LMFDB - Embedded newform 4000.2.f.c.3249.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4000,2,Mod(3249,4000)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4000, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4000.3249");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4000 = 2^{5} \cdot 5^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4000.f (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$31.9401608085$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 3 x^{19} + 6 x^{18} - 5 x^{17} - 3 x^{16} + 20 x^{15} - 28 x^{14} + 24 x^{13} + 16 x^{12} + \cdots + 1024 $ x^20 - 3x^19 + 6x^18 - 5x^17 - 3x^16 + 20x^15 - 28x^14 + 24x^13 + 16x^12 - 60x^11 + 116x^10 - 120x^9 + 64x^8 + 192x^7 - 448x^6 + 640x^5 - 192x^4 - 640x^3 + 1536x^2 - 1536*x + 1024
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	no (minimal twist has level 1000)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3249.16
Root			$0.627237 - 1.26751i$ of defining polynomial
Character	$\chi$	$=$	4000.3249
Dual form			4000.2.f.c.3249.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.69676 q^{3} +4.40040i q^{7} -0.120998 q^{9} +O(q^{10})$	$q+1.69676 q^{3} +4.40040i q^{7} -0.120998 q^{9} -0.720144i q^{11} +4.59595 q^{13} +2.38420i q^{17} +0.899742i q^{19} +7.46643i q^{21} +8.05029i q^{23} -5.29559 q^{27} -1.78194i q^{29} -9.07877 q^{31} -1.22191i q^{33} -4.74502 q^{37} +7.79823 q^{39} -2.82760 q^{41} +10.7770 q^{43} +4.99256i q^{47} -12.3635 q^{49} +4.04541i q^{51} -9.74069 q^{53} +1.52665i q^{57} -6.39947i q^{59} +0.635551i q^{61} -0.532441i q^{63} +1.73643 q^{67} +13.6594i q^{69} +8.41146 q^{71} -13.0770i q^{73} +3.16892 q^{77} +7.19828 q^{79} -8.62236 q^{81} -0.219648 q^{83} -3.02354i q^{87} -1.37331 q^{89} +20.2240i q^{91} -15.4045 q^{93} +8.41826i q^{97} +0.0871363i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 4 q^{3} + 12 q^{9}+O(q^{10}) $ 20 * q - 4 * q^3 + 12 * q^9	$ 20 q - 4 q^{3} + 12 q^{9} + 6 q^{13} + 8 q^{27} - 24 q^{31} + 18 q^{37} + 4 q^{39} + 22 q^{41} + 60 q^{43} - 6 q^{49} + 10 q^{53} + 40 q^{67} - 48 q^{71} + 24 q^{77} + 48 q^{79} - 28 q^{81} - 40 q^{83} + 22 q^{89} + 8 q^{93}+O(q^{100}) $ 20 * q - 4 * q^3 + 12 * q^9 + 6 * q^13 + 8 * q^27 - 24 * q^31 + 18 * q^37 + 4 * q^39 + 22 * q^41 + 60 * q^43 - 6 * q^49 + 10 * q^53 + 40 * q^67 - 48 * q^71 + 24 * q^77 + 48 * q^79 - 28 * q^81 - 40 * q^83 + 22 * q^89 + 8 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4000\mathbb{Z}\right)^\times$.

$n$	$1377$	$2501$	$2751$
$\chi(n)$	$-1$	$-1$	$1$

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
$3$	1.69676	0.979626	0.489813	−	0.871828i	$-0.337065\pi$
$3$	1.69676	0.979626	0.489813	+	0.871828i	$0.337065\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.40040i	1.66319i	0.555380	+	0.831597i	$0.312573\pi$
$7$	4.40040i	1.66319i	−0.555380	+	0.831597i	$0.687427\pi$
$8$	0	0
$9$	−0.120998	−0.0403328
$10$	0	0
$11$	− 0.720144i	− 0.217132i	−0.994089	−	0.108566i	$-0.965374\pi$
$11$	− 0.720144i	− 0.217132i	0.994089	−	0.108566i	$-0.0346258\pi$
$12$	0	0
$13$	4.59595	1.27469	0.637343	−	0.770580i	$-0.280033\pi$
$13$	4.59595	1.27469	0.637343	+	0.770580i	$0.280033\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.38420i	0.578252i	0.957291	+	0.289126i	$0.0933647\pi$
$17$	2.38420i	0.578252i	−0.957291	+	0.289126i	$0.906635\pi$
$18$	0	0
$19$	0.899742i	0.206415i	0.994660	+	0.103208i	$0.0329106\pi$
$19$	0.899742i	0.206415i	−0.994660	+	0.103208i	$0.967089\pi$
$20$	0	0
$21$	7.46643i	1.62931i
$22$	0	0
$23$	8.05029i	1.67860i	0.543668	+	0.839300i	$0.317035\pi$
$23$	8.05029i	1.67860i	−0.543668	+	0.839300i	$0.682965\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−5.29559	−1.01914
$28$	0	0
$29$	− 1.78194i	− 0.330899i	−0.986218	−	0.165449i	$-0.947093\pi$
$29$	− 1.78194i	− 0.330899i	0.986218	−	0.165449i	$-0.0529075\pi$
$30$	0	0
$31$	−9.07877	−1.63059	−0.815297	−	0.579042i	$-0.803427\pi$
$31$	−9.07877	−1.63059	−0.815297	+	0.579042i	$0.803427\pi$
$32$	0	0
$33$	− 1.22191i	− 0.212708i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−4.74502	−0.780076	−0.390038	−	0.920799i	$-0.627538\pi$
$37$	−4.74502	−0.780076	−0.390038	+	0.920799i	$0.627538\pi$
$38$	0	0
$39$	7.79823	1.24872
$40$	0	0
$41$	−2.82760	−0.441597	−0.220799	−	0.975319i	$-0.570866\pi$
$41$	−2.82760	−0.441597	−0.220799	+	0.975319i	$0.570866\pi$
$42$	0	0
$43$	10.7770	1.64347	0.821736	−	0.569868i	$-0.193006\pi$
$43$	10.7770	1.64347	0.821736	+	0.569868i	$0.193006\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.99256i	0.728240i	0.931352	+	0.364120i	$0.118630\pi$
$47$	4.99256i	0.728240i	−0.931352	+	0.364120i	$0.881370\pi$
$48$	0	0
$49$	−12.3635	−1.76621
$50$	0	0
$51$	4.04541i	0.566471i
$52$	0	0
$53$	−9.74069	−1.33799	−0.668994	−	0.743268i	$-0.733275\pi$
$53$	−9.74069	−1.33799	−0.668994	+	0.743268i	$0.733275\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.52665i	0.202210i
$58$	0	0
$59$	− 6.39947i	− 0.833141i	−0.909103	−	0.416570i	$-0.863232\pi$
