LMFDB - Embedded newform 8304.2.a.bf.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8304,2,Mod(1,8304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8304, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8304.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8304 = 2^{4} \cdot 3 \cdot 173 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8304.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:	$66.3077738385$
Analytic rank:	$1$
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} - 13x^{5} - 8x^{4} + 24x^{3} + 6x^{2} - 13x + 2 $ x^7 - 13x^5 - 8x^4 + 24x^3 + 6x^2 - 13*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2076)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.927549$ of defining polynomial
Character	$\chi$	$=$	8304.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -2.48191 q^{5} +1.43522 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -2.48191 q^{5} +1.43522 q^{7} +1.00000 q^{9} -0.681342 q^{11} -1.52521 q^{13} +2.48191 q^{15} -0.279003 q^{17} +1.75111 q^{19} -1.43522 q^{21} -1.39111 q^{23} +1.15987 q^{25} -1.00000 q^{27} +9.33280 q^{29} -10.3435 q^{31} +0.681342 q^{33} -3.56210 q^{35} +3.71073 q^{37} +1.52521 q^{39} +6.04248 q^{41} +0.878694 q^{43} -2.48191 q^{45} +0.0496203 q^{47} -4.94013 q^{49} +0.279003 q^{51} +1.77394 q^{53} +1.69103 q^{55} -1.75111 q^{57} -8.52253 q^{59} +9.90982 q^{61} +1.43522 q^{63} +3.78543 q^{65} +11.0284 q^{67} +1.39111 q^{69} +2.64124 q^{71} -8.14285 q^{73} -1.15987 q^{75} -0.977879 q^{77} -12.7570 q^{79} +1.00000 q^{81} -1.14936 q^{83} +0.692460 q^{85} -9.33280 q^{87} +11.9063 q^{89} -2.18902 q^{91} +10.3435 q^{93} -4.34610 q^{95} -7.87285 q^{97} -0.681342 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 7 q^{3} + 6 q^{5} - q^{7} + 7 q^{9}+O(q^{10}) $ 7 * q - 7 * q^3 + 6 * q^5 - q^7 + 7 * q^9	$ 7 q - 7 q^{3} + 6 q^{5} - q^{7} + 7 q^{9} - 4 q^{11} + 4 q^{13} - 6 q^{15} + 7 q^{17} - 9 q^{19} + q^{21} - 17 q^{23} + 5 q^{25} - 7 q^{27} + 6 q^{29} - 10 q^{31} + 4 q^{33} - 9 q^{35} + 6 q^{37} - 4 q^{39} + 7 q^{41} - 11 q^{43} + 6 q^{45} - 22 q^{47} - 8 q^{49} - 7 q^{51} + 20 q^{53} - 21 q^{55} + 9 q^{57} - 18 q^{59} - q^{63} - 2 q^{65} - 8 q^{67} + 17 q^{69} - 14 q^{71} - 14 q^{73} - 5 q^{75} + 13 q^{77} - q^{79} + 7 q^{81} - 7 q^{83} - 16 q^{85} - 6 q^{87} - 6 q^{89} - 10 q^{91} + 10 q^{93} - 31 q^{95} - 17 q^{97} - 4 q^{99}+O(q^{100}) $ 7 * q - 7 * q^3 + 6 * q^5 - q^7 + 7 * q^9 - 4 * q^11 + 4 * q^13 - 6 * q^15 + 7 * q^17 - 9 * q^19 + q^21 - 17 * q^23 + 5 * q^25 - 7 * q^27 + 6 * q^29 - 10 * q^31 + 4 * q^33 - 9 * q^35 + 6 * q^37 - 4 * q^39 + 7 * q^41 - 11 * q^43 + 6 * q^45 - 22 * q^47 - 8 * q^49 - 7 * q^51 + 20 * q^53 - 21 * q^55 + 9 * q^57 - 18 * q^59 - q^63 - 2 * q^65 - 8 * q^67 + 17 * q^69 - 14 * q^71 - 14 * q^73 - 5 * q^75 + 13 * q^77 - q^79 + 7 * q^81 - 7 * q^83 - 16 * q^85 - 6 * q^87 - 6 * q^89 - 10 * q^91 + 10 * q^93 - 31 * q^95 - 17 * q^97 - 4 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−2.48191	−1.10994	−0.554972	−	0.831869i	$-0.687271\pi$
$5$	−2.48191	−1.10994	−0.554972	+	0.831869i	$0.687271\pi$
$6$	0	0
$7$	1.43522	0.542464	0.271232	−	0.962514i	$-0.412569\pi$
$7$	1.43522	0.542464	0.271232	+	0.962514i	$0.412569\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.681342	−0.205432	−0.102716	−	0.994711i	$-0.532753\pi$
$11$	−0.681342	−0.205432	−0.102716	+	0.994711i	$0.532753\pi$
$12$	0	0
$13$	−1.52521	−0.423017	−0.211509	−	0.977376i	$-0.567838\pi$
$13$	−1.52521	−0.423017	−0.211509	+	0.977376i	$0.567838\pi$
$14$	0	0
$15$	2.48191	0.640826
$16$	0	0
$17$	−0.279003	−0.0676682	−0.0338341	−	0.999427i	$-0.510772\pi$
$17$	−0.279003	−0.0676682	−0.0338341	+	0.999427i	$0.510772\pi$
$18$	0	0
$19$	1.75111	0.401733	0.200866	−	0.979619i	$-0.435624\pi$
$19$	1.75111	0.401733	0.200866	+	0.979619i	$0.435624\pi$
$20$	0	0
$21$	−1.43522	−0.313192
$22$	0	0
$23$	−1.39111	−0.290066	−0.145033	−	0.989427i	$-0.546329\pi$
$23$	−1.39111	−0.290066	−0.145033	+	0.989427i	$0.546329\pi$
$24$	0	0
$25$	1.15987	0.231974
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	9.33280	1.73306	0.866529	−	0.499127i	$-0.166346\pi$
$29$	9.33280	1.73306	0.866529	+	0.499127i	$0.166346\pi$
$30$	0	0
$31$	−10.3435	−1.85775	−0.928876	−	0.370391i	$-0.879224\pi$
$31$	−10.3435	−1.85775	−0.928876	+	0.370391i	$0.879224\pi$
$32$	0	0
$33$	0.681342	0.118606
$34$	0	0
$35$	−3.56210	−0.602104
$36$	0	0
$37$	3.71073	0.610041	0.305020	−	0.952346i	$-0.401337\pi$
$37$	3.71073	0.610041	0.305020	+	0.952346i	$0.401337\pi$
$38$	0	0
$39$	1.52521	0.244229
$40$	0	0
$41$	6.04248	0.943677	0.471839	−	0.881685i	$-0.343590\pi$
$41$	6.04248	0.943677	0.471839	+	0.881685i	$0.343590\pi$
$42$	0	0
$43$	0.878694	0.134000	0.0669998	−	0.997753i	$-0.478657\pi$
$43$	0.878694	0.134000	0.0669998	+	0.997753i	$0.478657\pi$
$44$	0	0
$45$	−2.48191	−0.369981
$46$	0	0
$47$	0.0496203	0.00723787	0.00361893	−	0.999993i	$-0.498848\pi$
