LMFDB - Embedded newform 1875.2.a.c.1.2

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			1.2
Root			$1.61803$ 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)

$f(q)$	$=$	$q+1.61803 q^{2} +1.00000 q^{3} +0.618034 q^{4} +1.61803 q^{6} -2.00000 q^{7} -2.23607 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.61803 q^{2} +1.00000 q^{3} +0.618034 q^{4} +1.61803 q^{6} -2.00000 q^{7} -2.23607 q^{8} +1.00000 q^{9} -3.00000 q^{11} +0.618034 q^{12} -1.00000 q^{13} -3.23607 q^{14} -4.85410 q^{16} -4.23607 q^{17} +1.61803 q^{18} -6.70820 q^{19} -2.00000 q^{21} -4.85410 q^{22} +5.38197 q^{23} -2.23607 q^{24} -1.61803 q^{26} +1.00000 q^{27} -1.23607 q^{28} -3.61803 q^{29} +8.70820 q^{31} -3.38197 q^{32} -3.00000 q^{33} -6.85410 q^{34} +0.618034 q^{36} -2.00000 q^{37} -10.8541 q^{38} -1.00000 q^{39} -9.38197 q^{41} -3.23607 q^{42} -7.38197 q^{43} -1.85410 q^{44} +8.70820 q^{46} -4.76393 q^{47} -4.85410 q^{48} -3.00000 q^{49} -4.23607 q^{51} -0.618034 q^{52} +11.2361 q^{53} +1.61803 q^{54} +4.47214 q^{56} -6.70820 q^{57} -5.85410 q^{58} +3.94427 q^{59} +8.70820 q^{61} +14.0902 q^{62} -2.00000 q^{63} +4.23607 q^{64} -4.85410 q^{66} -13.1803 q^{67} -2.61803 q^{68} +5.38197 q^{69} +10.0902 q^{71} -2.23607 q^{72} +15.7082 q^{73} -3.23607 q^{74} -4.14590 q^{76} +6.00000 q^{77} -1.61803 q^{78} -9.14590 q^{79} +1.00000 q^{81} -15.1803 q^{82} +9.00000 q^{83} -1.23607 q^{84} -11.9443 q^{86} -3.61803 q^{87} +6.70820 q^{88} -11.1803 q^{89} +2.00000 q^{91} +3.32624 q^{92} +8.70820 q^{93} -7.70820 q^{94} -3.38197 q^{96} +3.85410 q^{97} -4.85410 q^{98} -3.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.61803	1.14412	0.572061	−	0.820211i	$-0.306144\pi$
$2$	1.61803	1.14412	0.572061	+	0.820211i	$0.306144\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0.618034	0.309017
$5$	0	0
$5$	0	0
$6$	1.61803	0.660560
$7$	−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	$0.649385\pi$
$12$	0.618034	0.178411
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	−3.23607	−0.864876
$15$	0	0
$16$	−4.85410	−1.21353
$17$	−4.23607	−1.02740	−0.513699	−	0.857971i	$-0.671725\pi$
$17$	−4.23607	−1.02740	−0.513699	+	0.857971i	$0.671725\pi$
$18$	1.61803	0.381374
$19$	−6.70820	−1.53897	−0.769484	−	0.638666i	$-0.779486\pi$
$19$	−6.70820	−1.53897	−0.769484	+	0.638666i	$0.779486\pi$
$20$	0	0
$21$	−2.00000	−0.436436
$22$	−4.85410	−1.03490
$23$	5.38197	1.12222	0.561109	−	0.827742i	$-0.310375\pi$
$23$	5.38197	1.12222	0.561109	+	0.827742i	$0.310375\pi$
$24$	−2.23607	−0.456435
$25$	0	0
$26$	−1.61803	−0.317323
$27$	1.00000	0.192450
$28$	−1.23607	−0.233595
$29$	−3.61803	−0.671852	−0.335926	−	0.941888i	$-0.609049\pi$
$29$	−3.61803	−0.671852	−0.335926	+	0.941888i	$0.609049\pi$
$30$	0	0
$31$	8.70820	1.56404	0.782020	−	0.623254i	$-0.214190\pi$
$31$	8.70820	1.56404	0.782020	+	0.623254i	$0.214190\pi$
$32$	−3.38197	−0.597853
$33$	−3.00000	−0.522233
$34$	−6.85410	−1.17547
$35$	0	0
$36$	0.618034	0.103006
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	−10.8541	−1.76077
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−9.38197	−1.46522	−0.732608	−	0.680650i	$-0.761697\pi$
$41$	−9.38197	−1.46522	−0.732608	+	0.680650i	$0.761697\pi$
$42$	−3.23607	−0.499336
$43$	−7.38197	−1.12574	−0.562870	−	0.826546i	$-0.690303\pi$
$43$	−7.38197	−1.12574	−0.562870	+	0.826546i	$0.690303\pi$
$44$	−1.85410	−0.279516
$45$	0	0
$46$	8.70820	1.28395
$47$	−4.76393	−0.694891	−0.347445	−	0.937700i	$-0.612951\pi$
$47$	−4.76393	−0.694891	−0.347445	+	0.937700i	$0.612951\pi$
$48$	−4.85410	−0.700629
$49$	−3.00000	−0.428571
$50$	0	0
$51$	−4.23607	−0.593168
$52$	−0.618034	−0.0857059
$53$	11.2361	1.54339	0.771696	−	0.635991i	$-0.219409\pi$
$53$	11.2361	1.54339	0.771696	+	0.635991i	$0.219409\pi$
$54$	1.61803	0.220187
$55$	0	0
$56$	4.47214	0.597614
$57$	−6.70820	−0.888523
$58$	−5.85410	−0.768681
$59$	3.94427	0.513500	0.256750	−	0.966478i	$-0.417348\pi$
$59$	3.94427	0.513500	0.256750	+	0.966478i	$0.417348\pi$
$60$	0	0
$61$	8.70820	1.11497	0.557486	−	0.830187i	$-0.311766\pi$
$61$	8.70820	1.11497	0.557486	+	0.830187i	$0.311766\pi$
$62$	14.0902	1.78945
$63$	−2.00000	−0.251976
$64$	4.23607	0.529508
$65$	0	0
$66$	−4.85410	−0.597499
$67$	−13.1803	−1.61023	−0.805117	−	0.593115i	$-0.797898\pi$
$67$	−13.1803	−1.61023	−0.805117	+	0.593115i	$0.797898\pi$
$68$	−2.61803	−0.317483
$69$	5.38197	0.647913
$70$	0	0
$71$	10.0902	1.19748	0.598741	−	0.800942i	$-0.295668\pi$
$71$	10.0902	1.19748	0.598741	+	0.800942i	$0.295668\pi$
$72$	−2.23607	−0.263523
$73$	15.7082	1.83851	0.919253	−	0.393667i	$-0.128794\pi$
$73$	15.7082	1.83851	0.919253	+	0.393667i	$0.128794\pi$
$74$	−3.23607	−0.376185
$75$	0	0
$76$	−4.14590	−0.475567
$77$	6.00000	0.683763
$78$	−1.61803	−0.183206
$79$	−9.14590	−1.02899	−0.514497	−	0.857492i	$-0.672021\pi$
$79$	−9.14590	−1.02899	−0.514497	+	0.857492i	$0.672021\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−15.1803	−1.67639
$83$	9.00000	0.987878	0.493939	−	0.869496i	$-0.335557\pi$
$83$	9.00000	0.987878	0.493939	+	0.869496i	$0.335557\pi$
$84$	−1.23607	−0.134866
$85$	0	0
$86$	−11.9443	−1.28798