$59$	− 6.39947i	− 0.833141i	0.909103	−	0.416570i	$-0.136768\pi$
$60$	0	0
$61$	0.635551i	0.0813740i	0.999172	+	0.0406870i	$0.0129547\pi$
$61$	0.635551i	0.0813740i	−0.999172	+	0.0406870i	$0.987045\pi$
$62$	0	0
$63$	− 0.532441i	− 0.0670812i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	1.73643	0.212138	0.106069	−	0.994359i	$-0.466174\pi$
$67$	1.73643	0.212138	0.106069	+	0.994359i	$0.466174\pi$
$68$	0	0
$69$	13.6594i	1.64440i
$70$	0	0
$71$	8.41146	0.998257	0.499128	−	0.866528i	$-0.333654\pi$
$71$	8.41146	0.998257	0.499128	+	0.866528i	$0.333654\pi$
$72$	0	0
$73$	− 13.0770i	− 1.53055i	−0.643705	−	0.765274i	$-0.722604\pi$
$73$	− 13.0770i	− 1.53055i	0.643705	−	0.765274i	$-0.277396\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.16892	0.361132
$78$	0	0
$79$	7.19828	0.809870	0.404935	−	0.914346i	$-0.367294\pi$
$79$	7.19828	0.809870	0.404935	+	0.914346i	$0.367294\pi$
$80$	0	0
$81$	−8.62236	−0.958040
$82$	0	0
$83$	−0.219648	−0.0241095	−0.0120548	−	0.999927i	$-0.503837\pi$
$83$	−0.219648	−0.0241095	−0.0120548	+	0.999927i	$0.503837\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 3.02354i	− 0.324157i
$88$	0	0
$89$	−1.37331	−0.145571	−0.0727853	−	0.997348i	$-0.523189\pi$
$89$	−1.37331	−0.145571	−0.0727853	+	0.997348i	$0.523189\pi$
$90$	0	0
$91$	20.2240i	2.12005i
$92$	0	0
$93$	−15.4045	−1.59737
$94$	0	0
$95$	0	0
$96$	0	0
$97$	8.41826i	0.854744i	0.904076	+	0.427372i	$0.140561\pi$
$97$	8.41826i	0.854744i	−0.904076	+	0.427372i	$0.859439\pi$
$98$	0	0
$99$	0.0871363i	0.00875752i
$100$	0	0
$101$	17.2981i	1.72122i	0.509264	+	0.860610i	$0.329918\pi$
$101$	17.2981i	1.72122i	−0.509264	+	0.860610i	$0.670082\pi$
$102$	0	0
$103$	9.40667i	0.926867i	0.886132	+	0.463433i	$0.153383\pi$
$103$	9.40667i	0.926867i	−0.886132	+	0.463433i	$0.846617\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	9.29949	0.899015	0.449508	−	0.893276i	$-0.351599\pi$
$107$	9.29949	0.899015	0.449508	+	0.893276i	$0.351599\pi$
$108$	0	0
$109$	− 19.0410i	− 1.82379i	−0.410418	−	0.911897i	$-0.634617\pi$
$109$	− 19.0410i	− 1.82379i	0.410418	−	0.911897i	$-0.365383\pi$
$110$	0	0
$111$	−8.05117	−0.764183
$112$	0	0
$113$	− 6.64364i	− 0.624981i	−0.949921	−	0.312491i	$-0.898837\pi$
$113$	− 6.64364i	− 0.624981i	0.949921	−	0.312491i	$-0.101163\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−0.556102	−0.0514116
$118$	0	0
$119$	−10.4914	−0.961746
$120$	0	0
$121$	10.4814	0.952854
$122$	0	0
$123$	−4.79777	−0.432600
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 2.37460i	− 0.210712i	−0.994435	−	0.105356i	$-0.966402\pi$
$127$	− 2.37460i	− 0.210712i	0.994435	−	0.105356i	$-0.0335982\pi$
$128$	0	0
$129$	18.2860	1.60999
$130$	0	0
$131$	13.4494i	1.17508i	0.809196	+	0.587538i	$0.199903\pi$
$131$	13.4494i	1.17508i	−0.809196	+	0.587538i	$0.800097\pi$
$132$	0	0
$133$	−3.95922	−0.343308
$134$	0	0
$135$	0	0
$136$	0	0
$137$	18.4004i	1.57205i	0.618194	+	0.786025i	$0.287864\pi$
$137$	18.4004i	1.57205i	−0.618194	+	0.786025i	$0.712136\pi$
$138$	0	0
$139$	9.62833i	0.816665i	0.912833	+	0.408332i	$0.133890\pi$
$139$	9.62833i	0.816665i	−0.912833	+	0.408332i	$0.866110\pi$
$140$	0	0
$141$	8.47119i	0.713403i
$142$	0	0
$143$	− 3.30974i	− 0.276775i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−20.9779	−1.73023
$148$	0	0
$149$	12.8658i	1.05401i	0.849862	+	0.527005i	$0.176685\pi$
$149$	12.8658i	1.05401i	−0.849862	+	0.527005i	$0.823315\pi$
$150$	0	0
$151$	−11.5898	−0.943167	−0.471583	−	0.881821i	$-0.656317\pi$
$151$	−11.5898	−0.943167	−0.471583	+	0.881821i	$0.656317\pi$
$152$	0	0
$153$	− 0.288484i	− 0.0233225i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−19.0105	−1.51721	−0.758603	−	0.651553i	$-0.774118\pi$
$157$	−19.0105	−1.51721	−0.758603	+	0.651553i	$0.774118\pi$
$158$	0	0
$159$	−16.5276	−1.31073
$160$	0	0
$161$	−35.4245	−2.79184
$162$	0	0
$163$	−14.3754	−1.12597	−0.562984	−	0.826468i	$-0.690347\pi$
$163$	−14.3754	−1.12597	−0.562984	+	0.826468i	$0.690347\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0.460864i	0.0356627i	0.999841	+	0.0178314i	$0.00567620\pi$
$167$	0.460864i	0.0356627i	−0.999841	+	0.0178314i	$0.994324\pi$
$168$	0	0
$169$	8.12272	0.624824
$170$	0	0
$171$	− 0.108867i	− 0.00832529i
$172$	0	0
$173$	−2.96448	−0.225385	−0.112693	−	0.993630i	$-0.535948\pi$
$173$	−2.96448	−0.225385	−0.112693	+	0.993630i	$0.535948\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 10.8584i	− 0.816166i
$178$	0	0
$179$	15.5047i	1.15888i	0.815016	+	0.579438i	$0.196728\pi$
$179$	15.5047i	1.15888i	−0.815016	+	0.579438i	$0.803272\pi$
$180$	0	0
$181$	− 6.57069i	− 0.488395i	−0.969725	−	0.244198i	$-0.921475\pi$
$181$	− 6.57069i	− 0.488395i	0.969725	−	0.244198i	$-0.0785246\pi$
$182$	0	0
$183$	1.07838i	0.0797161i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	1.71697	0.125557
$188$	0	0
$189$	− 23.3027i	− 1.69502i
$190$	0	0
$191$	−1.14661	−0.0829657	−0.0414829	−	0.999139i	$-0.513208\pi$