$47$	0.0496203	0.00723787	0.00361893	+	0.999993i	$0.498848\pi$
$48$	0	0
$49$	−4.94013	−0.705733
$50$	0	0
$51$	0.279003	0.0390682
$52$	0	0
$53$	1.77394	0.243670	0.121835	−	0.992550i	$-0.461122\pi$
$53$	1.77394	0.243670	0.121835	+	0.992550i	$0.461122\pi$
$54$	0	0
$55$	1.69103	0.228018
$56$	0	0
$57$	−1.75111	−0.231941
$58$	0	0
$59$	−8.52253	−1.10954	−0.554769	−	0.832004i	$-0.687194\pi$
$59$	−8.52253	−1.10954	−0.554769	+	0.832004i	$0.687194\pi$
$60$	0	0
$61$	9.90982	1.26882	0.634411	−	0.772996i	$-0.281243\pi$
$61$	9.90982	1.26882	0.634411	+	0.772996i	$0.281243\pi$
$62$	0	0
$63$	1.43522	0.180821
$64$	0	0
$65$	3.78543	0.469525
$66$	0	0
$67$	11.0284	1.34733	0.673667	−	0.739035i	$-0.264718\pi$
$67$	11.0284	1.34733	0.673667	+	0.739035i	$0.264718\pi$
$68$	0	0
$69$	1.39111	0.167470
$70$	0	0
$71$	2.64124	0.313457	0.156729	−	0.987642i	$-0.449905\pi$
$71$	2.64124	0.313457	0.156729	+	0.987642i	$0.449905\pi$
$72$	0	0
$73$	−8.14285	−0.953048	−0.476524	−	0.879161i	$-0.658104\pi$
$73$	−8.14285	−0.953048	−0.476524	+	0.879161i	$0.658104\pi$
$74$	0	0
$75$	−1.15987	−0.133930
$76$	0	0
$77$	−0.977879	−0.111440
$78$	0	0
$79$	−12.7570	−1.43527	−0.717636	−	0.696418i	$-0.754776\pi$
$79$	−12.7570	−1.43527	−0.717636	+	0.696418i	$0.754776\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−1.14936	−0.126159	−0.0630793	−	0.998009i	$-0.520092\pi$
$83$	−1.14936	−0.126159	−0.0630793	+	0.998009i	$0.520092\pi$
$84$	0	0
$85$	0.692460	0.0751078
$86$	0	0
$87$	−9.33280	−1.00058
$88$	0	0
$89$	11.9063	1.26207	0.631035	−	0.775754i	$-0.282630\pi$
$89$	11.9063	1.26207	0.631035	+	0.775754i	$0.282630\pi$
$90$	0	0
$91$	−2.18902	−0.229471
$92$	0	0
$93$	10.3435	1.07257
$94$	0	0
$95$	−4.34610	−0.445901
$96$	0	0
$97$	−7.87285	−0.799366	−0.399683	−	0.916653i	$-0.630880\pi$
$97$	−7.87285	−0.799366	−0.399683	+	0.916653i	$0.630880\pi$
$98$	0	0
$99$	−0.681342	−0.0684775
$100$	0	0
$101$	3.38676	0.336995	0.168497	−	0.985702i	$-0.446109\pi$
$101$	3.38676	0.336995	0.168497	+	0.985702i	$0.446109\pi$
$102$	0	0
$103$	−5.47934	−0.539895	−0.269948	−	0.962875i	$-0.587006\pi$
$103$	−5.47934	−0.539895	−0.269948	+	0.962875i	$0.587006\pi$
$104$	0	0
$105$	3.56210	0.347625
$106$	0	0
$107$	−12.1586	−1.17542	−0.587708	−	0.809073i	$-0.699970\pi$
$107$	−12.1586	−1.17542	−0.587708	+	0.809073i	$0.699970\pi$
$108$	0	0
$109$	7.12259	0.682220	0.341110	−	0.940023i	$-0.389197\pi$
$109$	7.12259	0.682220	0.341110	+	0.940023i	$0.389197\pi$
$110$	0	0
$111$	−3.71073	−0.352207
$112$	0	0
$113$	13.1009	1.23243	0.616213	−	0.787579i	$-0.288666\pi$
$113$	13.1009	1.23243	0.616213	+	0.787579i	$0.288666\pi$
$114$	0	0
$115$	3.45261	0.321957
$116$	0	0
$117$	−1.52521	−0.141006
$118$	0	0
$119$	−0.400432	−0.0367075
$120$	0	0
$121$	−10.5358	−0.957798
$122$	0	0
$123$	−6.04248	−0.544832
$124$	0	0
$125$	9.53086	0.852466
$126$	0	0
$127$	17.1402	1.52095	0.760474	−	0.649368i	$-0.224967\pi$
$127$	17.1402	1.52095	0.760474	+	0.649368i	$0.224967\pi$
$128$	0	0
$129$	−0.878694	−0.0773647
$130$	0	0
$131$	10.4105	0.909572	0.454786	−	0.890601i	$-0.349716\pi$
$131$	10.4105	0.909572	0.454786	+	0.890601i	$0.349716\pi$
$132$	0	0
$133$	2.51324	0.217926
$134$	0	0
$135$	2.48191	0.213609
$136$	0	0
$137$	−0.156779	−0.0133945	−0.00669726	−	0.999978i	$-0.502132\pi$
$137$	−0.156779	−0.0133945	−0.00669726	+	0.999978i	$0.502132\pi$
$138$	0	0
$139$	−7.73518	−0.656090	−0.328045	−	0.944662i	$-0.606390\pi$
$139$	−7.73518	−0.656090	−0.328045	+	0.944662i	$0.606390\pi$
$140$	0	0
$141$	−0.0496203	−0.00417878
$142$	0	0
$143$	1.03919	0.0869014
$144$	0	0
$145$	−23.1632	−1.92360
$146$	0	0
$147$	4.94013	0.407455
$148$	0	0
$149$	−3.45091	−0.282710	−0.141355	−	0.989959i	$-0.545146\pi$
$149$	−3.45091	−0.282710	−0.141355	+	0.989959i	$0.545146\pi$
$150$	0	0
$151$	−4.17687	−0.339909	−0.169954	−	0.985452i	$-0.554362\pi$
$151$	−4.17687	−0.339909	−0.169954	+	0.985452i	$0.554362\pi$
$152$	0	0
$153$	−0.279003	−0.0225561
$154$	0	0
$155$	25.6717	2.06200
$156$	0	0
$157$	7.82800	0.624743	0.312371	−	0.949960i	$-0.398877\pi$
$157$	7.82800	0.624743	0.312371	+	0.949960i	$0.398877\pi$
$158$	0	0
$159$	−1.77394	−0.140683
$160$	0	0
$161$	−1.99655	−0.157351
$162$	0	0
$163$	−6.70964	−0.525539	−0.262770	−	0.964859i	$-0.584636\pi$
$163$	−6.70964	−0.525539	−0.262770	+	0.964859i	$0.584636\pi$
$164$	0	0
$165$	−1.69103	−0.131646
$166$	0	0
$167$	−15.3699	−1.18936	−0.594678	−	0.803964i	$-0.702721\pi$
$167$	−15.3699	−1.18936	−0.594678	+	0.803964i	$0.702721\pi$
$168$	0	0
$169$	−10.6737	−0.821057
$170$	0	0
$171$	1.75111	0.133911
$172$	0	0
$173$	1.00000	0.0760286
$173$	1.00000	0.0760286
$174$	0	0
$175$	1.66467	0.125837
$176$	0	0
$177$	8.52253	0.640593
$178$	0	0
$179$	8.15708	0.609689	0.304844	−	0.952402i	$-0.401396\pi$
$179$	8.15708	0.609689	0.304844	+	0.952402i	$0.401396\pi$
$180$	0	0
$181$	0.732315	0.0544326	0.0272163	−	0.999630i	$-0.491336\pi$
