LMFDB - Embedded newform 8025.2.a.bc.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8025,2,Mod(1,8025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8025, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8025 = 3 \cdot 5^{2} \cdot 107 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8025.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:	$64.0799476221$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 13x^{8} + 26x^{7} + 51x^{6} - 101x^{5} - 65x^{4} + 126x^{3} + 5x^{2} - 10x + 1 $ x^10 - 2x^9 - 13x^8 + 26x^7 + 51x^6 - 101x^5 - 65x^4 + 126x^3 + 5x^2 - 10*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1605)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-1.67234$ of defining polynomial
Character	$\chi$	$=$	8025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.67234 q^{2} +1.00000 q^{3} +0.796721 q^{4} +1.67234 q^{6} -4.55690 q^{7} -2.01229 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.67234 q^{2} +1.00000 q^{3} +0.796721 q^{4} +1.67234 q^{6} -4.55690 q^{7} -2.01229 q^{8} +1.00000 q^{9} +0.631130 q^{11} +0.796721 q^{12} +6.53883 q^{13} -7.62069 q^{14} -4.95868 q^{16} +1.03632 q^{17} +1.67234 q^{18} -1.10775 q^{19} -4.55690 q^{21} +1.05546 q^{22} +1.82732 q^{23} -2.01229 q^{24} +10.9351 q^{26} +1.00000 q^{27} -3.63058 q^{28} -9.74810 q^{29} +9.43646 q^{31} -4.26801 q^{32} +0.631130 q^{33} +1.73308 q^{34} +0.796721 q^{36} -3.35748 q^{37} -1.85254 q^{38} +6.53883 q^{39} -4.44578 q^{41} -7.62069 q^{42} -10.4172 q^{43} +0.502834 q^{44} +3.05590 q^{46} -6.43490 q^{47} -4.95868 q^{48} +13.7654 q^{49} +1.03632 q^{51} +5.20962 q^{52} +6.62501 q^{53} +1.67234 q^{54} +9.16982 q^{56} -1.10775 q^{57} -16.3021 q^{58} -11.2581 q^{59} -7.65251 q^{61} +15.7810 q^{62} -4.55690 q^{63} +2.77979 q^{64} +1.05546 q^{66} -0.835690 q^{67} +0.825658 q^{68} +1.82732 q^{69} -11.8486 q^{71} -2.01229 q^{72} +7.03605 q^{73} -5.61484 q^{74} -0.882568 q^{76} -2.87600 q^{77} +10.9351 q^{78} +2.79085 q^{79} +1.00000 q^{81} -7.43486 q^{82} -7.53754 q^{83} -3.63058 q^{84} -17.4211 q^{86} -9.74810 q^{87} -1.27002 q^{88} -10.7848 q^{89} -29.7968 q^{91} +1.45586 q^{92} +9.43646 q^{93} -10.7613 q^{94} -4.26801 q^{96} +10.6767 q^{97} +23.0204 q^{98} +0.631130 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 2 q^{2} + 10 q^{3} + 10 q^{4} - 2 q^{6} + 10 q^{9}+O(q^{10}) $ 10 * q - 2 * q^2 + 10 * q^3 + 10 * q^4 - 2 * q^6 + 10 * q^9	$ 10 q - 2 q^{2} + 10 q^{3} + 10 q^{4} - 2 q^{6} + 10 q^{9} - 14 q^{11} + 10 q^{12} + 3 q^{13} - 16 q^{14} + 10 q^{16} - 8 q^{17} - 2 q^{18} - 19 q^{19} - 5 q^{22} - 4 q^{23} - 22 q^{26} + 10 q^{27} - 25 q^{29} - 2 q^{31} + 13 q^{32} - 14 q^{33} - 37 q^{34} + 10 q^{36} + 10 q^{37} + 13 q^{38} + 3 q^{39} - 31 q^{41} - 16 q^{42} - 62 q^{44} + 2 q^{46} + q^{47} + 10 q^{48} + 26 q^{49} - 8 q^{51} + 30 q^{52} - 9 q^{53} - 2 q^{54} - 63 q^{56} - 19 q^{57} - 30 q^{58} - 65 q^{59} + 12 q^{61} + 39 q^{62} - 2 q^{64} - 5 q^{66} + 10 q^{67} - 22 q^{68} - 4 q^{69} - 45 q^{71} - q^{73} + 19 q^{74} - 39 q^{76} + q^{77} - 22 q^{78} - 47 q^{79} + 10 q^{81} + 23 q^{82} - q^{83} + 12 q^{86} - 25 q^{87} + 8 q^{88} - 34 q^{89} - 26 q^{91} - 14 q^{92} - 2 q^{93} - 64 q^{94} + 13 q^{96} - 5 q^{97} + 51 q^{98} - 14 q^{99}+O(q^{100}) $ 10 * q - 2 * q^2 + 10 * q^3 + 10 * q^4 - 2 * q^6 + 10 * q^9 - 14 * q^11 + 10 * q^12 + 3 * q^13 - 16 * q^14 + 10 * q^16 - 8 * q^17 - 2 * q^18 - 19 * q^19 - 5 * q^22 - 4 * q^23 - 22 * q^26 + 10 * q^27 - 25 * q^29 - 2 * q^31 + 13 * q^32 - 14 * q^33 - 37 * q^34 + 10 * q^36 + 10 * q^37 + 13 * q^38 + 3 * q^39 - 31 * q^41 - 16 * q^42 - 62 * q^44 + 2 * q^46 + q^47 + 10 * q^48 + 26 * q^49 - 8 * q^51 + 30 * q^52 - 9 * q^53 - 2 * q^54 - 63 * q^56 - 19 * q^57 - 30 * q^58 - 65 * q^59 + 12 * q^61 + 39 * q^62 - 2 * q^64 - 5 * q^66 + 10 * q^67 - 22 * q^68 - 4 * q^69 - 45 * q^71 - q^73 + 19 * q^74 - 39 * q^76 + q^77 - 22 * q^78 - 47 * q^79 + 10 * q^81 + 23 * q^82 - q^83 + 12 * q^86 - 25 * q^87 + 8 * q^88 - 34 * q^89 - 26 * q^91 - 14 * q^92 - 2 * q^93 - 64 * q^94 + 13 * q^96 - 5 * q^97 + 51 * q^98 - 14 * 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 $

$2$	1.67234	1.18252	0.591261	−	0.806480i	$-0.298630\pi$
$2$	1.67234	1.18252	0.591261	+	0.806480i	$0.298630\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0.796721	0.398360
$5$	0	0
$5$	0	0
$6$	1.67234	0.682730
$7$	−4.55690	−1.72235	−0.861174	−	0.508310i	$-0.830270\pi$
$7$	−4.55690	−1.72235	−0.861174	+	0.508310i	$0.830270\pi$
$8$	−2.01229	−0.711453
$9$	1.00000	0.333333
$10$	0	0
$11$	0.631130	0.190293	0.0951464	−	0.995463i	$-0.469668\pi$
$11$	0.631130	0.190293	0.0951464	+	0.995463i	$0.469668\pi$
$12$	0.796721	0.229994
$13$	6.53883	1.81354	0.906772	−	0.421621i	$-0.138539\pi$
$13$	6.53883	1.81354	0.906772	+	0.421621i	$0.138539\pi$
$14$	−7.62069	−2.03672
$15$	0	0
$16$	−4.95868	−1.23967
$17$	1.03632	0.251345	0.125672	−	0.992072i	$-0.459891\pi$
$17$	1.03632	0.251345	0.125672	+	0.992072i	$0.459891\pi$
$18$	1.67234	0.394174
$19$	−1.10775	−0.254135	−0.127068	−	0.991894i	$-0.540557\pi$
$19$	−1.10775	−0.254135	−0.127068	+	0.991894i	$0.540557\pi$
$20$	0	0
$21$	−4.55690	−0.994398
$22$	1.05546	0.225026
$23$	1.82732	0.381022	0.190511	−	0.981685i	$-0.438985\pi$
$23$	1.82732	0.381022	0.190511	+	0.981685i	$0.438985\pi$
$24$	−2.01229	−0.410757
$25$	0	0
$26$	10.9351	2.14456
$27$	1.00000	0.192450
$28$	−3.63058	−0.686115
$29$	−9.74810	−1.81018	−0.905088	−	0.425224i	$-0.860195\pi$
$29$	−9.74810	−1.81018	−0.905088	+	0.425224i	$0.860195\pi$
$30$	0	0
$31$	9.43646	1.69484	0.847419	−	0.530925i	$-0.178155\pi$
$31$	9.43646	1.69484	0.847419	+	0.530925i	$0.178155\pi$
$32$	−4.26801	−0.754485
$33$	0.631130	0.109866
$34$	1.73308	0.297221
