LMFDB - Embedded newform 1755.2.a.u.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1755,2,Mod(1,1755)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1755, 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("1755.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1755 = 3^{3} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1755.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:	$14.0137455547$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 12x^{5} + 9x^{4} + 39x^{3} - 16x^{2} - 30x + 12 $ x^7 - x^6 - 12x^5 + 9x^4 + 39x^3 - 16x^2 - 30*x + 12
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-1.58727$ of defining polynomial
Character	$\chi$	$=$	1755.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.58727 q^{2} +0.519412 q^{4} -1.00000 q^{5} +2.72171 q^{7} -2.35009 q^{8} +O(q^{10})$	\(q+1.58727 q^{2} +0.519412 q^{4} -1.00000 q^{5} +2.72171 q^{7} -2.35009 q^{8} -1.58727 q^{10} +0.148791 q^{11} +1.00000 q^{13} +4.32008 q^{14} -4.76903 q^{16} +5.13965 q^{17} +7.28845 q^{19} -0.519412 q^{20} +0.236171 q^{22} -5.32732 q^{23} +1.00000 q^{25} +1.58727 q^{26} +1.41369 q^{28} -4.08391 q^{29} +8.19783 q^{31} -2.86955 q^{32} +8.15800 q^{34} -2.72171 q^{35} +7.47856 q^{37} +11.5687 q^{38} +2.35009 q^{40} +8.59642 q^{41} -9.95862 q^{43} +0.0772840 q^{44} -8.45587 q^{46} +0.909381 q^{47} +0.407728 q^{49} +1.58727 q^{50} +0.519412 q^{52} +9.31424 q^{53} -0.148791 q^{55} -6.39626 q^{56} -6.48225 q^{58} +11.4112 q^{59} -8.52754 q^{61} +13.0121 q^{62} +4.98333 q^{64} -1.00000 q^{65} +14.0748 q^{67} +2.66960 q^{68} -4.32008 q^{70} -0.424131 q^{71} +15.0360 q^{73} +11.8705 q^{74} +3.78570 q^{76} +0.404968 q^{77} -5.39188 q^{79} +4.76903 q^{80} +13.6448 q^{82} -4.32727 q^{83} -5.13965 q^{85} -15.8070 q^{86} -0.349673 q^{88} -10.1886 q^{89} +2.72171 q^{91} -2.76707 q^{92} +1.44343 q^{94} -7.28845 q^{95} -13.1693 q^{97} +0.647173 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - q^{2} + 11 q^{4} - 7 q^{5} + 2 q^{7} - 6 q^{8}+O(q^{10}) $ 7 * q - q^2 + 11 * q^4 - 7 * q^5 + 2 * q^7 - 6 * q^8	$ 7 q - q^{2} + 11 q^{4} - 7 q^{5} + 2 q^{7} - 6 q^{8} + q^{10} - q^{11} + 7 q^{13} + 12 q^{14} + 23 q^{16} - 11 q^{17} + 2 q^{19} - 11 q^{20} + 16 q^{22} + q^{23} + 7 q^{25} - q^{26} + 10 q^{28} + 4 q^{29} - 13 q^{32} + q^{34} - 2 q^{35} + 23 q^{37} + 15 q^{38} + 6 q^{40} - 2 q^{41} + 8 q^{43} - 10 q^{44} + 37 q^{46} - 2 q^{47} + 43 q^{49} - q^{50} + 11 q^{52} - 10 q^{53} + q^{55} + 68 q^{56} - 26 q^{58} + 13 q^{59} + 21 q^{61} - 9 q^{62} + 46 q^{64} - 7 q^{65} + 21 q^{67} - 53 q^{68} - 12 q^{70} + 10 q^{71} + 13 q^{73} + 68 q^{74} - 41 q^{76} - 6 q^{77} + 8 q^{79} - 23 q^{80} - 26 q^{82} + 4 q^{83} + 11 q^{85} + 12 q^{86} + 44 q^{88} + 27 q^{89} + 2 q^{91} + 9 q^{92} - 24 q^{94} - 2 q^{95} - 15 q^{97} - 7 q^{98}+O(q^{100}) $ 7 * q - q^2 + 11 * q^4 - 7 * q^5 + 2 * q^7 - 6 * q^8 + q^10 - q^11 + 7 * q^13 + 12 * q^14 + 23 * q^16 - 11 * q^17 + 2 * q^19 - 11 * q^20 + 16 * q^22 + q^23 + 7 * q^25 - q^26 + 10 * q^28 + 4 * q^29 - 13 * q^32 + q^34 - 2 * q^35 + 23 * q^37 + 15 * q^38 + 6 * q^40 - 2 * q^41 + 8 * q^43 - 10 * q^44 + 37 * q^46 - 2 * q^47 + 43 * q^49 - q^50 + 11 * q^52 - 10 * q^53 + q^55 + 68 * q^56 - 26 * q^58 + 13 * q^59 + 21 * q^61 - 9 * q^62 + 46 * q^64 - 7 * q^65 + 21 * q^67 - 53 * q^68 - 12 * q^70 + 10 * q^71 + 13 * q^73 + 68 * q^74 - 41 * q^76 - 6 * q^77 + 8 * q^79 - 23 * q^80 - 26 * q^82 + 4 * q^83 + 11 * q^85 + 12 * q^86 + 44 * q^88 + 27 * q^89 + 2 * q^91 + 9 * q^92 - 24 * q^94 - 2 * q^95 - 15 * q^97 - 7 * q^98

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 $

$2$	1.58727	1.12237	0.561183	−	0.827692i	$-0.310346\pi$
$2$	1.58727	1.12237	0.561183	+	0.827692i	$0.310346\pi$
$3$	0	0
$3$	0	0
$4$	0.519412	0.259706
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	2.72171	1.02871	0.514356	−	0.857577i	$-0.328031\pi$
$7$	2.72171	1.02871	0.514356	+	0.857577i	$0.328031\pi$
$8$	−2.35009	−0.830881
$9$	0	0
$10$	−1.58727	−0.501937
$11$	0.148791	0.0448623	0.0224311	−	0.999748i	$-0.492859\pi$
$11$	0.148791	0.0448623	0.0224311	+	0.999748i	$0.492859\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	4.32008	1.15459
$15$	0	0
$16$	−4.76903	−1.19226
$17$	5.13965	1.24655	0.623275	−	0.782003i	$-0.285802\pi$
$17$	5.13965	1.24655	0.623275	+	0.782003i	$0.285802\pi$
$18$	0	0
$19$	7.28845	1.67208	0.836042	−	0.548665i	$-0.184864\pi$
$19$	7.28845	1.67208	0.836042	+	0.548665i	$0.184864\pi$
$20$	−0.519412	−0.116144
$21$	0	0
$22$	0.236171	0.0503519
$23$	−5.32732	−1.11082	−0.555412	−	0.831576i	$-0.687439\pi$
$23$	−5.32732	−1.11082	−0.555412	+	0.831576i	$0.687439\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	1.58727	0.311288
$27$	0	0
$28$	1.41369	0.267162
$29$	−4.08391	−0.758363	−0.379182	−	0.925322i	$-0.623794\pi$
$29$	−4.08391	−0.758363	−0.379182	+	0.925322i	$0.623794\pi$
$30$	0	0
$31$	8.19783	1.47237	0.736187	−	0.676779i	$-0.236625\pi$
$31$	8.19783	1.47237	0.736187	+	0.676779i	$0.236625\pi$
$32$	−2.86955	−0.507270
$33$	0	0
$34$	8.15800	1.39908
$35$	−2.72171	−0.460054
$36$	0	0
$37$	7.47856	1.22947	0.614734	−	0.788735i	$-0.289264\pi$
$37$	7.47856	1.22947	0.614734	+	0.788735i	$0.289264\pi$
$38$	11.5687	1.87669
$39$	0	0
$40$	2.35009	0.371581
$41$	8.59642	1.34254	0.671268	−	0.741215i	$-0.265750\pi$
$41$	8.59642	1.34254	0.671268	+	0.741215i	$0.265750\pi$
$42$	0	0
