LMFDB - Embedded newform 4008.2.a.j.1.7

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:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 26x^{8} - 3x^{7} + 220x^{6} + 42x^{5} - 675x^{4} - 67x^{3} + 628x^{2} - 48x - 2 $ x^10 - 26x^8 - 3x^7 + 220x^6 + 42x^5 - 675x^4 - 67x^3 + 628x^2 - 48x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-1.39234$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +0.392343 q^{5} +0.0476950 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +0.392343 q^{5} +0.0476950 q^{7} +1.00000 q^{9} +1.67730 q^{11} -4.70371 q^{13} -0.392343 q^{15} +0.835685 q^{17} +0.998285 q^{19} -0.0476950 q^{21} +1.31016 q^{23} -4.84607 q^{25} -1.00000 q^{27} +6.86363 q^{29} -9.48974 q^{31} -1.67730 q^{33} +0.0187128 q^{35} +2.44944 q^{37} +4.70371 q^{39} +0.448099 q^{41} +9.51987 q^{43} +0.392343 q^{45} -3.80613 q^{47} -6.99773 q^{49} -0.835685 q^{51} -9.39965 q^{53} +0.658077 q^{55} -0.998285 q^{57} -3.24192 q^{59} +0.335380 q^{61} +0.0476950 q^{63} -1.84547 q^{65} -14.2570 q^{67} -1.31016 q^{69} +5.78446 q^{71} -2.80343 q^{73} +4.84607 q^{75} +0.0799990 q^{77} +15.3705 q^{79} +1.00000 q^{81} -7.43473 q^{83} +0.327875 q^{85} -6.86363 q^{87} -5.92823 q^{89} -0.224344 q^{91} +9.48974 q^{93} +0.391670 q^{95} +4.70012 q^{97} +1.67730 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 10 q^{3} - 10 q^{5} + q^{7} + 10 q^{9}+O(q^{10}) $ 10 * q - 10 * q^3 - 10 * q^5 + q^7 + 10 * q^9	$ 10 q - 10 q^{3} - 10 q^{5} + q^{7} + 10 q^{9} - q^{11} - 6 q^{13} + 10 q^{15} - 9 q^{17} + 2 q^{19} - q^{21} + 7 q^{23} + 12 q^{25} - 10 q^{27} - 13 q^{29} + 23 q^{31} + q^{33} + q^{35} - 6 q^{37} + 6 q^{39} - 12 q^{41} - 10 q^{45} + 10 q^{47} + 7 q^{49} + 9 q^{51} - 26 q^{53} + 11 q^{55} - 2 q^{57} - 10 q^{59} - 10 q^{61} + q^{63} - 22 q^{65} - 5 q^{67} - 7 q^{69} + 25 q^{71} - 8 q^{73} - 12 q^{75} - 46 q^{77} + 26 q^{79} + 10 q^{81} - 14 q^{83} + 9 q^{85} + 13 q^{87} - 31 q^{89} - 3 q^{91} - 23 q^{93} - 5 q^{95} - 32 q^{97} - q^{99}+O(q^{100}) $ 10 * q - 10 * q^3 - 10 * q^5 + q^7 + 10 * q^9 - q^11 - 6 * q^13 + 10 * q^15 - 9 * q^17 + 2 * q^19 - q^21 + 7 * q^23 + 12 * q^25 - 10 * q^27 - 13 * q^29 + 23 * q^31 + q^33 + q^35 - 6 * q^37 + 6 * q^39 - 12 * q^41 - 10 * q^45 + 10 * q^47 + 7 * q^49 + 9 * q^51 - 26 * q^53 + 11 * q^55 - 2 * q^57 - 10 * q^59 - 10 * q^61 + q^63 - 22 * q^65 - 5 * q^67 - 7 * q^69 + 25 * q^71 - 8 * q^73 - 12 * q^75 - 46 * q^77 + 26 * q^79 + 10 * q^81 - 14 * q^83 + 9 * q^85 + 13 * q^87 - 31 * q^89 - 3 * q^91 - 23 * q^93 - 5 * q^95 - 32 * q^97 - 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
$5$	0.392343	0.175461	0.0877305	−	0.996144i	$-0.472039\pi$
$5$	0.392343	0.175461	0.0877305	+	0.996144i	$0.472039\pi$
$6$	0	0
$7$	0.0476950	0.0180270	0.00901352	−	0.999959i	$-0.497131\pi$
$7$	0.0476950	0.0180270	0.00901352	+	0.999959i	$0.497131\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.67730	0.505726	0.252863	−	0.967502i	$-0.418628\pi$
$11$	1.67730	0.505726	0.252863	+	0.967502i	$0.418628\pi$
$12$	0	0
$13$	−4.70371	−1.30458	−0.652288	−	0.757971i	$-0.726191\pi$
$13$	−4.70371	−1.30458	−0.652288	+	0.757971i	$0.726191\pi$
$14$	0	0
$15$	−0.392343	−0.101302
$16$	0	0
$17$	0.835685	0.202683	0.101342	−	0.994852i	$-0.467686\pi$
$17$	0.835685	0.202683	0.101342	+	0.994852i	$0.467686\pi$
$18$	0	0
$19$	0.998285	0.229022	0.114511	−	0.993422i	$-0.463470\pi$
$19$	0.998285	0.229022	0.114511	+	0.993422i	$0.463470\pi$
$20$	0	0
$21$	−0.0476950	−0.0104079
$22$	0	0
$23$	1.31016	0.273187	0.136593	−	0.990627i	$-0.456385\pi$
$23$	1.31016	0.273187	0.136593	+	0.990627i	$0.456385\pi$
$24$	0	0
$25$	−4.84607	−0.969213
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	6.86363	1.27454	0.637272	−	0.770639i	$-0.280063\pi$
$29$	6.86363	1.27454	0.637272	+	0.770639i	$0.280063\pi$
$30$	0	0
$31$	−9.48974	−1.70441	−0.852204	−	0.523210i	$-0.824734\pi$
$31$	−9.48974	−1.70441	−0.852204	+	0.523210i	$0.824734\pi$
$32$	0	0
$33$	−1.67730	−0.291981
$34$	0	0
$35$	0.0187128	0.00316304
$36$	0	0
$37$	2.44944	0.402685	0.201343	−	0.979521i	$-0.435470\pi$
$37$	2.44944	0.402685	0.201343	+	0.979521i	$0.435470\pi$
$38$	0	0
$39$	4.70371	0.753197
$40$	0	0
$41$	0.448099	0.0699813	0.0349907	−	0.999388i	$-0.488860\pi$
$41$	0.448099	0.0699813	0.0349907	+	0.999388i	$0.488860\pi$
$42$	0	0
$43$	9.51987	1.45177	0.725883	−	0.687818i	$-0.241431\pi$
$43$	9.51987	1.45177	0.725883	+	0.687818i	$0.241431\pi$
$44$	0	0
$45$	0.392343	0.0584870
$46$	0	0
$47$	−3.80613	−0.555182	−0.277591	−	0.960699i	$-0.589536\pi$
$47$	−3.80613	−0.555182	−0.277591	+	0.960699i	$0.589536\pi$
$48$	0	0
$49$	−6.99773	−0.999675
$50$	0	0
$51$	−0.835685	−0.117019
$52$	0	0
$53$	−9.39965	−1.29114	−0.645571	−	0.763700i	$-0.723380\pi$
$53$	−9.39965	−1.29114	−0.645571	+	0.763700i	$0.723380\pi$
$54$	0	0
$55$	0.658077	0.0887351
$56$	0	0