$35$	0	0
$36$	0.796721	0.132787
$37$	−3.35748	−0.551966	−0.275983	−	0.961163i	$-0.589003\pi$
$37$	−3.35748	−0.551966	−0.275983	+	0.961163i	$0.589003\pi$
$38$	−1.85254	−0.300521
$39$	6.53883	1.04705
$40$	0	0
$41$	−4.44578	−0.694314	−0.347157	−	0.937807i	$-0.612853\pi$
$41$	−4.44578	−0.694314	−0.347157	+	0.937807i	$0.612853\pi$
$42$	−7.62069	−1.17590
$43$	−10.4172	−1.58860	−0.794302	−	0.607523i	$-0.792163\pi$
$43$	−10.4172	−1.58860	−0.794302	+	0.607523i	$0.792163\pi$
$44$	0.502834	0.0758051
$45$	0	0
$46$	3.05590	0.450568
$47$	−6.43490	−0.938627	−0.469314	−	0.883032i	$-0.655499\pi$
$47$	−6.43490	−0.938627	−0.469314	+	0.883032i	$0.655499\pi$
$48$	−4.95868	−0.715723
$49$	13.7654	1.96648
$50$	0	0
$51$	1.03632	0.145114
$52$	5.20962	0.722444
$53$	6.62501	0.910015	0.455008	−	0.890488i	$-0.349637\pi$
$53$	6.62501	0.910015	0.455008	+	0.890488i	$0.349637\pi$
$54$	1.67234	0.227577
$55$	0	0
$56$	9.16982	1.22537
$57$	−1.10775	−0.146725
$58$	−16.3021	−2.14058
$59$	−11.2581	−1.46568	−0.732840	−	0.680401i	$-0.761805\pi$
$59$	−11.2581	−1.46568	−0.732840	+	0.680401i	$0.761805\pi$
$60$	0	0
$61$	−7.65251	−0.979803	−0.489902	−	0.871778i	$-0.662967\pi$
$61$	−7.65251	−0.979803	−0.489902	+	0.871778i	$0.662967\pi$
$62$	15.7810	2.00418
$63$	−4.55690	−0.574116
$64$	2.77979	0.347474
$65$	0	0
$66$	1.05546	0.129919
$67$	−0.835690	−0.102096	−0.0510479	−	0.998696i	$-0.516256\pi$
$67$	−0.835690	−0.102096	−0.0510479	+	0.998696i	$0.516256\pi$
$68$	0.825658	0.100126
$69$	1.82732	0.219983
$70$	0	0
$71$	−11.8486	−1.40616	−0.703082	−	0.711109i	$-0.748193\pi$
$71$	−11.8486	−1.40616	−0.703082	+	0.711109i	$0.748193\pi$
$72$	−2.01229	−0.237151
$73$	7.03605	0.823507	0.411753	−	0.911295i	$-0.364916\pi$
$73$	7.03605	0.823507	0.411753	+	0.911295i	$0.364916\pi$
$74$	−5.61484	−0.652712
$75$	0	0
$76$	−0.882568	−0.101238
$77$	−2.87600	−0.327750
$78$	10.9351	1.23816
$79$	2.79085	0.313995	0.156997	−	0.987599i	$-0.449819\pi$
$79$	2.79085	0.313995	0.156997	+	0.987599i	$0.449819\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−7.43486	−0.821043
$83$	−7.53754	−0.827352	−0.413676	−	0.910424i	$-0.635755\pi$
$83$	−7.53754	−0.827352	−0.413676	+	0.910424i	$0.635755\pi$
$84$	−3.63058	−0.396129
$85$	0	0
$86$	−17.4211	−1.87856
$87$	−9.74810	−1.04511
$88$	−1.27002	−0.135384
$89$	−10.7848	−1.14319	−0.571595	−	0.820536i	$-0.693675\pi$
$89$	−10.7848	−1.14319	−0.571595	+	0.820536i	$0.693675\pi$
$90$	0	0
$91$	−29.7968	−3.12355
$92$	1.45586	0.151784
$93$	9.43646	0.978515
$94$	−10.7613	−1.10995
$95$	0	0
$96$	−4.26801	−0.435602
$97$	10.6767	1.08405	0.542027	−	0.840361i	$-0.317657\pi$
$97$	10.6767	1.08405	0.542027	+	0.840361i	$0.317657\pi$
$98$	23.0204	2.32541
$99$	0.631130	0.0634309
$100$	0	0
$101$	−4.65872	−0.463560	−0.231780	−	0.972768i	$-0.574455\pi$
$101$	−4.65872	−0.463560	−0.231780	+	0.972768i	$0.574455\pi$
$102$	1.73308	0.171601
$103$	−1.09394	−0.107789	−0.0538946	−	0.998547i	$-0.517163\pi$
$103$	−1.09394	−0.107789	−0.0538946	+	0.998547i	$0.517163\pi$
$104$	−13.1580	−1.29025
$105$	0	0
$106$	11.0793	1.07611
$107$	1.00000	0.0966736
$107$	1.00000	0.0966736
$108$	0.796721	0.0766645
$109$	−7.20013	−0.689647	−0.344824	−	0.938668i	$-0.612061\pi$
$109$	−7.20013	−0.689647	−0.344824	+	0.938668i	$0.612061\pi$
$110$	0	0
$111$	−3.35748	−0.318678
$112$	22.5962	2.13514
$113$	10.0102	0.941681	0.470840	−	0.882218i	$-0.343951\pi$
$113$	10.0102	0.941681	0.470840	+	0.882218i	$0.343951\pi$
$114$	−1.85254	−0.173506
$115$	0	0
$116$	−7.76651	−0.721103
$117$	6.53883	0.604515
$118$	−18.8274	−1.73320
$119$	−4.72241	−0.432903
$120$	0	0
$121$	−10.6017	−0.963789
$122$	−12.7976	−1.15864
$123$	−4.44578	−0.400863
$124$	7.51822	0.675156
$125$	0	0
$126$	−7.62069	−0.678905
$127$	−10.4523	−0.927490	−0.463745	−	0.885969i	$-0.653495\pi$
$127$	−10.4523	−0.927490	−0.463745	+	0.885969i	$0.653495\pi$
$128$	13.1848	1.16538
$129$	−10.4172	−0.917181
$130$	0	0
$131$	13.3551	1.16684	0.583422	−	0.812169i	$-0.301714\pi$
$131$	13.3551	1.16684	0.583422	+	0.812169i	$0.301714\pi$
$132$	0.502834	0.0437661
$133$	5.04791	0.437710
$134$	−1.39756	−0.120731
$135$	0	0
$136$	−2.08538	−0.178820
$137$	−22.5413	−1.92583	−0.962916	−	0.269802i	$-0.913042\pi$
$137$	−22.5413	−1.92583	−0.962916	+	0.269802i	$0.913042\pi$
$138$	3.05590	0.260135
$139$	−9.76123	−0.827937	−0.413968	−	0.910291i	$-0.635858\pi$
$139$	−9.76123	−0.827937	−0.413968	+	0.910291i	$0.635858\pi$
$140$	0	0
$141$	−6.43490	−0.541917
$142$	−19.8148	−1.66282
$143$	4.12685	0.345104
$144$	−4.95868	−0.413223
$145$	0	0
$146$	11.7667	0.973816
$147$	13.7654	1.13535
$148$	−2.67497	−0.219881
$149$	4.00713	0.328277	0.164139	−	0.986437i	$-0.447516\pi$
$149$	4.00713	0.328277	0.164139	+	0.986437i	$0.447516\pi$
$150$	0	0
$151$	3.12030	0.253926	0.126963	−	0.991907i	$-0.459477\pi$
$151$	3.12030	0.253926	0.126963	+	0.991907i	$0.459477\pi$
$152$	2.22912	0.180805
$153$	1.03632	0.0837816
$154$	−4.80965	−0.387572
$155$	0	0
$156$	5.20962	0.417103
$157$	11.4786	0.916092	0.458046	−	0.888928i	$-0.348549\pi$
$157$	11.4786	0.916092	0.458046	+	0.888928i	$0.348549\pi$
$158$	4.66724	0.371306
$159$	6.62501	0.525398
$160$	0	0
$161$	−8.32692	−0.656253
$162$	1.67234	0.131391
$163$	0.118992	0.00932018	0.00466009	−	0.999989i	$-0.498517\pi$
$163$	0.118992	0.00932018	0.00466009	+	0.999989i	$0.498517\pi$
$164$	−3.54205	−0.276587
$165$	0	0
$166$	−12.6053	−0.978363
$167$	−11.0224	−0.852939	−0.426469	−	0.904502i	$-0.640243\pi$
$167$	−11.0224	−0.852939	−0.426469	+	0.904502i	$0.640243\pi$
$168$	9.16982	0.707467
$169$	29.7563	2.28894
$170$	0	0
$171$	−1.10775	−0.0847118
$172$	−8.29958	−0.632837
$173$	−11.2386	−0.854455	−0.427227	−	0.904144i	$-0.640510\pi$
$173$	−11.2386	−0.854455	−0.427227	+	0.904144i	$0.640510\pi$
