LMFDB - Embedded newform 4008.2.a.l.1.9

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			1.9
Root			$2.31305$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +2.31305 q^{5} +4.58648 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +2.31305 q^{5} +4.58648 q^{7} +1.00000 q^{9} +3.21418 q^{11} -5.90839 q^{13} +2.31305 q^{15} +6.31037 q^{17} -7.08386 q^{19} +4.58648 q^{21} -1.12084 q^{23} +0.350210 q^{25} +1.00000 q^{27} -0.0956753 q^{29} +4.81707 q^{31} +3.21418 q^{33} +10.6088 q^{35} +3.60986 q^{37} -5.90839 q^{39} +8.59300 q^{41} -8.44278 q^{43} +2.31305 q^{45} +6.42753 q^{47} +14.0358 q^{49} +6.31037 q^{51} +2.56167 q^{53} +7.43455 q^{55} -7.08386 q^{57} +6.13741 q^{59} -5.66615 q^{61} +4.58648 q^{63} -13.6664 q^{65} -9.24652 q^{67} -1.12084 q^{69} -7.95267 q^{71} +14.9676 q^{73} +0.350210 q^{75} +14.7417 q^{77} +13.4092 q^{79} +1.00000 q^{81} -3.20203 q^{83} +14.5962 q^{85} -0.0956753 q^{87} -16.0148 q^{89} -27.0987 q^{91} +4.81707 q^{93} -16.3853 q^{95} +0.476438 q^{97} +3.21418 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	2.31305	1.03443	0.517214	−	0.855856i	$-0.326969\pi$
$5$	2.31305	1.03443	0.517214	+	0.855856i	$0.326969\pi$
$6$	0	0
$7$	4.58648	1.73353	0.866763	−	0.498720i	$-0.166196\pi$
$7$	4.58648	1.73353	0.866763	+	0.498720i	$0.166196\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.21418	0.969110	0.484555	−	0.874761i	$-0.338981\pi$
$11$	3.21418	0.969110	0.484555	+	0.874761i	$0.338981\pi$
$12$	0	0
$13$	−5.90839	−1.63869	−0.819346	−	0.573299i	$-0.805663\pi$
$13$	−5.90839	−1.63869	−0.819346	+	0.573299i	$0.805663\pi$
$14$	0	0
$15$	2.31305	0.597227
$16$	0	0
$17$	6.31037	1.53049	0.765244	−	0.643740i	$-0.222618\pi$
$17$	6.31037	1.53049	0.765244	+	0.643740i	$0.222618\pi$
$18$	0	0
$19$	−7.08386	−1.62515	−0.812574	−	0.582858i	$-0.801934\pi$
$19$	−7.08386	−1.62515	−0.812574	+	0.582858i	$0.801934\pi$
$20$	0	0
$21$	4.58648	1.00085
$22$	0	0
$23$	−1.12084	−0.233711	−0.116855	−	0.993149i	$-0.537281\pi$
$23$	−1.12084	−0.233711	−0.116855	+	0.993149i	$0.537281\pi$
$24$	0	0
$25$	0.350210	0.0700420
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−0.0956753	−0.0177665	−0.00888323	−	0.999961i	$-0.502828\pi$
$29$	−0.0956753	−0.0177665	−0.00888323	+	0.999961i	$0.502828\pi$
$30$	0	0
$31$	4.81707	0.865172	0.432586	−	0.901593i	$-0.357601\pi$
$31$	4.81707	0.865172	0.432586	+	0.901593i	$0.357601\pi$
$32$	0	0
$33$	3.21418	0.559516
$34$	0	0
$35$	10.6088	1.79321
$36$	0	0
$37$	3.60986	0.593458	0.296729	−	0.954962i	$-0.404104\pi$
$37$	3.60986	0.593458	0.296729	+	0.954962i	$0.404104\pi$
$38$	0	0
$39$	−5.90839	−0.946100
$40$	0	0
$41$	8.59300	1.34200	0.671001	−	0.741457i	$-0.265865\pi$
$41$	8.59300	1.34200	0.671001	+	0.741457i	$0.265865\pi$
$42$	0	0
$43$	−8.44278	−1.28751	−0.643756	−	0.765231i	$-0.722625\pi$
$43$	−8.44278	−1.28751	−0.643756	+	0.765231i	$0.722625\pi$
$44$	0	0
$45$	2.31305	0.344809
$46$	0	0
$47$	6.42753	0.937551	0.468776	−	0.883317i	$-0.344695\pi$
$47$	6.42753	0.937551	0.468776	+	0.883317i	$0.344695\pi$