$191$	−1.14661	−0.0829657	−0.0414829	+	0.999139i	$0.513208\pi$
$192$	0	0
$193$	− 15.5981i	− 1.12278i	−0.827552	−	0.561389i	$-0.810267\pi$
$193$	− 15.5981i	− 1.12278i	0.827552	−	0.561389i	$-0.189733\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	17.3637	1.23711	0.618557	−	0.785740i	$-0.287718\pi$
$197$	17.3637	1.23711	0.618557	+	0.785740i	$0.287718\pi$
$198$	0	0
$199$	−9.00089	−0.638056	−0.319028	−	0.947745i	$-0.603356\pi$
$199$	−9.00089	−0.638056	−0.319028	+	0.947745i	$0.603356\pi$
$200$	0	0
$201$	2.94630	0.207816
$202$	0	0
$203$	7.84127	0.550349
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 0.974071i	− 0.0677026i
$208$	0	0
$209$	0.647944	0.0448192
$210$	0	0
$211$	17.9912i	1.23856i	0.785169	+	0.619281i	$0.212576\pi$
$211$	17.9912i	1.23856i	−0.785169	+	0.619281i	$0.787424\pi$
$212$	0	0
$213$	14.2722	0.977918
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 39.9502i	− 2.71200i
$218$	0	0
$219$	− 22.1886i	− 1.49936i
$220$	0	0
$221$	10.9576i	0.737090i
$222$	0	0
$223$	6.79501i	0.455028i	0.973775	+	0.227514i	$0.0730597\pi$
$223$	6.79501i	0.455028i	−0.973775	+	0.227514i	$0.926940\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	26.0338	1.72792	0.863962	−	0.503557i	$-0.167976\pi$
$227$	26.0338	1.72792	0.863962	+	0.503557i	$0.167976\pi$
$228$	0	0
$229$	13.9303i	0.920539i	0.887779	+	0.460269i	$0.152247\pi$
$229$	13.9303i	0.920539i	−0.887779	+	0.460269i	$0.847753\pi$
$230$	0	0
$231$	5.37691	0.353774
$232$	0	0
$233$	12.5940i	0.825059i	0.910944	+	0.412529i	$0.135355\pi$
$233$	12.5940i	0.825059i	−0.910944	+	0.412529i	$0.864645\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	12.2138	0.793369
$238$	0	0
$239$	9.25499	0.598656	0.299328	−	0.954150i	$-0.403238\pi$
$239$	9.25499	0.598656	0.299328	+	0.954150i	$0.403238\pi$
$240$	0	0
$241$	−11.8958	−0.766278	−0.383139	−	0.923691i	$-0.625157\pi$
$241$	−11.8958	−0.766278	−0.383139	+	0.923691i	$0.625157\pi$
$242$	0	0
$243$	1.25667	0.0806157
$244$	0	0
$245$	0	0
$246$	0	0
$247$	4.13517i	0.263114i
$248$	0	0
$249$	−0.372691	−0.0236183
$250$	0	0
$251$	18.4410i	1.16399i	0.813193	+	0.581993i	$0.197727\pi$
$251$	18.4410i	1.16399i	−0.813193	+	0.581993i	$0.802273\pi$
$252$	0	0
$253$	5.79737	0.364477
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 4.04100i	− 0.252071i	−0.992026	−	0.126035i	$-0.959775\pi$
$257$	− 4.04100i	− 0.252071i	0.992026	−	0.126035i	$-0.0402253\pi$
$258$	0	0
$259$	− 20.8800i	− 1.29742i
$260$	0	0
$261$	0.215612i	0.0133461i
$262$	0	0
$263$	− 24.1904i	− 1.49164i	−0.666146	−	0.745821i	$-0.732057\pi$
$263$	− 24.1904i	− 1.49164i	0.666146	−	0.745821i	$-0.267943\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−2.33018	−0.142605
$268$	0	0
$269$	7.29938i	0.445051i	0.974927	+	0.222526i	$0.0714301\pi$
$269$	7.29938i	0.445051i	−0.974927	+	0.222526i	$0.928570\pi$
$270$	0	0
$271$	−7.82918	−0.475589	−0.237794	−	0.971316i	$-0.576424\pi$
$271$	−7.82918	−0.475589	−0.237794	+	0.971316i	$0.576424\pi$
$272$	0	0
$273$	34.3153i	2.07686i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	4.34266	0.260925	0.130463	−	0.991453i	$-0.458354\pi$
$277$	4.34266	0.260925	0.130463	+	0.991453i	$0.458354\pi$
$278$	0	0
$279$	1.09852	0.0657664
$280$	0	0
$281$	9.84024	0.587020	0.293510	−	0.955956i	$-0.405177\pi$
$281$	9.84024	0.587020	0.293510	+	0.955956i	$0.405177\pi$
$282$	0	0
$283$	24.8597	1.47776	0.738879	−	0.673839i	$-0.235356\pi$
$283$	24.8597	1.47776	0.738879	+	0.673839i	$0.235356\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 12.4426i	− 0.734462i
$288$	0	0
$289$	11.3156	0.665624
$290$	0	0
$291$	14.2838i	0.837330i
$292$	0	0
$293$	21.1433	1.23521	0.617603	−	0.786490i	$-0.288104\pi$
$293$	21.1433	1.23521	0.617603	+	0.786490i	$0.288104\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	3.81359i	0.221287i
$298$	0	0
$299$	36.9987i	2.13969i
$300$	0	0
$301$	47.4229i	2.73341i
$302$	0	0
$303$	29.3507i	1.68615i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	6.81511	0.388959	0.194479	−	0.980907i	$-0.437698\pi$
$307$	6.81511	0.388959	0.194479	+	0.980907i	$0.437698\pi$
$308$	0	0
$309$	15.9609i	0.907983i
$310$	0	0
$311$	3.23527	0.183455	0.0917276	−	0.995784i	$-0.470761\pi$
$311$	3.23527	0.183455	0.0917276	+	0.995784i	$0.470761\pi$
$312$	0	0
$313$	− 8.18203i	− 0.462476i	−0.972897	−	0.231238i	$-0.925722\pi$
$313$	− 8.18203i	− 0.462476i	0.972897	−	0.231238i	$-0.0742776\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	26.3642	1.48076	0.740380	−	0.672189i	$-0.234646\pi$
$317$	26.3642	1.48076	0.740380	+	0.672189i	$0.234646\pi$
$318$	0	0
$319$	−1.28326	−0.0718486
$320$	0	0
$321$	15.7790	0.880699
$322$	0	0
$323$	−2.14516	−0.119360
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 32.3080i	− 1.78664i
$328$	0	0
$329$	−21.9693	−1.21120
$330$	0	0
$331$	− 30.4387i	− 1.67306i	−0.547920	−	0.836531i	$-0.684580\pi$
$331$	− 30.4387i	− 1.67306i	0.547920	−	0.836531i	$-0.315420\pi$
$332$	0	0
$333$	0.574139	0.0314626
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 29.5259i	− 1.60838i	−0.594373	−	0.804190i	$-0.702600\pi$