$43$	−9.95862	−1.51867	−0.759337	−	0.650697i	$-0.774477\pi$
$43$	−9.95862	−1.51867	−0.759337	+	0.650697i	$0.774477\pi$
$44$	0.0772840	0.0116510
$45$	0	0
$46$	−8.45587	−1.24675
$47$	0.909381	0.132647	0.0663234	−	0.997798i	$-0.478873\pi$
$47$	0.909381	0.132647	0.0663234	+	0.997798i	$0.478873\pi$
$48$	0	0
$49$	0.407728	0.0582469
$50$	1.58727	0.224473
$51$	0	0
$52$	0.519412	0.0720294
$53$	9.31424	1.27941	0.639704	−	0.768621i	$-0.279057\pi$
$53$	9.31424	1.27941	0.639704	+	0.768621i	$0.279057\pi$
$54$	0	0
$55$	−0.148791	−0.0200630
$56$	−6.39626	−0.854737
$57$	0	0
$58$	−6.48225	−0.851161
$59$	11.4112	1.48561	0.742805	−	0.669508i	$-0.233495\pi$
$59$	11.4112	1.48561	0.742805	+	0.669508i	$0.233495\pi$
$60$	0	0
$61$	−8.52754	−1.09184	−0.545920	−	0.837837i	$-0.683820\pi$
$61$	−8.52754	−1.09184	−0.545920	+	0.837837i	$0.683820\pi$
$62$	13.0121	1.65254
$63$	0	0
$64$	4.98333	0.622916
$65$	−1.00000	−0.124035
$66$	0	0
$67$	14.0748	1.71951	0.859754	−	0.510709i	$-0.170617\pi$
$67$	14.0748	1.71951	0.859754	+	0.510709i	$0.170617\pi$
$68$	2.66960	0.323736
$69$	0	0
$70$	−4.32008	−0.516349
$71$	−0.424131	−0.0503351	−0.0251676	−	0.999683i	$-0.508012\pi$
$71$	−0.424131	−0.0503351	−0.0251676	+	0.999683i	$0.508012\pi$
$72$	0	0
$73$	15.0360	1.75983	0.879913	−	0.475136i	$-0.157601\pi$
$73$	15.0360	1.75983	0.879913	+	0.475136i	$0.157601\pi$
$74$	11.8705	1.37991
$75$	0	0
$76$	3.78570	0.434250
$77$	0.404968	0.0461503
$78$	0	0
$79$	−5.39188	−0.606634	−0.303317	−	0.952890i	$-0.598094\pi$
$79$	−5.39188	−0.606634	−0.303317	+	0.952890i	$0.598094\pi$
$80$	4.76903	0.533194
$81$	0	0
$82$	13.6448	1.50682
$83$	−4.32727	−0.474980	−0.237490	−	0.971390i	$-0.576325\pi$
$83$	−4.32727	−0.474980	−0.237490	+	0.971390i	$0.576325\pi$
$84$	0	0
$85$	−5.13965	−0.557474
$86$	−15.8070	−1.70451
$87$	0	0
$88$	−0.349673	−0.0372752
$89$	−10.1886	−1.07999	−0.539997	−	0.841667i	$-0.681575\pi$
$89$	−10.1886	−1.07999	−0.539997	+	0.841667i	$0.681575\pi$
$90$	0	0
$91$	2.72171	0.285313
$92$	−2.76707	−0.288487
$93$	0	0
$94$	1.44343	0.148878
$95$	−7.28845	−0.747779
$96$	0	0
$97$	−13.1693	−1.33714	−0.668572	−	0.743647i	$-0.733094\pi$
$97$	−13.1693	−1.33714	−0.668572	+	0.743647i	$0.733094\pi$
$98$	0.647173	0.0653743
$99$	0	0
$100$	0.519412	0.0519412
$101$	4.26890	0.424771	0.212386	−	0.977186i	$-0.431877\pi$
$101$	4.26890	0.424771	0.212386	+	0.977186i	$0.431877\pi$
$102$	0	0
$103$	2.05180	0.202169	0.101085	−	0.994878i	$-0.467769\pi$
$103$	2.05180	0.202169	0.101085	+	0.994878i	$0.467769\pi$
$104$	−2.35009	−0.230445
$105$	0	0
$106$	14.7842	1.43597
$107$	−20.3270	−1.96509	−0.982544	−	0.186031i	$-0.940438\pi$
$107$	−20.3270	−1.96509	−0.982544	+	0.186031i	$0.940438\pi$
$108$	0	0
$109$	−2.70260	−0.258862	−0.129431	−	0.991588i	$-0.541315\pi$
$109$	−2.70260	−0.258862	−0.129431	+	0.991588i	$0.541315\pi$
$110$	−0.236171	−0.0225181
$111$	0	0
$112$	−12.9799	−1.22649
$113$	−15.1099	−1.42142	−0.710708	−	0.703488i	$-0.751625\pi$
$113$	−15.1099	−1.42142	−0.710708	+	0.703488i	$0.751625\pi$
$114$	0	0
$115$	5.32732	0.496775
$116$	−2.12123	−0.196951
$117$	0	0
$118$	18.1126	1.66740
$119$	13.9887	1.28234
$120$	0	0
$121$	−10.9779	−0.997987
$122$	−13.5355	−1.22544
$123$	0	0
$124$	4.25805	0.382384
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	9.11899	0.809179	0.404590	−	0.914498i	$-0.367414\pi$
$127$	9.11899	0.809179	0.404590	+	0.914498i	$0.367414\pi$
$128$	13.6490	1.20641
$129$	0	0
$130$	−1.58727	−0.139212
$131$	−10.0428	−0.877447	−0.438724	−	0.898622i	$-0.644569\pi$
$131$	−10.0428	−0.877447	−0.438724	+	0.898622i	$0.644569\pi$
$132$	0	0
$133$	19.8371	1.72009
$134$	22.3404	1.92992
$135$	0	0
$136$	−12.0786	−1.03573
$137$	−17.7296	−1.51474	−0.757372	−	0.652983i	$-0.773517\pi$
$137$	−17.7296	−1.51474	−0.757372	+	0.652983i	$0.773517\pi$
$138$	0	0
$139$	7.14208	0.605784	0.302892	−	0.953025i	$-0.402048\pi$
$139$	7.14208	0.605784	0.302892	+	0.953025i	$0.402048\pi$
$140$	−1.41369	−0.119479
$141$	0	0
$142$	−0.673209	−0.0564945
$143$	0.148791	0.0124426
$144$	0	0
$145$	4.08391	0.339150
$146$	23.8660	1.97517
$147$	0	0
$148$	3.88445	0.319300
$149$	−5.90933	−0.484111	−0.242056	−	0.970262i	$-0.577822\pi$
$149$	−5.90933	−0.484111	−0.242056	+	0.970262i	$0.577822\pi$
$150$	0	0
$151$	−0.0283679	−0.00230855	−0.00115427	−	0.999999i	$-0.500367\pi$
$151$	−0.0283679	−0.00230855	−0.00115427	+	0.999999i	$0.500367\pi$
$152$	−17.1285	−1.38930
$153$	0	0
$154$	0.642791	0.0517976
$155$	−8.19783	−0.658465
$156$	0	0
$157$	17.0817	1.36327	0.681633	−	0.731694i	$-0.261270\pi$
$157$	17.0817	1.36327	0.681633	+	0.731694i	$0.261270\pi$
$158$	−8.55835	−0.680866
$159$	0	0
$160$	2.86955	0.226858
$161$	−14.4994	−1.14272
$162$	0	0
$163$	11.1693	0.874850	0.437425	−	0.899255i	$-0.355890\pi$
$163$	11.1693	0.874850	0.437425	+	0.899255i	$0.355890\pi$
$164$	4.46508	0.348664
$165$	0	0
$166$	−6.86853	−0.533101
$167$	0.191562	0.0148235	0.00741174	−	0.999973i	$-0.497641\pi$
$167$	0.191562	0.0148235	0.00741174	+	0.999973i	$0.497641\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−8.15800	−0.625690
$171$	0	0
$172$	−5.17262	−0.394409
$173$	19.8367	1.50815	0.754076	−	0.656787i	$-0.228085\pi$
$173$	19.8367	1.50815	0.754076	+	0.656787i	$0.228085\pi$
$174$	0	0
$175$	2.72171	0.205742
$176$	−0.709591	−0.0534875
$177$	0	0
$178$	−16.1721	−1.21215
$179$	4.74782	0.354869	0.177434	−	0.984133i	$-0.443220\pi$
