LMFDB - Embedded newform 2000.2.c.a.1249.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2000,2,Mod(1249,2000)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2000, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2000.1249");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2000 = 2^{4} \cdot 5^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2000.c (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:	$15.9700804043$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.12400.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 12x^{2} + 31 $ x^4 + 12*x^2 + 31
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 500)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.2
Root			$-2.86986i$ of defining polynomial
Character	$\chi$	$=$	2000.1249
Dual form			2000.2.c.a.1249.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.77367i q^{3} +4.64352i q^{7} -0.145898 q^{9} +O(q^{10})$	$q-1.77367i q^{3} +4.64352i q^{7} -0.145898 q^{9} -4.47214 q^{11} -5.73971i q^{13} +3.54734i q^{17} -1.23607 q^{19} +8.23607 q^{21} -1.09619i q^{23} -5.06223i q^{27} -4.61803 q^{29} -6.00000 q^{31} +7.93208i q^{33} -2.19237i q^{37} -10.1803 q^{39} -2.61803 q^{41} -3.96604i q^{43} -7.51338i q^{47} -14.5623 q^{49} +6.29180 q^{51} -11.4794i q^{53} +2.19237i q^{57} -12.4721 q^{59} +1.14590 q^{61} -0.677481i q^{63} +2.19237i q^{67} -1.94427 q^{69} -5.23607 q^{71} -5.73971i q^{73} -20.7665i q^{77} +9.70820 q^{79} -9.41641 q^{81} +6.83590i q^{83} +8.19086i q^{87} +11.3262 q^{89} +26.6525 q^{91} +10.6420i q^{93} +5.73971i q^{97} +0.652476 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 14 q^{9}+O(q^{10}) $ 4 * q - 14 * q^9	$ 4 q - 14 q^{9} + 4 q^{19} + 24 q^{21} - 14 q^{29} - 24 q^{31} + 4 q^{39} - 6 q^{41} - 18 q^{49} + 52 q^{51} - 32 q^{59} + 18 q^{61} + 28 q^{69} - 12 q^{71} + 12 q^{79} + 16 q^{81} + 14 q^{89} + 44 q^{91} - 60 q^{99}+O(q^{100}) $ 4 * q - 14 * q^9 + 4 * q^19 + 24 * q^21 - 14 * q^29 - 24 * q^31 + 4 * q^39 - 6 * q^41 - 18 * q^49 + 52 * q^51 - 32 * q^59 + 18 * q^61 + 28 * q^69 - 12 * q^71 + 12 * q^79 + 16 * q^81 + 14 * q^89 + 44 * q^91 - 60 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2000\mathbb{Z}\right)^\times$.

$n$	$501$	$751$	$1377$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	− 1.77367i	− 1.02403i	−0.858977	−	0.512014i	$-0.828900\pi$
$3$	− 1.77367i	− 1.02403i	0.858977	−	0.512014i	$-0.171100\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.64352i	1.75509i	0.479497	+	0.877543i	$0.340819\pi$
$7$	4.64352i	1.75509i	−0.479497	+	0.877543i	$0.659181\pi$
$8$	0	0
$9$	−0.145898	−0.0486327
$10$	0	0
$11$	−4.47214	−1.34840	−0.674200	−	0.738549i	$-0.735511\pi$
$11$	−4.47214	−1.34840	−0.674200	+	0.738549i	$0.735511\pi$
$12$	0	0
$13$	− 5.73971i	− 1.59191i	−0.605356	−	0.795955i	$-0.706969\pi$
$13$	− 5.73971i	− 1.59191i	0.605356	−	0.795955i	$-0.293031\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.54734i	0.860355i	0.902744	+	0.430178i	$0.141549\pi$
$17$	3.54734i	0.860355i	−0.902744	+	0.430178i	$0.858451\pi$
$18$	0	0
$19$	−1.23607	−0.283573	−0.141787	−	0.989897i	$-0.545285\pi$
$19$	−1.23607	−0.283573	−0.141787	+	0.989897i	$0.545285\pi$
$20$	0	0
$21$	8.23607	1.79726
$22$	0	0
$23$	− 1.09619i	− 0.228571i	−0.993448	−	0.114285i	$-0.963542\pi$
$23$	− 1.09619i	− 0.228571i	0.993448	−	0.114285i	$-0.0364578\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 5.06223i	− 0.974226i
$28$	0	0
$29$	−4.61803	−0.857547	−0.428774	−	0.903412i	$-0.641054\pi$
$29$	−4.61803	−0.857547	−0.428774	+	0.903412i	$0.641054\pi$
$30$	0	0
$31$	−6.00000	−1.07763	−0.538816	−	0.842424i	$-0.681128\pi$
$31$	−6.00000	−1.07763	−0.538816	+	0.842424i	$0.681128\pi$
$32$	0	0
$33$	7.93208i	1.38080i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 2.19237i	− 0.360424i	−0.983628	−	0.180212i	$-0.942322\pi$
$37$	− 2.19237i	− 0.360424i	0.983628	−	0.180212i	$-0.0576784\pi$
$38$	0	0
$39$	−10.1803	−1.63016
$40$	0	0
$41$	−2.61803	−0.408868	−0.204434	−	0.978880i	$-0.565535\pi$
$41$	−2.61803	−0.408868	−0.204434	+	0.978880i	$0.565535\pi$
$42$	0	0
$43$	− 3.96604i	− 0.604816i	−0.953179	−	0.302408i	$-0.902210\pi$
$43$	− 3.96604i	− 0.604816i	0.953179	−	0.302408i	$-0.0977905\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 7.51338i	− 1.09594i	−0.836498	−	0.547969i	$-0.815401\pi$
$47$	− 7.51338i	− 1.09594i	0.836498	−	0.547969i	$-0.184599\pi$
$48$	0	0
$49$	−14.5623	−2.08033
$50$	0	0
$51$	6.29180	0.881028
$52$	0	0
$53$	− 11.4794i	− 1.57682i	−0.615150	−	0.788410i	$-0.710905\pi$
$53$	− 11.4794i	− 1.57682i	0.615150	−	0.788410i	$-0.289095\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.19237i	0.290387i
$58$	0	0
$59$	−12.4721	−1.62373	−0.811867	−	0.583843i	$-0.801549\pi$
$59$	−12.4721	−1.62373	−0.811867	+	0.583843i	$0.801549\pi$
$60$	0	0
$61$	1.14590	0.146717	0.0733586	−	0.997306i	$-0.476628\pi$
$61$	1.14590	0.146717	0.0733586	+	0.997306i	$0.476628\pi$