$337$	− 29.5259i	− 1.60838i	0.594373	−	0.804190i	$-0.297400\pi$
$338$	0	0
$339$	− 11.2727i	− 0.612248i
$340$	0	0
$341$	6.53802i	0.354054i
$342$	0	0
$343$	− 23.6015i	− 1.27436i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−5.89868	−0.316658	−0.158329	−	0.987386i	$-0.550611\pi$
$347$	−5.89868	−0.316658	−0.158329	+	0.987386i	$0.550611\pi$
$348$	0	0
$349$	− 18.7429i	− 1.00329i	−0.865075	−	0.501643i	$-0.832729\pi$
$349$	− 18.7429i	− 1.00329i	0.865075	−	0.501643i	$-0.167271\pi$
$350$	0	0
$351$	−24.3383	−1.29908
$352$	0	0
$353$	25.1509i	1.33865i	0.742972	+	0.669323i	$0.233416\pi$
$353$	25.1509i	1.33865i	−0.742972	+	0.669323i	$0.766584\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−17.8014	−0.942151
$358$	0	0
$359$	1.12502	0.0593764	0.0296882	−	0.999559i	$-0.490549\pi$
$359$	1.12502	0.0593764	0.0296882	+	0.999559i	$0.490549\pi$
$360$	0	0
$361$	18.1905	0.957393
$362$	0	0
$363$	17.7844	0.933440
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 6.86079i	− 0.358130i	−0.983837	−	0.179065i	$-0.942693\pi$
$367$	− 6.86079i	− 0.358130i	0.983837	−	0.179065i	$-0.0573073\pi$
$368$	0	0
$369$	0.342135	0.0178108
$370$	0	0
$371$	− 42.8629i	− 2.22533i
$372$	0	0
$373$	7.89076	0.408568	0.204284	−	0.978912i	$-0.434513\pi$
$373$	7.89076	0.408568	0.204284	+	0.978912i	$0.434513\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 8.18972i	− 0.421792i
$378$	0	0
$379$	− 4.36955i	− 0.224449i	−0.993683	−	0.112224i	$-0.964202\pi$
$379$	− 4.36955i	− 0.224449i	0.993683	−	0.112224i	$-0.0357976\pi$
$380$	0	0
$381$	− 4.02913i	− 0.206419i
$382$	0	0
$383$	− 12.3406i	− 0.630574i	−0.948996	−	0.315287i	$-0.897899\pi$
$383$	− 12.3406i	− 0.630574i	0.948996	−	0.315287i	$-0.102101\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−1.30400	−0.0662858
$388$	0	0
$389$	− 18.1641i	− 0.920957i	−0.887671	−	0.460478i	$-0.847678\pi$
$389$	− 18.1641i	− 0.920957i	0.887671	−	0.460478i	$-0.152322\pi$
$390$	0	0
$391$	−19.1935	−0.970655
$392$	0	0
$393$	22.8204i	1.15114i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−19.0559	−0.956389	−0.478194	−	0.878254i	$-0.658709\pi$
$397$	−19.0559	−0.956389	−0.478194	+	0.878254i	$0.658709\pi$
$398$	0	0
$399$	−6.71786	−0.336314
$400$	0	0
$401$	−16.6574	−0.831833	−0.415917	−	0.909403i	$-0.636539\pi$
$401$	−16.6574	−0.831833	−0.415917	+	0.909403i	$0.636539\pi$
$402$	0	0
$403$	−41.7255	−2.07850
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3.41710i	0.169379i
$408$	0	0
$409$	25.0986	1.24105	0.620523	−	0.784188i	$-0.286920\pi$
$409$	25.0986	1.24105	0.620523	+	0.784188i	$0.286920\pi$
$410$	0	0
$411$	31.2211i	1.54002i
$412$	0	0
$413$	28.1602	1.38567
$414$	0	0
$415$	0	0
$416$	0	0
$417$	16.3370i	0.800026i
$418$	0	0
$419$	16.2968i	0.796149i	0.917353	+	0.398074i	$0.130321\pi$
$419$	16.2968i	0.796149i	−0.917353	+	0.398074i	$0.869679\pi$
$420$	0	0
$421$	− 13.3488i	− 0.650579i	−0.945614	−	0.325290i	$-0.894538\pi$
$421$	− 13.3488i	− 0.650579i	0.945614	−	0.325290i	$-0.105462\pi$
$422$	0	0
$423$	− 0.604092i	− 0.0293719i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−2.79668	−0.135341
$428$	0	0
$429$	− 5.61585i	− 0.271136i
$430$	0	0
$431$	26.1490	1.25955	0.629777	−	0.776776i	$-0.283146\pi$
$431$	26.1490	1.25955	0.629777	+	0.776776i	$0.283146\pi$
$432$	0	0
$433$	− 15.8230i	− 0.760403i	−0.924904	−	0.380201i	$-0.875855\pi$
$433$	− 15.8230i	− 0.760403i	0.924904	−	0.380201i	$-0.124145\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−7.24318	−0.346488
$438$	0	0
$439$	19.9204	0.950748	0.475374	−	0.879784i	$-0.342313\pi$
$439$	19.9204	0.950748	0.475374	+	0.879784i	$0.342313\pi$
$440$	0	0
$441$	1.49596	0.0712363
$442$	0	0
$443$	4.42692	0.210329	0.105165	−	0.994455i	$-0.466463\pi$
$443$	4.42692	0.210329	0.105165	+	0.994455i	$0.466463\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	21.8303i	1.03254i
$448$	0	0
$449$	−7.08837	−0.334521	−0.167260	−	0.985913i	$-0.553492\pi$
$449$	−7.08837	−0.334521	−0.167260	+	0.985913i	$0.553492\pi$
$450$	0	0
$451$	2.03628i	0.0958847i
$452$	0	0
$453$	−19.6652	−0.923951
$454$	0	0
$455$	0	0
$456$	0	0
$457$	27.7894i	1.29993i	0.759962	+	0.649967i	$0.225217\pi$
$457$	27.7894i	1.29993i	−0.759962	+	0.649967i	$0.774783\pi$
$458$	0	0
$459$	− 12.6257i	− 0.589319i
$460$	0	0
$461$	− 21.0565i	− 0.980701i	−0.871525	−	0.490350i	$-0.836869\pi$
$461$	− 21.0565i	− 0.980701i	0.871525	−	0.490350i	$-0.163131\pi$
$462$	0	0
$463$	− 30.0661i	− 1.39729i	−0.715468	−	0.698645i	$-0.753787\pi$
$463$	− 30.0661i	− 1.39729i	0.715468	−	0.698645i	$-0.246213\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−0.930499	−0.0430583	−0.0215292	−	0.999768i	$-0.506853\pi$
$467$	−0.930499	−0.0430583	−0.0215292	+	0.999768i	$0.506853\pi$
$468$	0	0
$469$	7.64096i	0.352827i
$470$	0	0
$471$	−32.2564	−1.48629
$472$	0	0
$473$	− 7.76097i	− 0.356850i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	1.17861	0.0539647
$478$	0	0