$57$	−0.998285	−0.132226
$58$	0	0
$59$	−3.24192	−0.422062	−0.211031	−	0.977479i	$-0.567682\pi$
$59$	−3.24192	−0.422062	−0.211031	+	0.977479i	$0.567682\pi$
$60$	0	0
$61$	0.335380	0.0429411	0.0214705	−	0.999769i	$-0.493165\pi$
$61$	0.335380	0.0429411	0.0214705	+	0.999769i	$0.493165\pi$
$62$	0	0
$63$	0.0476950	0.00600901
$64$	0	0
$65$	−1.84547	−0.228902
$66$	0	0
$67$	−14.2570	−1.74177	−0.870883	−	0.491491i	$-0.836452\pi$
$67$	−14.2570	−1.74177	−0.870883	+	0.491491i	$0.836452\pi$
$68$	0	0
$69$	−1.31016	−0.157724
$70$	0	0
$71$	5.78446	0.686489	0.343244	−	0.939246i	$-0.388474\pi$
$71$	5.78446	0.686489	0.343244	+	0.939246i	$0.388474\pi$
$72$	0	0
$73$	−2.80343	−0.328116	−0.164058	−	0.986451i	$-0.552458\pi$
$73$	−2.80343	−0.328116	−0.164058	+	0.986451i	$0.552458\pi$
$74$	0	0
$75$	4.84607	0.559576
$76$	0	0
$77$	0.0799990	0.00911674
$78$	0	0
$79$	15.3705	1.72932	0.864658	−	0.502362i	$-0.167535\pi$
$79$	15.3705	1.72932	0.864658	+	0.502362i	$0.167535\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−7.43473	−0.816068	−0.408034	−	0.912967i	$-0.633785\pi$
$83$	−7.43473	−0.816068	−0.408034	+	0.912967i	$0.633785\pi$
$84$	0	0
$85$	0.327875	0.0355630
$86$	0	0
$87$	−6.86363	−0.735859
$88$	0	0
$89$	−5.92823	−0.628391	−0.314195	−	0.949358i	$-0.601735\pi$
$89$	−5.92823	−0.628391	−0.314195	+	0.949358i	$0.601735\pi$
$90$	0	0
$91$	−0.224344	−0.0235176
$92$	0	0
$93$	9.48974	0.984040
$94$	0	0
$95$	0.391670	0.0401845
$96$	0	0
$97$	4.70012	0.477225	0.238612	−	0.971115i	$-0.423308\pi$
$97$	4.70012	0.477225	0.238612	+	0.971115i	$0.423308\pi$
$98$	0	0
$99$	1.67730	0.168575
$100$	0	0
$101$	−3.08547	−0.307015	−0.153508	−	0.988147i	$-0.549057\pi$
$101$	−3.08547	−0.307015	−0.153508	+	0.988147i	$0.549057\pi$
$102$	0	0
$103$	6.58913	0.649246	0.324623	−	0.945844i	$-0.394763\pi$
$103$	6.58913	0.649246	0.324623	+	0.945844i	$0.394763\pi$
$104$	0	0
$105$	−0.0187128	−0.00182618
$106$	0	0
$107$	10.7446	1.03872	0.519362	−	0.854555i	$-0.326170\pi$
$107$	10.7446	1.03872	0.519362	+	0.854555i	$0.326170\pi$
$108$	0	0
$109$	−5.37085	−0.514434	−0.257217	−	0.966354i	$-0.582806\pi$
$109$	−5.37085	−0.514434	−0.257217	+	0.966354i	$0.582806\pi$
$110$	0	0
$111$	−2.44944	−0.232491
$112$	0	0
$113$	−8.97790	−0.844570	−0.422285	−	0.906463i	$-0.638772\pi$
$113$	−8.97790	−0.844570	−0.422285	+	0.906463i	$0.638772\pi$
$114$	0	0
$115$	0.514030	0.0479336
$116$	0	0
$117$	−4.70371	−0.434859
$118$	0	0
$119$	0.0398580	0.00365378
$120$	0	0
$121$	−8.18665	−0.744241
$122$	0	0
$123$	−0.448099	−0.0404037
$124$	0	0
$125$	−3.86303	−0.345520
$126$	0	0
$127$	20.6485	1.83225	0.916127	−	0.400887i	$-0.131298\pi$
$127$	20.6485	1.83225	0.916127	+	0.400887i	$0.131298\pi$
$128$	0	0
$129$	−9.51987	−0.838178
$130$	0	0
$131$	−7.40239	−0.646750	−0.323375	−	0.946271i	$-0.604818\pi$
$131$	−7.40239	−0.646750	−0.323375	+	0.946271i	$0.604818\pi$
$132$	0	0
$133$	0.0476132	0.00412859
$134$	0	0
$135$	−0.392343	−0.0337675
$136$	0	0
$137$	−18.9040	−1.61508	−0.807539	−	0.589814i	$-0.799201\pi$
$137$	−18.9040	−1.61508	−0.807539	+	0.589814i	$0.799201\pi$
$138$	0	0
$139$	4.58198	0.388638	0.194319	−	0.980938i	$-0.437750\pi$
$139$	4.58198	0.388638	0.194319	+	0.980938i	$0.437750\pi$
$140$	0	0
$141$	3.80613	0.320534
$142$	0	0
$143$	−7.88955	−0.659758
$144$	0	0
$145$	2.69290	0.223633
$146$	0	0
$147$	6.99773	0.577163
$148$	0	0
$149$	−20.8271	−1.70622	−0.853112	−	0.521728i	$-0.825287\pi$
$149$	−20.8271	−1.70622	−0.853112	+	0.521728i	$0.825287\pi$
$150$	0	0
$151$	5.79277	0.471409	0.235704	−	0.971825i	$-0.424260\pi$
$151$	5.79277	0.471409	0.235704	+	0.971825i	$0.424260\pi$
$152$	0	0
$153$	0.835685	0.0675612
$154$	0	0
$155$	−3.72323	−0.299057
$156$	0	0
$157$	−12.6017	−1.00573	−0.502864	−	0.864366i	$-0.667720\pi$
$157$	−12.6017	−1.00573	−0.502864	+	0.864366i	$0.667720\pi$
$158$	0	0
$159$	9.39965	0.745441
$160$	0	0
$161$	0.0624880	0.00492474
$162$	0	0
$163$	−20.8695	−1.63463	−0.817313	−	0.576194i	$-0.804537\pi$
$163$	−20.8695	−1.63463	−0.817313	+	0.576194i	$0.804537\pi$
$164$	0	0
$165$	−0.658077	−0.0512313
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	9.12493	0.701918
$170$	0	0
$171$	0.998285	0.0763408
$172$	0	0
$173$	−16.6941	−1.26923	−0.634614	−	0.772829i	$-0.718841\pi$
$173$	−16.6941	−1.26923	−0.634614	+	0.772829i	$0.718841\pi$
$174$	0	0
$175$	−0.231133	−0.0174720
$176$	0	0
$177$	3.24192	0.243678
$178$	0	0
$179$	18.5138	1.38378	0.691892	−	0.722001i	$-0.256777\pi$
$179$	18.5138	1.38378	0.691892	+	0.722001i	$0.256777\pi$
$180$	0	0
$181$	−11.1554	−0.829177	−0.414588	−	0.910009i	$-0.636074\pi$
$181$	−11.1554	−0.829177	−0.414588	+	0.910009i	$0.636074\pi$
$182$	0	0
$183$	−0.335380	−0.0247920
$184$	0	0
$185$	0.961020	0.0706556
$186$	0	0
$187$	1.40170	0.102502
$188$	0	0
$189$	−0.0476950	−0.00346930
$190$	0	0
$191$	1.00647	0.0728259	0.0364129	−	0.999337i	$-0.488407\pi$
