 2 q^{42} - 17 q^{43} + 3 q^{44} + 4 q^{46} - 14 q^{47} - 3 q^{48} - 6 q^{49} - 4 q^{51} + q^{52} + 18 q^{53} + q^{54} - 5 q^{58} - 10 q^{59} + 4 q^{61} + 17 q^{62} - 4 q^{63} + 4 q^{64} - 3 q^{66} - 4 q^{67} - 3 q^{68} + 13 q^{69} + 9 q^{71} + 18 q^{73} - 2 q^{74} - 15 q^{76} + 12 q^{77} - q^{78} - 25 q^{79} + 2 q^{81} - 8 q^{82} + 18 q^{83} + 2 q^{84} - 6 q^{86} - 5 q^{87} + 4 q^{91} - 9 q^{92} + 4 q^{93} - 2 q^{94} - 9 q^{96} + q^{97} - 3 q^{98} - 6 q^{99}+O(q^{100}) $ 2 * q + q^2 + 2 * q^3 - q^4 + q^6 - 4 * q^7 + 2 * q^9 - 6 * q^11 - q^12 - 2 * q^13 - 2 * q^14 - 3 * q^16 - 4 * q^17 + q^18 - 4 * q^21 - 3 * q^22 + 13 * q^23 - q^26 + 2 * q^27 + 2 * q^28 - 5 * q^29 + 4 * q^31 - 9 * q^32 - 6 * q^33 - 7 * q^34 - q^36 - 4 * q^37 - 15 * q^38 - 2 * q^39 - 21 * q^41 - 2 * q^42 - 17 * q^43 + 3 * q^44 + 4 * q^46 - 14 * q^47 - 3 * q^48 - 6 * q^49 - 4 * q^51 + q^52 + 18 * q^53 + q^54 - 5 * q^58 - 10 * q^59 + 4 * q^61 + 17 * q^62 - 4 * q^63 + 4 * q^64 - 3 * q^66 - 4 * q^67 - 3 * q^68 + 13 * q^69 + 9 * q^71 + 18 * q^73 - 2 * q^74 - 15 * q^76 + 12 * q^77 - q^78 - 25 * q^79 + 2 * q^81 - 8 * q^82 + 18 * q^83 + 2 * q^84 - 6 * q^86 - 5 * q^87 + 4 * q^91 - 9 * q^92 + 4 * q^93 - 2 * q^94 - 9 * q^96 + q^97 - 3 * q^98 - 6 * 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 $

$2$	−0.618034	−0.437016	−0.218508	−	0.975835i	$-0.570119\pi$
$2$	−0.618034	−0.437016	−0.218508	+	0.975835i	$0.570119\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.61803	−0.809017
$5$	0	0
$5$	0	0
$6$	−0.618034	−0.252311
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	2.23607	0.790569
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	−1.61803	−0.467086
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	1.23607	0.330353
$15$	0	0
$16$	1.85410	0.463525
$17$	0.236068	0.0572549	0.0286274	−	0.999590i	$-0.490886\pi$
$17$	0.236068	0.0572549	0.0286274	+	0.999590i	$0.490886\pi$
$18$	−0.618034	−0.145672
$19$	6.70820	1.53897	0.769484	−	0.638666i	$-0.220514\pi$
$19$	6.70820	1.53897	0.769484	+	0.638666i	$0.220514\pi$
$20$	0	0
$21$	−2.00000	−0.436436
$22$	1.85410	0.395296
$23$	7.61803	1.58847	0.794235	−	0.607611i	$-0.207872\pi$
$23$	7.61803	1.58847	0.794235	+	0.607611i	$0.207872\pi$
$24$	2.23607	0.456435
$25$	0	0
$26$	0.618034	0.121206
$27$	1.00000	0.192450
$28$	3.23607	0.611559
$29$	−1.38197	−0.256625	−0.128312	−	0.991734i	$-0.540956\pi$
$29$	−1.38197	−0.256625	−0.128312	+	0.991734i	$0.540956\pi$
$30$	0	0
$31$	−4.70820	−0.845618	−0.422809	−	0.906219i	$-0.638956\pi$
$31$	−4.70820	−0.845618	−0.422809	+	0.906219i	$0.638956\pi$
$32$	−5.61803	−0.993137
$33$	−3.00000	−0.522233
$34$	−0.145898	−0.0250213
$35$	0	0
$36$	−1.61803	−0.269672
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	−4.14590	−0.672553
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−11.6180	−1.81443	−0.907216	−	0.420665i	$-0.861797\pi$
$41$	−11.6180	−1.81443	−0.907216	+	0.420665i	$0.861797\pi$
$42$	1.23607	0.190729
$43$	−9.61803	−1.46674	−0.733368	−	0.679832i	$-0.762053\pi$
$43$	−9.61803	−1.46674	−0.733368	+	0.679832i	$0.762053\pi$
$44$	4.85410	0.731783
$45$	0	0
$46$	−4.70820	−0.694187
$47$	−9.23607	−1.34722	−0.673609	−	0.739087i	$-0.735257\pi$
$47$	−9.23607	−1.34722	−0.673609	+	0.739087i	$0.735257\pi$
$48$	1.85410	0.267617
$49$	−3.00000	−0.428571
$50$	0	0
$51$	0.236068	0.0330561
$52$	1.61803	0.224381
$53$	6.76393	0.929098	0.464549	−	0.885548i	$-0.346217\pi$
$53$	6.76393	0.929098	0.464549	+	0.885548i	$0.346217\pi$
$54$	−0.618034	−0.0841038
$55$	0	0
$56$	−4.47214	−0.597614
$57$	6.70820	0.888523
$58$	0.854102	0.112149
$59$	−13.9443	−1.81539	−0.907695	−	0.419631i	$-0.862159\pi$
$59$	−13.9443	−1.81539	−0.907695	+	0.419631i	$0.862159\pi$
$60$	0	0
$61$	−4.70820	−0.602824	−0.301412	−	0.953494i	$-0.597458\pi$
$61$	−4.70820	−0.602824	−0.301412	+	0.953494i	$0.597458\pi$
$62$	2.90983	0.369549
$63$	−2.00000	−0.251976
$64$	−0.236068	−0.0295085
$65$	0	0
$66$	1.85410	0.228224
$67$	9.18034	1.12156	0.560779	−	0.827966i	$-0.310502\pi$
$67$	9.18034	1.12156	0.560779	+	0.827966i	$0.310502\pi$
$68$	−0.381966	−0.0463202
$69$	7.61803	0.917104
$70$	0	0
$71$	−1.09017	−0.129379	−0.0646897	−	0.997905i	$-0.520606\pi$
$71$	−1.09017	−0.129379	−0.0646897	+	0.997905i	$0.520606\pi$
$72$	2.23607	0.263523
$73$	2.29180	0.268234	0.134117	−	0.990965i	$-0.457180\pi$
$73$	2.29180	0.268234	0.134117	+	0.990965i	$0.457180\pi$
$74$	1.23607	0.143690
$75$	0	0
$76$	−10.8541	−1.24505
$77$	6.00000	0.683763
$78$	0.618034	0.0699786
$79$	−15.8541	−1.78373	−0.891863	−	0.452306i	$-0.850602\pi$
$79$	−15.8541	−1.78373	−0.891863	+	0.452306i	$0.850602\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	7.18034	0.792936
$83$	9.00000	0.987878	0.493939	−	0.869496i	$-0.335557\pi$
$83$	9.00000	0.987878	0.493939	+	0.869496i	$0.335557\pi$
$84$	3.23607	0.353084
$85$	0	0
$86$	5.94427	0.640987
$87$	−1.38197	−0.148162
$88$	−6.70820	−0.715097
$89$	11.1803	1.18511	0.592557	−	0.805529i	$-0.298119\pi$
$89$	11.1803	1.18511	0.592557	+	0.805529i	$0.298119\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	−12.3262	−1.28510
$93$	−4.70820	−0.488218
$94$	5.70820	0.588756
$95$	0	0
$96$	−5.61803	−0.573388
$97$	−2.85410	−0.289790	−0.144895	−	0.989447i	$-0.546284\pi$
$97$	−2.85410	−0.289790	−0.144895	+	0.989447i	$0.546284\pi$
$98$	1.85410	0.187293
$99$	−3.00000	−0.301511
$100$	0	0
