LMFDB - Embedded newform 4008.2.a.f.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4008,2,Mod(1,4008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4008, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4008 = 2^{3} \cdot 3 \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4008.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$32.0040411301$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.284897.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 6x^{3} + 5x^{2} + 10x + 3 $ x^5 - 2x^4 - 6x^3 + 5x^2 + 10x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.435916$ of defining polynomial
Character	$\chi$	$=$	4008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +2.07208 q^{5} +1.37406 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +2.07208 q^{5} +1.37406 q^{7} +1.00000 q^{9} -5.27463 q^{11} -6.01023 q^{13} +2.07208 q^{15} +0.202551 q^{17} -0.336870 q^{19} +1.37406 q^{21} -2.04334 q^{23} -0.706472 q^{25} +1.00000 q^{27} -6.05765 q^{29} -5.31798 q^{31} -5.27463 q^{33} +2.84717 q^{35} +1.31836 q^{37} -6.01023 q^{39} -11.1438 q^{41} -0.965489 q^{43} +2.07208 q^{45} +4.19640 q^{47} -5.11196 q^{49} +0.202551 q^{51} +8.35963 q^{53} -10.9295 q^{55} -0.336870 q^{57} +0.730755 q^{59} -10.2419 q^{61} +1.37406 q^{63} -12.4537 q^{65} +0.951843 q^{67} -2.04334 q^{69} +12.3009 q^{71} +12.2704 q^{73} -0.706472 q^{75} -7.24767 q^{77} -15.3159 q^{79} +1.00000 q^{81} +0.663677 q^{83} +0.419704 q^{85} -6.05765 q^{87} -10.6676 q^{89} -8.25842 q^{91} -5.31798 q^{93} -0.698022 q^{95} +2.07616 q^{97} -5.27463 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{3} + q^{5} - 4 q^{7} + 5 q^{9}+O(q^{10}) $ 5 * q + 5 * q^3 + q^5 - 4 * q^7 + 5 * q^9	$ 5 q + 5 q^{3} + q^{5} - 4 q^{7} + 5 q^{9} - 3 q^{11} - 14 q^{13} + q^{15} - 13 q^{17} - 2 q^{19} - 4 q^{21} - 5 q^{23} + 2 q^{25} + 5 q^{27} + 13 q^{29} + 2 q^{31} - 3 q^{33} - 12 q^{35} - 5 q^{37} - 14 q^{39} - 20 q^{41} - 20 q^{43} + q^{45} + q^{47} - 9 q^{49} - 13 q^{51} - 3 q^{53} - 3 q^{55} - 2 q^{57} - q^{59} - 34 q^{61} - 4 q^{63} - 22 q^{65} - 16 q^{67} - 5 q^{69} - 5 q^{71} - 12 q^{73} + 2 q^{75} - 8 q^{77} - 20 q^{79} + 5 q^{81} - 15 q^{83} - 27 q^{85} + 13 q^{87} - 48 q^{89} + 7 q^{91} + 2 q^{93} - 5 q^{95} - 21 q^{97} - 3 q^{99}+O(q^{100}) $ 5 * q + 5 * q^3 + q^5 - 4 * q^7 + 5 * q^9 - 3 * q^11 - 14 * q^13 + q^15 - 13 * q^17 - 2 * q^19 - 4 * q^21 - 5 * q^23 + 2 * q^25 + 5 * q^27 + 13 * q^29 + 2 * q^31 - 3 * q^33 - 12 * q^35 - 5 * q^37 - 14 * q^39 - 20 * q^41 - 20 * q^43 + q^45 + q^47 - 9 * q^49 - 13 * q^51 - 3 * q^53 - 3 * q^55 - 2 * q^57 - q^59 - 34 * q^61 - 4 * q^63 - 22 * q^65 - 16 * q^67 - 5 * q^69 - 5 * q^71 - 12 * q^73 + 2 * q^75 - 8 * q^77 - 20 * q^79 + 5 * q^81 - 15 * q^83 - 27 * q^85 + 13 * q^87 - 48 * q^89 + 7 * q^91 + 2 * q^93 - 5 * q^95 - 21 * q^97 - 3 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	2.07208	0.926664	0.463332	−	0.886185i	$-0.346654\pi$
$5$	2.07208	0.926664	0.463332	+	0.886185i	$0.346654\pi$
$6$	0	0
$7$	1.37406	0.519346	0.259673	−	0.965697i	$-0.416385\pi$
$7$	1.37406	0.519346	0.259673	+	0.965697i	$0.416385\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.27463	−1.59036	−0.795181	−	0.606372i	$-0.792624\pi$
$11$	−5.27463	−1.59036	−0.795181	+	0.606372i	$0.792624\pi$
$12$	0	0
$13$	−6.01023	−1.66694	−0.833469	−	0.552567i	$-0.813648\pi$
$13$	−6.01023	−1.66694	−0.833469	+	0.552567i	$0.813648\pi$
$14$	0	0
$15$	2.07208	0.535010
$16$	0	0
$17$	0.202551	0.0491260	0.0245630	−	0.999698i	$-0.492181\pi$
$17$	0.202551	0.0491260	0.0245630	+	0.999698i	$0.492181\pi$
$18$	0	0
$19$	−0.336870	−0.0772832	−0.0386416	−	0.999253i	$-0.512303\pi$
$19$	−0.336870	−0.0772832	−0.0386416	+	0.999253i	$0.512303\pi$
$20$	0	0
$21$	1.37406	0.299845
$22$	0	0
$23$	−2.04334	−0.426066	−0.213033	−	0.977045i	$-0.568334\pi$
$23$	−2.04334	−0.426066	−0.213033	+	0.977045i	$0.568334\pi$
$24$	0	0
$25$	−0.706472	−0.141294
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−6.05765	−1.12488	−0.562439	−	0.826839i	$-0.690137\pi$
$29$	−6.05765	−1.12488	−0.562439	+	0.826839i	$0.690137\pi$
$30$	0	0
$31$	−5.31798	−0.955137	−0.477568	−	0.878595i	$-0.658482\pi$
$31$	−5.31798	−0.955137	−0.477568	+	0.878595i	$0.658482\pi$
$32$	0	0
$33$	−5.27463	−0.918196
$34$	0	0
$35$	2.84717	0.481259
$36$	0	0
$37$	1.31836	0.216736	0.108368	−	0.994111i	$-0.465437\pi$
$37$	1.31836	0.216736	0.108368	+	0.994111i	$0.465437\pi$
$38$	0	0
$39$	−6.01023	−0.962407
$40$	0	0
$41$	−11.1438	−1.74037	−0.870183	−	0.492728i	$-0.836000\pi$
$41$	−11.1438	−1.74037	−0.870183	+	0.492728i	$0.836000\pi$
$42$	0	0
$43$	−0.965489	−0.147236	−0.0736179	−	0.997287i	$-0.523455\pi$
$43$	−0.965489	−0.147236	−0.0736179	+	0.997287i	$0.523455\pi$
$44$	0	0
$45$	2.07208	0.308888
$46$	0	0
$47$	4.19640	0.612108	0.306054	−	0.952014i	$-0.400991\pi$
$47$	4.19640	0.612108	0.306054	+	0.952014i	$0.400991\pi$