$48$	0	0
$49$	14.0358	2.00511
$50$	0	0
$51$	6.31037	0.883628
$52$	0	0
$53$	2.56167	0.351873	0.175936	−	0.984402i	$-0.443705\pi$
$53$	2.56167	0.351873	0.175936	+	0.984402i	$0.443705\pi$
$54$	0	0
$55$	7.43455	1.00248
$56$	0	0
$57$	−7.08386	−0.938280
$58$	0	0
$59$	6.13741	0.799022	0.399511	−	0.916728i	$-0.369180\pi$
$59$	6.13741	0.799022	0.399511	+	0.916728i	$0.369180\pi$
$60$	0	0
$61$	−5.66615	−0.725476	−0.362738	−	0.931891i	$-0.618158\pi$
$61$	−5.66615	−0.725476	−0.362738	+	0.931891i	$0.618158\pi$
$62$	0	0
$63$	4.58648	0.577842
$64$	0	0
$65$	−13.6664	−1.69511
$66$	0	0
$67$	−9.24652	−1.12964	−0.564821	−	0.825213i	$-0.691055\pi$
$67$	−9.24652	−1.12964	−0.564821	+	0.825213i	$0.691055\pi$
$68$	0	0
$69$	−1.12084	−0.134933
$70$	0	0
$71$	−7.95267	−0.943808	−0.471904	−	0.881650i	$-0.656433\pi$
$71$	−7.95267	−0.943808	−0.471904	+	0.881650i	$0.656433\pi$
$72$	0	0
$73$	14.9676	1.75182	0.875911	−	0.482473i	$-0.160261\pi$
$73$	14.9676	1.75182	0.875911	+	0.482473i	$0.160261\pi$
$74$	0	0
$75$	0.350210	0.0404388
$76$	0	0
$77$	14.7417	1.67998
$78$	0	0
$79$	13.4092	1.50866	0.754328	−	0.656498i	$-0.227963\pi$
$79$	13.4092	1.50866	0.754328	+	0.656498i	$0.227963\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−3.20203	−0.351468	−0.175734	−	0.984438i	$-0.556230\pi$
$83$	−3.20203	−0.351468	−0.175734	+	0.984438i	$0.556230\pi$
$84$	0	0
$85$	14.5962	1.58318
$86$	0	0
$87$	−0.0956753	−0.0102575
$88$	0	0
$89$	−16.0148	−1.69756	−0.848780	−	0.528746i	$-0.822662\pi$
$89$	−16.0148	−1.69756	−0.848780	+	0.528746i	$0.822662\pi$
$90$	0	0
$91$	−27.0987	−2.84072
$92$	0	0
$93$	4.81707	0.499507
$94$	0	0
$95$	−16.3853	−1.68110
$96$	0	0
$97$	0.476438	0.0483749	0.0241875	−	0.999707i	$-0.492300\pi$
$97$	0.476438	0.0483749	0.0241875	+	0.999707i	$0.492300\pi$
$98$	0	0
$99$	3.21418	0.323037
$100$	0	0
$101$	−15.3819	−1.53056	−0.765278	−	0.643699i	$-0.777399\pi$
$101$	−15.3819	−1.53056	−0.765278	+	0.643699i	$0.777399\pi$
$102$	0	0
$103$	8.09569	0.797692	0.398846	−	0.917018i	$-0.369411\pi$
$103$	8.09569	0.797692	0.398846	+	0.917018i	$0.369411\pi$
$104$	0	0
$105$	10.6088	1.03531
$106$	0	0
$107$	−2.87538	−0.277974	−0.138987	−	0.990294i	$-0.544385\pi$
$107$	−2.87538	−0.277974	−0.138987	+	0.990294i	$0.544385\pi$
$108$	0	0
$109$	3.95551	0.378869	0.189435	−	0.981893i	$-0.439335\pi$
$109$	3.95551	0.378869	0.189435	+	0.981893i	$0.439335\pi$
$110$	0	0
$111$	3.60986	0.342633
$112$	0	0
$113$	−12.7456	−1.19900	−0.599501	−	0.800374i	$-0.704634\pi$
$113$	−12.7456	−1.19900	−0.599501	+	0.800374i	$0.704634\pi$
$114$	0	0
$115$	−2.59255	−0.241757
$116$	0	0
$117$	−5.90839	−0.546231
$118$	0	0
$119$	28.9424	2.65314
$120$	0	0
$121$	−0.669077	−0.0608252
$122$	0	0
$123$	8.59300	0.774805
$124$	0	0
$125$	−10.7552	−0.961975
$126$	0	0
$127$	6.45367	0.572671	0.286335	−	0.958129i	$-0.407563\pi$
$127$	6.45367	0.572671	0.286335	+	0.958129i	$0.407563\pi$
$128$	0	0
$129$	−8.44278	−0.743345
$130$	0	0
$131$	8.69274	0.759488	0.379744	−	0.925092i	$-0.376012\pi$
$131$	8.69274	0.759488	0.379744	+	0.925092i	$0.376012\pi$