$181$	0.732315	0.0544326	0.0272163	+	0.999630i	$0.491336\pi$
$182$	0	0
$183$	−9.90982	−0.732555
$184$	0	0
$185$	−9.20970	−0.677111
$186$	0	0
$187$	0.190096	0.0139012
$188$	0	0
$189$	−1.43522	−0.104397
$190$	0	0
$191$	−6.71804	−0.486100	−0.243050	−	0.970014i	$-0.578148\pi$
$191$	−6.71804	−0.486100	−0.243050	+	0.970014i	$0.578148\pi$
$192$	0	0
$193$	−5.77120	−0.415420	−0.207710	−	0.978190i	$-0.566601\pi$
$193$	−5.77120	−0.415420	−0.207710	+	0.978190i	$0.566601\pi$
$194$	0	0
$195$	−3.78543	−0.271080
$196$	0	0
$197$	17.9065	1.27579	0.637893	−	0.770125i	$-0.279806\pi$
$197$	17.9065	1.27579	0.637893	+	0.770125i	$0.279806\pi$
$198$	0	0
$199$	14.3534	1.01748	0.508742	−	0.860919i	$-0.330111\pi$
$199$	14.3534	1.01748	0.508742	+	0.860919i	$0.330111\pi$
$200$	0	0
$201$	−11.0284	−0.777883
$202$	0	0
$203$	13.3947	0.940121
$204$	0	0
$205$	−14.9969	−1.04743
$206$	0	0
$207$	−1.39111	−0.0966888
$208$	0	0
$209$	−1.19311	−0.0825289
$210$	0	0
$211$	−15.6290	−1.07594	−0.537971	−	0.842963i	$-0.680809\pi$
$211$	−15.6290	−1.07594	−0.537971	+	0.842963i	$0.680809\pi$
$212$	0	0
$213$	−2.64124	−0.180975
$214$	0	0
$215$	−2.18084	−0.148732
$216$	0	0
$217$	−14.8453	−1.00776
$218$	0	0
$219$	8.14285	0.550243
$220$	0	0
$221$	0.425538	0.0286248
$222$	0	0
$223$	−11.7815	−0.788950	−0.394475	−	0.918907i	$-0.629073\pi$
$223$	−11.7815	−0.788950	−0.394475	+	0.918907i	$0.629073\pi$
$224$	0	0
$225$	1.15987	0.0773245
$226$	0	0
$227$	13.8804	0.921273	0.460637	−	0.887589i	$-0.347621\pi$
$227$	13.8804	0.921273	0.460637	+	0.887589i	$0.347621\pi$
$228$	0	0
$229$	−28.1884	−1.86274	−0.931370	−	0.364074i	$-0.881385\pi$
$229$	−28.1884	−1.86274	−0.931370	+	0.364074i	$0.881385\pi$
$230$	0	0
$231$	0.977879	0.0643397
$232$	0	0
$233$	−17.2620	−1.13087	−0.565435	−	0.824793i	$-0.691292\pi$
$233$	−17.2620	−1.13087	−0.565435	+	0.824793i	$0.691292\pi$
$234$	0	0
$235$	−0.123153	−0.00803362
$236$	0	0
$237$	12.7570	0.828655
$238$	0	0
$239$	−7.19829	−0.465619	−0.232809	−	0.972522i	$-0.574792\pi$
$239$	−7.19829	−0.465619	−0.232809	+	0.972522i	$0.574792\pi$
$240$	0	0
$241$	6.18787	0.398596	0.199298	−	0.979939i	$-0.436134\pi$
$241$	6.18787	0.398596	0.199298	+	0.979939i	$0.436134\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	12.2610	0.783323
$246$	0	0
$247$	−2.67081	−0.169940
$248$	0	0
$249$	1.14936	0.0728377
$250$	0	0
$251$	−11.4781	−0.724490	−0.362245	−	0.932083i	$-0.617990\pi$
$251$	−11.4781	−0.724490	−0.362245	+	0.932083i	$0.617990\pi$
$252$	0	0
$253$	0.947821	0.0595890
$254$	0	0
$255$	−0.692460	−0.0433635
$256$	0	0
$257$	−7.18265	−0.448041	−0.224021	−	0.974584i	$-0.571918\pi$
$257$	−7.18265	−0.448041	−0.224021	+	0.974584i	$0.571918\pi$
$258$	0	0
$259$	5.32574	0.330925
$260$	0	0
$261$	9.33280	0.577686
$262$	0	0
$263$	25.6948	1.58441	0.792204	−	0.610257i	$-0.208934\pi$
$263$	25.6948	1.58441	0.792204	+	0.610257i	$0.208934\pi$
$264$	0	0
$265$	−4.40277	−0.270460
$266$	0	0
$267$	−11.9063	−0.728656
$268$	0	0
$269$	17.3140	1.05565	0.527826	−	0.849353i	$-0.323007\pi$
$269$	17.3140	1.05565	0.527826	+	0.849353i	$0.323007\pi$
$270$	0	0
$271$	−5.95052	−0.361469	−0.180734	−	0.983532i	$-0.557847\pi$
$271$	−5.95052	−0.361469	−0.180734	+	0.983532i	$0.557847\pi$
$272$	0	0
$273$	2.18902	0.132485
$274$	0	0
$275$	−0.790267	−0.0476549
$276$	0	0
$277$	−13.1478	−0.789978	−0.394989	−	0.918686i	$-0.629252\pi$
$277$	−13.1478	−0.789978	−0.394989	+	0.918686i	$0.629252\pi$
$278$	0	0
$279$	−10.3435	−0.619251
$280$	0	0
$281$	−1.73336	−0.103404	−0.0517019	−	0.998663i	$-0.516465\pi$
$281$	−1.73336	−0.103404	−0.0517019	+	0.998663i	$0.516465\pi$
$282$	0	0
$283$	−19.6291	−1.16683	−0.583413	−	0.812176i	$-0.698283\pi$
$283$	−19.6291	−1.16683	−0.583413	+	0.812176i	$0.698283\pi$
$284$	0	0
$285$	4.34610	0.257441
$286$	0	0
$287$	8.67232	0.511911
$288$	0	0
$289$	−16.9222	−0.995421
$290$	0	0
$291$	7.87285	0.461514
$292$	0	0
$293$	−1.36314	−0.0796355	−0.0398178	−	0.999207i	$-0.512678\pi$
$293$	−1.36314	−0.0796355	−0.0398178	+	0.999207i	$0.512678\pi$
$294$	0	0
$295$	21.1521	1.23152
$296$	0	0
$297$	0.681342	0.0395355
$298$	0	0
$299$	2.12173	0.122703
$300$	0	0
$301$	1.26112	0.0726899
$302$	0	0
$303$	−3.38676	−0.194564
$304$	0	0
$305$	−24.5953	−1.40832
$306$	0	0
$307$	20.4507	1.16718	0.583590	−	0.812048i	$-0.301647\pi$
$307$	20.4507	1.16718	0.583590	+	0.812048i	$0.301647\pi$
$308$	0	0
$309$	5.47934	0.311709
$310$	0	0
$311$	−31.9816	−1.81351	−0.906756	−	0.421655i	$-0.861449\pi$
$311$	−31.9816	−1.81351	−0.906756	+	0.421655i	$0.861449\pi$
$312$	0	0
$313$	−5.37302	−0.303701	−0.151851	−	0.988403i	$-0.548523\pi$
$313$	−5.37302	−0.303701	−0.151851	+	0.988403i	$0.548523\pi$
$314$	0	0
$315$	−3.56210	−0.200701
$316$	0	0
$317$	−21.3702	−1.20027	−0.600135	−	0.799898i	$-0.704887\pi$
$317$	−21.3702	−1.20027	−0.600135	+	0.799898i	$0.704887\pi$
$318$	0	0
$319$	−6.35883	−0.356026
$320$	0	0
$321$	12.1586	0.678627
$322$	0	0
$323$	−0.488566	−0.0271845
$324$	0	0