$48$	0	0
$49$	−5.11196	−0.730279
$50$	0	0
$51$	0.202551	0.0283629
$52$	0	0
$53$	8.35963	1.14828	0.574142	−	0.818756i	$-0.305336\pi$
$53$	8.35963	1.14828	0.574142	+	0.818756i	$0.305336\pi$
$54$	0	0
$55$	−10.9295	−1.47373
$56$	0	0
$57$	−0.336870	−0.0446195
$58$	0	0
$59$	0.730755	0.0951362	0.0475681	−	0.998868i	$-0.484853\pi$
$59$	0.730755	0.0951362	0.0475681	+	0.998868i	$0.484853\pi$
$60$	0	0
$61$	−10.2419	−1.31134	−0.655671	−	0.755047i	$-0.727614\pi$
$61$	−10.2419	−1.31134	−0.655671	+	0.755047i	$0.727614\pi$
$62$	0	0
$63$	1.37406	0.173115
$64$	0	0
$65$	−12.4537	−1.54469
$66$	0	0
$67$	0.951843	0.116286	0.0581430	−	0.998308i	$-0.481482\pi$
$67$	0.951843	0.116286	0.0581430	+	0.998308i	$0.481482\pi$
$68$	0	0
$69$	−2.04334	−0.245989
$70$	0	0
$71$	12.3009	1.45984	0.729922	−	0.683531i	$-0.239556\pi$
$71$	12.3009	1.45984	0.729922	+	0.683531i	$0.239556\pi$
$72$	0	0
$73$	12.2704	1.43614	0.718072	−	0.695968i	$-0.245025\pi$
$73$	12.2704	1.43614	0.718072	+	0.695968i	$0.245025\pi$
$74$	0	0
$75$	−0.706472	−0.0815764
$76$	0	0
$77$	−7.24767	−0.825949
$78$	0	0
$79$	−15.3159	−1.72317	−0.861587	−	0.507610i	$-0.830529\pi$
$79$	−15.3159	−1.72317	−0.861587	+	0.507610i	$0.830529\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	0.663677	0.0728480	0.0364240	−	0.999336i	$-0.488403\pi$
$83$	0.663677	0.0728480	0.0364240	+	0.999336i	$0.488403\pi$
$84$	0	0
$85$	0.419704	0.0455232
$86$	0	0
$87$	−6.05765	−0.649448
$88$	0	0
$89$	−10.6676	−1.13076	−0.565381	−	0.824830i	$-0.691271\pi$
$89$	−10.6676	−1.13076	−0.565381	+	0.824830i	$0.691271\pi$
$90$	0	0
$91$	−8.25842	−0.865718
$92$	0	0
$93$	−5.31798	−0.551448
$94$	0	0
$95$	−0.698022	−0.0716155
$96$	0	0
$97$	2.07616	0.210802	0.105401	−	0.994430i	$-0.466387\pi$
$97$	2.07616	0.210802	0.105401	+	0.994430i	$0.466387\pi$
$98$	0	0
$99$	−5.27463	−0.530121
$100$	0	0
$101$	−17.3465	−1.72604	−0.863021	−	0.505168i	$-0.831430\pi$
$101$	−17.3465	−1.72604	−0.863021	+	0.505168i	$0.831430\pi$
$102$	0	0
$103$	16.8119	1.65653	0.828264	−	0.560338i	$-0.189329\pi$
$103$	16.8119	1.65653	0.828264	+	0.560338i	$0.189329\pi$
$104$	0	0
$105$	2.84717	0.277855
$106$	0	0
$107$	−3.15743	−0.305240	−0.152620	−	0.988285i	$-0.548771\pi$
$107$	−3.15743	−0.305240	−0.152620	+	0.988285i	$0.548771\pi$
$108$	0	0
$109$	−5.11105	−0.489550	−0.244775	−	0.969580i	$-0.578714\pi$
$109$	−5.11105	−0.489550	−0.244775	+	0.969580i	$0.578714\pi$
$110$	0	0
$111$	1.31836	0.125133
$112$	0	0
$113$	8.36231	0.786660	0.393330	−	0.919397i	$-0.371323\pi$
$113$	8.36231	0.786660	0.393330	+	0.919397i	$0.371323\pi$
$114$	0	0
$115$	−4.23397	−0.394820
$116$	0	0
$117$	−6.01023	−0.555646
$118$	0	0
$119$	0.278318	0.0255134
$120$	0	0
$121$	16.8218	1.52925
$122$	0	0
$123$	−11.1438	−1.00480
$124$	0	0
$125$	−11.8243	−1.05760
$126$	0	0
$127$	7.98111	0.708209	0.354104	−	0.935206i	$-0.384786\pi$
$127$	7.98111	0.708209	0.354104	+	0.935206i	$0.384786\pi$
$128$	0	0
$129$	−0.965489	−0.0850066
$130$	0	0
$131$	0.126390	0.0110428	0.00552138	−	0.999985i	$-0.498242\pi$
$131$	0.126390	0.0110428	0.00552138	+	0.999985i	$0.498242\pi$
$132$	0	0
$133$	−0.462879	−0.0401367
$134$	0	0
$135$	2.07208	0.178337
$136$	0	0
$137$	8.31975	0.710805	0.355402	−	0.934713i	$-0.384344\pi$
$137$	8.31975	0.710805	0.355402	+	0.934713i	$0.384344\pi$
$138$	0	0
$139$	2.22951	0.189105	0.0945525	−	0.995520i	$-0.469858\pi$
$139$	2.22951	0.189105	0.0945525	+	0.995520i	$0.469858\pi$
$140$	0	0
$141$	4.19640	0.353401
$142$	0	0
$143$	31.7018	2.65103
$144$	0	0
$145$	−12.5519	−1.04238
$146$	0	0
$147$	−5.11196	−0.421627
$148$	0	0
$149$	9.87451	0.808952	0.404476	−	0.914549i	$-0.367454\pi$
$149$	9.87451	0.808952	0.404476	+	0.914549i	$0.367454\pi$
$150$	0	0
$151$	−9.00556	−0.732862	−0.366431	−	0.930445i	$-0.619420\pi$
$151$	−9.00556	−0.732862	−0.366431	+	0.930445i	$0.619420\pi$
$152$	0	0
$153$	0.202551	0.0163753
$154$	0	0
$155$	−11.0193	−0.885091
$156$	0	0
$157$	−6.54956	−0.522712	−0.261356	−	0.965242i	$-0.584170\pi$
$157$	−6.54956	−0.522712	−0.261356	+	0.965242i	$0.584170\pi$
$158$	0	0
$159$	8.35963	0.662962
$160$	0	0
$161$	−2.80768	−0.221276
$162$	0	0
$163$	−2.22508	−0.174282	−0.0871409	−	0.996196i	$-0.527773\pi$
$163$	−2.22508	−0.174282	−0.0871409	+	0.996196i	$0.527773\pi$
$164$	0	0
$165$	−10.9295	−0.850859
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	23.1228	1.77868
$170$	0	0
$171$	−0.336870	−0.0257611
$172$	0	0
$173$	5.46198	0.415266	0.207633	−	0.978207i	$-0.433424\pi$
$173$	5.46198	0.415266	0.207633	+	0.978207i	$0.433424\pi$
$174$	0	0
$175$	−0.970736	−0.0733807
$176$	0	0
$177$	0.730755	0.0549269
$178$	0	0
$179$	25.3449	1.89437	0.947183	−	0.320693i	$-0.103916\pi$
$179$	25.3449	1.89437	0.947183	+	0.320693i	$0.103916\pi$
$180$	0	0
$181$	−4.47384	−0.332538	−0.166269	−	0.986080i	$-0.553172\pi$
$181$	−4.47384	−0.332538	−0.166269	+	0.986080i	$0.553172\pi$
$182$	0	0