$101$	−11.6180	−1.15604	−0.578019	−	0.816023i	$-0.696174\pi$
$101$	−11.6180	−1.15604	−0.578019	+	0.816023i	$0.696174\pi$
$102$	−0.145898	−0.0144461
$103$	12.4164	1.22343	0.611713	−	0.791080i	$-0.290481\pi$
$103$	12.4164	1.22343	0.611713	+	0.791080i	$0.290481\pi$
$104$	−2.23607	−0.219265
$105$	0	0
$106$	−4.18034	−0.406031
$107$	−7.85410	−0.759285	−0.379642	−	0.925133i	$-0.623953\pi$
$107$	−7.85410	−0.759285	−0.379642	+	0.925133i	$0.623953\pi$
$108$	−1.61803	−0.155695
$109$	−10.8541	−1.03963	−0.519817	−	0.854278i	$-0.674000\pi$
$109$	−10.8541	−1.03963	−0.519817	+	0.854278i	$0.674000\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	−3.70820	−0.350392
$113$	−8.23607	−0.774784	−0.387392	−	0.921915i	$-0.626624\pi$
$113$	−8.23607	−0.774784	−0.387392	+	0.921915i	$0.626624\pi$
$114$	−4.14590	−0.388299
$115$	0	0
$116$	2.23607	0.207614
$117$	−1.00000	−0.0924500
$118$	8.61803	0.793354
$119$	−0.472136	−0.0432806
$120$	0	0
$121$	−2.00000	−0.181818
$122$	2.90983	0.263444
$123$	−11.6180	−1.04756
$124$	7.61803	0.684120
$125$	0	0
$126$	1.23607	0.110118
$127$	−17.6525	−1.56640	−0.783202	−	0.621767i	$-0.786415\pi$
$127$	−17.6525	−1.56640	−0.783202	+	0.621767i	$0.786415\pi$
$128$	11.3820	1.00603
$129$	−9.61803	−0.846821
$130$	0	0
$131$	8.18034	0.714720	0.357360	−	0.933967i	$-0.383677\pi$
$131$	8.18034	0.714720	0.357360	+	0.933967i	$0.383677\pi$
$132$	4.85410	0.422495
$133$	−13.4164	−1.16335
$134$	−5.67376	−0.490138
$135$	0	0
$136$	0.527864	0.0452640
$137$	20.5623	1.75676	0.878378	−	0.477966i	$-0.158626\pi$
$137$	20.5623	1.75676	0.878378	+	0.477966i	$0.158626\pi$
$138$	−4.70820	−0.400789
$139$	−13.4164	−1.13796	−0.568982	−	0.822350i	$-0.692663\pi$
$139$	−13.4164	−1.13796	−0.568982	+	0.822350i	$0.692663\pi$
$140$	0	0
$141$	−9.23607	−0.777817
$142$	0.673762	0.0565409
$143$	3.00000	0.250873
$144$	1.85410	0.154508
$145$	0	0
$146$	−1.41641	−0.117223
$147$	−3.00000	−0.247436
$148$	3.23607	0.266003
$149$	−1.90983	−0.156459	−0.0782297	−	0.996935i	$-0.524927\pi$
$149$	−1.90983	−0.156459	−0.0782297	+	0.996935i	$0.524927\pi$
$150$	0	0
$151$	−4.38197	−0.356599	−0.178300	−	0.983976i	$-0.557060\pi$
$151$	−4.38197	−0.356599	−0.178300	+	0.983976i	$0.557060\pi$
$152$	15.0000	1.21666
$153$	0.236068	0.0190850
$154$	−3.70820	−0.298816
$155$	0	0
$156$	1.61803	0.129546
$157$	3.85410	0.307591	0.153795	−	0.988103i	$-0.450850\pi$
$157$	3.85410	0.307591	0.153795	+	0.988103i	$0.450850\pi$
$158$	9.79837	0.779517
$159$	6.76393	0.536415
$160$	0	0
$161$	−15.2361	−1.20077
$162$	−0.618034	−0.0485573
$163$	−15.2705	−1.19608	−0.598039	−	0.801467i	$-0.704053\pi$
$163$	−15.2705	−1.19608	−0.598039	+	0.801467i	$0.704053\pi$
$164$	18.7984	1.46791
$165$	0	0
$166$	−5.56231	−0.431719
$167$	−6.79837	−0.526074	−0.263037	−	0.964786i	$-0.584724\pi$
$167$	−6.79837	−0.526074	−0.263037	+	0.964786i	$0.584724\pi$
$168$	−4.47214	−0.345033
$169$	−12.0000	−0.923077
$170$	0	0
$171$	6.70820	0.512989
$172$	15.5623	1.18661
$173$	12.0902	0.919199	0.459599	−	0.888126i	$-0.347993\pi$
$173$	12.0902	0.919199	0.459599	+	0.888126i	$0.347993\pi$
$174$	0.854102	0.0647493
$175$	0	0
$176$	−5.56231	−0.419275
$177$	−13.9443	−1.04812
$178$	−6.90983	−0.517914
$179$	15.6525	1.16992	0.584960	−	0.811062i	$-0.301110\pi$
$179$	15.6525	1.16992	0.584960	+	0.811062i	$0.301110\pi$
$180$	0	0
$181$	−3.52786	−0.262224	−0.131112	−	0.991368i	$-0.541855\pi$
$181$	−3.52786	−0.262224	−0.131112	+	0.991368i	$0.541855\pi$
$182$	−1.23607	−0.0916235
$183$	−4.70820	−0.348040
$184$	17.0344	1.25580
$185$	0	0
$186$	2.90983	0.213359
$187$	−0.708204	−0.0517890
$188$	14.9443	1.08992
$189$	−2.00000	−0.145479
$190$	0	0
$191$	1.67376	0.121109	0.0605546	−	0.998165i	$-0.480713\pi$
$191$	1.67376	0.121109	0.0605546	+	0.998165i	$0.480713\pi$
$192$	−0.236068	−0.0170367
$193$	−11.0000	−0.791797	−0.395899	−	0.918294i	$-0.629567\pi$
$193$	−11.0000	−0.791797	−0.395899	+	0.918294i	$0.629567\pi$
$194$	1.76393	0.126643
$195$	0	0
$196$	4.85410	0.346722
$197$	11.0902	0.790142	0.395071	−	0.918651i	$-0.370720\pi$
$197$	11.0902	0.790142	0.395071	+	0.918651i	$0.370720\pi$
$198$	1.85410	0.131765
$199$	−1.70820	−0.121091	−0.0605457	−	0.998165i	$-0.519284\pi$
$199$	−1.70820	−0.121091	−0.0605457	+	0.998165i	$0.519284\pi$
$200$	0	0
$201$	9.18034	0.647531
$202$	7.18034	0.505207
$203$	2.76393	0.193990
$204$	−0.381966	−0.0267430
$205$	0	0
$206$	−7.67376	−0.534656
$207$	7.61803	0.529490
$208$	−1.85410	−0.128559
$209$	−20.1246	−1.39205
$210$	0	0
$211$	−3.00000	−0.206529	−0.103264	−	0.994654i	$-0.532929\pi$
$211$	−3.00000	−0.206529	−0.103264	+	0.994654i	$0.532929\pi$
$212$	−10.9443	−0.751656
$213$	−1.09017	−0.0746972
$214$	4.85410	0.331820
$215$	0	0
$216$	2.23607	0.152145
$217$	9.41641	0.639227
$218$	6.70820	0.454337
$219$	2.29180	0.154865
$220$	0	0
$221$	−0.236068	−0.0158797
$222$	1.23607	0.0829595
$223$	−16.8541	−1.12863	−0.564317	−	0.825558i	$-0.690860\pi$
$223$	−16.8541	−1.12863	−0.564317	+	0.825558i	$0.690860\pi$
$224$	11.2361	0.750741
$225$	0	0
$226$	5.09017	0.338593
$227$	10.2361	0.679392	0.339696	−	0.940535i	$-0.389676\pi$
$227$	10.2361	0.679392	0.339696	+	0.940535i	$0.389676\pi$
$228$	−10.8541	−0.718830
$229$	6.18034	0.408408	0.204204	−	0.978928i	$-0.434539\pi$
$229$	6.18034	0.408408	0.204204	+	0.978928i	$0.434539\pi$
$230$	0	0
$231$	6.00000	0.394771
$232$	−3.09017	−0.202880