$179$	4.74782	0.354869	0.177434	+	0.984133i	$0.443220\pi$
$180$	0	0
$181$	−12.1070	−0.899903	−0.449952	−	0.893053i	$-0.648559\pi$
$181$	−12.1070	−0.899903	−0.449952	+	0.893053i	$0.648559\pi$
$182$	4.32008	0.320226
$183$	0	0
$184$	12.5197	0.922962
$185$	−7.47856	−0.549834
$186$	0	0
$187$	0.764736	0.0559231
$188$	0.472343	0.0344491
$189$	0	0
$190$	−11.5687	−0.839282
$191$	12.0984	0.875409	0.437704	−	0.899119i	$-0.355792\pi$
$191$	12.0984	0.875409	0.437704	+	0.899119i	$0.355792\pi$
$192$	0	0
$193$	−19.5765	−1.40915	−0.704574	−	0.709630i	$-0.748862\pi$
$193$	−19.5765	−1.40915	−0.704574	+	0.709630i	$0.748862\pi$
$194$	−20.9032	−1.50077
$195$	0	0
$196$	0.211779	0.0151271
$197$	−9.12242	−0.649946	−0.324973	−	0.945723i	$-0.605355\pi$
$197$	−9.12242	−0.649946	−0.324973	+	0.945723i	$0.605355\pi$
$198$	0	0
$199$	2.75289	0.195148	0.0975738	−	0.995228i	$-0.468892\pi$
$199$	2.75289	0.195148	0.0975738	+	0.995228i	$0.468892\pi$
$200$	−2.35009	−0.166176
$201$	0	0
$202$	6.77587	0.476749
$203$	−11.1152	−0.780137
$204$	0	0
$205$	−8.59642	−0.600400
$206$	3.25675	0.226908
$207$	0	0
$208$	−4.76903	−0.330673
$209$	1.08446	0.0750135
$210$	0	0
$211$	−17.2239	−1.18574	−0.592870	−	0.805298i	$-0.702005\pi$
$211$	−17.2239	−1.18574	−0.592870	+	0.805298i	$0.702005\pi$
$212$	4.83792	0.332270
$213$	0	0
$214$	−32.2644	−2.20555
$215$	9.95862	0.679172
$216$	0	0
$217$	22.3121	1.51465
$218$	−4.28975	−0.290538
$219$	0	0
$220$	−0.0772840	−0.00521048
$221$	5.13965	0.345731
$222$	0	0
$223$	−20.4625	−1.37027	−0.685136	−	0.728415i	$-0.740257\pi$
$223$	−20.4625	−1.37027	−0.685136	+	0.728415i	$0.740257\pi$
$224$	−7.81010	−0.521834
$225$	0	0
$226$	−23.9833	−1.59535
$227$	−5.41799	−0.359605	−0.179802	−	0.983703i	$-0.557546\pi$
$227$	−5.41799	−0.359605	−0.179802	+	0.983703i	$0.557546\pi$
$228$	0	0
$229$	25.2989	1.67180	0.835898	−	0.548885i	$-0.184948\pi$
$229$	25.2989	1.67180	0.835898	+	0.548885i	$0.184948\pi$
$230$	8.45587	0.557564
$231$	0	0
$232$	9.59755	0.630110
$233$	−0.669918	−0.0438878	−0.0219439	−	0.999759i	$-0.506986\pi$
$233$	−0.669918	−0.0438878	−0.0219439	+	0.999759i	$0.506986\pi$
$234$	0	0
$235$	−0.909381	−0.0593215
$236$	5.92710	0.385821
$237$	0	0
$238$	22.2037	1.43925
$239$	0.685837	0.0443631	0.0221815	−	0.999754i	$-0.492939\pi$
$239$	0.685837	0.0443631	0.0221815	+	0.999754i	$0.492939\pi$
$240$	0	0
$241$	3.87745	0.249769	0.124884	−	0.992171i	$-0.460144\pi$
$241$	3.87745	0.249769	0.124884	+	0.992171i	$0.460144\pi$
$242$	−17.4248	−1.12011
$243$	0	0
$244$	−4.42930	−0.283557
$245$	−0.407728	−0.0260488
$246$	0	0
$247$	7.28845	0.463753
$248$	−19.2656	−1.22337
$249$	0	0
$250$	−1.58727	−0.100387
$251$	9.79079	0.617989	0.308995	−	0.951064i	$-0.400007\pi$
$251$	9.79079	0.617989	0.308995	+	0.951064i	$0.400007\pi$
$252$	0	0
$253$	−0.792659	−0.0498341
$254$	14.4743	0.908195
$255$	0	0
$256$	11.6979	0.731117
$257$	−24.0206	−1.49836	−0.749180	−	0.662366i	$-0.769552\pi$
$257$	−24.0206	−1.49836	−0.749180	+	0.662366i	$0.769552\pi$
$258$	0	0
$259$	20.3545	1.26477
$260$	−0.519412	−0.0322125
$261$	0	0
$262$	−15.9407	−0.984817
$263$	−26.8326	−1.65457	−0.827284	−	0.561784i	$-0.810115\pi$
$263$	−26.8326	−1.65457	−0.827284	+	0.561784i	$0.810115\pi$
$264$	0	0
$265$	−9.31424	−0.572169
$266$	31.4867	1.93057
$267$	0	0
$268$	7.31060	0.446566
$269$	−15.1846	−0.925823	−0.462911	−	0.886405i	$-0.653195\pi$
$269$	−15.1846	−0.925823	−0.462911	+	0.886405i	$0.653195\pi$
$270$	0	0
$271$	−15.1758	−0.921867	−0.460934	−	0.887435i	$-0.652485\pi$
$271$	−15.1758	−0.921867	−0.460934	+	0.887435i	$0.652485\pi$
$272$	−24.5112	−1.48621
$273$	0	0
$274$	−28.1416	−1.70010
$275$	0.148791	0.00897246
$276$	0	0
$277$	12.0428	0.723584	0.361792	−	0.932259i	$-0.382165\pi$
$277$	12.0428	0.723584	0.361792	+	0.932259i	$0.382165\pi$
$278$	11.3364	0.679911
$279$	0	0
$280$	6.39626	0.382250
$281$	−32.0618	−1.91264	−0.956322	−	0.292315i	$-0.905574\pi$
$281$	−32.0618	−1.91264	−0.956322	+	0.292315i	$0.905574\pi$
$282$	0	0
$283$	−12.2711	−0.729441	−0.364721	−	0.931117i	$-0.618835\pi$
$283$	−12.2711	−0.729441	−0.364721	+	0.931117i	$0.618835\pi$
$284$	−0.220299	−0.0130723
$285$	0	0
$286$	0.236171	0.0139651
$287$	23.3970	1.38108
$288$	0	0
$289$	9.41605	0.553885
$290$	6.48225	0.380651
$291$	0	0
$292$	7.80985	0.457037
$293$	2.67880	0.156497	0.0782487	−	0.996934i	$-0.475067\pi$
$293$	2.67880	0.156497	0.0782487	+	0.996934i	$0.475067\pi$
$294$	0	0
$295$	−11.4112	−0.664385
$296$	−17.5753	−1.02154
$297$	0	0
$298$	−9.37967	−0.543350
$299$	−5.32732	−0.308087
$300$	0	0
$301$	−27.1045	−1.56228
$302$	−0.0450274	−0.00259104
$303$	0	0
$304$	−34.7589	−1.99356
$305$	8.52754	0.488286
$306$	0	0
$307$	−8.49069	−0.484589	−0.242295	−	0.970203i	$-0.577900\pi$
$307$	−8.49069	−0.484589	−0.242295	+	0.970203i	$0.577900\pi$
$308$	0.210345	0.0119855
$309$	0	0
$310$	−13.0121	−0.739039
$311$	7.91577	0.448862	0.224431	−	0.974490i	$-0.427948\pi$
$311$	7.91577	0.448862	0.224431	+	0.974490i	$0.427948\pi$
$312$	0	0
$313$	29.4822	1.66643	0.833216	−	0.552947i	$-0.186497\pi$
$313$	29.4822	1.66643	0.833216	+	0.552947i	$0.186497\pi$
$314$	27.1131	1.53008
$315$	0	0
$316$	−2.80061	−0.157546
$317$	34.6412	1.94564	0.972822	−	0.231554i	$-0.0743808\pi$
$317$	34.6412	1.94564	0.972822	+	0.231554i	$0.0743808\pi$
$318$	0	0
$319$	−0.607651	−0.0340219
$320$	−4.98333	−0.278577
$321$	0	0
$322$	−23.0145	−1.28255
$323$	37.4601	2.08434