$325$	−1.76904	−0.0981288
$326$	0	0
$327$	−7.12259	−0.393880
$328$	0	0
$329$	0.0712163	0.00392628
$330$	0	0
$331$	−8.40384	−0.461917	−0.230958	−	0.972964i	$-0.574186\pi$
$331$	−8.40384	−0.461917	−0.230958	+	0.972964i	$0.574186\pi$
$332$	0	0
$333$	3.71073	0.203347
$334$	0	0
$335$	−27.3715	−1.49546
$336$	0	0
$337$	25.5299	1.39070	0.695351	−	0.718670i	$-0.255249\pi$
$337$	25.5299	1.39070	0.695351	+	0.718670i	$0.255249\pi$
$338$	0	0
$339$	−13.1009	−0.711542
$340$	0	0
$341$	7.04748	0.381642
$342$	0	0
$343$	−17.1368	−0.925298
$344$	0	0
$345$	−3.45261	−0.185882
$346$	0	0
$347$	3.09610	0.166208	0.0831038	−	0.996541i	$-0.473517\pi$
$347$	3.09610	0.166208	0.0831038	+	0.996541i	$0.473517\pi$
$348$	0	0
$349$	−15.6155	−0.835878	−0.417939	−	0.908475i	$-0.637247\pi$
$349$	−15.6155	−0.835878	−0.417939	+	0.908475i	$0.637247\pi$
$350$	0	0
$351$	1.52521	0.0814097
$352$	0	0
$353$	−6.64415	−0.353632	−0.176816	−	0.984244i	$-0.556580\pi$
$353$	−6.64415	−0.353632	−0.176816	+	0.984244i	$0.556580\pi$
$354$	0	0
$355$	−6.55531	−0.347920
$356$	0	0
$357$	0.400432	0.0211931
$358$	0	0
$359$	6.75590	0.356563	0.178281	−	0.983980i	$-0.442946\pi$
$359$	6.75590	0.356563	0.178281	+	0.983980i	$0.442946\pi$
$360$	0	0
$361$	−15.9336	−0.838611
$362$	0	0
$363$	10.5358	0.552985
$364$	0	0
$365$	20.2098	1.05783
$366$	0	0
$367$	−22.1165	−1.15447	−0.577236	−	0.816577i	$-0.695869\pi$
$367$	−22.1165	−1.15447	−0.577236	+	0.816577i	$0.695869\pi$
$368$	0	0
$369$	6.04248	0.314559
$370$	0	0
$371$	2.54601	0.132182
$372$	0	0
$373$	−5.74185	−0.297302	−0.148651	−	0.988890i	$-0.547493\pi$
$373$	−5.74185	−0.297302	−0.148651	+	0.988890i	$0.547493\pi$
$374$	0	0
$375$	−9.53086	−0.492171
$376$	0	0
$377$	−14.2345	−0.733113
$378$	0	0
$379$	4.14059	0.212688	0.106344	−	0.994329i	$-0.466086\pi$
$379$	4.14059	0.212688	0.106344	+	0.994329i	$0.466086\pi$
$380$	0	0
$381$	−17.1402	−0.878120
$382$	0	0
$383$	−19.0688	−0.974368	−0.487184	−	0.873299i	$-0.661976\pi$
$383$	−19.0688	−0.974368	−0.487184	+	0.873299i	$0.661976\pi$
$384$	0	0
$385$	2.42701	0.123692
$386$	0	0
$387$	0.878694	0.0446665
$388$	0	0
$389$	14.1819	0.719052	0.359526	−	0.933135i	$-0.382938\pi$
$389$	14.1819	0.719052	0.359526	+	0.933135i	$0.382938\pi$
$390$	0	0
$391$	0.388124	0.0196283
$392$	0	0
$393$	−10.4105	−0.525142
$394$	0	0
$395$	31.6617	1.59307
$396$	0	0
$397$	−10.4078	−0.522352	−0.261176	−	0.965291i	$-0.584110\pi$
$397$	−10.4078	−0.522352	−0.261176	+	0.965291i	$0.584110\pi$
$398$	0	0
$399$	−2.51324	−0.125819
$400$	0	0
$401$	−20.6073	−1.02908	−0.514540	−	0.857466i	$-0.672037\pi$
$401$	−20.6073	−1.02908	−0.514540	+	0.857466i	$0.672037\pi$
$402$	0	0
$403$	15.7760	0.785861
$404$	0	0
$405$	−2.48191	−0.123327
$406$	0	0
$407$	−2.52828	−0.125322
$408$	0	0
$409$	−11.3668	−0.562052	−0.281026	−	0.959700i	$-0.590675\pi$
$409$	−11.3668	−0.562052	−0.281026	+	0.959700i	$0.590675\pi$
$410$	0	0
$411$	0.156779	0.00773333
$412$	0	0
$413$	−12.2317	−0.601885
$414$	0	0
$415$	2.85261	0.140029
$416$	0	0
$417$	7.73518	0.378794
$418$	0	0
$419$	−37.2742	−1.82096	−0.910482	−	0.413548i	$-0.864289\pi$
$419$	−37.2742	−1.82096	−0.910482	+	0.413548i	$0.864289\pi$
$420$	0	0
$421$	13.9062	0.677748	0.338874	−	0.940832i	$-0.389954\pi$
$421$	13.9062	0.677748	0.338874	+	0.940832i	$0.389954\pi$
$422$	0	0
$423$	0.0496203	0.00241262
$424$	0	0
$425$	−0.323607	−0.0156972
$426$	0	0
$427$	14.2228	0.688291
$428$	0	0
$429$	−1.03919	−0.0501725
$430$	0	0
$431$	23.4972	1.13182	0.565910	−	0.824467i	$-0.308525\pi$
$431$	23.4972	1.13182	0.565910	+	0.824467i	$0.308525\pi$
$432$	0	0
$433$	−5.36934	−0.258034	−0.129017	−	0.991642i	$-0.541182\pi$
$433$	−5.36934	−0.258034	−0.129017	+	0.991642i	$0.541182\pi$
$434$	0	0
$435$	23.1632	1.11059
$436$	0	0
$437$	−2.43599	−0.116529
$438$	0	0
$439$	−10.5541	−0.503722	−0.251861	−	0.967763i	$-0.581043\pi$
$439$	−10.5541	−0.503722	−0.251861	+	0.967763i	$0.581043\pi$
$440$	0	0
$441$	−4.94013	−0.235244
$442$	0	0
$443$	−26.3074	−1.24990	−0.624952	−	0.780663i	$-0.714881\pi$
$443$	−26.3074	−1.24990	−0.624952	+	0.780663i	$0.714881\pi$
$444$	0	0
$445$	−29.5505	−1.40083
$446$	0	0
$447$	3.45091	0.163223
$448$	0	0
$449$	−23.3335	−1.10118	−0.550588	−	0.834777i	$-0.685596\pi$
$449$	−23.3335	−1.10118	−0.550588	+	0.834777i	$0.685596\pi$
$450$	0	0
$451$	−4.11700	−0.193862
$452$	0	0
$453$	4.17687	0.196246
$454$	0	0
$455$	5.43294	0.254700
$456$	0	0
$457$	−8.96259	−0.419252	−0.209626	−	0.977782i	$-0.567225\pi$
$457$	−8.96259	−0.419252	−0.209626	+	0.977782i	$0.567225\pi$
$458$	0	0
$459$	0.279003	0.0130227
$460$	0	0
$461$	−15.1080	−0.703651	−0.351826	−	0.936066i	$-0.614439\pi$
$461$	−15.1080	−0.703651	−0.351826	+	0.936066i	$0.614439\pi$
$462$	0	0
$463$	−9.66054	−0.448964	−0.224482	−	0.974478i	$-0.572069\pi$
$463$	−9.66054	−0.448964	−0.224482	+	0.974478i	$0.572069\pi$
$464$	0	0
$465$	−25.6717	−1.19050
$466$	0	0
$467$	−4.63374	−0.214424	−0.107212	−	0.994236i	$-0.534192\pi$