$233$	−12.1803	−0.797961	−0.398980	−	0.916959i	$-0.630636\pi$
$233$	−12.1803	−0.797961	−0.398980	+	0.916959i	$0.630636\pi$
$234$	0.618034	0.0404021
$235$	0	0
$236$	22.5623	1.46868
$237$	−15.8541	−1.02983
$238$	0.291796	0.0189143
$239$	23.6180	1.52772	0.763862	−	0.645380i	$-0.223301\pi$
$239$	23.6180	1.52772	0.763862	+	0.645380i	$0.223301\pi$
$240$	0	0
$241$	−8.32624	−0.536340	−0.268170	−	0.963372i	$-0.586419\pi$
$241$	−8.32624	−0.536340	−0.268170	+	0.963372i	$0.586419\pi$
$242$	1.23607	0.0794575
$243$	1.00000	0.0641500
$244$	7.61803	0.487695
$245$	0	0
$246$	7.18034	0.457802
$247$	−6.70820	−0.426833
$248$	−10.5279	−0.668520
$249$	9.00000	0.570352
$250$	0	0
$251$	27.9787	1.76600	0.883000	−	0.469372i	$-0.155520\pi$
$251$	27.9787	1.76600	0.883000	+	0.469372i	$0.155520\pi$
$252$	3.23607	0.203853
$253$	−22.8541	−1.43683
$254$	10.9098	0.684544
$255$	0	0
$256$	−6.56231	−0.410144
$257$	−20.2148	−1.26096	−0.630482	−	0.776204i	$-0.717143\pi$
$257$	−20.2148	−1.26096	−0.630482	+	0.776204i	$0.717143\pi$
$258$	5.94427	0.370074
$259$	4.00000	0.248548
$260$	0	0
$261$	−1.38197	−0.0855415
$262$	−5.05573	−0.312344
$263$	25.5066	1.57280	0.786401	−	0.617716i	$-0.211942\pi$
$263$	25.5066	1.57280	0.786401	+	0.617716i	$0.211942\pi$
$264$	−6.70820	−0.412861
$265$	0	0
$266$	8.29180	0.508403
$267$	11.1803	0.684226
$268$	−14.8541	−0.907359
$269$	−29.4721	−1.79695	−0.898474	−	0.439027i	$-0.855323\pi$
$269$	−29.4721	−1.79695	−0.898474	+	0.439027i	$0.855323\pi$
$270$	0	0
$271$	15.4164	0.936480	0.468240	−	0.883601i	$-0.344888\pi$
$271$	15.4164	0.936480	0.468240	+	0.883601i	$0.344888\pi$
$272$	0.437694	0.0265391
$273$	2.00000	0.121046
$274$	−12.7082	−0.767731
$275$	0	0
$276$	−12.3262	−0.741952
$277$	−30.9443	−1.85926	−0.929631	−	0.368493i	$-0.879874\pi$
$277$	−30.9443	−1.85926	−0.929631	+	0.368493i	$0.879874\pi$
$278$	8.29180	0.497309
$279$	−4.70820	−0.281873
$280$	0	0
$281$	8.18034	0.487998	0.243999	−	0.969775i	$-0.421541\pi$
$281$	8.18034	0.487998	0.243999	+	0.969775i	$0.421541\pi$
$282$	5.70820	0.339919
$283$	15.7082	0.933756	0.466878	−	0.884322i	$-0.345379\pi$
$283$	15.7082	0.933756	0.466878	+	0.884322i	$0.345379\pi$
$284$	1.76393	0.104670
$285$	0	0
$286$	−1.85410	−0.109635
$287$	23.2361	1.37158
$288$	−5.61803	−0.331046
$289$	−16.9443	−0.996722
$290$	0	0
$291$	−2.85410	−0.167310
$292$	−3.70820	−0.217006
$293$	9.32624	0.544845	0.272422	−	0.962178i	$-0.412175\pi$
$293$	9.32624	0.544845	0.272422	+	0.962178i	$0.412175\pi$
$294$	1.85410	0.108133
$295$	0	0
$296$	−4.47214	−0.259938
$297$	−3.00000	−0.174078
$298$	1.18034	0.0683753
$299$	−7.61803	−0.440562
$300$	0	0
$301$	19.2361	1.10875
$302$	2.70820	0.155840
$303$	−11.6180	−0.667439
$304$	12.4377	0.713351
$305$	0	0
$306$	−0.145898	−0.00834044
$307$	2.14590	0.122473	0.0612364	−	0.998123i	$-0.480496\pi$
$307$	2.14590	0.122473	0.0612364	+	0.998123i	$0.480496\pi$
$308$	−9.70820	−0.553176
$309$	12.4164	0.706345
$310$	0	0
$311$	−22.4721	−1.27428	−0.637139	−	0.770749i	$-0.719882\pi$
$311$	−22.4721	−1.27428	−0.637139	+	0.770749i	$0.719882\pi$
$312$	−2.23607	−0.126592
$313$	15.7082	0.887880	0.443940	−	0.896056i	$-0.353580\pi$
$313$	15.7082	0.887880	0.443940	+	0.896056i	$0.353580\pi$
$314$	−2.38197	−0.134422
$315$	0	0
$316$	25.6525	1.44306
$317$	0.437694	0.0245833	0.0122917	−	0.999924i	$-0.496087\pi$
$317$	0.437694	0.0245833	0.0122917	+	0.999924i	$0.496087\pi$
$318$	−4.18034	−0.234422
$319$	4.14590	0.232126
$320$	0	0
$321$	−7.85410	−0.438373
$322$	9.41641	0.524756
$323$	1.58359	0.0881134
$324$	−1.61803	−0.0898908
$325$	0	0
$326$	9.43769	0.522706
$327$	−10.8541	−0.600233
$328$	−25.9787	−1.43443
$329$	18.4721	1.01840
$330$	0	0
$331$	29.6869	1.63174	0.815870	−	0.578235i	$-0.196258\pi$
$331$	29.6869	1.63174	0.815870	+	0.578235i	$0.196258\pi$
$332$	−14.5623	−0.799210
$333$	−2.00000	−0.109599
$334$	4.20163	0.229903
$335$	0	0
$336$	−3.70820	−0.202299
$337$	−10.8197	−0.589384	−0.294692	−	0.955592i	$-0.595217\pi$
$337$	−10.8197	−0.589384	−0.294692	+	0.955592i	$0.595217\pi$
$338$	7.41641	0.403399
$339$	−8.23607	−0.447322
$340$	0	0
$341$	14.1246	0.764891
$342$	−4.14590	−0.224184
$343$	20.0000	1.07990
$344$	−21.5066	−1.15956
$345$	0	0
$346$	−7.47214	−0.401705
$347$	−21.2705	−1.14186	−0.570930	−	0.820998i	$-0.693417\pi$
$347$	−21.2705	−1.14186	−0.570930	+	0.820998i	$0.693417\pi$
$348$	2.23607	0.119866
$349$	2.76393	0.147950	0.0739749	−	0.997260i	$-0.476432\pi$
$349$	2.76393	0.147950	0.0739749	+	0.997260i	$0.476432\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	16.8541	0.898327
$353$	−14.6180	−0.778039	−0.389020	−	0.921229i	$-0.627186\pi$
$353$	−14.6180	−0.778039	−0.389020	+	0.921229i	$0.627186\pi$
$354$	8.61803	0.458043
$355$	0	0
$356$	−18.0902	−0.958777
$357$	−0.472136	−0.0249881
$358$	−9.67376	−0.511274
$359$	6.05573	0.319609	0.159805	−	0.987149i	$-0.448914\pi$
$359$	6.05573	0.319609	0.159805	+	0.987149i	$0.448914\pi$
$360$	0	0
$361$	26.0000	1.36842
$362$	2.18034	0.114596
$363$	−2.00000	−0.104973
$364$	−3.23607	−0.169616
$365$	0	0
$366$	2.90983	0.152099
$367$	−21.4721	−1.12084	−0.560418	−	0.828210i	$-0.689360\pi$
$367$	−21.4721	−1.12084	−0.560418	+	0.828210i	$0.689360\pi$
$368$	14.1246	0.736296
$369$	−11.6180	−0.604811
$370$	0	0
$371$	−13.5279	−0.702332
$372$	7.61803	0.394977
$373$	−9.41641	−0.487563	−0.243782	−	0.969830i	$-0.578388\pi$
