LMFDB - Embedded newform 4008.2.a.l.1.5

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:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} - 31 x^{10} + 131 x^{9} + 309 x^{8} - 1453 x^{7} - 1072 x^{6} + 6350 x^{5} + \cdots + 3008 $ x^12 - 4x^11 - 31x^10 + 131x^9 + 309x^8 - 1453x^7 - 1072x^6 + 6350x^5 + 1411x^4 - 11022x^3 - 2450x^2 + 6960*x + 3008
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.716151$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -0.716151 q^{5} -3.33002 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -0.716151 q^{5} -3.33002 q^{7} +1.00000 q^{9} -1.47099 q^{11} -1.07152 q^{13} -0.716151 q^{15} +4.68991 q^{17} -1.60112 q^{19} -3.33002 q^{21} +5.96922 q^{23} -4.48713 q^{25} +1.00000 q^{27} +4.73151 q^{29} +8.82346 q^{31} -1.47099 q^{33} +2.38479 q^{35} -10.4028 q^{37} -1.07152 q^{39} -2.92019 q^{41} -8.38527 q^{43} -0.716151 q^{45} +6.24255 q^{47} +4.08900 q^{49} +4.68991 q^{51} -9.50501 q^{53} +1.05345 q^{55} -1.60112 q^{57} -4.74996 q^{59} +7.09947 q^{61} -3.33002 q^{63} +0.767367 q^{65} +13.6608 q^{67} +5.96922 q^{69} +13.6859 q^{71} +7.30688 q^{73} -4.48713 q^{75} +4.89844 q^{77} +12.8036 q^{79} +1.00000 q^{81} +1.24092 q^{83} -3.35869 q^{85} +4.73151 q^{87} -7.67545 q^{89} +3.56816 q^{91} +8.82346 q^{93} +1.14664 q^{95} +13.4461 q^{97} -1.47099 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{3} + 4 q^{5} + 11 q^{7} + 12 q^{9}+O(q^{10}) $ 12 * q + 12 * q^3 + 4 * q^5 + 11 * q^7 + 12 * q^9	$ 12 q + 12 q^{3} + 4 q^{5} + 11 q^{7} + 12 q^{9} + q^{11} + 8 q^{13} + 4 q^{15} + 3 q^{17} + 12 q^{19} + 11 q^{21} + 7 q^{23} + 18 q^{25} + 12 q^{27} + 5 q^{29} + 33 q^{31} + q^{33} + 15 q^{35} + 8 q^{37} + 8 q^{39} - 6 q^{41} + 16 q^{43} + 4 q^{45} + 18 q^{47} + 25 q^{49} + 3 q^{51} + 20 q^{53} + 39 q^{55} + 12 q^{57} + 4 q^{59} + 10 q^{61} + 11 q^{63} + 9 q^{67} + 7 q^{69} + 11 q^{71} + 22 q^{73} + 18 q^{75} + 24 q^{77} + 56 q^{79} + 12 q^{81} + 26 q^{83} + 15 q^{85} + 5 q^{87} - 15 q^{89} + 11 q^{91} + 33 q^{93} + 3 q^{95} + 8 q^{97} + q^{99}+O(q^{100}) $ 12 * q + 12 * q^3 + 4 * q^5 + 11 * q^7 + 12 * q^9 + q^11 + 8 * q^13 + 4 * q^15 + 3 * q^17 + 12 * q^19 + 11 * q^21 + 7 * q^23 + 18 * q^25 + 12 * q^27 + 5 * q^29 + 33 * q^31 + q^33 + 15 * q^35 + 8 * q^37 + 8 * q^39 - 6 * q^41 + 16 * q^43 + 4 * q^45 + 18 * q^47 + 25 * q^49 + 3 * q^51 + 20 * q^53 + 39 * q^55 + 12 * q^57 + 4 * q^59 + 10 * q^61 + 11 * q^63 + 9 * q^67 + 7 * q^69 + 11 * q^71 + 22 * q^73 + 18 * q^75 + 24 * q^77 + 56 * q^79 + 12 * q^81 + 26 * q^83 + 15 * q^85 + 5 * q^87 - 15 * q^89 + 11 * q^91 + 33 * q^93 + 3 * q^95 + 8 * 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
$5$	−0.716151	−0.320272	−0.160136	−	0.987095i	$-0.551193\pi$
$5$	−0.716151	−0.320272	−0.160136	+	0.987095i	$0.551193\pi$
$6$	0	0
$7$	−3.33002	−1.25863	−0.629314	−	0.777151i	$-0.716664\pi$
$7$	−3.33002	−1.25863	−0.629314	+	0.777151i	$0.716664\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.47099	−0.443522	−0.221761	−	0.975101i	$-0.571180\pi$
$11$	−1.47099	−0.443522	−0.221761	+	0.975101i	$0.571180\pi$
$12$	0	0
$13$	−1.07152	−0.297185	−0.148593	−	0.988899i	$-0.547474\pi$
$13$	−1.07152	−0.297185	−0.148593	+	0.988899i	$0.547474\pi$
$14$	0	0
$15$	−0.716151	−0.184909
$16$	0	0
$17$	4.68991	1.13747	0.568735	−	0.822520i	$-0.307433\pi$
$17$	4.68991	1.13747	0.568735	+	0.822520i	$0.307433\pi$
$18$	0	0
$19$	−1.60112	−0.367321	−0.183661	−	0.982990i	$-0.558795\pi$
$19$	−1.60112	−0.367321	−0.183661	+	0.982990i	$0.558795\pi$
$20$	0	0
$21$	−3.33002	−0.726669
$22$	0	0
$23$	5.96922	1.24467	0.622334	−	0.782752i	$-0.286185\pi$
$23$	5.96922	1.24467	0.622334	+	0.782752i	$0.286185\pi$
$24$	0	0
$25$	−4.48713	−0.897426
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	4.73151	0.878620	0.439310	−	0.898336i	$-0.355223\pi$
$29$	4.73151	0.878620	0.439310	+	0.898336i	$0.355223\pi$
$30$	0	0
$31$	8.82346	1.58474	0.792370	−	0.610041i	$-0.208847\pi$
$31$	8.82346	1.58474	0.792370	+	0.610041i	$0.208847\pi$
$32$	0	0
$33$	−1.47099	−0.256067
$34$	0	0
$35$	2.38479	0.403104
$36$	0	0
$37$	−10.4028	−1.71021	−0.855106	−	0.518453i	$-0.826508\pi$
$37$	−10.4028	−1.71021	−0.855106	+	0.518453i	$0.826508\pi$
$38$	0	0
$39$	−1.07152	−0.171580
$40$	0	0
$41$	−2.92019	−0.456058	−0.228029	−	0.973654i	$-0.573228\pi$
$41$	−2.92019	−0.456058	−0.228029	+	0.973654i	$0.573228\pi$
$42$	0	0
$43$	−8.38527	−1.27874	−0.639371	−	0.768899i	$-0.720805\pi$
$43$	−8.38527	−1.27874	−0.639371	+	0.768899i	$0.720805\pi$
$44$	0	0
$45$	−0.716151	−0.106757
$46$	0	0
$47$	6.24255	0.910569	0.455285	−	0.890346i	$-0.349538\pi$
$47$	6.24255	0.910569	0.455285	+	0.890346i	$0.349538\pi$
$48$	0	0
$49$	4.08900	0.584143
$50$	0	0
$51$	4.68991	0.656719
$52$	0	0
$53$	−9.50501	−1.30561	−0.652807	−	0.757524i	$-0.726409\pi$