$87$	−3.61803	−0.387894
$88$	6.70820	0.715097
$89$	−11.1803	−1.18511	−0.592557	−	0.805529i	$-0.701881\pi$
$89$	−11.1803	−1.18511	−0.592557	+	0.805529i	$0.701881\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	3.32624	0.346784
$93$	8.70820	0.902999
$94$	−7.70820	−0.795041
$95$	0	0
$96$	−3.38197	−0.345170
$97$	3.85410	0.391325	0.195662	−	0.980671i	$-0.437314\pi$
$97$	3.85410	0.391325	0.195662	+	0.980671i	$0.437314\pi$
$98$	−4.85410	−0.490338
$99$	−3.00000	−0.301511
$100$	0	0
$101$	−9.38197	−0.933541	−0.466770	−	0.884379i	$-0.654583\pi$
$101$	−9.38197	−0.933541	−0.466770	+	0.884379i	$0.654583\pi$
$102$	−6.85410	−0.678657
$103$	−14.4164	−1.42049	−0.710245	−	0.703954i	$-0.751416\pi$
$103$	−14.4164	−1.42049	−0.710245	+	0.703954i	$0.751416\pi$
$104$	2.23607	0.219265
$105$	0	0
$106$	18.1803	1.76583
$107$	−1.14590	−0.110778	−0.0553891	−	0.998465i	$-0.517640\pi$
$107$	−1.14590	−0.110778	−0.0553891	+	0.998465i	$0.517640\pi$
$108$	0.618034	0.0594703
$109$	−4.14590	−0.397105	−0.198553	−	0.980090i	$-0.563624\pi$
$109$	−4.14590	−0.397105	−0.198553	+	0.980090i	$0.563624\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	9.70820	0.917339
$113$	−3.76393	−0.354081	−0.177040	−	0.984204i	$-0.556652\pi$
$113$	−3.76393	−0.354081	−0.177040	+	0.984204i	$0.556652\pi$
$114$	−10.8541	−1.01658
$115$	0	0
$116$	−2.23607	−0.207614
$117$	−1.00000	−0.0924500
$118$	6.38197	0.587508
$119$	8.47214	0.776639
$120$	0	0
$121$	−2.00000	−0.181818
$122$	14.0902	1.27566
$123$	−9.38197	−0.845943
$124$	5.38197	0.483315
$125$	0	0
$126$	−3.23607	−0.288292
$127$	13.6525	1.21146	0.605731	−	0.795670i	$-0.292881\pi$
$127$	13.6525	1.21146	0.605731	+	0.795670i	$0.292881\pi$
$128$	13.6180	1.20368
$129$	−7.38197	−0.649946
$130$	0	0
$131$	−14.1803	−1.23894	−0.619471	−	0.785020i	$-0.712653\pi$
$131$	−14.1803	−1.23894	−0.619471	+	0.785020i	$0.712653\pi$
$132$	−1.85410	−0.161379
$133$	13.4164	1.16335
$134$	−21.3262	−1.84231
$135$	0	0
$136$	9.47214	0.812229
$137$	0.437694	0.0373947	0.0186974	−	0.999825i	$-0.494048\pi$
$137$	0.437694	0.0373947	0.0186974	+	0.999825i	$0.494048\pi$
$138$	8.70820	0.741292
$139$	13.4164	1.13796	0.568982	−	0.822350i	$-0.307337\pi$
$139$	13.4164	1.13796	0.568982	+	0.822350i	$0.307337\pi$
$140$	0	0
$141$	−4.76393	−0.401195
$142$	16.3262	1.37007
$143$	3.00000	0.250873
$144$	−4.85410	−0.404508
$145$	0	0
$146$	25.4164	2.10348
$147$	−3.00000	−0.247436
$148$	−1.23607	−0.101604
$149$	−13.0902	−1.07239	−0.536194	−	0.844095i	$-0.680139\pi$
$149$	−13.0902	−1.07239	−0.536194	+	0.844095i	$0.680139\pi$
$150$	0	0
$151$	−6.61803	−0.538568	−0.269284	−	0.963061i	$-0.586787\pi$
$151$	−6.61803	−0.538568	−0.269284	+	0.963061i	$0.586787\pi$
$152$	15.0000	1.21666
$153$	−4.23607	−0.342466
$154$	9.70820	0.782309
$155$	0	0
$156$	−0.618034	−0.0494823
$157$	−2.85410	−0.227782	−0.113891	−	0.993493i	$-0.536331\pi$
$157$	−2.85410	−0.227782	−0.113891	+	0.993493i	$0.536331\pi$
$158$	−14.7984	−1.17730
$159$	11.2361	0.891078
$160$	0	0
$161$	−10.7639	−0.848317
$162$	1.61803	0.127125
$163$	18.2705	1.43106	0.715528	−	0.698584i	$-0.246186\pi$
$163$	18.2705	1.43106	0.715528	+	0.698584i	$0.246186\pi$
$164$	−5.79837	−0.452777
$165$	0	0
$166$	14.5623	1.13025
$167$	17.7984	1.37728	0.688640	−	0.725104i	$-0.258208\pi$
$167$	17.7984	1.37728	0.688640	+	0.725104i	$0.258208\pi$
$168$	4.47214	0.345033
$169$	−12.0000	−0.923077
$170$	0	0
$171$	−6.70820	−0.512989
$172$	−4.56231	−0.347873
$173$	0.909830	0.0691731	0.0345865	−	0.999402i	$-0.488989\pi$
$173$	0.909830	0.0691731	0.0345865	+	0.999402i	$0.488989\pi$
$174$	−5.85410	−0.443798
$175$	0	0
$176$	14.5623	1.09768
$177$	3.94427	0.296470
$178$	−18.0902	−1.35592
$179$	−15.6525	−1.16992	−0.584960	−	0.811062i	$-0.698890\pi$
$179$	−15.6525	−1.16992	−0.584960	+	0.811062i	$0.698890\pi$
$180$	0	0
$181$	−12.4721	−0.927047	−0.463523	−	0.886085i	$-0.653415\pi$
$181$	−12.4721	−0.927047	−0.463523	+	0.886085i	$0.653415\pi$
$182$	3.23607	0.239873
$183$	8.70820	0.643729
$184$	−12.0344	−0.887191
$185$	0	0
$186$	14.0902	1.03314
$187$	12.7082	0.929316
$188$	−2.94427	−0.214733
$189$	−2.00000	−0.145479
$190$	0	0
$191$	17.3262	1.25368	0.626841	−	0.779147i	$-0.284347\pi$
$191$	17.3262	1.25368	0.626841	+	0.779147i	$0.284347\pi$
$192$	4.23607	0.305712
$193$	−11.0000	−0.791797	−0.395899	−	0.918294i	$-0.629567\pi$
$193$	−11.0000	−0.791797	−0.395899	+	0.918294i	$0.629567\pi$
$194$	6.23607	0.447724
$195$	0	0
$196$	−1.85410	−0.132436
$197$	−0.0901699	−0.00642434	−0.00321217	−	0.999995i	$-0.501022\pi$
$197$	−0.0901699	−0.00642434	−0.00321217	+	0.999995i	$0.501022\pi$
$198$	−4.85410	−0.344966
$199$	11.7082	0.829973	0.414986	−	0.909828i	$-0.363786\pi$
$199$	11.7082	0.829973	0.414986	+	0.909828i	$0.363786\pi$
$200$	0	0
$201$	−13.1803	−0.929669
$202$	−15.1803	−1.06808
$203$	7.23607	0.507872
$204$	−2.61803	−0.183299
$205$	0	0
$206$	−23.3262	−1.62522
$207$	5.38197	0.374072
$208$	4.85410	0.336571
$209$	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	$0.532929\pi$
$212$	6.94427	0.476935
$213$	10.0902	0.691367
$214$	−1.85410	−0.126744
$215$	0	0
$216$	−2.23607	−0.152145
$217$	−17.4164	−1.18230