$373$	−9.41641	−0.487563	−0.243782	+	0.969830i	$0.578388\pi$
$374$	0.437694	0.0226326
$375$	0	0
$376$	−20.6525	−1.06507
$377$	1.38197	0.0711749
$378$	1.23607	0.0635765
$379$	−11.3820	−0.584652	−0.292326	−	0.956319i	$-0.594429\pi$
$379$	−11.3820	−0.584652	−0.292326	+	0.956319i	$0.594429\pi$
$380$	0	0
$381$	−17.6525	−0.904364
$382$	−1.03444	−0.0529266
$383$	22.9443	1.17240	0.586199	−	0.810167i	$-0.300624\pi$
$383$	22.9443	1.17240	0.586199	+	0.810167i	$0.300624\pi$
$384$	11.3820	0.580834
$385$	0	0
$386$	6.79837	0.346028
$387$	−9.61803	−0.488912
$388$	4.61803	0.234445
$389$	−30.6525	−1.55414	−0.777071	−	0.629413i	$-0.783295\pi$
$389$	−30.6525	−1.55414	−0.777071	+	0.629413i	$0.783295\pi$
$390$	0	0
$391$	1.79837	0.0909477
$392$	−6.70820	−0.338815
$393$	8.18034	0.412644
$394$	−6.85410	−0.345305
$395$	0	0
$396$	4.85410	0.243928
$397$	−11.4721	−0.575770	−0.287885	−	0.957665i	$-0.592952\pi$
$397$	−11.4721	−0.575770	−0.287885	+	0.957665i	$0.592952\pi$
$398$	1.05573	0.0529189
$399$	−13.4164	−0.671660
$400$	0	0
$401$	2.72949	0.136304	0.0681521	−	0.997675i	$-0.478290\pi$
$401$	2.72949	0.136304	0.0681521	+	0.997675i	$0.478290\pi$
$402$	−5.67376	−0.282982
$403$	4.70820	0.234532
$404$	18.7984	0.935254
$405$	0	0
$406$	−1.70820	−0.0847767
$407$	6.00000	0.297409
$408$	0.527864	0.0261332
$409$	35.1246	1.73680	0.868400	−	0.495864i	$-0.165149\pi$
$409$	35.1246	1.73680	0.868400	+	0.495864i	$0.165149\pi$
$410$	0	0
$411$	20.5623	1.01426
$412$	−20.0902	−0.989772
$413$	27.8885	1.37231
$414$	−4.70820	−0.231396
$415$	0	0
$416$	5.61803	0.275447
$417$	−13.4164	−0.657004
$418$	12.4377	0.608348
$419$	−15.3262	−0.748736	−0.374368	−	0.927280i	$-0.622140\pi$
$419$	−15.3262	−0.748736	−0.374368	+	0.927280i	$0.622140\pi$
$420$	0	0
$421$	14.3607	0.699897	0.349948	−	0.936769i	$-0.386199\pi$
$421$	14.3607	0.699897	0.349948	+	0.936769i	$0.386199\pi$
$422$	1.85410	0.0902563
$423$	−9.23607	−0.449073
$424$	15.1246	0.734516
$425$	0	0
$426$	0.673762	0.0326439
$427$	9.41641	0.455692
$428$	12.7082	0.614274
$429$	3.00000	0.144841
$430$	0	0
$431$	34.2361	1.64909	0.824547	−	0.565794i	$-0.191430\pi$
$431$	34.2361	1.64909	0.824547	+	0.565794i	$0.191430\pi$
$432$	1.85410	0.0892055
$433$	−5.47214	−0.262974	−0.131487	−	0.991318i	$-0.541975\pi$
$433$	−5.47214	−0.262974	−0.131487	+	0.991318i	$0.541975\pi$
$434$	−5.81966	−0.279353
$435$	0	0
$436$	17.5623	0.841082
$437$	51.1033	2.44460
$438$	−1.41641	−0.0676786
$439$	−2.96556	−0.141538	−0.0707692	−	0.997493i	$-0.522545\pi$
$439$	−2.96556	−0.141538	−0.0707692	+	0.997493i	$0.522545\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	0.145898	0.00693966
$443$	7.41641	0.352364	0.176182	−	0.984358i	$-0.443625\pi$
$443$	7.41641	0.352364	0.176182	+	0.984358i	$0.443625\pi$
$444$	3.23607	0.153577
$445$	0	0
$446$	10.4164	0.493231
$447$	−1.90983	−0.0903319
$448$	0.472136	0.0223063
$449$	−21.5066	−1.01496	−0.507479	−	0.861664i	$-0.669423\pi$
$449$	−21.5066	−1.01496	−0.507479	+	0.861664i	$0.669423\pi$
$450$	0	0
$451$	34.8541	1.64122
$452$	13.3262	0.626814
$453$	−4.38197	−0.205883
$454$	−6.32624	−0.296905
$455$	0	0
$456$	15.0000	0.702439
$457$	25.8885	1.21102	0.605508	−	0.795840i	$-0.292970\pi$
$457$	25.8885	1.21102	0.605508	+	0.795840i	$0.292970\pi$
$458$	−3.81966	−0.178481
$459$	0.236068	0.0110187
$460$	0	0
$461$	3.18034	0.148123	0.0740616	−	0.997254i	$-0.476404\pi$
$461$	3.18034	0.148123	0.0740616	+	0.997254i	$0.476404\pi$
$462$	−3.70820	−0.172521
$463$	26.6869	1.24025	0.620123	−	0.784505i	$-0.287083\pi$
$463$	26.6869	1.24025	0.620123	+	0.784505i	$0.287083\pi$
$464$	−2.56231	−0.118952
$465$	0	0
$466$	7.52786	0.348722
$467$	16.4164	0.759661	0.379830	−	0.925056i	$-0.375982\pi$
$467$	16.4164	0.759661	0.379830	+	0.925056i	$0.375982\pi$
$468$	1.61803	0.0747936
$469$	−18.3607	−0.847817
$470$	0	0
$471$	3.85410	0.177588
$472$	−31.1803	−1.43519
$473$	28.8541	1.32671
$474$	9.79837	0.450054
$475$	0	0
$476$	0.763932	0.0350148
$477$	6.76393	0.309699
$478$	−14.5967	−0.667640
$479$	−19.7984	−0.904611	−0.452305	−	0.891863i	$-0.649398\pi$
$479$	−19.7984	−0.904611	−0.452305	+	0.891863i	$0.649398\pi$
$480$	0	0
$481$	2.00000	0.0911922
$482$	5.14590	0.234389
$483$	−15.2361	−0.693265
$484$	3.23607	0.147094
$485$	0	0
$486$	−0.618034	−0.0280346
$487$	14.3820	0.651709	0.325855	−	0.945420i	$-0.394348\pi$
$487$	14.3820	0.651709	0.325855	+	0.945420i	$0.394348\pi$
$488$	−10.5279	−0.476574
$489$	−15.2705	−0.690556
$490$	0	0
$491$	6.67376	0.301183	0.150591	−	0.988596i	$-0.451882\pi$
$491$	6.67376	0.301183	0.150591	+	0.988596i	$0.451882\pi$
$492$	18.7984	0.847496
$493$	−0.326238	−0.0146930
$494$	4.14590	0.186533
$495$	0	0
$496$	−8.72949	−0.391966
$497$	2.18034	0.0978016
$498$	−5.56231	−0.249253
$499$	−15.0000	−0.671492	−0.335746	−	0.941953i	$-0.608988\pi$
$499$	−15.0000	−0.671492	−0.335746	+	0.941953i	$0.608988\pi$
$500$	0	0
$501$	−6.79837	−0.303729
$502$	−17.2918	−0.771771
$503$	−33.0344	−1.47293	−0.736466	−	0.676474i	$-0.763507\pi$
$503$	−33.0344	−1.47293	−0.736466	+	0.676474i	$0.763507\pi$
$504$	−4.47214	−0.199205
$505$	0	0
$506$	14.1246	0.627916
$507$	−12.0000	−0.532939
$508$	28.5623	1.26725
$509$	−2.88854	−0.128032	−0.0640162	−	0.997949i	$-0.520391\pi$
$509$	−2.88854	−0.128032	−0.0640162	+	0.997949i	$0.520391\pi$
$510$	0	0
$511$	−4.58359	−0.202766
$512$	−18.7082	−0.826794
$513$	6.70820	0.296174
$514$	12.4934	0.551061