$62$	0	0
$63$	− 0.677481i	− 0.0853546i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	2.19237i	0.267841i	0.990992	+	0.133921i	$0.0427567\pi$
$67$	2.19237i	0.267841i	−0.990992	+	0.133921i	$0.957243\pi$
$68$	0	0
$69$	−1.94427	−0.234063
$70$	0	0
$71$	−5.23607	−0.621407	−0.310703	−	0.950507i	$-0.600565\pi$
$71$	−5.23607	−0.621407	−0.310703	+	0.950507i	$0.600565\pi$
$72$	0	0
$73$	− 5.73971i	− 0.671782i	−0.941901	−	0.335891i	$-0.890963\pi$
$73$	− 5.73971i	− 0.671782i	0.941901	−	0.335891i	$-0.109037\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 20.7665i	− 2.36656i
$78$	0	0
$79$	9.70820	1.09226	0.546129	−	0.837701i	$-0.316101\pi$
$79$	9.70820	1.09226	0.546129	+	0.837701i	$0.316101\pi$
$80$	0	0
$81$	−9.41641	−1.04627
$82$	0	0
$83$	6.83590i	0.750337i	0.926957	+	0.375169i	$0.122415\pi$
$83$	6.83590i	0.750337i	−0.926957	+	0.375169i	$0.877585\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	8.19086i	0.878152i
$88$	0	0
$89$	11.3262	1.20058	0.600289	−	0.799783i	$-0.295052\pi$
$89$	11.3262	1.20058	0.600289	+	0.799783i	$0.295052\pi$
$90$	0	0
$91$	26.6525	2.79394
$92$	0	0
$93$	10.6420i	1.10352i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	5.73971i	0.582779i	0.956604	+	0.291390i	$0.0941176\pi$
$97$	5.73971i	0.582779i	−0.956604	+	0.291390i	$0.905882\pi$
$98$	0	0
$99$	0.652476	0.0655763
$100$	0	0
$101$	−19.3262	−1.92303	−0.961516	−	0.274748i	$-0.911406\pi$
$101$	−19.3262	−1.92303	−0.961516	+	0.274748i	$0.911406\pi$
$102$	0	0
$103$	− 9.28705i	− 0.915080i	−0.889189	−	0.457540i	$-0.848731\pi$
$103$	− 9.28705i	− 0.915080i	0.889189	−	0.457540i	$-0.151269\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0.418706i	0.0404779i	0.999795	+	0.0202389i	$0.00644269\pi$
$107$	0.418706i	0.0404779i	−0.999795	+	0.0202389i	$0.993557\pi$
$108$	0	0
$109$	−13.3820	−1.28176	−0.640880	−	0.767641i	$-0.721430\pi$
$109$	−13.3820	−1.28176	−0.640880	+	0.767641i	$0.721430\pi$
$110$	0	0
$111$	−3.88854	−0.369084
$112$	0	0
$113$	17.2191i	1.61984i	0.586541	+	0.809920i	$0.300489\pi$
$113$	17.2191i	1.61984i	−0.586541	+	0.809920i	$0.699511\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0.837412i	0.0774188i
$118$	0	0
$119$	−16.4721	−1.51000
$120$	0	0
$121$	9.00000	0.818182
$122$	0	0
$123$	4.64352i	0.418692i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 14.6080i	− 1.29625i	−0.761532	−	0.648127i	$-0.775552\pi$
$127$	− 14.6080i	− 1.29625i	0.761532	−	0.648127i	$-0.224448\pi$
$128$	0	0
$129$	−7.03444	−0.619348
$130$	0	0
$131$	7.23607	0.632218	0.316109	−	0.948723i	$-0.397623\pi$
$131$	7.23607	0.632218	0.316109	+	0.948723i	$0.397623\pi$
$132$	0	0
$133$	− 5.73971i	− 0.497696i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	15.0268i	1.28382i	0.766779	+	0.641911i	$0.221858\pi$
$137$	15.0268i	1.28382i	−0.766779	+	0.641911i	$0.778142\pi$
$138$	0	0
$139$	−2.29180	−0.194388	−0.0971938	−	0.995265i	$-0.530987\pi$
$139$	−2.29180	−0.194388	−0.0971938	+	0.995265i	$0.530987\pi$
$140$	0	0
$141$	−13.3262	−1.12227
$142$	0	0
$143$	25.6688i	2.14653i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	25.8287i	2.13031i
$148$	0	0
$149$	−9.32624	−0.764035	−0.382018	−	0.924155i	$-0.624771\pi$
$149$	−9.32624	−0.764035	−0.382018	+	0.924155i	$0.624771\pi$
$150$	0	0
$151$	1.70820	0.139012	0.0695058	−	0.997582i	$-0.477858\pi$
$151$	1.70820	0.139012	0.0695058	+	0.997582i	$0.477858\pi$
$152$	0	0
$153$	− 0.517549i	− 0.0418414i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	15.0268i	1.19927i	0.800275	+	0.599633i	$0.204687\pi$
$157$	15.0268i	1.19927i	−0.800275	+	0.599633i	$0.795313\pi$
$158$	0	0
$159$	−20.3607	−1.61471
$160$	0	0
$161$	5.09017	0.401162
$162$	0	0
$163$	15.2855i	1.19726i	0.801027	+	0.598628i	$0.204287\pi$
$163$	15.2855i	1.19726i	−0.801027	+	0.598628i	$0.795713\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.77367i	0.137251i	0.997643	+	0.0686253i	$0.0218613\pi$
$167$	1.77367i	0.137251i	−0.997643	+	0.0686253i	$0.978139\pi$
$168$	0	0
$169$	−19.9443	−1.53417
$170$	0	0
$171$	0.180340	0.0137909
$172$	0	0
$173$	22.1214i	1.68186i	0.541143	+	0.840931i	$0.317992\pi$
$173$	22.1214i	1.68186i	−0.541143	+	0.840931i	$0.682008\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	22.1214i	1.66275i
$178$	0	0
$179$	3.52786	0.263685	0.131842	−	0.991271i	$-0.457911\pi$
$179$	3.52786	0.263685	0.131842	+	0.991271i	$0.457911\pi$
$180$	0	0
$181$	2.43769	0.181192	0.0905962	−	0.995888i	$-0.471123\pi$
$181$	2.43769	0.181192	0.0905962	+	0.995888i	$0.471123\pi$
$182$	0	0
$183$	− 2.03244i	− 0.150242i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 15.8642i	− 1.16010i
$188$	0	0
$189$	23.5066	1.70985
$190$	0	0
$191$	−8.65248	−0.626071	−0.313036	−	0.949741i	$-0.601346\pi$
$191$	−8.65248	−0.626071	−0.313036	+	0.949741i	$0.601346\pi$
$192$	0	0
$193$	− 9.28705i	− 0.668496i	−0.942485	−	0.334248i	$-0.891518\pi$