$191$	1.00647	0.0728259	0.0364129	+	0.999337i	$0.488407\pi$
$192$	0	0
$193$	−0.00263902	−0.000189961 0	−9.49805e−5	−	1.00000i	$-0.500030\pi$
$193$	−0.00263902	−0.000189961 0	−9.49805e−5		1.00000i	$0.500030\pi$
$194$	0	0
$195$	1.84547	0.132157
$196$	0	0
$197$	3.29685	0.234891	0.117445	−	0.993079i	$-0.462529\pi$
$197$	3.29685	0.234891	0.117445	+	0.993079i	$0.462529\pi$
$198$	0	0
$199$	−2.00744	−0.142304	−0.0711519	−	0.997465i	$-0.522668\pi$
$199$	−2.00744	−0.142304	−0.0711519	+	0.997465i	$0.522668\pi$
$200$	0	0
$201$	14.2570	1.00561
$202$	0	0
$203$	0.327361	0.0229763
$204$	0	0
$205$	0.175808	0.0122790
$206$	0	0
$207$	1.31016	0.0910622
$208$	0	0
$209$	1.67443	0.115823
$210$	0	0
$211$	−23.1716	−1.59520	−0.797598	−	0.603189i	$-0.793897\pi$
$211$	−23.1716	−1.59520	−0.797598	+	0.603189i	$0.793897\pi$
$212$	0	0
$213$	−5.78446	−0.396344
$214$	0	0
$215$	3.73505	0.254728
$216$	0	0
$217$	−0.452614	−0.0307254
$218$	0	0
$219$	2.80343	0.189438
$220$	0	0
$221$	−3.93083	−0.264416
$222$	0	0
$223$	11.6448	0.779791	0.389895	−	0.920859i	$-0.372511\pi$
$223$	11.6448	0.779791	0.389895	+	0.920859i	$0.372511\pi$
$224$	0	0
$225$	−4.84607	−0.323071
$226$	0	0
$227$	−22.8199	−1.51461	−0.757306	−	0.653060i	$-0.773485\pi$
$227$	−22.8199	−1.51461	−0.757306	+	0.653060i	$0.773485\pi$
$228$	0	0
$229$	13.7218	0.906759	0.453380	−	0.891318i	$-0.350218\pi$
$229$	13.7218	0.906759	0.453380	+	0.891318i	$0.350218\pi$
$230$	0	0
$231$	−0.0799990	−0.00526355
$232$	0	0
$233$	1.63184	0.106905	0.0534525	−	0.998570i	$-0.482977\pi$
$233$	1.63184	0.106905	0.0534525	+	0.998570i	$0.482977\pi$
$234$	0	0
$235$	−1.49331	−0.0974127
$236$	0	0
$237$	−15.3705	−0.998420
$238$	0	0
$239$	15.6884	1.01480	0.507400	−	0.861711i	$-0.330607\pi$
$239$	15.6884	1.01480	0.507400	+	0.861711i	$0.330607\pi$
$240$	0	0
$241$	1.42558	0.0918297	0.0459149	−	0.998945i	$-0.485380\pi$
$241$	1.42558	0.0918297	0.0459149	+	0.998945i	$0.485380\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−2.74551	−0.175404
$246$	0	0
$247$	−4.69565	−0.298777
$248$	0	0
$249$	7.43473	0.471157
$250$	0	0
$251$	−20.7834	−1.31183	−0.655917	−	0.754833i	$-0.727718\pi$
$251$	−20.7834	−1.31183	−0.655917	+	0.754833i	$0.727718\pi$
$252$	0	0
$253$	2.19753	0.138158
$254$	0	0
$255$	−0.327875	−0.0205323
$256$	0	0
$257$	−19.0163	−1.18620	−0.593101	−	0.805128i	$-0.702097\pi$
$257$	−19.0163	−1.18620	−0.593101	+	0.805128i	$0.702097\pi$
$258$	0	0
$259$	0.116826	0.00725922
$260$	0	0
$261$	6.86363	0.424848
$262$	0	0
$263$	−7.63989	−0.471096	−0.235548	−	0.971863i	$-0.575688\pi$
$263$	−7.63989	−0.471096	−0.235548	+	0.971863i	$0.575688\pi$
$264$	0	0
$265$	−3.68788	−0.226545
$266$	0	0
$267$	5.92823	0.362802
$268$	0	0
$269$	−27.2370	−1.66067	−0.830336	−	0.557264i	$-0.811851\pi$
$269$	−27.2370	−1.66067	−0.830336	+	0.557264i	$0.811851\pi$
$270$	0	0
$271$	−6.54938	−0.397847	−0.198923	−	0.980015i	$-0.563744\pi$
$271$	−6.54938	−0.397847	−0.198923	+	0.980015i	$0.563744\pi$
$272$	0	0
$273$	0.224344	0.0135779
$274$	0	0
$275$	−8.12832	−0.490156
$276$	0	0
$277$	−14.1616	−0.850889	−0.425445	−	0.904984i	$-0.639882\pi$
$277$	−14.1616	−0.850889	−0.425445	+	0.904984i	$0.639882\pi$
$278$	0	0
$279$	−9.48974	−0.568136
$280$	0	0
$281$	20.7817	1.23973	0.619867	−	0.784707i	$-0.287187\pi$
$281$	20.7817	1.23973	0.619867	+	0.784707i	$0.287187\pi$
$282$	0	0
$283$	−16.9536	−1.00779	−0.503894	−	0.863766i	$-0.668100\pi$
$283$	−16.9536	−1.00779	−0.503894	+	0.863766i	$0.668100\pi$
$284$	0	0
$285$	−0.391670	−0.0232005
$286$	0	0
$287$	0.0213721	0.00126156
$288$	0	0
$289$	−16.3016	−0.958919
$290$	0	0
$291$	−4.70012	−0.275526
$292$	0	0
$293$	−2.31142	−0.135035	−0.0675173	−	0.997718i	$-0.521508\pi$
$293$	−2.31142	−0.135035	−0.0675173	+	0.997718i	$0.521508\pi$
$294$	0	0
$295$	−1.27194	−0.0740554
$296$	0	0
$297$	−1.67730	−0.0973270
$298$	0	0
$299$	−6.16261	−0.356393
$300$	0	0
$301$	0.454051	0.0261710
$302$	0	0
$303$	3.08547	0.177255
$304$	0	0
$305$	0.131584	0.00753448
$306$	0	0
$307$	−26.6433	−1.52062	−0.760308	−	0.649563i	$-0.774952\pi$
$307$	−26.6433	−1.52062	−0.760308	+	0.649563i	$0.774952\pi$
$308$	0	0
$309$	−6.58913	−0.374842
$310$	0	0
$311$	−2.71532	−0.153972	−0.0769859	−	0.997032i	$-0.524530\pi$
$311$	−2.71532	−0.153972	−0.0769859	+	0.997032i	$0.524530\pi$
$312$	0	0
$313$	−6.54601	−0.370002	−0.185001	−	0.982738i	$-0.559229\pi$
$313$	−6.54601	−0.370002	−0.185001	+	0.982738i	$0.559229\pi$
$314$	0	0
$315$	0.0187128	0.00105435
$316$	0	0
$317$	−19.4676	−1.09341	−0.546704	−	0.837326i	$-0.684118\pi$
$317$	−19.4676	−1.09341	−0.546704	+	0.837326i	$0.684118\pi$
$318$	0	0
$319$	11.5124	0.644570
$320$	0	0
$321$	−10.7446	−0.599707
$322$	0	0
$323$	0.834252	0.0464190
$324$	0	0
$325$	22.7945	1.26441
$326$	0	0
$327$	5.37085	0.297009
$328$	0	0
$329$	−0.181534	−0.0100083
$330$	0	0
$331$	11.3987	0.626530	0.313265	−	0.949666i	$-0.398577\pi$