$174$	−16.3021	−1.23586
$175$	0	0
$176$	−3.12957	−0.235900
$177$	−11.2581	−0.846210
$178$	−18.0359	−1.35185
$179$	−11.0644	−0.826992	−0.413496	−	0.910506i	$-0.635692\pi$
$179$	−11.0644	−0.826992	−0.413496	+	0.910506i	$0.635692\pi$
$180$	0	0
$181$	−18.9856	−1.41119	−0.705593	−	0.708617i	$-0.749319\pi$
$181$	−18.9856	−1.41119	−0.705593	+	0.708617i	$0.749319\pi$
$182$	−49.8304	−3.69367
$183$	−7.65251	−0.565690
$184$	−3.67710	−0.271079
$185$	0	0
$186$	15.7810	1.15712
$187$	0.654053	0.0478291
$188$	−5.12682	−0.373912
$189$	−4.55690	−0.331466
$190$	0	0
$191$	−20.1191	−1.45577	−0.727884	−	0.685700i	$-0.759496\pi$
$191$	−20.1191	−1.45577	−0.727884	+	0.685700i	$0.759496\pi$
$192$	2.77979	0.200614
$193$	16.3149	1.17437	0.587185	−	0.809453i	$-0.300236\pi$
$193$	16.3149	1.17437	0.587185	+	0.809453i	$0.300236\pi$
$194$	17.8550	1.28192
$195$	0	0
$196$	10.9672	0.783369
$197$	−8.47291	−0.603670	−0.301835	−	0.953360i	$-0.597599\pi$
$197$	−8.47291	−0.603670	−0.301835	+	0.953360i	$0.597599\pi$
$198$	1.05546	0.0750085
$199$	−8.42429	−0.597182	−0.298591	−	0.954381i	$-0.596517\pi$
$199$	−8.42429	−0.597182	−0.298591	+	0.954381i	$0.596517\pi$
$200$	0	0
$201$	−0.835690	−0.0589450
$202$	−7.79097	−0.548171
$203$	44.4212	3.11775
$204$	0.825658	0.0578077
$205$	0	0
$206$	−1.82944	−0.127463
$207$	1.82732	0.127007
$208$	−32.4239	−2.24820
$209$	−0.699134	−0.0483601
$210$	0	0
$211$	−22.9865	−1.58246	−0.791228	−	0.611521i	$-0.790558\pi$
$211$	−22.9865	−1.58246	−0.791228	+	0.611521i	$0.790558\pi$
$212$	5.27828	0.362514
$213$	−11.8486	−0.811849
$214$	1.67234	0.114319
$215$	0	0
$216$	−2.01229	−0.136919
$217$	−43.0010	−2.91910
$218$	−12.0411	−0.815524
$219$	7.03605	0.475452
$220$	0	0
$221$	6.77632	0.455825
$222$	−5.61484	−0.376844
$223$	−0.466572	−0.0312440	−0.0156220	−	0.999878i	$-0.504973\pi$
$223$	−0.466572	−0.0312440	−0.0156220	+	0.999878i	$0.504973\pi$
$224$	19.4489	1.29949
$225$	0	0
$226$	16.7405	1.11356
$227$	5.43209	0.360541	0.180270	−	0.983617i	$-0.442303\pi$
$227$	5.43209	0.360541	0.180270	+	0.983617i	$0.442303\pi$
$228$	−0.882568	−0.0584495
$229$	18.1138	1.19699	0.598496	−	0.801126i	$-0.295765\pi$
$229$	18.1138	1.19699	0.598496	+	0.801126i	$0.295765\pi$
$230$	0	0
$231$	−2.87600	−0.189227
$232$	19.6160	1.28785
$233$	20.7380	1.35859	0.679296	−	0.733864i	$-0.262285\pi$
$233$	20.7380	1.35859	0.679296	+	0.733864i	$0.262285\pi$
$234$	10.9351	0.714853
$235$	0	0
$236$	−8.96956	−0.583869
$237$	2.79085	0.181285
$238$	−7.89748	−0.511918
$239$	16.5068	1.06773	0.533867	−	0.845568i	$-0.320738\pi$
$239$	16.5068	1.06773	0.533867	+	0.845568i	$0.320738\pi$
$240$	0	0
$241$	−24.1059	−1.55280	−0.776399	−	0.630241i	$-0.782956\pi$
$241$	−24.1059	−1.55280	−0.776399	+	0.630241i	$0.782956\pi$
$242$	−17.7296	−1.13970
$243$	1.00000	0.0641500
$244$	−6.09691	−0.390315
$245$	0	0
$246$	−7.43486	−0.474029
$247$	−7.24339	−0.460886
$248$	−18.9889	−1.20580
$249$	−7.53754	−0.477672
$250$	0	0
$251$	−12.7402	−0.804154	−0.402077	−	0.915606i	$-0.631712\pi$
$251$	−12.7402	−0.804154	−0.402077	+	0.915606i	$0.631712\pi$
$252$	−3.63058	−0.228705
$253$	1.15328	0.0725058
$254$	−17.4798	−1.09678
$255$	0	0
$256$	16.4898	1.03062
$257$	15.0987	0.941832	0.470916	−	0.882178i	$-0.343923\pi$
$257$	15.0987	0.941832	0.470916	+	0.882178i	$0.343923\pi$
$258$	−17.4211	−1.08459
$259$	15.2997	0.950677
$260$	0	0
$261$	−9.74810	−0.603392
$262$	22.3343	1.37982
$263$	1.09168	0.0673162	0.0336581	−	0.999433i	$-0.489284\pi$
$263$	1.09168	0.0673162	0.0336581	+	0.999433i	$0.489284\pi$
$264$	−1.27002	−0.0781641
$265$	0	0
$266$	8.44183	0.517602
$267$	−10.7848	−0.660021
$268$	−0.665811	−0.0406709
$269$	−23.0272	−1.40399	−0.701996	−	0.712181i	$-0.747707\pi$
$269$	−23.0272	−1.40399	−0.701996	+	0.712181i	$0.747707\pi$
$270$	0	0
$271$	−3.07898	−0.187035	−0.0935173	−	0.995618i	$-0.529811\pi$
$271$	−3.07898	−0.187035	−0.0935173	+	0.995618i	$0.529811\pi$
$272$	−5.13878	−0.311584
$273$	−29.7968	−1.80338
$274$	−37.6967	−2.27734
$275$	0	0
$276$	1.45586	0.0876327
$277$	27.5290	1.65406	0.827029	−	0.562159i	$-0.190029\pi$
$277$	27.5290	1.65406	0.827029	+	0.562159i	$0.190029\pi$
$278$	−16.3241	−0.979054
$279$	9.43646	0.564946
$280$	0	0
$281$	2.65372	0.158308	0.0791538	−	0.996862i	$-0.474778\pi$
$281$	2.65372	0.158308	0.0791538	+	0.996862i	$0.474778\pi$
$282$	−10.7613	−0.640829
$283$	22.2178	1.32071	0.660356	−	0.750953i	$-0.270406\pi$
$283$	22.2178	1.32071	0.660356	+	0.750953i	$0.270406\pi$
$284$	−9.43999	−0.560160
$285$	0	0
$286$	6.90149	0.408094
$287$	20.2590	1.19585
$288$	−4.26801	−0.251495
$289$	−15.9260	−0.936826
$290$	0	0
$291$	10.6767	0.625878
$292$	5.60577	0.328053
$293$	−1.41665	−0.0827614	−0.0413807	−	0.999143i	$-0.513176\pi$
$293$	−1.41665	−0.0827614	−0.0413807	+	0.999143i	$0.513176\pi$
$294$	23.0204	1.34258
$295$	0	0
$296$	6.75622	0.392698
$297$	0.631130	0.0366219
$298$	6.70129	0.388195
$299$	11.9485	0.691001
$300$	0	0
$301$	47.4701	2.73613
$302$	5.21820	0.300274
$303$	−4.65872	−0.267637
$304$	5.49298	0.315044
$305$	0	0
$306$	1.73308	0.0990736
$307$	−12.5909	−0.718601	−0.359300	−	0.933222i	$-0.616985\pi$
$307$	−12.5909	−0.718601	−0.359300	+	0.933222i	$0.616985\pi$
$308$	−2.29137	−0.130563
$309$	−1.09394	−0.0622321
$310$	0	0
$311$	−15.2282	−0.863512	−0.431756	−	0.901990i	$-0.642106\pi$
$311$	−15.2282	−0.863512	−0.431756	+	0.901990i	$0.642106\pi$
$312$	−13.1580	−0.744927
$313$	4.70555	0.265973	0.132987	−	0.991118i	$-0.457543\pi$
$313$	4.70555	0.265973	0.132987	+	0.991118i	$0.457543\pi$
$314$	19.1961	1.08330
$315$	0	0
$316$	2.22353	0.125083
$317$	12.1677	0.683408	0.341704	−	0.939808i	$-0.388996\pi$
$317$	12.1677	0.683408	0.341704	+	0.939808i	$0.388996\pi$