$132$	0	0
$133$	−32.4900	−2.81724
$134$	0	0
$135$	2.31305	0.199076
$136$	0	0
$137$	8.18280	0.699104	0.349552	−	0.936917i	$-0.386334\pi$
$137$	8.18280	0.699104	0.349552	+	0.936917i	$0.386334\pi$
$138$	0	0
$139$	−8.00849	−0.679271	−0.339636	−	0.940557i	$-0.610304\pi$
$139$	−8.00849	−0.679271	−0.339636	+	0.940557i	$0.610304\pi$
$140$	0	0
$141$	6.42753	0.541295
$142$	0	0
$143$	−18.9906	−1.58807
$144$	0	0
$145$	−0.221302	−0.0183781
$146$	0	0
$147$	14.0358	1.15765
$148$	0	0
$149$	−12.2148	−1.00067	−0.500336	−	0.865831i	$-0.666790\pi$
$149$	−12.2148	−1.00067	−0.500336	+	0.865831i	$0.666790\pi$
$150$	0	0
$151$	14.8534	1.20875	0.604377	−	0.796698i	$-0.293422\pi$
$151$	14.8534	1.20875	0.604377	+	0.796698i	$0.293422\pi$
$152$	0	0
$153$	6.31037	0.510163
$154$	0	0
$155$	11.1421	0.894958
$156$	0	0
$157$	−14.3909	−1.14852	−0.574259	−	0.818674i	$-0.694710\pi$
$157$	−14.3909	−1.14852	−0.574259	+	0.818674i	$0.694710\pi$
$158$	0	0
$159$	2.56167	0.203154
$160$	0	0
$161$	−5.14070	−0.405144
$162$	0	0
$163$	9.84431	0.771066	0.385533	−	0.922694i	$-0.374018\pi$
$163$	9.84431	0.771066	0.385533	+	0.922694i	$0.374018\pi$
$164$	0	0
$165$	7.43455	0.578779
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	21.9091	1.68531
$170$	0	0
$171$	−7.08386	−0.541716
$172$	0	0
$173$	−21.4914	−1.63396	−0.816981	−	0.576665i	$-0.804354\pi$
$173$	−21.4914	−1.63396	−0.816981	+	0.576665i	$0.804354\pi$
$174$	0	0
$175$	1.60623	0.121420
$176$	0	0
$177$	6.13741	0.461316
$178$	0	0
$179$	−12.7394	−0.952190	−0.476095	−	0.879394i	$-0.657948\pi$
$179$	−12.7394	−0.952190	−0.476095	+	0.879394i	$0.657948\pi$
$180$	0	0
$181$	16.1252	1.19858	0.599288	−	0.800534i	$-0.295451\pi$
$181$	16.1252	1.19858	0.599288	+	0.800534i	$0.295451\pi$
$182$	0	0
$183$	−5.66615	−0.418854
$184$	0	0
$185$	8.34980	0.613890
$186$	0	0
$187$	20.2826	1.48321
$188$	0	0
$189$	4.58648	0.333617
$190$	0	0
$191$	−3.06940	−0.222094	−0.111047	−	0.993815i	$-0.535420\pi$
$191$	−3.06940	−0.222094	−0.111047	+	0.993815i	$0.535420\pi$
$192$	0	0
$193$	−20.3544	−1.46514	−0.732572	−	0.680689i	$-0.761680\pi$
$193$	−20.3544	−1.46514	−0.732572	+	0.680689i	$0.761680\pi$
$194$	0	0
$195$	−13.6664	−0.978672
$196$	0	0
$197$	−6.35545	−0.452807	−0.226404	−	0.974034i	$-0.572697\pi$
$197$	−6.35545	−0.452807	−0.226404	+	0.974034i	$0.572697\pi$
$198$	0	0
$199$	−12.8698	−0.912315	−0.456158	−	0.889899i	$-0.650775\pi$
$199$	−12.8698	−0.912315	−0.456158	+	0.889899i	$0.650775\pi$
$200$	0	0
$201$	−9.24652	−0.652200
$202$	0	0
$203$	−0.438813	−0.0307986
$204$	0	0
$205$	19.8761	1.38820
$206$	0	0
$207$	−1.12084	−0.0779035
$208$	0	0
$209$	−22.7688	−1.57495
$210$	0	0
$211$	12.5752	0.865714	0.432857	−	0.901463i	$-0.357506\pi$
$211$	12.5752	0.865714	0.432857	+	0.901463i	$0.357506\pi$
$212$	0	0
$213$	−7.95267	−0.544908
$214$	0	0
$215$	−19.5286	−1.33184
$216$	0	0
$217$	22.0934	1.49980
$218$	0	0
$219$	14.9676	1.01141
$220$	0	0
$221$	−37.2841	−2.50800
$222$	0	0
$223$	−13.4007	−0.897375	−0.448688	−	0.893689i	$-0.648108\pi$