$193$	− 9.28705i	− 0.668496i	0.942485	−	0.334248i	$-0.108482\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 10.6420i	− 0.758212i	−0.925353	−	0.379106i	$-0.876232\pi$
$197$	− 10.6420i	− 0.758212i	0.925353	−	0.379106i	$-0.123768\pi$
$198$	0	0
$199$	19.7082	1.39708	0.698539	−	0.715572i	$-0.253834\pi$
$199$	19.7082	1.39708	0.698539	+	0.715572i	$0.253834\pi$
$200$	0	0
$201$	3.88854	0.274277
$202$	0	0
$203$	− 21.4439i	− 1.50507i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0.159932i	0.0111160i
$208$	0	0
$209$	5.52786	0.382370
$210$	0	0
$211$	−2.29180	−0.157774	−0.0788869	−	0.996884i	$-0.525137\pi$
$211$	−2.29180	−0.157774	−0.0788869	+	0.996884i	$0.525137\pi$
$212$	0	0
$213$	9.28705i	0.636338i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 27.8611i	− 1.89134i
$218$	0	0
$219$	−10.1803	−0.687924
$220$	0	0
$221$	20.3607	1.36961
$222$	0	0
$223$	− 16.8004i	− 1.12504i	−0.826784	−	0.562520i	$-0.809832\pi$
$223$	− 16.8004i	− 1.12504i	0.826784	−	0.562520i	$-0.190168\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	6.83590i	0.453714i	0.973928	+	0.226857i	$0.0728451\pi$
$227$	6.83590i	0.453714i	−0.973928	+	0.226857i	$0.927155\pi$
$228$	0	0
$229$	−4.85410	−0.320768	−0.160384	−	0.987055i	$-0.551273\pi$
$229$	−4.85410	−0.320768	−0.160384	+	0.987055i	$0.551273\pi$
$230$	0	0
$231$	−36.8328	−2.42342
$232$	0	0
$233$	− 19.4115i	− 1.27169i	−0.771817	−	0.635845i	$-0.780652\pi$
$233$	− 19.4115i	− 1.27169i	0.771817	−	0.635845i	$-0.219348\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 17.2191i	− 1.11850i
$238$	0	0
$239$	14.2918	0.924459	0.462230	−	0.886760i	$-0.347050\pi$
$239$	14.2918	0.924459	0.462230	+	0.886760i	$0.347050\pi$
$240$	0	0
$241$	20.3820	1.31292	0.656459	−	0.754362i	$-0.272054\pi$
$241$	20.3820	1.31292	0.656459	+	0.754362i	$0.272054\pi$
$242$	0	0
$243$	1.51489i	0.0971805i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	7.09467i	0.451423i
$248$	0	0
$249$	12.1246	0.768366
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	4.90230i	0.308205i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 14.1893i	− 0.885107i	−0.896742	−	0.442553i	$-0.854073\pi$
$257$	− 14.1893i	− 0.885107i	0.896742	−	0.442553i	$-0.145927\pi$
$258$	0	0
$259$	10.1803	0.632576
$260$	0	0
$261$	0.673762	0.0417048
$262$	0	0
$263$	1.77367i	0.109369i	0.998504	+	0.0546845i	$0.0174153\pi$
$263$	1.77367i	0.109369i	−0.998504	+	0.0546845i	$0.982585\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 20.0890i	− 1.22943i
$268$	0	0
$269$	10.9443	0.667284	0.333642	−	0.942700i	$-0.391722\pi$
$269$	10.9443	0.667284	0.333642	+	0.942700i	$0.391722\pi$
$270$	0	0
$271$	−5.52786	−0.335794	−0.167897	−	0.985805i	$-0.553698\pi$
$271$	−5.52786	−0.335794	−0.167897	+	0.985805i	$0.553698\pi$
$272$	0	0
$273$	− 47.2726i	− 2.86107i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 19.4115i	− 1.16632i	−0.812356	−	0.583162i	$-0.801815\pi$
$277$	− 19.4115i	− 1.16632i	0.812356	−	0.583162i	$-0.198185\pi$
$278$	0	0
$279$	0.875388	0.0524081
$280$	0	0
$281$	−11.0344	−0.658260	−0.329130	−	0.944285i	$-0.606755\pi$
$281$	−11.0344	−0.658260	−0.329130	+	0.944285i	$0.606755\pi$
$282$	0	0
$283$	− 9.28705i	− 0.552058i	−0.961149	−	0.276029i	$-0.910981\pi$
$283$	− 9.28705i	− 0.552058i	0.961149	−	0.276029i	$-0.0890185\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 12.1569i	− 0.717599i
$288$	0	0
$289$	4.41641	0.259789
$290$	0	0
$291$	10.1803	0.596782
$292$	0	0
$293$	12.8344i	0.749793i	0.927067	+	0.374896i	$0.122322\pi$
$293$	12.8344i	0.749793i	−0.927067	+	0.374896i	$0.877678\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	22.6390i	1.31365i
$298$	0	0
$299$	−6.29180	−0.363864
$300$	0	0
$301$	18.4164	1.06150
$302$	0	0
$303$	34.2783i	1.96924i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 3.28856i	− 0.187688i	−0.995587	−	0.0938441i	$-0.970084\pi$
$307$	− 3.28856i	− 0.187688i	0.995587	−	0.0938441i	$-0.0299155\pi$
$308$	0	0
$309$	−16.4721	−0.937067
$310$	0	0
$311$	−25.4164	−1.44123	−0.720616	−	0.693334i	$-0.756141\pi$
$311$	−25.4164	−1.44123	−0.720616	+	0.693334i	$0.756141\pi$
$312$	0	0
$313$	18.5741i	1.04987i	0.851142	+	0.524935i	$0.175910\pi$
$313$	18.5741i	1.04987i	−0.851142	+	0.524935i	$0.824090\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	16.3817i	0.920089i	0.887896	+	0.460044i	$0.152167\pi$
$317$	16.3817i	0.920089i	−0.887896	+	0.460044i	$0.847833\pi$
$318$	0	0
$319$	20.6525	1.15632
$320$	0	0
$321$	0.742646	0.0414504
$322$	0	0
$323$	− 4.38475i	− 0.243974i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	23.7352i	1.31256i
$328$	0	0
$329$	34.8885	1.92347
$330$	0	0
$331$	10.2918	0.565688	0.282844	−	0.959166i	$-0.408722\pi$
$331$	10.2918	0.565688	0.282844	+	0.959166i	$0.408722\pi$
$332$	0	0
$333$	0.319863i	0.0175284i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 27.8611i	− 1.51769i	−0.651270	−	0.758846i	$-0.725763\pi$
$337$	− 27.8611i	− 1.51769i	0.651270	−	0.758846i	$-0.274237\pi$