$318$	11.0793	0.621295
$319$	−6.15232	−0.344464
$320$	0	0
$321$	1.00000	0.0558146
$322$	−13.9254	−0.776034
$323$	−1.14799	−0.0638756
$324$	0.796721	0.0442623
$325$	0	0
$326$	0.198995	0.0110213
$327$	−7.20013	−0.398168
$328$	8.94621	0.493972
$329$	29.3232	1.61664
$330$	0	0
$331$	7.45359	0.409686	0.204843	−	0.978795i	$-0.434332\pi$
$331$	7.45359	0.409686	0.204843	+	0.978795i	$0.434332\pi$
$332$	−6.00531	−0.329584
$333$	−3.35748	−0.183989
$334$	−18.4332	−1.00862
$335$	0	0
$336$	22.5962	1.23272
$337$	−21.1841	−1.15397	−0.576987	−	0.816754i	$-0.695772\pi$
$337$	−21.1841	−1.15397	−0.576987	+	0.816754i	$0.695772\pi$
$338$	49.7626	2.70673
$339$	10.0102	0.543680
$340$	0	0
$341$	5.95563	0.322515
$342$	−1.85254	−0.100174
$343$	−30.8292	−1.66462
$344$	20.9624	1.13022
$345$	0	0
$346$	−18.7948	−1.01041
$347$	18.3959	0.987543	0.493772	−	0.869592i	$-0.335618\pi$
$347$	18.3959	0.987543	0.493772	+	0.869592i	$0.335618\pi$
$348$	−7.76651	−0.416329
$349$	−27.6646	−1.48085	−0.740426	−	0.672138i	$-0.765376\pi$
$349$	−27.6646	−1.48085	−0.740426	+	0.672138i	$0.765376\pi$
$350$	0	0
$351$	6.53883	0.349017
$352$	−2.69367	−0.143573
$353$	−18.2846	−0.973190	−0.486595	−	0.873628i	$-0.661761\pi$
$353$	−18.2846	−0.973190	−0.486595	+	0.873628i	$0.661761\pi$
$354$	−18.8274	−1.00066
$355$	0	0
$356$	−8.59251	−0.455402
$357$	−4.72241	−0.249937
$358$	−18.5034	−0.977937
$359$	5.97321	0.315254	0.157627	−	0.987499i	$-0.449616\pi$
$359$	5.97321	0.315254	0.157627	+	0.987499i	$0.449616\pi$
$360$	0	0
$361$	−17.7729	−0.935415
$362$	−31.7503	−1.66876
$363$	−10.6017	−0.556444
$364$	−23.7397	−1.24430
$365$	0	0
$366$	−12.7976	−0.668941
$367$	−16.1541	−0.843237	−0.421619	−	0.906773i	$-0.638538\pi$
$367$	−16.1541	−0.843237	−0.421619	+	0.906773i	$0.638538\pi$
$368$	−9.06109	−0.472342
$369$	−4.44578	−0.231438
$370$	0	0
$371$	−30.1895	−1.56736
$372$	7.51822	0.389802
$373$	6.58257	0.340833	0.170416	−	0.985372i	$-0.445489\pi$
$373$	6.58257	0.340833	0.170416	+	0.985372i	$0.445489\pi$
$374$	1.09380	0.0565590
$375$	0	0
$376$	12.9489	0.667789
$377$	−63.7411	−3.28284
$378$	−7.62069	−0.391966
$379$	27.8520	1.43066	0.715329	−	0.698787i	$-0.246277\pi$
$379$	27.8520	1.43066	0.715329	+	0.698787i	$0.246277\pi$
$380$	0	0
$381$	−10.4523	−0.535486
$382$	−33.6460	−1.72148
$383$	13.1985	0.674411	0.337206	−	0.941431i	$-0.390518\pi$
$383$	13.1985	0.674411	0.337206	+	0.941431i	$0.390518\pi$
$384$	13.1848	0.672833
$385$	0	0
$386$	27.2840	1.38872
$387$	−10.4172	−0.529535
$388$	8.50634	0.431844
$389$	−1.73205	−0.0878184	−0.0439092	−	0.999036i	$-0.513981\pi$
$389$	−1.73205	−0.0878184	−0.0439092	+	0.999036i	$0.513981\pi$
$390$	0	0
$391$	1.89369	0.0957680
$392$	−27.6999	−1.39906
$393$	13.3551	0.673677
$394$	−14.1696	−0.713854
$395$	0	0
$396$	0.502834	0.0252684
$397$	−32.7900	−1.64568	−0.822842	−	0.568270i	$-0.807613\pi$
$397$	−32.7900	−1.64568	−0.822842	+	0.568270i	$0.807613\pi$
$398$	−14.0883	−0.706181
$399$	5.04791	0.252712
$400$	0	0
$401$	−10.4631	−0.522503	−0.261251	−	0.965271i	$-0.584135\pi$
$401$	−10.4631	−0.522503	−0.261251	+	0.965271i	$0.584135\pi$
$402$	−1.39756	−0.0697038
$403$	61.7034	3.07366
$404$	−3.71170	−0.184664
$405$	0	0
$406$	74.2873	3.68682
$407$	−2.11900	−0.105035
$408$	−2.08538	−0.103242
$409$	38.9159	1.92427	0.962133	−	0.272581i	$-0.0878772\pi$
$409$	38.9159	1.92427	0.962133	+	0.272581i	$0.0878772\pi$
$410$	0	0
$411$	−22.5413	−1.11188
$412$	−0.871565	−0.0429389
$413$	51.3021	2.52441
$414$	3.05590	0.150189
$415$	0	0
$416$	−27.9078	−1.36829
$417$	−9.76123	−0.478009
$418$	−1.16919	−0.0571870
$419$	−1.90730	−0.0931777	−0.0465889	−	0.998914i	$-0.514835\pi$
$419$	−1.90730	−0.0931777	−0.0465889	+	0.998914i	$0.514835\pi$
$420$	0	0
$421$	14.3920	0.701424	0.350712	−	0.936483i	$-0.385940\pi$
$421$	14.3920	0.701424	0.350712	+	0.936483i	$0.385940\pi$
$422$	−38.4413	−1.87129
$423$	−6.43490	−0.312876
$424$	−13.3315	−0.647433
$425$	0	0
$426$	−19.8148	−0.960030
$427$	34.8718	1.68756
$428$	0.796721	0.0385110
$429$	4.12685	0.199246
$430$	0	0
$431$	31.2083	1.50325	0.751624	−	0.659591i	$-0.229271\pi$
$431$	31.2083	1.50325	0.751624	+	0.659591i	$0.229271\pi$
$432$	−4.95868	−0.238574
$433$	0.692015	0.0332561	0.0166281	−	0.999862i	$-0.494707\pi$
$433$	0.692015	0.0332561	0.0166281	+	0.999862i	$0.494707\pi$
$434$	−71.9123	−3.45190
$435$	0	0
$436$	−5.73649	−0.274728
$437$	−2.02421	−0.0968313
$438$	11.7667	0.562233
$439$	−23.1138	−1.10316	−0.551580	−	0.834122i	$-0.685975\pi$
$439$	−23.1138	−1.10316	−0.551580	+	0.834122i	$0.685975\pi$
$440$	0	0
$441$	13.7654	0.655494
$442$	11.3323	0.539023
$443$	−29.6487	−1.40865	−0.704327	−	0.709876i	$-0.748751\pi$
$443$	−29.6487	−1.40865	−0.704327	+	0.709876i	$0.748751\pi$
$444$	−2.67497	−0.126949
$445$	0	0
$446$	−0.780267	−0.0369467
$447$	4.00713	0.189531
$448$	−12.6672	−0.598471
$449$	−15.7349	−0.742578	−0.371289	−	0.928517i	$-0.621084\pi$
$449$	−15.7349	−0.742578	−0.371289	+	0.928517i	$0.621084\pi$
$450$	0	0
$451$	−2.80586	−0.132123
$452$	7.97534	0.375128
$453$	3.12030	0.146604
$454$	9.08431	0.426348
$455$	0	0
$456$	2.22912	0.104388
$457$	16.9670	0.793682	0.396841	−	0.917887i	$-0.370107\pi$
$457$	16.9670	0.793682	0.396841	+	0.917887i	$0.370107\pi$
$458$	30.2924	1.41547
$459$	1.03632	0.0483713
$460$	0	0
$461$	20.5408	0.956681	0.478341	−	0.878174i	$-0.341238\pi$
$461$	20.5408	0.956681	0.478341	+	0.878174i	$0.341238\pi$
$462$	−4.80965	−0.223765
$463$	−25.6216	−1.19074	−0.595368	−	0.803453i	$-0.702994\pi$
$463$	−25.6216	−1.19074	−0.595368	+	0.803453i	$0.702994\pi$
$464$	48.3377	2.24402
$465$	0	0
$466$	34.6810	1.60657
$467$	22.8478	1.05727	0.528634	−	0.848850i	$-0.322704\pi$
$467$	22.8478	1.05727	0.528634	+	0.848850i	$0.322704\pi$