$331$	11.3987	0.626530	0.313265	+	0.949666i	$0.398577\pi$
$332$	0	0
$333$	2.44944	0.134228
$334$	0	0
$335$	−5.59361	−0.305612
$336$	0	0
$337$	9.46084	0.515365	0.257682	−	0.966230i	$-0.417041\pi$
$337$	9.46084	0.515365	0.257682	+	0.966230i	$0.417041\pi$
$338$	0	0
$339$	8.97790	0.487613
$340$	0	0
$341$	−15.9172	−0.861963
$342$	0	0
$343$	−0.667622	−0.0360482
$344$	0	0
$345$	−0.514030	−0.0276745
$346$	0	0
$347$	−18.1189	−0.972674	−0.486337	−	0.873771i	$-0.661667\pi$
$347$	−18.1189	−0.972674	−0.486337	+	0.873771i	$0.661667\pi$
$348$	0	0
$349$	7.17579	0.384111	0.192056	−	0.981384i	$-0.438485\pi$
$349$	7.17579	0.384111	0.192056	+	0.981384i	$0.438485\pi$
$350$	0	0
$351$	4.70371	0.251066
$352$	0	0
$353$	32.7788	1.74464	0.872319	−	0.488938i	$-0.162616\pi$
$353$	32.7788	1.74464	0.872319	+	0.488938i	$0.162616\pi$
$354$	0	0
$355$	2.26949	0.120452
$356$	0	0
$357$	−0.0398580	−0.00210951
$358$	0	0
$359$	25.9833	1.37135	0.685674	−	0.727909i	$-0.259508\pi$
$359$	25.9833	1.37135	0.685674	+	0.727909i	$0.259508\pi$
$360$	0	0
$361$	−18.0034	−0.947549
$362$	0	0
$363$	8.18665	0.429688
$364$	0	0
$365$	−1.09990	−0.0575716
$366$	0	0
$367$	0.622748	0.0325072	0.0162536	−	0.999868i	$-0.494826\pi$
$367$	0.622748	0.0325072	0.0162536	+	0.999868i	$0.494826\pi$
$368$	0	0
$369$	0.448099	0.0233271
$370$	0	0
$371$	−0.448317	−0.0232754
$372$	0	0
$373$	25.3579	1.31298	0.656492	−	0.754333i	$-0.272040\pi$
$373$	25.3579	1.31298	0.656492	+	0.754333i	$0.272040\pi$
$374$	0	0
$375$	3.86303	0.199486
$376$	0	0
$377$	−32.2846	−1.66274
$378$	0	0
$379$	21.1584	1.08684	0.543418	−	0.839462i	$-0.317130\pi$
$379$	21.1584	1.08684	0.543418	+	0.839462i	$0.317130\pi$
$380$	0	0
$381$	−20.6485	−1.05785
$382$	0	0
$383$	−25.0508	−1.28004	−0.640019	−	0.768359i	$-0.721073\pi$
$383$	−25.0508	−1.28004	−0.640019	+	0.768359i	$0.721073\pi$
$384$	0	0
$385$	0.0313870	0.00159963
$386$	0	0
$387$	9.51987	0.483922
$388$	0	0
$389$	−25.1811	−1.27673	−0.638367	−	0.769732i	$-0.720390\pi$
$389$	−25.1811	−1.27673	−0.638367	+	0.769732i	$0.720390\pi$
$390$	0	0
$391$	1.09488	0.0553704
$392$	0	0
$393$	7.40239	0.373401
$394$	0	0
$395$	6.03050	0.303427
$396$	0	0
$397$	16.5383	0.830035	0.415017	−	0.909813i	$-0.363775\pi$
$397$	16.5383	0.830035	0.415017	+	0.909813i	$0.363775\pi$
$398$	0	0
$399$	−0.0476132	−0.00238364
$400$	0	0
$401$	−29.5026	−1.47329	−0.736645	−	0.676280i	$-0.763591\pi$
$401$	−29.5026	−1.47329	−0.736645	+	0.676280i	$0.763591\pi$
$402$	0	0
$403$	44.6370	2.22353
$404$	0	0
$405$	0.392343	0.0194957
$406$	0	0
$407$	4.10845	0.203648
$408$	0	0
$409$	10.9493	0.541410	0.270705	−	0.962662i	$-0.412743\pi$
$409$	10.9493	0.541410	0.270705	+	0.962662i	$0.412743\pi$
$410$	0	0
$411$	18.9040	0.932466
$412$	0	0
$413$	−0.154624	−0.00760853
$414$	0	0
$415$	−2.91696	−0.143188
$416$	0	0
$417$	−4.58198	−0.224380
$418$	0	0
$419$	19.1069	0.933433	0.466716	−	0.884407i	$-0.345437\pi$
$419$	19.1069	0.933433	0.466716	+	0.884407i	$0.345437\pi$
$420$	0	0
$421$	18.8853	0.920414	0.460207	−	0.887812i	$-0.347775\pi$
$421$	18.8853	0.920414	0.460207	+	0.887812i	$0.347775\pi$
$422$	0	0
$423$	−3.80613	−0.185061
$424$	0	0
$425$	−4.04979	−0.196444
$426$	0	0
$427$	0.0159960	0.000774100 0
$428$	0	0
$429$	7.88955	0.380911
$430$	0	0
$431$	1.54122	0.0742381	0.0371191	−	0.999311i	$-0.488182\pi$
$431$	1.54122	0.0742381	0.0371191	+	0.999311i	$0.488182\pi$
$432$	0	0
$433$	−1.63875	−0.0787535	−0.0393768	−	0.999224i	$-0.512537\pi$
$433$	−1.63875	−0.0787535	−0.0393768	+	0.999224i	$0.512537\pi$
$434$	0	0
$435$	−2.69290	−0.129114
$436$	0	0
$437$	1.30791	0.0625658
$438$	0	0
$439$	−27.9964	−1.33620	−0.668098	−	0.744073i	$-0.732891\pi$
$439$	−27.9964	−1.33620	−0.668098	+	0.744073i	$0.732891\pi$
$440$	0	0
$441$	−6.99773	−0.333225
$442$	0	0
$443$	−17.8603	−0.848568	−0.424284	−	0.905529i	$-0.639474\pi$
$443$	−17.8603	−0.848568	−0.424284	+	0.905529i	$0.639474\pi$
$444$	0	0
$445$	−2.32590	−0.110258
$446$	0	0
$447$	20.8271	0.985088
$448$	0	0
$449$	−23.9248	−1.12908	−0.564540	−	0.825406i	$-0.690946\pi$
$449$	−23.9248	−1.12908	−0.564540	+	0.825406i	$0.690946\pi$
$450$	0	0
$451$	0.751598	0.0353914
$452$	0	0
$453$	−5.79277	−0.272168
$454$	0	0
$455$	−0.0880196	−0.00412642
$456$	0	0
$457$	30.8005	1.44079	0.720393	−	0.693566i	$-0.243962\pi$
$457$	30.8005	1.44079	0.720393	+	0.693566i	$0.243962\pi$
$458$	0	0
$459$	−0.835685	−0.0390065
$460$	0	0
$461$	28.1069	1.30907	0.654534	−	0.756032i	$-0.272865\pi$
$461$	28.1069	1.30907	0.654534	+	0.756032i	$0.272865\pi$
$462$	0	0
$463$	9.82630	0.456667	0.228333	−	0.973583i	$-0.426672\pi$
$463$	9.82630	0.456667	0.228333	+	0.973583i	$0.426672\pi$
$464$	0	0
$465$	3.72323	0.172661
$466$	0	0
$467$	−3.50796	−0.162329	−0.0811646	−	0.996701i	$-0.525864\pi$
$467$	−3.50796	−0.162329	−0.0811646	+	0.996701i	$0.525864\pi$
$468$	0	0