$338$	0	0
$339$	30.5410	1.65876
$340$	0	0
$341$	26.8328	1.45308
$342$	0	0
$343$	− 35.1157i	− 1.89607i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 26.2474i	− 1.40903i	−0.709687	−	0.704517i	$-0.751164\pi$
$347$	− 26.2474i	− 1.40903i	0.709687	−	0.704517i	$-0.248836\pi$
$348$	0	0
$349$	25.9787	1.39061	0.695304	−	0.718715i	$-0.255270\pi$
$349$	25.9787	1.39061	0.695304	+	0.718715i	$0.255270\pi$
$350$	0	0
$351$	−29.0557	−1.55088
$352$	0	0
$353$	5.73971i	0.305494i	0.988265	+	0.152747i	$0.0488120\pi$
$353$	5.73971i	0.305494i	−0.988265	+	0.152747i	$0.951188\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	29.2161i	1.54628i
$358$	0	0
$359$	−14.9443	−0.788729	−0.394364	−	0.918954i	$-0.629035\pi$
$359$	−14.9443	−0.788729	−0.394364	+	0.918954i	$0.629035\pi$
$360$	0	0
$361$	−17.4721	−0.919586
$362$	0	0
$363$	− 15.9630i	− 0.837841i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	3.28856i	0.171662i	0.996310	+	0.0858308i	$0.0273544\pi$
$367$	3.28856i	0.171662i	−0.996310	+	0.0858308i	$0.972646\pi$
$368$	0	0
$369$	0.381966	0.0198844
$370$	0	0
$371$	53.3050	2.76746
$372$	0	0
$373$	6.57712i	0.340550i	0.985397	+	0.170275i	$0.0544657\pi$
$373$	6.57712i	0.340550i	−0.985397	+	0.170275i	$0.945534\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	26.5062i	1.36514i
$378$	0	0
$379$	24.3607	1.25132	0.625662	−	0.780094i	$-0.284829\pi$
$379$	24.3607	1.25132	0.625662	+	0.780094i	$0.284829\pi$
$380$	0	0
$381$	−25.9098	−1.32740
$382$	0	0
$383$	− 23.3775i	− 1.19454i	−0.802041	−	0.597268i	$-0.796253\pi$
$383$	− 23.3775i	− 1.19454i	0.802041	−	0.597268i	$-0.203747\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0.578638i	0.0294138i
$388$	0	0
$389$	−12.2148	−0.619314	−0.309657	−	0.950848i	$-0.600214\pi$
$389$	−12.2148	−0.619314	−0.309657	+	0.950848i	$0.600214\pi$
$390$	0	0
$391$	3.88854	0.196652
$392$	0	0
$393$	− 12.8344i	− 0.647409i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	21.6039i	1.08427i	0.840292	+	0.542134i	$0.182383\pi$
$397$	21.6039i	1.08427i	−0.840292	+	0.542134i	$0.817617\pi$
$398$	0	0
$399$	−10.1803	−0.509654
$400$	0	0
$401$	−20.4508	−1.02127	−0.510633	−	0.859799i	$-0.670589\pi$
$401$	−20.4508	−1.02127	−0.510633	+	0.859799i	$0.670589\pi$
$402$	0	0
$403$	34.4383i	1.71549i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	9.80460i	0.485996i
$408$	0	0
$409$	−19.6180	−0.970049	−0.485025	−	0.874500i	$-0.661189\pi$
$409$	−19.6180	−0.970049	−0.485025	+	0.874500i	$0.661189\pi$
$410$	0	0
$411$	26.6525	1.31467
$412$	0	0
$413$	− 57.9147i	− 2.84979i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	4.06489i	0.199058i
$418$	0	0
$419$	−16.3607	−0.799272	−0.399636	−	0.916674i	$-0.630863\pi$
$419$	−16.3607	−0.799272	−0.399636	+	0.916674i	$0.630863\pi$
$420$	0	0
$421$	−8.79837	−0.428807	−0.214403	−	0.976745i	$-0.568781\pi$
$421$	−8.79837	−0.428807	−0.214403	+	0.976745i	$0.568781\pi$
$422$	0	0
$423$	1.09619i	0.0532984i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	5.32100i	0.257501i
$428$	0	0
$429$	45.5279	2.19811
$430$	0	0
$431$	−23.2361	−1.11924	−0.559621	−	0.828749i	$-0.689053\pi$
$431$	−23.2361	−1.11924	−0.559621	+	0.828749i	$0.689053\pi$
$432$	0	0
$433$	11.9970i	0.576538i	0.957550	+	0.288269i	$0.0930797\pi$
$433$	11.9970i	0.576538i	−0.957550	+	0.288269i	$0.906920\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.35496i	0.0648166i
$438$	0	0
$439$	23.1246	1.10368	0.551839	−	0.833951i	$-0.313926\pi$
$439$	23.1246	1.10368	0.551839	+	0.833951i	$0.313926\pi$
$440$	0	0
$441$	2.12461	0.101172
$442$	0	0
$443$	− 9.02827i	− 0.428946i	−0.976730	−	0.214473i	$-0.931197\pi$
$443$	− 9.02827i	− 0.428946i	0.976730	−	0.214473i	$-0.0688034\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	16.5416i	0.782393i
$448$	0	0
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	0	0
$451$	11.7082	0.551318
$452$	0	0
$453$	− 3.02979i	− 0.142352i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 37.1482i	− 1.73772i	−0.495059	−	0.868859i	$-0.664854\pi$
$457$	− 37.1482i	− 1.73772i	0.495059	−	0.868859i	$-0.335146\pi$
$458$	0	0
$459$	17.9574	0.838181
$460$	0	0
$461$	−29.7984	−1.38785	−0.693924	−	0.720048i	$-0.744120\pi$
$461$	−29.7984	−1.38785	−0.693924	+	0.720048i	$0.744120\pi$
$462$	0	0
$463$	21.8627i	1.01604i	0.861344	+	0.508022i	$0.169623\pi$
$463$	21.8627i	1.01604i	−0.861344	+	0.508022i	$0.830377\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	31.1497i	1.44144i	0.693228	+	0.720718i	$0.256188\pi$
$467$	31.1497i	1.44144i	−0.693228	+	0.720718i	$0.743812\pi$
$468$	0	0
$469$	−10.1803	−0.470084
$470$	0	0
$471$	26.6525	1.22808
$472$	0	0
$473$	17.7367i	0.815533i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	1.67482i	0.0766850i
$478$	0	0
$479$	14.2918	0.653009	0.326504	−	0.945196i	$-0.394129\pi$
$479$	14.2918	0.653009	0.326504	+	0.945196i	$0.394129\pi$
$480$	0	0
$481$	−12.5836	−0.573762
$482$	0	0