$183$	−10.2419	−0.757103
$184$	0	0
$185$	2.73174	0.200842
$186$	0	0
$187$	−1.06839	−0.0781281
$188$	0	0
$189$	1.37406	0.0999483
$190$	0	0
$191$	23.2392	1.68153	0.840764	−	0.541402i	$-0.182106\pi$
$191$	23.2392	1.68153	0.840764	+	0.541402i	$0.182106\pi$
$192$	0	0
$193$	7.47719	0.538220	0.269110	−	0.963110i	$-0.413271\pi$
$193$	7.47719	0.538220	0.269110	+	0.963110i	$0.413271\pi$
$194$	0	0
$195$	−12.4537	−0.891827
$196$	0	0
$197$	17.5011	1.24690	0.623451	−	0.781862i	$-0.285730\pi$
$197$	17.5011	1.24690	0.623451	+	0.781862i	$0.285730\pi$
$198$	0	0
$199$	22.2943	1.58040	0.790199	−	0.612850i	$-0.209977\pi$
$199$	22.2943	1.58040	0.790199	+	0.612850i	$0.209977\pi$
$200$	0	0
$201$	0.951843	0.0671378
$202$	0	0
$203$	−8.32358	−0.584201
$204$	0	0
$205$	−23.0908	−1.61273
$206$	0	0
$207$	−2.04334	−0.142022
$208$	0	0
$209$	1.77686	0.122908
$210$	0	0
$211$	−12.0997	−0.832975	−0.416487	−	0.909142i	$-0.636739\pi$
$211$	−12.0997	−0.832975	−0.416487	+	0.909142i	$0.636739\pi$
$212$	0	0
$213$	12.3009	0.842841
$214$	0	0
$215$	−2.00057	−0.136438
$216$	0	0
$217$	−7.30723	−0.496047
$218$	0	0
$219$	12.2704	0.829159
$220$	0	0
$221$	−1.21738	−0.0818899
$222$	0	0
$223$	4.28101	0.286678	0.143339	−	0.989674i	$-0.454216\pi$
$223$	4.28101	0.286678	0.143339	+	0.989674i	$0.454216\pi$
$224$	0	0
$225$	−0.706472	−0.0470981
$226$	0	0
$227$	6.47975	0.430076	0.215038	−	0.976606i	$-0.431012\pi$
$227$	6.47975	0.430076	0.215038	+	0.976606i	$0.431012\pi$
$228$	0	0
$229$	15.3238	1.01263	0.506314	−	0.862349i	$-0.331008\pi$
$229$	15.3238	1.01263	0.506314	+	0.862349i	$0.331008\pi$
$230$	0	0
$231$	−7.24767	−0.476862
$232$	0	0
$233$	−3.54194	−0.232040	−0.116020	−	0.993247i	$-0.537014\pi$
$233$	−3.54194	−0.232040	−0.116020	+	0.993247i	$0.537014\pi$
$234$	0	0
$235$	8.69529	0.567218
$236$	0	0
$237$	−15.3159	−0.994875
$238$	0	0
$239$	−13.3809	−0.865537	−0.432768	−	0.901505i	$-0.642463\pi$
$239$	−13.3809	−0.865537	−0.432768	+	0.901505i	$0.642463\pi$
$240$	0	0
$241$	−28.4045	−1.82969	−0.914846	−	0.403803i	$-0.867688\pi$
$241$	−28.4045	−1.82969	−0.914846	+	0.403803i	$0.867688\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−10.5924	−0.676723
$246$	0	0
$247$	2.02466	0.128826
$248$	0	0
$249$	0.663677	0.0420588
$250$	0	0
$251$	−3.32320	−0.209758	−0.104879	−	0.994485i	$-0.533446\pi$
$251$	−3.32320	−0.209758	−0.104879	+	0.994485i	$0.533446\pi$
$252$	0	0
$253$	10.7779	0.677600
$254$	0	0
$255$	0.419704	0.0262829
$256$	0	0
$257$	5.17474	0.322791	0.161396	−	0.986890i	$-0.448400\pi$
$257$	5.17474	0.322791	0.161396	+	0.986890i	$0.448400\pi$
$258$	0	0
$259$	1.81150	0.112561
$260$	0	0
$261$	−6.05765	−0.374959
$262$	0	0
$263$	−4.72275	−0.291217	−0.145609	−	0.989342i	$-0.546514\pi$
$263$	−4.72275	−0.291217	−0.145609	+	0.989342i	$0.546514\pi$
$264$	0	0
$265$	17.3218	1.06407
$266$	0	0
$267$	−10.6676	−0.652846
$268$	0	0
$269$	−5.74547	−0.350307	−0.175154	−	0.984541i	$-0.556042\pi$
$269$	−5.74547	−0.350307	−0.175154	+	0.984541i	$0.556042\pi$
$270$	0	0
$271$	−29.7842	−1.80926	−0.904632	−	0.426194i	$-0.859854\pi$
$271$	−29.7842	−1.80926	−0.904632	+	0.426194i	$0.859854\pi$
$272$	0	0
$273$	−8.25842	−0.499822
$274$	0	0
$275$	3.72638	0.224709
$276$	0	0
$277$	−28.1486	−1.69128	−0.845642	−	0.533751i	$-0.820782\pi$
$277$	−28.1486	−1.69128	−0.845642	+	0.533751i	$0.820782\pi$
$278$	0	0
$279$	−5.31798	−0.318379
$280$	0	0
$281$	1.00907	0.0601962	0.0300981	−	0.999547i	$-0.490418\pi$
$281$	1.00907	0.0601962	0.0300981	+	0.999547i	$0.490418\pi$
$282$	0	0
$283$	4.30214	0.255736	0.127868	−	0.991791i	$-0.459187\pi$
$283$	4.30214	0.255736	0.127868	+	0.991791i	$0.459187\pi$
$284$	0	0
$285$	−0.698022	−0.0413472
$286$	0	0
$287$	−15.3122	−0.903853
$288$	0	0
$289$	−16.9590	−0.997587
$290$	0	0
$291$	2.07616	0.121707
$292$	0	0
$293$	−3.15536	−0.184338	−0.0921691	−	0.995743i	$-0.529380\pi$
$293$	−3.15536	−0.184338	−0.0921691	+	0.995743i	$0.529380\pi$
$294$	0	0
$295$	1.51418	0.0881592
$296$	0	0
$297$	−5.27463	−0.306065
$298$	0	0
$299$	12.2810	0.710226
$300$	0	0
$301$	−1.32664	−0.0764663
$302$	0	0
$303$	−17.3465	−0.996531
$304$	0	0
$305$	−21.2221	−1.21517
$306$	0	0
$307$	−6.89668	−0.393615	−0.196807	−	0.980442i	$-0.563057\pi$
$307$	−6.89668	−0.393615	−0.196807	+	0.980442i	$0.563057\pi$
$308$	0	0
$309$	16.8119	0.956397
$310$	0	0
$311$	−1.13302	−0.0642477	−0.0321239	−	0.999484i	$-0.510227\pi$
$311$	−1.13302	−0.0642477	−0.0321239	+	0.999484i	$0.510227\pi$
$312$	0	0
$313$	−10.1280	−0.572466	−0.286233	−	0.958160i	$-0.592403\pi$
$313$	−10.1280	−0.572466	−0.286233	+	0.958160i	$0.592403\pi$
$314$	0	0
$315$	2.84717	0.160420
$316$	0	0
$317$	−15.8272	−0.888945	−0.444473	−	0.895792i	$-0.646609\pi$
$317$	−15.8272	−0.888945	−0.444473	+	0.895792i	$0.646609\pi$
$318$	0	0
$319$	31.9519	1.78896
$320$	0	0
$321$	−3.15743	−0.176231
$322$	0	0
$323$	−0.0682334	−0.00379661
$324$	0	0
$325$	4.24606	0.235529