$223$	−13.4007	−0.897375	−0.448688	+	0.893689i	$0.648108\pi$
$224$	0	0
$225$	0.350210	0.0233473
$226$	0	0
$227$	24.2416	1.60897	0.804486	−	0.593972i	$-0.202441\pi$
$227$	24.2416	1.60897	0.804486	+	0.593972i	$0.202441\pi$
$228$	0	0
$229$	1.58387	0.104665	0.0523325	−	0.998630i	$-0.483334\pi$
$229$	1.58387	0.104665	0.0523325	+	0.998630i	$0.483334\pi$
$230$	0	0
$231$	14.7417	0.969936
$232$	0	0
$233$	18.0966	1.18554	0.592772	−	0.805370i	$-0.298033\pi$
$233$	18.0966	1.18554	0.592772	+	0.805370i	$0.298033\pi$
$234$	0	0
$235$	14.8672	0.969829
$236$	0	0
$237$	13.4092	0.871023
$238$	0	0
$239$	14.8153	0.958321	0.479161	−	0.877727i	$-0.340941\pi$
$239$	14.8153	0.958321	0.479161	+	0.877727i	$0.340941\pi$
$240$	0	0
$241$	−28.7654	−1.85294	−0.926472	−	0.376363i	$-0.877175\pi$
$241$	−28.7654	−1.85294	−0.926472	+	0.376363i	$0.877175\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	32.4655	2.07415
$246$	0	0
$247$	41.8542	2.66312
$248$	0	0
$249$	−3.20203	−0.202920
$250$	0	0
$251$	24.2143	1.52839	0.764196	−	0.644985i	$-0.223136\pi$
$251$	24.2143	1.52839	0.764196	+	0.644985i	$0.223136\pi$
$252$	0	0
$253$	−3.60257	−0.226491
$254$	0	0
$255$	14.5962	0.914050
$256$	0	0
$257$	−0.609749	−0.0380351	−0.0190175	−	0.999819i	$-0.506054\pi$
$257$	−0.609749	−0.0380351	−0.0190175	+	0.999819i	$0.506054\pi$
$258$	0	0
$259$	16.5566	1.02878
$260$	0	0
$261$	−0.0956753	−0.00592216
$262$	0	0
$263$	−15.6789	−0.966801	−0.483401	−	0.875399i	$-0.660599\pi$
$263$	−15.6789	−0.966801	−0.483401	+	0.875399i	$0.660599\pi$
$264$	0	0
$265$	5.92528	0.363987
$266$	0	0
$267$	−16.0148	−0.980087
$268$	0	0
$269$	−21.4228	−1.30617	−0.653087	−	0.757283i	$-0.726526\pi$
$269$	−21.4228	−1.30617	−0.653087	+	0.757283i	$0.726526\pi$
$270$	0	0
$271$	−12.2126	−0.741864	−0.370932	−	0.928660i	$-0.620962\pi$
$271$	−12.2126	−0.741864	−0.370932	+	0.928660i	$0.620962\pi$
$272$	0	0
$273$	−27.0987	−1.64009
$274$	0	0
$275$	1.12564	0.0678784
$276$	0	0
$277$	14.4046	0.865489	0.432744	−	0.901517i	$-0.357545\pi$
$277$	14.4046	0.865489	0.432744	+	0.901517i	$0.357545\pi$
$278$	0	0
$279$	4.81707	0.288391
$280$	0	0
$281$	4.73061	0.282204	0.141102	−	0.989995i	$-0.454935\pi$
$281$	4.73061	0.282204	0.141102	+	0.989995i	$0.454935\pi$
$282$	0	0
$283$	−18.9197	−1.12466	−0.562329	−	0.826914i	$-0.690094\pi$
$283$	−18.9197	−1.12466	−0.562329	+	0.826914i	$0.690094\pi$
$284$	0	0
$285$	−16.3853	−0.970583
$286$	0	0
$287$	39.4116	2.32639
$288$	0	0
$289$	22.8207	1.34239
$290$	0	0
$291$	0.476438	0.0279293
$292$	0	0
$293$	4.27544	0.249774	0.124887	−	0.992171i	$-0.460143\pi$
$293$	4.27544	0.249774	0.124887	+	0.992171i	$0.460143\pi$
$294$	0	0
$295$	14.1961	0.826531
$296$	0	0
$297$	3.21418	0.186505
$298$	0	0
$299$	6.62234	0.382980
$300$	0	0
$301$	−38.7226	−2.23193
$302$	0	0
$303$	−15.3819	−0.883667
$304$	0	0
$305$	−13.1061	−0.750453
$306$	0	0
$307$	20.2247	1.15429	0.577143	−	0.816643i	$-0.304167\pi$
$307$	20.2247	1.15429	0.577143	+	0.816643i	$0.304167\pi$
$308$	0	0
$309$	8.09569	0.460548
$310$	0	0
$311$	−19.1435	−1.08553	−0.542763	−	0.839886i	$-0.682622\pi$