$324$	0	0
$325$	1.00000	0.0554700
$326$	17.7287	0.981902
$327$	0	0
$328$	−20.2023	−1.11549
$329$	2.47507	0.136455
$330$	0	0
$331$	−10.7393	−0.590284	−0.295142	−	0.955453i	$-0.595367\pi$
$331$	−10.7393	−0.590284	−0.295142	+	0.955453i	$0.595367\pi$
$332$	−2.24763	−0.123355
$333$	0	0
$334$	0.304059	0.0166374
$335$	−14.0748	−0.768987
$336$	0	0
$337$	13.4707	0.733795	0.366898	−	0.930261i	$-0.380420\pi$
$337$	13.4707	0.733795	0.366898	+	0.930261i	$0.380420\pi$
$338$	1.58727	0.0863359
$339$	0	0
$340$	−2.66960	−0.144779
$341$	1.21977	0.0660540
$342$	0	0
$343$	−17.9423	−0.968792
$344$	23.4036	1.26184
$345$	0	0
$346$	31.4860	1.69270
$347$	−20.5192	−1.10153	−0.550763	−	0.834662i	$-0.685663\pi$
$347$	−20.5192	−1.10153	−0.550763	+	0.834662i	$0.685663\pi$
$348$	0	0
$349$	−14.2303	−0.761728	−0.380864	−	0.924631i	$-0.624373\pi$
$349$	−14.2303	−0.761728	−0.380864	+	0.924631i	$0.624373\pi$
$350$	4.32008	0.230918
$351$	0	0
$352$	−0.426964	−0.0227573
$353$	−11.3635	−0.604815	−0.302408	−	0.953179i	$-0.597790\pi$
$353$	−11.3635	−0.604815	−0.302408	+	0.953179i	$0.597790\pi$
$354$	0	0
$355$	0.424131	0.0225106
$356$	−5.29210	−0.280481
$357$	0	0
$358$	7.53606	0.398293
$359$	−13.8717	−0.732122	−0.366061	−	0.930591i	$-0.619294\pi$
$359$	−13.8717	−0.732122	−0.366061	+	0.930591i	$0.619294\pi$
$360$	0	0
$361$	34.1214	1.79587
$362$	−19.2170	−1.01002
$363$	0	0
$364$	1.41369	0.0740975
$365$	−15.0360	−0.787018
$366$	0	0
$367$	−9.36985	−0.489102	−0.244551	−	0.969636i	$-0.578641\pi$
$367$	−9.36985	−0.489102	−0.244551	+	0.969636i	$0.578641\pi$
$368$	25.4062	1.32439
$369$	0	0
$370$	−11.8705	−0.617116
$371$	25.3507	1.31614
$372$	0	0
$373$	−6.13529	−0.317673	−0.158837	−	0.987305i	$-0.550774\pi$
$373$	−6.13529	−0.317673	−0.158837	+	0.987305i	$0.550774\pi$
$374$	1.21384	0.0627661
$375$	0	0
$376$	−2.13712	−0.110214
$377$	−4.08391	−0.210332
$378$	0	0
$379$	19.8315	1.01868	0.509338	−	0.860567i	$-0.329890\pi$
$379$	19.8315	1.01868	0.509338	+	0.860567i	$0.329890\pi$
$380$	−3.78570	−0.194202
$381$	0	0
$382$	19.2034	0.982529
$383$	15.1263	0.772915	0.386458	−	0.922307i	$-0.373699\pi$
$383$	15.1263	0.772915	0.386458	+	0.922307i	$0.373699\pi$
$384$	0	0
$385$	−0.404968	−0.0206391
$386$	−31.0731	−1.58158
$387$	0	0
$388$	−6.84031	−0.347264
$389$	14.1223	0.716029	0.358014	−	0.933716i	$-0.383454\pi$
$389$	14.1223	0.716029	0.358014	+	0.933716i	$0.383454\pi$
$390$	0	0
$391$	−27.3806	−1.38470
$392$	−0.958197	−0.0483963
$393$	0	0
$394$	−14.4797	−0.729477
$395$	5.39188	0.271295
$396$	0	0
$397$	−6.49729	−0.326090	−0.163045	−	0.986619i	$-0.552132\pi$
$397$	−6.49729	−0.326090	−0.163045	+	0.986619i	$0.552132\pi$
$398$	4.36957	0.219027
$399$	0	0
$400$	−4.76903	−0.238452
$401$	22.6659	1.13188	0.565940	−	0.824446i	$-0.308513\pi$
$401$	22.6659	1.13188	0.565940	+	0.824446i	$0.308513\pi$
$402$	0	0
$403$	8.19783	0.408363
$404$	2.21731	0.110316
$405$	0	0
$406$	−17.6428	−0.875599
$407$	1.11274	0.0551567
$408$	0	0
$409$	2.24304	0.110911	0.0554556	−	0.998461i	$-0.482339\pi$
$409$	2.24304	0.110911	0.0554556	+	0.998461i	$0.482339\pi$
$410$	−13.6448	−0.673869
$411$	0	0
$412$	1.06573	0.0525046
$413$	31.0580	1.52826
$414$	0	0
$415$	4.32727	0.212417
$416$	−2.86955	−0.140691
$417$	0	0
$418$	1.72132	0.0841926
$419$	−2.37775	−0.116161	−0.0580803	−	0.998312i	$-0.518498\pi$
$419$	−2.37775	−0.116161	−0.0580803	+	0.998312i	$0.518498\pi$
$420$	0	0
$421$	17.1808	0.837340	0.418670	−	0.908138i	$-0.362496\pi$
$421$	17.1808	0.837340	0.418670	+	0.908138i	$0.362496\pi$
$422$	−27.3389	−1.33083
$423$	0	0
$424$	−21.8893	−1.06304
$425$	5.13965	0.249310
$426$	0	0
$427$	−23.2095	−1.12319
$428$	−10.5581	−0.510345
$429$	0	0
$430$	15.8070	0.762280
$431$	7.91789	0.381391	0.190696	−	0.981649i	$-0.438926\pi$
$431$	7.91789	0.381391	0.190696	+	0.981649i	$0.438926\pi$
$432$	0	0
$433$	−17.8848	−0.859489	−0.429744	−	0.902951i	$-0.641396\pi$
$433$	−17.8848	−0.859489	−0.429744	+	0.902951i	$0.641396\pi$
$434$	35.4153	1.69999
$435$	0	0
$436$	−1.40376	−0.0672280
$437$	−38.8279	−1.85739
$438$	0	0
$439$	19.7789	0.943996	0.471998	−	0.881600i	$-0.343533\pi$
$439$	19.7789	0.943996	0.471998	+	0.881600i	$0.343533\pi$
$440$	0.349673	0.0166700
$441$	0	0
$442$	8.15800	0.388036
$443$	1.26535	0.0601186	0.0300593	−	0.999548i	$-0.490430\pi$
$443$	1.26535	0.0601186	0.0300593	+	0.999548i	$0.490430\pi$
$444$	0	0
$445$	10.1886	0.482988
$446$	−32.4795	−1.53795
$447$	0	0
$448$	13.5632	0.640801
$449$	33.7102	1.59088	0.795442	−	0.606030i	$-0.207239\pi$
$449$	33.7102	1.59088	0.795442	+	0.606030i	$0.207239\pi$
$450$	0	0
$451$	1.27907	0.0602292
$452$	−7.84823	−0.369150
$453$	0	0
$454$	−8.59979	−0.403608
$455$	−2.72171	−0.127596
$456$	0	0
$457$	−6.43844	−0.301178	−0.150589	−	0.988596i	$-0.548117\pi$
$457$	−6.43844	−0.301178	−0.150589	+	0.988596i	$0.548117\pi$
$458$	40.1560	1.87637
$459$	0	0
$460$	2.76707	0.129015
$461$	20.2556	0.943398	0.471699	−	0.881760i	$-0.343641\pi$
$461$	20.2556	0.943398	0.471699	+	0.881760i	$0.343641\pi$
$462$	0	0
$463$	7.85389	0.365001	0.182501	−	0.983206i	$-0.441581\pi$
$463$	7.85389	0.365001	0.182501	+	0.983206i	$0.441581\pi$
$464$	19.4763	0.904165
$465$	0	0
$466$	−1.06334	−0.0492582
$467$	−3.45708	−0.159975	−0.0799873	−	0.996796i	$-0.525488\pi$
$467$	−3.45708	−0.159975	−0.0799873	+	0.996796i	$0.525488\pi$
$468$	0	0
$469$	38.3075	1.76888
$470$	−1.44343	−0.0665804
$471$	0	0