$479$	−12.3798	−0.565648	−0.282824	−	0.959172i	$-0.591271\pi$
$479$	−12.3798	−0.565648	−0.282824	+	0.959172i	$0.591271\pi$
$480$	0	0
$481$	−21.8079	−0.994353
$482$	0	0
$483$	−60.1069	−2.73496
$484$	0	0
$485$	0	0
$486$	0	0
$487$	30.0601i	1.36215i	0.732212	+	0.681076i	$0.238488\pi$
$487$	30.0601i	1.36215i	−0.732212	+	0.681076i	$0.761512\pi$
$488$	0	0
$489$	−24.3916	−1.10303
$490$	0	0
$491$	− 7.77369i	− 0.350822i	−0.984495	−	0.175411i	$-0.943875\pi$
$491$	− 7.77369i	− 0.350822i	0.984495	−	0.175411i	$-0.0561253\pi$
$492$	0	0
$493$	4.24851	0.191343
$494$	0	0
$495$	0	0
$496$	0	0
$497$	37.0138i	1.66029i
$498$	0	0
$499$	3.02886i	0.135590i	0.997699	+	0.0677952i	$0.0215964\pi$
$499$	3.02886i	0.135590i	−0.997699	+	0.0677952i	$0.978404\pi$
$500$	0	0
$501$	0.781976i	0.0349361i
$502$	0	0
$503$	− 18.5947i	− 0.829096i	−0.910027	−	0.414548i	$-0.863940\pi$
$503$	− 18.5947i	− 0.829096i	0.910027	−	0.414548i	$-0.136060\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	13.7823	0.612094
$508$	0	0
$509$	35.8471i	1.58889i	0.607334	+	0.794446i	$0.292239\pi$
$509$	35.8471i	1.58889i	−0.607334	+	0.794446i	$0.707761\pi$
$510$	0	0
$511$	57.5440	2.54560
$512$	0	0
$513$	− 4.76467i	− 0.210365i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	3.59537	0.158124
$518$	0	0
$519$	−5.03002	−0.220794
$520$	0	0
$521$	8.00169	0.350560	0.175280	−	0.984519i	$-0.443917\pi$
$521$	8.00169	0.350560	0.175280	+	0.984519i	$0.443917\pi$
$522$	0	0
$523$	−23.5586	−1.03015	−0.515073	−	0.857146i	$-0.672235\pi$
$523$	−23.5586	−1.03015	−0.515073	+	0.857146i	$0.672235\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 21.6456i	− 0.942895i
$528$	0	0
$529$	−41.8071	−1.81770
$530$	0	0
$531$	0.774326i	0.0336029i
$532$	0	0
$533$	−12.9955	−0.562898
$534$	0	0
$535$	0	0
$536$	0	0
$537$	26.3078i	1.13527i
$538$	0	0
$539$	8.90350i	0.383501i
$540$	0	0
$541$	− 14.5520i	− 0.625638i	−0.949813	−	0.312819i	$-0.898727\pi$
$541$	− 14.5520i	− 0.625638i	0.949813	−	0.312819i	$-0.101273\pi$
$542$	0	0
$543$	− 11.1489i	− 0.478445i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−7.00697	−0.299597	−0.149798	−	0.988717i	$-0.547862\pi$
$547$	−7.00697	−0.299597	−0.149798	+	0.988717i	$0.547862\pi$
$548$	0	0
$549$	− 0.0769007i	− 0.00328204i
$550$	0	0
$551$	1.60329	0.0683025
$552$	0	0
$553$	31.6753i	1.34697i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	30.8696	1.30799	0.653993	−	0.756501i	$-0.273093\pi$
$557$	30.8696	1.30799	0.653993	+	0.756501i	$0.273093\pi$
$558$	0	0
$559$	49.5304	2.09491
$560$	0	0
$561$	2.91328	0.122999
$562$	0	0
$563$	−19.5281	−0.823011	−0.411505	−	0.911407i	$-0.634997\pi$
$563$	−19.5281	−0.823011	−0.411505	+	0.911407i	$0.634997\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 37.9418i	− 1.59341i
$568$	0	0
$569$	2.43721	0.102173	0.0510866	−	0.998694i	$-0.483732\pi$
$569$	2.43721	0.102173	0.0510866	+	0.998694i	$0.483732\pi$
$570$	0	0
$571$	3.60054i	0.150678i	0.997158	+	0.0753389i	$0.0240039\pi$
$571$	3.60054i	0.150678i	−0.997158	+	0.0753389i	$0.975996\pi$
$572$	0	0
$573$	−1.94552	−0.0812754
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 22.5193i	− 0.937490i	−0.883334	−	0.468745i	$-0.844706\pi$
$577$	− 22.5193i	− 0.937490i	0.883334	−	0.468745i	$-0.155294\pi$
$578$	0	0
$579$	− 26.4663i	− 1.09990i
$580$	0	0
$581$	− 0.966540i	− 0.0400988i
$582$	0	0
$583$	7.01470i	0.290519i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−10.2171	−0.421703	−0.210851	−	0.977518i	$-0.567624\pi$
$587$	−10.2171	−0.421703	−0.210851	+	0.977518i	$0.567624\pi$
$588$	0	0
$589$	− 8.16855i	− 0.336579i
$590$	0	0
$591$	29.4621	1.21191
$592$	0	0
$593$	− 45.8240i	− 1.88177i	−0.338733	−	0.940883i	$-0.609998\pi$
$593$	− 45.8240i	− 1.88177i	0.338733	−	0.940883i	$-0.390002\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−15.2724	−0.625056
$598$	0	0
$599$	46.4420	1.89757	0.948785	−	0.315923i	$-0.102314\pi$
$599$	46.4420	1.89757	0.948785	+	0.315923i	$0.102314\pi$
$600$	0	0
$601$	27.7642	1.13253	0.566263	−	0.824225i	$-0.308389\pi$
$601$	27.7642	1.13253	0.566263	+	0.824225i	$0.308389\pi$
$602$	0	0
$603$	−0.210105	−0.00855612
$604$	0	0
$605$	0	0
$606$	0	0
$607$	15.2919i	0.620681i	0.950625	+	0.310340i	$0.100443\pi$
$607$	15.2919i	0.620681i	−0.950625	+	0.310340i	$0.899557\pi$
$608$	0	0
$609$	13.3048	0.539136
$610$	0	0
$611$	22.9455i	0.928277i
$612$	0	0
$613$	−19.6201	−0.792448	−0.396224	−	0.918154i	$-0.629680\pi$
$613$	−19.6201	−0.792448	−0.396224	+	0.918154i	$0.629680\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 11.2105i	− 0.451316i	−0.974207	−	0.225658i	$-0.927547\pi$
$617$	− 11.2105i	− 0.451316i	0.974207	−	0.225658i	$-0.0724532\pi$
$618$	0	0
$619$	− 18.3211i	− 0.736388i	−0.929749	−	0.368194i	$-0.879976\pi$
$619$	− 18.3211i	− 0.736388i	0.929749	−	0.368194i	$-0.120024\pi$
$620$	0	0
$621$	− 42.6310i	− 1.71072i
$622$	0	0
$623$	− 6.04311i	− 0.242112i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	1.09941	0.0439061
$628$	0	0
$629$	− 11.3131i	− 0.451081i
$630$	0	0
