LMFDB - Embedded newform 4000.2.c.h.1249.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4000,2,Mod(1249,4000)]

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, 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("4000.1249");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4000 = 2^{5} \cdot 5^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4000.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:	$31.9401608085$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} - 10 x^{14} + 40 x^{13} + 147 x^{12} - 378 x^{11} - 845 x^{10} + 1620 x^{9} + \cdots + 10324 $ x^16 - 4x^15 - 10x^14 + 40x^13 + 147x^12 - 378x^11 - 845x^10 + 1620x^9 + 4271x^8 - 6478x^7 - 14916x^6 + 16548x^5 + 51265x^4 + 19946x^3 - 18890x^2 - 2972*x + 10324
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.2
Root			$-2.11393 - 1.66741i$ of defining polynomial
Character	$\chi$	$=$	4000.1249
Dual form			4000.2.c.h.1249.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.33481i q^{3} +1.78602i q^{7} -8.12097 q^{9} +O(q^{10})$	$q-3.33481i q^{3} +1.78602i q^{7} -8.12097 q^{9} +4.05479 q^{11} +4.78297i q^{13} -4.40100i q^{17} -6.94463 q^{19} +5.95604 q^{21} +2.78101i q^{23} +17.0775i q^{27} -5.16493 q^{29} +2.06103 q^{31} -13.5220i q^{33} -5.33800i q^{37} +15.9503 q^{39} -2.50293 q^{41} +3.73821i q^{43} +10.0717i q^{47} +3.81014 q^{49} -14.6765 q^{51} +12.4590i q^{53} +23.1590i q^{57} -0.658848 q^{59} +5.75217 q^{61} -14.5042i q^{63} +8.73065i q^{67} +9.27414 q^{69} +12.2000 q^{71} -5.41276i q^{73} +7.24193i q^{77} +2.04146 q^{79} +32.5872 q^{81} -1.77989i q^{83} +17.2241i q^{87} +16.7771 q^{89} -8.54247 q^{91} -6.87314i q^{93} -6.68690i q^{97} -32.9288 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 36 q^{9}+O(q^{10}) $ 16 * q - 36 * q^9	$ 16 q - 36 q^{9} + 36 q^{21} - 48 q^{29} + 36 q^{41} - 52 q^{49} + 44 q^{61} - 64 q^{69} + 120 q^{81} - 20 q^{89}+O(q^{100}) $ 16 * q - 36 * q^9 + 36 * q^21 - 48 * q^29 + 36 * q^41 - 52 * q^49 + 44 * q^61 - 64 * q^69 + 120 * q^81 - 20 * q^89

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
$3$	− 3.33481i	− 1.92535i	−0.270652	−	0.962677i	$-0.587239\pi$
$3$	− 3.33481i	− 1.92535i	0.270652	−	0.962677i	$-0.412761\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.78602i	0.675052i	0.941316	+	0.337526i	$0.109590\pi$
$7$	1.78602i	0.675052i	−0.941316	+	0.337526i	$0.890410\pi$
$8$	0	0
$9$	−8.12097	−2.70699
$10$	0	0
$11$	4.05479	1.22257	0.611283	−	0.791412i	$-0.290654\pi$
$11$	4.05479	1.22257	0.611283	+	0.791412i	$0.290654\pi$
$12$	0	0
$13$	4.78297i	1.32656i	0.748373	+	0.663278i	$0.230835\pi$
$13$	4.78297i	1.32656i	−0.748373	+	0.663278i	$0.769165\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 4.40100i	− 1.06740i	−0.845674	−	0.533700i	$-0.820801\pi$
$17$	− 4.40100i	− 1.06740i	0.845674	−	0.533700i	$-0.179199\pi$
$18$	0	0
$19$	−6.94463	−1.59321	−0.796604	−	0.604502i	$-0.793372\pi$
$19$	−6.94463	−1.59321	−0.796604	+	0.604502i	$0.793372\pi$
$20$	0	0
$21$	5.95604	1.29971
$22$	0	0
$23$	2.78101i	0.579880i	0.957045	+	0.289940i	$0.0936354\pi$
$23$	2.78101i	0.579880i	−0.957045	+	0.289940i	$0.906365\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	17.0775i	3.28656i
$28$	0	0
$29$	−5.16493	−0.959104	−0.479552	−	0.877514i	$-0.659201\pi$
$29$	−5.16493	−0.959104	−0.479552	+	0.877514i	$0.659201\pi$
$30$	0	0
$31$	2.06103	0.370171	0.185086	−	0.982722i	$-0.440744\pi$
$31$	2.06103	0.370171	0.185086	+	0.982722i	$0.440744\pi$
$32$	0	0
$33$	− 13.5220i	− 2.35387i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 5.33800i	− 0.877562i	−0.898594	−	0.438781i	$-0.855410\pi$
$37$	− 5.33800i	− 0.877562i	0.898594	−	0.438781i	$-0.144590\pi$
$38$	0	0
$39$	15.9503	2.55409
$40$	0	0
$41$	−2.50293	−0.390892	−0.195446	−	0.980714i	$-0.562616\pi$
$41$	−2.50293	−0.390892	−0.195446	+	0.980714i	$0.562616\pi$
$42$	0	0
$43$	3.73821i	0.570072i	0.958517	+	0.285036i	$0.0920056\pi$
$43$	3.73821i	0.570072i	−0.958517	+	0.285036i	$0.907994\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	10.0717i	1.46911i	0.678550	+	0.734554i	$0.262609\pi$
$47$	10.0717i	1.46911i	−0.678550	+	0.734554i	$0.737391\pi$
$48$	0	0
$49$	3.81014	0.544305
$50$	0	0
$51$	−14.6765	−2.05512
$52$	0	0
$53$	12.4590i	1.71137i	0.517496	+	0.855685i	$0.326864\pi$
$53$	12.4590i	1.71137i	−0.517496	+	0.855685i	$0.673136\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	23.1590i	3.06749i
$58$	0	0
$59$	−0.658848	−0.0857747	−0.0428873	−	0.999080i	$-0.513656\pi$
$59$	−0.658848	−0.0857747	−0.0428873	+	0.999080i	$0.513656\pi$
$60$	0	0
$61$	5.75217	0.736490	0.368245	−	0.929729i	$-0.379959\pi$
$61$	5.75217	0.736490	0.368245	+	0.929729i	$0.379959\pi$
$62$	0	0
$63$	− 14.5042i	− 1.82736i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	8.73065i	1.06662i	0.845920	+	0.533309i	$0.179052\pi$