$53$	−9.50501	−1.30561	−0.652807	+	0.757524i	$0.726409\pi$
$54$	0	0
$55$	1.05345	0.142048
$56$	0	0
$57$	−1.60112	−0.212073
$58$	0	0
$59$	−4.74996	−0.618393	−0.309196	−	0.950998i	$-0.600060\pi$
$59$	−4.74996	−0.618393	−0.309196	+	0.950998i	$0.600060\pi$
$60$	0	0
$61$	7.09947	0.908993	0.454497	−	0.890748i	$-0.349819\pi$
$61$	7.09947	0.908993	0.454497	+	0.890748i	$0.349819\pi$
$62$	0	0
$63$	−3.33002	−0.419543
$64$	0	0
$65$	0.767367	0.0951802
$66$	0	0
$67$	13.6608	1.66893	0.834466	−	0.551059i	$-0.185776\pi$
$67$	13.6608	1.66893	0.834466	+	0.551059i	$0.185776\pi$
$68$	0	0
$69$	5.96922	0.718610
$70$	0	0
$71$	13.6859	1.62422	0.812109	−	0.583505i	$-0.198319\pi$
$71$	13.6859	1.62422	0.812109	+	0.583505i	$0.198319\pi$
$72$	0	0
$73$	7.30688	0.855205	0.427603	−	0.903967i	$-0.359358\pi$
$73$	7.30688	0.855205	0.427603	+	0.903967i	$0.359358\pi$
$74$	0	0
$75$	−4.48713	−0.518129
$76$	0	0
$77$	4.89844	0.558229
$78$	0	0
$79$	12.8036	1.44052	0.720259	−	0.693705i	$-0.244023\pi$
$79$	12.8036	1.44052	0.720259	+	0.693705i	$0.244023\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	1.24092	0.136209	0.0681045	−	0.997678i	$-0.478305\pi$
$83$	1.24092	0.136209	0.0681045	+	0.997678i	$0.478305\pi$
$84$	0	0
$85$	−3.35869	−0.364301
$86$	0	0
$87$	4.73151	0.507271
$88$	0	0
$89$	−7.67545	−0.813596	−0.406798	−	0.913518i	$-0.633355\pi$
$89$	−7.67545	−0.813596	−0.406798	+	0.913518i	$0.633355\pi$
$90$	0	0
$91$	3.56816	0.374045
$92$	0	0
$93$	8.82346	0.914950
$94$	0	0
$95$	1.14664	0.117643
$96$	0	0
$97$	13.4461	1.36525	0.682623	−	0.730771i	$-0.260839\pi$
$97$	13.4461	1.36525	0.682623	+	0.730771i	$0.260839\pi$
$98$	0	0
$99$	−1.47099	−0.147841
$100$	0	0
$101$	−0.372387	−0.0370539	−0.0185269	−	0.999828i	$-0.505898\pi$
$101$	−0.372387	−0.0370539	−0.0185269	+	0.999828i	$0.505898\pi$
$102$	0	0
$103$	7.26876	0.716213	0.358106	−	0.933681i	$-0.383423\pi$
$103$	7.26876	0.716213	0.358106	+	0.933681i	$0.383423\pi$
$104$	0	0
$105$	2.38479	0.232732
$106$	0	0
$107$	7.75739	0.749935	0.374968	−	0.927038i	$-0.377654\pi$
$107$	7.75739	0.749935	0.374968	+	0.927038i	$0.377654\pi$
$108$	0	0
$109$	9.49121	0.909094	0.454547	−	0.890723i	$-0.349801\pi$
$109$	9.49121	0.909094	0.454547	+	0.890723i	$0.349801\pi$
$110$	0	0
$111$	−10.4028	−0.987392
$112$	0	0
$113$	2.74341	0.258078	0.129039	−	0.991639i	$-0.458811\pi$
$113$	2.74341	0.258078	0.129039	+	0.991639i	$0.458811\pi$
$114$	0	0
$115$	−4.27486	−0.398633
$116$	0	0
$117$	−1.07152	−0.0990617
$118$	0	0
$119$	−15.6175	−1.43165
$120$	0	0
$121$	−8.83617	−0.803289
$122$	0	0
$123$	−2.92019	−0.263305
$124$	0	0
$125$	6.79422	0.607693
$126$	0	0
$127$	15.0695	1.33720	0.668600	−	0.743622i	$-0.266894\pi$
$127$	15.0695	1.33720	0.668600	+	0.743622i	$0.266894\pi$
$128$	0	0
$129$	−8.38527	−0.738282
$130$	0	0
$131$	−8.64719	−0.755508	−0.377754	−	0.925906i	$-0.623304\pi$
$131$	−8.64719	−0.755508	−0.377754	+	0.925906i	$0.623304\pi$
$132$	0	0
$133$	5.33174	0.462321
$134$	0	0
$135$	−0.716151	−0.0616365
$136$	0	0
$137$	13.9735	1.19383	0.596917	−	0.802303i	$-0.296392\pi$
$137$	13.9735	1.19383	0.596917	+	0.802303i	$0.296392\pi$
$138$	0	0
$139$	9.76531	0.828283	0.414141	−	0.910213i	$-0.364082\pi$
$139$	9.76531	0.828283	0.414141	+	0.910213i	$0.364082\pi$
$140$	0	0
$141$	6.24255	0.525717
$142$	0	0
$143$	1.57619	0.131808
$144$	0	0
$145$	−3.38848	−0.281398
$146$	0	0
$147$	4.08900	0.337255
$148$	0	0
$149$	12.6591	1.03708	0.518538	−	0.855054i	$-0.326476\pi$
$149$	12.6591	1.03708	0.518538	+	0.855054i	$0.326476\pi$
$150$	0	0
$151$	−10.2773	−0.836352	−0.418176	−	0.908366i	$-0.637331\pi$
$151$	−10.2773	−0.836352	−0.418176	+	0.908366i	$0.637331\pi$
$152$	0	0
$153$	4.68991	0.379157
$154$	0	0
$155$	−6.31893	−0.507549
$156$	0	0
$157$	17.3312	1.38318	0.691590	−	0.722291i	$-0.256911\pi$
$157$	17.3312	1.38318	0.691590	+	0.722291i	$0.256911\pi$
$158$	0	0
$159$	−9.50501	−0.753797
$160$	0	0
$161$	−19.8776	−1.56657
$162$	0	0
$163$	6.39092	0.500576	0.250288	−	0.968171i	$-0.419475\pi$
$163$	6.39092	0.500576	0.250288	+	0.968171i	$0.419475\pi$
$164$	0	0
$165$	1.05345	0.0820113
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	−11.8519	−0.911681
$170$	0	0
$171$	−1.60112	−0.122440
$172$	0	0
$173$	23.0601	1.75323	0.876614	−	0.481195i	$-0.159797\pi$
$173$	23.0601	1.75323	0.876614	+	0.481195i	$0.159797\pi$
$174$	0	0
$175$	14.9422	1.12952
$176$	0	0
$177$	−4.74996	−0.357029
$178$	0	0
$179$	−1.56721	−0.117139	−0.0585693	−	0.998283i	$-0.518654\pi$
$179$	−1.56721	−0.117139	−0.0585693	+	0.998283i	$0.518654\pi$
$180$	0	0
$181$	−9.66369	−0.718297	−0.359148	−	0.933281i	$-0.616933\pi$
$181$	−9.66369	−0.718297	−0.359148	+	0.933281i	$0.616933\pi$
$182$	0	0
$183$	7.09947	0.524808
$184$	0	0
$185$	7.44999	0.547734
$186$	0	0
$187$	−6.89884	−0.504493
$188$	0	0
$189$	−3.33002	−0.242223
$190$	0	0
$191$	15.1174	1.09386	0.546930	−	0.837179i	$-0.315796\pi$