$631$	7.54155	0.300225	0.150112	−	0.988669i	$-0.452036\pi$
$631$	7.54155	0.300225	0.150112	+	0.988669i	$0.452036\pi$
$632$	0	0
$633$	30.5267i	1.21333i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−56.8220	−2.25137
$638$	0	0
$639$	−1.01777	−0.0402625
$640$	0	0
$641$	−19.5415	−0.771843	−0.385921	−	0.922532i	$-0.626116\pi$
$641$	−19.5415	−0.771843	−0.385921	+	0.922532i	$0.626116\pi$
$642$	0	0
$643$	17.4075	0.686484	0.343242	−	0.939247i	$-0.388475\pi$
$643$	17.4075	0.686484	0.343242	+	0.939247i	$0.388475\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	43.0753i	1.69346i	0.532020	+	0.846732i	$0.321433\pi$
$647$	43.0753i	1.69346i	−0.532020	+	0.846732i	$0.678567\pi$
$648$	0	0
$649$	−4.60855	−0.180901
$650$	0	0
$651$	− 67.7860i	− 2.65674i
$652$	0	0
$653$	−6.98379	−0.273297	−0.136648	−	0.990620i	$-0.543633\pi$
$653$	−6.98379	−0.273297	−0.136648	+	0.990620i	$0.543633\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	1.58230i	0.0617312i
$658$	0	0
$659$	− 5.78732i	− 0.225442i	−0.993627	−	0.112721i	$-0.964043\pi$
$659$	− 5.78732i	− 0.225442i	0.993627	−	0.112721i	$-0.0359566\pi$
$660$	0	0
$661$	− 29.6124i	− 1.15179i	−0.817523	−	0.575895i	$-0.804654\pi$
$661$	− 29.6124i	− 1.15179i	0.817523	−	0.575895i	$-0.195346\pi$
$662$	0	0
$663$	18.5925i	0.722073i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	14.3452	0.555447
$668$	0	0
$669$	11.5295i	0.445757i
$670$	0	0
$671$	0.457689	0.0176689
$672$	0	0
$673$	39.5316i	1.52383i	0.647677	+	0.761915i	$0.275741\pi$
$673$	39.5316i	1.52383i	−0.647677	+	0.761915i	$0.724259\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	13.4716	0.517757	0.258879	−	0.965910i	$-0.416647\pi$
$677$	13.4716	0.517757	0.258879	+	0.965910i	$0.416647\pi$
$678$	0	0
$679$	−37.0437	−1.42161
$680$	0	0
$681$	44.1732	1.69272
$682$	0	0
$683$	41.8857	1.60271	0.801356	−	0.598187i	$-0.204112\pi$
$683$	41.8857	1.60271	0.801356	+	0.598187i	$0.204112\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	23.6364i	0.901784i
$688$	0	0
$689$	−44.7677	−1.70551
$690$	0	0
$691$	− 34.8109i	− 1.32427i	−0.749386	−	0.662134i	$-0.769651\pi$
$691$	− 34.8109i	− 1.32427i	0.749386	−	0.662134i	$-0.230349\pi$
$692$	0	0
$693$	−0.383434	−0.0145655
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 6.74156i	− 0.255355i
$698$	0	0
$699$	21.3690i	0.808249i
$700$	0	0
$701$	− 42.3521i	− 1.59962i	−0.600257	−	0.799808i	$-0.704935\pi$
$701$	− 42.3521i	− 1.59962i	0.600257	−	0.799808i	$-0.295065\pi$
$702$	0	0
$703$	− 4.26930i	− 0.161020i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−76.1183	−2.86272
$708$	0	0
$709$	− 12.6239i	− 0.474102i	−0.971497	−	0.237051i	$-0.923819\pi$
$709$	− 12.6239i	− 0.474102i	0.971497	−	0.237051i	$-0.0761809\pi$
$710$	0	0
$711$	−0.870980	−0.0326643
$712$	0	0
$713$	− 73.0867i	− 2.73712i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	15.7035	0.586459
$718$	0	0
$719$	−21.8304	−0.814136	−0.407068	−	0.913398i	$-0.633449\pi$
$719$	−21.8304	−0.814136	−0.407068	+	0.913398i	$0.633449\pi$
$720$	0	0
$721$	−41.3931	−1.54156
$722$	0	0
$723$	−20.1844	−0.750666
$724$	0	0
$725$	0	0
$726$	0	0
$727$	32.2227i	1.19507i	0.801841	+	0.597537i	$0.203854\pi$
$727$	32.2227i	1.19507i	−0.801841	+	0.597537i	$0.796146\pi$
$728$	0	0
$729$	27.9994	1.03701
$730$	0	0
$731$	25.6944i	0.950342i
$732$	0	0
$733$	22.6294	0.835837	0.417918	−	0.908485i	$-0.362760\pi$
$733$	22.6294	0.835837	0.417918	+	0.908485i	$0.362760\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 1.25048i	− 0.0460619i
$738$	0	0
$739$	26.3553i	0.969494i	0.874654	+	0.484747i	$0.161088\pi$
$739$	26.3553i	0.969494i	−0.874654	+	0.484747i	$0.838912\pi$
$740$	0	0
$741$	7.01639i	0.257754i
$742$	0	0
$743$	− 0.548881i	− 0.0201365i	−0.999949	−	0.0100682i	$-0.996795\pi$
$743$	− 0.548881i	− 0.0201365i	0.999949	−	0.0100682i	$-0.00320487\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0.0265771	0.000972404 0
$748$	0	0
$749$	40.9214i	1.49524i
$750$	0	0
$751$	−22.2625	−0.812371	−0.406185	−	0.913791i	$-0.633141\pi$
$751$	−22.2625	−0.812371	−0.406185	+	0.913791i	$0.633141\pi$
$752$	0	0
$753$	31.2900i	1.14027i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	8.81385	0.320345	0.160172	−	0.987089i	$-0.448795\pi$
$757$	8.81385	0.320345	0.160172	+	0.987089i	$0.448795\pi$
$758$	0	0
$759$	9.83675	0.357052
$760$	0	0
$761$	4.81932	0.174700	0.0873500	−	0.996178i	$-0.472160\pi$
$761$	4.81932	0.174700	0.0873500	+	0.996178i	$0.472160\pi$
$762$	0	0
$763$	83.7879	3.03332
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 29.4116i	− 1.06199i
$768$	0	0
$769$	28.5713	1.03031	0.515153	−	0.857098i	$-0.327735\pi$
$769$	28.5713	1.03031	0.515153	+	0.857098i	$0.327735\pi$
$770$	0	0
$771$	− 6.85661i	− 0.246935i
$772$	0	0
$773$	9.95026	0.357886	0.178943	−	0.983859i	$-0.442732\pi$
$773$	9.95026	0.357886	0.178943	+	0.983859i	$0.442732\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 35.4283i	− 1.27098i
$778$	0	0
$779$	− 2.54411i	− 0.0911523i
$780$	0	0