$468$	5.20962	0.240815
$469$	3.80816	0.175844
$470$	0	0
$471$	11.4786	0.528906
$472$	22.6546	1.04276
$473$	−6.57459	−0.302300
$474$	4.66724	0.214374
$475$	0	0
$476$	−3.76245	−0.172451
$477$	6.62501	0.303338
$478$	27.6049	1.26262
$479$	−8.05999	−0.368271	−0.184135	−	0.982901i	$-0.558948\pi$
$479$	−8.05999	−0.368271	−0.184135	+	0.982901i	$0.558948\pi$
$480$	0	0
$481$	−21.9540	−1.00101
$482$	−40.3133	−1.83622
$483$	−8.32692	−0.378888
$484$	−8.44658	−0.383935
$485$	0	0
$486$	1.67234	0.0758589
$487$	−1.55981	−0.0706816	−0.0353408	−	0.999375i	$-0.511252\pi$
$487$	−1.55981	−0.0706816	−0.0353408	+	0.999375i	$0.511252\pi$
$488$	15.3991	0.697084
$489$	0.118992	0.00538101
$490$	0	0
$491$	16.2614	0.733865	0.366932	−	0.930248i	$-0.380408\pi$
$491$	16.2614	0.733865	0.366932	+	0.930248i	$0.380408\pi$
$492$	−3.54205	−0.159688
$493$	−10.1022	−0.454978
$494$	−12.1134	−0.545008
$495$	0	0
$496$	−46.7924	−2.10104
$497$	53.9927	2.42190
$498$	−12.6053	−0.564858
$499$	−2.10827	−0.0943793	−0.0471897	−	0.998886i	$-0.515027\pi$
$499$	−2.10827	−0.0943793	−0.0471897	+	0.998886i	$0.515027\pi$
$500$	0	0
$501$	−11.0224	−0.492444
$502$	−21.3059	−0.950931
$503$	38.4265	1.71335	0.856676	−	0.515856i	$-0.172526\pi$
$503$	38.4265	1.71335	0.856676	+	0.515856i	$0.172526\pi$
$504$	9.16982	0.408456
$505$	0	0
$506$	1.92867	0.0857398
$507$	29.7563	1.32152
$508$	−8.32755	−0.369475
$509$	34.3033	1.52047	0.760233	−	0.649650i	$-0.225085\pi$
$509$	34.3033	1.52047	0.760233	+	0.649650i	$0.225085\pi$
$510$	0	0
$511$	−32.0626	−1.41837
$512$	1.20708	0.0533459
$513$	−1.10775	−0.0489084
$514$	25.2502	1.11374
$515$	0	0
$516$	−8.29958	−0.365369
$517$	−4.06126	−0.178614
$518$	25.5863	1.12420
$519$	−11.2386	−0.493320
$520$	0	0
$521$	19.8882	0.871318	0.435659	−	0.900112i	$-0.356515\pi$
$521$	19.8882	0.871318	0.435659	+	0.900112i	$0.356515\pi$
$522$	−16.3021	−0.713525
$523$	12.8596	0.562313	0.281156	−	0.959662i	$-0.409282\pi$
$523$	12.8596	0.562313	0.281156	+	0.959662i	$0.409282\pi$
$524$	10.6403	0.464824
$525$	0	0
$526$	1.82567	0.0796029
$527$	9.77920	0.425989
$528$	−3.12957	−0.136197
$529$	−19.6609	−0.854822
$530$	0	0
$531$	−11.2581	−0.488560
$532$	4.02178	0.174366
$533$	−29.0702	−1.25917
$534$	−18.0359	−0.780491
$535$	0	0
$536$	1.68165	0.0726363
$537$	−11.0644	−0.477464
$538$	−38.5092	−1.66025
$539$	8.68774	0.374207
$540$	0	0
$541$	29.7636	1.27964	0.639818	−	0.768527i	$-0.279010\pi$
$541$	29.7636	1.27964	0.639818	+	0.768527i	$0.279010\pi$
$542$	−5.14910	−0.221173
$543$	−18.9856	−0.814749
$544$	−4.42303	−0.189636
$545$	0	0
$546$	−49.8304	−2.13254
$547$	−29.8622	−1.27681	−0.638407	−	0.769699i	$-0.720406\pi$
$547$	−29.8622	−1.27681	−0.638407	+	0.769699i	$0.720406\pi$
$548$	−17.9591	−0.767175
$549$	−7.65251	−0.326601
$550$	0	0
$551$	10.7985	0.460030
$552$	−3.67710	−0.156508
$553$	−12.7176	−0.540808
$554$	46.0379	1.95596
$555$	0	0
$556$	−7.77697	−0.329817
$557$	10.5558	0.447262	0.223631	−	0.974674i	$-0.428209\pi$
$557$	10.5558	0.447262	0.223631	+	0.974674i	$0.428209\pi$
$558$	15.7810	0.668062
$559$	−68.1161	−2.88101
$560$	0	0
$561$	0.654053	0.0276141
$562$	4.43792	0.187202
$563$	9.29829	0.391876	0.195938	−	0.980616i	$-0.437225\pi$
$563$	9.29829	0.391876	0.195938	+	0.980616i	$0.437225\pi$
$564$	−5.12682	−0.215878
$565$	0	0
$566$	37.1557	1.56177
$567$	−4.55690	−0.191372
$568$	23.8427	1.00042
$569$	10.6897	0.448136	0.224068	−	0.974573i	$-0.428066\pi$
$569$	10.6897	0.448136	0.224068	+	0.974573i	$0.428066\pi$
$570$	0	0
$571$	15.2899	0.639863	0.319932	−	0.947441i	$-0.396340\pi$
$571$	15.2899	0.639863	0.319932	+	0.947441i	$0.396340\pi$
$572$	3.28795	0.137476
$573$	−20.1191	−0.840489
$574$	33.8799	1.41412
$575$	0	0
$576$	2.77979	0.115825
$577$	22.0161	0.916542	0.458271	−	0.888813i	$-0.348469\pi$
$577$	22.0161	0.916542	0.458271	+	0.888813i	$0.348469\pi$
$578$	−26.6338	−1.10782
$579$	16.3149	0.678022
$580$	0	0
$581$	34.3478	1.42499
$582$	17.8550	0.740116
$583$	4.18124	0.173169
$584$	−14.1586	−0.585886
$585$	0	0
$586$	−2.36912	−0.0978672
$587$	−35.0432	−1.44639	−0.723193	−	0.690646i	$-0.757326\pi$
$587$	−35.0432	−1.44639	−0.723193	+	0.690646i	$0.757326\pi$
$588$	10.9672	0.452278
$589$	−10.4532	−0.430718
$590$	0	0
$591$	−8.47291	−0.348529
$592$	16.6486	0.684255
$593$	−20.3761	−0.836746	−0.418373	−	0.908275i	$-0.637399\pi$
$593$	−20.3761	−0.836746	−0.418373	+	0.908275i	$0.637399\pi$
$594$	1.05546	0.0433062
$595$	0	0
$596$	3.19257	0.130773
$597$	−8.42429	−0.344783
$598$	19.9820	0.817125
$599$	21.9988	0.898845	0.449423	−	0.893319i	$-0.351630\pi$
$599$	21.9988	0.898845	0.449423	+	0.893319i	$0.351630\pi$
$600$	0	0
$601$	12.8842	0.525559	0.262780	−	0.964856i	$-0.415361\pi$
$601$	12.8842	0.525559	0.262780	+	0.964856i	$0.415361\pi$
$602$	79.3861	3.23554
$603$	−0.835690	−0.0340319
$604$	2.48601	0.101154
$605$	0	0
$606$	−7.79097	−0.316487
$607$	36.8736	1.49665	0.748327	−	0.663330i	$-0.230857\pi$
$607$	36.8736	1.49665	0.748327	+	0.663330i	$0.230857\pi$
$608$	4.72789	0.191741
$609$	44.4212	1.80004
$610$	0	0
$611$	−42.0767	−1.70224
$612$	0.825658	0.0333753
$613$	16.7139	0.675067	0.337533	−	0.941314i	$-0.390407\pi$
$613$	16.7139	0.675067	0.337533	+	0.941314i	$0.390407\pi$
$614$	−21.0563	−0.849762
$615$	0	0
$616$	5.78735	0.233179
$617$	23.2618	0.936487	0.468243	−	0.883600i	$-0.344887\pi$
$617$	23.2618	0.936487	0.468243	+	0.883600i	$0.344887\pi$
$618$	−1.82944	−0.0735909
$619$	−2.58844	−0.104038	−0.0520190	−	0.998646i	$-0.516566\pi$
$619$	−2.58844	−0.104038	−0.0520190	+	0.998646i	$0.516566\pi$
$620$	0	0
$621$	1.82732	0.0733278
$622$	−25.4667	−1.02112
$623$	49.1455	1.96897
$624$	−32.4239	−1.29800
$625$	0	0
$626$	7.86927	0.314519
$627$	−0.699134	−0.0279207
$628$	9.14524	0.364935