$191$	15.1174	1.09386	0.546930	+	0.837179i	$0.315796\pi$
$192$	0	0
$193$	14.9243	1.07427	0.537137	−	0.843495i	$-0.319506\pi$
$193$	14.9243	1.07427	0.537137	+	0.843495i	$0.319506\pi$
$194$	0	0
$195$	0.767367	0.0549523
$196$	0	0
$197$	−13.7458	−0.979350	−0.489675	−	0.871905i	$-0.662885\pi$
$197$	−13.7458	−0.979350	−0.489675	+	0.871905i	$0.662885\pi$
$198$	0	0
$199$	9.45882	0.670518	0.335259	−	0.942126i	$-0.391176\pi$
$199$	9.45882	0.670518	0.335259	+	0.942126i	$0.391176\pi$
$200$	0	0
$201$	13.6608	0.963558
$202$	0	0
$203$	−15.7560	−1.10586
$204$	0	0
$205$	2.09130	0.146063
$206$	0	0
$207$	5.96922	0.414889
$208$	0	0
$209$	2.35523	0.162915
$210$	0	0
$211$	19.5293	1.34445	0.672227	−	0.740345i	$-0.265338\pi$
$211$	19.5293	1.34445	0.672227	+	0.740345i	$0.265338\pi$
$212$	0	0
$213$	13.6859	0.937743
$214$	0	0
$215$	6.00512	0.409546
$216$	0	0
$217$	−29.3822	−1.99460
$218$	0	0
$219$	7.30688	0.493753
$220$	0	0
$221$	−5.02532	−0.338039
$222$	0	0
$223$	5.37221	0.359750	0.179875	−	0.983690i	$-0.442431\pi$
$223$	5.37221	0.359750	0.179875	+	0.983690i	$0.442431\pi$
$224$	0	0
$225$	−4.48713	−0.299142
$226$	0	0
$227$	−27.0420	−1.79484	−0.897421	−	0.441174i	$-0.854562\pi$
$227$	−27.0420	−1.79484	−0.897421	+	0.441174i	$0.854562\pi$
$228$	0	0
$229$	−12.6136	−0.833527	−0.416764	−	0.909015i	$-0.636836\pi$
$229$	−12.6136	−0.833527	−0.416764	+	0.909015i	$0.636836\pi$
$230$	0	0
$231$	4.89844	0.322293
$232$	0	0
$233$	25.6565	1.68082	0.840408	−	0.541954i	$-0.182315\pi$
$233$	25.6565	1.68082	0.840408	+	0.541954i	$0.182315\pi$
$234$	0	0
$235$	−4.47061	−0.291630
$236$	0	0
$237$	12.8036	0.831684
$238$	0	0
$239$	2.74058	0.177273	0.0886367	−	0.996064i	$-0.471749\pi$
$239$	2.74058	0.177273	0.0886367	+	0.996064i	$0.471749\pi$
$240$	0	0
$241$	−19.6085	−1.26310	−0.631548	−	0.775337i	$-0.717580\pi$
$241$	−19.6085	−1.26310	−0.631548	+	0.775337i	$0.717580\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−2.92834	−0.187085
$246$	0	0
$247$	1.71562	0.109162
$248$	0	0
$249$	1.24092	0.0786403
$250$	0	0
$251$	−15.4602	−0.975842	−0.487921	−	0.872888i	$-0.662245\pi$
$251$	−15.4602	−0.975842	−0.487921	+	0.872888i	$0.662245\pi$
$252$	0	0
$253$	−8.78069	−0.552037
$254$	0	0
$255$	−3.35869	−0.210329
$256$	0	0
$257$	−28.2550	−1.76250	−0.881250	−	0.472650i	$-0.843298\pi$
$257$	−28.2550	−1.76250	−0.881250	+	0.472650i	$0.843298\pi$
$258$	0	0
$259$	34.6415	2.15252
$260$	0	0
$261$	4.73151	0.292873
$262$	0	0
$263$	−0.0685960	−0.00422981	−0.00211490	−	0.999998i	$-0.500673\pi$
$263$	−0.0685960	−0.00422981	−0.00211490	+	0.999998i	$0.500673\pi$
$264$	0	0
$265$	6.80702	0.418152
$266$	0	0
$267$	−7.67545	−0.469730
$268$	0	0
$269$	−16.3857	−0.999053	−0.499526	−	0.866299i	$-0.666493\pi$
$269$	−16.3857	−0.999053	−0.499526	+	0.866299i	$0.666493\pi$
$270$	0	0
$271$	−8.90565	−0.540980	−0.270490	−	0.962723i	$-0.587186\pi$
$271$	−8.90565	−0.540980	−0.270490	+	0.962723i	$0.587186\pi$
$272$	0	0
$273$	3.56816	0.215955
$274$	0	0
$275$	6.60054	0.398028
$276$	0	0
$277$	−10.7421	−0.645430	−0.322715	−	0.946496i	$-0.604596\pi$
$277$	−10.7421	−0.645430	−0.322715	+	0.946496i	$0.604596\pi$
$278$	0	0
$279$	8.82346	0.528247
$280$	0	0
$281$	−20.4669	−1.22095	−0.610475	−	0.792035i	$-0.709022\pi$
$281$	−20.4669	−1.22095	−0.610475	+	0.792035i	$0.709022\pi$
$282$	0	0
$283$	0.746921	0.0443998	0.0221999	−	0.999754i	$-0.492933\pi$
$283$	0.746921	0.0443998	0.0221999	+	0.999754i	$0.492933\pi$
$284$	0	0
$285$	1.14664	0.0679212
$286$	0	0
$287$	9.72429	0.574007
$288$	0	0
$289$	4.99528	0.293840
$290$	0	0
$291$	13.4461	0.788225
$292$	0	0
$293$	23.0541	1.34683	0.673417	−	0.739263i	$-0.264826\pi$
$293$	23.0541	1.34683	0.673417	+	0.739263i	$0.264826\pi$
$294$	0	0
$295$	3.40169	0.198054
$296$	0	0
$297$	−1.47099	−0.0853558
$298$	0	0
$299$	−6.39611	−0.369897
$300$	0	0
$301$	27.9231	1.60946
$302$	0	0
$303$	−0.372387	−0.0213931
$304$	0	0
$305$	−5.08429	−0.291126
$306$	0	0
$307$	−20.1854	−1.15204	−0.576021	−	0.817435i	$-0.695395\pi$
$307$	−20.1854	−1.15204	−0.576021	+	0.817435i	$0.695395\pi$
$308$	0	0
$309$	7.26876	0.413506
$310$	0	0
$311$	−18.6993	−1.06034	−0.530171	−	0.847891i	$-0.677872\pi$
$311$	−18.6993	−1.06034	−0.530171	+	0.847891i	$0.677872\pi$
$312$	0	0
$313$	−33.3300	−1.88392	−0.941962	−	0.335719i	$-0.891021\pi$
$313$	−33.3300	−1.88392	−0.941962	+	0.335719i	$0.891021\pi$
$314$	0	0
$315$	2.38479	0.134368
$316$	0	0
$317$	−13.7905	−0.774553	−0.387277	−	0.921964i	$-0.626584\pi$
$317$	−13.7905	−0.774553	−0.387277	+	0.921964i	$0.626584\pi$
$318$	0	0
$319$	−6.96003	−0.389687
$320$	0	0
$321$	7.75739	0.432975
$322$	0	0
$323$	−7.50910	−0.417817
$324$	0	0
$325$	4.80803	0.266701
$326$	0	0
$327$	9.49121	0.524865
$328$	0	0
$329$	−20.7878	−1.14607
$330$	0	0
$331$	7.26826	0.399500	0.199750	−	0.979847i	$-0.435987\pi$