$467$	−4.63374	−0.214424	−0.107212	+	0.994236i	$0.534192\pi$
$468$	0	0
$469$	15.8282	0.730880
$470$	0	0
$471$	−7.82800	−0.360695
$472$	0	0
$473$	−0.598691	−0.0275278
$474$	0	0
$475$	2.03106	0.0931914
$476$	0	0
$477$	1.77394	0.0812233
$478$	0	0
$479$	34.7190	1.58635	0.793177	−	0.608992i	$-0.208426\pi$
$479$	34.7190	1.58635	0.793177	+	0.608992i	$0.208426\pi$
$480$	0	0
$481$	−5.65965	−0.258058
$482$	0	0
$483$	1.99655	0.0908464
$484$	0	0
$485$	19.5397	0.887251
$486$	0	0
$487$	41.5416	1.88243	0.941215	−	0.337807i	$-0.109685\pi$
$487$	41.5416	1.88243	0.941215	+	0.337807i	$0.109685\pi$
$488$	0	0
$489$	6.70964	0.303420
$490$	0	0
$491$	−14.1828	−0.640061	−0.320030	−	0.947407i	$-0.603693\pi$
$491$	−14.1828	−0.640061	−0.320030	+	0.947407i	$0.603693\pi$
$492$	0	0
$493$	−2.60388	−0.117273
$494$	0	0
$495$	1.69103	0.0760061
$496$	0	0
$497$	3.79077	0.170039
$498$	0	0
$499$	−22.1668	−0.992322	−0.496161	−	0.868231i	$-0.665257\pi$
$499$	−22.1668	−0.992322	−0.496161	+	0.868231i	$0.665257\pi$
$500$	0	0
$501$	15.3699	0.686675
$502$	0	0
$503$	−41.0454	−1.83012	−0.915062	−	0.403312i	$-0.867859\pi$
$503$	−41.0454	−1.83012	−0.915062	+	0.403312i	$0.867859\pi$
$504$	0	0
$505$	−8.40562	−0.374045
$506$	0	0
$507$	10.6737	0.474037
$508$	0	0
$509$	−14.5980	−0.647044	−0.323522	−	0.946221i	$-0.604867\pi$
$509$	−14.5980	−0.647044	−0.323522	+	0.946221i	$0.604867\pi$
$510$	0	0
$511$	−11.6868	−0.516994
$512$	0	0
$513$	−1.75111	−0.0773135
$514$	0	0
$515$	13.5992	0.599253
$516$	0	0
$517$	−0.0338084	−0.00148689
$518$	0	0
$519$	−1.00000	−0.0438951
$520$	0	0
$521$	8.43037	0.369341	0.184671	−	0.982800i	$-0.440878\pi$
$521$	8.43037	0.369341	0.184671	+	0.982800i	$0.440878\pi$
$522$	0	0
$523$	−9.28192	−0.405870	−0.202935	−	0.979192i	$-0.565048\pi$
$523$	−9.28192	−0.405870	−0.202935	+	0.979192i	$0.565048\pi$
$524$	0	0
$525$	−1.66467	−0.0726522
$526$	0	0
$527$	2.88587	0.125711
$528$	0	0
$529$	−21.0648	−0.915862
$530$	0	0
$531$	−8.52253	−0.369846
$532$	0	0
$533$	−9.21605	−0.399192
$534$	0	0
$535$	30.1765	1.30464
$536$	0	0
$537$	−8.15708	−0.352004
$538$	0	0
$539$	3.36592	0.144980
$540$	0	0
$541$	3.03579	0.130519	0.0652593	−	0.997868i	$-0.479213\pi$
$541$	3.03579	0.130519	0.0652593	+	0.997868i	$0.479213\pi$
$542$	0	0
$543$	−0.732315	−0.0314267
$544$	0	0
$545$	−17.6776	−0.757226
$546$	0	0
$547$	39.4320	1.68599	0.842995	−	0.537922i	$-0.180790\pi$
$547$	39.4320	1.68599	0.842995	+	0.537922i	$0.180790\pi$
$548$	0	0
$549$	9.90982	0.422941
$550$	0	0
$551$	16.3428	0.696226
$552$	0	0
$553$	−18.3091	−0.778584
$554$	0	0
$555$	9.20970	0.390930
$556$	0	0
$557$	−13.2582	−0.561769	−0.280885	−	0.959742i	$-0.590628\pi$
$557$	−13.2582	−0.561769	−0.280885	+	0.959742i	$0.590628\pi$
$558$	0	0
$559$	−1.34019	−0.0566841
$560$	0	0
$561$	−0.190096	−0.00802588
$562$	0	0
$563$	−36.1732	−1.52452	−0.762259	−	0.647272i	$-0.775910\pi$
$563$	−36.1732	−1.52452	−0.762259	+	0.647272i	$0.775910\pi$
$564$	0	0
$565$	−32.5152	−1.36792
$566$	0	0
$567$	1.43522	0.0602738
$568$	0	0
$569$	19.6245	0.822703	0.411352	−	0.911477i	$-0.365057\pi$
$569$	19.6245	0.822703	0.411352	+	0.911477i	$0.365057\pi$
$570$	0	0
$571$	−38.1890	−1.59816	−0.799079	−	0.601226i	$-0.794679\pi$
$571$	−38.1890	−1.59816	−0.799079	+	0.601226i	$0.794679\pi$
$572$	0	0
$573$	6.71804	0.280650
$574$	0	0
$575$	−1.61350	−0.0672877
$576$	0	0
$577$	11.0417	0.459672	0.229836	−	0.973229i	$-0.426181\pi$
$577$	11.0417	0.459672	0.229836	+	0.973229i	$0.426181\pi$
$578$	0	0
$579$	5.77120	0.239843
$580$	0	0
$581$	−1.64959	−0.0684365
$582$	0	0
$583$	−1.20866	−0.0500577
$584$	0	0
$585$	3.78543	0.156508
$586$	0	0
$587$	0.758988	0.0313268	0.0156634	−	0.999877i	$-0.495014\pi$
$587$	0.758988	0.0313268	0.0156634	+	0.999877i	$0.495014\pi$
$588$	0	0
$589$	−18.1127	−0.746320
$590$	0	0
$591$	−17.9065	−0.736576
$592$	0	0
$593$	−17.7909	−0.730585	−0.365292	−	0.930893i	$-0.619031\pi$
$593$	−17.7909	−0.730585	−0.365292	+	0.930893i	$0.619031\pi$
$594$	0	0
$595$	0.993835	0.0407433
$596$	0	0
$597$	−14.3534	−0.587445
$598$	0	0
$599$	22.7424	0.929230	0.464615	−	0.885513i	$-0.346193\pi$
$599$	22.7424	0.929230	0.464615	+	0.885513i	$0.346193\pi$
$600$	0	0
$601$	−44.7532	−1.82552	−0.912760	−	0.408496i	$-0.866053\pi$
$601$	−44.7532	−1.82552	−0.912760	+	0.408496i	$0.866053\pi$
$602$	0	0
$603$	11.0284	0.449111
$604$	0	0
$605$	26.1488	1.06310
$606$	0	0
$607$	−11.6371	−0.472334	−0.236167	−	0.971713i	$-0.575891\pi$
$607$	−11.6371	−0.472334	−0.236167	+	0.971713i	$0.575891\pi$
$608$	0	0
$609$	−13.3947	−0.542779
$610$	0	0
$611$	−0.0756814	−0.00306174
$612$	0	0
$613$	−27.9193	−1.12765	−0.563825	−	0.825895i	$-0.690671\pi$
$613$	−27.9193	−1.12765	−0.563825	+	0.825895i	$0.690671\pi$
$614$	0	0
$615$	14.9969	0.604733
$616$	0	0
$617$	26.9603	1.08538	0.542690	−	0.839933i	$-0.317406\pi$
$617$	26.9603	1.08538	0.542690	+	0.839933i	$0.317406\pi$
$618$	0	0
$619$	−22.4075	−0.900634	−0.450317	−	0.892869i	$-0.648689\pi$