$629$	−3.47942	−0.138734
$630$	0	0
$631$	13.8647	0.551945	0.275972	−	0.961166i	$-0.411000\pi$
$631$	13.8647	0.551945	0.275972	+	0.961166i	$0.411000\pi$
$632$	−5.61600	−0.223392
$633$	−22.9865	−0.913632
$634$	20.3486	0.808146
$635$	0	0
$636$	5.27828	0.209298
$637$	90.0094	3.56630
$638$	−10.2888	−0.407336
$639$	−11.8486	−0.468721
$640$	0	0
$641$	23.6150	0.932736	0.466368	−	0.884591i	$-0.345562\pi$
$641$	23.6150	0.932736	0.466368	+	0.884591i	$0.345562\pi$
$642$	1.67234	0.0660020
$643$	44.2789	1.74619	0.873095	−	0.487550i	$-0.162109\pi$
$643$	44.2789	1.74619	0.873095	+	0.487550i	$0.162109\pi$
$644$	−6.63423	−0.261425
$645$	0	0
$646$	−1.91982	−0.0755344
$647$	28.7018	1.12838	0.564192	−	0.825644i	$-0.309188\pi$
$647$	28.7018	1.12838	0.564192	+	0.825644i	$0.309188\pi$
$648$	−2.01229	−0.0790503
$649$	−7.10532	−0.278908
$650$	0	0
$651$	−43.0010	−1.68534
$652$	0.0948035	0.00371279
$653$	−39.8201	−1.55828	−0.779141	−	0.626849i	$-0.784344\pi$
$653$	−39.8201	−1.55828	−0.779141	+	0.626849i	$0.784344\pi$
$654$	−12.0411	−0.470843
$655$	0	0
$656$	22.0452	0.860720
$657$	7.03605	0.274502
$658$	49.0384	1.91172
$659$	27.9667	1.08943	0.544714	−	0.838622i	$-0.316638\pi$
$659$	27.9667	1.08943	0.544714	+	0.838622i	$0.316638\pi$
$660$	0	0
$661$	−6.59539	−0.256531	−0.128265	−	0.991740i	$-0.540941\pi$
$661$	−6.59539	−0.256531	−0.128265	+	0.991740i	$0.540941\pi$
$662$	12.4649	0.484463
$663$	6.77632	0.263171
$664$	15.1677	0.588622
$665$	0	0
$666$	−5.61484	−0.217571
$667$	−17.8129	−0.689718
$668$	−8.78177	−0.339777
$669$	−0.466572	−0.0180387
$670$	0	0
$671$	−4.82973	−0.186450
$672$	19.4489	0.750258
$673$	15.4837	0.596854	0.298427	−	0.954432i	$-0.403538\pi$
$673$	15.4837	0.596854	0.298427	+	0.954432i	$0.403538\pi$
$674$	−35.4271	−1.36460
$675$	0	0
$676$	23.7074	0.911825
$677$	42.1391	1.61954	0.809770	−	0.586748i	$-0.199592\pi$
$677$	42.1391	1.61954	0.809770	+	0.586748i	$0.199592\pi$
$678$	16.7405	0.642914
$679$	−48.6526	−1.86712
$680$	0	0
$681$	5.43209	0.208158
$682$	9.95984	0.381382
$683$	14.9558	0.572269	0.286134	−	0.958189i	$-0.407630\pi$
$683$	14.9558	0.572269	0.286134	+	0.958189i	$0.407630\pi$
$684$	−0.882568	−0.0337458
$685$	0	0
$686$	−51.5568	−1.96845
$687$	18.1138	0.691084
$688$	51.6554	1.96934
$689$	43.3198	1.65035
$690$	0	0
$691$	−24.3232	−0.925297	−0.462648	−	0.886542i	$-0.653101\pi$
$691$	−24.3232	−0.925297	−0.462648	+	0.886542i	$0.653101\pi$
$692$	−8.95403	−0.340381
$693$	−2.87600	−0.109250
$694$	30.7642	1.16779
$695$	0	0
$696$	19.6160	0.743543
$697$	−4.60726	−0.174512
$698$	−46.2646	−1.75114
$699$	20.7380	0.784384
$700$	0	0
$701$	45.8311	1.73102	0.865508	−	0.500895i	$-0.166996\pi$
$701$	45.8311	1.73102	0.865508	+	0.500895i	$0.166996\pi$
$702$	10.9351	0.412720
$703$	3.71925	0.140274
$704$	1.75441	0.0661217
$705$	0	0
$706$	−30.5780	−1.15082
$707$	21.2294	0.798412
$708$	−8.96956	−0.337097
$709$	4.16109	0.156273	0.0781365	−	0.996943i	$-0.475103\pi$
$709$	4.16109	0.156273	0.0781365	+	0.996943i	$0.475103\pi$
$710$	0	0
$711$	2.79085	0.104665
$712$	21.7022	0.813326
$713$	17.2434	0.645771
$714$	−7.89748	−0.295556
$715$	0	0
$716$	−8.81523	−0.329441
$717$	16.5068	0.616457
$718$	9.98924	0.372795
$719$	−25.4272	−0.948274	−0.474137	−	0.880451i	$-0.657240\pi$
$719$	−25.4272	−0.948274	−0.474137	+	0.880451i	$0.657240\pi$
$720$	0	0
$721$	4.98498	0.185650
$722$	−29.7223	−1.10615
$723$	−24.1059	−0.896509
$724$	−15.1262	−0.562161
$725$	0	0
$726$	−17.7296	−0.658007
$727$	−49.2952	−1.82826	−0.914129	−	0.405424i	$-0.867124\pi$
$727$	−49.2952	−1.82826	−0.914129	+	0.405424i	$0.867124\pi$
$728$	59.9599	2.22226
$729$	1.00000	0.0370370
$730$	0	0
$731$	−10.7955	−0.399287
$732$	−6.09691	−0.225348
$733$	21.9873	0.812118	0.406059	−	0.913847i	$-0.366903\pi$
$733$	21.9873	0.812118	0.406059	+	0.913847i	$0.366903\pi$
$734$	−27.0151	−0.997147
$735$	0	0
$736$	−7.79902	−0.287476
$737$	−0.527429	−0.0194281
$738$	−7.43486	−0.273681
$739$	29.2864	1.07732	0.538660	−	0.842523i	$-0.318931\pi$
$739$	29.2864	1.07732	0.538660	+	0.842523i	$0.318931\pi$
$740$	0	0
$741$	−7.24339	−0.266093
$742$	−50.4872	−1.85344
$743$	−31.8867	−1.16981	−0.584905	−	0.811102i	$-0.698868\pi$
$743$	−31.8867	−1.16981	−0.584905	+	0.811102i	$0.698868\pi$
$744$	−18.9889	−0.696167
$745$	0	0
$746$	11.0083	0.403042
$747$	−7.53754	−0.275784
$748$	0.521098	0.0190532
$749$	−4.55690	−0.166506
$750$	0	0
$751$	35.7793	1.30561	0.652803	−	0.757528i	$-0.273593\pi$
$751$	35.7793	1.30561	0.652803	+	0.757528i	$0.273593\pi$
$752$	31.9086	1.16359
$753$	−12.7402	−0.464279
$754$	−106.597	−3.88203
$755$	0	0
$756$	−3.63058	−0.132043
$757$	45.8560	1.66666	0.833332	−	0.552772i	$-0.186430\pi$
$757$	45.8560	1.66666	0.833332	+	0.552772i	$0.186430\pi$
$758$	46.5779	1.69179
$759$	1.15328	0.0418612
$760$	0	0
$761$	0.426643	0.0154658	0.00773290	−	0.999970i	$-0.497539\pi$
$761$	0.426643	0.0154658	0.00773290	+	0.999970i	$0.497539\pi$
$762$	−17.4798	−0.633225
$763$	32.8103	1.18781
$764$	−16.0293	−0.579921
$765$	0	0
$766$	22.0724	0.797506
$767$	−73.6148	−2.65807
$768$	16.4898	0.595026
$769$	21.6710	0.781477	0.390739	−	0.920502i	$-0.372220\pi$
$769$	21.6710	0.781477	0.390739	+	0.920502i	$0.372220\pi$
$770$	0	0
$771$	15.0987	0.543767
$772$	12.9984	0.467822
$773$	12.3939	0.445778	0.222889	−	0.974844i	$-0.428451\pi$
$773$	12.3939	0.445778	0.222889	+	0.974844i	$0.428451\pi$
$774$	−17.4211	−0.626187
$775$	0	0
$776$	−21.4846	−0.771252
$777$	15.2997	0.548874
$778$	−2.89657	−0.103847
$779$	4.92482	0.176450
$780$	0	0
$781$	−7.47797	−0.267583
$782$	3.16689	0.113248
$783$	−9.74810	−0.348369
$784$	−68.2580	−2.43779
$785$	0	0
$786$	22.3343	0.796639
$787$	18.4278	0.656880	0.328440	−	0.944525i	$-0.393477\pi$