$218$	−6.70820	−0.454337
$219$	15.7082	1.06146
$220$	0	0
$221$	4.23607	0.284949
$222$	−3.23607	−0.217191
$223$	−10.1459	−0.679420	−0.339710	−	0.940530i	$-0.610329\pi$
$223$	−10.1459	−0.679420	−0.339710	+	0.940530i	$0.610329\pi$
$224$	6.76393	0.451934
$225$	0	0
$226$	−6.09017	−0.405112
$227$	5.76393	0.382566	0.191283	−	0.981535i	$-0.438735\pi$
$227$	5.76393	0.382566	0.191283	+	0.981535i	$0.438735\pi$
$228$	−4.14590	−0.274569
$229$	−16.1803	−1.06923	−0.534613	−	0.845097i	$-0.679543\pi$
$229$	−16.1803	−1.06923	−0.534613	+	0.845097i	$0.679543\pi$
$230$	0	0
$231$	6.00000	0.394771
$232$	8.09017	0.531146
$233$	10.1803	0.666936	0.333468	−	0.942761i	$-0.391781\pi$
$233$	10.1803	0.666936	0.333468	+	0.942761i	$0.391781\pi$
$234$	−1.61803	−0.105774
$235$	0	0
$236$	2.43769	0.158680
$237$	−9.14590	−0.594090
$238$	13.7082	0.888571
$239$	21.3820	1.38308	0.691542	−	0.722336i	$-0.256932\pi$
$239$	21.3820	1.38308	0.691542	+	0.722336i	$0.256932\pi$
$240$	0	0
$241$	7.32624	0.471924	0.235962	−	0.971762i	$-0.424176\pi$
$241$	7.32624	0.471924	0.235962	+	0.971762i	$0.424176\pi$
$242$	−3.23607	−0.208022
$243$	1.00000	0.0641500
$244$	5.38197	0.344545
$245$	0	0
$246$	−15.1803	−0.967863
$247$	6.70820	0.426833
$248$	−19.4721	−1.23648
$249$	9.00000	0.570352
$250$	0	0
$251$	−18.9787	−1.19793	−0.598963	−	0.800777i	$-0.704420\pi$
$251$	−18.9787	−1.19793	−0.598963	+	0.800777i	$0.704420\pi$
$252$	−1.23607	−0.0778650
$253$	−16.1459	−1.01508
$254$	22.0902	1.38606
$255$	0	0
$256$	13.5623	0.847644
$257$	31.2148	1.94712	0.973562	−	0.228422i	$-0.0733565\pi$
$257$	31.2148	1.94712	0.973562	+	0.228422i	$0.0733565\pi$
$258$	−11.9443	−0.743618
$259$	4.00000	0.248548
$260$	0	0
$261$	−3.61803	−0.223951
$262$	−22.9443	−1.41750
$263$	−12.5066	−0.771189	−0.385594	−	0.922668i	$-0.626004\pi$
$263$	−12.5066	−0.771189	−0.385594	+	0.922668i	$0.626004\pi$
$264$	6.70820	0.412861
$265$	0	0
$266$	21.7082	1.33102
$267$	−11.1803	−0.684226
$268$	−8.14590	−0.497590
$269$	−20.5279	−1.25161	−0.625803	−	0.779981i	$-0.715229\pi$
$269$	−20.5279	−1.25161	−0.625803	+	0.779981i	$0.715229\pi$
$270$	0	0
$271$	−11.4164	−0.693497	−0.346749	−	0.937958i	$-0.612714\pi$
$271$	−11.4164	−0.693497	−0.346749	+	0.937958i	$0.612714\pi$
$272$	20.5623	1.24677
$273$	2.00000	0.121046
$274$	0.708204	0.0427842
$275$	0	0
$276$	3.32624	0.200216
$277$	−13.0557	−0.784443	−0.392221	−	0.919871i	$-0.628293\pi$
$277$	−13.0557	−0.784443	−0.392221	+	0.919871i	$0.628293\pi$
$278$	21.7082	1.30197
$279$	8.70820	0.521347
$280$	0	0
$281$	−14.1803	−0.845928	−0.422964	−	0.906146i	$-0.639010\pi$
$281$	−14.1803	−0.845928	−0.422964	+	0.906146i	$0.639010\pi$
$282$	−7.70820	−0.459017
$283$	2.29180	0.136233	0.0681166	−	0.997677i	$-0.478301\pi$
$283$	2.29180	0.136233	0.0681166	+	0.997677i	$0.478301\pi$
$284$	6.23607	0.370043
$285$	0	0
$286$	4.85410	0.287029
$287$	18.7639	1.10760
$288$	−3.38197	−0.199284
$289$	0.944272	0.0555454
$290$	0	0
$291$	3.85410	0.225931
$292$	9.70820	0.568130
$293$	−6.32624	−0.369583	−0.184791	−	0.982778i	$-0.559161\pi$
$293$	−6.32624	−0.369583	−0.184791	+	0.982778i	$0.559161\pi$
$294$	−4.85410	−0.283097
$295$	0	0
$296$	4.47214	0.259938
$297$	−3.00000	−0.174078
$298$	−21.1803	−1.22694
$299$	−5.38197	−0.311247
$300$	0	0
$301$	14.7639	0.850979
$302$	−10.7082	−0.616188
$303$	−9.38197	−0.538980
$304$	32.5623	1.86758
$305$	0	0
$306$	−6.85410	−0.391823
$307$	8.85410	0.505330	0.252665	−	0.967554i	$-0.418693\pi$
$307$	8.85410	0.505330	0.252665	+	0.967554i	$0.418693\pi$
$308$	3.70820	0.211295
$309$	−14.4164	−0.820121
$310$	0	0
$311$	−13.5279	−0.767095	−0.383547	−	0.923521i	$-0.625298\pi$
$311$	−13.5279	−0.767095	−0.383547	+	0.923521i	$0.625298\pi$
$312$	2.23607	0.126592
$313$	2.29180	0.129540	0.0647700	−	0.997900i	$-0.479369\pi$
$313$	2.29180	0.129540	0.0647700	+	0.997900i	$0.479369\pi$
$314$	−4.61803	−0.260611
$315$	0	0
$316$	−5.65248	−0.317977
$317$	20.5623	1.15489	0.577447	−	0.816428i	$-0.304049\pi$
$317$	20.5623	1.15489	0.577447	+	0.816428i	$0.304049\pi$
$318$	18.1803	1.01950
$319$	10.8541	0.607713
$320$	0	0
$321$	−1.14590	−0.0639578
$322$	−17.4164	−0.970578
$323$	28.4164	1.58113
$324$	0.618034	0.0343352
$325$	0	0
$326$	29.5623	1.63730
$327$	−4.14590	−0.229269
$328$	20.9787	1.15836
$329$	9.52786	0.525288
$330$	0	0
$331$	−30.6869	−1.68671	−0.843353	−	0.537360i	$-0.819422\pi$
$331$	−30.6869	−1.68671	−0.843353	+	0.537360i	$0.819422\pi$
$332$	5.56231	0.305271
$333$	−2.00000	−0.109599
$334$	28.7984	1.57578
$335$	0	0
$336$	9.70820	0.529626
$337$	−33.1803	−1.80745	−0.903724	−	0.428115i	$-0.859178\pi$
$337$	−33.1803	−1.80745	−0.903724	+	0.428115i	$0.859178\pi$
$338$	−19.4164	−1.05611
$339$	−3.76393	−0.204429
$340$	0	0
$341$	−26.1246	−1.41473
$342$	−10.8541	−0.586923
$343$	20.0000	1.07990
$344$	16.5066	0.889975
$345$	0	0
$346$	1.47214	0.0791425
$347$	12.2705	0.658715	0.329358	−	0.944205i	$-0.393168\pi$
$347$	12.2705	0.658715	0.329358	+	0.944205i	$0.393168\pi$
$348$	−2.23607	−0.119866
$349$	7.23607	0.387338	0.193669	−	0.981067i	$-0.437961\pi$
$349$	7.23607	0.387338	0.193669	+	0.981067i	$0.437961\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	10.1459	0.540778