$311$	−19.1435	−1.08553	−0.542763	+	0.839886i	$0.682622\pi$
$312$	0	0
$313$	−27.2504	−1.54028	−0.770141	−	0.637874i	$-0.779814\pi$
$313$	−27.2504	−1.54028	−0.770141	+	0.637874i	$0.779814\pi$
$314$	0	0
$315$	10.6088	0.597736
$316$	0	0
$317$	5.80299	0.325928	0.162964	−	0.986632i	$-0.447895\pi$
$317$	5.80299	0.325928	0.162964	+	0.986632i	$0.447895\pi$
$318$	0	0
$319$	−0.307517	−0.0172177
$320$	0	0
$321$	−2.87538	−0.160488
$322$	0	0
$323$	−44.7017	−2.48727
$324$	0	0
$325$	−2.06918	−0.114777
$326$	0	0
$327$	3.95551	0.218740
$328$	0	0
$329$	29.4797	1.62527
$330$	0	0
$331$	−10.6624	−0.586056	−0.293028	−	0.956104i	$-0.594663\pi$
$331$	−10.6624	−0.586056	−0.293028	+	0.956104i	$0.594663\pi$
$332$	0	0
$333$	3.60986	0.197819
$334$	0	0
$335$	−21.3877	−1.16853
$336$	0	0
$337$	−23.0252	−1.25426	−0.627132	−	0.778913i	$-0.715771\pi$
$337$	−23.0252	−1.25426	−0.627132	+	0.778913i	$0.715771\pi$
$338$	0	0
$339$	−12.7456	−0.692244
$340$	0	0
$341$	15.4829	0.838447
$342$	0	0
$343$	32.2695	1.74239
$344$	0	0
$345$	−2.59255	−0.139578
$346$	0	0
$347$	−10.3802	−0.557238	−0.278619	−	0.960402i	$-0.589877\pi$
$347$	−10.3802	−0.557238	−0.278619	+	0.960402i	$0.589877\pi$
$348$	0	0
$349$	31.7466	1.69935	0.849677	−	0.527303i	$-0.176797\pi$
$349$	31.7466	1.69935	0.849677	+	0.527303i	$0.176797\pi$
$350$	0	0
$351$	−5.90839	−0.315367
$352$	0	0
$353$	29.5012	1.57019	0.785094	−	0.619376i	$-0.212615\pi$
$353$	29.5012	1.57019	0.785094	+	0.619376i	$0.212615\pi$
$354$	0	0
$355$	−18.3949	−0.976302
$356$	0	0
$357$	28.9424	1.53179
$358$	0	0
$359$	36.5925	1.93128	0.965639	−	0.259888i	$-0.0836855\pi$
$359$	36.5925	1.93128	0.965639	+	0.259888i	$0.0836855\pi$
$360$	0	0
$361$	31.1810	1.64111
$362$	0	0
$363$	−0.669077	−0.0351174
$364$	0	0
$365$	34.6208	1.81213
$366$	0	0
$367$	13.2131	0.689718	0.344859	−	0.938655i	$-0.387927\pi$
$367$	13.2131	0.689718	0.344859	+	0.938655i	$0.387927\pi$
$368$	0	0
$369$	8.59300	0.447334
$370$	0	0
$371$	11.7491	0.609980
$372$	0	0
$373$	10.4903	0.543169	0.271585	−	0.962415i	$-0.412452\pi$
$373$	10.4903	0.543169	0.271585	+	0.962415i	$0.412452\pi$
$374$	0	0
$375$	−10.7552	−0.555396
$376$	0	0
$377$	0.565287	0.0291138
$378$	0	0
$379$	18.9819	0.975035	0.487517	−	0.873113i	$-0.337903\pi$
$379$	18.9819	0.975035	0.487517	+	0.873113i	$0.337903\pi$
$380$	0	0
$381$	6.45367	0.330632
$382$	0	0
$383$	−7.61342	−0.389028	−0.194514	−	0.980900i	$-0.562313\pi$
$383$	−7.61342	−0.389028	−0.194514	+	0.980900i	$0.562313\pi$
$384$	0	0
$385$	34.0984	1.73782
$386$	0	0
$387$	−8.44278	−0.429170
$388$	0	0
$389$	11.1854	0.567124	0.283562	−	0.958954i	$-0.408484\pi$
$389$	11.1854	0.567124	0.283562	+	0.958954i	$0.408484\pi$
$390$	0	0
$391$	−7.07289	−0.357691
$392$	0	0
$393$	8.69274	0.438491
$394$	0	0
$395$	31.0162	1.56060
$396$	0	0
$397$	14.4789	0.726675	0.363338	−	0.931658i	$-0.381637\pi$
$397$	14.4789	0.726675	0.363338	+	0.931658i	$0.381637\pi$
$398$	0	0
$399$	−32.4900	−1.62653
$400$	0	0
$401$	−18.3950	−0.918602	−0.459301	−	0.888281i	$-0.651900\pi$