$67$	8.73065i	1.06662i	−0.845920	+	0.533309i	$0.820948\pi$
$68$	0	0
$69$	9.27414	1.11647
$70$	0	0
$71$	12.2000	1.44787	0.723936	−	0.689867i	$-0.242331\pi$
$71$	12.2000	1.44787	0.723936	+	0.689867i	$0.242331\pi$
$72$	0	0
$73$	− 5.41276i	− 0.633516i	−0.948506	−	0.316758i	$-0.897406\pi$
$73$	− 5.41276i	− 0.633516i	0.948506	−	0.316758i	$-0.102594\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	7.24193i	0.825295i
$78$	0	0
$79$	2.04146	0.229683	0.114841	−	0.993384i	$-0.463364\pi$
$79$	2.04146	0.229683	0.114841	+	0.993384i	$0.463364\pi$
$80$	0	0
$81$	32.5872	3.62080
$82$	0	0
$83$	− 1.77989i	− 0.195368i	−0.995217	−	0.0976842i	$-0.968857\pi$
$83$	− 1.77989i	− 0.195368i	0.995217	−	0.0976842i	$-0.0311435\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	17.2241i	1.84661i
$88$	0	0
$89$	16.7771	1.77837	0.889183	−	0.457552i	$-0.151274\pi$
$89$	16.7771	1.77837	0.889183	+	0.457552i	$0.151274\pi$
$90$	0	0
$91$	−8.54247	−0.895494
$92$	0	0
$93$	− 6.87314i	− 0.712711i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 6.68690i	− 0.678952i	−0.940615	−	0.339476i	$-0.889750\pi$
$97$	− 6.68690i	− 0.678952i	0.940615	−	0.339476i	$-0.110250\pi$
$98$	0	0
$99$	−32.9288	−3.30947
$100$	0	0
$101$	2.63120	0.261814	0.130907	−	0.991395i	$-0.458211\pi$
$101$	2.63120	0.261814	0.130907	+	0.991395i	$0.458211\pi$
$102$	0	0
$103$	11.7489i	1.15765i	0.815451	+	0.578826i	$0.196489\pi$
$103$	11.7489i	1.15765i	−0.815451	+	0.578826i	$0.803511\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.55492i	0.150320i	0.997171	+	0.0751600i	$0.0239467\pi$
$107$	1.55492i	0.150320i	−0.997171	+	0.0751600i	$0.976053\pi$
$108$	0	0
$109$	16.4662	1.57718	0.788590	−	0.614920i	$-0.210812\pi$
$109$	16.4662	1.57718	0.788590	+	0.614920i	$0.210812\pi$
$110$	0	0
$111$	−17.8012	−1.68962
$112$	0	0
$113$	13.4069i	1.26121i	0.776103	+	0.630606i	$0.217194\pi$
$113$	13.4069i	1.26121i	−0.776103	+	0.630606i	$0.782806\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 38.8423i	− 3.59097i
$118$	0	0
$119$	7.86027	0.720550
$120$	0	0
$121$	5.44134	0.494667
$122$	0	0
$123$	8.34681i	0.752607i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 7.14407i	− 0.633934i	−0.948437	−	0.316967i	$-0.897336\pi$
$127$	− 7.14407i	− 0.633934i	0.948437	−	0.316967i	$-0.102664\pi$
$128$	0	0
$129$	12.4662	1.09759
$130$	0	0
$131$	6.45575	0.564041	0.282021	−	0.959408i	$-0.408995\pi$
$131$	6.45575	0.564041	0.282021	+	0.959408i	$0.408995\pi$
$132$	0	0
$133$	− 12.4032i	− 1.07550i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	14.1364i	1.20775i	0.797078	+	0.603876i	$0.206378\pi$
$137$	14.1364i	1.20775i	−0.797078	+	0.603876i	$0.793622\pi$
$138$	0	0
$139$	−5.83468	−0.494891	−0.247446	−	0.968902i	$-0.579591\pi$
$139$	−5.83468	−0.494891	−0.247446	+	0.968902i	$0.579591\pi$
$140$	0	0
$141$	33.5872	2.82855
$142$	0	0
$143$	19.3939i	1.62180i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 12.7061i	− 1.04798i
$148$	0	0
$149$	−9.16493	−0.750820	−0.375410	−	0.926859i	$-0.622498\pi$
$149$	−9.16493	−0.750820	−0.375410	+	0.926859i	$0.622498\pi$
$150$	0	0
$151$	0.743331	0.0604914	0.0302457	−	0.999542i	$-0.490371\pi$
$151$	0.743331	0.0604914	0.0302457	+	0.999542i	$0.490371\pi$
$152$	0	0
$153$	35.7404i	2.88944i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	7.04983i	0.562638i	0.959614	+	0.281319i	$0.0907719\pi$
$157$	7.04983i	0.562638i	−0.959614	+	0.281319i	$0.909228\pi$
$158$	0	0
$159$	41.5483	3.29500
$160$	0	0
$161$	−4.96693	−0.391449
$162$	0	0
$163$	14.8503i	1.16316i	0.813489	+	0.581581i	$0.197566\pi$
$163$	14.8503i	1.16316i	−0.813489	+	0.581581i	$0.802434\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	12.7439i	0.986150i	0.869987	+	0.493075i	$0.164127\pi$
$167$	12.7439i	0.986150i	−0.869987	+	0.493075i	$0.835873\pi$
$168$	0	0
$169$	−9.87676	−0.759751
$170$	0	0
$171$	56.3971	4.31280
$172$	0	0
$173$	− 5.94287i	− 0.451828i	−0.974147	−	0.225914i	$-0.927463\pi$
$173$	− 5.94287i	− 0.451828i	0.974147	−	0.225914i	$-0.0725368\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	2.19713i	0.165147i
$178$	0	0
$179$	4.74908	0.354963	0.177481	−	0.984124i	$-0.443205\pi$
$179$	4.74908	0.354963	0.177481	+	0.984124i	$0.443205\pi$
$180$	0	0
$181$	−17.8079	−1.32365	−0.661824	−	0.749659i	$-0.730217\pi$
$181$	−17.8079	−1.32365	−0.661824	+	0.749659i	$0.730217\pi$
$182$	0	0
$183$	− 19.1824i	− 1.41800i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 17.8451i	− 1.30497i
$188$	0	0
$189$	−30.5007	−2.21860
$190$	0	0
$191$	−17.0458	−1.23339	−0.616696	−	0.787202i	$-0.711529\pi$
$191$	−17.0458	−1.23339	−0.616696	+	0.787202i	$0.711529\pi$
$192$	0	0
$193$	− 14.9171i	− 1.07376i	−0.843660	−	0.536878i	$-0.819603\pi$
$193$	− 14.9171i	− 1.07376i	0.843660	−	0.536878i	$-0.180397\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 4.87314i	− 0.347197i	−0.984817	−	0.173598i	$-0.944461\pi$