$326$	0	0
$327$	−5.11105	−0.282642
$328$	0	0
$329$	5.76611	0.317896
$330$	0	0
$331$	−9.06680	−0.498356	−0.249178	−	0.968458i	$-0.580160\pi$
$331$	−9.06680	−0.498356	−0.249178	+	0.968458i	$0.580160\pi$
$332$	0	0
$333$	1.31836	0.0722455
$334$	0	0
$335$	1.97230	0.107758
$336$	0	0
$337$	26.3569	1.43575	0.717876	−	0.696171i	$-0.245114\pi$
$337$	26.3569	1.43575	0.717876	+	0.696171i	$0.245114\pi$
$338$	0	0
$339$	8.36231	0.454178
$340$	0	0
$341$	28.0504	1.51901
$342$	0	0
$343$	−16.6426	−0.898614
$344$	0	0
$345$	−4.23397	−0.227949
$346$	0	0
$347$	−30.4435	−1.63429	−0.817146	−	0.576430i	$-0.804445\pi$
$347$	−30.4435	−1.63429	−0.817146	+	0.576430i	$0.804445\pi$
$348$	0	0
$349$	−21.7753	−1.16560	−0.582801	−	0.812615i	$-0.698043\pi$
$349$	−21.7753	−1.16560	−0.582801	+	0.812615i	$0.698043\pi$
$350$	0	0
$351$	−6.01023	−0.320802
$352$	0	0
$353$	−13.4066	−0.713560	−0.356780	−	0.934188i	$-0.616125\pi$
$353$	−13.4066	−0.713560	−0.356780	+	0.934188i	$0.616125\pi$
$354$	0	0
$355$	25.4884	1.35278
$356$	0	0
$357$	0.278318	0.0147302
$358$	0	0
$359$	5.78786	0.305471	0.152736	−	0.988267i	$-0.451192\pi$
$359$	5.78786	0.305471	0.152736	+	0.988267i	$0.451192\pi$
$360$	0	0
$361$	−18.8865	−0.994027
$362$	0	0
$363$	16.8218	0.882914
$364$	0	0
$365$	25.4253	1.33082
$366$	0	0
$367$	−11.1654	−0.582828	−0.291414	−	0.956597i	$-0.594126\pi$
$367$	−11.1654	−0.582828	−0.291414	+	0.956597i	$0.594126\pi$
$368$	0	0
$369$	−11.1438	−0.580122
$370$	0	0
$371$	11.4866	0.596357
$372$	0	0
$373$	−11.7084	−0.606238	−0.303119	−	0.952953i	$-0.598028\pi$
$373$	−11.7084	−0.606238	−0.303119	+	0.952953i	$0.598028\pi$
$374$	0	0
$375$	−11.8243	−0.610603
$376$	0	0
$377$	36.4078	1.87510
$378$	0	0
$379$	−8.73464	−0.448668	−0.224334	−	0.974512i	$-0.572021\pi$
$379$	−8.73464	−0.448668	−0.224334	+	0.974512i	$0.572021\pi$
$380$	0	0
$381$	7.98111	0.408884
$382$	0	0
$383$	−30.7953	−1.57356	−0.786782	−	0.617230i	$-0.788255\pi$
$383$	−30.7953	−1.57356	−0.786782	+	0.617230i	$0.788255\pi$
$384$	0	0
$385$	−15.0178	−0.765377
$386$	0	0
$387$	−0.965489	−0.0490786
$388$	0	0
$389$	−9.79229	−0.496489	−0.248244	−	0.968697i	$-0.579854\pi$
$389$	−9.79229	−0.496489	−0.248244	+	0.968697i	$0.579854\pi$
$390$	0	0
$391$	−0.413882	−0.0209309
$392$	0	0
$393$	0.126390	0.00637555
$394$	0	0
$395$	−31.7358	−1.59680
$396$	0	0
$397$	−28.2801	−1.41934	−0.709669	−	0.704535i	$-0.751156\pi$
$397$	−28.2801	−1.41934	−0.709669	+	0.704535i	$0.751156\pi$
$398$	0	0
$399$	−0.462879	−0.0231730
$400$	0	0
$401$	−21.4063	−1.06898	−0.534490	−	0.845175i	$-0.679496\pi$
$401$	−21.4063	−1.06898	−0.534490	+	0.845175i	$0.679496\pi$
$402$	0	0
$403$	31.9623	1.59215
$404$	0	0
$405$	2.07208	0.102963
$406$	0	0
$407$	−6.95385	−0.344690
$408$	0	0
$409$	−15.5676	−0.769769	−0.384885	−	0.922965i	$-0.625759\pi$
$409$	−15.5676	−0.769769	−0.384885	+	0.922965i	$0.625759\pi$
$410$	0	0
$411$	8.31975	0.410383
$412$	0	0
$413$	1.00410	0.0494086
$414$	0	0
$415$	1.37519	0.0675056
$416$	0	0
$417$	2.22951	0.109180
$418$	0	0
$419$	−6.33403	−0.309438	−0.154719	−	0.987959i	$-0.549447\pi$
$419$	−6.33403	−0.309438	−0.154719	+	0.987959i	$0.549447\pi$
$420$	0	0
$421$	6.61588	0.322438	0.161219	−	0.986919i	$-0.448457\pi$
$421$	6.61588	0.322438	0.161219	+	0.986919i	$0.448457\pi$
$422$	0	0
$423$	4.19640	0.204036
$424$	0	0
$425$	−0.143097	−0.00694122
$426$	0	0
$427$	−14.0730	−0.681040
$428$	0	0
$429$	31.7018	1.53058
$430$	0	0
$431$	12.5186	0.602999	0.301499	−	0.953466i	$-0.402513\pi$
$431$	12.5186	0.602999	0.301499	+	0.953466i	$0.402513\pi$
$432$	0	0
$433$	−23.6289	−1.13553	−0.567767	−	0.823189i	$-0.692192\pi$
$433$	−23.6289	−1.13553	−0.567767	+	0.823189i	$0.692192\pi$
$434$	0	0
$435$	−12.5519	−0.601820
$436$	0	0
$437$	0.688340	0.0329278
$438$	0	0
$439$	28.0247	1.33755	0.668774	−	0.743466i	$-0.266819\pi$
$439$	28.0247	1.33755	0.668774	+	0.743466i	$0.266819\pi$
$440$	0	0
$441$	−5.11196	−0.243426
$442$	0	0
$443$	33.1995	1.57735	0.788677	−	0.614808i	$-0.210766\pi$
$443$	33.1995	1.57735	0.788677	+	0.614808i	$0.210766\pi$
$444$	0	0
$445$	−22.1041	−1.04784
$446$	0	0
$447$	9.87451	0.467048
$448$	0	0
$449$	20.3396	0.959885	0.479942	−	0.877300i	$-0.340658\pi$
$449$	20.3396	0.959885	0.479942	+	0.877300i	$0.340658\pi$
$450$	0	0
$451$	58.7794	2.76781
$452$	0	0
$453$	−9.00556	−0.423118
$454$	0	0
$455$	−17.1121	−0.802229
$456$	0	0
$457$	−4.58262	−0.214366	−0.107183	−	0.994239i	$-0.534183\pi$
$457$	−4.58262	−0.214366	−0.107183	+	0.994239i	$0.534183\pi$
$458$	0	0
$459$	0.202551	0.00945429
$460$	0	0
$461$	37.0853	1.72723	0.863616	−	0.504150i	$-0.168194\pi$
$461$	37.0853	1.72723	0.863616	+	0.504150i	$0.168194\pi$
$462$	0	0
$463$	28.7461	1.33594	0.667972	−	0.744187i	$-0.267163\pi$
$463$	28.7461	1.33594	0.667972	+	0.744187i	$0.267163\pi$
$464$	0	0
$465$	−11.0193	−0.511007
$466$	0	0
$467$	7.43392	0.344001	0.172000	−	0.985097i	$-0.444977\pi$