$515$	0	0
$516$	15.5623	0.685092
$517$	27.7082	1.21861
$518$	−2.47214	−0.108619
$519$	12.0902	0.530700
$520$	0	0
$521$	28.9098	1.26656	0.633281	−	0.773922i	$-0.281708\pi$
$521$	28.9098	1.26656	0.633281	+	0.773922i	$0.281708\pi$
$522$	0.854102	0.0373830
$523$	−18.5623	−0.811673	−0.405836	−	0.913946i	$-0.633020\pi$
$523$	−18.5623	−0.811673	−0.405836	+	0.913946i	$0.633020\pi$
$524$	−13.2361	−0.578220
$525$	0	0
$526$	−15.7639	−0.687340
$527$	−1.11146	−0.0484158
$528$	−5.56231	−0.242068
$529$	35.0344	1.52324
$530$	0	0
$531$	−13.9443	−0.605130
$532$	21.7082	0.941170
$533$	11.6180	0.503233
$534$	−6.90983	−0.299018
$535$	0	0
$536$	20.5279	0.886669
$537$	15.6525	0.675454
$538$	18.2148	0.785295
$539$	9.00000	0.387657
$540$	0	0
$541$	−39.7082	−1.70719	−0.853595	−	0.520938i	$-0.825582\pi$
$541$	−39.7082	−1.70719	−0.853595	+	0.520938i	$0.825582\pi$
$542$	−9.52786	−0.409257
$543$	−3.52786	−0.151395
$544$	−1.32624	−0.0568620
$545$	0	0
$546$	−1.23607	−0.0528988
$547$	11.2918	0.482802	0.241401	−	0.970425i	$-0.422393\pi$
$547$	11.2918	0.482802	0.241401	+	0.970425i	$0.422393\pi$
$548$	−33.2705	−1.42125
$549$	−4.70820	−0.200941
$550$	0	0
$551$	−9.27051	−0.394937
$552$	17.0344	0.725034
$553$	31.7082	1.34837
$554$	19.1246	0.812527
$555$	0	0
$556$	21.7082	0.920633
$557$	−6.34752	−0.268953	−0.134477	−	0.990917i	$-0.542935\pi$
$557$	−6.34752	−0.268953	−0.134477	+	0.990917i	$0.542935\pi$
$558$	2.90983	0.123183
$559$	9.61803	0.406799
$560$	0	0
$561$	−0.708204	−0.0299004
$562$	−5.05573	−0.213263
$563$	9.00000	0.379305	0.189652	−	0.981851i	$-0.439264\pi$
$563$	9.00000	0.379305	0.189652	+	0.981851i	$0.439264\pi$
$564$	14.9443	0.629267
$565$	0	0
$566$	−9.70820	−0.408066
$567$	−2.00000	−0.0839921
$568$	−2.43769	−0.102283
$569$	−4.14590	−0.173805	−0.0869025	−	0.996217i	$-0.527697\pi$
$569$	−4.14590	−0.173805	−0.0869025	+	0.996217i	$0.527697\pi$
$570$	0	0
$571$	2.12461	0.0889122	0.0444561	−	0.999011i	$-0.485845\pi$
$571$	2.12461	0.0889122	0.0444561	+	0.999011i	$0.485845\pi$
$572$	−4.85410	−0.202960
$573$	1.67376	0.0699224
$574$	−14.3607	−0.599403
$575$	0	0
$576$	−0.236068	−0.00983617
$577$	37.2705	1.55159	0.775796	−	0.630984i	$-0.217349\pi$
$577$	37.2705	1.55159	0.775796	+	0.630984i	$0.217349\pi$
$578$	10.4721	0.435583
$579$	−11.0000	−0.457144
$580$	0	0
$581$	−18.0000	−0.746766
$582$	1.76393	0.0731173
$583$	−20.2918	−0.840400
$584$	5.12461	0.212058
$585$	0	0
$586$	−5.76393	−0.238106
$587$	−23.3050	−0.961898	−0.480949	−	0.876748i	$-0.659708\pi$
$587$	−23.3050	−0.961898	−0.480949	+	0.876748i	$0.659708\pi$
$588$	4.85410	0.200180
$589$	−31.5836	−1.30138
$590$	0	0
$591$	11.0902	0.456189
$592$	−3.70820	−0.152406
$593$	15.3820	0.631662	0.315831	−	0.948816i	$-0.397717\pi$
$593$	15.3820	0.631662	0.315831	+	0.948816i	$0.397717\pi$
$594$	1.85410	0.0760747
$595$	0	0
$596$	3.09017	0.126578
$597$	−1.70820	−0.0699121
$598$	4.70820	0.192533
$599$	5.72949	0.234101	0.117050	−	0.993126i	$-0.462656\pi$
$599$	5.72949	0.234101	0.117050	+	0.993126i	$0.462656\pi$
$600$	0	0
$601$	−11.2918	−0.460602	−0.230301	−	0.973119i	$-0.573971\pi$
$601$	−11.2918	−0.460602	−0.230301	+	0.973119i	$0.573971\pi$
$602$	−11.8885	−0.484541
$603$	9.18034	0.373852
$604$	7.09017	0.288495
$605$	0	0
$606$	7.18034	0.291681
$607$	−16.1459	−0.655342	−0.327671	−	0.944792i	$-0.606264\pi$
$607$	−16.1459	−0.655342	−0.327671	+	0.944792i	$0.606264\pi$
$608$	−37.6869	−1.52841
$609$	2.76393	0.112000
$610$	0	0
$611$	9.23607	0.373651
$612$	−0.381966	−0.0154401
$613$	−46.1246	−1.86296	−0.931478	−	0.363798i	$-0.881480\pi$
$613$	−46.1246	−1.86296	−0.931478	+	0.363798i	$0.881480\pi$
$614$	−1.32624	−0.0535226
$615$	0	0
$616$	13.4164	0.540562
$617$	20.7639	0.835924	0.417962	−	0.908464i	$-0.362744\pi$
$617$	20.7639	0.835924	0.417962	+	0.908464i	$0.362744\pi$
$618$	−7.67376	−0.308684
$619$	0.729490	0.0293207	0.0146603	−	0.999893i	$-0.495333\pi$
$619$	0.729490	0.0293207	0.0146603	+	0.999893i	$0.495333\pi$
$620$	0	0
$621$	7.61803	0.305701
$622$	13.8885	0.556880
$623$	−22.3607	−0.895862
$624$	−1.85410	−0.0742235
$625$	0	0
$626$	−9.70820	−0.388018
$627$	−20.1246	−0.803700
$628$	−6.23607	−0.248846
$629$	−0.472136	−0.0188253
$630$	0	0
$631$	−15.2361	−0.606538	−0.303269	−	0.952905i	$-0.598078\pi$
$631$	−15.2361	−0.606538	−0.303269	+	0.952905i	$0.598078\pi$
$632$	−35.4508	−1.41016
$633$	−3.00000	−0.119239
$634$	−0.270510	−0.0107433
$635$	0	0
$636$	−10.9443	−0.433969
$637$	3.00000	0.118864
$638$	−2.56231	−0.101443
$639$	−1.09017	−0.0431265
$640$	0	0
$641$	−7.67376	−0.303095	−0.151548	−	0.988450i	$-0.548426\pi$
$641$	−7.67376	−0.303095	−0.151548	+	0.988450i	$0.548426\pi$
$642$	4.85410	0.191576
$643$	17.0902	0.673971	0.336985	−	0.941510i	$-0.390593\pi$
$643$	17.0902	0.673971	0.336985	+	0.941510i	$0.390593\pi$
$644$	24.6525	0.971444
$645$	0	0
$646$	−0.978714	−0.0385070
$647$	10.0344	0.394495	0.197247	−	0.980354i	$-0.436800\pi$
$647$	10.0344	0.394495	0.197247	+	0.980354i	$0.436800\pi$
$648$	2.23607	0.0878410
$649$	41.8328	1.64208
$650$	0	0
$651$	9.41641	0.369058
$652$	24.7082	0.967648
$653$	−1.65248	−0.0646664	−0.0323332	−	0.999477i	$-0.510294\pi$
$653$	−1.65248	−0.0646664	−0.0323332	+	0.999477i	$0.510294\pi$
$654$	6.70820	0.262312
$655$	0	0
$656$	−21.5410	−0.841036
$657$	2.29180	0.0894115
$658$	−11.4164	−0.445058
$659$	2.23607	0.0871048	0.0435524	−	0.999051i	$-0.486132\pi$