$472$	−26.8173	−1.23436
$473$	−1.48176	−0.0681312
$474$	0	0
$475$	7.28845	0.334417
$476$	7.26588	0.333031
$477$	0	0
$478$	1.08861	0.0497916
$479$	−4.13577	−0.188968	−0.0944840	−	0.995526i	$-0.530120\pi$
$479$	−4.13577	−0.188968	−0.0944840	+	0.995526i	$0.530120\pi$
$480$	0	0
$481$	7.47856	0.340993
$482$	6.15454	0.280332
$483$	0	0
$484$	−5.70203	−0.259183
$485$	13.1693	0.597989
$486$	0	0
$487$	−12.7528	−0.577884	−0.288942	−	0.957347i	$-0.593303\pi$
$487$	−12.7528	−0.577884	−0.288942	+	0.957347i	$0.593303\pi$
$488$	20.0405	0.907189
$489$	0	0
$490$	−0.647173	−0.0292363
$491$	42.9952	1.94034	0.970172	−	0.242417i	$-0.0779404\pi$
$491$	42.9952	1.94034	0.970172	+	0.242417i	$0.0779404\pi$
$492$	0	0
$493$	−20.9899	−0.945337
$494$	11.5687	0.520500
$495$	0	0
$496$	−39.0957	−1.75545
$497$	−1.15436	−0.0517803
$498$	0	0
$499$	−27.9517	−1.25129	−0.625645	−	0.780108i	$-0.715164\pi$
$499$	−27.9517	−1.25129	−0.625645	+	0.780108i	$0.715164\pi$
$500$	−0.519412	−0.0232288
$501$	0	0
$502$	15.5406	0.693610
$503$	−16.8939	−0.753260	−0.376630	−	0.926364i	$-0.622917\pi$
$503$	−16.8939	−0.753260	−0.376630	+	0.926364i	$0.622917\pi$
$504$	0	0
$505$	−4.26890	−0.189963
$506$	−1.25816	−0.0559321
$507$	0	0
$508$	4.73651	0.210149
$509$	−7.85910	−0.348349	−0.174174	−	0.984715i	$-0.555726\pi$
$509$	−7.85910	−0.348349	−0.174174	+	0.984715i	$0.555726\pi$
$510$	0	0
$511$	40.9236	1.81035
$512$	−8.73031	−0.385829
$513$	0	0
$514$	−38.1270	−1.68171
$515$	−2.05180	−0.0904129
$516$	0	0
$517$	0.135308	0.00595084
$518$	32.3080	1.41953
$519$	0	0
$520$	2.35009	0.103058
$521$	−15.5192	−0.679909	−0.339954	−	0.940442i	$-0.610412\pi$
$521$	−15.5192	−0.679909	−0.339954	+	0.940442i	$0.610412\pi$
$522$	0	0
$523$	−39.7180	−1.73675	−0.868374	−	0.495909i	$-0.834835\pi$
$523$	−39.7180	−1.73675	−0.868374	+	0.495909i	$0.834835\pi$
$524$	−5.21637	−0.227878
$525$	0	0
$526$	−42.5904	−1.85703
$527$	42.1340	1.83539
$528$	0	0
$529$	5.38035	0.233928
$530$	−14.7842	−0.642183
$531$	0	0
$532$	10.3036	0.446718
$533$	8.59642	0.372352
$534$	0	0
$535$	20.3270	0.878814
$536$	−33.0769	−1.42871
$537$	0	0
$538$	−24.1020	−1.03911
$539$	0.0606665	0.00261309
$540$	0	0
$541$	−30.2705	−1.30143	−0.650716	−	0.759321i	$-0.725531\pi$
$541$	−30.2705	−1.30143	−0.650716	+	0.759321i	$0.725531\pi$
$542$	−24.0881	−1.03467
$543$	0	0
$544$	−14.7485	−0.632337
$545$	2.70260	0.115767
$546$	0	0
$547$	−12.4150	−0.530828	−0.265414	−	0.964135i	$-0.585509\pi$
$547$	−12.4150	−0.530828	−0.265414	+	0.964135i	$0.585509\pi$
$548$	−9.20898	−0.393388
$549$	0	0
$550$	0.236171	0.0100704
$551$	−29.7654	−1.26805
$552$	0	0
$553$	−14.6752	−0.624052
$554$	19.1152	0.812127
$555$	0	0
$556$	3.70968	0.157326
$557$	−31.0672	−1.31636	−0.658180	−	0.752860i	$-0.728674\pi$
$557$	−31.0672	−1.31636	−0.658180	+	0.752860i	$0.728674\pi$
$558$	0	0
$559$	−9.95862	−0.421205
$560$	12.9799	0.548503
$561$	0	0
$562$	−50.8905	−2.14669
$563$	7.24278	0.305247	0.152623	−	0.988284i	$-0.451228\pi$
$563$	7.24278	0.305247	0.152623	+	0.988284i	$0.451228\pi$
$564$	0	0
$565$	15.1099	0.635676
$566$	−19.4775	−0.818700
$567$	0	0
$568$	0.996746	0.0418225
$569$	4.04327	0.169503	0.0847513	−	0.996402i	$-0.472990\pi$
$569$	4.04327	0.169503	0.0847513	+	0.996402i	$0.472990\pi$
$570$	0	0
$571$	16.6455	0.696591	0.348295	−	0.937385i	$-0.386761\pi$
$571$	16.6455	0.696591	0.348295	+	0.937385i	$0.386761\pi$
$572$	0.0772840	0.00323140
$573$	0	0
$574$	37.1372	1.55008
$575$	−5.32732	−0.222165
$576$	0	0
$577$	−2.56771	−0.106895	−0.0534476	−	0.998571i	$-0.517021\pi$
$577$	−2.56771	−0.106895	−0.0534476	+	0.998571i	$0.517021\pi$
$578$	14.9458	0.621662
$579$	0	0
$580$	2.12123	0.0880793
$581$	−11.7776	−0.488617
$582$	0	0
$583$	1.38588	0.0573972
$584$	−35.3358	−1.46221
$585$	0	0
$586$	4.25197	0.175647
$587$	22.5595	0.931128	0.465564	−	0.885014i	$-0.345851\pi$
$587$	22.5595	0.931128	0.465564	+	0.885014i	$0.345851\pi$
$588$	0	0
$589$	59.7494	2.46193
$590$	−18.1126	−0.745683
$591$	0	0
$592$	−35.6655	−1.46584
$593$	24.5779	1.00929	0.504647	−	0.863326i	$-0.331623\pi$
$593$	24.5779	1.00929	0.504647	+	0.863326i	$0.331623\pi$
$594$	0	0
$595$	−13.9887	−0.573480
$596$	−3.06937	−0.125726
$597$	0	0
$598$	−8.45587	−0.345786
$599$	38.5309	1.57433	0.787166	−	0.616741i	$-0.211548\pi$
$599$	38.5309	1.57433	0.787166	+	0.616741i	$0.211548\pi$
$600$	0	0
$601$	1.38309	0.0564175	0.0282088	−	0.999602i	$-0.491020\pi$
$601$	1.38309	0.0564175	0.0282088	+	0.999602i	$0.491020\pi$
$602$	−43.0220	−1.75345
$603$	0	0
$604$	−0.0147346	−0.000599544 0
$605$	10.9779	0.446314
$606$	0	0
$607$	−39.6980	−1.61129	−0.805647	−	0.592396i	$-0.798182\pi$
$607$	−39.6980	−1.61129	−0.805647	+	0.592396i	$0.798182\pi$
$608$	−20.9146	−0.848198
$609$	0	0
$610$	13.5355	0.548035
$611$	0.909381	0.0367896
$612$	0	0
$613$	26.1959	1.05804	0.529021	−	0.848609i	$-0.322559\pi$
$613$	26.1959	1.05804	0.529021	+	0.848609i	$0.322559\pi$
$614$	−13.4770	−0.543886
$615$	0	0
$616$	−0.951709	−0.0383454
$617$	13.4836	0.542831	0.271415	−	0.962462i	$-0.412508\pi$
$617$	13.4836	0.542831	0.271415	+	0.962462i	$0.412508\pi$
$618$	0	0
$619$	0.579475	0.0232911	0.0116455	−	0.999932i	$-0.496293\pi$
$619$	0.579475	0.0232911	0.0116455	+	0.999932i	$0.496293\pi$
$620$	−4.25805	−0.171007
$621$	0	0
$622$	12.5644	0.503788
$623$	−27.7306	−1.11100
$624$	0	0
$625$	1.00000	0.0400000
$626$	46.7961	1.87035
$627$	0	0