$467$	7.43392	0.344001	0.172000	+	0.985097i	$0.444977\pi$
$468$	0	0
$469$	1.30789	0.0603928
$470$	0	0
$471$	−6.54956	−0.301788
$472$	0	0
$473$	5.09260	0.234158
$474$	0	0
$475$	0.237989	0.0109197
$476$	0	0
$477$	8.35963	0.382761
$478$	0	0
$479$	37.3938	1.70857	0.854284	−	0.519806i	$-0.173996\pi$
$479$	37.3938	1.70857	0.854284	+	0.519806i	$0.173996\pi$
$480$	0	0
$481$	−7.92362	−0.361286
$482$	0	0
$483$	−2.80768	−0.127754
$484$	0	0
$485$	4.30198	0.195343
$486$	0	0
$487$	17.5735	0.796333	0.398167	−	0.917313i	$-0.369647\pi$
$487$	17.5735	0.796333	0.398167	+	0.917313i	$0.369647\pi$
$488$	0	0
$489$	−2.22508	−0.100622
$490$	0	0
$491$	3.26494	0.147345	0.0736723	−	0.997283i	$-0.476528\pi$
$491$	3.26494	0.147345	0.0736723	+	0.997283i	$0.476528\pi$
$492$	0	0
$493$	−1.22699	−0.0552607
$494$	0	0
$495$	−10.9295	−0.491244
$496$	0	0
$497$	16.9021	0.758164
$498$	0	0
$499$	−26.5936	−1.19049	−0.595246	−	0.803543i	$-0.702945\pi$
$499$	−26.5936	−1.19049	−0.595246	+	0.803543i	$0.702945\pi$
$500$	0	0
$501$	1.00000	0.0446767
$502$	0	0
$503$	22.9368	1.02270	0.511351	−	0.859372i	$-0.329145\pi$
$503$	22.9368	1.02270	0.511351	+	0.859372i	$0.329145\pi$
$504$	0	0
$505$	−35.9434	−1.59946
$506$	0	0
$507$	23.1228	1.02692
$508$	0	0
$509$	−33.2388	−1.47328	−0.736642	−	0.676283i	$-0.763590\pi$
$509$	−33.2388	−1.47328	−0.736642	+	0.676283i	$0.763590\pi$
$510$	0	0
$511$	16.8603	0.745857
$512$	0	0
$513$	−0.336870	−0.0148732
$514$	0	0
$515$	34.8357	1.53504
$516$	0	0
$517$	−22.1345	−0.973473
$518$	0	0
$519$	5.46198	0.239754
$520$	0	0
$521$	−4.46558	−0.195641	−0.0978204	−	0.995204i	$-0.531187\pi$
$521$	−4.46558	−0.195641	−0.0978204	+	0.995204i	$0.531187\pi$
$522$	0	0
$523$	1.08095	0.0472668	0.0236334	−	0.999721i	$-0.492477\pi$
$523$	1.08095	0.0472668	0.0236334	+	0.999721i	$0.492477\pi$
$524$	0	0
$525$	−0.970736	−0.0423664
$526$	0	0
$527$	−1.07716	−0.0469220
$528$	0	0
$529$	−18.8248	−0.818468
$530$	0	0
$531$	0.730755	0.0317121
$532$	0	0
$533$	66.9767	2.90108
$534$	0	0
$535$	−6.54246	−0.282855
$536$	0	0
$537$	25.3449	1.09371
$538$	0	0
$539$	26.9637	1.16141
$540$	0	0
$541$	−16.5141	−0.709998	−0.354999	−	0.934867i	$-0.615519\pi$
$541$	−16.5141	−0.709998	−0.354999	+	0.934867i	$0.615519\pi$
$542$	0	0
$543$	−4.47384	−0.191991
$544$	0	0
$545$	−10.5905	−0.453648
$546$	0	0
$547$	0.00160491	6.86211e−5 0	3.43105e−5	−	1.00000i	$-0.499989\pi$
$547$	0.00160491	6.86211e−5 0	3.43105e−5		1.00000i	$0.499989\pi$
$548$	0	0
$549$	−10.2419	−0.437114
$550$	0	0
$551$	2.04064	0.0869341
$552$	0	0
$553$	−21.0450	−0.894924
$554$	0	0
$555$	2.73174	0.115956
$556$	0	0
$557$	19.6063	0.830747	0.415374	−	0.909651i	$-0.363651\pi$
$557$	19.6063	0.830747	0.415374	+	0.909651i	$0.363651\pi$
$558$	0	0
$559$	5.80281	0.245433
$560$	0	0
$561$	−1.06839	−0.0451073
$562$	0	0
$563$	−11.3368	−0.477790	−0.238895	−	0.971045i	$-0.576785\pi$
$563$	−11.3368	−0.477790	−0.238895	+	0.971045i	$0.576785\pi$
$564$	0	0
$565$	17.3274	0.728969
$566$	0	0
$567$	1.37406	0.0577051
$568$	0	0
$569$	−35.0972	−1.47135	−0.735676	−	0.677333i	$-0.763136\pi$
$569$	−35.0972	−1.47135	−0.735676	+	0.677333i	$0.763136\pi$
$570$	0	0
$571$	−14.7322	−0.616525	−0.308262	−	0.951301i	$-0.599748\pi$
$571$	−14.7322	−0.616525	−0.308262	+	0.951301i	$0.599748\pi$
$572$	0	0
$573$	23.2392	0.970830
$574$	0	0
$575$	1.44356	0.0602008
$576$	0	0
$577$	12.0436	0.501383	0.250691	−	0.968067i	$-0.419342\pi$
$577$	12.0436	0.501383	0.250691	+	0.968067i	$0.419342\pi$
$578$	0	0
$579$	7.47719	0.310741
$580$	0	0
$581$	0.911932	0.0378333
$582$	0	0
$583$	−44.0940	−1.82619
$584$	0	0
$585$	−12.4537	−0.514897
$586$	0	0
$587$	−39.3028	−1.62220	−0.811101	−	0.584906i	$-0.801131\pi$
$587$	−39.3028	−1.62220	−0.811101	+	0.584906i	$0.801131\pi$
$588$	0	0
$589$	1.79146	0.0738160
$590$	0	0
$591$	17.5011	0.719900
$592$	0	0
$593$	−15.1187	−0.620849	−0.310424	−	0.950598i	$-0.600471\pi$
$593$	−15.1187	−0.620849	−0.310424	+	0.950598i	$0.600471\pi$
$594$	0	0
$595$	0.576698	0.0236423
$596$	0	0
$597$	22.2943	0.912444
$598$	0	0
$599$	12.0427	0.492053	0.246027	−	0.969263i	$-0.420875\pi$
$599$	12.0427	0.492053	0.246027	+	0.969263i	$0.420875\pi$
$600$	0	0
$601$	31.6066	1.28926	0.644630	−	0.764495i	$-0.277012\pi$
$601$	31.6066	1.28926	0.644630	+	0.764495i	$0.277012\pi$
$602$	0	0
$603$	0.951843	0.0387620
$604$	0	0
$605$	34.8561	1.41710
$606$	0	0
$607$	37.1053	1.50606	0.753028	−	0.657988i	$-0.228592\pi$
$607$	37.1053	1.50606	0.753028	+	0.657988i	$0.228592\pi$
$608$	0	0
$609$	−8.32358	−0.337288
$610$	0	0
$611$	−25.2213	−1.02035
$612$	0	0
$613$	10.6070	0.428414	0.214207	−	0.976788i	$-0.431283\pi$
$613$	10.6070	0.428414	0.214207	+	0.976788i	$0.431283\pi$
$614$	0	0
$615$	−23.0908	−0.931113
$616$	0	0
$617$	−16.2374	−0.653693	−0.326847	−	0.945077i	$-0.605986\pi$
$617$	−16.2374	−0.653693	−0.326847	+	0.945077i	$0.605986\pi$
$618$	0	0