$483$	− 9.02827i	− 0.410801i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 5.32100i	− 0.241118i	−0.992706	−	0.120559i	$-0.961531\pi$
$487$	− 5.32100i	− 0.241118i	0.992706	−	0.120559i	$-0.0384686\pi$
$488$	0	0
$489$	27.1115	1.22602
$490$	0	0
$491$	−6.18034	−0.278915	−0.139457	−	0.990228i	$-0.544536\pi$
$491$	−6.18034	−0.278915	−0.139457	+	0.990228i	$0.544536\pi$
$492$	0	0
$493$	− 16.3817i	− 0.737795i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 24.3138i	− 1.09062i
$498$	0	0
$499$	−39.1246	−1.75146	−0.875729	−	0.482803i	$-0.839619\pi$
$499$	−39.1246	−1.75146	−0.875729	+	0.482803i	$0.839619\pi$
$500$	0	0
$501$	3.14590	0.140548
$502$	0	0
$503$	− 35.2146i	− 1.57014i	−0.619407	−	0.785070i	$-0.712627\pi$
$503$	− 35.2146i	− 1.57014i	0.619407	−	0.785070i	$-0.287373\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	35.3745i	1.57104i
$508$	0	0
$509$	10.5836	0.469109	0.234555	−	0.972103i	$-0.424637\pi$
$509$	10.5836	0.469109	0.234555	+	0.972103i	$0.424637\pi$
$510$	0	0
$511$	26.6525	1.17904
$512$	0	0
$513$	6.25726i	0.276265i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	33.6008i	1.47776i
$518$	0	0
$519$	39.2361	1.72227
$520$	0	0
$521$	−12.9098	−0.565590	−0.282795	−	0.959180i	$-0.591262\pi$
$521$	−12.9098	−0.565590	−0.282795	+	0.959180i	$0.591262\pi$
$522$	0	0
$523$	7.51338i	0.328537i	0.986416	+	0.164269i	$0.0525264\pi$
$523$	7.51338i	0.328537i	−0.986416	+	0.164269i	$0.947474\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 21.2840i	− 0.927146i
$528$	0	0
$529$	21.7984	0.947755
$530$	0	0
$531$	1.81966	0.0789665
$532$	0	0
$533$	15.0268i	0.650881i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 6.25726i	− 0.270021i
$538$	0	0
$539$	65.1246	2.80512
$540$	0	0
$541$	33.2705	1.43041	0.715205	−	0.698914i	$-0.246333\pi$
$541$	33.2705	1.43041	0.715205	+	0.698914i	$0.246333\pi$
$542$	0	0
$543$	− 4.32366i	− 0.185546i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 24.5726i	− 1.05065i	−0.850902	−	0.525324i	$-0.823944\pi$
$547$	− 24.5726i	− 1.05065i	0.850902	−	0.525324i	$-0.176056\pi$
$548$	0	0
$549$	−0.167184	−0.00713525
$550$	0	0
$551$	5.70820	0.243178
$552$	0	0
$553$	45.0803i	1.91701i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 23.7963i	− 1.00828i	−0.863622	−	0.504140i	$-0.831810\pi$
$557$	− 23.7963i	− 1.00828i	0.863622	−	0.504140i	$-0.168190\pi$
$558$	0	0
$559$	−22.7639	−0.962812
$560$	0	0
$561$	−28.1378	−1.18798
$562$	0	0
$563$	13.6718i	0.576197i	0.957601	+	0.288099i	$0.0930231\pi$
$563$	13.6718i	0.576197i	−0.957601	+	0.288099i	$0.906977\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 43.7253i	− 1.83629i
$568$	0	0
$569$	26.9098	1.12812	0.564059	−	0.825734i	$-0.309239\pi$
$569$	26.9098	1.12812	0.564059	+	0.825734i	$0.309239\pi$
$570$	0	0
$571$	44.3607	1.85644	0.928218	−	0.372036i	$-0.121340\pi$
$571$	44.3607	1.85644	0.928218	+	0.372036i	$0.121340\pi$
$572$	0	0
$573$	15.3466i	0.641114i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 2.70992i	− 0.112816i	−0.998408	−	0.0564078i	$-0.982035\pi$
$577$	− 2.70992i	− 0.112816i	0.998408	−	0.0564078i	$-0.0179647\pi$
$578$	0	0
$579$	−16.4721	−0.684559
$580$	0	0
$581$	−31.7426	−1.31691
$582$	0	0
$583$	51.3375i	2.12618i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	34.9558i	1.44278i	0.692529	+	0.721390i	$0.256497\pi$
$587$	34.9558i	1.44278i	−0.692529	+	0.721390i	$0.743503\pi$
$588$	0	0
$589$	7.41641	0.305588
$590$	0	0
$591$	−18.8754	−0.776430
$592$	0	0
$593$	13.6718i	0.561433i	0.959791	+	0.280717i	$0.0905721\pi$
$593$	13.6718i	0.561433i	−0.959791	+	0.280717i	$0.909428\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 34.9558i	− 1.43065i
$598$	0	0
$599$	19.5967	0.800701	0.400351	−	0.916362i	$-0.368888\pi$
$599$	19.5967	0.800701	0.400351	+	0.916362i	$0.368888\pi$
$600$	0	0
$601$	42.6869	1.74124	0.870618	−	0.491960i	$-0.163719\pi$
$601$	42.6869	1.74124	0.870618	+	0.491960i	$0.163719\pi$
$602$	0	0
$603$	− 0.319863i	− 0.0130258i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 27.8611i	− 1.13085i	−0.824800	−	0.565424i	$-0.808712\pi$
$607$	− 27.8611i	− 1.13085i	0.824800	−	0.565424i	$-0.191288\pi$
$608$	0	0
$609$	−38.0344	−1.54123
$610$	0	0
$611$	−43.1246	−1.74464
$612$	0	0
$613$	− 2.70992i	− 0.109453i	−0.998501	−	0.0547264i	$-0.982571\pi$
$613$	− 2.70992i	− 0.109453i	0.998501	−	0.0547264i	$-0.0174287\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 22.1214i	− 0.890575i	−0.895388	−	0.445288i	$-0.853101\pi$
$617$	− 22.1214i	− 0.890575i	0.895388	−	0.445288i	$-0.146899\pi$
$618$	0	0
$619$	−2.76393	−0.111092	−0.0555459	−	0.998456i	$-0.517690\pi$
$619$	−2.76393	−0.111092	−0.0555459	+	0.998456i	$0.517690\pi$
$620$	0	0
$621$	−5.54915	−0.222680
$622$	0	0
$623$	52.5936i	2.10712i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 9.80460i	− 0.391558i
$628$	0	0
$629$	7.77709	0.310093
$630$	0	0
$631$	−28.8328	−1.14782	−0.573908	−	0.818920i	$-0.694573\pi$