$628$	8.87242	0.354048
$629$	38.4372	1.53259
$630$	0	0
$631$	−4.13320	−0.164540	−0.0822700	−	0.996610i	$-0.526217\pi$
$631$	−4.13320	−0.164540	−0.0822700	+	0.996610i	$0.526217\pi$
$632$	12.6714	0.504041
$633$	0	0
$634$	54.9848	2.18373
$635$	−9.11899	−0.361876
$636$	0	0
$637$	0.407728	0.0161548
$638$	−0.964503	−0.0381850
$639$	0	0
$640$	−13.6490	−0.539523
$641$	14.2807	0.564055	0.282028	−	0.959406i	$-0.408993\pi$
$641$	14.2807	0.564055	0.282028	+	0.959406i	$0.408993\pi$
$642$	0	0
$643$	30.2165	1.19162	0.595812	−	0.803124i	$-0.296830\pi$
$643$	30.2165	1.19162	0.595812	+	0.803124i	$0.296830\pi$
$644$	−7.53118	−0.296770
$645$	0	0
$646$	59.4591	2.33939
$647$	41.3380	1.62516	0.812582	−	0.582847i	$-0.198061\pi$
$647$	41.3380	1.62516	0.812582	+	0.582847i	$0.198061\pi$
$648$	0	0
$649$	1.69789	0.0666478
$650$	1.58727	0.0622577
$651$	0	0
$652$	5.80149	0.227204
$653$	−37.1484	−1.45373	−0.726864	−	0.686781i	$-0.759023\pi$
$653$	−37.1484	−1.45373	−0.726864	+	0.686781i	$0.759023\pi$
$654$	0	0
$655$	10.0428	0.392406
$656$	−40.9966	−1.60065
$657$	0	0
$658$	3.92860	0.153153
$659$	−12.1784	−0.474405	−0.237203	−	0.971460i	$-0.576230\pi$
$659$	−12.1784	−0.474405	−0.237203	+	0.971460i	$0.576230\pi$
$660$	0	0
$661$	−1.46279	−0.0568960	−0.0284480	−	0.999595i	$-0.509057\pi$
$661$	−1.46279	−0.0568960	−0.0284480	+	0.999595i	$0.509057\pi$
$662$	−17.0461	−0.662514
$663$	0	0
$664$	10.1695	0.394652
$665$	−19.8371	−0.769249
$666$	0	0
$667$	21.7563	0.842408
$668$	0.0994994	0.00384975
$669$	0	0
$670$	−22.3404	−0.863085
$671$	−1.26882	−0.0489824
$672$	0	0
$673$	−14.7570	−0.568841	−0.284421	−	0.958700i	$-0.591801\pi$
$673$	−14.7570	−0.568841	−0.284421	+	0.958700i	$0.591801\pi$
$674$	21.3816	0.823587
$675$	0	0
$676$	0.519412	0.0199774
$677$	−9.32282	−0.358305	−0.179153	−	0.983821i	$-0.557336\pi$
$677$	−9.32282	−0.358305	−0.179153	+	0.983821i	$0.557336\pi$
$678$	0	0
$679$	−35.8432	−1.37554
$680$	12.0786	0.463195
$681$	0	0
$682$	1.93609	0.0741368
$683$	−27.5508	−1.05420	−0.527100	−	0.849803i	$-0.676721\pi$
$683$	−27.5508	−1.05420	−0.527100	+	0.849803i	$0.676721\pi$
$684$	0	0
$685$	17.7296	0.677414
$686$	−28.4792	−1.08734
$687$	0	0
$688$	47.4930	1.81065
$689$	9.31424	0.354844
$690$	0	0
$691$	25.3662	0.964977	0.482489	−	0.875902i	$-0.339733\pi$
$691$	25.3662	0.964977	0.482489	+	0.875902i	$0.339733\pi$
$692$	10.3034	0.391676
$693$	0	0
$694$	−32.5693	−1.23632
$695$	−7.14208	−0.270915
$696$	0	0
$697$	44.1826	1.67354
$698$	−22.5872	−0.854938
$699$	0	0
$700$	1.41369	0.0534325
$701$	27.6490	1.04429	0.522144	−	0.852857i	$-0.325132\pi$
$701$	27.6490	1.04429	0.522144	+	0.852857i	$0.325132\pi$
$702$	0	0
$703$	54.5071	2.05577
$704$	0.741477	0.0279455
$705$	0	0
$706$	−18.0368	−0.678824
$707$	11.6187	0.436967
$708$	0	0
$709$	−2.19024	−0.0822563	−0.0411281	−	0.999154i	$-0.513095\pi$
$709$	−2.19024	−0.0822563	−0.0411281	+	0.999154i	$0.513095\pi$
$710$	0.673209	0.0252651
$711$	0	0
$712$	23.9442	0.897346
$713$	−43.6725	−1.63555
$714$	0	0
$715$	−0.148791	−0.00556448
$716$	2.46607	0.0921615
$717$	0	0
$718$	−22.0181	−0.821709
$719$	2.86622	0.106892	0.0534460	−	0.998571i	$-0.482980\pi$
$719$	2.86622	0.106892	0.0534460	+	0.998571i	$0.482980\pi$
$720$	0	0
$721$	5.58440	0.207974
$722$	54.1598	2.01562
$723$	0	0
$724$	−6.28849	−0.233710
$725$	−4.08391	−0.151673
$726$	0	0
$727$	20.7762	0.770548	0.385274	−	0.922802i	$-0.374107\pi$
$727$	20.7762	0.770548	0.385274	+	0.922802i	$0.374107\pi$
$728$	−6.39626	−0.237061
$729$	0	0
$730$	−23.8660	−0.883322
$731$	−51.1839	−1.89310
$732$	0	0
$733$	−23.4892	−0.867593	−0.433796	−	0.901011i	$-0.642826\pi$
$733$	−23.4892	−0.867593	−0.433796	+	0.901011i	$0.642826\pi$
$734$	−14.8724	−0.548952
$735$	0	0
$736$	15.2870	0.563487
$737$	2.09421	0.0771410
$738$	0	0
$739$	35.2534	1.29682	0.648409	−	0.761292i	$-0.275435\pi$
$739$	35.2534	1.29682	0.648409	+	0.761292i	$0.275435\pi$
$740$	−3.88445	−0.142795
$741$	0	0
$742$	40.2383	1.47719
$743$	−14.9958	−0.550143	−0.275071	−	0.961424i	$-0.588702\pi$
$743$	−14.9958	−0.550143	−0.275071	+	0.961424i	$0.588702\pi$
$744$	0	0
$745$	5.90933	0.216501
$746$	−9.73833	−0.356546
$747$	0	0
$748$	0.397213	0.0145235
$749$	−55.3244	−2.02151
$750$	0	0
$751$	27.0570	0.987324	0.493662	−	0.869654i	$-0.335658\pi$
$751$	27.0570	0.987324	0.493662	+	0.869654i	$0.335658\pi$
$752$	−4.33687	−0.158149
$753$	0	0
$754$	−6.48225	−0.236070
$755$	0.0283679	0.00103241
$756$	0	0
$757$	16.2326	0.589983	0.294992	−	0.955500i	$-0.404683\pi$
$757$	16.2326	0.589983	0.294992	+	0.955500i	$0.404683\pi$
$758$	31.4779	1.14333
$759$	0	0
$760$	17.1285	0.621315
$761$	25.6904	0.931276	0.465638	−	0.884975i	$-0.345825\pi$
$761$	25.6904	0.931276	0.465638	+	0.884975i	$0.345825\pi$
$762$	0	0
$763$	−7.35571	−0.266295
$764$	6.28404	0.227349
$765$	0	0
$766$	24.0094	0.867494
$767$	11.4112	0.412034
$768$	0	0
$769$	−43.9565	−1.58511	−0.792556	−	0.609799i	$-0.791250\pi$
$769$	−43.9565	−1.58511	−0.792556	+	0.609799i	$0.791250\pi$
$770$	−0.642791	−0.0231646
$771$	0	0
$772$	−10.1683	−0.365964
$773$	31.2043	1.12234	0.561170	−	0.827701i	$-0.310352\pi$
$773$	31.2043	1.12234	0.561170	+	0.827701i	$0.310352\pi$
$774$	0	0
$775$	8.19783	0.294475
$776$	30.9491	1.11101
$777$	0	0
$778$	22.4158	0.803646
$779$	62.6545	2.24483
$780$	0	0
$781$	−0.0631071	−0.00225815
$782$	−43.4603	−1.55414
$783$	0	0
$784$	−1.94447	−0.0694454
$785$	−17.0817	−0.609671