$469$	−0.679986	−0.0313989
$470$	0	0
$471$	12.6017	0.580657
$472$	0	0
$473$	15.9677	0.734196
$474$	0	0
$475$	−4.83776	−0.221971
$476$	0	0
$477$	−9.39965	−0.430380
$478$	0	0
$479$	15.9493	0.728742	0.364371	−	0.931254i	$-0.381284\pi$
$479$	15.9493	0.728742	0.364371	+	0.931254i	$0.381284\pi$
$480$	0	0
$481$	−11.5215	−0.525334
$482$	0	0
$483$	−0.0624880	−0.00284330
$484$	0	0
$485$	1.84406	0.0837343
$486$	0	0
$487$	−34.8701	−1.58012	−0.790058	−	0.613032i	$-0.789950\pi$
$487$	−34.8701	−1.58012	−0.790058	+	0.613032i	$0.789950\pi$
$488$	0	0
$489$	20.8695	0.943751
$490$	0	0
$491$	3.03356	0.136902	0.0684512	−	0.997654i	$-0.478194\pi$
$491$	3.03356	0.136902	0.0684512	+	0.997654i	$0.478194\pi$
$492$	0	0
$493$	5.73584	0.258329
$494$	0	0
$495$	0.658077	0.0295784
$496$	0	0
$497$	0.275890	0.0123754
$498$	0	0
$499$	8.84557	0.395982	0.197991	−	0.980204i	$-0.436558\pi$
$499$	8.84557	0.395982	0.197991	+	0.980204i	$0.436558\pi$
$500$	0	0
$501$	1.00000	0.0446767
$502$	0	0
$503$	−10.9142	−0.486642	−0.243321	−	0.969946i	$-0.578237\pi$
$503$	−10.9142	−0.486642	−0.243321	+	0.969946i	$0.578237\pi$
$504$	0	0
$505$	−1.21056	−0.0538692
$506$	0	0
$507$	−9.12493	−0.405252
$508$	0	0
$509$	−26.6881	−1.18293	−0.591466	−	0.806330i	$-0.701450\pi$
$509$	−26.6881	−1.18293	−0.591466	+	0.806330i	$0.701450\pi$
$510$	0	0
$511$	−0.133710	−0.00591496
$512$	0	0
$513$	−0.998285	−0.0440754
$514$	0	0
$515$	2.58519	0.113917
$516$	0	0
$517$	−6.38404	−0.280770
$518$	0	0
$519$	16.6941	0.732790
$520$	0	0
$521$	42.5402	1.86372	0.931860	−	0.362818i	$-0.118185\pi$
$521$	42.5402	1.86372	0.931860	+	0.362818i	$0.118185\pi$
$522$	0	0
$523$	−3.07635	−0.134520	−0.0672598	−	0.997735i	$-0.521426\pi$
$523$	−3.07635	−0.134520	−0.0672598	+	0.997735i	$0.521426\pi$
$524$	0	0
$525$	0.231133	0.0100875
$526$	0	0
$527$	−7.93044	−0.345455
$528$	0	0
$529$	−21.2835	−0.925369
$530$	0	0
$531$	−3.24192	−0.140687
$532$	0	0
$533$	−2.10773	−0.0912960
$534$	0	0
$535$	4.21558	0.182255
$536$	0	0
$537$	−18.5138	−0.798928
$538$	0	0
$539$	−11.7373	−0.505562
$540$	0	0
$541$	40.0473	1.72177	0.860883	−	0.508803i	$-0.169912\pi$
$541$	40.0473	1.72177	0.860883	+	0.508803i	$0.169912\pi$
$542$	0	0
$543$	11.1554	0.478725
$544$	0	0
$545$	−2.10721	−0.0902631
$546$	0	0
$547$	−45.1866	−1.93204	−0.966019	−	0.258471i	$-0.916782\pi$
$547$	−45.1866	−1.93204	−0.966019	+	0.258471i	$0.916782\pi$
$548$	0	0
$549$	0.335380	0.0143137
$550$	0	0
$551$	6.85186	0.291899
$552$	0	0
$553$	0.733096	0.0311744
$554$	0	0
$555$	−0.961020	−0.0407930
$556$	0	0
$557$	23.6465	1.00193	0.500967	−	0.865466i	$-0.332978\pi$
$557$	23.6465	1.00193	0.500967	+	0.865466i	$0.332978\pi$
$558$	0	0
$559$	−44.7788	−1.89394
$560$	0	0
$561$	−1.40170	−0.0591797
$562$	0	0
$563$	33.9807	1.43211	0.716057	−	0.698041i	$-0.245945\pi$
$563$	33.9807	1.43211	0.716057	+	0.698041i	$0.245945\pi$
$564$	0	0
$565$	−3.52241	−0.148189
$566$	0	0
$567$	0.0476950	0.00200300
$568$	0	0
$569$	10.1043	0.423593	0.211797	−	0.977314i	$-0.432069\pi$
$569$	10.1043	0.423593	0.211797	+	0.977314i	$0.432069\pi$
$570$	0	0
$571$	45.2277	1.89272	0.946359	−	0.323116i	$-0.104730\pi$
$571$	45.2277	1.89272	0.946359	+	0.323116i	$0.104730\pi$
$572$	0	0
$573$	−1.00647	−0.0420460
$574$	0	0
$575$	−6.34911	−0.264776
$576$	0	0
$577$	−21.0191	−0.875035	−0.437517	−	0.899210i	$-0.644142\pi$
$577$	−21.0191	−0.875035	−0.437517	+	0.899210i	$0.644142\pi$
$578$	0	0
$579$	0.00263902	0.000109674 0
$580$	0	0
$581$	−0.354600	−0.0147113
$582$	0	0
$583$	−15.7661	−0.652964
$584$	0	0
$585$	−1.84547	−0.0763007
$586$	0	0
$587$	12.5907	0.519673	0.259837	−	0.965653i	$-0.416331\pi$
$587$	12.5907	0.519673	0.259837	+	0.965653i	$0.416331\pi$
$588$	0	0
$589$	−9.47347	−0.390347
$590$	0	0
$591$	−3.29685	−0.135614
$592$	0	0
$593$	18.6770	0.766973	0.383486	−	0.923547i	$-0.374723\pi$
$593$	18.6770	0.766973	0.383486	+	0.923547i	$0.374723\pi$
$594$	0	0
$595$	0.0156380	0.000641096 0
$596$	0	0
$597$	2.00744	0.0821591
$598$	0	0
$599$	43.2696	1.76795	0.883975	−	0.467535i	$-0.154858\pi$
$599$	43.2696	1.76795	0.883975	+	0.467535i	$0.154858\pi$
$600$	0	0
$601$	18.4237	0.751520	0.375760	−	0.926717i	$-0.377382\pi$
$601$	18.4237	0.751520	0.375760	+	0.926717i	$0.377382\pi$
$602$	0	0
$603$	−14.2570	−0.580588
$604$	0	0
$605$	−3.21197	−0.130585
$606$	0	0
$607$	12.4290	0.504478	0.252239	−	0.967665i	$-0.418833\pi$
$607$	12.4290	0.504478	0.252239	+	0.967665i	$0.418833\pi$
$608$	0	0
$609$	−0.327361	−0.0132653
$610$	0	0
$611$	17.9030	0.724276
$612$	0	0
$613$	−3.08597	−0.124641	−0.0623205	−	0.998056i	$-0.519850\pi$
$613$	−3.08597	−0.124641	−0.0623205	+	0.998056i	$0.519850\pi$
$614$	0	0
$615$	−0.175808	−0.00708928
$616$	0	0
$617$	5.89338	0.237259	0.118629	−	0.992939i	$-0.462150\pi$
$617$	5.89338	0.237259	0.118629	+	0.992939i	$0.462150\pi$
$618$	0	0
$619$	11.0381	0.443657	0.221829	−	0.975086i	$-0.428797\pi$