$619$	−36.8237	−1.48007	−0.740035	−	0.672569i	$-0.765191\pi$
$619$	−36.8237	−1.48007	−0.740035	+	0.672569i	$0.765191\pi$
$620$	0	0
$621$	−2.04334	−0.0819965
$622$	0	0
$623$	−14.6579	−0.587257
$624$	0	0
$625$	−20.9685	−0.838741
$626$	0	0
$627$	1.77686	0.0709611
$628$	0	0
$629$	0.267035	0.0106474
$630$	0	0
$631$	12.8100	0.509958	0.254979	−	0.966947i	$-0.417931\pi$
$631$	12.8100	0.509958	0.254979	+	0.966947i	$0.417931\pi$
$632$	0	0
$633$	−12.0997	−0.480918
$634$	0	0
$635$	16.5375	0.656271
$636$	0	0
$637$	30.7240	1.21733
$638$	0	0
$639$	12.3009	0.486615
$640$	0	0
$641$	12.6972	0.501509	0.250754	−	0.968051i	$-0.419321\pi$
$641$	12.6972	0.501509	0.250754	+	0.968051i	$0.419321\pi$
$642$	0	0
$643$	11.3118	0.446096	0.223048	−	0.974807i	$-0.428399\pi$
$643$	11.3118	0.446096	0.223048	+	0.974807i	$0.428399\pi$
$644$	0	0
$645$	−2.00057	−0.0787725
$646$	0	0
$647$	−16.1608	−0.635348	−0.317674	−	0.948200i	$-0.602902\pi$
$647$	−16.1608	−0.635348	−0.317674	+	0.948200i	$0.602902\pi$
$648$	0	0
$649$	−3.85446	−0.151301
$650$	0	0
$651$	−7.30723	−0.286393
$652$	0	0
$653$	−28.4137	−1.11191	−0.555957	−	0.831211i	$-0.687648\pi$
$653$	−28.4137	−1.11191	−0.555957	+	0.831211i	$0.687648\pi$
$654$	0	0
$655$	0.261891	0.0102329
$656$	0	0
$657$	12.2704	0.478715
$658$	0	0
$659$	43.1667	1.68154	0.840768	−	0.541396i	$-0.182104\pi$
$659$	43.1667	1.68154	0.840768	+	0.541396i	$0.182104\pi$
$660$	0	0
$661$	−40.3262	−1.56851	−0.784254	−	0.620440i	$-0.786954\pi$
$661$	−40.3262	−1.56851	−0.784254	+	0.620440i	$0.786954\pi$
$662$	0	0
$663$	−1.21738	−0.0472792
$664$	0	0
$665$	−0.959125	−0.0371933
$666$	0	0
$667$	12.3778	0.479272
$668$	0	0
$669$	4.28101	0.165514
$670$	0	0
$671$	54.0223	2.08551
$672$	0	0
$673$	−16.7667	−0.646310	−0.323155	−	0.946346i	$-0.604744\pi$
$673$	−16.7667	−0.646310	−0.323155	+	0.946346i	$0.604744\pi$
$674$	0	0
$675$	−0.706472	−0.0271921
$676$	0	0
$677$	37.1987	1.42966	0.714832	−	0.699297i	$-0.246503\pi$
$677$	37.1987	1.42966	0.714832	+	0.699297i	$0.246503\pi$
$678$	0	0
$679$	2.85277	0.109479
$680$	0	0
$681$	6.47975	0.248305
$682$	0	0
$683$	21.4255	0.819825	0.409913	−	0.912125i	$-0.365559\pi$
$683$	21.4255	0.819825	0.409913	+	0.912125i	$0.365559\pi$
$684$	0	0
$685$	17.2392	0.658677
$686$	0	0
$687$	15.3238	0.584641
$688$	0	0
$689$	−50.2433	−1.91412
$690$	0	0
$691$	−17.1907	−0.653966	−0.326983	−	0.945030i	$-0.606032\pi$
$691$	−17.1907	−0.653966	−0.326983	+	0.945030i	$0.606032\pi$
$692$	0	0
$693$	−7.24767	−0.275316
$694$	0	0
$695$	4.61974	0.175237
$696$	0	0
$697$	−2.25719	−0.0854972
$698$	0	0
$699$	−3.54194	−0.133968
$700$	0	0
$701$	18.6527	0.704504	0.352252	−	0.935905i	$-0.385416\pi$
$701$	18.6527	0.704504	0.352252	+	0.935905i	$0.385416\pi$
$702$	0	0
$703$	−0.444114	−0.0167501
$704$	0	0
$705$	8.69529	0.327484
$706$	0	0
$707$	−23.8352	−0.896413
$708$	0	0
$709$	−33.3235	−1.25149	−0.625744	−	0.780028i	$-0.715205\pi$
$709$	−33.3235	−1.25149	−0.625744	+	0.780028i	$0.715205\pi$
$710$	0	0
$711$	−15.3159	−0.574391
$712$	0	0
$713$	10.8664	0.406951
$714$	0	0
$715$	65.6887	2.45662
$716$	0	0
$717$	−13.3809	−0.499718
$718$	0	0
$719$	33.9117	1.26469	0.632346	−	0.774686i	$-0.282092\pi$
$719$	33.9117	1.26469	0.632346	+	0.774686i	$0.282092\pi$
$720$	0	0
$721$	23.1006	0.860312
$722$	0	0
$723$	−28.4045	−1.05637
$724$	0	0
$725$	4.27956	0.158939
$726$	0	0
$727$	−32.3637	−1.20030	−0.600152	−	0.799886i	$-0.704893\pi$
$727$	−32.3637	−1.20030	−0.600152	+	0.799886i	$0.704893\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−0.195561	−0.00723310
$732$	0	0
$733$	4.01849	0.148426	0.0742131	−	0.997242i	$-0.476355\pi$
$733$	4.01849	0.148426	0.0742131	+	0.997242i	$0.476355\pi$
$734$	0	0
$735$	−10.5924	−0.390706
$736$	0	0
$737$	−5.02062	−0.184937
$738$	0	0
$739$	−3.69570	−0.135948	−0.0679742	−	0.997687i	$-0.521654\pi$
$739$	−3.69570	−0.135948	−0.0679742	+	0.997687i	$0.521654\pi$
$740$	0	0
$741$	2.02466	0.0743779
$742$	0	0
$743$	32.3327	1.18617	0.593086	−	0.805139i	$-0.297909\pi$
$743$	32.3327	1.18617	0.593086	+	0.805139i	$0.297909\pi$
$744$	0	0
$745$	20.4608	0.749626
$746$	0	0
$747$	0.663677	0.0242827
$748$	0	0
$749$	−4.33851	−0.158526
$750$	0	0
$751$	2.13842	0.0780322	0.0390161	−	0.999239i	$-0.487578\pi$
$751$	2.13842	0.0780322	0.0390161	+	0.999239i	$0.487578\pi$
$752$	0	0
$753$	−3.32320	−0.121104
$754$	0	0
$755$	−18.6603	−0.679116
$756$	0	0
$757$	−27.1166	−0.985570	−0.492785	−	0.870151i	$-0.664021\pi$
$757$	−27.1166	−0.985570	−0.492785	+	0.870151i	$0.664021\pi$
$758$	0	0
$759$	10.7779	0.391212
$760$	0	0
$761$	17.7022	0.641705	0.320852	−	0.947129i	$-0.396031\pi$
$761$	17.7022	0.641705	0.320852	+	0.947129i	$0.396031\pi$
$762$	0	0
$763$	−7.02290	−0.254246
$764$	0	0
$765$	0.419704	0.0151744
$766$	0	0
$767$	−4.39200	−0.158586
$768$	0	0
$769$	−21.7313	−0.783651	−0.391825	−	0.920040i	$-0.628156\pi$
$769$	−21.7313	−0.783651	−0.391825	+	0.920040i	$0.628156\pi$