$787$	18.4278	0.656880	0.328440	+	0.944525i	$0.393477\pi$
$788$	−6.75055	−0.240478
$789$	1.09168	0.0388650
$790$	0	0
$791$	−45.6155	−1.62190
$792$	−1.27002	−0.0451281
$793$	−50.0384	−1.77692
$794$	−54.8361	−1.94606
$795$	0	0
$796$	−6.71181	−0.237894
$797$	−32.0019	−1.13356	−0.566782	−	0.823868i	$-0.691812\pi$
$797$	−32.0019	−1.13356	−0.566782	+	0.823868i	$0.691812\pi$
$798$	8.44183	0.298837
$799$	−6.66863	−0.235919
$800$	0	0
$801$	−10.7848	−0.381064
$802$	−17.4979	−0.617871
$803$	4.44066	0.156707
$804$	−0.665811	−0.0234814
$805$	0	0
$806$	103.189	3.63468
$807$	−23.0272	−0.810595
$808$	9.37471	0.329801
$809$	−20.7941	−0.731080	−0.365540	−	0.930796i	$-0.619116\pi$
$809$	−20.7941	−0.731080	−0.365540	+	0.930796i	$0.619116\pi$
$810$	0	0
$811$	−37.0803	−1.30207	−0.651033	−	0.759049i	$-0.725664\pi$
$811$	−37.0803	−1.30207	−0.651033	+	0.759049i	$0.725664\pi$
$812$	35.3913	1.24199
$813$	−3.07898	−0.107984
$814$	−3.54369	−0.124206
$815$	0	0
$816$	−5.13878	−0.179893
$817$	11.5396	0.403721
$818$	65.0806	2.27549
$819$	−29.7968	−1.04118
$820$	0	0
$821$	17.1669	0.599128	0.299564	−	0.954076i	$-0.403159\pi$
$821$	17.1669	0.599128	0.299564	+	0.954076i	$0.403159\pi$
$822$	−37.6967	−1.31482
$823$	−9.22948	−0.321720	−0.160860	−	0.986977i	$-0.551427\pi$
$823$	−9.22948	−0.321720	−0.160860	+	0.986977i	$0.551427\pi$
$824$	2.20133	0.0766869
$825$	0	0
$826$	85.7945	2.98517
$827$	−42.9199	−1.49247	−0.746235	−	0.665682i	$-0.768141\pi$
$827$	−42.9199	−1.49247	−0.746235	+	0.665682i	$0.768141\pi$
$828$	1.45586	0.0505947
$829$	49.1283	1.70630	0.853149	−	0.521667i	$-0.174690\pi$
$829$	49.1283	1.70630	0.853149	+	0.521667i	$0.174690\pi$
$830$	0	0
$831$	27.5290	0.954971
$832$	18.1766	0.630159
$833$	14.2653	0.494265
$834$	−16.3241	−0.565257
$835$	0	0
$836$	−0.557015	−0.0192648
$837$	9.43646	0.326172
$838$	−3.18965	−0.110185
$839$	−9.19913	−0.317589	−0.158795	−	0.987312i	$-0.550761\pi$
$839$	−9.19913	−0.317589	−0.158795	+	0.987312i	$0.550761\pi$
$840$	0	0
$841$	66.0255	2.27674
$842$	24.0684	0.829450
$843$	2.65372	0.0913989
$844$	−18.3138	−0.630388
$845$	0	0
$846$	−10.7613	−0.369983
$847$	48.3108	1.65998
$848$	−32.8513	−1.12812
$849$	22.2178	0.762513
$850$	0	0
$851$	−6.13518	−0.210311
$852$	−9.43999	−0.323409
$853$	−17.7852	−0.608953	−0.304476	−	0.952520i	$-0.598481\pi$
$853$	−17.7852	−0.608953	−0.304476	+	0.952520i	$0.598481\pi$
$854$	58.3174	1.99558
$855$	0	0
$856$	−2.01229	−0.0687787
$857$	−48.0971	−1.64297	−0.821483	−	0.570234i	$-0.806853\pi$
$857$	−48.0971	−1.64297	−0.821483	+	0.570234i	$0.806853\pi$
$858$	6.90149	0.235613
$859$	−11.7704	−0.401601	−0.200800	−	0.979632i	$-0.564354\pi$
$859$	−11.7704	−0.401601	−0.200800	+	0.979632i	$0.564354\pi$
$860$	0	0
$861$	20.2590	0.690425
$862$	52.1908	1.77763
$863$	7.17069	0.244093	0.122047	−	0.992524i	$-0.461054\pi$
$863$	7.17069	0.244093	0.122047	+	0.992524i	$0.461054\pi$
$864$	−4.26801	−0.145201
$865$	0	0
$866$	1.15729	0.0393261
$867$	−15.9260	−0.540877
$868$	−34.2598	−1.16285
$869$	1.76139	0.0597509
$870$	0	0
$871$	−5.46443	−0.185155
$872$	14.4888	0.490651
$873$	10.6767	0.361351
$874$	−3.38517	−0.114505
$875$	0	0
$876$	5.60577	0.189401
$877$	−20.9768	−0.708338	−0.354169	−	0.935181i	$-0.615236\pi$
$877$	−20.9768	−0.708338	−0.354169	+	0.935181i	$0.615236\pi$
$878$	−38.6541	−1.30451
$879$	−1.41665	−0.0477823
$880$	0	0
$881$	−4.75718	−0.160273	−0.0801367	−	0.996784i	$-0.525536\pi$
$881$	−4.75718	−0.160273	−0.0801367	+	0.996784i	$0.525536\pi$
$882$	23.0204	0.775137
$883$	8.43619	0.283900	0.141950	−	0.989874i	$-0.454663\pi$
$883$	8.43619	0.283900	0.141950	+	0.989874i	$0.454663\pi$
$884$	5.39884	0.181583
$885$	0	0
$886$	−49.5828	−1.66577
$887$	−32.5050	−1.09141	−0.545705	−	0.837977i	$-0.683738\pi$
$887$	−32.5050	−1.09141	−0.545705	+	0.837977i	$0.683738\pi$
$888$	6.75622	0.226724
$889$	47.6300	1.59746
$890$	0	0
$891$	0.631130	0.0211436
$892$	−0.371728	−0.0124464
$893$	7.12827	0.238539
$894$	6.70129	0.224125
$895$	0	0
$896$	−60.0818	−2.00719
$897$	11.9485	0.398950
$898$	−26.3142	−0.878115
$899$	−91.9875	−3.06796
$900$	0	0
$901$	6.86564	0.228728
$902$	−4.69236	−0.156238
$903$	47.4701	1.57971
$904$	−20.1435	−0.669961
$905$	0	0
$906$	5.21820	0.173363
$907$	16.9947	0.564300	0.282150	−	0.959370i	$-0.408952\pi$
$907$	16.9947	0.564300	0.282150	+	0.959370i	$0.408952\pi$
$908$	4.32786	0.143625
$909$	−4.65872	−0.154520
$910$	0	0
$911$	2.44604	0.0810409	0.0405205	−	0.999179i	$-0.487098\pi$
$911$	2.44604	0.0810409	0.0405205	+	0.999179i	$0.487098\pi$
$912$	5.49298	0.181891
$913$	−4.75716	−0.157439
$914$	28.3745	0.938547
$915$	0	0
$916$	14.4316	0.476834
$917$	−60.8580	−2.00971
$918$	1.73308	0.0572002
$919$	−50.3411	−1.66060	−0.830300	−	0.557317i	$-0.811831\pi$
$919$	−50.3411	−1.66060	−0.830300	+	0.557317i	$0.811831\pi$
$920$	0	0
$921$	−12.5909	−0.414884
$922$	34.3512	1.13130
$923$	−77.4756	−2.55014
$924$	−2.29137	−0.0753804
$925$	0	0
$926$	−42.8480	−1.40807
$927$	−1.09394	−0.0359297
$928$	41.6050	1.36575
$929$	−47.8739	−1.57069	−0.785345	−	0.619058i	$-0.787515\pi$
$929$	−47.8739	−1.57069	−0.785345	+	0.619058i	$0.787515\pi$
$930$	0	0
$931$	−15.2486	−0.499753
$932$	16.5224	0.541209
$933$	−15.2282	−0.498549
$934$	38.2092	1.25024
$935$	0	0
$936$	−13.1580	−0.430084
$937$	−0.907269	−0.0296392	−0.0148196	−	0.999890i	$-0.504717\pi$
$937$	−0.907269	−0.0296392	−0.0148196	+	0.999890i	$0.504717\pi$
$938$	6.36853	0.207940
$939$	4.70555	0.153560
$940$	0	0
$941$	27.2717	0.889031	0.444515	−	0.895771i	$-0.353376\pi$
$941$	27.2717	0.889031	0.444515	+	0.895771i	$0.353376\pi$
$942$	19.1961	0.625444
$943$	−8.12386	−0.264549
$944$	55.8253	1.81696
$945$	0	0
$946$	−10.9949	−0.357477
$947$	−5.49476	−0.178556	−0.0892779	−	0.996007i	$-0.528456\pi$