$781$	− 6.05746i	− 0.216753i
$782$	0	0
$783$	9.43645i	0.337231i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	49.1601	1.75237	0.876185	−	0.481975i	$-0.160081\pi$
$787$	49.1601	1.75237	0.876185	+	0.481975i	$0.160081\pi$
$788$	0	0
$789$	− 41.0453i	− 1.46125i
$790$	0	0
$791$	29.2347	1.03947
$792$	0	0
$793$	2.92096i	0.103726i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−45.6344	−1.61645	−0.808227	−	0.588871i	$-0.799573\pi$
$797$	−45.6344	−1.61645	−0.808227	+	0.588871i	$0.799573\pi$
$798$	0	0
$799$	−11.9032	−0.421107
$800$	0	0
$801$	0.166168	0.00587127
$802$	0	0
$803$	−9.41733	−0.332330
$804$	0	0
$805$	0	0
$806$	0	0
$807$	12.3853i	0.435984i
$808$	0	0
$809$	13.2646	0.466360	0.233180	−	0.972434i	$-0.425087\pi$
$809$	13.2646	0.466360	0.233180	+	0.972434i	$0.425087\pi$
$810$	0	0
$811$	− 31.0646i	− 1.09083i	−0.838167	−	0.545413i	$-0.816373\pi$
$811$	− 31.0646i	− 1.09083i	0.838167	−	0.545413i	$-0.183627\pi$
$812$	0	0
$813$	−13.2843	−0.465899
$814$	0	0
$815$	0	0
$816$	0	0
$817$	9.69649i	0.339237i
$818$	0	0
$819$	− 2.44707i	− 0.0855075i
$820$	0	0
$821$	16.9551i	0.591738i	0.955229	+	0.295869i	$0.0956092\pi$
$821$	16.9551i	0.591738i	−0.955229	+	0.295869i	$0.904391\pi$
$822$	0	0
$823$	35.5910i	1.24063i	0.784354	+	0.620313i	$0.212994\pi$
$823$	35.5910i	1.24063i	−0.784354	+	0.620313i	$0.787006\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−11.4594	−0.398481	−0.199240	−	0.979951i	$-0.563847\pi$
$827$	−11.4594	−0.398481	−0.199240	+	0.979951i	$0.563847\pi$
$828$	0	0
$829$	− 3.98218i	− 0.138307i	−0.997606	−	0.0691534i	$-0.977970\pi$
$829$	− 3.98218i	− 0.138307i	0.997606	−	0.0691534i	$-0.0220298\pi$
$830$	0	0
$831$	7.36847	0.255609
$832$	0	0
$833$	− 29.4770i	− 1.02132i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	48.0774	1.66180
$838$	0	0
$839$	−54.2269	−1.87212	−0.936060	−	0.351840i	$-0.885556\pi$
$839$	−54.2269	−1.87212	−0.936060	+	0.351840i	$0.885556\pi$
$840$	0	0
$841$	25.8247	0.890506
$842$	0	0
$843$	16.6966	0.575060
$844$	0	0
$845$	0	0
$846$	0	0
$847$	46.1223i	1.58478i
$848$	0	0
$849$	42.1810	1.44765
$850$	0	0
$851$	− 38.1988i	− 1.30944i
$852$	0	0
$853$	28.9671	0.991814	0.495907	−	0.868376i	$-0.334836\pi$
$853$	28.9671	0.991814	0.495907	+	0.868376i	$0.334836\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	49.1740i	1.67975i	0.542779	+	0.839876i	$0.317372\pi$
$857$	49.1740i	1.67975i	−0.542779	+	0.839876i	$0.682628\pi$
$858$	0	0
$859$	− 11.8879i	− 0.405610i	−0.979219	−	0.202805i	$-0.934994\pi$
$859$	− 11.8879i	− 0.405610i	0.979219	−	0.202805i	$-0.0650057\pi$
$860$	0	0
$861$	− 21.1121i	− 0.719498i
$862$	0	0
$863$	2.10569i	0.0716784i	0.999358	+	0.0358392i	$0.0114104\pi$
$863$	2.10569i	0.0716784i	−0.999358	+	0.0358392i	$0.988590\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	19.1999	0.652063
$868$	0	0
$869$	− 5.18380i	− 0.175848i
$870$	0	0
$871$	7.98052	0.270409
$872$	0	0
$873$	− 1.01859i	− 0.0344742i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	35.4248	1.19621	0.598105	−	0.801418i	$-0.295921\pi$
$877$	35.4248	1.19621	0.598105	+	0.801418i	$0.295921\pi$
$878$	0	0
$879$	35.8752	1.21004
$880$	0	0
$881$	35.4798	1.19534	0.597672	−	0.801741i	$-0.296093\pi$
$881$	35.4798	1.19534	0.597672	+	0.801741i	$0.296093\pi$
$882$	0	0
$883$	−3.31152	−0.111442	−0.0557208	−	0.998446i	$-0.517746\pi$
$883$	−3.31152	−0.111442	−0.0557208	+	0.998446i	$0.517746\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 9.76373i	− 0.327834i	−0.986474	−	0.163917i	$-0.947587\pi$
$887$	− 9.76373i	− 0.327834i	0.986474	−	0.163917i	$-0.0524129\pi$
$888$	0	0
$889$	10.4492	0.350454
$890$	0	0
$891$	6.20935i	0.208021i
$892$	0	0
$893$	−4.49202	−0.150320
$894$	0	0
$895$	0	0
$896$	0	0
$897$	62.7779i	2.09609i
$898$	0	0
$899$	16.1779i	0.539562i
$900$	0	0
$901$	− 23.2237i	− 0.773694i
$902$	0	0
$903$	80.4655i	2.67772i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−23.9575	−0.795495	−0.397747	−	0.917495i	$-0.630208\pi$
$907$	−23.9575	−0.795495	−0.397747	+	0.917495i	$0.630208\pi$
$908$	0	0
$909$	− 2.09304i	− 0.0694216i
$910$	0	0
$911$	−8.70638	−0.288455	−0.144228	−	0.989545i	$-0.546070\pi$
$911$	−8.70638	−0.288455	−0.144228	+	0.989545i	$0.546070\pi$
$912$	0	0
$913$	0.158178i	0.00523494i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−59.1825	−1.95438
$918$	0	0
$919$	−28.2017	−0.930289	−0.465144	−	0.885235i	$-0.653998\pi$
$919$	−28.2017	−0.930289	−0.465144	+	0.885235i	$0.653998\pi$
$920$	0	0
$921$	11.5636	0.381034
$922$	0	0
$923$	38.6586	1.27246
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 1.13819i	− 0.0373831i
$928$	0	0
$929$	12.8660	0.422119	0.211059	−	0.977473i	$-0.432309\pi$
$929$	12.8660	0.422119	0.211059	+	0.977473i	$0.432309\pi$
$930$	0	0
$931$	− 11.1240i	− 0.364573i
$932$	0	0
$933$	5.48948	0.179718
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 8.91407i	− 0.291210i	−0.989343	−	0.145605i	$-0.953487\pi$
$937$	− 8.91407i	− 0.291210i	0.989343	−	0.145605i	$-0.0465129\pi$
$938$	0	0