$353$	−12.3820	−0.659026	−0.329513	−	0.944151i	$-0.606884\pi$
$353$	−12.3820	−0.659026	−0.329513	+	0.944151i	$0.606884\pi$
$354$	6.38197	0.339198
$355$	0	0
$356$	−6.90983	−0.366220
$357$	8.47214	0.448393
$358$	−25.3262	−1.33853
$359$	23.9443	1.26373	0.631865	−	0.775078i	$-0.282290\pi$
$359$	23.9443	1.26373	0.631865	+	0.775078i	$0.282290\pi$
$360$	0	0
$361$	26.0000	1.36842
$362$	−20.1803	−1.06066
$363$	−2.00000	−0.104973
$364$	1.23607	0.0647876
$365$	0	0
$366$	14.0902	0.736505
$367$	−12.5279	−0.653949	−0.326975	−	0.945033i	$-0.606029\pi$
$367$	−12.5279	−0.653949	−0.326975	+	0.945033i	$0.606029\pi$
$368$	−26.1246	−1.36184
$369$	−9.38197	−0.488406
$370$	0	0
$371$	−22.4721	−1.16670
$372$	5.38197	0.279042
$373$	17.4164	0.901787	0.450894	−	0.892578i	$-0.351105\pi$
$373$	17.4164	0.901787	0.450894	+	0.892578i	$0.351105\pi$
$374$	20.5623	1.06325
$375$	0	0
$376$	10.6525	0.549359
$377$	3.61803	0.186338
$378$	−3.23607	−0.166445
$379$	−13.6180	−0.699511	−0.349756	−	0.936841i	$-0.613735\pi$
$379$	−13.6180	−0.699511	−0.349756	+	0.936841i	$0.613735\pi$
$380$	0	0
$381$	13.6525	0.699438
$382$	28.0344	1.43437
$383$	5.05573	0.258336	0.129168	−	0.991623i	$-0.458769\pi$
$383$	5.05573	0.258336	0.129168	+	0.991623i	$0.458769\pi$
$384$	13.6180	0.694942
$385$	0	0
$386$	−17.7984	−0.905913
$387$	−7.38197	−0.375246
$388$	2.38197	0.120926
$389$	0.652476	0.0330818	0.0165409	−	0.999863i	$-0.494735\pi$
$389$	0.652476	0.0330818	0.0165409	+	0.999863i	$0.494735\pi$
$390$	0	0
$391$	−22.7984	−1.15296
$392$	6.70820	0.338815
$393$	−14.1803	−0.715304
$394$	−0.145898	−0.00735024
$395$	0	0
$396$	−1.85410	−0.0931721
$397$	−2.52786	−0.126870	−0.0634349	−	0.997986i	$-0.520206\pi$
$397$	−2.52786	−0.126870	−0.0634349	+	0.997986i	$0.520206\pi$
$398$	18.9443	0.949591
$399$	13.4164	0.671660
$400$	0	0
$401$	36.2705	1.81126	0.905631	−	0.424066i	$-0.139397\pi$
$401$	36.2705	1.81126	0.905631	+	0.424066i	$0.139397\pi$
$402$	−21.3262	−1.06366
$403$	−8.70820	−0.433787
$404$	−5.79837	−0.288480
$405$	0	0
$406$	11.7082	0.581068
$407$	6.00000	0.297409
$408$	9.47214	0.468941
$409$	−5.12461	−0.253396	−0.126698	−	0.991941i	$-0.540438\pi$
$409$	−5.12461	−0.253396	−0.126698	+	0.991941i	$0.540438\pi$
$410$	0	0
$411$	0.437694	0.0215899
$412$	−8.90983	−0.438956
$413$	−7.88854	−0.388170
$414$	8.70820	0.427985
$415$	0	0
$416$	3.38197	0.165815
$417$	13.4164	0.657004
$418$	32.5623	1.59267
$419$	0.326238	0.0159378	0.00796888	−	0.999968i	$-0.497463\pi$
$419$	0.326238	0.0159378	0.00796888	+	0.999968i	$0.497463\pi$
$420$	0	0
$421$	−30.3607	−1.47969	−0.739844	−	0.672778i	$-0.765101\pi$
$421$	−30.3607	−1.47969	−0.739844	+	0.672778i	$0.765101\pi$
$422$	−4.85410	−0.236294
$423$	−4.76393	−0.231630
$424$	−25.1246	−1.22016
$425$	0	0
$426$	16.3262	0.791009
$427$	−17.4164	−0.842839
$428$	−0.708204	−0.0342323
$429$	3.00000	0.144841
$430$	0	0
$431$	29.7639	1.43368	0.716839	−	0.697239i	$-0.245588\pi$
$431$	29.7639	1.43368	0.716839	+	0.697239i	$0.245588\pi$
$432$	−4.85410	−0.233543
$433$	3.47214	0.166860	0.0834301	−	0.996514i	$-0.473412\pi$
$433$	3.47214	0.166860	0.0834301	+	0.996514i	$0.473412\pi$
$434$	−28.1803	−1.35270
$435$	0	0
$436$	−2.56231	−0.122712
$437$	−36.1033	−1.72706
$438$	25.4164	1.21444
$439$	−32.0344	−1.52892	−0.764460	−	0.644671i	$-0.776994\pi$
$439$	−32.0344	−1.52892	−0.764460	+	0.644671i	$0.776994\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	6.85410	0.326016
$443$	−19.4164	−0.922501	−0.461251	−	0.887270i	$-0.652599\pi$
$443$	−19.4164	−0.922501	−0.461251	+	0.887270i	$0.652599\pi$
$444$	−1.23607	−0.0586612
$445$	0	0
$446$	−16.4164	−0.777339
$447$	−13.0902	−0.619144
$448$	−8.47214	−0.400271
$449$	16.5066	0.778994	0.389497	−	0.921028i	$-0.372649\pi$
$449$	16.5066	0.778994	0.389497	+	0.921028i	$0.372649\pi$
$450$	0	0
$451$	28.1459	1.32534
$452$	−2.32624	−0.109417
$453$	−6.61803	−0.310942
$454$	9.32624	0.437702
$455$	0	0
$456$	15.0000	0.702439
$457$	−9.88854	−0.462567	−0.231283	−	0.972886i	$-0.574292\pi$
$457$	−9.88854	−0.462567	−0.231283	+	0.972886i	$0.574292\pi$
$458$	−26.1803	−1.22333
$459$	−4.23607	−0.197723
$460$	0	0
$461$	−19.1803	−0.893317	−0.446659	−	0.894704i	$-0.647386\pi$
$461$	−19.1803	−0.893317	−0.446659	+	0.894704i	$0.647386\pi$
$462$	9.70820	0.451667
$463$	−33.6869	−1.56556	−0.782782	−	0.622296i	$-0.786200\pi$
$463$	−33.6869	−1.56556	−0.782782	+	0.622296i	$0.786200\pi$
$464$	17.5623	0.815310
$465$	0	0
$466$	16.4721	0.763057
$467$	−10.4164	−0.482014	−0.241007	−	0.970523i	$-0.577478\pi$
$467$	−10.4164	−0.482014	−0.241007	+	0.970523i	$0.577478\pi$
$468$	−0.618034	−0.0285686
$469$	26.3607	1.21722
$470$	0	0
$471$	−2.85410	−0.131510
$472$	−8.81966	−0.405958
$473$	22.1459	1.01827
$474$	−14.7984	−0.679712
$475$	0	0
$476$	5.23607	0.239995
$477$	11.2361	0.514464
$478$	34.5967	1.58242
$479$	4.79837	0.219243	0.109622	−	0.993973i	$-0.465036\pi$
$479$	4.79837	0.219243	0.109622	+	0.993973i	$0.465036\pi$
$480$	0	0
$481$	2.00000	0.0911922
$482$	11.8541	0.539940
$483$	−10.7639	−0.489776
$484$	−1.23607	−0.0561849
$485$	0	0
$486$	1.61803	0.0733955