$947$	−5.49476	−0.178556	−0.0892779	+	0.996007i	$0.528456\pi$
$948$	2.22353	0.0722168
$949$	46.0075	1.49347
$950$	0	0
$951$	12.1677	0.394566
$952$	9.50288	0.307990
$953$	15.2648	0.494475	0.247238	−	0.968955i	$-0.420477\pi$
$953$	15.2648	0.494475	0.247238	+	0.968955i	$0.420477\pi$
$954$	11.0793	0.358705
$955$	0	0
$956$	13.1513	0.425343
$957$	−6.15232	−0.198876
$958$	−13.4791	−0.435488
$959$	102.718	3.31695
$960$	0	0
$961$	58.0468	1.87248
$962$	−36.7145	−1.18372
$963$	1.00000	0.0322245
$964$	−19.2057	−0.618574
$965$	0	0
$966$	−13.9254	−0.448044
$967$	−55.0700	−1.77093	−0.885465	−	0.464705i	$-0.846160\pi$
$967$	−55.0700	−1.77093	−0.885465	+	0.464705i	$0.846160\pi$
$968$	21.3337	0.685690
$969$	−1.14799	−0.0368786
$970$	0	0
$971$	26.1952	0.840644	0.420322	−	0.907375i	$-0.361917\pi$
$971$	26.1952	0.840644	0.420322	+	0.907375i	$0.361917\pi$
$972$	0.796721	0.0255548
$973$	44.4810	1.42599
$974$	−2.60853	−0.0835826
$975$	0	0
$976$	37.9463	1.21463
$977$	5.77309	0.184698	0.0923488	−	0.995727i	$-0.470563\pi$
$977$	5.77309	0.184698	0.0923488	+	0.995727i	$0.470563\pi$
$978$	0.198995	0.00636317
$979$	−6.80663	−0.217541
$980$	0	0
$981$	−7.20013	−0.229882
$982$	27.1945	0.867812
$983$	−24.2688	−0.774055	−0.387027	−	0.922068i	$-0.626498\pi$
$983$	−24.2688	−0.774055	−0.387027	+	0.922068i	$0.626498\pi$
$984$	8.94621	0.285195
$985$	0	0
$986$	−16.8942	−0.538022
$987$	29.3232	0.933369
$988$	−5.77096	−0.183599
$989$	−19.0355	−0.605294
$990$	0	0
$991$	−6.41227	−0.203693	−0.101846	−	0.994800i	$-0.532475\pi$
$991$	−6.41227	−0.203693	−0.101846	+	0.994800i	$0.532475\pi$
$992$	−40.2749	−1.27873
$993$	7.45359	0.236532
$994$	90.2942	2.86396
$995$	0	0
$996$	−6.00531	−0.190286
$997$	49.3275	1.56222	0.781109	−	0.624394i	$-0.214654\pi$
$997$	49.3275	1.56222	0.781109	+	0.624394i	$0.214654\pi$
$998$	−3.52575	−0.111606
$999$	−3.35748	−0.106226

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8025.2.a.bc.1.8		10
5.4	even	2		1605.2.a.k.1.3	✓	10
15.14	odd	2		4815.2.a.p.1.8		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1605.2.a.k.1.3	✓	10	5.4	even	2
4815.2.a.p.1.8		10	15.14	odd	2
8025.2.a.bc.1.8		10	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8025 = 3 \cdot 5^{2} \cdot 107 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8025.a (trivial)

\(f(q)\)	\(=\)	\(q+1.67234 q^{2} +1.00000 q^{3} +0.796721 q^{4} +1.67234 q^{6} -4.55690 q^{7} -2.01229 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.67234 q^{2} +1.00000 q^{3} +0.796721 q^{4} +1.67234 q^{6} -4.55690 q^{7} -2.01229 q^{8} +1.00000 q^{9} +0.631130 q^{11} +0.796721 q^{12} +6.53883 q^{13} -7.62069 q^{14} -4.95868 q^{16} +1.03632 q^{17} +1.67234 q^{18} -1.10775 q^{19} -4.55690 q^{21} +1.05546 q^{22} +1.82732 q^{23} -2.01229 q^{24} +10.9351 q^{26} +1.00000 q^{27} -3.63058 q^{28} -9.74810 q^{29} +9.43646 q^{31} -4.26801 q^{32} +0.631130 q^{33} +1.73308 q^{34} +0.796721 q^{36} -3.35748 q^{37} -1.85254 q^{38} +6.53883 q^{39} -4.44578 q^{41} -7.62069 q^{42} -10.4172 q^{43} +0.502834 q^{44} +3.05590 q^{46} -6.43490 q^{47} -4.95868 q^{48} +13.7654 q^{49} +1.03632 q^{51} +5.20962 q^{52} +6.62501 q^{53} +1.67234 q^{54} +9.16982 q^{56} -1.10775 q^{57} -16.3021 q^{58} -11.2581 q^{59} -7.65251 q^{61} +15.7810 q^{62} -4.55690 q^{63} +2.77979 q^{64} +1.05546 q^{66} -0.835690 q^{67} +0.825658 q^{68} +1.82732 q^{69} -11.8486 q^{71} -2.01229 q^{72} +7.03605 q^{73} -5.61484 q^{74} -0.882568 q^{76} -2.87600 q^{77} +10.9351 q^{78} +2.79085 q^{79} +1.00000 q^{81} -7.43486 q^{82} -7.53754 q^{83} -3.63058 q^{84} -17.4211 q^{86} -9.74810 q^{87} -1.27002 q^{88} -10.7848 q^{89} -29.7968 q^{91} +1.45586 q^{92} +9.43646 q^{93} -10.7613 q^{94} -4.26801 q^{96} +10.6767 q^{97} +23.0204 q^{98} +0.631130 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 2 q^{2} + 10 q^{3} + 10 q^{4} - 2 q^{6} + 10 q^{9}+O(q^{10}) \) 10 * q - 2 * q^2 + 10 * q^3 + 10 * q^4 - 2 * q^6 + 10 * q^9	\( 10 q - 2 q^{2} + 10 q^{3} + 10 q^{4} - 2 q^{6} + 10 q^{9} - 14 q^{11} + 10 q^{12} + 3 q^{13} - 16 q^{14} + 10 q^{16} - 8 q^{17} - 2 q^{18} - 19 q^{19} - 5 q^{22} - 4 q^{23} - 22 q^{26} + 10 q^{27} - 25 q^{29} - 2 q^{31} + 13 q^{32} - 14 q^{33} - 37 q^{34} + 10 q^{36} + 10 q^{37} + 13 q^{38} + 3 q^{39} - 31 q^{41} - 16 q^{42} - 62 q^{44} + 2 q^{46} + q^{47} + 10 q^{48} + 26 q^{49} - 8 q^{51} + 30 q^{52} - 9 q^{53} - 2 q^{54} - 63 q^{56} - 19 q^{57} - 30 q^{58} - 65 q^{59} + 12 q^{61} + 39 q^{62} - 2 q^{64} - 5 q^{66} + 10 q^{67} - 22 q^{68} - 4 q^{69} - 45 q^{71} - q^{73} + 19 q^{74} - 39 q^{76} + q^{77} - 22 q^{78} - 47 q^{79} + 10 q^{81} + 23 q^{82} - q^{83} + 12 q^{86} - 25 q^{87} + 8 q^{88} - 34 q^{89} - 26 q^{91} - 14 q^{92} - 2 q^{93} - 64 q^{94} + 13 q^{96} - 5 q^{97} + 51 q^{98} - 14 q^{99}+O(q^{100}) \) 10 * q - 2 * q^2 + 10 * q^3 + 10 * q^4 - 2 * q^6 + 10 * q^9 - 14 * q^11 + 10 * q^12 + 3 * q^13 - 16 * q^14 + 10 * q^16 - 8 * q^17 - 2 * q^18 - 19 * q^19 - 5 * q^22 - 4 * q^23 - 22 * q^26 + 10 * q^27 - 25 * q^29 - 2 * q^31 + 13 * q^32 - 14 * q^33 - 37 * q^34 + 10 * q^36 + 10 * q^37 + 13 * q^38 + 3 * q^39 - 31 * q^41 - 16 * q^42 - 62 * q^44 + 2 * q^46 + q^47 + 10 * q^48 + 26 * q^49 - 8 * q^51 + 30 * q^52 - 9 * q^53 - 2 * q^54 - 63 * q^56 - 19 * q^57 - 30 * q^58 - 65 * q^59 + 12 * q^61 + 39 * q^62 - 2 * q^64 - 5 * q^66 + 10 * q^67 - 22 * q^68 - 4 * q^69 - 45 * q^71 - q^73 + 19 * q^74 - 39 * q^76 + q^77 - 22 * q^78 - 47 * q^79 + 10 * q^81 + 23 * q^82 - q^83 + 12 * q^86 - 25 * q^87 + 8 * q^88 - 34 * q^89 - 26 * q^91 - 14 * q^92 - 2 * q^93 - 64 * q^94 + 13 * q^96 - 5 * q^97 + 51 * q^98 - 14 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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