$331$	7.26826	0.399500	0.199750	+	0.979847i	$0.435987\pi$
$332$	0	0
$333$	−10.4028	−0.570071
$334$	0	0
$335$	−9.78319	−0.534513
$336$	0	0
$337$	−6.75359	−0.367891	−0.183946	−	0.982936i	$-0.558887\pi$
$337$	−6.75359	−0.367891	−0.183946	+	0.982936i	$0.558887\pi$
$338$	0	0
$339$	2.74341	0.149002
$340$	0	0
$341$	−12.9793	−0.702866
$342$	0	0
$343$	9.69367	0.523409
$344$	0	0
$345$	−4.27486	−0.230151
$346$	0	0
$347$	1.61353	0.0866191	0.0433095	−	0.999062i	$-0.486210\pi$
$347$	1.61353	0.0866191	0.0433095	+	0.999062i	$0.486210\pi$
$348$	0	0
$349$	5.36893	0.287392	0.143696	−	0.989622i	$-0.454101\pi$
$349$	5.36893	0.287392	0.143696	+	0.989622i	$0.454101\pi$
$350$	0	0
$351$	−1.07152	−0.0571933
$352$	0	0
$353$	−4.70610	−0.250480	−0.125240	−	0.992126i	$-0.539970\pi$
$353$	−4.70610	−0.250480	−0.125240	+	0.992126i	$0.539970\pi$
$354$	0	0
$355$	−9.80118	−0.520192
$356$	0	0
$357$	−15.6175	−0.826565
$358$	0	0
$359$	−21.9268	−1.15725	−0.578625	−	0.815594i	$-0.696410\pi$
$359$	−21.9268	−1.15725	−0.578625	+	0.815594i	$0.696410\pi$
$360$	0	0
$361$	−16.4364	−0.865075
$362$	0	0
$363$	−8.83617	−0.463779
$364$	0	0
$365$	−5.23283	−0.273899
$366$	0	0
$367$	23.5650	1.23008	0.615040	−	0.788496i	$-0.289140\pi$
$367$	23.5650	1.23008	0.615040	+	0.788496i	$0.289140\pi$
$368$	0	0
$369$	−2.92019	−0.152019
$370$	0	0
$371$	31.6518	1.64328
$372$	0	0
$373$	0.522216	0.0270393	0.0135197	−	0.999909i	$-0.495696\pi$
$373$	0.522216	0.0270393	0.0135197	+	0.999909i	$0.495696\pi$
$374$	0	0
$375$	6.79422	0.350852
$376$	0	0
$377$	−5.06989	−0.261113
$378$	0	0
$379$	30.0093	1.54147	0.770737	−	0.637153i	$-0.219888\pi$
$379$	30.0093	1.54147	0.770737	+	0.637153i	$0.219888\pi$
$380$	0	0
$381$	15.0695	0.772033
$382$	0	0
$383$	11.1272	0.568572	0.284286	−	0.958740i	$-0.408244\pi$
$383$	11.1272	0.568572	0.284286	+	0.958740i	$0.408244\pi$
$384$	0	0
$385$	−3.50802	−0.178785
$386$	0	0
$387$	−8.38527	−0.426247
$388$	0	0
$389$	25.8762	1.31198	0.655989	−	0.754771i	$-0.272252\pi$
$389$	25.8762	1.31198	0.655989	+	0.754771i	$0.272252\pi$
$390$	0	0
$391$	27.9951	1.41577
$392$	0	0
$393$	−8.64719	−0.436193
$394$	0	0
$395$	−9.16932	−0.461358
$396$	0	0
$397$	−16.1106	−0.808569	−0.404284	−	0.914633i	$-0.632479\pi$
$397$	−16.1106	−0.808569	−0.404284	+	0.914633i	$0.632479\pi$
$398$	0	0
$399$	5.33174	0.266921
$400$	0	0
$401$	30.7106	1.53361	0.766807	−	0.641878i	$-0.221844\pi$
$401$	30.7106	1.53361	0.766807	+	0.641878i	$0.221844\pi$
$402$	0	0
$403$	−9.45448	−0.470961
$404$	0	0
$405$	−0.716151	−0.0355858
$406$	0	0
$407$	15.3025	0.758516
$408$	0	0
$409$	21.7712	1.07652	0.538259	−	0.842779i	$-0.319082\pi$
$409$	21.7712	1.07652	0.538259	+	0.842779i	$0.319082\pi$
$410$	0	0
$411$	13.9735	0.689260
$412$	0	0
$413$	15.8175	0.778326
$414$	0	0
$415$	−0.888688	−0.0436240
$416$	0	0
$417$	9.76531	0.478209
$418$	0	0
$419$	−1.38933	−0.0678734	−0.0339367	−	0.999424i	$-0.510804\pi$
$419$	−1.38933	−0.0678734	−0.0339367	+	0.999424i	$0.510804\pi$
$420$	0	0
$421$	29.4498	1.43530	0.717648	−	0.696406i	$-0.245219\pi$
$421$	29.4498	1.43530	0.717648	+	0.696406i	$0.245219\pi$
$422$	0	0
$423$	6.24255	0.303523
$424$	0	0
$425$	−21.0442	−1.02080
$426$	0	0
$427$	−23.6413	−1.14408
$428$	0	0
$429$	1.57619	0.0760994
$430$	0	0
$431$	−14.1214	−0.680202	−0.340101	−	0.940389i	$-0.610461\pi$
$431$	−14.1214	−0.680202	−0.340101	+	0.940389i	$0.610461\pi$
$432$	0	0
$433$	−32.8372	−1.57806	−0.789028	−	0.614357i	$-0.789415\pi$
$433$	−32.8372	−1.57806	−0.789028	+	0.614357i	$0.789415\pi$
$434$	0	0
$435$	−3.38848	−0.162465
$436$	0	0
$437$	−9.55742	−0.457193
$438$	0	0
$439$	37.4769	1.78868	0.894338	−	0.447392i	$-0.147647\pi$
$439$	37.4769	1.78868	0.894338	+	0.447392i	$0.147647\pi$
$440$	0	0
$441$	4.08900	0.194714
$442$	0	0
$443$	8.76333	0.416358	0.208179	−	0.978091i	$-0.433246\pi$
$443$	8.76333	0.416358	0.208179	+	0.978091i	$0.433246\pi$
$444$	0	0
$445$	5.49678	0.260572
$446$	0	0
$447$	12.6591	0.598757
$448$	0	0
$449$	−8.19198	−0.386603	−0.193302	−	0.981139i	$-0.561920\pi$
$449$	−8.19198	−0.386603	−0.193302	+	0.981139i	$0.561920\pi$
$450$	0	0
$451$	4.29559	0.202271
$452$	0	0
$453$	−10.2773	−0.482868
$454$	0	0
$455$	−2.55534	−0.119796
$456$	0	0
$457$	26.6671	1.24743	0.623717	−	0.781651i	$-0.285622\pi$
$457$	26.6671	1.24743	0.623717	+	0.781651i	$0.285622\pi$
$458$	0	0
$459$	4.68991	0.218906
$460$	0	0
$461$	−27.6394	−1.28730	−0.643648	−	0.765322i	$-0.722580\pi$
$461$	−27.6394	−1.28730	−0.643648	+	0.765322i	$0.722580\pi$
$462$	0	0
$463$	−39.7468	−1.84719	−0.923595	−	0.383371i	$-0.874763\pi$
$463$	−39.7468	−1.84719	−0.923595	+	0.383371i	$0.874763\pi$
$464$	0	0
$465$	−6.31893	−0.293033
$466$	0	0
$467$	−3.99234	−0.184743	−0.0923717	−	0.995725i	$-0.529445\pi$
$467$	−3.99234	−0.184743	−0.0923717	+	0.995725i	$0.529445\pi$
$468$	0	0
$469$	−45.4907	−2.10056
$470$	0	0