$786$	0	0
$787$	−36.0147	−1.28379	−0.641893	−	0.766794i	$-0.721851\pi$
$787$	−36.0147	−1.28379	−0.641893	+	0.766794i	$0.721851\pi$
$788$	−4.73829	−0.168795
$789$	0	0
$790$	8.55835	0.304492
$791$	−41.1247	−1.46223
$792$	0	0
$793$	−8.52754	−0.302822
$794$	−10.3129	−0.365992
$795$	0	0
$796$	1.42989	0.0506809
$797$	−1.57272	−0.0557087	−0.0278544	−	0.999612i	$-0.508867\pi$
$797$	−1.57272	−0.0557087	−0.0278544	+	0.999612i	$0.508867\pi$
$798$	0	0
$799$	4.67390	0.165351
$800$	−2.86955	−0.101454
$801$	0	0
$802$	35.9768	1.27038
$803$	2.23722	0.0789498
$804$	0	0
$805$	14.4994	0.511038
$806$	13.0121	0.458333
$807$	0	0
$808$	−10.0323	−0.352934
$809$	21.9914	0.773175	0.386588	−	0.922253i	$-0.373654\pi$
$809$	21.9914	0.773175	0.386588	+	0.922253i	$0.373654\pi$
$810$	0	0
$811$	−38.0124	−1.33480	−0.667398	−	0.744701i	$-0.732592\pi$
$811$	−38.0124	−1.33480	−0.667398	+	0.744701i	$0.732592\pi$
$812$	−5.77338	−0.202606
$813$	0	0
$814$	1.76622	0.0619060
$815$	−11.1693	−0.391245
$816$	0	0
$817$	−72.5828	−2.53935
$818$	3.56030	0.124483
$819$	0	0
$820$	−4.46508	−0.155927
$821$	45.2954	1.58082	0.790411	−	0.612577i	$-0.209867\pi$
$821$	45.2954	1.58082	0.790411	+	0.612577i	$0.209867\pi$
$822$	0	0
$823$	−41.5250	−1.44747	−0.723735	−	0.690078i	$-0.757576\pi$
$823$	−41.5250	−1.44747	−0.723735	+	0.690078i	$0.757576\pi$
$824$	−4.82190	−0.167979
$825$	0	0
$826$	49.2972	1.71527
$827$	−46.2851	−1.60949	−0.804746	−	0.593619i	$-0.797699\pi$
$827$	−46.2851	−1.60949	−0.804746	+	0.593619i	$0.797699\pi$
$828$	0	0
$829$	13.6273	0.473297	0.236648	−	0.971595i	$-0.423951\pi$
$829$	13.6273	0.473297	0.236648	+	0.971595i	$0.423951\pi$
$830$	6.86853	0.238410
$831$	0	0
$832$	4.98333	0.172766
$833$	2.09558	0.0726076
$834$	0	0
$835$	−0.191562	−0.00662927
$836$	0.563280	0.0194814
$837$	0	0
$838$	−3.77412	−0.130375
$839$	−29.8161	−1.02937	−0.514683	−	0.857381i	$-0.672090\pi$
$839$	−29.8161	−1.02937	−0.514683	+	0.857381i	$0.672090\pi$
$840$	0	0
$841$	−12.3217	−0.424885
$842$	27.2705	0.939803
$843$	0	0
$844$	−8.94628	−0.307944
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	−29.8786	−1.02664
$848$	−44.4199	−1.52539
$849$	0	0
$850$	8.15800	0.279817
$851$	−39.8407	−1.36572
$852$	0	0
$853$	54.1175	1.85295	0.926474	−	0.376359i	$-0.122824\pi$
$853$	54.1175	1.85295	0.926474	+	0.376359i	$0.122824\pi$
$854$	−36.8397	−1.26063
$855$	0	0
$856$	47.7703	1.63275
$857$	−34.6965	−1.18521	−0.592606	−	0.805493i	$-0.701901\pi$
$857$	−34.6965	−1.18521	−0.592606	+	0.805493i	$0.701901\pi$
$858$	0	0
$859$	−30.2055	−1.03060	−0.515300	−	0.857010i	$-0.672319\pi$
$859$	−30.2055	−1.03060	−0.515300	+	0.857010i	$0.672319\pi$
$860$	5.17262	0.176385
$861$	0	0
$862$	12.5678	0.428061
$863$	3.57660	0.121749	0.0608745	−	0.998145i	$-0.480611\pi$
$863$	3.57660	0.121749	0.0608745	+	0.998145i	$0.480611\pi$
$864$	0	0
$865$	−19.8367	−0.674466
$866$	−28.3879	−0.964661
$867$	0	0
$868$	11.5892	0.393363
$869$	−0.802266	−0.0272150
$870$	0	0
$871$	14.0748	0.476906
$872$	6.35135	0.215084
$873$	0	0
$874$	−61.6302	−2.08467
$875$	−2.72171	−0.0920107
$876$	0	0
$877$	16.7090	0.564223	0.282111	−	0.959382i	$-0.408965\pi$
$877$	16.7090	0.564223	0.282111	+	0.959382i	$0.408965\pi$
$878$	31.3944	1.05951
$879$	0	0
$880$	0.709591	0.0239203
$881$	29.0005	0.977050	0.488525	−	0.872550i	$-0.337535\pi$
$881$	29.0005	0.977050	0.488525	+	0.872550i	$0.337535\pi$
$882$	0	0
$883$	−53.3398	−1.79503	−0.897513	−	0.440988i	$-0.854628\pi$
$883$	−53.3398	−1.79503	−0.897513	+	0.440988i	$0.854628\pi$
$884$	2.66960	0.0897882
$885$	0	0
$886$	2.00845	0.0674751
$887$	−0.501362	−0.0168341	−0.00841705	−	0.999965i	$-0.502679\pi$
$887$	−0.501362	−0.0168341	−0.00841705	+	0.999965i	$0.502679\pi$
$888$	0	0
$889$	24.8193	0.832412
$890$	16.1721	0.542089
$891$	0	0
$892$	−10.6285	−0.355868
$893$	6.62797	0.221797
$894$	0	0
$895$	−4.74782	−0.158702
$896$	37.1486	1.24105
$897$	0	0
$898$	53.5071	1.78555
$899$	−33.4792	−1.11659
$900$	0	0
$901$	47.8720	1.59485
$902$	2.03023	0.0675992
$903$	0	0
$904$	35.5095	1.18103
$905$	12.1070	0.402449
$906$	0	0
$907$	−41.0257	−1.36223	−0.681117	−	0.732174i	$-0.738506\pi$
$907$	−41.0257	−1.36223	−0.681117	+	0.732174i	$0.738506\pi$
$908$	−2.81417	−0.0933914
$909$	0	0
$910$	−4.32008	−0.143209
$911$	−10.0752	−0.333807	−0.166904	−	0.985973i	$-0.553377\pi$
$911$	−10.0752	−0.333807	−0.166904	+	0.985973i	$0.553377\pi$
$912$	0	0
$913$	−0.643860	−0.0213087
$914$	−10.2195	−0.338032
$915$	0	0
$916$	13.1405	0.434175
$917$	−27.3338	−0.902640
$918$	0	0
$919$	−2.68571	−0.0885933	−0.0442966	−	0.999018i	$-0.514105\pi$
$919$	−2.68571	−0.0885933	−0.0442966	+	0.999018i	$0.514105\pi$
$920$	−12.5197	−0.412761
$921$	0	0
$922$	32.1510	1.05884
$923$	−0.424131	−0.0139605
$924$	0	0
$925$	7.47856	0.245893
$926$	12.4662	0.409665
$927$	0	0
$928$	11.7190	0.384695
$929$	42.6254	1.39850	0.699248	−	0.714880i	$-0.253519\pi$
$929$	42.6254	1.39850	0.699248	+	0.714880i	$0.253519\pi$
$930$	0	0
$931$	2.97171	0.0973937
$932$	−0.347963	−0.0113979
$933$	0	0
$934$	−5.48731	−0.179550
$935$	−0.764736	−0.0250096
$936$	0	0
$937$	−6.27262	−0.204918	−0.102459	−	0.994737i	$-0.532671\pi$
$937$	−6.27262	−0.204918	−0.102459	+	0.994737i	$0.532671\pi$
$938$	60.8042	1.98533
$939$	0	0
$940$	−0.472343	−0.0154061
$941$	1.80587	0.0588697	0.0294348	−	0.999567i	$-0.490629\pi$
$941$	1.80587	0.0588697	0.0294348	+	0.999567i	$0.490629\pi$