$659$	2.23607	0.0871048	0.0435524	+	0.999051i	$0.486132\pi$
$660$	0	0
$661$	−30.8885	−1.20143	−0.600713	−	0.799465i	$-0.705116\pi$
$661$	−30.8885	−1.20143	−0.600713	+	0.799465i	$0.705116\pi$
$662$	−18.3475	−0.713097
$663$	−0.236068	−0.00916812
$664$	20.1246	0.780986
$665$	0	0
$666$	1.23607	0.0478967
$667$	−10.5279	−0.407641
$668$	11.0000	0.425603
$669$	−16.8541	−0.651617
$670$	0	0
$671$	14.1246	0.545275
$672$	11.2361	0.433441
$673$	24.7771	0.955087	0.477543	−	0.878608i	$-0.341527\pi$
$673$	24.7771	0.955087	0.477543	+	0.878608i	$0.341527\pi$
$674$	6.68692	0.257570
$675$	0	0
$676$	19.4164	0.746785
$677$	−9.11146	−0.350182	−0.175091	−	0.984552i	$-0.556022\pi$
$677$	−9.11146	−0.350182	−0.175091	+	0.984552i	$0.556022\pi$
$678$	5.09017	0.195487
$679$	5.70820	0.219061
$680$	0	0
$681$	10.2361	0.392247
$682$	−8.72949	−0.334269
$683$	48.5967	1.85950	0.929751	−	0.368188i	$-0.120022\pi$
$683$	48.5967	1.85950	0.929751	+	0.368188i	$0.120022\pi$
$684$	−10.8541	−0.415017
$685$	0	0
$686$	−12.3607	−0.471933
$687$	6.18034	0.235795
$688$	−17.8328	−0.679870
$689$	−6.76393	−0.257685
$690$	0	0
$691$	3.90983	0.148737	0.0743685	−	0.997231i	$-0.476306\pi$
$691$	3.90983	0.148737	0.0743685	+	0.997231i	$0.476306\pi$
$692$	−19.5623	−0.743647
$693$	6.00000	0.227921
$694$	13.1459	0.499011
$695$	0	0
$696$	−3.09017	−0.117133
$697$	−2.74265	−0.103885
$698$	−1.70820	−0.0646565
$699$	−12.1803	−0.460703
$700$	0	0
$701$	−17.3475	−0.655207	−0.327603	−	0.944815i	$-0.606241\pi$
$701$	−17.3475	−0.655207	−0.327603	+	0.944815i	$0.606241\pi$
$702$	0.618034	0.0233262
$703$	−13.4164	−0.506009
$704$	0.708204	0.0266914
$705$	0	0
$706$	9.03444	0.340016
$707$	23.2361	0.873882
$708$	22.5623	0.847943
$709$	−29.7984	−1.11910	−0.559551	−	0.828796i	$-0.689026\pi$
$709$	−29.7984	−1.11910	−0.559551	+	0.828796i	$0.689026\pi$
$710$	0	0
$711$	−15.8541	−0.594575
$712$	25.0000	0.936915
$713$	−35.8673	−1.34324
$714$	0.291796	0.0109202
$715$	0	0
$716$	−25.3262	−0.946486
$717$	23.6180	0.882032
$718$	−3.74265	−0.139674
$719$	5.12461	0.191116	0.0955579	−	0.995424i	$-0.469536\pi$
$719$	5.12461	0.191116	0.0955579	+	0.995424i	$0.469536\pi$
$720$	0	0
$721$	−24.8328	−0.924822
$722$	−16.0689	−0.598022
$723$	−8.32624	−0.309656
$724$	5.70820	0.212144
$725$	0	0
$726$	1.23607	0.0458748
$727$	−34.5623	−1.28184	−0.640922	−	0.767606i	$-0.721448\pi$
$727$	−34.5623	−1.28184	−0.640922	+	0.767606i	$0.721448\pi$
$728$	4.47214	0.165748
$729$	1.00000	0.0370370
$730$	0	0
$731$	−2.27051	−0.0839778
$732$	7.61803	0.281571
$733$	−16.8541	−0.622520	−0.311260	−	0.950325i	$-0.600751\pi$
$733$	−16.8541	−0.622520	−0.311260	+	0.950325i	$0.600751\pi$
$734$	13.2705	0.489823
$735$	0	0
$736$	−42.7984	−1.57757
$737$	−27.5410	−1.01449
$738$	7.18034	0.264312
$739$	11.7082	0.430693	0.215347	−	0.976538i	$-0.430912\pi$
$739$	11.7082	0.430693	0.215347	+	0.976538i	$0.430912\pi$
$740$	0	0
$741$	−6.70820	−0.246432
$742$	8.36068	0.306930
$743$	−25.4721	−0.934482	−0.467241	−	0.884130i	$-0.654752\pi$
$743$	−25.4721	−0.934482	−0.467241	+	0.884130i	$0.654752\pi$
$744$	−10.5279	−0.385970
$745$	0	0
$746$	5.81966	0.213073
$747$	9.00000	0.329293
$748$	1.14590	0.0418982
$749$	15.7082	0.573965
$750$	0	0
$751$	28.7082	1.04758	0.523789	−	0.851848i	$-0.324518\pi$
$751$	28.7082	1.04758	0.523789	+	0.851848i	$0.324518\pi$
$752$	−17.1246	−0.624470
$753$	27.9787	1.01960
$754$	−0.854102	−0.0311046
$755$	0	0
$756$	3.23607	0.117695
$757$	−1.27051	−0.0461775	−0.0230887	−	0.999733i	$-0.507350\pi$
$757$	−1.27051	−0.0461775	−0.0230887	+	0.999733i	$0.507350\pi$
$758$	7.03444	0.255502
$759$	−22.8541	−0.829551
$760$	0	0
$761$	−19.1803	−0.695287	−0.347643	−	0.937627i	$-0.613018\pi$
$761$	−19.1803	−0.695287	−0.347643	+	0.937627i	$0.613018\pi$
$762$	10.9098	0.395221
$763$	21.7082	0.785890
$764$	−2.70820	−0.0979794
$765$	0	0
$766$	−14.1803	−0.512357
$767$	13.9443	0.503498
$768$	−6.56231	−0.236797
$769$	26.3050	0.948581	0.474290	−	0.880368i	$-0.342705\pi$
$769$	26.3050	0.948581	0.474290	+	0.880368i	$0.342705\pi$
$770$	0	0
$771$	−20.2148	−0.728018
$772$	17.7984	0.640577
$773$	29.7771	1.07101	0.535504	−	0.844533i	$-0.320122\pi$
$773$	29.7771	1.07101	0.535504	+	0.844533i	$0.320122\pi$
$774$	5.94427	0.213662
$775$	0	0
$776$	−6.38197	−0.229099
$777$	4.00000	0.143499
$778$	18.9443	0.679185
$779$	−77.9361	−2.79235
$780$	0	0
$781$	3.27051	0.117028
$782$	−1.11146	−0.0397456
$783$	−1.38197	−0.0493874
$784$	−5.56231	−0.198654
$785$	0	0
$786$	−5.05573	−0.180332
$787$	23.8541	0.850307	0.425153	−	0.905121i	$-0.360220\pi$
$787$	23.8541	0.850307	0.425153	+	0.905121i	$0.360220\pi$
$788$	−17.9443	−0.639238
$789$	25.5066	0.908058
$790$	0	0
$791$	16.4721	0.585682
$792$	−6.70820	−0.238366
$793$	4.70820	0.167193
$794$	7.09017	0.251621
$795$	0	0
$796$	2.76393	0.0979650
$797$	46.0132	1.62987	0.814935	−	0.579553i	$-0.196773\pi$
$797$	46.0132	1.62987	0.814935	+	0.579553i	$0.196773\pi$
$798$	8.29180	0.293526
$799$	−2.18034	−0.0771349
$800$	0	0
$801$	11.1803	0.395038
$802$	−1.68692	−0.0595671
$803$	−6.87539	−0.242627
$804$	−14.8541	−0.523864
$805$	0	0
$806$	−2.90983	−0.102494
$807$	−29.4721	−1.03747
$808$	−25.9787	−0.913928
$809$	24.9230	0.876246	0.438123	−	0.898915i	$-0.355644\pi$
$809$	24.9230	0.876246	0.438123	+	0.898915i	$0.355644\pi$
$810$	0	0
$811$	37.7771	1.32653	0.663266	−	0.748383i	$-0.269170\pi$