$197$	− 4.87314i	− 0.347197i	0.984817	−	0.173598i	$-0.0555394\pi$
$198$	0	0
$199$	−22.2165	−1.57489	−0.787443	−	0.616387i	$-0.788596\pi$
$199$	−22.2165	−1.57489	−0.787443	+	0.616387i	$0.788596\pi$
$200$	0	0
$201$	29.1151	2.05362
$202$	0	0
$203$	− 9.22466i	− 0.647445i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 22.5845i	− 1.56973i
$208$	0	0
$209$	−28.1590	−1.94780
$210$	0	0
$211$	−20.5211	−1.41273	−0.706365	−	0.707847i	$-0.749666\pi$
$211$	−20.5211	−1.41273	−0.706365	+	0.707847i	$0.749666\pi$
$212$	0	0
$213$	− 40.6846i	− 2.78767i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	3.68103i	0.249885i
$218$	0	0
$219$	−18.0505	−1.21974
$220$	0	0
$221$	21.0498	1.41596
$222$	0	0
$223$	− 14.2927i	− 0.957108i	−0.878058	−	0.478554i	$-0.841161\pi$
$223$	− 14.2927i	− 0.957108i	0.878058	−	0.478554i	$-0.158839\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 9.14614i	− 0.607051i	−0.952823	−	0.303525i	$-0.901836\pi$
$227$	− 9.14614i	− 0.607051i	0.952823	−	0.303525i	$-0.0981637\pi$
$228$	0	0
$229$	−16.1268	−1.06569	−0.532846	−	0.846212i	$-0.678877\pi$
$229$	−16.1268	−1.06569	−0.532846	+	0.846212i	$0.678877\pi$
$230$	0	0
$231$	24.1505	1.58899
$232$	0	0
$233$	2.54327i	0.166615i	0.996524	+	0.0833077i	$0.0265484\pi$
$233$	2.54327i	0.166615i	−0.996524	+	0.0833077i	$0.973452\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 6.80790i	− 0.442220i
$238$	0	0
$239$	25.3020	1.63665	0.818325	−	0.574755i	$-0.194903\pi$
$239$	25.3020	1.63665	0.818325	+	0.574755i	$0.194903\pi$
$240$	0	0
$241$	−2.66786	−0.171852	−0.0859261	−	0.996302i	$-0.527385\pi$
$241$	−2.66786	−0.171852	−0.0859261	+	0.996302i	$0.527385\pi$
$242$	0	0
$243$	− 57.4398i	− 3.68477i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 33.2159i	− 2.11348i
$248$	0	0
$249$	−5.93560	−0.376153
$250$	0	0
$251$	−24.7005	−1.55908	−0.779541	−	0.626351i	$-0.784548\pi$
$251$	−24.7005	−1.55908	−0.779541	+	0.626351i	$0.784548\pi$
$252$	0	0
$253$	11.2764i	0.708942i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 3.87900i	− 0.241965i	−0.992655	−	0.120983i	$-0.961395\pi$
$257$	− 3.87900i	− 0.241965i	0.992655	−	0.120983i	$-0.0386046\pi$
$258$	0	0
$259$	9.53377	0.592400
$260$	0	0
$261$	41.9442	2.59628
$262$	0	0
$263$	24.9876i	1.54080i	0.637560	+	0.770401i	$0.279944\pi$
$263$	24.9876i	1.54080i	−0.637560	+	0.770401i	$0.720056\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 55.9484i	− 3.42398i
$268$	0	0
$269$	−12.5161	−0.763120	−0.381560	−	0.924344i	$-0.624613\pi$
$269$	−12.5161	−0.763120	−0.381560	+	0.924344i	$0.624613\pi$
$270$	0	0
$271$	10.6890	0.649309	0.324654	−	0.945833i	$-0.394752\pi$
$271$	10.6890	0.649309	0.324654	+	0.945833i	$0.394752\pi$
$272$	0	0
$273$	28.4875i	1.72414i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 25.8423i	− 1.55272i	−0.630292	−	0.776358i	$-0.717065\pi$
$277$	− 25.8423i	− 1.55272i	0.630292	−	0.776358i	$-0.282935\pi$
$278$	0	0
$279$	−16.7375	−1.00205
$280$	0	0
$281$	−2.30493	−0.137501	−0.0687504	−	0.997634i	$-0.521901\pi$
$281$	−2.30493	−0.137501	−0.0687504	+	0.997634i	$0.521901\pi$
$282$	0	0
$283$	23.3632i	1.38880i	0.719589	+	0.694400i	$0.244330\pi$
$283$	23.3632i	1.38880i	−0.719589	+	0.694400i	$0.755670\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 4.47029i	− 0.263873i
$288$	0	0
$289$	−2.36880	−0.139341
$290$	0	0
$291$	−22.2995	−1.30722
$292$	0	0
$293$	− 7.51247i	− 0.438883i	−0.975626	−	0.219442i	$-0.929576\pi$
$293$	− 7.51247i	− 0.438883i	0.975626	−	0.219442i	$-0.0704236\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	69.2455i	4.01803i
$298$	0	0
$299$	−13.3015	−0.769243
$300$	0	0
$301$	−6.67652	−0.384828
$302$	0	0
$303$	− 8.77456i	− 0.504085i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	19.4708i	1.11126i	0.831430	+	0.555630i	$0.187523\pi$
$307$	19.4708i	1.11126i	−0.831430	+	0.555630i	$0.812477\pi$
$308$	0	0
$309$	39.1803	2.22889
$310$	0	0
$311$	19.5223	1.10701	0.553505	−	0.832846i	$-0.313290\pi$
$311$	19.5223	1.10701	0.553505	+	0.832846i	$0.313290\pi$
$312$	0	0
$313$	− 12.9443i	− 0.731654i	−0.930683	−	0.365827i	$-0.880786\pi$
$313$	− 12.9443i	− 0.731654i	0.930683	−	0.365827i	$-0.119214\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.69052i	0.0949493i	0.998872	+	0.0474746i	$0.0151173\pi$
$317$	1.69052i	0.0949493i	−0.998872	+	0.0474746i	$0.984883\pi$
$318$	0	0
$319$	−20.9427	−1.17257
$320$	0	0
$321$	5.18537	0.289419
$322$	0	0
$323$	30.5633i	1.70059i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 54.9118i	− 3.03663i
$328$	0	0
$329$	−17.9882	−0.991724
$330$	0	0
$331$	26.6197	1.46315	0.731575	−	0.681760i	$-0.238785\pi$
$331$	26.6197	1.46315	0.731575	+	0.681760i	$0.238785\pi$
$332$	0	0
$333$	43.3497i	2.37555i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	23.3453i	1.27170i	0.771813	+	0.635849i	$0.219350\pi$