$471$	17.3312	0.798579
$472$	0	0
$473$	12.3347	0.567149
$474$	0	0
$475$	7.18442	0.329644
$476$	0	0
$477$	−9.50501	−0.435205
$478$	0	0
$479$	−20.4282	−0.933390	−0.466695	−	0.884418i	$-0.654555\pi$
$479$	−20.4282	−0.933390	−0.466695	+	0.884418i	$0.654555\pi$
$480$	0	0
$481$	11.1468	0.508250
$482$	0	0
$483$	−19.8776	−0.904462
$484$	0	0
$485$	−9.62944	−0.437251
$486$	0	0
$487$	−29.7212	−1.34680	−0.673399	−	0.739280i	$-0.735166\pi$
$487$	−29.7212	−1.34680	−0.673399	+	0.739280i	$0.735166\pi$
$488$	0	0
$489$	6.39092	0.289007
$490$	0	0
$491$	−28.6530	−1.29309	−0.646546	−	0.762875i	$-0.723787\pi$
$491$	−28.6530	−1.29309	−0.646546	+	0.762875i	$0.723787\pi$
$492$	0	0
$493$	22.1904	0.999404
$494$	0	0
$495$	1.05345	0.0473493
$496$	0	0
$497$	−45.5743	−2.04429
$498$	0	0
$499$	−15.6063	−0.698636	−0.349318	−	0.937004i	$-0.613587\pi$
$499$	−15.6063	−0.698636	−0.349318	+	0.937004i	$0.613587\pi$
$500$	0	0
$501$	1.00000	0.0446767
$502$	0	0
$503$	21.5905	0.962671	0.481335	−	0.876536i	$-0.340152\pi$
$503$	21.5905	0.962671	0.481335	+	0.876536i	$0.340152\pi$
$504$	0	0
$505$	0.266685	0.0118673
$506$	0	0
$507$	−11.8519	−0.526359
$508$	0	0
$509$	−32.2493	−1.42942	−0.714712	−	0.699419i	$-0.753442\pi$
$509$	−32.2493	−1.42942	−0.714712	+	0.699419i	$0.753442\pi$
$510$	0	0
$511$	−24.3320	−1.07638
$512$	0	0
$513$	−1.60112	−0.0706910
$514$	0	0
$515$	−5.20553	−0.229383
$516$	0	0
$517$	−9.18276	−0.403857
$518$	0	0
$519$	23.0601	1.01223
$520$	0	0
$521$	−5.29471	−0.231965	−0.115983	−	0.993251i	$-0.537002\pi$
$521$	−5.29471	−0.231965	−0.115983	+	0.993251i	$0.537002\pi$
$522$	0	0
$523$	30.7766	1.34577	0.672884	−	0.739748i	$-0.265055\pi$
$523$	30.7766	1.34577	0.672884	+	0.739748i	$0.265055\pi$
$524$	0	0
$525$	14.9422	0.652131
$526$	0	0
$527$	41.3812	1.80260
$528$	0	0
$529$	12.6316	0.549199
$530$	0	0
$531$	−4.74996	−0.206131
$532$	0	0
$533$	3.12903	0.135534
$534$	0	0
$535$	−5.55546	−0.240184
$536$	0	0
$537$	−1.56721	−0.0676300
$538$	0	0
$539$	−6.01490	−0.259080
$540$	0	0
$541$	−4.15500	−0.178637	−0.0893187	−	0.996003i	$-0.528469\pi$
$541$	−4.15500	−0.178637	−0.0893187	+	0.996003i	$0.528469\pi$
$542$	0	0
$543$	−9.66369	−0.414709
$544$	0	0
$545$	−6.79714	−0.291158
$546$	0	0
$547$	33.6839	1.44022	0.720110	−	0.693860i	$-0.244091\pi$
$547$	33.6839	1.44022	0.720110	+	0.693860i	$0.244091\pi$
$548$	0	0
$549$	7.09947	0.302998
$550$	0	0
$551$	−7.57570	−0.322736
$552$	0	0
$553$	−42.6362	−1.81308
$554$	0	0
$555$	7.44999	0.316234
$556$	0	0
$557$	−42.2475	−1.79008	−0.895042	−	0.445983i	$-0.852854\pi$
$557$	−42.2475	−1.79008	−0.895042	+	0.445983i	$0.852854\pi$
$558$	0	0
$559$	8.98495	0.380023
$560$	0	0
$561$	−6.89884	−0.291269
$562$	0	0
$563$	−0.759500	−0.0320091	−0.0160046	−	0.999872i	$-0.505095\pi$
$563$	−0.759500	−0.0320091	−0.0160046	+	0.999872i	$0.505095\pi$
$564$	0	0
$565$	−1.96470	−0.0826554
$566$	0	0
$567$	−3.33002	−0.139848
$568$	0	0
$569$	37.2088	1.55987	0.779937	−	0.625858i	$-0.215251\pi$
$569$	37.2088	1.55987	0.779937	+	0.625858i	$0.215251\pi$
$570$	0	0
$571$	21.3536	0.893621	0.446811	−	0.894629i	$-0.352560\pi$
$571$	21.3536	0.893621	0.446811	+	0.894629i	$0.352560\pi$
$572$	0	0
$573$	15.1174	0.631540
$574$	0	0
$575$	−26.7846	−1.11700
$576$	0	0
$577$	0.837781	0.0348773	0.0174386	−	0.999848i	$-0.494449\pi$
$577$	0.837781	0.0348773	0.0174386	+	0.999848i	$0.494449\pi$
$578$	0	0
$579$	14.9243	0.620233
$580$	0	0
$581$	−4.13229	−0.171436
$582$	0	0
$583$	13.9818	0.579068
$584$	0	0
$585$	0.767367	0.0317267
$586$	0	0
$587$	9.10109	0.375642	0.187821	−	0.982203i	$-0.439857\pi$
$587$	9.10109	0.375642	0.187821	+	0.982203i	$0.439857\pi$
$588$	0	0
$589$	−14.1274	−0.582109
$590$	0	0
$591$	−13.7458	−0.565428
$592$	0	0
$593$	−32.8995	−1.35102	−0.675510	−	0.737351i	$-0.736076\pi$
$593$	−32.8995	−1.35102	−0.675510	+	0.737351i	$0.736076\pi$
$594$	0	0
$595$	11.1845	0.458519
$596$	0	0
$597$	9.45882	0.387124
$598$	0	0
$599$	14.9054	0.609018	0.304509	−	0.952509i	$-0.401508\pi$
$599$	14.9054	0.609018	0.304509	+	0.952509i	$0.401508\pi$
$600$	0	0
$601$	45.6503	1.86212	0.931058	−	0.364871i	$-0.118887\pi$
$601$	45.6503	1.86212	0.931058	+	0.364871i	$0.118887\pi$
$602$	0	0
$603$	13.6608	0.556311
$604$	0	0
$605$	6.32803	0.257271
$606$	0	0
$607$	−7.14482	−0.289999	−0.145000	−	0.989432i	$-0.546318\pi$
$607$	−7.14482	−0.289999	−0.145000	+	0.989432i	$0.546318\pi$
$608$	0	0
$609$	−15.7560	−0.638466
$610$	0	0
$611$	−6.68899	−0.270608
$612$	0	0
$613$	12.0707	0.487530	0.243765	−	0.969834i	$-0.421617\pi$
$613$	12.0707	0.487530	0.243765	+	0.969834i	$0.421617\pi$
$614$	0	0
$615$	2.09130	0.0843294
$616$	0	0
$617$	−18.6455	−0.750640	−0.375320	−	0.926895i	$-0.622467\pi$
$617$	−18.6455	−0.750640	−0.375320	+	0.926895i	$0.622467\pi$
$618$	0	0
$619$	46.6989	1.87699	0.938495	−	0.345294i	$-0.112221\pi$