$487$	16.6180	0.753035	0.376518	−	0.926410i	$-0.377121\pi$
$487$	16.6180	0.753035	0.376518	+	0.926410i	$0.377121\pi$
$488$	−19.4721	−0.881462
$489$	18.2705	0.826221
$490$	0	0
$491$	22.3262	1.00757	0.503785	−	0.863829i	$-0.331941\pi$
$491$	22.3262	1.00757	0.503785	+	0.863829i	$0.331941\pi$
$492$	−5.79837	−0.261411
$493$	15.3262	0.690259
$494$	10.8541	0.488349
$495$	0	0
$496$	−42.2705	−1.89800
$497$	−20.1803	−0.905212
$498$	14.5623	0.652553
$499$	−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
$501$	17.7984	0.795173
$502$	−30.7082	−1.37057
$503$	−3.96556	−0.176815	−0.0884077	−	0.996084i	$-0.528178\pi$
$503$	−3.96556	−0.176815	−0.0884077	+	0.996084i	$0.528178\pi$
$504$	4.47214	0.199205
$505$	0	0
$506$	−26.1246	−1.16138
$507$	−12.0000	−0.532939
$508$	8.43769	0.374362
$509$	32.8885	1.45776	0.728880	−	0.684642i	$-0.240041\pi$
$509$	32.8885	1.45776	0.728880	+	0.684642i	$0.240041\pi$
$510$	0	0
$511$	−31.4164	−1.38978
$512$	−5.29180	−0.233867
$513$	−6.70820	−0.296174
$514$	50.5066	2.22775
$515$	0	0
$516$	−4.56231	−0.200844
$517$	14.2918	0.628552
$518$	6.47214	0.284369
$519$	0.909830	0.0399371
$520$	0	0
$521$	40.0902	1.75638	0.878191	−	0.478310i	$-0.158750\pi$
$521$	40.0902	1.75638	0.878191	+	0.478310i	$0.158750\pi$
$522$	−5.85410	−0.256227
$523$	1.56231	0.0683149	0.0341574	−	0.999416i	$-0.489125\pi$
$523$	1.56231	0.0683149	0.0341574	+	0.999416i	$0.489125\pi$
$524$	−8.76393	−0.382854
$525$	0	0
$526$	−20.2361	−0.882334
$527$	−36.8885	−1.60689
$528$	14.5623	0.633743
$529$	5.96556	0.259372
$530$	0	0
$531$	3.94427	0.171167
$532$	8.29180	0.359495
$533$	9.38197	0.406378
$534$	−18.0902	−0.782838
$535$	0	0
$536$	29.4721	1.27300
$537$	−15.6525	−0.675454
$538$	−33.2148	−1.43199
$539$	9.00000	0.387657
$540$	0	0
$541$	−26.2918	−1.13037	−0.565186	−	0.824963i	$-0.691196\pi$
$541$	−26.2918	−1.13037	−0.565186	+	0.824963i	$0.691196\pi$
$542$	−18.4721	−0.793446
$543$	−12.4721	−0.535231
$544$	14.3262	0.614232
$545$	0	0
$546$	3.23607	0.138491
$547$	24.7082	1.05645	0.528223	−	0.849106i	$-0.322858\pi$
$547$	24.7082	1.05645	0.528223	+	0.849106i	$0.322858\pi$
$548$	0.270510	0.0115556
$549$	8.70820	0.371657
$550$	0	0
$551$	24.2705	1.03396
$552$	−12.0344	−0.512220
$553$	18.2918	0.777846
$554$	−21.1246	−0.897499
$555$	0	0
$556$	8.29180	0.351650
$557$	−37.6525	−1.59539	−0.797693	−	0.603063i	$-0.793947\pi$
$557$	−37.6525	−1.59539	−0.797693	+	0.603063i	$0.793947\pi$
$558$	14.0902	0.596484
$559$	7.38197	0.312224
$560$	0	0
$561$	12.7082	0.536541
$562$	−22.9443	−0.967846
$563$	9.00000	0.379305	0.189652	−	0.981851i	$-0.439264\pi$
$563$	9.00000	0.379305	0.189652	+	0.981851i	$0.439264\pi$
$564$	−2.94427	−0.123976
$565$	0	0
$566$	3.70820	0.155867
$567$	−2.00000	−0.0839921
$568$	−22.5623	−0.946693
$569$	−10.8541	−0.455028	−0.227514	−	0.973775i	$-0.573060\pi$
$569$	−10.8541	−0.455028	−0.227514	+	0.973775i	$0.573060\pi$
$570$	0	0
$571$	−38.1246	−1.59547	−0.797733	−	0.603011i	$-0.793967\pi$
$571$	−38.1246	−1.59547	−0.797733	+	0.603011i	$0.793967\pi$
$572$	1.85410	0.0775239
$573$	17.3262	0.723814
$574$	30.3607	1.26723
$575$	0	0
$576$	4.23607	0.176503
$577$	3.72949	0.155261	0.0776304	−	0.996982i	$-0.475265\pi$
$577$	3.72949	0.155261	0.0776304	+	0.996982i	$0.475265\pi$
$578$	1.52786	0.0635508
$579$	−11.0000	−0.457144
$580$	0	0
$581$	−18.0000	−0.746766
$582$	6.23607	0.258493
$583$	−33.7082	−1.39605
$584$	−35.1246	−1.45347
$585$	0	0
$586$	−10.2361	−0.422848
$587$	39.3050	1.62229	0.811144	−	0.584846i	$-0.198845\pi$
$587$	39.3050	1.62229	0.811144	+	0.584846i	$0.198845\pi$
$588$	−1.85410	−0.0764619
$589$	−58.4164	−2.40701
$590$	0	0
$591$	−0.0901699	−0.00370910
$592$	9.70820	0.399005
$593$	17.6180	0.723486	0.361743	−	0.932278i	$-0.382182\pi$
$593$	17.6180	0.723486	0.361743	+	0.932278i	$0.382182\pi$
$594$	−4.85410	−0.199166
$595$	0	0
$596$	−8.09017	−0.331386
$597$	11.7082	0.479185
$598$	−8.70820	−0.356105
$599$	39.2705	1.60455	0.802275	−	0.596955i	$-0.203623\pi$
$599$	39.2705	1.60455	0.802275	+	0.596955i	$0.203623\pi$
$600$	0	0
$601$	−24.7082	−1.00787	−0.503934	−	0.863742i	$-0.668115\pi$
$601$	−24.7082	−1.00787	−0.503934	+	0.863742i	$0.668115\pi$
$602$	23.8885	0.973624
$603$	−13.1803	−0.536745
$604$	−4.09017	−0.166427
$605$	0	0
$606$	−15.1803	−0.616659
$607$	−22.8541	−0.927619	−0.463810	−	0.885935i	$-0.653518\pi$
$607$	−22.8541	−0.927619	−0.463810	+	0.885935i	$0.653518\pi$
$608$	22.6869	0.920076
$609$	7.23607	0.293220
$610$	0	0
$611$	4.76393	0.192728
$612$	−2.61803	−0.105828
$613$	−5.87539	−0.237305	−0.118652	−	0.992936i	$-0.537857\pi$
$613$	−5.87539	−0.237305	−0.118652	+	0.992936i	$0.537857\pi$
$614$	14.3262	0.578160
$615$	0	0
$616$	−13.4164	−0.540562
$617$	25.2361	1.01597	0.507983	−	0.861367i	$-0.330391\pi$
$617$	25.2361	1.01597	0.507983	+	0.861367i	$0.330391\pi$
$618$	−23.3262	−0.938319
$619$	34.2705	1.37745	0.688724	−	0.725024i	$-0.258171\pi$
$619$	34.2705	1.37745	0.688724	+	0.725024i	$0.258171\pi$
$620$	0	0
$621$	5.38197	0.215971
$622$	−21.8885	−0.877651
$623$	22.3607	0.895862
$624$	4.85410	0.194320
$625$	0	0
$626$	3.70820	0.148210
$627$	20.1246	0.803700
$628$	−1.76393	−0.0703886
$629$	8.47214	0.337806