$631$	−28.8328	−1.14782	−0.573908	+	0.818920i	$0.694573\pi$
$632$	0	0
$633$	4.06489i	0.161565i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	83.5834i	3.31170i
$638$	0	0
$639$	0.763932	0.0302207
$640$	0	0
$641$	25.7426	1.01677	0.508387	−	0.861129i	$-0.330242\pi$
$641$	25.7426	1.01677	0.508387	+	0.861129i	$0.330242\pi$
$642$	0	0
$643$	39.5993i	1.56165i	0.624753	+	0.780823i	$0.285200\pi$
$643$	39.5993i	1.56165i	−0.624753	+	0.780823i	$0.714800\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 23.4764i	− 0.922952i	−0.887153	−	0.461476i	$-0.847320\pi$
$647$	− 23.4764i	− 0.922952i	0.887153	−	0.461476i	$-0.152680\pi$
$648$	0	0
$649$	55.7771	2.18944
$650$	0	0
$651$	−49.4164	−1.93678
$652$	0	0
$653$	− 39.3406i	− 1.53952i	−0.638336	−	0.769758i	$-0.720377\pi$
$653$	− 39.3406i	− 1.53952i	0.638336	−	0.769758i	$-0.279623\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0.837412i	0.0326706i
$658$	0	0
$659$	20.0689	0.781773	0.390886	−	0.920439i	$-0.372169\pi$
$659$	20.0689	0.781773	0.390886	+	0.920439i	$0.372169\pi$
$660$	0	0
$661$	6.61803	0.257412	0.128706	−	0.991683i	$-0.458918\pi$
$661$	6.61803	0.257412	0.128706	+	0.991683i	$0.458918\pi$
$662$	0	0
$663$	− 36.1131i	− 1.40252i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	5.06223i	0.196010i
$668$	0	0
$669$	−29.7984	−1.15207
$670$	0	0
$671$	−5.12461	−0.197833
$672$	0	0
$673$	27.0237i	1.04169i	0.853652	+	0.520844i	$0.174383\pi$
$673$	27.0237i	1.04169i	−0.853652	+	0.520844i	$0.825617\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	7.09467i	0.272670i	0.990663	+	0.136335i	$0.0435324\pi$
$677$	7.09467i	0.272670i	−0.990663	+	0.136335i	$0.956468\pi$
$678$	0	0
$679$	−26.6525	−1.02283
$680$	0	0
$681$	12.1246	0.464616
$682$	0	0
$683$	− 1.77367i	− 0.0678675i	−0.999424	−	0.0339338i	$-0.989196\pi$
$683$	− 1.77367i	− 0.0678675i	0.999424	−	0.0339338i	$-0.0108035\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	8.60957i	0.328475i
$688$	0	0
$689$	−65.8885	−2.51015
$690$	0	0
$691$	33.7082	1.28232	0.641160	−	0.767407i	$-0.278453\pi$
$691$	33.7082	1.28232	0.641160	+	0.767407i	$0.278453\pi$
$692$	0	0
$693$	3.02979i	0.115092i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 9.28705i	− 0.351772i
$698$	0	0
$699$	−34.4296	−1.30225
$700$	0	0
$701$	−7.88854	−0.297946	−0.148973	−	0.988841i	$-0.547597\pi$
$701$	−7.88854	−0.297946	−0.148973	+	0.988841i	$0.547597\pi$
$702$	0	0
$703$	2.70992i	0.102207i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 89.7418i	− 3.37509i
$708$	0	0
$709$	−4.49342	−0.168754	−0.0843770	−	0.996434i	$-0.526890\pi$
$709$	−4.49342	−0.168754	−0.0843770	+	0.996434i	$0.526890\pi$
$710$	0	0
$711$	−1.41641	−0.0531194
$712$	0	0
$713$	6.57712i	0.246315i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 25.3489i	− 0.946672i
$718$	0	0
$719$	−47.0132	−1.75329	−0.876647	−	0.481133i	$-0.840225\pi$
$719$	−47.0132	−1.75329	−0.876647	+	0.481133i	$0.840225\pi$
$720$	0	0
$721$	43.1246	1.60604
$722$	0	0
$723$	− 36.1508i	− 1.34446i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	29.1173i	1.07990i	0.841697	+	0.539950i	$0.181557\pi$
$727$	29.1173i	1.07990i	−0.841697	+	0.539950i	$0.818443\pi$
$728$	0	0
$729$	−25.5623	−0.946752
$730$	0	0
$731$	14.0689	0.520356
$732$	0	0
$733$	15.5443i	0.574142i	0.957909	+	0.287071i	$0.0926816\pi$
$733$	15.5443i	0.574142i	−0.957909	+	0.287071i	$0.907318\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 9.80460i	− 0.361157i
$738$	0	0
$739$	−18.2918	−0.672875	−0.336437	−	0.941706i	$-0.609222\pi$
$739$	−18.2918	−0.672875	−0.336437	+	0.941706i	$0.609222\pi$
$740$	0	0
$741$	12.5836	0.462270
$742$	0	0
$743$	− 16.3817i	− 0.600987i	−0.953784	−	0.300493i	$-0.902849\pi$
$743$	− 16.3817i	− 0.600987i	0.953784	−	0.300493i	$-0.0971514\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 0.997344i	− 0.0364909i
$748$	0	0
$749$	−1.94427	−0.0710421
$750$	0	0
$751$	2.29180	0.0836288	0.0418144	−	0.999125i	$-0.486686\pi$
$751$	2.29180	0.0836288	0.0418144	+	0.999125i	$0.486686\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	4.06489i	0.147741i	0.997268	+	0.0738704i	$0.0235351\pi$
$757$	4.06489i	0.147741i	−0.997268	+	0.0738704i	$0.976465\pi$
$758$	0	0
$759$	8.69505	0.315610
$760$	0	0
$761$	36.2705	1.31480	0.657402	−	0.753540i	$-0.271655\pi$
$761$	36.2705	1.31480	0.657402	+	0.753540i	$0.271655\pi$
$762$	0	0
$763$	− 62.1395i	− 2.24960i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	71.5864i	2.58484i
$768$	0	0
$769$	−14.4377	−0.520637	−0.260318	−	0.965523i	$-0.583827\pi$
$769$	−14.4377	−0.520637	−0.260318	+	0.965523i	$0.583827\pi$
$770$	0	0
$771$	−25.1672	−0.906374
$772$	0	0
$773$	− 9.80460i	− 0.352647i	−0.984332	−	0.176323i	$-0.943580\pi$
$773$	− 9.80460i	− 0.352647i	0.984332	−	0.176323i	$-0.0564204\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 18.0565i	− 0.647775i
$778$	0	0
$779$	3.23607	0.115944