$811$	37.7771	1.32653	0.663266	+	0.748383i	$0.269170\pi$
$812$	−4.47214	−0.156941
$813$	15.4164	0.540677
$814$	−3.70820	−0.129972
$815$	0	0
$816$	0.437694	0.0153224
$817$	−64.5197	−2.25726
$818$	−21.7082	−0.759010
$819$	2.00000	0.0698857
$820$	0	0
$821$	−11.9443	−0.416858	−0.208429	−	0.978038i	$-0.566835\pi$
$821$	−11.9443	−0.416858	−0.208429	+	0.978038i	$0.566835\pi$
$822$	−12.7082	−0.443250
$823$	−8.43769	−0.294120	−0.147060	−	0.989128i	$-0.546981\pi$
$823$	−8.43769	−0.294120	−0.147060	+	0.989128i	$0.546981\pi$
$824$	27.7639	0.967202
$825$	0	0
$826$	−17.2361	−0.599720
$827$	2.02129	0.0702870	0.0351435	−	0.999382i	$-0.488811\pi$
$827$	2.02129	0.0702870	0.0351435	+	0.999382i	$0.488811\pi$
$828$	−12.3262	−0.428366
$829$	−9.87539	−0.342986	−0.171493	−	0.985185i	$-0.554859\pi$
$829$	−9.87539	−0.342986	−0.171493	+	0.985185i	$0.554859\pi$
$830$	0	0
$831$	−30.9443	−1.07344
$832$	0.236068	0.00818418
$833$	−0.708204	−0.0245378
$834$	8.29180	0.287121
$835$	0	0
$836$	32.5623	1.12619
$837$	−4.70820	−0.162739
$838$	9.47214	0.327210
$839$	48.2148	1.66456	0.832280	−	0.554356i	$-0.187035\pi$
$839$	48.2148	1.66456	0.832280	+	0.554356i	$0.187035\pi$
$840$	0	0
$841$	−27.0902	−0.934144
$842$	−8.87539	−0.305866
$843$	8.18034	0.281746
$844$	4.85410	0.167085
$845$	0	0
$846$	5.70820	0.196252
$847$	4.00000	0.137442
$848$	12.5410	0.430660
$849$	15.7082	0.539104
$850$	0	0
$851$	−15.2361	−0.522286
$852$	1.76393	0.0604313
$853$	53.3951	1.82821	0.914107	−	0.405473i	$-0.132893\pi$
$853$	53.3951	1.82821	0.914107	+	0.405473i	$0.132893\pi$
$854$	−5.81966	−0.199145
$855$	0	0
$856$	−17.5623	−0.600267
$857$	26.9443	0.920399	0.460199	−	0.887816i	$-0.347778\pi$
$857$	26.9443	0.920399	0.460199	+	0.887816i	$0.347778\pi$
$858$	−1.85410	−0.0632980
$859$	−25.1246	−0.857241	−0.428620	−	0.903485i	$-0.641000\pi$
$859$	−25.1246	−0.857241	−0.428620	+	0.903485i	$0.641000\pi$
$860$	0	0
$861$	23.2361	0.791883
$862$	−21.1591	−0.720680
$863$	−45.0689	−1.53416	−0.767081	−	0.641550i	$-0.778292\pi$
$863$	−45.0689	−1.53416	−0.767081	+	0.641550i	$0.778292\pi$
$864$	−5.61803	−0.191129
$865$	0	0
$866$	3.38197	0.114924
$867$	−16.9443	−0.575458
$868$	−15.2361	−0.517146
$869$	47.5623	1.61344
$870$	0	0
$871$	−9.18034	−0.311064
$872$	−24.2705	−0.821903
$873$	−2.85410	−0.0965967
$874$	−31.5836	−1.06833
$875$	0	0
$876$	−3.70820	−0.125289
$877$	2.87539	0.0970950	0.0485475	−	0.998821i	$-0.484541\pi$
$877$	2.87539	0.0970950	0.0485475	+	0.998821i	$0.484541\pi$
$878$	1.83282	0.0618545
$879$	9.32624	0.314566
$880$	0	0
$881$	3.90983	0.131726	0.0658628	−	0.997829i	$-0.479020\pi$
$881$	3.90983	0.131726	0.0658628	+	0.997829i	$0.479020\pi$
$882$	1.85410	0.0624309
$883$	53.7984	1.81046	0.905230	−	0.424923i	$-0.139699\pi$
$883$	53.7984	1.81046	0.905230	+	0.424923i	$0.139699\pi$
$884$	0.381966	0.0128469
$885$	0	0
$886$	−4.58359	−0.153989
$887$	−21.9230	−0.736102	−0.368051	−	0.929806i	$-0.619975\pi$
$887$	−21.9230	−0.736102	−0.368051	+	0.929806i	$0.619975\pi$
$888$	−4.47214	−0.150075
$889$	35.3050	1.18409
$890$	0	0
$891$	−3.00000	−0.100504
$892$	27.2705	0.913084
$893$	−61.9574	−2.07333
$894$	1.18034	0.0394765
$895$	0	0
$896$	−22.7639	−0.760490
$897$	−7.61803	−0.254359
$898$	13.2918	0.443553
$899$	6.50658	0.217007
$900$	0	0
$901$	1.59675	0.0531954
$902$	−21.5410	−0.717238
$903$	19.2361	0.640136
$904$	−18.4164	−0.612521
$905$	0	0
$906$	2.70820	0.0899740
$907$	−17.0000	−0.564476	−0.282238	−	0.959344i	$-0.591077\pi$
$907$	−17.0000	−0.564476	−0.282238	+	0.959344i	$0.591077\pi$
$908$	−16.5623	−0.549639
$909$	−11.6180	−0.385346
$910$	0	0
$911$	−20.8885	−0.692068	−0.346034	−	0.938222i	$-0.612472\pi$
$911$	−20.8885	−0.692068	−0.346034	+	0.938222i	$0.612472\pi$
$912$	12.4377	0.411853
$913$	−27.0000	−0.893570
$914$	−16.0000	−0.529233
$915$	0	0
$916$	−10.0000	−0.330409
$917$	−16.3607	−0.540277
$918$	−0.145898	−0.00481535
$919$	−5.00000	−0.164935	−0.0824674	−	0.996594i	$-0.526280\pi$
$919$	−5.00000	−0.164935	−0.0824674	+	0.996594i	$0.526280\pi$
$920$	0	0
$921$	2.14590	0.0707097
$922$	−1.96556	−0.0647322
$923$	1.09017	0.0358834
$924$	−9.70820	−0.319376
$925$	0	0
$926$	−16.4934	−0.542007
$927$	12.4164	0.407808
$928$	7.76393	0.254864
$929$	19.5967	0.642948	0.321474	−	0.946918i	$-0.395822\pi$
$929$	19.5967	0.642948	0.321474	+	0.946918i	$0.395822\pi$
$930$	0	0
$931$	−20.1246	−0.659558
$932$	19.7082	0.645564
$933$	−22.4721	−0.735705
$934$	−10.1459	−0.331984
$935$	0	0
$936$	−2.23607	−0.0730882
$937$	16.4164	0.536301	0.268150	−	0.963377i	$-0.413588\pi$
$937$	16.4164	0.536301	0.268150	+	0.963377i	$0.413588\pi$
$938$	11.3475	0.370510
$939$	15.7082	0.512618
$940$	0	0
$941$	11.0213	0.359284	0.179642	−	0.983732i	$-0.442506\pi$
$941$	11.0213	0.359284	0.179642	+	0.983732i	$0.442506\pi$
$942$	−2.38197	−0.0776086
$943$	−88.5066	−2.88217
$944$	−25.8541	−0.841479
$945$	0	0
$946$	−17.8328	−0.579795
$947$	29.8328	0.969436	0.484718	−	0.874670i	$-0.338922\pi$
$947$	29.8328	0.969436	0.484718	+	0.874670i	$0.338922\pi$
$948$	25.6525	0.833154
$949$	−2.29180	−0.0743948
$950$	0	0
$951$	0.437694	0.0141932
$952$	−1.05573	−0.0342163
$953$	−59.9443	−1.94179	−0.970893	−	0.239515i	$-0.923012\pi$
$953$	−59.9443	−1.94179	−0.970893	+	0.239515i	$0.923012\pi$
$954$	−4.18034	−0.135344
$955$	0	0
$956$	−38.2148	−1.23595
$957$	4.14590	0.134018
$958$	12.2361	0.395329