$770$	0	0
$771$	5.17474	0.186364
$772$	0	0
$773$	5.91086	0.212599	0.106299	−	0.994334i	$-0.466100\pi$
$773$	5.91086	0.212599	0.106299	+	0.994334i	$0.466100\pi$
$774$	0	0
$775$	3.75700	0.134956
$776$	0	0
$777$	1.81150	0.0649873
$778$	0	0
$779$	3.75400	0.134501
$780$	0	0
$781$	−64.8825	−2.32168
$782$	0	0
$783$	−6.05765	−0.216483
$784$	0	0
$785$	−13.5712	−0.484378
$786$	0	0
$787$	−39.2576	−1.39938	−0.699692	−	0.714445i	$-0.746679\pi$
$787$	−39.2576	−1.39938	−0.699692	+	0.714445i	$0.746679\pi$
$788$	0	0
$789$	−4.72275	−0.168134
$790$	0	0
$791$	11.4903	0.408549
$792$	0	0
$793$	61.5562	2.18592
$794$	0	0
$795$	17.3218	0.614342
$796$	0	0
$797$	48.4430	1.71594	0.857970	−	0.513700i	$-0.171725\pi$
$797$	48.4430	1.71594	0.857970	+	0.513700i	$0.171725\pi$
$798$	0	0
$799$	0.849987	0.0300704
$800$	0	0
$801$	−10.6676	−0.376921
$802$	0	0
$803$	−64.7220	−2.28399
$804$	0	0
$805$	−5.81774	−0.205048
$806$	0	0
$807$	−5.74547	−0.202250
$808$	0	0
$809$	−26.7053	−0.938910	−0.469455	−	0.882956i	$-0.655550\pi$
$809$	−26.7053	−0.938910	−0.469455	+	0.882956i	$0.655550\pi$
$810$	0	0
$811$	−51.0389	−1.79222	−0.896108	−	0.443836i	$-0.853617\pi$
$811$	−51.0389	−1.79222	−0.896108	+	0.443836i	$0.853617\pi$
$812$	0	0
$813$	−29.7842	−1.04458
$814$	0	0
$815$	−4.61055	−0.161501
$816$	0	0
$817$	0.325244	0.0113788
$818$	0	0
$819$	−8.25842	−0.288573
$820$	0	0
$821$	50.2885	1.75508	0.877540	−	0.479504i	$-0.159183\pi$
$821$	50.2885	1.75508	0.877540	+	0.479504i	$0.159183\pi$
$822$	0	0
$823$	−21.0372	−0.733310	−0.366655	−	0.930357i	$-0.619497\pi$
$823$	−21.0372	−0.733310	−0.366655	+	0.930357i	$0.619497\pi$
$824$	0	0
$825$	3.72638	0.129736
$826$	0	0
$827$	9.90656	0.344485	0.172242	−	0.985055i	$-0.444899\pi$
$827$	9.90656	0.344485	0.172242	+	0.985055i	$0.444899\pi$
$828$	0	0
$829$	−27.9584	−0.971035	−0.485518	−	0.874227i	$-0.661369\pi$
$829$	−27.9584	−0.971035	−0.485518	+	0.874227i	$0.661369\pi$
$830$	0	0
$831$	−28.1486	−0.976463
$832$	0	0
$833$	−1.03543	−0.0358757
$834$	0	0
$835$	2.07208	0.0717074
$836$	0	0
$837$	−5.31798	−0.183816
$838$	0	0
$839$	23.6314	0.815848	0.407924	−	0.913016i	$-0.366253\pi$
$839$	23.6314	0.815848	0.407924	+	0.913016i	$0.366253\pi$
$840$	0	0
$841$	7.69510	0.265348
$842$	0	0
$843$	1.00907	0.0347543
$844$	0	0
$845$	47.9125	1.64824
$846$	0	0
$847$	23.1141	0.794211
$848$	0	0
$849$	4.30214	0.147649
$850$	0	0
$851$	−2.69385	−0.0923441
$852$	0	0
$853$	24.1256	0.826045	0.413023	−	0.910721i	$-0.364473\pi$
$853$	24.1256	0.826045	0.413023	+	0.910721i	$0.364473\pi$
$854$	0	0
$855$	−0.698022	−0.0238718
$856$	0	0
$857$	−31.9576	−1.09165	−0.545825	−	0.837899i	$-0.683784\pi$
$857$	−31.9576	−1.09165	−0.545825	+	0.837899i	$0.683784\pi$
$858$	0	0
$859$	56.0883	1.91371	0.956854	−	0.290569i	$-0.0938445\pi$
$859$	56.0883	1.91371	0.956854	+	0.290569i	$0.0938445\pi$
$860$	0	0
$861$	−15.3122	−0.521840
$862$	0	0
$863$	49.9022	1.69869	0.849344	−	0.527839i	$-0.176998\pi$
$863$	49.9022	1.69869	0.849344	+	0.527839i	$0.176998\pi$
$864$	0	0
$865$	11.3177	0.384812
$866$	0	0
$867$	−16.9590	−0.575957
$868$	0	0
$869$	80.7858	2.74047
$870$	0	0
$871$	−5.72079	−0.193842
$872$	0	0
$873$	2.07616	0.0702674
$874$	0	0
$875$	−16.2473	−0.549259
$876$	0	0
$877$	28.1153	0.949388	0.474694	−	0.880151i	$-0.342559\pi$
$877$	28.1153	0.949388	0.474694	+	0.880151i	$0.342559\pi$
$878$	0	0
$879$	−3.15536	−0.106428
$880$	0	0
$881$	48.3928	1.63039	0.815197	−	0.579184i	$-0.196629\pi$
$881$	48.3928	1.63039	0.815197	+	0.579184i	$0.196629\pi$
$882$	0	0
$883$	−35.0305	−1.17887	−0.589435	−	0.807816i	$-0.700649\pi$
$883$	−35.0305	−1.17887	−0.589435	+	0.807816i	$0.700649\pi$
$884$	0	0
$885$	1.51418	0.0508988
$886$	0	0
$887$	−18.6796	−0.627200	−0.313600	−	0.949555i	$-0.601535\pi$
$887$	−18.6796	−0.627200	−0.313600	+	0.949555i	$0.601535\pi$
$888$	0	0
$889$	10.9665	0.367806
$890$	0	0
$891$	−5.27463	−0.176707
$892$	0	0
$893$	−1.41364	−0.0473057
$894$	0	0
$895$	52.5167	1.75544
$896$	0	0
$897$	12.2810	0.410049
$898$	0	0
$899$	32.2144	1.07441
$900$	0	0
$901$	1.69325	0.0564105
$902$	0	0
$903$	−1.32664	−0.0441479
$904$	0	0
$905$	−9.27018	−0.308151
$906$	0	0
$907$	−40.5486	−1.34639	−0.673197	−	0.739463i	$-0.735080\pi$
$907$	−40.5486	−1.34639	−0.673197	+	0.739463i	$0.735080\pi$
$908$	0	0
$909$	−17.3465	−0.575347
$910$	0	0
$911$	21.2864	0.705252	0.352626	−	0.935764i	$-0.385289\pi$
$911$	21.2864	0.705252	0.352626	+	0.935764i	$0.385289\pi$
$912$	0	0
$913$	−3.50065	−0.115855
$914$	0	0
$915$	−21.2221	−0.701580
$916$	0	0
$917$	0.173668	0.00573502
$918$	0	0
$919$	16.0390	0.529077	0.264538	−	0.964375i	$-0.414780\pi$
$919$	16.0390	0.529077	0.264538	+	0.964375i	$0.414780\pi$
$920$	0	0
$921$	−6.89668	−0.227253
$922$	0	0
$923$	−73.9310	−2.43347
$924$	0	0
$925$	−0.931382	−0.0306237
$926$	0	0
$927$	16.8119	0.552176
$928$	0	0
$929$	12.7224	0.417409	0.208704	−	0.977979i	$-0.433075\pi$