$780$	0	0
$781$	23.4164	0.837905
$782$	0	0
$783$	23.3775i	0.835445i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	44.1440i	1.57356i	0.617231	+	0.786782i	$0.288254\pi$
$787$	44.1440i	1.57356i	−0.617231	+	0.786782i	$0.711746\pi$
$788$	0	0
$789$	3.14590	0.111997
$790$	0	0
$791$	−79.9574	−2.84296
$792$	0	0
$793$	− 6.57712i	− 0.233560i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 10.1245i	− 0.358627i	−0.983792	−	0.179313i	$-0.942612\pi$
$797$	− 10.1245i	− 0.358627i	0.983792	−	0.179313i	$-0.0573876\pi$
$798$	0	0
$799$	26.6525	0.942897
$800$	0	0
$801$	−1.65248	−0.0583874
$802$	0	0
$803$	25.6688i	0.905831i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 19.4115i	− 0.683317i
$808$	0	0
$809$	7.90983	0.278095	0.139047	−	0.990286i	$-0.455596\pi$
$809$	7.90983	0.278095	0.139047	+	0.990286i	$0.455596\pi$
$810$	0	0
$811$	−0.832816	−0.0292441	−0.0146221	−	0.999893i	$-0.504655\pi$
$811$	−0.832816	−0.0292441	−0.0146221	+	0.999893i	$0.504655\pi$
$812$	0	0
$813$	9.80460i	0.343862i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	4.90230i	0.171510i
$818$	0	0
$819$	−3.88854	−0.135877
$820$	0	0
$821$	−27.2148	−0.949802	−0.474901	−	0.880039i	$-0.657516\pi$
$821$	−27.2148	−0.949802	−0.474901	+	0.880039i	$0.657516\pi$
$822$	0	0
$823$	18.0565i	0.629412i	0.949189	+	0.314706i	$0.101906\pi$
$823$	18.0565i	0.629412i	−0.949189	+	0.314706i	$0.898094\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	20.7665i	0.722121i	0.932542	+	0.361060i	$0.117585\pi$
$827$	20.7665i	0.722121i	−0.932542	+	0.361060i	$0.882415\pi$
$828$	0	0
$829$	23.2016	0.805826	0.402913	−	0.915238i	$-0.367998\pi$
$829$	23.2016	0.805826	0.402913	+	0.915238i	$0.367998\pi$
$830$	0	0
$831$	−34.4296	−1.19435
$832$	0	0
$833$	− 51.6574i	− 1.78982i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	30.3734i	1.04986i
$838$	0	0
$839$	−15.2361	−0.526007	−0.263004	−	0.964795i	$-0.584713\pi$
$839$	−15.2361	−0.526007	−0.263004	+	0.964795i	$0.584713\pi$
$840$	0	0
$841$	−7.67376	−0.264612
$842$	0	0
$843$	19.5714i	0.674076i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	41.7917i	1.43598i
$848$	0	0
$849$	−16.4721	−0.565322
$850$	0	0
$851$	−2.40325	−0.0823824
$852$	0	0
$853$	− 25.6688i	− 0.878882i	−0.898272	−	0.439441i	$-0.855177\pi$
$853$	− 25.6688i	− 0.878882i	0.898272	−	0.439441i	$-0.144823\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	45.5978i	1.55759i	0.627277	+	0.778796i	$0.284169\pi$
$857$	45.5978i	1.55759i	−0.627277	+	0.778796i	$0.715831\pi$
$858$	0	0
$859$	−30.8328	−1.05200	−0.526001	−	0.850484i	$-0.676309\pi$
$859$	−30.8328	−1.05200	−0.526001	+	0.850484i	$0.676309\pi$
$860$	0	0
$861$	−21.5623	−0.734841
$862$	0	0
$863$	− 26.7650i	− 0.911090i	−0.890213	−	0.455545i	$-0.849444\pi$
$863$	− 26.7650i	− 0.911090i	0.890213	−	0.455545i	$-0.150556\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 7.83324i	− 0.266031i
$868$	0	0
$869$	−43.4164	−1.47280
$870$	0	0
$871$	12.5836	0.426379
$872$	0	0
$873$	− 0.837412i	− 0.0283421i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 6.57712i	− 0.222094i	−0.993815	−	0.111047i	$-0.964580\pi$
$877$	− 6.57712i	− 0.222094i	0.993815	−	0.111047i	$-0.0354204\pi$
$878$	0	0
$879$	22.7639	0.767808
$880$	0	0
$881$	−3.97871	−0.134046	−0.0670231	−	0.997751i	$-0.521350\pi$
$881$	−3.97871	−0.134046	−0.0670231	+	0.997751i	$0.521350\pi$
$882$	0	0
$883$	13.9306i	0.468801i	0.972140	+	0.234400i	$0.0753127\pi$
$883$	13.9306i	0.468801i	−0.972140	+	0.234400i	$0.924687\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	3.28856i	0.110419i	0.998475	+	0.0552095i	$0.0175827\pi$
$887$	3.28856i	0.110419i	−0.998475	+	0.0552095i	$0.982417\pi$
$888$	0	0
$889$	67.8328	2.27504
$890$	0	0
$891$	42.1115	1.41079
$892$	0	0
$893$	9.28705i	0.310779i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	11.1596i	0.372607i
$898$	0	0
$899$	27.7082	0.924120
$900$	0	0
$901$	40.7214	1.35663
$902$	0	0
$903$	− 32.6646i	− 1.08701i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	58.1734i	1.93162i	0.259257	+	0.965808i	$0.416522\pi$
$907$	58.1734i	1.93162i	−0.259257	+	0.965808i	$0.583478\pi$
$908$	0	0
$909$	2.81966	0.0935222
$910$	0	0
$911$	44.8328	1.48538	0.742689	−	0.669637i	$-0.233550\pi$
$911$	44.8328	1.48538	0.742689	+	0.669637i	$0.233550\pi$
$912$	0	0
$913$	− 30.5711i	− 1.01175i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	33.6008i	1.10960i
$918$	0	0
$919$	−13.2361	−0.436618	−0.218309	−	0.975880i	$-0.570054\pi$
$919$	−13.2361	−0.436618	−0.218309	+	0.975880i	$0.570054\pi$
$920$	0	0
$921$	−5.83282	−0.192198
$922$	0	0
$923$	30.0535i	0.989223i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	1.35496i	0.0445028i
$928$	0	0
$929$	−29.5066	−0.968079	−0.484040	−	0.875046i	$-0.660831\pi$
$929$	−29.5066	−0.968079	−0.484040	+	0.875046i	$0.660831\pi$
$930$	0	0
$931$	18.0000	0.589926
$932$	0	0