$619$	46.6989	1.87699	0.938495	+	0.345294i	$0.112221\pi$
$620$	0	0
$621$	5.96922	0.239537
$622$	0	0
$623$	25.5594	1.02401
$624$	0	0
$625$	17.5700	0.702798
$626$	0	0
$627$	2.35523	0.0940590
$628$	0	0
$629$	−48.7883	−1.94532
$630$	0	0
$631$	37.4683	1.49159	0.745794	−	0.666177i	$-0.232070\pi$
$631$	37.4683	1.49159	0.745794	+	0.666177i	$0.232070\pi$
$632$	0	0
$633$	19.5293	0.776221
$634$	0	0
$635$	−10.7920	−0.428269
$636$	0	0
$637$	−4.38143	−0.173599
$638$	0	0
$639$	13.6859	0.541406
$640$	0	0
$641$	19.3368	0.763757	0.381878	−	0.924213i	$-0.375277\pi$
$641$	19.3368	0.763757	0.381878	+	0.924213i	$0.375277\pi$
$642$	0	0
$643$	35.6061	1.40417	0.702083	−	0.712095i	$-0.252253\pi$
$643$	35.6061	1.40417	0.702083	+	0.712095i	$0.252253\pi$
$644$	0	0
$645$	6.00512	0.236451
$646$	0	0
$647$	−24.6666	−0.969743	−0.484872	−	0.874585i	$-0.661134\pi$
$647$	−24.6666	−0.969743	−0.484872	+	0.874585i	$0.661134\pi$
$648$	0	0
$649$	6.98717	0.274271
$650$	0	0
$651$	−29.3822	−1.15158
$652$	0	0
$653$	37.6448	1.47315	0.736577	−	0.676353i	$-0.236441\pi$
$653$	37.6448	1.47315	0.736577	+	0.676353i	$0.236441\pi$
$654$	0	0
$655$	6.19269	0.241968
$656$	0	0
$657$	7.30688	0.285068
$658$	0	0
$659$	−15.0705	−0.587063	−0.293532	−	0.955949i	$-0.594831\pi$
$659$	−15.0705	−0.587063	−0.293532	+	0.955949i	$0.594831\pi$
$660$	0	0
$661$	−15.4489	−0.600892	−0.300446	−	0.953799i	$-0.597136\pi$
$661$	−15.4489	−0.600892	−0.300446	+	0.953799i	$0.597136\pi$
$662$	0	0
$663$	−5.02532	−0.195167
$664$	0	0
$665$	−3.81833	−0.148069
$666$	0	0
$667$	28.2434	1.09359
$668$	0	0
$669$	5.37221	0.207702
$670$	0	0
$671$	−10.4433	−0.403158
$672$	0	0
$673$	−19.7469	−0.761187	−0.380593	−	0.924743i	$-0.624280\pi$
$673$	−19.7469	−0.761187	−0.380593	+	0.924743i	$0.624280\pi$
$674$	0	0
$675$	−4.48713	−0.172710
$676$	0	0
$677$	45.4498	1.74678	0.873388	−	0.487025i	$-0.161918\pi$
$677$	45.4498	1.74678	0.873388	+	0.487025i	$0.161918\pi$
$678$	0	0
$679$	−44.7757	−1.71834
$680$	0	0
$681$	−27.0420	−1.03625
$682$	0	0
$683$	−13.5432	−0.518217	−0.259108	−	0.965848i	$-0.583429\pi$
$683$	−13.5432	−0.518217	−0.259108	+	0.965848i	$0.583429\pi$
$684$	0	0
$685$	−10.0071	−0.382352
$686$	0	0
$687$	−12.6136	−0.481237
$688$	0	0
$689$	10.1848	0.388009
$690$	0	0
$691$	−10.4424	−0.397250	−0.198625	−	0.980076i	$-0.563648\pi$
$691$	−10.4424	−0.397250	−0.198625	+	0.980076i	$0.563648\pi$
$692$	0	0
$693$	4.89844	0.186076
$694$	0	0
$695$	−6.99344	−0.265276
$696$	0	0
$697$	−13.6955	−0.518752
$698$	0	0
$699$	25.6565	0.970420
$700$	0	0
$701$	−6.26217	−0.236519	−0.118259	−	0.992983i	$-0.537731\pi$
$701$	−6.26217	−0.236519	−0.118259	+	0.992983i	$0.537731\pi$
$702$	0	0
$703$	16.6561	0.628198
$704$	0	0
$705$	−4.47061	−0.168373
$706$	0	0
$707$	1.24005	0.0466371
$708$	0	0
$709$	−38.7936	−1.45692	−0.728462	−	0.685086i	$-0.759765\pi$
$709$	−38.7936	−1.45692	−0.728462	+	0.685086i	$0.759765\pi$
$710$	0	0
$711$	12.8036	0.480173
$712$	0	0
$713$	52.6692	1.97248
$714$	0	0
$715$	−1.12879	−0.0422145
$716$	0	0
$717$	2.74058	0.102349
$718$	0	0
$719$	26.4655	0.986997	0.493498	−	0.869747i	$-0.335718\pi$
$719$	26.4655	0.986997	0.493498	+	0.869747i	$0.335718\pi$
$720$	0	0
$721$	−24.2051	−0.901445
$722$	0	0
$723$	−19.6085	−0.729249
$724$	0	0
$725$	−21.2309	−0.788496
$726$	0	0
$727$	43.0571	1.59690	0.798450	−	0.602062i	$-0.205654\pi$
$727$	43.0571	1.59690	0.798450	+	0.602062i	$0.205654\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−39.3262	−1.45453
$732$	0	0
$733$	−21.9645	−0.811276	−0.405638	−	0.914034i	$-0.632951\pi$
$733$	−21.9645	−0.811276	−0.405638	+	0.914034i	$0.632951\pi$
$734$	0	0
$735$	−2.92834	−0.108014
$736$	0	0
$737$	−20.0950	−0.740208
$738$	0	0
$739$	16.2983	0.599542	0.299771	−	0.954011i	$-0.403090\pi$
$739$	16.2983	0.599542	0.299771	+	0.954011i	$0.403090\pi$
$740$	0	0
$741$	1.71562	0.0630250
$742$	0	0
$743$	32.0904	1.17728	0.588642	−	0.808394i	$-0.299663\pi$
$743$	32.0904	1.17728	0.588642	+	0.808394i	$0.299663\pi$
$744$	0	0
$745$	−9.06585	−0.332147
$746$	0	0
$747$	1.24092	0.0454030
$748$	0	0
$749$	−25.8322	−0.943889
$750$	0	0
$751$	−20.8716	−0.761616	−0.380808	−	0.924654i	$-0.624354\pi$
$751$	−20.8716	−0.761616	−0.380808	+	0.924654i	$0.624354\pi$
$752$	0	0
$753$	−15.4602	−0.563403
$754$	0	0
$755$	7.36008	0.267861
$756$	0	0
$757$	4.00358	0.145512	0.0727562	−	0.997350i	$-0.476820\pi$
$757$	4.00358	0.145512	0.0727562	+	0.997350i	$0.476820\pi$
$758$	0	0
$759$	−8.78069	−0.318719
$760$	0	0
$761$	−42.4521	−1.53889	−0.769443	−	0.638716i	$-0.779466\pi$
$761$	−42.4521	−1.53889	−0.769443	+	0.638716i	$0.779466\pi$
$762$	0	0
$763$	−31.6059	−1.14421
$764$	0	0
$765$	−3.35869	−0.121434
$766$	0	0
$767$	5.08966	0.183777
$768$	0	0
$769$	8.42722	0.303893	0.151947	−	0.988389i	$-0.451446\pi$
$769$	8.42722	0.303893	0.151947	+	0.988389i	$0.451446\pi$
$770$	0	0
$771$	−28.2550	−1.01758