$619$	11.0381	0.443657	0.221829	+	0.975086i	$0.428797\pi$
$620$	0	0
$621$	−1.31016	−0.0525748
$622$	0	0
$623$	−0.282747	−0.0113280
$624$	0	0
$625$	22.7147	0.908588
$626$	0	0
$627$	−1.67443	−0.0668702
$628$	0	0
$629$	2.04696	0.0816177
$630$	0	0
$631$	44.0298	1.75280	0.876400	−	0.481584i	$-0.159938\pi$
$631$	44.0298	1.75280	0.876400	+	0.481584i	$0.159938\pi$
$632$	0	0
$633$	23.1716	0.920987
$634$	0	0
$635$	8.10127	0.321489
$636$	0	0
$637$	32.9153	1.30415
$638$	0	0
$639$	5.78446	0.228830
$640$	0	0
$641$	−30.4393	−1.20228	−0.601140	−	0.799144i	$-0.705286\pi$
$641$	−30.4393	−1.20228	−0.601140	+	0.799144i	$0.705286\pi$
$642$	0	0
$643$	11.1979	0.441601	0.220801	−	0.975319i	$-0.429133\pi$
$643$	11.1979	0.441601	0.220801	+	0.975319i	$0.429133\pi$
$644$	0	0
$645$	−3.73505	−0.147067
$646$	0	0
$647$	−3.77009	−0.148218	−0.0741089	−	0.997250i	$-0.523611\pi$
$647$	−3.77009	−0.148218	−0.0741089	+	0.997250i	$0.523611\pi$
$648$	0	0
$649$	−5.43769	−0.213448
$650$	0	0
$651$	0.452614	0.0177393
$652$	0	0
$653$	−44.9124	−1.75756	−0.878780	−	0.477227i	$-0.841642\pi$
$653$	−44.9124	−1.75756	−0.878780	+	0.477227i	$0.841642\pi$
$654$	0	0
$655$	−2.90427	−0.113479
$656$	0	0
$657$	−2.80343	−0.109372
$658$	0	0
$659$	0.530432	0.0206627	0.0103313	−	0.999947i	$-0.496711\pi$
$659$	0.530432	0.0206627	0.0103313	+	0.999947i	$0.496711\pi$
$660$	0	0
$661$	−37.7146	−1.46693	−0.733464	−	0.679728i	$-0.762098\pi$
$661$	−37.7146	−1.46693	−0.733464	+	0.679728i	$0.762098\pi$
$662$	0	0
$663$	3.93083	0.152661
$664$	0	0
$665$	0.0186807	0.000724407 0
$666$	0	0
$667$	8.99244	0.348189
$668$	0	0
$669$	−11.6448	−0.450212
$670$	0	0
$671$	0.562535	0.0217164
$672$	0	0
$673$	14.2368	0.548787	0.274393	−	0.961618i	$-0.411523\pi$
$673$	14.2368	0.548787	0.274393	+	0.961618i	$0.411523\pi$
$674$	0	0
$675$	4.84607	0.186525
$676$	0	0
$677$	−27.2571	−1.04758	−0.523789	−	0.851848i	$-0.675482\pi$
$677$	−27.2571	−1.04758	−0.523789	+	0.851848i	$0.675482\pi$
$678$	0	0
$679$	0.224172	0.00860294
$680$	0	0
$681$	22.8199	0.874461
$682$	0	0
$683$	−28.3603	−1.08518	−0.542589	−	0.839998i	$-0.682556\pi$
$683$	−28.3603	−1.08518	−0.542589	+	0.839998i	$0.682556\pi$
$684$	0	0
$685$	−7.41685	−0.283383
$686$	0	0
$687$	−13.7218	−0.523518
$688$	0	0
$689$	44.2133	1.68439
$690$	0	0
$691$	32.5658	1.23886	0.619430	−	0.785052i	$-0.287364\pi$
$691$	32.5658	1.23886	0.619430	+	0.785052i	$0.287364\pi$
$692$	0	0
$693$	0.0799990	0.00303891
$694$	0	0
$695$	1.79770	0.0681908
$696$	0	0
$697$	0.374470	0.0141841
$698$	0	0
$699$	−1.63184	−0.0617217
$700$	0	0
$701$	−34.2700	−1.29436	−0.647179	−	0.762338i	$-0.724051\pi$
$701$	−34.2700	−1.29436	−0.647179	+	0.762338i	$0.724051\pi$
$702$	0	0
$703$	2.44524	0.0922240
$704$	0	0
$705$	1.49331	0.0562412
$706$	0	0
$707$	−0.147161	−0.00553457
$708$	0	0
$709$	17.4855	0.656680	0.328340	−	0.944560i	$-0.393511\pi$
$709$	17.4855	0.656680	0.328340	+	0.944560i	$0.393511\pi$
$710$	0	0
$711$	15.3705	0.576438
$712$	0	0
$713$	−12.4331	−0.465622
$714$	0	0
$715$	−3.09541	−0.115762
$716$	0	0
$717$	−15.6884	−0.585895
$718$	0	0
$719$	−15.3333	−0.571836	−0.285918	−	0.958254i	$-0.592298\pi$
$719$	−15.3333	−0.571836	−0.285918	+	0.958254i	$0.592298\pi$
$720$	0	0
$721$	0.314269	0.0117040
$722$	0	0
$723$	−1.42558	−0.0530179
$724$	0	0
$725$	−33.2616	−1.23531
$726$	0	0
$727$	−20.8931	−0.774881	−0.387440	−	0.921895i	$-0.626641\pi$
$727$	−20.8931	−0.774881	−0.387440	+	0.921895i	$0.626641\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	7.95562	0.294249
$732$	0	0
$733$	−40.9819	−1.51370	−0.756851	−	0.653587i	$-0.773263\pi$
$733$	−40.9819	−1.51370	−0.756851	+	0.653587i	$0.773263\pi$
$734$	0	0
$735$	2.74551	0.101269
$736$	0	0
$737$	−23.9132	−0.880856
$738$	0	0
$739$	−4.74579	−0.174577	−0.0872883	−	0.996183i	$-0.527820\pi$
$739$	−4.74579	−0.174577	−0.0872883	+	0.996183i	$0.527820\pi$
$740$	0	0
$741$	4.69565	0.172499
$742$	0	0
$743$	10.2902	0.377509	0.188755	−	0.982024i	$-0.439555\pi$
$743$	10.2902	0.377509	0.188755	+	0.982024i	$0.439555\pi$
$744$	0	0
$745$	−8.17136	−0.299376
$746$	0	0
$747$	−7.43473	−0.272023
$748$	0	0
$749$	0.512466	0.0187251
$750$	0	0
$751$	−34.7969	−1.26976	−0.634878	−	0.772612i	$-0.718950\pi$
$751$	−34.7969	−1.26976	−0.634878	+	0.772612i	$0.718950\pi$
$752$	0	0
$753$	20.7834	0.757387
$754$	0	0
$755$	2.27275	0.0827138
$756$	0	0
$757$	24.1181	0.876588	0.438294	−	0.898832i	$-0.355583\pi$
$757$	24.1181	0.876588	0.438294	+	0.898832i	$0.355583\pi$
$758$	0	0
$759$	−2.19753	−0.0797653
$760$	0	0
$761$	22.7164	0.823470	0.411735	−	0.911304i	$-0.364923\pi$
$761$	22.7164	0.823470	0.411735	+	0.911304i	$0.364923\pi$
$762$	0	0
$763$	−0.256163	−0.00927372
$764$	0	0
$765$	0.327875	0.0118543
$766$	0	0
$767$	15.2491	0.550612
$768$	0	0
$769$	42.8534	1.54533	0.772667	−	0.634811i	$-0.218922\pi$
$769$	42.8534	1.54533	0.772667	+	0.634811i	$0.218922\pi$