$337$	23.3453i	1.27170i	−0.771813	+	0.635849i	$0.780650\pi$
$338$	0	0
$339$	44.7094	2.42828
$340$	0	0
$341$	8.35704	0.452559
$342$	0	0
$343$	19.3071i	1.04249i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	20.8473i	1.11914i	0.828782	+	0.559572i	$0.189034\pi$
$347$	20.8473i	1.11914i	−0.828782	+	0.559572i	$0.810966\pi$
$348$	0	0
$349$	34.9728	1.87205	0.936025	−	0.351932i	$-0.114475\pi$
$349$	34.9728	1.87205	0.936025	+	0.351932i	$0.114475\pi$
$350$	0	0
$351$	−81.6809	−4.35980
$352$	0	0
$353$	− 31.8695i	− 1.69624i	−0.529804	−	0.848120i	$-0.677735\pi$
$353$	− 31.8695i	− 1.69624i	0.529804	−	0.848120i	$-0.322265\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 26.2125i	− 1.38731i
$358$	0	0
$359$	−14.2488	−0.752020	−0.376010	−	0.926616i	$-0.622704\pi$
$359$	−14.2488	−0.752020	−0.376010	+	0.926616i	$0.622704\pi$
$360$	0	0
$361$	29.2279	1.53831
$362$	0	0
$363$	− 18.1458i	− 0.952410i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	5.45931i	0.284974i	0.989797	+	0.142487i	$0.0455099\pi$
$367$	5.45931i	0.284974i	−0.989797	+	0.142487i	$0.954490\pi$
$368$	0	0
$369$	20.3262	1.05814
$370$	0	0
$371$	−22.2520	−1.15526
$372$	0	0
$373$	21.6071i	1.11877i	0.828907	+	0.559387i	$0.188963\pi$
$373$	21.6071i	1.11877i	−0.828907	+	0.559387i	$0.811037\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 24.7037i	− 1.27230i
$378$	0	0
$379$	0.937644	0.0481635	0.0240818	−	0.999710i	$-0.492334\pi$
$379$	0.937644	0.0481635	0.0240818	+	0.999710i	$0.492334\pi$
$380$	0	0
$381$	−23.8241	−1.22055
$382$	0	0
$383$	− 1.03060i	− 0.0526610i	−0.999653	−	0.0263305i	$-0.991618\pi$
$383$	− 1.03060i	− 0.0526610i	0.999653	−	0.0263305i	$-0.00838223\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 30.3579i	− 1.54318i
$388$	0	0
$389$	−6.15907	−0.312277	−0.156139	−	0.987735i	$-0.549905\pi$
$389$	−6.15907	−0.312277	−0.156139	+	0.987735i	$0.549905\pi$
$390$	0	0
$391$	12.2392	0.618964
$392$	0	0
$393$	− 21.5287i	− 1.08598i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	10.4794i	0.525946i	0.964803	+	0.262973i	$0.0847031\pi$
$397$	10.4794i	0.525946i	−0.964803	+	0.262973i	$0.915297\pi$
$398$	0	0
$399$	−41.3625	−2.07071
$400$	0	0
$401$	−8.42455	−0.420702	−0.210351	−	0.977626i	$-0.567461\pi$
$401$	−8.42455	−0.420702	−0.210351	+	0.977626i	$0.567461\pi$
$402$	0	0
$403$	9.85782i	0.491053i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 21.6445i	− 1.07288i
$408$	0	0
$409$	23.0199	1.13826	0.569130	−	0.822248i	$-0.307280\pi$
$409$	23.0199	1.13826	0.569130	+	0.822248i	$0.307280\pi$
$410$	0	0
$411$	47.1421	2.32535
$412$	0	0
$413$	− 1.17671i	− 0.0579023i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	19.4576i	0.952841i
$418$	0	0
$419$	−21.1995	−1.03566	−0.517832	−	0.855482i	$-0.673261\pi$
$419$	−21.1995	−1.03566	−0.517832	+	0.855482i	$0.673261\pi$
$420$	0	0
$421$	−24.7412	−1.20581	−0.602907	−	0.797811i	$-0.705991\pi$
$421$	−24.7412	−1.20581	−0.602907	+	0.797811i	$0.705991\pi$
$422$	0	0
$423$	− 81.7919i	− 3.97686i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	10.2735i	0.497169i
$428$	0	0
$429$	64.6751	3.12254
$430$	0	0
$431$	−8.12915	−0.391567	−0.195784	−	0.980647i	$-0.562725\pi$
$431$	−8.12915	−0.391567	−0.195784	+	0.980647i	$0.562725\pi$
$432$	0	0
$433$	19.6656i	0.945068i	0.881312	+	0.472534i	$0.156661\pi$
$433$	19.6656i	0.945068i	−0.881312	+	0.472534i	$0.843339\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 19.3131i	− 0.923869i
$438$	0	0
$439$	24.3362	1.16150	0.580752	−	0.814080i	$-0.302758\pi$
$439$	24.3362	1.16150	0.580752	+	0.814080i	$0.302758\pi$
$440$	0	0
$441$	−30.9420	−1.47343
$442$	0	0
$443$	− 2.57922i	− 0.122543i	−0.998121	−	0.0612713i	$-0.980485\pi$
$443$	− 2.57922i	− 0.122543i	0.998121	−	0.0612713i	$-0.0195155\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	30.5633i	1.44560i
$448$	0	0
$449$	−6.48530	−0.306060	−0.153030	−	0.988222i	$-0.548903\pi$
$449$	−6.48530	−0.306060	−0.153030	+	0.988222i	$0.548903\pi$
$450$	0	0
$451$	−10.1489	−0.477892
$452$	0	0
$453$	− 2.47887i	− 0.116467i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 11.0820i	− 0.518394i	−0.965824	−	0.259197i	$-0.916542\pi$
$457$	− 11.0820i	− 0.518394i	0.965824	−	0.259197i	$-0.0834579\pi$
$458$	0	0
$459$	75.1579	3.50807
$460$	0	0
$461$	−37.0380	−1.72503	−0.862517	−	0.506029i	$-0.831113\pi$
$461$	−37.0380	−1.72503	−0.862517	+	0.506029i	$0.831113\pi$
$462$	0	0
$463$	− 15.0285i	− 0.698435i	−0.937042	−	0.349217i	$-0.886447\pi$
$463$	− 15.0285i	− 0.698435i	0.937042	−	0.349217i	$-0.113553\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0.633156i	0.0292990i	0.999893	+	0.0146495i	$0.00466324\pi$
$467$	0.633156i	0.0292990i	−0.999893	+	0.0146495i	$0.995337\pi$
$468$	0	0
$469$	−15.5931	−0.720023
$470$	0	0
$471$	23.5099	1.08328
$472$	0	0
$473$	15.1577i	0.696951i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 101.179i	− 4.63266i