$630$	0	0
$631$	−10.7639	−0.428505	−0.214253	−	0.976778i	$-0.568732\pi$
$631$	−10.7639	−0.428505	−0.214253	+	0.976778i	$0.568732\pi$
$632$	20.4508	0.813491
$633$	−3.00000	−0.119239
$634$	33.2705	1.32134
$635$	0	0
$636$	6.94427	0.275358
$637$	3.00000	0.118864
$638$	17.5623	0.695298
$639$	10.0902	0.399161
$640$	0	0
$641$	−23.3262	−0.921331	−0.460666	−	0.887574i	$-0.652389\pi$
$641$	−23.3262	−0.921331	−0.460666	+	0.887574i	$0.652389\pi$
$642$	−1.85410	−0.0731756
$643$	5.90983	0.233061	0.116530	−	0.993187i	$-0.462823\pi$
$643$	5.90983	0.233061	0.116530	+	0.993187i	$0.462823\pi$
$644$	−6.65248	−0.262144
$645$	0	0
$646$	45.9787	1.80901
$647$	−19.0344	−0.748321	−0.374161	−	0.927364i	$-0.622069\pi$
$647$	−19.0344	−0.748321	−0.374161	+	0.927364i	$0.622069\pi$
$648$	−2.23607	−0.0878410
$649$	−11.8328	−0.464479
$650$	0	0
$651$	−17.4164	−0.682603
$652$	11.2918	0.442221
$653$	29.6525	1.16039	0.580196	−	0.814477i	$-0.302976\pi$
$653$	29.6525	1.16039	0.580196	+	0.814477i	$0.302976\pi$
$654$	−6.70820	−0.262312
$655$	0	0
$656$	45.5410	1.77808
$657$	15.7082	0.612835
$658$	15.4164	0.600994
$659$	−2.23607	−0.0871048	−0.0435524	−	0.999051i	$-0.513868\pi$
$659$	−2.23607	−0.0871048	−0.0435524	+	0.999051i	$0.513868\pi$
$660$	0	0
$661$	4.88854	0.190142	0.0950712	−	0.995470i	$-0.469692\pi$
$661$	4.88854	0.190142	0.0950712	+	0.995470i	$0.469692\pi$
$662$	−49.6525	−1.92980
$663$	4.23607	0.164515
$664$	−20.1246	−0.780986
$665$	0	0
$666$	−3.23607	−0.125395
$667$	−19.4721	−0.753964
$668$	11.0000	0.425603
$669$	−10.1459	−0.392263
$670$	0	0
$671$	−26.1246	−1.00853
$672$	6.76393	0.260924
$673$	−46.7771	−1.80312	−0.901562	−	0.432650i	$-0.857579\pi$
$673$	−46.7771	−1.80312	−0.901562	+	0.432650i	$0.857579\pi$
$674$	−53.6869	−2.06794
$675$	0	0
$676$	−7.41641	−0.285246
$677$	−44.8885	−1.72521	−0.862603	−	0.505881i	$-0.831168\pi$
$677$	−44.8885	−1.72521	−0.862603	+	0.505881i	$0.831168\pi$
$678$	−6.09017	−0.233892
$679$	−7.70820	−0.295814
$680$	0	0
$681$	5.76393	0.220874
$682$	−42.2705	−1.61862
$683$	−0.596748	−0.0228339	−0.0114170	−	0.999935i	$-0.503634\pi$
$683$	−0.596748	−0.0228339	−0.0114170	+	0.999935i	$0.503634\pi$
$684$	−4.14590	−0.158522
$685$	0	0
$686$	32.3607	1.23554
$687$	−16.1803	−0.617318
$688$	35.8328	1.36611
$689$	−11.2361	−0.428060
$690$	0	0
$691$	15.0902	0.574057	0.287029	−	0.957922i	$-0.407333\pi$
$691$	15.0902	0.574057	0.287029	+	0.957922i	$0.407333\pi$
$692$	0.562306	0.0213757
$693$	6.00000	0.227921
$694$	19.8541	0.753651
$695$	0	0
$696$	8.09017	0.306657
$697$	39.7426	1.50536
$698$	11.7082	0.443162
$699$	10.1803	0.385056
$700$	0	0
$701$	−48.6525	−1.83758	−0.918789	−	0.394748i	$-0.870832\pi$
$701$	−48.6525	−1.83758	−0.918789	+	0.394748i	$0.870832\pi$
$702$	−1.61803	−0.0610688
$703$	13.4164	0.506009
$704$	−12.7082	−0.478958
$705$	0	0
$706$	−20.0344	−0.754006
$707$	18.7639	0.705690
$708$	2.43769	0.0916142
$709$	−5.20163	−0.195351	−0.0976756	−	0.995218i	$-0.531141\pi$
$709$	−5.20163	−0.195351	−0.0976756	+	0.995218i	$0.531141\pi$
$710$	0	0
$711$	−9.14590	−0.342998
$712$	25.0000	0.936915
$713$	46.8673	1.75519
$714$	13.7082	0.513017
$715$	0	0
$716$	−9.67376	−0.361525
$717$	21.3820	0.798524
$718$	38.7426	1.44586
$719$	−35.1246	−1.30993	−0.654963	−	0.755661i	$-0.727316\pi$
$719$	−35.1246	−1.30993	−0.654963	+	0.755661i	$0.727316\pi$
$720$	0	0
$721$	28.8328	1.07379
$722$	42.0689	1.56564
$723$	7.32624	0.272466
$724$	−7.70820	−0.286473
$725$	0	0
$726$	−3.23607	−0.120102
$727$	−14.4377	−0.535464	−0.267732	−	0.963493i	$-0.586274\pi$
$727$	−14.4377	−0.535464	−0.267732	+	0.963493i	$0.586274\pi$
$728$	−4.47214	−0.165748
$729$	1.00000	0.0370370
$730$	0	0
$731$	31.2705	1.15658
$732$	5.38197	0.198923
$733$	−10.1459	−0.374747	−0.187374	−	0.982289i	$-0.559998\pi$
$733$	−10.1459	−0.374747	−0.187374	+	0.982289i	$0.559998\pi$
$734$	−20.2705	−0.748198
$735$	0	0
$736$	−18.2016	−0.670921
$737$	39.5410	1.45651
$738$	−15.1803	−0.558796
$739$	−1.70820	−0.0628373	−0.0314186	−	0.999506i	$-0.510003\pi$
$739$	−1.70820	−0.0628373	−0.0314186	+	0.999506i	$0.510003\pi$
$740$	0	0
$741$	6.70820	0.246432
$742$	−36.3607	−1.33484
$743$	−16.5279	−0.606349	−0.303174	−	0.952935i	$-0.598046\pi$
$743$	−16.5279	−0.606349	−0.303174	+	0.952935i	$0.598046\pi$
$744$	−19.4721	−0.713883
$745$	0	0
$746$	28.1803	1.03176
$747$	9.00000	0.329293
$748$	7.85410	0.287174
$749$	2.29180	0.0837404
$750$	0	0
$751$	15.2918	0.558006	0.279003	−	0.960290i	$-0.409996\pi$
$751$	15.2918	0.558006	0.279003	+	0.960290i	$0.409996\pi$
$752$	23.1246	0.843268
$753$	−18.9787	−0.691623
$754$	5.85410	0.213194
$755$	0	0
$756$	−1.23607	−0.0449554
$757$	32.2705	1.17289	0.586446	−	0.809988i	$-0.300527\pi$
$757$	32.2705	1.17289	0.586446	+	0.809988i	$0.300527\pi$
$758$	−22.0344	−0.800327
$759$	−16.1459	−0.586059
$760$	0	0
$761$	3.18034	0.115287	0.0576436	−	0.998337i	$-0.481641\pi$
$761$	3.18034	0.115287	0.0576436	+	0.998337i	$0.481641\pi$
$762$	22.0902	0.800242
$763$	8.29180	0.300183
$764$	10.7082	0.387409
$765$	0	0
$766$	8.18034	0.295568
$767$	−3.94427	−0.142419
$768$	13.5623	0.489388
$769$	−36.3050	−1.30919	−0.654595	−	0.755980i	$-0.727161\pi$