$401$	−18.3950	−0.918602	−0.459301	+	0.888281i	$0.651900\pi$
$402$	0	0
$403$	−28.4611	−1.41775
$404$	0	0
$405$	2.31305	0.114936
$406$	0	0
$407$	11.6027	0.575126
$408$	0	0
$409$	−31.8381	−1.57429	−0.787146	−	0.616767i	$-0.788442\pi$
$409$	−31.8381	−1.57429	−0.787146	+	0.616767i	$0.788442\pi$
$410$	0	0
$411$	8.18280	0.403628
$412$	0	0
$413$	28.1491	1.38513
$414$	0	0
$415$	−7.40646	−0.363569
$416$	0	0
$417$	−8.00849	−0.392177
$418$	0	0
$419$	−35.5931	−1.73883	−0.869417	−	0.494078i	$-0.835506\pi$
$419$	−35.5931	−1.73883	−0.869417	+	0.494078i	$0.835506\pi$
$420$	0	0
$421$	−27.2164	−1.32644	−0.663222	−	0.748423i	$-0.730812\pi$
$421$	−27.2164	−1.32644	−0.663222	+	0.748423i	$0.730812\pi$
$422$	0	0
$423$	6.42753	0.312517
$424$	0	0
$425$	2.20995	0.107198
$426$	0	0
$427$	−25.9877	−1.25763
$428$	0	0
$429$	−18.9906	−0.916875
$430$	0	0
$431$	11.7305	0.565036	0.282518	−	0.959262i	$-0.408830\pi$
$431$	11.7305	0.565036	0.282518	+	0.959262i	$0.408830\pi$
$432$	0	0
$433$	−6.60527	−0.317429	−0.158715	−	0.987325i	$-0.550735\pi$
$433$	−6.60527	−0.317429	−0.158715	+	0.987325i	$0.550735\pi$
$434$	0	0
$435$	−0.221302	−0.0106106
$436$	0	0
$437$	7.93985	0.379814
$438$	0	0
$439$	−8.99747	−0.429426	−0.214713	−	0.976677i	$-0.568882\pi$
$439$	−8.99747	−0.429426	−0.214713	+	0.976677i	$0.568882\pi$
$440$	0	0
$441$	14.0358	0.668371
$442$	0	0
$443$	−11.4240	−0.542771	−0.271385	−	0.962471i	$-0.587482\pi$
$443$	−11.4240	−0.542771	−0.271385	+	0.962471i	$0.587482\pi$
$444$	0	0
$445$	−37.0430	−1.75600
$446$	0	0
$447$	−12.2148	−0.577738
$448$	0	0
$449$	−8.51165	−0.401690	−0.200845	−	0.979623i	$-0.564369\pi$
$449$	−8.51165	−0.401690	−0.200845	+	0.979623i	$0.564369\pi$
$450$	0	0
$451$	27.6194	1.30055
$452$	0	0
$453$	14.8534	0.697875
$454$	0	0
$455$	−62.6807	−2.93852
$456$	0	0
$457$	8.37503	0.391767	0.195884	−	0.980627i	$-0.437243\pi$
$457$	8.37503	0.391767	0.195884	+	0.980627i	$0.437243\pi$
$458$	0	0
$459$	6.31037	0.294543
$460$	0	0
$461$	−14.6981	−0.684557	−0.342279	−	0.939599i	$-0.611199\pi$
$461$	−14.6981	−0.684557	−0.342279	+	0.939599i	$0.611199\pi$
$462$	0	0
$463$	−8.96089	−0.416448	−0.208224	−	0.978081i	$-0.566768\pi$
$463$	−8.96089	−0.416448	−0.208224	+	0.978081i	$0.566768\pi$
$464$	0	0
$465$	11.1421	0.516704
$466$	0	0
$467$	13.5404	0.626576	0.313288	−	0.949658i	$-0.398570\pi$
$467$	13.5404	0.626576	0.313288	+	0.949658i	$0.398570\pi$
$468$	0	0
$469$	−42.4090	−1.95827
$470$	0	0
$471$	−14.3909	−0.663097
$472$	0	0
$473$	−27.1366	−1.24774
$474$	0	0
$475$	−2.48084	−0.113829
$476$	0	0
$477$	2.56167	0.117291
$478$	0	0
$479$	−8.28706	−0.378646	−0.189323	−	0.981915i	$-0.560629\pi$
$479$	−8.28706	−0.378646	−0.189323	+	0.981915i	$0.560629\pi$
$480$	0	0
$481$	−21.3285	−0.972495
$482$	0	0
$483$	−5.14070	−0.233910
$484$	0	0
$485$	1.10203	0.0500404
$486$	0	0
$487$	−36.4042	−1.64963	−0.824815	−	0.565402i	$-0.808721\pi$
$487$	−36.4042	−1.64963	−0.824815	+	0.565402i	$0.808721\pi$
$488$	0	0
$489$	9.84431	0.445175
$490$	0	0
$491$	25.7273	1.16105	0.580527	−	0.814241i	$-0.302846\pi$