$478$	0	0
$479$	−1.21499	−0.0555144	−0.0277572	−	0.999615i	$-0.508837\pi$
$479$	−1.21499	−0.0555144	−0.0277572	+	0.999615i	$0.508837\pi$
$480$	0	0
$481$	25.5315	1.16414
$482$	0	0
$483$	16.5638i	0.753678i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	9.87838i	0.447632i	0.974631	+	0.223816i	$0.0718515\pi$
$487$	9.87838i	0.447632i	−0.974631	+	0.223816i	$0.928149\pi$
$488$	0	0
$489$	49.5228	2.23950
$490$	0	0
$491$	−1.26035	−0.0568788	−0.0284394	−	0.999596i	$-0.509054\pi$
$491$	−1.26035	−0.0568788	−0.0284394	+	0.999596i	$0.509054\pi$
$492$	0	0
$493$	22.7309i	1.02375i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	21.7894i	0.977388i
$498$	0	0
$499$	32.0660	1.43547	0.717736	−	0.696315i	$-0.245178\pi$
$499$	32.0660	1.43547	0.717736	+	0.696315i	$0.245178\pi$
$500$	0	0
$501$	42.4984	1.89869
$502$	0	0
$503$	33.6134i	1.49875i	0.662148	+	0.749373i	$0.269645\pi$
$503$	33.6134i	1.49875i	−0.662148	+	0.749373i	$0.730355\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	32.9371i	1.46279i
$508$	0	0
$509$	−33.3021	−1.47609	−0.738046	−	0.674751i	$-0.764251\pi$
$509$	−33.3021	−1.47609	−0.738046	+	0.674751i	$0.764251\pi$
$510$	0	0
$511$	9.66730	0.427656
$512$	0	0
$513$	− 118.597i	− 5.23617i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	40.8386i	1.79608i
$518$	0	0
$519$	−19.8183	−0.869929
$520$	0	0
$521$	4.09690	0.179489	0.0897443	−	0.995965i	$-0.471395\pi$
$521$	4.09690	0.179489	0.0897443	+	0.995965i	$0.471395\pi$
$522$	0	0
$523$	14.2414i	0.622735i	0.950290	+	0.311368i	$0.100787\pi$
$523$	14.2414i	0.622735i	−0.950290	+	0.311368i	$0.899213\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 9.07058i	− 0.395121i
$528$	0	0
$529$	15.2660	0.663739
$530$	0	0
$531$	5.35048	0.232191
$532$	0	0
$533$	− 11.9714i	− 0.518541i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 15.8373i	− 0.683430i
$538$	0	0
$539$	15.4493	0.665449
$540$	0	0
$541$	2.80200	0.120467	0.0602337	−	0.998184i	$-0.480815\pi$
$541$	2.80200	0.120467	0.0602337	+	0.998184i	$0.480815\pi$
$542$	0	0
$543$	59.3859i	2.54849i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 8.85292i	− 0.378523i	−0.981927	−	0.189262i	$-0.939391\pi$
$547$	− 8.85292i	− 0.378523i	0.981927	−	0.189262i	$-0.0606094\pi$
$548$	0	0
$549$	−46.7132	−1.99367
$550$	0	0
$551$	35.8685	1.52805
$552$	0	0
$553$	3.64609i	0.155048i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 4.44721i	− 0.188434i	−0.995552	−	0.0942171i	$-0.969965\pi$
$557$	− 4.44721i	− 0.188434i	0.995552	−	0.0942171i	$-0.0300348\pi$
$558$	0	0
$559$	−17.8797	−0.756233
$560$	0	0
$561$	−59.5102	−2.51252
$562$	0	0
$563$	39.3846i	1.65986i	0.557866	+	0.829931i	$0.311620\pi$
$563$	39.3846i	1.65986i	−0.557866	+	0.829931i	$0.688380\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	58.2014i	2.44423i
$568$	0	0
$569$	2.48079	0.104000	0.0520001	−	0.998647i	$-0.483440\pi$
$569$	2.48079	0.104000	0.0520001	+	0.998647i	$0.483440\pi$
$570$	0	0
$571$	4.32601	0.181038	0.0905190	−	0.995895i	$-0.471147\pi$
$571$	4.32601	0.181038	0.0905190	+	0.995895i	$0.471147\pi$
$572$	0	0
$573$	56.8445i	2.37472i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	5.38643i	0.224240i	0.993695	+	0.112120i	$0.0357641\pi$
$577$	5.38643i	0.224240i	−0.993695	+	0.112120i	$0.964236\pi$
$578$	0	0
$579$	−49.7457	−2.06736
$580$	0	0
$581$	3.17892	0.131884
$582$	0	0
$583$	50.5185i	2.09226i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	2.63539i	0.108774i	0.998520	+	0.0543871i	$0.0173205\pi$
$587$	2.63539i	0.108774i	−0.998520	+	0.0543871i	$0.982679\pi$
$588$	0	0
$589$	−14.3131	−0.589760
$590$	0	0
$591$	−16.2510	−0.668476
$592$	0	0
$593$	10.8849i	0.446988i	0.974705	+	0.223494i	$0.0717463\pi$
$593$	10.8849i	0.446988i	−0.974705	+	0.223494i	$0.928254\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	74.0879i	3.03221i
$598$	0	0
$599$	−35.9152	−1.46746	−0.733729	−	0.679443i	$-0.762222\pi$
$599$	−35.9152	−1.46746	−0.733729	+	0.679443i	$0.762222\pi$
$600$	0	0
$601$	−25.5505	−1.04223	−0.521114	−	0.853487i	$-0.674483\pi$
$601$	−25.5505	−1.04223	−0.521114	+	0.853487i	$0.674483\pi$
$602$	0	0
$603$	− 70.9013i	− 2.88732i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	12.3993i	0.503270i	0.967822	+	0.251635i	$0.0809683\pi$
$607$	12.3993i	0.503270i	−0.967822	+	0.251635i	$0.919032\pi$
$608$	0	0
$609$	−30.7625	−1.24656
$610$	0	0
$611$	−48.1726	−1.94885
$612$	0	0
$613$	− 1.45088i	− 0.0586005i	−0.999571	−	0.0293003i	$-0.990672\pi$
$613$	− 1.45088i	− 0.0586005i	0.999571	−	0.0293003i	$-0.00932790\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 8.74714i	− 0.352147i	−0.984377	−	0.176073i	$-0.943660\pi$
$617$	− 8.74714i	− 0.352147i	0.984377	−	0.176073i	$-0.0563396\pi$
$618$	0	0
$619$	48.2337	1.93868	0.969338	−	0.245730i	$-0.0790278\pi$
$619$	48.2337	1.93868	0.969338	+	0.245730i	$0.0790278\pi$