$770$	0	0
$771$	19.0163	0.684855
$772$	0	0
$773$	51.2076	1.84181	0.920905	−	0.389786i	$-0.127451\pi$
$773$	51.2076	1.84181	0.920905	+	0.389786i	$0.127451\pi$
$774$	0	0
$775$	45.9879	1.65194
$776$	0	0
$777$	−0.116826	−0.00419111
$778$	0	0
$779$	0.447331	0.0160273
$780$	0	0
$781$	9.70229	0.347175
$782$	0	0
$783$	−6.86363	−0.245286
$784$	0	0
$785$	−4.94420	−0.176466
$786$	0	0
$787$	−4.78898	−0.170709	−0.0853543	−	0.996351i	$-0.527202\pi$
$787$	−4.78898	−0.170709	−0.0853543	+	0.996351i	$0.527202\pi$
$788$	0	0
$789$	7.63989	0.271987
$790$	0	0
$791$	−0.428201	−0.0152251
$792$	0	0
$793$	−1.57753	−0.0560199
$794$	0	0
$795$	3.68788	0.130796
$796$	0	0
$797$	−9.75603	−0.345576	−0.172788	−	0.984959i	$-0.555278\pi$
$797$	−9.75603	−0.345576	−0.172788	+	0.984959i	$0.555278\pi$
$798$	0	0
$799$	−3.18073	−0.112526
$800$	0	0
$801$	−5.92823	−0.209464
$802$	0	0
$803$	−4.70220	−0.165937
$804$	0	0
$805$	0.0245167	0.000864100 0
$806$	0	0
$807$	27.2370	0.958789
$808$	0	0
$809$	24.1859	0.850332	0.425166	−	0.905115i	$-0.360216\pi$
$809$	24.1859	0.850332	0.425166	+	0.905115i	$0.360216\pi$
$810$	0	0
$811$	46.5662	1.63516	0.817581	−	0.575814i	$-0.195315\pi$
$811$	46.5662	1.63516	0.817581	+	0.575814i	$0.195315\pi$
$812$	0	0
$813$	6.54938	0.229697
$814$	0	0
$815$	−8.18799	−0.286813
$816$	0	0
$817$	9.50355	0.332487
$818$	0	0
$819$	−0.224344	−0.00783921
$820$	0	0
$821$	41.0430	1.43241	0.716205	−	0.697890i	$-0.245878\pi$
$821$	41.0430	1.43241	0.716205	+	0.697890i	$0.245878\pi$
$822$	0	0
$823$	37.0327	1.29088	0.645439	−	0.763811i	$-0.276674\pi$
$823$	37.0327	1.29088	0.645439	+	0.763811i	$0.276674\pi$
$824$	0	0
$825$	8.12832	0.282992
$826$	0	0
$827$	−15.3486	−0.533724	−0.266862	−	0.963735i	$-0.585987\pi$
$827$	−15.3486	−0.533724	−0.266862	+	0.963735i	$0.585987\pi$
$828$	0	0
$829$	49.5804	1.72200	0.861000	−	0.508605i	$-0.169839\pi$
$829$	49.5804	1.72200	0.861000	+	0.508605i	$0.169839\pi$
$830$	0	0
$831$	14.1616	0.491261
$832$	0	0
$833$	−5.84790	−0.202618
$834$	0	0
$835$	−0.392343	−0.0135776
$836$	0	0
$837$	9.48974	0.328013
$838$	0	0
$839$	6.18421	0.213503	0.106751	−	0.994286i	$-0.465955\pi$
$839$	6.18421	0.213503	0.106751	+	0.994286i	$0.465955\pi$
$840$	0	0
$841$	18.1095	0.624464
$842$	0	0
$843$	−20.7817	−0.715760
$844$	0	0
$845$	3.58010	0.123159
$846$	0	0
$847$	−0.390463	−0.0134165
$848$	0	0
$849$	16.9536	0.581846
$850$	0	0
$851$	3.20915	0.110008
$852$	0	0
$853$	−5.43706	−0.186161	−0.0930807	−	0.995659i	$-0.529671\pi$
$853$	−5.43706	−0.186161	−0.0930807	+	0.995659i	$0.529671\pi$
$854$	0	0
$855$	0.391670	0.0133948
$856$	0	0
$857$	−27.2410	−0.930533	−0.465267	−	0.885171i	$-0.654042\pi$
$857$	−27.2410	−0.930533	−0.465267	+	0.885171i	$0.654042\pi$
$858$	0	0
$859$	−14.1977	−0.484419	−0.242210	−	0.970224i	$-0.577872\pi$
$859$	−14.1977	−0.484419	−0.242210	+	0.970224i	$0.577872\pi$
$860$	0	0
$861$	−0.0213721	−0.000728360 0
$862$	0	0
$863$	3.59454	0.122359	0.0611797	−	0.998127i	$-0.480514\pi$
$863$	3.59454	0.122359	0.0611797	+	0.998127i	$0.480514\pi$
$864$	0	0
$865$	−6.54980	−0.222700
$866$	0	0
$867$	16.3016	0.553632
$868$	0	0
$869$	25.7810	0.874559
$870$	0	0
$871$	67.0607	2.27226
$872$	0	0
$873$	4.70012	0.159075
$874$	0	0
$875$	−0.184247	−0.00622870
$876$	0	0
$877$	15.7663	0.532389	0.266195	−	0.963919i	$-0.414234\pi$
$877$	15.7663	0.532389	0.266195	+	0.963919i	$0.414234\pi$
$878$	0	0
$879$	2.31142	0.0779623
$880$	0	0
$881$	−43.1684	−1.45438	−0.727190	−	0.686436i	$-0.759174\pi$
$881$	−43.1684	−1.45438	−0.727190	+	0.686436i	$0.759174\pi$
$882$	0	0
$883$	57.9427	1.94993	0.974963	−	0.222365i	$-0.0713777\pi$
$883$	57.9427	1.94993	0.974963	+	0.222365i	$0.0713777\pi$
$884$	0	0
$885$	1.27194	0.0427559
$886$	0	0
$887$	57.4395	1.92863	0.964314	−	0.264760i	$-0.0852929\pi$
$887$	57.4395	1.92863	0.964314	+	0.264760i	$0.0852929\pi$
$888$	0	0
$889$	0.984829	0.0330301
$890$	0	0
$891$	1.67730	0.0561918
$892$	0	0
$893$	−3.79961	−0.127149
$894$	0	0
$895$	7.26374	0.242800
$896$	0	0
$897$	6.16261	0.205763
$898$	0	0
$899$	−65.1341	−2.17234
$900$	0	0
$901$	−7.85515	−0.261693
$902$	0	0
$903$	−0.454051	−0.0151099
$904$	0	0
$905$	−4.37675	−0.145488
$906$	0	0
$907$	−49.5615	−1.64566	−0.822831	−	0.568286i	$-0.807607\pi$
$907$	−49.5615	−1.64566	−0.822831	+	0.568286i	$0.807607\pi$
$908$	0	0
$909$	−3.08547	−0.102338
$910$	0	0
$911$	30.6908	1.01683	0.508415	−	0.861112i	$-0.330232\pi$
$911$	30.6908	1.01683	0.508415	+	0.861112i	$0.330232\pi$
$912$	0	0
$913$	−12.4703	−0.412707
$914$	0	0
$915$	−0.131584	−0.00435003
$916$	0	0
$917$	−0.353057	−0.0116590
$918$	0	0
$919$	−29.4304	−0.970819	−0.485409	−	0.874287i	$-0.661329\pi$
$919$	−29.4304	−0.970819	−0.485409	+	0.874287i	$0.661329\pi$
$920$	0	0
$921$	26.6433	0.877928
$922$	0	0
$923$	−27.2084	−0.895577
$924$	0	0
$925$	−11.8702	−0.390288
$926$	0	0
$927$	6.58913	0.216415