$939$	− 13.8830i	− 0.453053i
$940$	0	0
$941$	− 29.6233i	− 0.965691i	−0.875706	−	0.482845i	$-0.839603\pi$
$941$	− 29.6233i	− 0.965691i	0.875706	−	0.482845i	$-0.160397\pi$
$942$	0	0
$943$	− 22.7630i	− 0.741265i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−0.0952553	−0.00309538	−0.00154769	−	0.999999i	$-0.500493\pi$
$947$	−0.0952553	−0.00309538	−0.00154769	+	0.999999i	$0.500493\pi$
$948$	0	0
$949$	− 60.1012i	− 1.95097i
$950$	0	0
$951$	44.7337	1.45059
$952$	0	0
$953$	34.3248i	1.11189i	0.831219	+	0.555945i	$0.187643\pi$
$953$	34.3248i	1.11189i	−0.831219	+	0.555945i	$0.812357\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−2.17738	−0.0703848
$958$	0	0
$959$	−80.9690	−2.61463
$960$	0	0
$961$	51.4240	1.65884
$962$	0	0
$963$	−1.12522	−0.0362598
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 7.19999i	− 0.231536i	−0.993276	−	0.115768i	$-0.963067\pi$
$967$	− 7.19999i	− 0.231536i	0.993276	−	0.115768i	$-0.0369329\pi$
$968$	0	0
$969$	−3.63983	−0.116928
$970$	0	0
$971$	− 53.7682i	− 1.72550i	−0.505627	−	0.862752i	$-0.668739\pi$
$971$	− 53.7682i	− 1.72550i	0.505627	−	0.862752i	$-0.331261\pi$
$972$	0	0
$973$	−42.3685	−1.35827
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 10.4880i	− 0.335541i	−0.985826	−	0.167770i	$-0.946343\pi$
$977$	− 10.4880i	− 0.335541i	0.985826	−	0.167770i	$-0.0536567\pi$
$978$	0	0
$979$	0.988982i	0.0316080i
$980$	0	0
$981$	2.30393i	0.0735587i
$982$	0	0
$983$	22.0512i	0.703324i	0.936127	+	0.351662i	$0.114383\pi$
$983$	22.0512i	0.703324i	−0.936127	+	0.351662i	$0.885617\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−37.2766	−1.18653
$988$	0	0
$989$	86.7577i	2.75873i
$990$	0	0
$991$	−4.31045	−0.136926	−0.0684630	−	0.997654i	$-0.521810\pi$
$991$	−4.31045	−0.136926	−0.0684630	+	0.997654i	$0.521810\pi$
$992$	0	0
$993$	− 51.6472i	− 1.63897i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−12.3284	−0.390444	−0.195222	−	0.980759i	$-0.562543\pi$
$997$	−12.3284	−0.390444	−0.195222	+	0.980759i	$0.562543\pi$
$998$	0	0
$999$	25.1277	0.795005

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4000.2.f.c.3249.16		20
4.3	odd	2		1000.2.f.c.749.7		20
5.2	odd	4		4000.2.d.c.2001.39		40
5.3	odd	4		4000.2.d.c.2001.2		40
5.4	even	2		4000.2.f.d.3249.5		20
8.3	odd	2		1000.2.f.d.749.13		20
8.5	even	2		4000.2.f.d.3249.6		20
20.3	even	4		1000.2.d.c.501.8	yes	40
20.7	even	4		1000.2.d.c.501.33	yes	40
20.19	odd	2		1000.2.f.d.749.14		20
40.3	even	4		1000.2.d.c.501.7	✓	40
40.13	odd	4		4000.2.d.c.2001.1		40
40.19	odd	2		1000.2.f.c.749.8		20
40.27	even	4		1000.2.d.c.501.34	yes	40
40.29	even	2	inner	4000.2.f.c.3249.15		20
40.37	odd	4		4000.2.d.c.2001.40		40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1000.2.d.c.501.7	✓	40	40.3	even	4
1000.2.d.c.501.8	yes	40	20.3	even	4
1000.2.d.c.501.33	yes	40	20.7	even	4
1000.2.d.c.501.34	yes	40	40.27	even	4
1000.2.f.c.749.7		20	4.3	odd	2
1000.2.f.c.749.8		20	40.19	odd	2
1000.2.f.d.749.13		20	8.3	odd	2
1000.2.f.d.749.14		20	20.19	odd	2
4000.2.d.c.2001.1		40	40.13	odd	4
4000.2.d.c.2001.2		40	5.3	odd	4
4000.2.d.c.2001.39		40	5.2	odd	4
4000.2.d.c.2001.40		40	40.37	odd	4
4000.2.f.c.3249.15		20	40.29	even	2	inner
4000.2.f.c.3249.16		20	1.1	even	1	trivial
4000.2.f.d.3249.5		20	5.4	even	2
4000.2.f.d.3249.6		20	8.5	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:	\( N \)	\(=\)	\( 4000 = 2^{5} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4000.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.69676 q^{3} +4.40040i q^{7} -0.120998 q^{9} +O(q^{10})\)	\(q+1.69676 q^{3} +4.40040i q^{7} -0.120998 q^{9} -0.720144i q^{11} +4.59595 q^{13} +2.38420i q^{17} +0.899742i q^{19} +7.46643i q^{21} +8.05029i q^{23} -5.29559 q^{27} -1.78194i q^{29} -9.07877 q^{31} -1.22191i q^{33} -4.74502 q^{37} +7.79823 q^{39} -2.82760 q^{41} +10.7770 q^{43} +4.99256i q^{47} -12.3635 q^{49} +4.04541i q^{51} -9.74069 q^{53} +1.52665i q^{57} -6.39947i q^{59} +0.635551i q^{61} -0.532441i q^{63} +1.73643 q^{67} +13.6594i q^{69} +8.41146 q^{71} -13.0770i q^{73} +3.16892 q^{77} +7.19828 q^{79} -8.62236 q^{81} -0.219648 q^{83} -3.02354i q^{87} -1.37331 q^{89} +20.2240i q^{91} -15.4045 q^{93} +8.41826i q^{97} +0.0871363i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 4 q^{3} + 12 q^{9}+O(q^{10}) \) 20 * q - 4 * q^3 + 12 * q^9	\( 20 q - 4 q^{3} + 12 q^{9} + 6 q^{13} + 8 q^{27} - 24 q^{31} + 18 q^{37} + 4 q^{39} + 22 q^{41} + 60 q^{43} - 6 q^{49} + 10 q^{53} + 40 q^{67} - 48 q^{71} + 24 q^{77} + 48 q^{79} - 28 q^{81} - 40 q^{83} + 22 q^{89} + 8 q^{93}+O(q^{100}) \) 20 * q - 4 * q^3 + 12 * q^9 + 6 * q^13 + 8 * q^27 - 24 * q^31 + 18 * q^37 + 4 * q^39 + 22 * q^41 + 60 * q^43 - 6 * q^49 + 10 * q^53 + 40 * q^67 - 48 * q^71 + 24 * q^77 + 48 * q^79 - 28 * q^81 - 40 * q^83 + 22 * q^89 + 8 * q^93

Introduction

L-functions

Modular forms

Varieties

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Database

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