$619$	−22.4075	−0.900634	−0.450317	+	0.892869i	$0.648689\pi$
$620$	0	0
$621$	1.39111	0.0558233
$622$	0	0
$623$	17.0883	0.684627
$624$	0	0
$625$	−29.4540	−1.17816
$626$	0	0
$627$	1.19311	0.0476481
$628$	0	0
$629$	−1.03531	−0.0412803
$630$	0	0
$631$	17.2444	0.686488	0.343244	−	0.939246i	$-0.388474\pi$
$631$	17.2444	0.686488	0.343244	+	0.939246i	$0.388474\pi$
$632$	0	0
$633$	15.6290	0.621195
$634$	0	0
$635$	−42.5404	−1.68817
$636$	0	0
$637$	7.53473	0.298537
$638$	0	0
$639$	2.64124	0.104486
$640$	0	0
$641$	10.4400	0.412353	0.206177	−	0.978515i	$-0.433898\pi$
$641$	10.4400	0.412353	0.206177	+	0.978515i	$0.433898\pi$
$642$	0	0
$643$	−32.1639	−1.26842	−0.634211	−	0.773160i	$-0.718675\pi$
$643$	−32.1639	−1.26842	−0.634211	+	0.773160i	$0.718675\pi$
$644$	0	0
$645$	2.18084	0.0858704
$646$	0	0
$647$	13.7743	0.541524	0.270762	−	0.962646i	$-0.412724\pi$
$647$	13.7743	0.541524	0.270762	+	0.962646i	$0.412724\pi$
$648$	0	0
$649$	5.80676	0.227935
$650$	0	0
$651$	14.8453	0.581832
$652$	0	0
$653$	−20.7680	−0.812715	−0.406358	−	0.913714i	$-0.633201\pi$
$653$	−20.7680	−0.812715	−0.406358	+	0.913714i	$0.633201\pi$
$654$	0	0
$655$	−25.8380	−1.00957
$656$	0	0
$657$	−8.14285	−0.317683
$658$	0	0
$659$	34.0650	1.32699	0.663493	−	0.748183i	$-0.269073\pi$
$659$	34.0650	1.32699	0.663493	+	0.748183i	$0.269073\pi$
$660$	0	0
$661$	−26.7379	−1.03998	−0.519991	−	0.854172i	$-0.674065\pi$
$661$	−26.7379	−1.03998	−0.519991	+	0.854172i	$0.674065\pi$
$662$	0	0
$663$	−0.425538	−0.0165265
$664$	0	0
$665$	−6.23763	−0.241885
$666$	0	0
$667$	−12.9830	−0.502702
$668$	0	0
$669$	11.7815	0.455500
$670$	0	0
$671$	−6.75198	−0.260657
$672$	0	0
$673$	2.05014	0.0790271	0.0395136	−	0.999219i	$-0.487419\pi$
$673$	2.05014	0.0790271	0.0395136	+	0.999219i	$0.487419\pi$
$674$	0	0
$675$	−1.15987	−0.0446433
$676$	0	0
$677$	−35.9310	−1.38094	−0.690470	−	0.723361i	$-0.742596\pi$
$677$	−35.9310	−1.38094	−0.690470	+	0.723361i	$0.742596\pi$
$678$	0	0
$679$	−11.2993	−0.433627
$680$	0	0
$681$	−13.8804	−0.531897
$682$	0	0
$683$	−24.4303	−0.934799	−0.467400	−	0.884046i	$-0.654809\pi$
$683$	−24.4303	−0.934799	−0.467400	+	0.884046i	$0.654809\pi$
$684$	0	0
$685$	0.389111	0.0148672
$686$	0	0
$687$	28.1884	1.07545
$688$	0	0
$689$	−2.70564	−0.103077
$690$	0	0
$691$	−1.37585	−0.0523400	−0.0261700	−	0.999658i	$-0.508331\pi$
$691$	−1.37585	−0.0523400	−0.0261700	+	0.999658i	$0.508331\pi$
$692$	0	0
$693$	−0.977879	−0.0371465
$694$	0	0
$695$	19.1980	0.728222
$696$	0	0
$697$	−1.68587	−0.0638569
$698$	0	0
$699$	17.2620	0.652908
$700$	0	0
$701$	7.38286	0.278847	0.139423	−	0.990233i	$-0.455475\pi$
$701$	7.38286	0.278847	0.139423	+	0.990233i	$0.455475\pi$
$702$	0	0
$703$	6.49791	0.245073
$704$	0	0
$705$	0.123153	0.00463821
$706$	0	0
$707$	4.86076	0.182808
$708$	0	0
$709$	22.9236	0.860915	0.430457	−	0.902611i	$-0.358352\pi$
$709$	22.9236	0.860915	0.430457	+	0.902611i	$0.358352\pi$
$710$	0	0
$711$	−12.7570	−0.478424
$712$	0	0
$713$	14.3890	0.538871
$714$	0	0
$715$	−2.57917	−0.0964556
$716$	0	0
$717$	7.19829	0.268825
$718$	0	0
$719$	−19.5412	−0.728764	−0.364382	−	0.931250i	$-0.618720\pi$
$719$	−19.5412	−0.728764	−0.364382	+	0.931250i	$0.618720\pi$
$720$	0	0
$721$	−7.86408	−0.292874
$722$	0	0
$723$	−6.18787	−0.230129
$724$	0	0
$725$	10.8248	0.402024
$726$	0	0
$727$	8.11765	0.301067	0.150533	−	0.988605i	$-0.451901\pi$
$727$	8.11765	0.301067	0.150533	+	0.988605i	$0.451901\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−0.245158	−0.00906750
$732$	0	0
$733$	32.4930	1.20016	0.600078	−	0.799941i	$-0.295136\pi$
$733$	32.4930	1.20016	0.600078	+	0.799941i	$0.295136\pi$
$734$	0	0
$735$	−12.2610	−0.452252
$736$	0	0
$737$	−7.51411	−0.276786
$738$	0	0
$739$	−5.99186	−0.220414	−0.110207	−	0.993909i	$-0.535151\pi$
$739$	−5.99186	−0.220414	−0.110207	+	0.993909i	$0.535151\pi$
$740$	0	0
$741$	2.67081	0.0981148
$742$	0	0
$743$	−12.0964	−0.443773	−0.221886	−	0.975073i	$-0.571221\pi$
$743$	−12.0964	−0.443773	−0.221886	+	0.975073i	$0.571221\pi$
$744$	0	0
$745$	8.56485	0.313792
$746$	0	0
$747$	−1.14936	−0.0420529
$748$	0	0
$749$	−17.4503	−0.637621
$750$	0	0
$751$	−3.20726	−0.117035	−0.0585173	−	0.998286i	$-0.518637\pi$
$751$	−3.20726	−0.117035	−0.0585173	+	0.998286i	$0.518637\pi$
$752$	0	0
$753$	11.4781	0.418284
$754$	0	0
$755$	10.3666	0.377279
$756$	0	0
$757$	−20.8002	−0.755995	−0.377997	−	0.925807i	$-0.623387\pi$
$757$	−20.8002	−0.755995	−0.377997	+	0.925807i	$0.623387\pi$
$758$	0	0
$759$	−0.947821	−0.0344037
$760$	0	0
$761$	17.2629	0.625781	0.312891	−	0.949789i	$-0.398703\pi$
$761$	17.2629	0.625781	0.312891	+	0.949789i	$0.398703\pi$
$762$	0	0
$763$	10.2225	0.370080
$764$	0	0
$765$	0.692460	0.0250359
$766$	0	0
$767$	12.9986	0.469354
$768$	0	0
$769$	15.8252	0.570670	0.285335	−	0.958428i	$-0.407895\pi$
$769$	15.8252	0.570670	0.285335	+	0.958428i	$0.407895\pi$
$770$	0	0
$771$	7.18265	0.258677