$769$	−36.3050	−1.30919	−0.654595	+	0.755980i	$0.727161\pi$
$770$	0	0
$771$	31.2148	1.12417
$772$	−6.79837	−0.244679
$773$	−41.7771	−1.50262	−0.751309	−	0.659951i	$-0.770577\pi$
$773$	−41.7771	−1.50262	−0.751309	+	0.659951i	$0.770577\pi$
$774$	−11.9443	−0.429328
$775$	0	0
$776$	−8.61803	−0.309369
$777$	4.00000	0.143499
$778$	1.05573	0.0378497
$779$	62.9361	2.25492
$780$	0	0
$781$	−30.2705	−1.08316
$782$	−36.8885	−1.31913
$783$	−3.61803	−0.129298
$784$	14.5623	0.520082
$785$	0	0
$786$	−22.9443	−0.818395
$787$	17.1459	0.611185	0.305593	−	0.952162i	$-0.401145\pi$
$787$	17.1459	0.611185	0.305593	+	0.952162i	$0.401145\pi$
$788$	−0.0557281	−0.00198523
$789$	−12.5066	−0.445246
$790$	0	0
$791$	7.52786	0.267660
$792$	6.70820	0.238366
$793$	−8.70820	−0.309237
$794$	−4.09017	−0.145155
$795$	0	0
$796$	7.23607	0.256476
$797$	−30.0132	−1.06312	−0.531560	−	0.847020i	$-0.678394\pi$
$797$	−30.0132	−1.06312	−0.531560	+	0.847020i	$0.678394\pi$
$798$	21.7082	0.768462
$799$	20.1803	0.713929
$800$	0	0
$801$	−11.1803	−0.395038
$802$	58.6869	2.07231
$803$	−47.1246	−1.66299
$804$	−8.14590	−0.287284
$805$	0	0
$806$	−14.0902	−0.496305
$807$	−20.5279	−0.722615
$808$	20.9787	0.738029
$809$	−39.9230	−1.40362	−0.701809	−	0.712365i	$-0.747624\pi$
$809$	−39.9230	−1.40362	−0.701809	+	0.712365i	$0.747624\pi$
$810$	0	0
$811$	−33.7771	−1.18607	−0.593037	−	0.805175i	$-0.702071\pi$
$811$	−33.7771	−1.18607	−0.593037	+	0.805175i	$0.702071\pi$
$812$	4.47214	0.156941
$813$	−11.4164	−0.400391
$814$	9.70820	0.340272
$815$	0	0
$816$	20.5623	0.719825
$817$	49.5197	1.73248
$818$	−8.29180	−0.289916
$819$	2.00000	0.0698857
$820$	0	0
$821$	5.94427	0.207457	0.103728	−	0.994606i	$-0.466923\pi$
$821$	5.94427	0.207457	0.103728	+	0.994606i	$0.466923\pi$
$822$	0.708204	0.0247014
$823$	−28.5623	−0.995619	−0.497810	−	0.867286i	$-0.665862\pi$
$823$	−28.5623	−0.995619	−0.497810	+	0.867286i	$0.665862\pi$
$824$	32.2361	1.12300
$825$	0	0
$826$	−12.7639	−0.444114
$827$	48.9787	1.70316	0.851578	−	0.524227i	$-0.175646\pi$
$827$	48.9787	1.70316	0.851578	+	0.524227i	$0.175646\pi$
$828$	3.32624	0.115595
$829$	−50.1246	−1.74090	−0.870450	−	0.492257i	$-0.836172\pi$
$829$	−50.1246	−1.74090	−0.870450	+	0.492257i	$0.836172\pi$
$830$	0	0
$831$	−13.0557	−0.452898
$832$	−4.23607	−0.146859
$833$	12.7082	0.440313
$834$	21.7082	0.751694
$835$	0	0
$836$	12.4377	0.430167
$837$	8.70820	0.301000
$838$	0.527864	0.0182348
$839$	−3.21478	−0.110987	−0.0554933	−	0.998459i	$-0.517673\pi$
$839$	−3.21478	−0.110987	−0.0554933	+	0.998459i	$0.517673\pi$
$840$	0	0
$841$	−15.9098	−0.548615
$842$	−49.1246	−1.69295
$843$	−14.1803	−0.488397
$844$	−1.85410	−0.0638208
$845$	0	0
$846$	−7.70820	−0.265014
$847$	4.00000	0.137442
$848$	−54.5410	−1.87295
$849$	2.29180	0.0786542
$850$	0	0
$851$	−10.7639	−0.368983
$852$	6.23607	0.213644
$853$	−20.3951	−0.698316	−0.349158	−	0.937064i	$-0.613532\pi$
$853$	−20.3951	−0.698316	−0.349158	+	0.937064i	$0.613532\pi$
$854$	−28.1803	−0.964311
$855$	0	0
$856$	2.56231	0.0875778
$857$	9.05573	0.309338	0.154669	−	0.987966i	$-0.450569\pi$
$857$	9.05573	0.309338	0.154669	+	0.987966i	$0.450569\pi$
$858$	4.85410	0.165716
$859$	15.1246	0.516045	0.258023	−	0.966139i	$-0.416929\pi$
$859$	15.1246	0.516045	0.258023	+	0.966139i	$0.416929\pi$
$860$	0	0
$861$	18.7639	0.639473
$862$	48.1591	1.64030
$863$	13.0689	0.444870	0.222435	−	0.974948i	$-0.428599\pi$
$863$	13.0689	0.444870	0.222435	+	0.974948i	$0.428599\pi$
$864$	−3.38197	−0.115057
$865$	0	0
$866$	5.61803	0.190909
$867$	0.944272	0.0320692
$868$	−10.7639	−0.365352
$869$	27.4377	0.930760
$870$	0	0
$871$	13.1803	0.446599
$872$	9.27051	0.313939
$873$	3.85410	0.130442
$874$	−58.4164	−1.97596
$875$	0	0
$876$	9.70820	0.328010
$877$	43.1246	1.45621	0.728107	−	0.685463i	$-0.240400\pi$
$877$	43.1246	1.45621	0.728107	+	0.685463i	$0.240400\pi$
$878$	−51.8328	−1.74927
$879$	−6.32624	−0.213379
$880$	0	0
$881$	15.0902	0.508401	0.254200	−	0.967152i	$-0.418188\pi$
$881$	15.0902	0.508401	0.254200	+	0.967152i	$0.418188\pi$
$882$	−4.85410	−0.163446
$883$	29.2016	0.982713	0.491356	−	0.870959i	$-0.336501\pi$
$883$	29.2016	0.982713	0.491356	+	0.870959i	$0.336501\pi$
$884$	2.61803	0.0880540
$885$	0	0
$886$	−31.4164	−1.05545
$887$	42.9230	1.44121	0.720606	−	0.693344i	$-0.243864\pi$
$887$	42.9230	1.44121	0.720606	+	0.693344i	$0.243864\pi$
$888$	4.47214	0.150075
$889$	−27.3050	−0.915779
$890$	0	0
$891$	−3.00000	−0.100504
$892$	−6.27051	−0.209952
$893$	31.9574	1.06941
$894$	−21.1803	−0.708377
$895$	0	0
$896$	−27.2361	−0.909893
$897$	−5.38197	−0.179699
$898$	26.7082	0.891264
$899$	−31.5066	−1.05080
$900$	0	0
$901$	−47.5967	−1.58568
$902$	45.5410	1.51635
$903$	14.7639	0.491313
$904$	8.41641	0.279926
$905$	0	0
$906$	−10.7082	−0.355756
$907$	−17.0000	−0.564476	−0.282238	−	0.959344i	$-0.591077\pi$
$907$	−17.0000	−0.564476	−0.282238	+	0.959344i	$0.591077\pi$
$908$	3.56231	0.118219
$909$	−9.38197	−0.311180
$910$	0	0
$911$	14.8885	0.493279	0.246640	−	0.969107i	$-0.420674\pi$
$911$	14.8885	0.493279	0.246640	+	0.969107i	$0.420674\pi$
$912$	32.5623	1.07825
$913$	−27.0000	−0.893570