$929$	12.7224	0.417409	0.208704	+	0.977979i	$0.433075\pi$
$930$	0	0
$931$	1.72206	0.0564383
$932$	0	0
$933$	−1.13302	−0.0370935
$934$	0	0
$935$	−2.21378	−0.0723984
$936$	0	0
$937$	17.0752	0.557822	0.278911	−	0.960317i	$-0.410026\pi$
$937$	17.0752	0.557822	0.278911	+	0.960317i	$0.410026\pi$
$938$	0	0
$939$	−10.1280	−0.330513
$940$	0	0
$941$	29.7023	0.968267	0.484134	−	0.874994i	$-0.339135\pi$
$941$	29.7023	0.968267	0.484134	+	0.874994i	$0.339135\pi$
$942$	0	0
$943$	22.7706	0.741512
$944$	0	0
$945$	2.84717	0.0926184
$946$	0	0
$947$	−1.64205	−0.0533594	−0.0266797	−	0.999644i	$-0.508493\pi$
$947$	−1.64205	−0.0533594	−0.0266797	+	0.999644i	$0.508493\pi$
$948$	0	0
$949$	−73.7481	−2.39396
$950$	0	0
$951$	−15.8272	−0.513233
$952$	0	0
$953$	−4.67237	−0.151353	−0.0756764	−	0.997132i	$-0.524112\pi$
$953$	−4.67237	−0.151353	−0.0756764	+	0.997132i	$0.524112\pi$
$954$	0	0
$955$	48.1535	1.55821
$956$	0	0
$957$	31.9519	1.03286
$958$	0	0
$959$	11.4319	0.369154
$960$	0	0
$961$	−2.71913	−0.0877138
$962$	0	0
$963$	−3.15743	−0.101747
$964$	0	0
$965$	15.4933	0.498749
$966$	0	0
$967$	−2.49406	−0.0802037	−0.0401018	−	0.999196i	$-0.512768\pi$
$967$	−2.49406	−0.0802037	−0.0401018	+	0.999196i	$0.512768\pi$
$968$	0	0
$969$	−0.0682334	−0.00219197
$970$	0	0
$971$	43.9326	1.40986	0.704931	−	0.709275i	$-0.250978\pi$
$971$	43.9326	1.40986	0.704931	+	0.709275i	$0.250978\pi$
$972$	0	0
$973$	3.06349	0.0982110
$974$	0	0
$975$	4.24606	0.135983
$976$	0	0
$977$	44.8137	1.43372	0.716859	−	0.697218i	$-0.245579\pi$
$977$	44.8137	1.43372	0.716859	+	0.697218i	$0.245579\pi$
$978$	0	0
$979$	56.2676	1.79832
$980$	0	0
$981$	−5.11105	−0.163183
$982$	0	0
$983$	34.9940	1.11613	0.558067	−	0.829796i	$-0.311543\pi$
$983$	34.9940	1.11613	0.558067	+	0.829796i	$0.311543\pi$
$984$	0	0
$985$	36.2638	1.15546
$986$	0	0
$987$	5.76611	0.183537
$988$	0	0
$989$	1.97282	0.0627322
$990$	0	0
$991$	−26.5450	−0.843230	−0.421615	−	0.906775i	$-0.638537\pi$
$991$	−26.5450	−0.843230	−0.421615	+	0.906775i	$0.638537\pi$
$992$	0	0
$993$	−9.06680	−0.287726
$994$	0	0
$995$	46.1956	1.46450
$996$	0	0
$997$	51.9221	1.64439	0.822195	−	0.569206i	$-0.192749\pi$
$997$	51.9221	1.64439	0.822195	+	0.569206i	$0.192749\pi$
$998$	0	0
$999$	1.31836	0.0417110

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4008.2.a.f.1.4	✓	5
4.3	odd	2		8016.2.a.s.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4008.2.a.f.1.4	✓	5	1.1	even	1	trivial
8016.2.a.s.1.4		5	4.3	odd	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:	\( N \)	\(=\)	\( 4008 = 2^{3} \cdot 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4008.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +2.07208 q^{5} +1.37406 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +2.07208 q^{5} +1.37406 q^{7} +1.00000 q^{9} -5.27463 q^{11} -6.01023 q^{13} +2.07208 q^{15} +0.202551 q^{17} -0.336870 q^{19} +1.37406 q^{21} -2.04334 q^{23} -0.706472 q^{25} +1.00000 q^{27} -6.05765 q^{29} -5.31798 q^{31} -5.27463 q^{33} +2.84717 q^{35} +1.31836 q^{37} -6.01023 q^{39} -11.1438 q^{41} -0.965489 q^{43} +2.07208 q^{45} +4.19640 q^{47} -5.11196 q^{49} +0.202551 q^{51} +8.35963 q^{53} -10.9295 q^{55} -0.336870 q^{57} +0.730755 q^{59} -10.2419 q^{61} +1.37406 q^{63} -12.4537 q^{65} +0.951843 q^{67} -2.04334 q^{69} +12.3009 q^{71} +12.2704 q^{73} -0.706472 q^{75} -7.24767 q^{77} -15.3159 q^{79} +1.00000 q^{81} +0.663677 q^{83} +0.419704 q^{85} -6.05765 q^{87} -10.6676 q^{89} -8.25842 q^{91} -5.31798 q^{93} -0.698022 q^{95} +2.07616 q^{97} -5.27463 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{3} + q^{5} - 4 q^{7} + 5 q^{9}+O(q^{10}) \) 5 * q + 5 * q^3 + q^5 - 4 * q^7 + 5 * q^9	\( 5 q + 5 q^{3} + q^{5} - 4 q^{7} + 5 q^{9} - 3 q^{11} - 14 q^{13} + q^{15} - 13 q^{17} - 2 q^{19} - 4 q^{21} - 5 q^{23} + 2 q^{25} + 5 q^{27} + 13 q^{29} + 2 q^{31} - 3 q^{33} - 12 q^{35} - 5 q^{37} - 14 q^{39} - 20 q^{41} - 20 q^{43} + q^{45} + q^{47} - 9 q^{49} - 13 q^{51} - 3 q^{53} - 3 q^{55} - 2 q^{57} - q^{59} - 34 q^{61} - 4 q^{63} - 22 q^{65} - 16 q^{67} - 5 q^{69} - 5 q^{71} - 12 q^{73} + 2 q^{75} - 8 q^{77} - 20 q^{79} + 5 q^{81} - 15 q^{83} - 27 q^{85} + 13 q^{87} - 48 q^{89} + 7 q^{91} + 2 q^{93} - 5 q^{95} - 21 q^{97} - 3 q^{99}+O(q^{100}) \) 5 * q + 5 * q^3 + q^5 - 4 * q^7 + 5 * q^9 - 3 * q^11 - 14 * q^13 + q^15 - 13 * q^17 - 2 * q^19 - 4 * q^21 - 5 * q^23 + 2 * q^25 + 5 * q^27 + 13 * q^29 + 2 * q^31 - 3 * q^33 - 12 * q^35 - 5 * q^37 - 14 * q^39 - 20 * q^41 - 20 * q^43 + q^45 + q^47 - 9 * q^49 - 13 * q^51 - 3 * q^53 - 3 * q^55 - 2 * q^57 - q^59 - 34 * q^61 - 4 * q^63 - 22 * q^65 - 16 * q^67 - 5 * q^69 - 5 * q^71 - 12 * q^73 + 2 * q^75 - 8 * q^77 - 20 * q^79 + 5 * q^81 - 15 * q^83 - 27 * q^85 + 13 * q^87 - 48 * q^89 + 7 * q^91 + 2 * q^93 - 5 * q^95 - 21 * q^97 - 3 * q^99

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