$620$	0	0
$621$	−47.4925	−1.90581
$622$	0	0
$623$	29.9642i	1.20049i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	93.9051i	3.75021i
$628$	0	0
$629$	−23.4925	−0.936709
$630$	0	0
$631$	45.4575	1.80963	0.904817	−	0.425801i	$-0.140008\pi$
$631$	45.4575	1.80963	0.904817	+	0.425801i	$0.140008\pi$
$632$	0	0
$633$	68.4340i	2.72001i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	18.2238i	0.722052i
$638$	0	0
$639$	−99.0757	−3.91937
$640$	0	0
$641$	37.9442	1.49871	0.749353	−	0.662170i	$-0.230364\pi$
$641$	37.9442	1.49871	0.749353	+	0.662170i	$0.230364\pi$
$642$	0	0
$643$	− 20.4638i	− 0.807012i	−0.914977	−	0.403506i	$-0.867791\pi$
$643$	− 20.4638i	− 0.807012i	0.914977	−	0.403506i	$-0.132209\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	33.7257i	1.32589i	0.748666	+	0.662947i	$0.230695\pi$
$647$	33.7257i	1.32589i	−0.748666	+	0.662947i	$0.769305\pi$
$648$	0	0
$649$	−2.67149	−0.104865
$650$	0	0
$651$	12.2755	0.481117
$652$	0	0
$653$	− 10.2243i	− 0.400106i	−0.979785	−	0.200053i	$-0.935889\pi$
$653$	− 10.2243i	− 0.400106i	0.979785	−	0.200053i	$-0.0641114\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	43.9569i	1.71492i
$658$	0	0
$659$	20.2742	0.789772	0.394886	−	0.918730i	$-0.370784\pi$
$659$	20.2742	0.789772	0.394886	+	0.918730i	$0.370784\pi$
$660$	0	0
$661$	0.329864	0.0128302	0.00641510	−	0.999979i	$-0.497958\pi$
$661$	0.329864	0.0128302	0.00641510	+	0.999979i	$0.497958\pi$
$662$	0	0
$663$	− 70.1972i	− 2.72623i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 14.3637i	− 0.556165i
$668$	0	0
$669$	−47.6633	−1.84277
$670$	0	0
$671$	23.3238	0.900407
$672$	0	0
$673$	− 29.3960i	− 1.13313i	−0.824017	−	0.566566i	$-0.808272\pi$
$673$	− 29.3960i	− 1.13313i	0.824017	−	0.566566i	$-0.191728\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.05346i	0.0404876i	0.999795	+	0.0202438i	$0.00644425\pi$
$677$	1.05346i	0.0404876i	−0.999795	+	0.0202438i	$0.993556\pi$
$678$	0	0
$679$	11.9429	0.458327
$680$	0	0
$681$	−30.5007	−1.16879
$682$	0	0
$683$	− 19.5661i	− 0.748675i	−0.927293	−	0.374337i	$-0.877870\pi$
$683$	− 19.5661i	− 0.748675i	0.927293	−	0.374337i	$-0.122130\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	53.7799i	2.05183i
$688$	0	0
$689$	−59.5908	−2.27023
$690$	0	0
$691$	−46.2129	−1.75802	−0.879011	−	0.476802i	$-0.841796\pi$
$691$	−46.2129	−1.75802	−0.879011	+	0.476802i	$0.841796\pi$
$692$	0	0
$693$	− 58.8115i	− 2.23406i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	11.0154i	0.417238i
$698$	0	0
$699$	8.48133	0.320793
$700$	0	0
$701$	−19.2134	−0.725679	−0.362840	−	0.931852i	$-0.618193\pi$
$701$	−19.2134	−0.725679	−0.362840	+	0.931852i	$0.618193\pi$
$702$	0	0
$703$	37.0704i	1.39814i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	4.69937i	0.176738i
$708$	0	0
$709$	4.96644	0.186519	0.0932594	−	0.995642i	$-0.470271\pi$
$709$	4.96644	0.186519	0.0932594	+	0.995642i	$0.470271\pi$
$710$	0	0
$711$	−16.5787	−0.621748
$712$	0	0
$713$	5.73173i	0.214655i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 84.3774i	− 3.15113i
$718$	0	0
$719$	2.76975	0.103294	0.0516471	−	0.998665i	$-0.483553\pi$
$719$	2.76975	0.103294	0.0516471	+	0.998665i	$0.483553\pi$
$720$	0	0
$721$	−20.9837	−0.781475
$722$	0	0
$723$	8.89683i	0.330877i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	33.9826i	1.26035i	0.776455	+	0.630173i	$0.217016\pi$
$727$	33.9826i	1.26035i	−0.776455	+	0.630173i	$0.782984\pi$
$728$	0	0
$729$	−93.7893	−3.47368
$730$	0	0
$731$	16.4519	0.608495
$732$	0	0
$733$	− 9.75576i	− 0.360337i	−0.983636	−	0.180169i	$-0.942336\pi$
$733$	− 9.75576i	− 0.360337i	0.983636	−	0.180169i	$-0.0576644\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	35.4010i	1.30401i
$738$	0	0
$739$	3.98140	0.146458	0.0732291	−	0.997315i	$-0.476670\pi$
$739$	3.98140	0.146458	0.0732291	+	0.997315i	$0.476670\pi$
$740$	0	0
$741$	−110.769	−4.06920
$742$	0	0
$743$	− 39.5319i	− 1.45029i	−0.688599	−	0.725143i	$-0.741774\pi$
$743$	− 39.5319i	− 1.45029i	0.688599	−	0.725143i	$-0.258226\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	14.4544i	0.528860i
$748$	0	0
$749$	−2.77712	−0.101474
$750$	0	0
$751$	2.99037	0.109120	0.0545600	−	0.998510i	$-0.482624\pi$
$751$	2.99037	0.109120	0.0545600	+	0.998510i	$0.482624\pi$
$752$	0	0
$753$	82.3715i	3.00178i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 14.4685i	− 0.525866i	−0.964814	−	0.262933i	$-0.915310\pi$
$757$	− 14.4685i	− 0.525866i	0.964814	−	0.262933i	$-0.0846898\pi$
$758$	0	0
$759$	37.6047	1.36496
$760$	0	0
$761$	−7.76482	−0.281474	−0.140737	−	0.990047i	$-0.544947\pi$
$761$	−7.76482	−0.281474	−0.140737	+	0.990047i	$0.544947\pi$
$762$	0	0
$763$	29.4090i	1.06468i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 3.15125i	− 0.113785i
$768$	0	0
$769$	41.2823	1.48868	0.744338	−	0.667803i	$-0.232765\pi$