$772$	0	0
$773$	37.4061	1.34540	0.672702	−	0.739914i	$-0.265134\pi$
$773$	37.4061	1.34540	0.672702	+	0.739914i	$0.265134\pi$
$774$	0	0
$775$	−39.5920	−1.42219
$776$	0	0
$777$	34.6415	1.24276
$778$	0	0
$779$	4.67557	0.167520
$780$	0	0
$781$	−20.1319	−0.720376
$782$	0	0
$783$	4.73151	0.169090
$784$	0	0
$785$	−12.4117	−0.442994
$786$	0	0
$787$	13.8673	0.494316	0.247158	−	0.968975i	$-0.420503\pi$
$787$	13.8673	0.494316	0.247158	+	0.968975i	$0.420503\pi$
$788$	0	0
$789$	−0.0685960	−0.00244208
$790$	0	0
$791$	−9.13560	−0.324825
$792$	0	0
$793$	−7.60719	−0.270139
$794$	0	0
$795$	6.80702	0.241420
$796$	0	0
$797$	−28.2739	−1.00151	−0.500756	−	0.865588i	$-0.666945\pi$
$797$	−28.2739	−1.00151	−0.500756	+	0.865588i	$0.666945\pi$
$798$	0	0
$799$	29.2770	1.03575
$800$	0	0
$801$	−7.67545	−0.271199
$802$	0	0
$803$	−10.7484	−0.379302
$804$	0	0
$805$	14.2354	0.501730
$806$	0	0
$807$	−16.3857	−0.576803
$808$	0	0
$809$	45.5606	1.60183	0.800913	−	0.598781i	$-0.204348\pi$
$809$	45.5606	1.60183	0.800913	+	0.598781i	$0.204348\pi$
$810$	0	0
$811$	−39.1207	−1.37371	−0.686856	−	0.726794i	$-0.741010\pi$
$811$	−39.1207	−1.37371	−0.686856	+	0.726794i	$0.741010\pi$
$812$	0	0
$813$	−8.90565	−0.312335
$814$	0	0
$815$	−4.57686	−0.160321
$816$	0	0
$817$	13.4258	0.469709
$818$	0	0
$819$	3.56816	0.124682
$820$	0	0
$821$	9.50395	0.331690	0.165845	−	0.986152i	$-0.446965\pi$
$821$	9.50395	0.331690	0.165845	+	0.986152i	$0.446965\pi$
$822$	0	0
$823$	−36.2413	−1.26329	−0.631646	−	0.775257i	$-0.717620\pi$
$823$	−36.2413	−1.26329	−0.631646	+	0.775257i	$0.717620\pi$
$824$	0	0
$825$	6.60054	0.229801
$826$	0	0
$827$	28.0245	0.974508	0.487254	−	0.873260i	$-0.337999\pi$
$827$	28.0245	0.974508	0.487254	+	0.873260i	$0.337999\pi$
$828$	0	0
$829$	−37.6916	−1.30908	−0.654542	−	0.756026i	$-0.727138\pi$
$829$	−37.6916	−1.30908	−0.654542	+	0.756026i	$0.727138\pi$
$830$	0	0
$831$	−10.7421	−0.372639
$832$	0	0
$833$	19.1771	0.664446
$834$	0	0
$835$	−0.716151	−0.0247834
$836$	0	0
$837$	8.82346	0.304983
$838$	0	0
$839$	−13.3962	−0.462490	−0.231245	−	0.972896i	$-0.574280\pi$
$839$	−13.3962	−0.462490	−0.231245	+	0.972896i	$0.574280\pi$
$840$	0	0
$841$	−6.61279	−0.228027
$842$	0	0
$843$	−20.4669	−0.704916
$844$	0	0
$845$	8.48772	0.291986
$846$	0	0
$847$	29.4246	1.01104
$848$	0	0
$849$	0.746921	0.0256343
$850$	0	0
$851$	−62.0967	−2.12865
$852$	0	0
$853$	2.22324	0.0761223	0.0380611	−	0.999275i	$-0.487882\pi$
$853$	2.22324	0.0761223	0.0380611	+	0.999275i	$0.487882\pi$
$854$	0	0
$855$	1.14664	0.0392143
$856$	0	0
$857$	−44.5709	−1.52251	−0.761257	−	0.648451i	$-0.775417\pi$
$857$	−44.5709	−1.52251	−0.761257	+	0.648451i	$0.775417\pi$
$858$	0	0
$859$	9.70998	0.331300	0.165650	−	0.986185i	$-0.447028\pi$
$859$	9.70998	0.331300	0.165650	+	0.986185i	$0.447028\pi$
$860$	0	0
$861$	9.72429	0.331403
$862$	0	0
$863$	22.7865	0.775662	0.387831	−	0.921731i	$-0.373224\pi$
$863$	22.7865	0.775662	0.387831	+	0.921731i	$0.373224\pi$
$864$	0	0
$865$	−16.5145	−0.561511
$866$	0	0
$867$	4.99528	0.169649
$868$	0	0
$869$	−18.8340	−0.638901
$870$	0	0
$871$	−14.6378	−0.495982
$872$	0	0
$873$	13.4461	0.455082
$874$	0	0
$875$	−22.6248	−0.764859
$876$	0	0
$877$	18.2439	0.616051	0.308026	−	0.951378i	$-0.400332\pi$
$877$	18.2439	0.616051	0.308026	+	0.951378i	$0.400332\pi$
$878$	0	0
$879$	23.0541	0.777595
$880$	0	0
$881$	38.2386	1.28829	0.644145	−	0.764904i	$-0.277213\pi$
$881$	38.2386	1.28829	0.644145	+	0.764904i	$0.277213\pi$
$882$	0	0
$883$	−23.0956	−0.777228	−0.388614	−	0.921401i	$-0.627046\pi$
$883$	−23.0956	−0.777228	−0.388614	+	0.921401i	$0.627046\pi$
$884$	0	0
$885$	3.40169	0.114347
$886$	0	0
$887$	13.8961	0.466584	0.233292	−	0.972407i	$-0.425050\pi$
$887$	13.8961	0.466584	0.233292	+	0.972407i	$0.425050\pi$
$888$	0	0
$889$	−50.1816	−1.68304
$890$	0	0
$891$	−1.47099	−0.0492802
$892$	0	0
$893$	−9.99505	−0.334471
$894$	0	0
$895$	1.12236	0.0375163
$896$	0	0
$897$	−6.39611	−0.213560
$898$	0	0
$899$	41.7483	1.39238
$900$	0	0
$901$	−44.5777	−1.48510
$902$	0	0
$903$	27.9231	0.929222
$904$	0	0
$905$	6.92066	0.230051
$906$	0	0
$907$	−52.7844	−1.75268	−0.876339	−	0.481694i	$-0.840022\pi$
$907$	−52.7844	−1.75268	−0.876339	+	0.481694i	$0.840022\pi$
$908$	0	0
$909$	−0.372387	−0.0123513
$910$	0	0
$911$	57.2105	1.89547	0.947735	−	0.319059i	$-0.103367\pi$
$911$	57.2105	1.89547	0.947735	+	0.319059i	$0.103367\pi$
$912$	0	0
$913$	−1.82539	−0.0604116
$914$	0	0
$915$	−5.08429	−0.168081
$916$	0	0
$917$	28.7953	0.950903
$918$	0	0
$919$	−28.2303	−0.931230	−0.465615	−	0.884987i	$-0.654167\pi$
$919$	−28.2303	−0.931230	−0.465615	+	0.884987i	$0.654167\pi$
$920$	0	0
$921$	−20.1854	−0.665131
$922$	0	0
$923$	−14.6647	−0.482693
$924$	0	0
$925$	46.6788	1.53479
$926$	0	0
$927$	7.26876	0.238738
$928$	0	0
$929$	−44.0655	−1.44574	−0.722871	−	0.690983i	$-0.757178\pi$