$942$	0	0
$943$	−45.7959	−1.49132
$944$	−54.4203	−1.77123
$945$	0	0
$946$	−2.35194	−0.0764682
$947$	15.3035	0.497296	0.248648	−	0.968594i	$-0.420014\pi$
$947$	15.3035	0.497296	0.248648	+	0.968594i	$0.420014\pi$
$948$	0	0
$949$	15.0360	0.488088
$950$	11.5687	0.375338
$951$	0	0
$952$	−32.8746	−1.06547
$953$	8.96284	0.290335	0.145167	−	0.989407i	$-0.453628\pi$
$953$	8.96284	0.290335	0.145167	+	0.989407i	$0.453628\pi$
$954$	0	0
$955$	−12.0984	−0.391495
$956$	0.356232	0.0115214
$957$	0	0
$958$	−6.56456	−0.212091
$959$	−48.2550	−1.55823
$960$	0	0
$961$	36.2044	1.16788
$962$	11.8705	0.382719
$963$	0	0
$964$	2.01399	0.0648663
$965$	19.5765	0.630191
$966$	0	0
$967$	−31.6792	−1.01874	−0.509368	−	0.860549i	$-0.670121\pi$
$967$	−31.6792	−1.01874	−0.509368	+	0.860549i	$0.670121\pi$
$968$	25.7989	0.829209
$969$	0	0
$970$	20.9032	0.671163
$971$	−17.2002	−0.551979	−0.275990	−	0.961161i	$-0.589006\pi$
$971$	−17.2002	−0.551979	−0.275990	+	0.961161i	$0.589006\pi$
$972$	0	0
$973$	19.4387	0.623176
$974$	−20.2420	−0.648597
$975$	0	0
$976$	40.6681	1.30176
$977$	−25.6536	−0.820732	−0.410366	−	0.911921i	$-0.634599\pi$
$977$	−25.6536	−0.820732	−0.410366	+	0.911921i	$0.634599\pi$
$978$	0	0
$979$	−1.51598	−0.0484510
$980$	−0.211779	−0.00676503
$981$	0	0
$982$	68.2447	2.17778
$983$	1.92999	0.0615571	0.0307786	−	0.999526i	$-0.490201\pi$
$983$	1.92999	0.0615571	0.0307786	+	0.999526i	$0.490201\pi$
$984$	0	0
$985$	9.12242	0.290665
$986$	−33.3165	−1.06101
$987$	0	0
$988$	3.78570	0.120439
$989$	53.0527	1.68698
$990$	0	0
$991$	−44.6495	−1.41834	−0.709169	−	0.705039i	$-0.750930\pi$
$991$	−44.6495	−1.41834	−0.709169	+	0.705039i	$0.750930\pi$
$992$	−23.5241	−0.746890
$993$	0	0
$994$	−1.83228	−0.0581165
$995$	−2.75289	−0.0872726
$996$	0	0
$997$	25.2211	0.798760	0.399380	−	0.916785i	$-0.369225\pi$
$997$	25.2211	0.798760	0.399380	+	0.916785i	$0.369225\pi$
$998$	−44.3668	−1.40441
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1755.2.a.u.1.6	✓	7
3.2	odd	2		1755.2.a.v.1.2	yes	7
5.4	even	2		8775.2.a.bx.1.2		7
15.14	odd	2		8775.2.a.bw.1.6		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1755.2.a.u.1.6	✓	7	1.1	even	1	trivial
1755.2.a.v.1.2	yes	7	3.2	odd	2
8775.2.a.bw.1.6		7	15.14	odd	2
8775.2.a.bx.1.2		7	5.4	even	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:	\( N \)	\(=\)	\( 1755 = 3^{3} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1755.a (trivial)

\(f(q)\)	\(=\)	\(q+1.58727 q^{2} +0.519412 q^{4} -1.00000 q^{5} +2.72171 q^{7} -2.35009 q^{8} +O(q^{10})\)	\(q+1.58727 q^{2} +0.519412 q^{4} -1.00000 q^{5} +2.72171 q^{7} -2.35009 q^{8} -1.58727 q^{10} +0.148791 q^{11} +1.00000 q^{13} +4.32008 q^{14} -4.76903 q^{16} +5.13965 q^{17} +7.28845 q^{19} -0.519412 q^{20} +0.236171 q^{22} -5.32732 q^{23} +1.00000 q^{25} +1.58727 q^{26} +1.41369 q^{28} -4.08391 q^{29} +8.19783 q^{31} -2.86955 q^{32} +8.15800 q^{34} -2.72171 q^{35} +7.47856 q^{37} +11.5687 q^{38} +2.35009 q^{40} +8.59642 q^{41} -9.95862 q^{43} +0.0772840 q^{44} -8.45587 q^{46} +0.909381 q^{47} +0.407728 q^{49} +1.58727 q^{50} +0.519412 q^{52} +9.31424 q^{53} -0.148791 q^{55} -6.39626 q^{56} -6.48225 q^{58} +11.4112 q^{59} -8.52754 q^{61} +13.0121 q^{62} +4.98333 q^{64} -1.00000 q^{65} +14.0748 q^{67} +2.66960 q^{68} -4.32008 q^{70} -0.424131 q^{71} +15.0360 q^{73} +11.8705 q^{74} +3.78570 q^{76} +0.404968 q^{77} -5.39188 q^{79} +4.76903 q^{80} +13.6448 q^{82} -4.32727 q^{83} -5.13965 q^{85} -15.8070 q^{86} -0.349673 q^{88} -10.1886 q^{89} +2.72171 q^{91} -2.76707 q^{92} +1.44343 q^{94} -7.28845 q^{95} -13.1693 q^{97} +0.647173 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - q^{2} + 11 q^{4} - 7 q^{5} + 2 q^{7} - 6 q^{8}+O(q^{10}) \) 7 * q - q^2 + 11 * q^4 - 7 * q^5 + 2 * q^7 - 6 * q^8	\( 7 q - q^{2} + 11 q^{4} - 7 q^{5} + 2 q^{7} - 6 q^{8} + q^{10} - q^{11} + 7 q^{13} + 12 q^{14} + 23 q^{16} - 11 q^{17} + 2 q^{19} - 11 q^{20} + 16 q^{22} + q^{23} + 7 q^{25} - q^{26} + 10 q^{28} + 4 q^{29} - 13 q^{32} + q^{34} - 2 q^{35} + 23 q^{37} + 15 q^{38} + 6 q^{40} - 2 q^{41} + 8 q^{43} - 10 q^{44} + 37 q^{46} - 2 q^{47} + 43 q^{49} - q^{50} + 11 q^{52} - 10 q^{53} + q^{55} + 68 q^{56} - 26 q^{58} + 13 q^{59} + 21 q^{61} - 9 q^{62} + 46 q^{64} - 7 q^{65} + 21 q^{67} - 53 q^{68} - 12 q^{70} + 10 q^{71} + 13 q^{73} + 68 q^{74} - 41 q^{76} - 6 q^{77} + 8 q^{79} - 23 q^{80} - 26 q^{82} + 4 q^{83} + 11 q^{85} + 12 q^{86} + 44 q^{88} + 27 q^{89} + 2 q^{91} + 9 q^{92} - 24 q^{94} - 2 q^{95} - 15 q^{97} - 7 q^{98}+O(q^{100}) \) 7 * q - q^2 + 11 * q^4 - 7 * q^5 + 2 * q^7 - 6 * q^8 + q^10 - q^11 + 7 * q^13 + 12 * q^14 + 23 * q^16 - 11 * q^17 + 2 * q^19 - 11 * q^20 + 16 * q^22 + q^23 + 7 * q^25 - q^26 + 10 * q^28 + 4 * q^29 - 13 * q^32 + q^34 - 2 * q^35 + 23 * q^37 + 15 * q^38 + 6 * q^40 - 2 * q^41 + 8 * q^43 - 10 * q^44 + 37 * q^46 - 2 * q^47 + 43 * q^49 - q^50 + 11 * q^52 - 10 * q^53 + q^55 + 68 * q^56 - 26 * q^58 + 13 * q^59 + 21 * q^61 - 9 * q^62 + 46 * q^64 - 7 * q^65 + 21 * q^67 - 53 * q^68 - 12 * q^70 + 10 * q^71 + 13 * q^73 + 68 * q^74 - 41 * q^76 - 6 * q^77 + 8 * q^79 - 23 * q^80 - 26 * q^82 + 4 * q^83 + 11 * q^85 + 12 * q^86 + 44 * q^88 + 27 * q^89 + 2 * q^91 + 9 * q^92 - 24 * q^94 - 2 * q^95 - 15 * q^97 - 7 * q^98

Introduction

L-functions

Modular forms

Varieties

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