$769$	41.2823	1.48868	0.744338	+	0.667803i	$0.232765\pi$
$770$	0	0
$771$	−12.9357	−0.465869
$772$	0	0
$773$	− 15.3856i	− 0.553381i	−0.960959	−	0.276691i	$-0.910762\pi$
$773$	− 15.3856i	− 0.553381i	0.960959	−	0.276691i	$-0.0892377\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 31.7933i	− 1.14058i
$778$	0	0
$779$	17.3819	0.622773
$780$	0	0
$781$	49.4684	1.77012
$782$	0	0
$783$	− 88.2039i	− 3.15215i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 30.3700i	− 1.08257i	−0.840838	−	0.541287i	$-0.817937\pi$
$787$	− 30.3700i	− 1.08257i	0.840838	−	0.541287i	$-0.182063\pi$
$788$	0	0
$789$	83.3289	2.96659
$790$	0	0
$791$	−23.9449	−0.851383
$792$	0	0
$793$	27.5124i	0.976995i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 11.9375i	− 0.422848i	−0.977394	−	0.211424i	$-0.932190\pi$
$797$	− 11.9375i	− 0.422848i	0.977394	−	0.211424i	$-0.0678100\pi$
$798$	0	0
$799$	44.3255	1.56812
$800$	0	0
$801$	−136.246	−4.81402
$802$	0	0
$803$	− 21.9476i	− 0.774515i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	41.7388i	1.46928i
$808$	0	0
$809$	−9.92975	−0.349111	−0.174556	−	0.984647i	$-0.555849\pi$
$809$	−9.92975	−0.349111	−0.174556	+	0.984647i	$0.555849\pi$
$810$	0	0
$811$	−35.0971	−1.23243	−0.616213	−	0.787580i	$-0.711334\pi$
$811$	−35.0971	−1.23243	−0.616213	+	0.787580i	$0.711334\pi$
$812$	0	0
$813$	− 35.6457i	− 1.25015i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 25.9605i	− 0.908244i
$818$	0	0
$819$	69.3731	2.42409
$820$	0	0
$821$	13.2442	0.462224	0.231112	−	0.972927i	$-0.425764\pi$
$821$	13.2442	0.462224	0.231112	+	0.972927i	$0.425764\pi$
$822$	0	0
$823$	10.3502i	0.360786i	0.983595	+	0.180393i	$0.0577370\pi$
$823$	10.3502i	0.360786i	−0.983595	+	0.180393i	$0.942263\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	26.7372i	0.929743i	0.885378	+	0.464871i	$0.153899\pi$
$827$	26.7372i	0.929743i	−0.885378	+	0.464871i	$0.846101\pi$
$828$	0	0
$829$	31.2632	1.08582	0.542909	−	0.839792i	$-0.317323\pi$
$829$	31.2632	1.08582	0.542909	+	0.839792i	$0.317323\pi$
$830$	0	0
$831$	−86.1793	−2.98953
$832$	0	0
$833$	− 16.7684i	− 0.580991i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	35.1971i	1.21659i
$838$	0	0
$839$	−12.7498	−0.440173	−0.220087	−	0.975480i	$-0.570634\pi$
$839$	−12.7498	−0.440173	−0.220087	+	0.975480i	$0.570634\pi$
$840$	0	0
$841$	−2.32348	−0.0801200
$842$	0	0
$843$	7.68652i	0.264738i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	9.71833i	0.333926i
$848$	0	0
$849$	77.9120	2.67393
$850$	0	0
$851$	14.8450	0.508881
$852$	0	0
$853$	33.5497i	1.14872i	0.818603	+	0.574359i	$0.194749\pi$
$853$	33.5497i	1.14872i	−0.818603	+	0.574359i	$0.805251\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 8.07787i	− 0.275935i	−0.990437	−	0.137967i	$-0.955943\pi$
$857$	− 8.07787i	− 0.275935i	0.990437	−	0.137967i	$-0.0440569\pi$
$858$	0	0
$859$	−49.1730	−1.67776	−0.838880	−	0.544316i	$-0.816789\pi$
$859$	−49.1730	−1.67776	−0.838880	+	0.544316i	$0.816789\pi$
$860$	0	0
$861$	−14.9076	−0.508048
$862$	0	0
$863$	13.7683i	0.468680i	0.972155	+	0.234340i	$0.0752929\pi$
$863$	13.7683i	0.468680i	−0.972155	+	0.234340i	$0.924707\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	7.89950i	0.268281i
$868$	0	0
$869$	8.27771	0.280802
$870$	0	0
$871$	−41.7584	−1.41493
$872$	0	0
$873$	54.3041i	1.83791i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 19.4377i	− 0.656364i	−0.944614	−	0.328182i	$-0.893564\pi$
$877$	− 19.4377i	− 0.656364i	0.944614	−	0.328182i	$-0.106436\pi$
$878$	0	0
$879$	−25.0527	−0.845006
$880$	0	0
$881$	6.89720	0.232373	0.116186	−	0.993227i	$-0.462933\pi$
$881$	6.89720	0.232373	0.116186	+	0.993227i	$0.462933\pi$
$882$	0	0
$883$	18.1775i	0.611721i	0.952076	+	0.305861i	$0.0989442\pi$
$883$	18.1775i	0.611721i	−0.952076	+	0.305861i	$0.901056\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	16.0469i	0.538801i	0.963028	+	0.269400i	$0.0868255\pi$
$887$	16.0469i	0.538801i	−0.963028	+	0.269400i	$0.913174\pi$
$888$	0	0
$889$	12.7595	0.427938
$890$	0	0
$891$	132.134	4.42667
$892$	0	0
$893$	− 69.9442i	− 2.34059i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	44.3579i	1.48107i
$898$	0	0
$899$	−10.6451	−0.355033
$900$	0	0
$901$	54.8319	1.82672
$902$	0	0
$903$	22.2649i	0.740931i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 4.16580i	− 0.138323i	−0.997605	−	0.0691615i	$-0.977968\pi$
$907$	− 4.16580i	− 0.138323i	0.997605	−	0.0691615i	$-0.0220324\pi$
$908$	0	0
$909$	−21.3679	−0.708729
$910$	0	0
$911$	39.3846	1.30487	0.652435	−	0.757845i	$-0.273748\pi$
$911$	39.3846	1.30487	0.652435	+	0.757845i	$0.273748\pi$
$912$	0	0
$913$	− 7.21708i	− 0.238851i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	11.5301i	0.380757i
$918$	0	0
$919$	−50.6597	−1.67111	−0.835555	−	0.549406i	$-0.814854\pi$
$919$	−50.6597	−1.67111	−0.835555	+	0.549406i	$0.814854\pi$