$928$	0	0
$929$	−3.45739	−0.113433	−0.0567166	−	0.998390i	$-0.518063\pi$
$929$	−3.45739	−0.113433	−0.0567166	+	0.998390i	$0.518063\pi$
$930$	0	0
$931$	−6.98572	−0.228948
$932$	0	0
$933$	2.71532	0.0888957
$934$	0	0
$935$	0.549946	0.0179851
$936$	0	0
$937$	−2.19174	−0.0716010	−0.0358005	−	0.999359i	$-0.511398\pi$
$937$	−2.19174	−0.0716010	−0.0358005	+	0.999359i	$0.511398\pi$
$938$	0	0
$939$	6.54601	0.213621
$940$	0	0
$941$	37.7444	1.23043	0.615216	−	0.788358i	$-0.289069\pi$
$941$	37.7444	1.23043	0.615216	+	0.788358i	$0.289069\pi$
$942$	0	0
$943$	0.587081	0.0191180
$944$	0	0
$945$	−0.0187128	−0.000608727 0
$946$	0	0
$947$	−49.2053	−1.59896	−0.799479	−	0.600694i	$-0.794891\pi$
$947$	−49.2053	−1.59896	−0.799479	+	0.600694i	$0.794891\pi$
$948$	0	0
$949$	13.1865	0.428053
$950$	0	0
$951$	19.4676	0.631280
$952$	0	0
$953$	23.3028	0.754851	0.377426	−	0.926040i	$-0.376809\pi$
$953$	23.3028	0.754851	0.377426	+	0.926040i	$0.376809\pi$
$954$	0	0
$955$	0.394882	0.0127781
$956$	0	0
$957$	−11.5124	−0.372143
$958$	0	0
$959$	−0.901628	−0.0291151
$960$	0	0
$961$	59.0552	1.90501
$962$	0	0
$963$	10.7446	0.346241
$964$	0	0
$965$	−0.00103540	−3.33307e−5 0
$966$	0	0
$967$	1.90814	0.0613616	0.0306808	−	0.999529i	$-0.490232\pi$
$967$	1.90814	0.0613616	0.0306808	+	0.999529i	$0.490232\pi$
$968$	0	0
$969$	−0.834252	−0.0268000
$970$	0	0
$971$	−25.9471	−0.832682	−0.416341	−	0.909209i	$-0.636688\pi$
$971$	−25.9471	−0.832682	−0.416341	+	0.909209i	$0.636688\pi$
$972$	0	0
$973$	0.218538	0.00700599
$974$	0	0
$975$	−22.7945	−0.730009
$976$	0	0
$977$	−27.6096	−0.883311	−0.441655	−	0.897185i	$-0.645609\pi$
$977$	−27.6096	−0.883311	−0.441655	+	0.897185i	$0.645609\pi$
$978$	0	0
$979$	−9.94343	−0.317793
$980$	0	0
$981$	−5.37085	−0.171478
$982$	0	0
$983$	11.9349	0.380665	0.190332	−	0.981720i	$-0.439043\pi$
$983$	11.9349	0.380665	0.190332	+	0.981720i	$0.439043\pi$
$984$	0	0
$985$	1.29350	0.0412142
$986$	0	0
$987$	0.181534	0.00577828
$988$	0	0
$989$	12.4725	0.396603
$990$	0	0
$991$	26.4396	0.839881	0.419940	−	0.907552i	$-0.362051\pi$
$991$	26.4396	0.839881	0.419940	+	0.907552i	$0.362051\pi$
$992$	0	0
$993$	−11.3987	−0.361727
$994$	0	0
$995$	−0.787605	−0.0249688
$996$	0	0
$997$	44.4393	1.40741	0.703704	−	0.710494i	$-0.251528\pi$
$997$	44.4393	1.40741	0.703704	+	0.710494i	$0.251528\pi$
$998$	0	0
$999$	−2.44944	−0.0774969

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4008.2.a.j.1.7	✓	10
4.3	odd	2		8016.2.a.bd.1.7		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4008.2.a.j.1.7	✓	10	1.1	even	1	trivial
8016.2.a.bd.1.7		10	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +0.392343 q^{5} +0.0476950 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +0.392343 q^{5} +0.0476950 q^{7} +1.00000 q^{9} +1.67730 q^{11} -4.70371 q^{13} -0.392343 q^{15} +0.835685 q^{17} +0.998285 q^{19} -0.0476950 q^{21} +1.31016 q^{23} -4.84607 q^{25} -1.00000 q^{27} +6.86363 q^{29} -9.48974 q^{31} -1.67730 q^{33} +0.0187128 q^{35} +2.44944 q^{37} +4.70371 q^{39} +0.448099 q^{41} +9.51987 q^{43} +0.392343 q^{45} -3.80613 q^{47} -6.99773 q^{49} -0.835685 q^{51} -9.39965 q^{53} +0.658077 q^{55} -0.998285 q^{57} -3.24192 q^{59} +0.335380 q^{61} +0.0476950 q^{63} -1.84547 q^{65} -14.2570 q^{67} -1.31016 q^{69} +5.78446 q^{71} -2.80343 q^{73} +4.84607 q^{75} +0.0799990 q^{77} +15.3705 q^{79} +1.00000 q^{81} -7.43473 q^{83} +0.327875 q^{85} -6.86363 q^{87} -5.92823 q^{89} -0.224344 q^{91} +9.48974 q^{93} +0.391670 q^{95} +4.70012 q^{97} +1.67730 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 10 q^{3} - 10 q^{5} + q^{7} + 10 q^{9}+O(q^{10}) \) 10 * q - 10 * q^3 - 10 * q^5 + q^7 + 10 * q^9	\( 10 q - 10 q^{3} - 10 q^{5} + q^{7} + 10 q^{9} - q^{11} - 6 q^{13} + 10 q^{15} - 9 q^{17} + 2 q^{19} - q^{21} + 7 q^{23} + 12 q^{25} - 10 q^{27} - 13 q^{29} + 23 q^{31} + q^{33} + q^{35} - 6 q^{37} + 6 q^{39} - 12 q^{41} - 10 q^{45} + 10 q^{47} + 7 q^{49} + 9 q^{51} - 26 q^{53} + 11 q^{55} - 2 q^{57} - 10 q^{59} - 10 q^{61} + q^{63} - 22 q^{65} - 5 q^{67} - 7 q^{69} + 25 q^{71} - 8 q^{73} - 12 q^{75} - 46 q^{77} + 26 q^{79} + 10 q^{81} - 14 q^{83} + 9 q^{85} + 13 q^{87} - 31 q^{89} - 3 q^{91} - 23 q^{93} - 5 q^{95} - 32 q^{97} - q^{99}+O(q^{100}) \) 10 * q - 10 * q^3 - 10 * q^5 + q^7 + 10 * q^9 - q^11 - 6 * q^13 + 10 * q^15 - 9 * q^17 + 2 * q^19 - q^21 + 7 * q^23 + 12 * q^25 - 10 * q^27 - 13 * q^29 + 23 * q^31 + q^33 + q^35 - 6 * q^37 + 6 * q^39 - 12 * q^41 - 10 * q^45 + 10 * q^47 + 7 * q^49 + 9 * q^51 - 26 * q^53 + 11 * q^55 - 2 * q^57 - 10 * q^59 - 10 * q^61 + q^63 - 22 * q^65 - 5 * q^67 - 7 * q^69 + 25 * q^71 - 8 * q^73 - 12 * q^75 - 46 * q^77 + 26 * q^79 + 10 * q^81 - 14 * q^83 + 9 * q^85 + 13 * q^87 - 31 * q^89 - 3 * q^91 - 23 * q^93 - 5 * q^95 - 32 * q^97 - q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more