$491$	25.7273	1.16105	0.580527	+	0.814241i	$0.302846\pi$
$492$	0	0
$493$	−0.603746	−0.0271914
$494$	0	0
$495$	7.43455	0.334158
$496$	0	0
$497$	−36.4747	−1.63612
$498$	0	0
$499$	−34.6775	−1.55238	−0.776190	−	0.630499i	$-0.782850\pi$
$499$	−34.6775	−1.55238	−0.776190	+	0.630499i	$0.782850\pi$
$500$	0	0
$501$	1.00000	0.0446767
$502$	0	0
$503$	−12.6211	−0.562748	−0.281374	−	0.959598i	$-0.590790\pi$
$503$	−12.6211	−0.562748	−0.281374	+	0.959598i	$0.590790\pi$
$504$	0	0
$505$	−35.5792	−1.58325
$506$	0	0
$507$	21.9091	0.973017
$508$	0	0
$509$	29.2360	1.29586	0.647931	−	0.761699i	$-0.275635\pi$
$509$	29.2360	1.29586	0.647931	+	0.761699i	$0.275635\pi$
$510$	0	0
$511$	68.6484	3.03683
$512$	0	0
$513$	−7.08386	−0.312760
$514$	0	0
$515$	18.7258	0.825155
$516$	0	0
$517$	20.6592	0.908591
$518$	0	0
$519$	−21.4914	−0.943368
$520$	0	0
$521$	−29.0135	−1.27110	−0.635551	−	0.772059i	$-0.719227\pi$
$521$	−29.0135	−1.27110	−0.635551	+	0.772059i	$0.719227\pi$
$522$	0	0
$523$	13.5844	0.594003	0.297001	−	0.954877i	$-0.404013\pi$
$523$	13.5844	0.594003	0.297001	+	0.954877i	$0.404013\pi$
$524$	0	0
$525$	1.60623	0.0701016
$526$	0	0
$527$	30.3975	1.32414
$528$	0	0
$529$	−21.7437	−0.945379
$530$	0	0
$531$	6.13741	0.266341
$532$	0	0
$533$	−50.7708	−2.19913
$534$	0	0
$535$	−6.65091	−0.287544
$536$	0	0
$537$	−12.7394	−0.549747
$538$	0	0
$539$	45.1135	1.94318
$540$	0	0
$541$	29.2472	1.25743	0.628717	−	0.777634i	$-0.283581\pi$
$541$	29.2472	1.25743	0.628717	+	0.777634i	$0.283581\pi$
$542$	0	0
$543$	16.1252	0.691998
$544$	0	0
$545$	9.14930	0.391913
$546$	0	0
$547$	−15.7016	−0.671353	−0.335676	−	0.941977i	$-0.608965\pi$
$547$	−15.7016	−0.671353	−0.335676	+	0.941977i	$0.608965\pi$
$548$	0	0
$549$	−5.66615	−0.241825
$550$	0	0
$551$	0.677751	0.0288731
$552$	0	0
$553$	61.5011	2.61529
$554$	0	0
$555$	8.34980	0.354429
$556$	0	0
$557$	−22.6588	−0.960083	−0.480041	−	0.877246i	$-0.659378\pi$
$557$	−22.6588	−0.960083	−0.480041	+	0.877246i	$0.659378\pi$
$558$	0	0
$559$	49.8832	2.10984
$560$	0	0
$561$	20.2826	0.856333
$562$	0	0
$563$	−34.2384	−1.44298	−0.721488	−	0.692427i	$-0.756542\pi$
$563$	−34.2384	−1.44298	−0.721488	+	0.692427i	$0.756542\pi$
$564$	0	0
$565$	−29.4812	−1.24028
$566$	0	0
$567$	4.58648	0.192614
$568$	0	0
$569$	−35.2105	−1.47610	−0.738050	−	0.674746i	$-0.764253\pi$
$569$	−35.2105	−1.47610	−0.738050	+	0.674746i	$0.764253\pi$
$570$	0	0
$571$	38.6184	1.61613	0.808065	−	0.589093i	$-0.200515\pi$
$571$	38.6184	1.61613	0.808065	+	0.589093i	$0.200515\pi$
$572$	0	0
$573$	−3.06940	−0.128226
$574$	0	0
$575$	−0.392528	−0.0163696
$576$	0	0
$577$	−42.9247	−1.78698	−0.893490	−	0.449084i	$-0.851750\pi$
$577$	−42.9247	−1.78698	−0.893490	+	0.449084i	$0.851750\pi$
$578$	0	0
$579$	−20.3544	−0.845901
$580$	0	0
$581$	−14.6860	−0.609280
$582$	0	0
$583$	8.23366	0.341003
$584$	0	0
$585$	−13.6664	−0.565037
$586$	0	0
$587$	16.7904	0.693013	0.346507	−	0.938048i	$-0.387368\pi$
$587$	16.7904	0.693013	0.346507	+	0.938048i	$0.387368\pi$
$588$	0	0
$589$	−34.1235	−1.40603
$590$	0	0