$920$	0	0
$921$	64.9316	2.13957
$922$	0	0
$923$	58.3521i	1.92068i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 95.4123i	− 3.13375i
$928$	0	0
$929$	−11.1283	−0.365107	−0.182553	−	0.983196i	$-0.558436\pi$
$929$	−11.1283	−0.365107	−0.182553	+	0.983196i	$0.558436\pi$
$930$	0	0
$931$	−26.4600	−0.867191
$932$	0	0
$933$	− 65.1033i	− 2.13139i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 32.4472i	− 1.06000i	−0.847997	−	0.530001i	$-0.822191\pi$
$937$	− 32.4472i	− 1.06000i	0.847997	−	0.530001i	$-0.177809\pi$
$938$	0	0
$939$	−43.1667	−1.40869
$940$	0	0
$941$	−34.7793	−1.13377	−0.566887	−	0.823796i	$-0.691852\pi$
$941$	−34.7793	−1.13377	−0.566887	+	0.823796i	$0.691852\pi$
$942$	0	0
$943$	− 6.96068i	− 0.226671i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 22.9403i	− 0.745459i	−0.927940	−	0.372729i	$-0.878422\pi$
$947$	− 22.9403i	− 0.745459i	0.927940	−	0.372729i	$-0.121578\pi$
$948$	0	0
$949$	25.8891	0.840394
$950$	0	0
$951$	5.63758	0.182811
$952$	0	0
$953$	41.1854i	1.33412i	0.745002	+	0.667062i	$0.232449\pi$
$953$	41.1854i	1.33412i	−0.745002	+	0.667062i	$0.767551\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	69.8400i	2.25761i
$958$	0	0
$959$	−25.2478	−0.815295
$960$	0	0
$961$	−26.7522	−0.862973
$962$	0	0
$963$	− 12.6275i	− 0.406914i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	28.4819i	0.915916i	0.888974	+	0.457958i	$0.151419\pi$
$967$	28.4819i	0.915916i	−0.888974	+	0.457958i	$0.848581\pi$
$968$	0	0
$969$	101.923	3.27424
$970$	0	0
$971$	−50.1014	−1.60783	−0.803916	−	0.594743i	$-0.797254\pi$
$971$	−50.1014	−1.60783	−0.803916	+	0.594743i	$0.797254\pi$
$972$	0	0
$973$	− 10.4209i	− 0.334077i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	14.6370i	0.468280i	0.972203	+	0.234140i	$0.0752275\pi$
$977$	14.6370i	0.468280i	−0.972203	+	0.234140i	$0.924773\pi$
$978$	0	0
$979$	68.0275	2.17417
$980$	0	0
$981$	−133.722	−4.26941
$982$	0	0
$983$	19.1817i	0.611800i	0.952064	+	0.305900i	$0.0989574\pi$
$983$	19.1817i	0.611800i	−0.952064	+	0.305900i	$0.901043\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	59.9874i	1.90942i
$988$	0	0
$989$	−10.3960	−0.330574
$990$	0	0
$991$	7.30268	0.231977	0.115989	−	0.993251i	$-0.462996\pi$
$991$	7.30268	0.231977	0.115989	+	0.993251i	$0.462996\pi$
$992$	0	0
$993$	− 88.7717i	− 2.81708i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 57.4816i	− 1.82046i	−0.414104	−	0.910230i	$-0.635905\pi$
$997$	− 57.4816i	− 1.82046i	0.414104	−	0.910230i	$-0.364095\pi$
$998$	0	0
$999$	91.1595	2.88416

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4000.2.c.h.1249.2		16
4.3	odd	2	inner	4000.2.c.h.1249.15		16
5.2	odd	4		4000.2.a.r.1.1	yes	8
5.3	odd	4		4000.2.a.q.1.8	yes	8
5.4	even	2	inner	4000.2.c.h.1249.16		16
20.3	even	4		4000.2.a.q.1.1	✓	8
20.7	even	4		4000.2.a.r.1.8	yes	8
20.19	odd	2	inner	4000.2.c.h.1249.1		16
40.3	even	4		8000.2.a.cb.1.8		8
40.13	odd	4		8000.2.a.cb.1.1		8
40.27	even	4		8000.2.a.ca.1.1		8
40.37	odd	4		8000.2.a.ca.1.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4000.2.a.q.1.1	✓	8	20.3	even	4
4000.2.a.q.1.8	yes	8	5.3	odd	4
4000.2.a.r.1.1	yes	8	5.2	odd	4
4000.2.a.r.1.8	yes	8	20.7	even	4
4000.2.c.h.1249.1		16	20.19	odd	2	inner
4000.2.c.h.1249.2		16	1.1	even	1	trivial
4000.2.c.h.1249.15		16	4.3	odd	2	inner
4000.2.c.h.1249.16		16	5.4	even	2	inner
8000.2.a.ca.1.1		8	40.27	even	4
8000.2.a.ca.1.8		8	40.37	odd	4
8000.2.a.cb.1.1		8	40.13	odd	4
8000.2.a.cb.1.8		8	40.3	even	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:	\( N \)	\(=\)	\( 4000 = 2^{5} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4000.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.33481i q^{3} +1.78602i q^{7} -8.12097 q^{9} +O(q^{10})\)	\(q-3.33481i q^{3} +1.78602i q^{7} -8.12097 q^{9} +4.05479 q^{11} +4.78297i q^{13} -4.40100i q^{17} -6.94463 q^{19} +5.95604 q^{21} +2.78101i q^{23} +17.0775i q^{27} -5.16493 q^{29} +2.06103 q^{31} -13.5220i q^{33} -5.33800i q^{37} +15.9503 q^{39} -2.50293 q^{41} +3.73821i q^{43} +10.0717i q^{47} +3.81014 q^{49} -14.6765 q^{51} +12.4590i q^{53} +23.1590i q^{57} -0.658848 q^{59} +5.75217 q^{61} -14.5042i q^{63} +8.73065i q^{67} +9.27414 q^{69} +12.2000 q^{71} -5.41276i q^{73} +7.24193i q^{77} +2.04146 q^{79} +32.5872 q^{81} -1.77989i q^{83} +17.2241i q^{87} +16.7771 q^{89} -8.54247 q^{91} -6.87314i q^{93} -6.68690i q^{97} -32.9288 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 36 q^{9}+O(q^{10}) \) 16 * q - 36 * q^9	\( 16 q - 36 q^{9} + 36 q^{21} - 48 q^{29} + 36 q^{41} - 52 q^{49} + 44 q^{61} - 64 q^{69} + 120 q^{81} - 20 q^{89}+O(q^{100}) \) 16 * q - 36 * q^9 + 36 * q^21 - 48 * q^29 + 36 * q^41 - 52 * q^49 + 44 * q^61 - 64 * q^69 + 120 * q^81 - 20 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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