$959$	−41.1246	−1.32798
$960$	0	0
$961$	−8.83282	−0.284930
$962$	−1.23607	−0.0398524
$963$	−7.85410	−0.253095
$964$	13.4721	0.433908
$965$	0	0
$966$	9.41641	0.302968
$967$	−8.58359	−0.276030	−0.138015	−	0.990430i	$-0.544072\pi$
$967$	−8.58359	−0.276030	−0.138015	+	0.990430i	$0.544072\pi$
$968$	−4.47214	−0.143740
$969$	1.58359	0.0508723
$970$	0	0
$971$	−5.88854	−0.188972	−0.0944862	−	0.995526i	$-0.530121\pi$
$971$	−5.88854	−0.188972	−0.0944862	+	0.995526i	$0.530121\pi$
$972$	−1.61803	−0.0518985
$973$	26.8328	0.860221
$974$	−8.88854	−0.284807
$975$	0	0
$976$	−8.72949	−0.279424
$977$	−6.34752	−0.203075	−0.101538	−	0.994832i	$-0.532376\pi$
$977$	−6.34752	−0.203075	−0.101538	+	0.994832i	$0.532376\pi$
$978$	9.43769	0.301784
$979$	−33.5410	−1.07198
$980$	0	0
$981$	−10.8541	−0.346545
$982$	−4.12461	−0.131622
$983$	−22.3820	−0.713874	−0.356937	−	0.934128i	$-0.616179\pi$
$983$	−22.3820	−0.713874	−0.356937	+	0.934128i	$0.616179\pi$
$984$	−25.9787	−0.828171
$985$	0	0
$986$	0.201626	0.00642108
$987$	18.4721	0.587975
$988$	10.8541	0.345315
$989$	−73.2705	−2.32987
$990$	0	0
$991$	32.0000	1.01651	0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000	1.01651	0.508257	+	0.861206i	$0.330290\pi$
$992$	26.4508	0.839815
$993$	29.6869	0.942086
$994$	−1.34752	−0.0427409
$995$	0	0
$996$	−14.5623	−0.461424
$997$	−61.0689	−1.93407	−0.967035	−	0.254642i	$-0.918042\pi$
$997$	−61.0689	−1.93407	−0.967035	+	0.254642i	$0.918042\pi$
$998$	9.27051	0.293453
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1875.2.a.c.1.1	yes	2
3.2	odd	2		5625.2.a.b.1.2		2
5.2	odd	4		1875.2.b.a.1249.2		4
5.3	odd	4		1875.2.b.a.1249.3		4
5.4	even	2		1875.2.a.b.1.2	✓	2
15.14	odd	2		5625.2.a.g.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1875.2.a.b.1.2	✓	2	5.4	even	2
1875.2.a.c.1.1	yes	2	1.1	even	1	trivial
1875.2.b.a.1249.2		4	5.2	odd	4
1875.2.b.a.1249.3		4	5.3	odd	4
5625.2.a.b.1.2		2	3.2	odd	2
5625.2.a.g.1.1		2	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:	\( N \)	\(=\)	\( 1875 = 3 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1875.a (trivial)

\(f(q)\)	\(=\)	\(q-0.618034 q^{2} +1.00000 q^{3} -1.61803 q^{4} -0.618034 q^{6} -2.00000 q^{7} +2.23607 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.618034 q^{2} +1.00000 q^{3} -1.61803 q^{4} -0.618034 q^{6} -2.00000 q^{7} +2.23607 q^{8} +1.00000 q^{9} -3.00000 q^{11} -1.61803 q^{12} -1.00000 q^{13} +1.23607 q^{14} +1.85410 q^{16} +0.236068 q^{17} -0.618034 q^{18} +6.70820 q^{19} -2.00000 q^{21} +1.85410 q^{22} +7.61803 q^{23} +2.23607 q^{24} +0.618034 q^{26} +1.00000 q^{27} +3.23607 q^{28} -1.38197 q^{29} -4.70820 q^{31} -5.61803 q^{32} -3.00000 q^{33} -0.145898 q^{34} -1.61803 q^{36} -2.00000 q^{37} -4.14590 q^{38} -1.00000 q^{39} -11.6180 q^{41} +1.23607 q^{42} -9.61803 q^{43} +4.85410 q^{44} -4.70820 q^{46} -9.23607 q^{47} +1.85410 q^{48} -3.00000 q^{49} +0.236068 q^{51} +1.61803 q^{52} +6.76393 q^{53} -0.618034 q^{54} -4.47214 q^{56} +6.70820 q^{57} +0.854102 q^{58} -13.9443 q^{59} -4.70820 q^{61} +2.90983 q^{62} -2.00000 q^{63} -0.236068 q^{64} +1.85410 q^{66} +9.18034 q^{67} -0.381966 q^{68} +7.61803 q^{69} -1.09017 q^{71} +2.23607 q^{72} +2.29180 q^{73} +1.23607 q^{74} -10.8541 q^{76} +6.00000 q^{77} +0.618034 q^{78} -15.8541 q^{79} +1.00000 q^{81} +7.18034 q^{82} +9.00000 q^{83} +3.23607 q^{84} +5.94427 q^{86} -1.38197 q^{87} -6.70820 q^{88} +11.1803 q^{89} +2.00000 q^{91} -12.3262 q^{92} -4.70820 q^{93} +5.70820 q^{94} -5.61803 q^{96} -2.85410 q^{97} +1.85410 q^{98} -3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} + 2 q^{3} - q^{4} + q^{6} - 4 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + q^2 + 2 * q^3 - q^4 + q^6 - 4 * q^7 + 2 * q^9	\( 2 q + q^{2} + 2 q^{3} - q^{4} + q^{6} - 4 q^{7} + 2 q^{9} - 6 q^{11} - q^{12} - 2 q^{13} - 2 q^{14} - 3 q^{16} - 4 q^{17} + q^{18} - 4 q^{21} - 3 q^{22} + 13 q^{23} - q^{26} + 2 q^{27} + 2 q^{28} - 5 q^{29} + 4 q^{31} - 9 q^{32} - 6 q^{33} - 7 q^{34} - q^{36} - 4 q^{37} - 15 q^{38} - 2 q^{39} - 21 q^{41} - 2 q^{42} - 17 q^{43} + 3 q^{44} + 4 q^{46} - 14 q^{47} - 3 q^{48} - 6 q^{49} - 4 q^{51} + q^{52} + 18 q^{53} + q^{54} - 5 q^{58} - 10 q^{59} + 4 q^{61} + 17 q^{62} - 4 q^{63} + 4 q^{64} - 3 q^{66} - 4 q^{67} - 3 q^{68} + 13 q^{69} + 9 q^{71} + 18 q^{73} - 2 q^{74} - 15 q^{76} + 12 q^{77} - q^{78} - 25 q^{79} + 2 q^{81} - 8 q^{82} + 18 q^{83} + 2 q^{84} - 6 q^{86} - 5 q^{87} + 4 q^{91} - 9 q^{92} + 4 q^{93} - 2 q^{94} - 9 q^{96} + q^{97} - 3 q^{98} - 6 q^{99}+O(q^{100}) \) 2 * q + q^2 + 2 * q^3 - q^4 + q^6 - 4 * q^7 + 2 * q^9 - 6 * q^11 - q^12 - 2 * q^13 - 2 * q^14 - 3 * q^16 - 4 * q^17 + q^18 - 4 * q^21 - 3 * q^22 + 13 * q^23 - q^26 + 2 * q^27 + 2 * q^28 - 5 * q^29 + 4 * q^31 - 9 * q^32 - 6 * q^33 - 7 * q^34 - q^36 - 4 * q^37 - 15 * q^38 - 2 * q^39 - 21 * q^41 - 2 * q^42 - 17 * q^43 + 3 * q^44 + 4 * q^46 - 14 * q^47 - 3 * q^48 - 6 * q^49 - 4 * q^51 + q^52 + 18 * q^53 + q^54 - 5 * q^58 - 10 * q^59 + 4 * q^61 + 17 * q^62 - 4 * q^63 + 4 * q^64 - 3 * q^66 - 4 * q^67 - 3 * q^68 + 13 * q^69 + 9 * q^71 + 18 * q^73 - 2 * q^74 - 15 * q^76 + 12 * q^77 - q^78 - 25 * q^79 + 2 * q^81 - 8 * q^82 + 18 * q^83 + 2 * q^84 - 6 * q^86 - 5 * q^87 + 4 * q^91 - 9 * q^92 + 4 * q^93 - 2 * q^94 - 9 * q^96 + q^97 - 3 * q^98 - 6 * q^99

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