$772$	0	0
$773$	54.5518	1.96209	0.981046	−	0.193774i	$-0.0620729\pi$
$773$	54.5518	1.96209	0.981046	+	0.193774i	$0.0620729\pi$
$774$	0	0
$775$	−11.9971	−0.430949
$776$	0	0
$777$	−5.32574	−0.191060
$778$	0	0
$779$	10.5811	0.379106
$780$	0	0
$781$	−1.79959	−0.0643942
$782$	0	0
$783$	−9.33280	−0.333527
$784$	0	0
$785$	−19.4284	−0.693429
$786$	0	0
$787$	4.81735	0.171720	0.0858600	−	0.996307i	$-0.472636\pi$
$787$	4.81735	0.171720	0.0858600	+	0.996307i	$0.472636\pi$
$788$	0	0
$789$	−25.6948	−0.914758
$790$	0	0
$791$	18.8027	0.668547
$792$	0	0
$793$	−15.1146	−0.536734
$794$	0	0
$795$	4.40277	0.156150
$796$	0	0
$797$	37.8920	1.34220	0.671102	−	0.741365i	$-0.265821\pi$
$797$	37.8920	1.34220	0.671102	+	0.741365i	$0.265821\pi$
$798$	0	0
$799$	−0.0138442	−0.000489773 0
$800$	0	0
$801$	11.9063	0.420690
$802$	0	0
$803$	5.54807	0.195787
$804$	0	0
$805$	4.95526	0.174650
$806$	0	0
$807$	−17.3140	−0.609481
$808$	0	0
$809$	15.5207	0.545678	0.272839	−	0.962060i	$-0.412037\pi$
$809$	15.5207	0.545678	0.272839	+	0.962060i	$0.412037\pi$
$810$	0	0
$811$	26.0297	0.914025	0.457012	−	0.889460i	$-0.348920\pi$
$811$	26.0297	0.914025	0.457012	+	0.889460i	$0.348920\pi$
$812$	0	0
$813$	5.95052	0.208694
$814$	0	0
$815$	16.6527	0.583319
$816$	0	0
$817$	1.53869	0.0538320
$818$	0	0
$819$	−2.18902	−0.0764905
$820$	0	0
$821$	−14.0069	−0.488843	−0.244422	−	0.969669i	$-0.578598\pi$
$821$	−14.0069	−0.488843	−0.244422	+	0.969669i	$0.578598\pi$
$822$	0	0
$823$	29.1935	1.01762	0.508812	−	0.860878i	$-0.330085\pi$
$823$	29.1935	1.01762	0.508812	+	0.860878i	$0.330085\pi$
$824$	0	0
$825$	0.790267	0.0275136
$826$	0	0
$827$	13.0146	0.452561	0.226281	−	0.974062i	$-0.427343\pi$
$827$	13.0146	0.452561	0.226281	+	0.974062i	$0.427343\pi$
$828$	0	0
$829$	−22.4724	−0.780500	−0.390250	−	0.920709i	$-0.627611\pi$
$829$	−22.4724	−0.780500	−0.390250	+	0.920709i	$0.627611\pi$
$830$	0	0
$831$	13.1478	0.456094
$832$	0	0
$833$	1.37831	0.0477557
$834$	0	0
$835$	38.1466	1.32012
$836$	0	0
$837$	10.3435	0.357525
$838$	0	0
$839$	3.82802	0.132158	0.0660789	−	0.997814i	$-0.478951\pi$
$839$	3.82802	0.132158	0.0660789	+	0.997814i	$0.478951\pi$
$840$	0	0
$841$	58.1012	2.00349
$842$	0	0
$843$	1.73336	0.0597002
$844$	0	0
$845$	26.4912	0.911326
$846$	0	0
$847$	−15.1212	−0.519571
$848$	0	0
$849$	19.6291	0.673667
$850$	0	0
$851$	−5.16204	−0.176952
$852$	0	0
$853$	17.5199	0.599872	0.299936	−	0.953959i	$-0.403035\pi$
$853$	17.5199	0.599872	0.299936	+	0.953959i	$0.403035\pi$
$854$	0	0
$855$	−4.34610	−0.148634
$856$	0	0
$857$	−33.8377	−1.15588	−0.577938	−	0.816081i	$-0.696142\pi$
$857$	−33.8377	−1.15588	−0.577938	+	0.816081i	$0.696142\pi$
$858$	0	0
$859$	−13.9298	−0.475279	−0.237640	−	0.971353i	$-0.576374\pi$
$859$	−13.9298	−0.475279	−0.237640	+	0.971353i	$0.576374\pi$
$860$	0	0
$861$	−8.67232	−0.295552
$862$	0	0
$863$	−22.4902	−0.765576	−0.382788	−	0.923836i	$-0.625036\pi$
$863$	−22.4902	−0.765576	−0.382788	+	0.923836i	$0.625036\pi$
$864$	0	0
$865$	−2.48191	−0.0843874
$866$	0	0
$867$	16.9222	0.574707
$868$	0	0
$869$	8.69187	0.294852
$870$	0	0
$871$	−16.8206	−0.569945
$872$	0	0
$873$	−7.87285	−0.266455
$874$	0	0
$875$	13.6789	0.462432
$876$	0	0
$877$	35.1740	1.18774	0.593870	−	0.804561i	$-0.297599\pi$
$877$	35.1740	1.18774	0.593870	+	0.804561i	$0.297599\pi$
$878$	0	0
$879$	1.36314	0.0459776
$880$	0	0
$881$	10.2301	0.344661	0.172331	−	0.985039i	$-0.444870\pi$
$881$	10.2301	0.344661	0.172331	+	0.985039i	$0.444870\pi$
$882$	0	0
$883$	−10.6336	−0.357849	−0.178924	−	0.983863i	$-0.557262\pi$
$883$	−10.6336	−0.357849	−0.178924	+	0.983863i	$0.557262\pi$
$884$	0	0
$885$	−21.1521	−0.711021
$886$	0	0
$887$	−17.9319	−0.602095	−0.301047	−	0.953609i	$-0.597336\pi$
$887$	−17.9319	−0.602095	−0.301047	+	0.953609i	$0.597336\pi$
$888$	0	0
$889$	24.6001	0.825059
$890$	0	0
$891$	−0.681342	−0.0228258
$892$	0	0
$893$	0.0868908	0.00290769
$894$	0	0
$895$	−20.2451	−0.676720
$896$	0	0
$897$	−2.12173	−0.0708426
$898$	0	0
$899$	−96.5341	−3.21959
$900$	0	0
$901$	−0.494936	−0.0164887
$902$	0	0
$903$	−1.26112	−0.0419675
$904$	0	0
$905$	−1.81754	−0.0604171
$906$	0	0
$907$	−5.12998	−0.170338	−0.0851691	−	0.996367i	$-0.527143\pi$
$907$	−5.12998	−0.170338	−0.0851691	+	0.996367i	$0.527143\pi$
$908$	0	0
$909$	3.38676	0.112332
$910$	0	0
$911$	−40.7888	−1.35139	−0.675696	−	0.737180i	$-0.736157\pi$
$911$	−40.7888	−1.35139	−0.675696	+	0.737180i	$0.736157\pi$
$912$	0	0
$913$	0.783107	0.0259171
$914$	0	0
$915$	24.5953	0.813095
$916$	0	0
$917$	14.9414	0.493410
$918$	0	0
$919$	−39.1346	−1.29093	−0.645466	−	0.763789i	$-0.723337\pi$
$919$	−39.1346	−1.29093	−0.645466	+	0.763789i	$0.723337\pi$
$920$	0	0
$921$	−20.4507	−0.673872
$922$	0	0
$923$	−4.02844	−0.132598
$924$	0	0
$925$	4.30396	0.141513
$926$	0	0
$927$	−5.47934	−0.179965
$928$	0	0
$929$	−34.9075	−1.14528	−0.572638	−	0.819808i	$-0.694080\pi$
$929$	−34.9075	−1.14528	−0.572638	+	0.819808i	$0.694080\pi$