$914$	−16.0000	−0.529233
$915$	0	0
$916$	−10.0000	−0.330409
$917$	28.3607	0.936552
$918$	−6.85410	−0.226219
$919$	−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
$921$	8.85410	0.291753
$922$	−31.0344	−1.02206
$923$	−10.0902	−0.332122
$924$	3.70820	0.121991
$925$	0	0
$926$	−54.5066	−1.79120
$927$	−14.4164	−0.473497
$928$	12.2361	0.401669
$929$	−29.5967	−0.971038	−0.485519	−	0.874226i	$-0.661369\pi$
$929$	−29.5967	−0.971038	−0.485519	+	0.874226i	$0.661369\pi$
$930$	0	0
$931$	20.1246	0.659558
$932$	6.29180	0.206095
$933$	−13.5279	−0.442882
$934$	−16.8541	−0.551483
$935$	0	0
$936$	2.23607	0.0730882
$937$	−10.4164	−0.340289	−0.170145	−	0.985419i	$-0.554423\pi$
$937$	−10.4164	−0.340289	−0.170145	+	0.985419i	$0.554423\pi$
$938$	42.6525	1.39265
$939$	2.29180	0.0747899
$940$	0	0
$941$	57.9787	1.89005	0.945026	−	0.326995i	$-0.106036\pi$
$941$	57.9787	1.89005	0.945026	+	0.326995i	$0.106036\pi$
$942$	−4.61803	−0.150464
$943$	−50.4934	−1.64429
$944$	−19.1459	−0.623146
$945$	0	0
$946$	35.8328	1.16503
$947$	−23.8328	−0.774462	−0.387231	−	0.921983i	$-0.626568\pi$
$947$	−23.8328	−0.774462	−0.387231	+	0.921983i	$0.626568\pi$
$948$	−5.65248	−0.183584
$949$	−15.7082	−0.509910
$950$	0	0
$951$	20.5623	0.666778
$952$	−18.9443	−0.613987
$953$	−42.0557	−1.36232	−0.681159	−	0.732135i	$-0.738524\pi$
$953$	−42.0557	−1.36232	−0.681159	+	0.732135i	$0.738524\pi$
$954$	18.1803	0.588610
$955$	0	0
$956$	13.2148	0.427397
$957$	10.8541	0.350863
$958$	7.76393	0.250841
$959$	−0.875388	−0.0282678
$960$	0	0
$961$	44.8328	1.44622
$962$	3.23607	0.104335
$963$	−1.14590	−0.0369260
$964$	4.52786	0.145833
$965$	0	0
$966$	−17.4164	−0.560364
$967$	−35.4164	−1.13891	−0.569457	−	0.822021i	$-0.692847\pi$
$967$	−35.4164	−1.13891	−0.569457	+	0.822021i	$0.692847\pi$
$968$	4.47214	0.143740
$969$	28.4164	0.912867
$970$	0	0
$971$	29.8885	0.959169	0.479585	−	0.877496i	$-0.340787\pi$
$971$	29.8885	0.959169	0.479585	+	0.877496i	$0.340787\pi$
$972$	0.618034	0.0198234
$973$	−26.8328	−0.860221
$974$	26.8885	0.861565
$975$	0	0
$976$	−42.2705	−1.35305
$977$	−37.6525	−1.20461	−0.602305	−	0.798266i	$-0.705751\pi$
$977$	−37.6525	−1.20461	−0.602305	+	0.798266i	$0.705751\pi$
$978$	29.5623	0.945298
$979$	33.5410	1.07198
$980$	0	0
$981$	−4.14590	−0.132368
$982$	36.1246	1.15278
$983$	−24.6180	−0.785193	−0.392597	−	0.919711i	$-0.628423\pi$
$983$	−24.6180	−0.785193	−0.392597	+	0.919711i	$0.628423\pi$
$984$	20.9787	0.668777
$985$	0	0
$986$	24.7984	0.789741
$987$	9.52786	0.303275
$988$	4.14590	0.131899
$989$	−39.7295	−1.26332
$990$	0	0
$991$	32.0000	1.01651	0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000	1.01651	0.508257	+	0.861206i	$0.330290\pi$
$992$	−29.4508	−0.935065
$993$	−30.6869	−0.973820
$994$	−32.6525	−1.03567
$995$	0	0
$996$	5.56231	0.176248
$997$	−2.93112	−0.0928294	−0.0464147	−	0.998922i	$-0.514780\pi$
$997$	−2.93112	−0.0928294	−0.0464147	+	0.998922i	$0.514780\pi$
$998$	−24.2705	−0.768270
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1875.2.a.c.1.2	yes	2
3.2	odd	2		5625.2.a.b.1.1		2
5.2	odd	4		1875.2.b.a.1249.4		4
5.3	odd	4		1875.2.b.a.1249.1		4
5.4	even	2		1875.2.a.b.1.1	✓	2
15.14	odd	2		5625.2.a.g.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1875.2.a.b.1.1	✓	2	5.4	even	2
1875.2.a.c.1.2	yes	2	1.1	even	1	trivial
1875.2.b.a.1249.1		4	5.3	odd	4
1875.2.b.a.1249.4		4	5.2	odd	4
5625.2.a.b.1.1		2	3.2	odd	2
5625.2.a.g.1.2		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)

\(f(q)\)	\(=\)	\(q+1.61803 q^{2} +1.00000 q^{3} +0.618034 q^{4} +1.61803 q^{6} -2.00000 q^{7} -2.23607 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.61803 q^{2} +1.00000 q^{3} +0.618034 q^{4} +1.61803 q^{6} -2.00000 q^{7} -2.23607 q^{8} +1.00000 q^{9} -3.00000 q^{11} +0.618034 q^{12} -1.00000 q^{13} -3.23607 q^{14} -4.85410 q^{16} -4.23607 q^{17} +1.61803 q^{18} -6.70820 q^{19} -2.00000 q^{21} -4.85410 q^{22} +5.38197 q^{23} -2.23607 q^{24} -1.61803 q^{26} +1.00000 q^{27} -1.23607 q^{28} -3.61803 q^{29} +8.70820 q^{31} -3.38197 q^{32} -3.00000 q^{33} -6.85410 q^{34} +0.618034 q^{36} -2.00000 q^{37} -10.8541 q^{38} -1.00000 q^{39} -9.38197 q^{41} -3.23607 q^{42} -7.38197 q^{43} -1.85410 q^{44} +8.70820 q^{46} -4.76393 q^{47} -4.85410 q^{48} -3.00000 q^{49} -4.23607 q^{51} -0.618034 q^{52} +11.2361 q^{53} +1.61803 q^{54} +4.47214 q^{56} -6.70820 q^{57} -5.85410 q^{58} +3.94427 q^{59} +8.70820 q^{61} +14.0902 q^{62} -2.00000 q^{63} +4.23607 q^{64} -4.85410 q^{66} -13.1803 q^{67} -2.61803 q^{68} +5.38197 q^{69} +10.0902 q^{71} -2.23607 q^{72} +15.7082 q^{73} -3.23607 q^{74} -4.14590 q^{76} +6.00000 q^{77} -1.61803 q^{78} -9.14590 q^{79} +1.00000 q^{81} -15.1803 q^{82} +9.00000 q^{83} -1.23607 q^{84} -11.9443 q^{86} -3.61803 q^{87} +6.70820 q^{88} -11.1803 q^{89} +2.00000 q^{91} +3.32624 q^{92} +8.70820 q^{93} -7.70820 q^{94} -3.38197 q^{96} +3.85410 q^{97} -4.85410 q^{98} -3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} + 2 q^{3} - q^{4} + q^{6} - 4 q^{7} + 2 q^{9}+O(q^{10}) \) 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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