$933$	45.0803i	1.47586i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 18.5741i	− 0.606789i	−0.952865	−	0.303395i	$-0.901880\pi$
$937$	− 18.5741i	− 0.606789i	0.952865	−	0.303395i	$-0.0981200\pi$
$938$	0	0
$939$	32.9443	1.07510
$940$	0	0
$941$	−1.41641	−0.0461736	−0.0230868	−	0.999733i	$-0.507349\pi$
$941$	−1.41641	−0.0461736	−0.0230868	+	0.999733i	$0.507349\pi$
$942$	0	0
$943$	2.86986i	0.0934553i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 15.9630i	− 0.518728i	−0.965780	−	0.259364i	$-0.916487\pi$
$947$	− 15.9630i	− 0.518728i	0.965780	−	0.259364i	$-0.0835130\pi$
$948$	0	0
$949$	−32.9443	−1.06942
$950$	0	0
$951$	29.0557	0.942197
$952$	0	0
$953$	30.0535i	0.973529i	0.873533	+	0.486764i	$0.161823\pi$
$953$	30.0535i	0.973529i	−0.873533	+	0.486764i	$0.838177\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 36.6306i	− 1.18410i
$958$	0	0
$959$	−69.7771	−2.25322
$960$	0	0
$961$	5.00000	0.161290
$962$	0	0
$963$	− 0.0610884i	− 0.00196855i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 29.9547i	− 0.963277i	−0.876370	−	0.481639i	$-0.840042\pi$
$967$	− 29.9547i	− 0.963277i	0.876370	−	0.481639i	$-0.159958\pi$
$968$	0	0
$969$	−7.77709	−0.249836
$970$	0	0
$971$	−43.2361	−1.38751	−0.693756	−	0.720210i	$-0.744045\pi$
$971$	−43.2361	−1.38751	−0.693756	+	0.720210i	$0.744045\pi$
$972$	0	0
$973$	− 10.6420i	− 0.341167i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	30.8909i	0.988288i	0.869380	+	0.494144i	$0.164518\pi$
$977$	30.8909i	0.988288i	−0.869380	+	0.494144i	$0.835482\pi$
$978$	0	0
$979$	−50.6525	−1.61886
$980$	0	0
$981$	1.95240	0.0623354
$982$	0	0
$983$	− 27.8611i	− 0.888632i	−0.895870	−	0.444316i	$-0.853447\pi$
$983$	− 27.8611i	− 0.888632i	0.895870	−	0.444316i	$-0.146553\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 61.8807i	− 1.96968i
$988$	0	0
$989$	−4.34752	−0.138243
$990$	0	0
$991$	39.7771	1.26356	0.631780	−	0.775147i	$-0.282324\pi$
$991$	39.7771	1.26356	0.631780	+	0.775147i	$0.282324\pi$
$992$	0	0
$993$	− 18.2542i	− 0.579280i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 29.2161i	− 0.925283i	−0.886545	−	0.462642i	$-0.846902\pi$
$997$	− 29.2161i	− 0.925283i	0.886545	−	0.462642i	$-0.153098\pi$
$998$	0	0
$999$	−11.0983	−0.351135

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2000.2.c.a.1249.2		4
4.3	odd	2		500.2.c.a.249.3		4
5.2	odd	4		2000.2.a.p.1.2		4
5.3	odd	4		2000.2.a.p.1.3		4
5.4	even	2	inner	2000.2.c.a.1249.3		4
12.11	even	2		4500.2.d.e.4249.1		4
20.3	even	4		500.2.a.c.1.2	✓	4
20.7	even	4		500.2.a.c.1.3	yes	4
20.19	odd	2		500.2.c.a.249.2		4
40.3	even	4		8000.2.a.bl.1.3		4
40.13	odd	4		8000.2.a.bm.1.2		4
40.27	even	4		8000.2.a.bl.1.2		4
40.37	odd	4		8000.2.a.bm.1.3		4
60.23	odd	4		4500.2.a.q.1.1		4
60.47	odd	4		4500.2.a.q.1.4		4
60.59	even	2		4500.2.d.e.4249.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
500.2.a.c.1.2	✓	4	20.3	even	4
500.2.a.c.1.3	yes	4	20.7	even	4
500.2.c.a.249.2		4	20.19	odd	2
500.2.c.a.249.3		4	4.3	odd	2
2000.2.a.p.1.2		4	5.2	odd	4
2000.2.a.p.1.3		4	5.3	odd	4
2000.2.c.a.1249.2		4	1.1	even	1	trivial
2000.2.c.a.1249.3		4	5.4	even	2	inner
4500.2.a.q.1.1		4	60.23	odd	4
4500.2.a.q.1.4		4	60.47	odd	4
4500.2.d.e.4249.1		4	12.11	even	2
4500.2.d.e.4249.4		4	60.59	even	2
8000.2.a.bl.1.2		4	40.27	even	4
8000.2.a.bl.1.3		4	40.3	even	4
8000.2.a.bm.1.2		4	40.13	odd	4
8000.2.a.bm.1.3		4	40.37	odd	4

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:	\( N \)	\(=\)	\( 2000 = 2^{4} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2000.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.77367i q^{3} +4.64352i q^{7} -0.145898 q^{9} +O(q^{10})\)	\(q-1.77367i q^{3} +4.64352i q^{7} -0.145898 q^{9} -4.47214 q^{11} -5.73971i q^{13} +3.54734i q^{17} -1.23607 q^{19} +8.23607 q^{21} -1.09619i q^{23} -5.06223i q^{27} -4.61803 q^{29} -6.00000 q^{31} +7.93208i q^{33} -2.19237i q^{37} -10.1803 q^{39} -2.61803 q^{41} -3.96604i q^{43} -7.51338i q^{47} -14.5623 q^{49} +6.29180 q^{51} -11.4794i q^{53} +2.19237i q^{57} -12.4721 q^{59} +1.14590 q^{61} -0.677481i q^{63} +2.19237i q^{67} -1.94427 q^{69} -5.23607 q^{71} -5.73971i q^{73} -20.7665i q^{77} +9.70820 q^{79} -9.41641 q^{81} +6.83590i q^{83} +8.19086i q^{87} +11.3262 q^{89} +26.6525 q^{91} +10.6420i q^{93} +5.73971i q^{97} +0.652476 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 14 q^{9}+O(q^{10}) \) 4 * q - 14 * q^9	\( 4 q - 14 q^{9} + 4 q^{19} + 24 q^{21} - 14 q^{29} - 24 q^{31} + 4 q^{39} - 6 q^{41} - 18 q^{49} + 52 q^{51} - 32 q^{59} + 18 q^{61} + 28 q^{69} - 12 q^{71} + 12 q^{79} + 16 q^{81} + 14 q^{89} + 44 q^{91} - 60 q^{99}+O(q^{100}) \) 4 * q - 14 * q^9 + 4 * q^19 + 24 * q^21 - 14 * q^29 - 24 * q^31 + 4 * q^39 - 6 * q^41 - 18 * q^49 + 52 * q^51 - 32 * q^59 + 18 * q^61 + 28 * q^69 - 12 * q^71 + 12 * q^79 + 16 * q^81 + 14 * q^89 + 44 * q^91 - 60 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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