$929$	−44.0655	−1.44574	−0.722871	+	0.690983i	$0.757178\pi$
$930$	0	0
$931$	−6.54697	−0.214568
$932$	0	0
$933$	−18.6993	−0.612189
$934$	0	0
$935$	4.94061	0.161575
$936$	0	0
$937$	20.3892	0.666087	0.333043	−	0.942912i	$-0.391924\pi$
$937$	20.3892	0.666087	0.333043	+	0.942912i	$0.391924\pi$
$938$	0	0
$939$	−33.3300	−1.08768
$940$	0	0
$941$	30.8213	1.00475	0.502373	−	0.864651i	$-0.332460\pi$
$941$	30.8213	1.00475	0.502373	+	0.864651i	$0.332460\pi$
$942$	0	0
$943$	−17.4313	−0.567641
$944$	0	0
$945$	2.38479	0.0775773
$946$	0	0
$947$	−38.8708	−1.26313	−0.631566	−	0.775322i	$-0.717587\pi$
$947$	−38.8708	−1.26313	−0.631566	+	0.775322i	$0.717587\pi$
$948$	0	0
$949$	−7.82943	−0.254154
$950$	0	0
$951$	−13.7905	−0.447188
$952$	0	0
$953$	−19.3760	−0.627649	−0.313824	−	0.949481i	$-0.601610\pi$
$953$	−19.3760	−0.627649	−0.313824	+	0.949481i	$0.601610\pi$
$954$	0	0
$955$	−10.8264	−0.350333
$956$	0	0
$957$	−6.96003	−0.224986
$958$	0	0
$959$	−46.5318	−1.50259
$960$	0	0
$961$	46.8534	1.51140
$962$	0	0
$963$	7.75739	0.249978
$964$	0	0
$965$	−10.6880	−0.344060
$966$	0	0
$967$	−24.4967	−0.787761	−0.393880	−	0.919162i	$-0.628868\pi$
$967$	−24.4967	−0.787761	−0.393880	+	0.919162i	$0.628868\pi$
$968$	0	0
$969$	−7.50910	−0.241227
$970$	0	0
$971$	−14.0223	−0.449996	−0.224998	−	0.974359i	$-0.572238\pi$
$971$	−14.0223	−0.449996	−0.224998	+	0.974359i	$0.572238\pi$
$972$	0	0
$973$	−32.5186	−1.04250
$974$	0	0
$975$	4.80803	0.153980
$976$	0	0
$977$	−20.1355	−0.644191	−0.322096	−	0.946707i	$-0.604387\pi$
$977$	−20.1355	−0.644191	−0.322096	+	0.946707i	$0.604387\pi$
$978$	0	0
$979$	11.2905	0.360847
$980$	0	0
$981$	9.49121	0.303031
$982$	0	0
$983$	36.0705	1.15047	0.575235	−	0.817988i	$-0.304911\pi$
$983$	36.0705	1.15047	0.575235	+	0.817988i	$0.304911\pi$
$984$	0	0
$985$	9.84409	0.313659
$986$	0	0
$987$	−20.7878	−0.661682
$988$	0	0
$989$	−50.0535	−1.59161
$990$	0	0
$991$	−28.6331	−0.909561	−0.454780	−	0.890604i	$-0.650282\pi$
$991$	−28.6331	−0.909561	−0.454780	+	0.890604i	$0.650282\pi$
$992$	0	0
$993$	7.26826	0.230651
$994$	0	0
$995$	−6.77395	−0.214749
$996$	0	0
$997$	19.1826	0.607518	0.303759	−	0.952749i	$-0.401758\pi$
$997$	19.1826	0.607518	0.303759	+	0.952749i	$0.401758\pi$
$998$	0	0
$999$	−10.4028	−0.329131

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4008.2.a.l.1.5	✓	12
4.3	odd	2		8016.2.a.bf.1.5		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4008.2.a.l.1.5	✓	12	1.1	even	1	trivial
8016.2.a.bf.1.5		12	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -0.716151 q^{5} -3.33002 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -0.716151 q^{5} -3.33002 q^{7} +1.00000 q^{9} -1.47099 q^{11} -1.07152 q^{13} -0.716151 q^{15} +4.68991 q^{17} -1.60112 q^{19} -3.33002 q^{21} +5.96922 q^{23} -4.48713 q^{25} +1.00000 q^{27} +4.73151 q^{29} +8.82346 q^{31} -1.47099 q^{33} +2.38479 q^{35} -10.4028 q^{37} -1.07152 q^{39} -2.92019 q^{41} -8.38527 q^{43} -0.716151 q^{45} +6.24255 q^{47} +4.08900 q^{49} +4.68991 q^{51} -9.50501 q^{53} +1.05345 q^{55} -1.60112 q^{57} -4.74996 q^{59} +7.09947 q^{61} -3.33002 q^{63} +0.767367 q^{65} +13.6608 q^{67} +5.96922 q^{69} +13.6859 q^{71} +7.30688 q^{73} -4.48713 q^{75} +4.89844 q^{77} +12.8036 q^{79} +1.00000 q^{81} +1.24092 q^{83} -3.35869 q^{85} +4.73151 q^{87} -7.67545 q^{89} +3.56816 q^{91} +8.82346 q^{93} +1.14664 q^{95} +13.4461 q^{97} -1.47099 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{3} + 4 q^{5} + 11 q^{7} + 12 q^{9}+O(q^{10}) \) 12 * q + 12 * q^3 + 4 * q^5 + 11 * q^7 + 12 * q^9	\( 12 q + 12 q^{3} + 4 q^{5} + 11 q^{7} + 12 q^{9} + q^{11} + 8 q^{13} + 4 q^{15} + 3 q^{17} + 12 q^{19} + 11 q^{21} + 7 q^{23} + 18 q^{25} + 12 q^{27} + 5 q^{29} + 33 q^{31} + q^{33} + 15 q^{35} + 8 q^{37} + 8 q^{39} - 6 q^{41} + 16 q^{43} + 4 q^{45} + 18 q^{47} + 25 q^{49} + 3 q^{51} + 20 q^{53} + 39 q^{55} + 12 q^{57} + 4 q^{59} + 10 q^{61} + 11 q^{63} + 9 q^{67} + 7 q^{69} + 11 q^{71} + 22 q^{73} + 18 q^{75} + 24 q^{77} + 56 q^{79} + 12 q^{81} + 26 q^{83} + 15 q^{85} + 5 q^{87} - 15 q^{89} + 11 q^{91} + 33 q^{93} + 3 q^{95} + 8 q^{97} + q^{99}+O(q^{100}) \) 12 * q + 12 * q^3 + 4 * q^5 + 11 * q^7 + 12 * q^9 + q^11 + 8 * q^13 + 4 * q^15 + 3 * q^17 + 12 * q^19 + 11 * q^21 + 7 * q^23 + 18 * q^25 + 12 * q^27 + 5 * q^29 + 33 * q^31 + q^33 + 15 * q^35 + 8 * q^37 + 8 * q^39 - 6 * q^41 + 16 * q^43 + 4 * q^45 + 18 * q^47 + 25 * q^49 + 3 * q^51 + 20 * q^53 + 39 * q^55 + 12 * q^57 + 4 * q^59 + 10 * q^61 + 11 * q^63 + 9 * q^67 + 7 * q^69 + 11 * q^71 + 22 * q^73 + 18 * q^75 + 24 * q^77 + 56 * q^79 + 12 * q^81 + 26 * q^83 + 15 * q^85 + 5 * q^87 - 15 * q^89 + 11 * q^91 + 33 * q^93 + 3 * q^95 + 8 * q^97 + q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

Properties

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