$930$	0	0
$931$	−8.65073	−0.283516
$932$	0	0
$933$	31.9816	1.04703
$934$	0	0
$935$	−0.471802	−0.0154296
$936$	0	0
$937$	35.8603	1.17151	0.585753	−	0.810490i	$-0.300799\pi$
$937$	35.8603	1.17151	0.585753	+	0.810490i	$0.300799\pi$
$938$	0	0
$939$	5.37302	0.175342
$940$	0	0
$941$	−25.4726	−0.830381	−0.415191	−	0.909734i	$-0.636285\pi$
$941$	−25.4726	−0.830381	−0.415191	+	0.909734i	$0.636285\pi$
$942$	0	0
$943$	−8.40575	−0.273729
$944$	0	0
$945$	3.56210	0.115875
$946$	0	0
$947$	−3.21400	−0.104441	−0.0522205	−	0.998636i	$-0.516630\pi$
$947$	−3.21400	−0.104441	−0.0522205	+	0.998636i	$0.516630\pi$
$948$	0	0
$949$	12.4196	0.403156
$950$	0	0
$951$	21.3702	0.692977
$952$	0	0
$953$	−32.0818	−1.03923	−0.519616	−	0.854400i	$-0.673925\pi$
$953$	−32.0818	−1.03923	−0.519616	+	0.854400i	$0.673925\pi$
$954$	0	0
$955$	16.6736	0.539544
$956$	0	0
$957$	6.35883	0.205552
$958$	0	0
$959$	−0.225013	−0.00726604
$960$	0	0
$961$	75.9885	2.45124
$962$	0	0
$963$	−12.1586	−0.391805
$964$	0	0
$965$	14.3236	0.461093
$966$	0	0
$967$	1.86204	0.0598793	0.0299397	−	0.999552i	$-0.490468\pi$
$967$	1.86204	0.0598793	0.0299397	+	0.999552i	$0.490468\pi$
$968$	0	0
$969$	0.488566	0.0156950
$970$	0	0
$971$	11.6501	0.373869	0.186935	−	0.982372i	$-0.440145\pi$
$971$	11.6501	0.373869	0.186935	+	0.982372i	$0.440145\pi$
$972$	0	0
$973$	−11.1017	−0.355905
$974$	0	0
$975$	1.76904	0.0566547
$976$	0	0
$977$	13.3631	0.427523	0.213761	−	0.976886i	$-0.431428\pi$
$977$	13.3631	0.427523	0.213761	+	0.976886i	$0.431428\pi$
$978$	0	0
$979$	−8.11229	−0.259270
$980$	0	0
$981$	7.12259	0.227407
$982$	0	0
$983$	21.4462	0.684029	0.342014	−	0.939695i	$-0.388891\pi$
$983$	21.4462	0.684029	0.342014	+	0.939695i	$0.388891\pi$
$984$	0	0
$985$	−44.4423	−1.41605
$986$	0	0
$987$	−0.0712163	−0.00226684
$988$	0	0
$989$	−1.22236	−0.0388688
$990$	0	0
$991$	−44.7582	−1.42179	−0.710895	−	0.703298i	$-0.751710\pi$
$991$	−44.7582	−1.42179	−0.710895	+	0.703298i	$0.751710\pi$
$992$	0	0
$993$	8.40384	0.266688
$994$	0	0
$995$	−35.6238	−1.12935
$996$	0	0
$997$	18.8389	0.596633	0.298316	−	0.954467i	$-0.403575\pi$
$997$	18.8389	0.596633	0.298316	+	0.954467i	$0.403575\pi$
$998$	0	0
$999$	−3.71073	−0.117402

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8304.2.a.bf.1.1		7
4.3	odd	2		2076.2.a.g.1.1	✓	7
12.11	even	2		6228.2.a.m.1.7		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2076.2.a.g.1.1	✓	7	4.3	odd	2
6228.2.a.m.1.7		7	12.11	even	2
8304.2.a.bf.1.1		7	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:	\( N \)	\(=\)	\( 8304 = 2^{4} \cdot 3 \cdot 173 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8304.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -2.48191 q^{5} +1.43522 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -2.48191 q^{5} +1.43522 q^{7} +1.00000 q^{9} -0.681342 q^{11} -1.52521 q^{13} +2.48191 q^{15} -0.279003 q^{17} +1.75111 q^{19} -1.43522 q^{21} -1.39111 q^{23} +1.15987 q^{25} -1.00000 q^{27} +9.33280 q^{29} -10.3435 q^{31} +0.681342 q^{33} -3.56210 q^{35} +3.71073 q^{37} +1.52521 q^{39} +6.04248 q^{41} +0.878694 q^{43} -2.48191 q^{45} +0.0496203 q^{47} -4.94013 q^{49} +0.279003 q^{51} +1.77394 q^{53} +1.69103 q^{55} -1.75111 q^{57} -8.52253 q^{59} +9.90982 q^{61} +1.43522 q^{63} +3.78543 q^{65} +11.0284 q^{67} +1.39111 q^{69} +2.64124 q^{71} -8.14285 q^{73} -1.15987 q^{75} -0.977879 q^{77} -12.7570 q^{79} +1.00000 q^{81} -1.14936 q^{83} +0.692460 q^{85} -9.33280 q^{87} +11.9063 q^{89} -2.18902 q^{91} +10.3435 q^{93} -4.34610 q^{95} -7.87285 q^{97} -0.681342 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 7 q^{3} + 6 q^{5} - q^{7} + 7 q^{9}+O(q^{10}) \) 7 * q - 7 * q^3 + 6 * q^5 - q^7 + 7 * q^9	\( 7 q - 7 q^{3} + 6 q^{5} - q^{7} + 7 q^{9} - 4 q^{11} + 4 q^{13} - 6 q^{15} + 7 q^{17} - 9 q^{19} + q^{21} - 17 q^{23} + 5 q^{25} - 7 q^{27} + 6 q^{29} - 10 q^{31} + 4 q^{33} - 9 q^{35} + 6 q^{37} - 4 q^{39} + 7 q^{41} - 11 q^{43} + 6 q^{45} - 22 q^{47} - 8 q^{49} - 7 q^{51} + 20 q^{53} - 21 q^{55} + 9 q^{57} - 18 q^{59} - q^{63} - 2 q^{65} - 8 q^{67} + 17 q^{69} - 14 q^{71} - 14 q^{73} - 5 q^{75} + 13 q^{77} - q^{79} + 7 q^{81} - 7 q^{83} - 16 q^{85} - 6 q^{87} - 6 q^{89} - 10 q^{91} + 10 q^{93} - 31 q^{95} - 17 q^{97} - 4 q^{99}+O(q^{100}) \) 7 * q - 7 * q^3 + 6 * q^5 - q^7 + 7 * q^9 - 4 * q^11 + 4 * q^13 - 6 * q^15 + 7 * q^17 - 9 * q^19 + q^21 - 17 * q^23 + 5 * q^25 - 7 * q^27 + 6 * q^29 - 10 * q^31 + 4 * q^33 - 9 * q^35 + 6 * q^37 - 4 * q^39 + 7 * q^41 - 11 * q^43 + 6 * q^45 - 22 * q^47 - 8 * q^49 - 7 * q^51 + 20 * q^53 - 21 * q^55 + 9 * q^57 - 18 * q^59 - q^63 - 2 * q^65 - 8 * q^67 + 17 * q^69 - 14 * q^71 - 14 * q^73 - 5 * q^75 + 13 * q^77 - q^79 + 7 * q^81 - 7 * q^83 - 16 * q^85 - 6 * q^87 - 6 * q^89 - 10 * q^91 + 10 * q^93 - 31 * q^95 - 17 * q^97 - 4 * q^99

Introduction

L-functions

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