$591$	−6.35545	−0.261428
$592$	0	0
$593$	−1.96900	−0.0808573	−0.0404287	−	0.999182i	$-0.512872\pi$
$593$	−1.96900	−0.0808573	−0.0404287	+	0.999182i	$0.512872\pi$
$594$	0	0
$595$	66.9452	2.74449
$596$	0	0
$597$	−12.8698	−0.526725
$598$	0	0
$599$	1.97393	0.0806528	0.0403264	−	0.999187i	$-0.487160\pi$
$599$	1.97393	0.0806528	0.0403264	+	0.999187i	$0.487160\pi$
$600$	0	0
$601$	21.9942	0.897160	0.448580	−	0.893743i	$-0.351930\pi$
$601$	21.9942	0.897160	0.448580	+	0.893743i	$0.351930\pi$
$602$	0	0
$603$	−9.24652	−0.376548
$604$	0	0
$605$	−1.54761	−0.0629193
$606$	0	0
$607$	13.8801	0.563378	0.281689	−	0.959506i	$-0.409105\pi$
$607$	13.8801	0.563378	0.281689	+	0.959506i	$0.409105\pi$
$608$	0	0
$609$	−0.438813	−0.0177816
$610$	0	0
$611$	−37.9763	−1.53636
$612$	0	0
$613$	35.7685	1.44468	0.722338	−	0.691540i	$-0.243067\pi$
$613$	35.7685	1.44468	0.722338	+	0.691540i	$0.243067\pi$
$614$	0	0
$615$	19.8761	0.801480
$616$	0	0
$617$	−21.5812	−0.868825	−0.434413	−	0.900714i	$-0.643044\pi$
$617$	−21.5812	−0.868825	−0.434413	+	0.900714i	$0.643044\pi$
$618$	0	0
$619$	35.1766	1.41387	0.706933	−	0.707280i	$-0.250078\pi$
$619$	35.1766	1.41387	0.706933	+	0.707280i	$0.250078\pi$
$620$	0	0
$621$	−1.12084	−0.0449776
$622$	0	0
$623$	−73.4513	−2.94277
$624$	0	0
$625$	−26.6284	−1.06514
$626$	0	0
$627$	−22.7688	−0.909297
$628$	0	0
$629$	22.7796	0.908281
$630$	0	0
$631$	28.6537	1.14068	0.570342	−	0.821407i	$-0.306811\pi$
$631$	28.6537	1.14068	0.570342	+	0.821407i	$0.306811\pi$
$632$	0	0
$633$	12.5752	0.499820
$634$	0	0
$635$	14.9277	0.592387
$636$	0	0
$637$	−82.9290	−3.28577
$638$	0	0
$639$	−7.95267	−0.314603
$640$	0	0
$641$	6.25764	0.247162	0.123581	−	0.992334i	$-0.460562\pi$
$641$	6.25764	0.247162	0.123581	+	0.992334i	$0.460562\pi$
$642$	0	0
$643$	−3.84712	−0.151716	−0.0758578	−	0.997119i	$-0.524170\pi$
$643$	−3.84712	−0.151716	−0.0758578	+	0.997119i	$0.524170\pi$
$644$	0	0
$645$	−19.5286	−0.768937
$646$	0	0
$647$	−24.4438	−0.960986	−0.480493	−	0.876998i	$-0.659542\pi$
$647$	−24.4438	−0.960986	−0.480493	+	0.876998i	$0.659542\pi$
$648$	0	0
$649$	19.7267	0.774341
$650$	0	0
$651$	22.0934	0.865909
$652$	0	0
$653$	47.9659	1.87705	0.938525	−	0.345212i	$-0.112193\pi$
$653$	47.9659	1.87705	0.938525	+	0.345212i	$0.112193\pi$
$654$	0	0
$655$	20.1068	0.785636
$656$	0	0
$657$	14.9676	0.583940
$658$	0	0
$659$	14.0165	0.546006	0.273003	−	0.962013i	$-0.411983\pi$
$659$	14.0165	0.546006	0.273003	+	0.962013i	$0.411983\pi$
$660$	0	0
$661$	−16.7905	−0.653075	−0.326538	−	0.945184i	$-0.605882\pi$
$661$	−16.7905	−0.653075	−0.326538	+	0.945184i	$0.605882\pi$
$662$	0	0
$663$	−37.2841	−1.44799
$664$	0	0
$665$	−75.1510	−2.91423
$666$	0	0
$667$	0.107236	0.00415221
$668$	0	0
$669$	−13.4007	−0.518100
$670$	0	0
$671$	−18.2120	−0.703066
$672$	0	0
$673$	−51.0281	−1.96699	−0.983494	−	0.180939i	$-0.942086\pi$
$673$	−51.0281	−1.96699	−0.983494	+	0.180939i	$0.942086\pi$
$674$	0	0
$675$	0.350210	0.0134796
$676$	0	0
$677$	24.3164	0.934556	0.467278	−	0.884111i	$-0.345235\pi$
$677$	24.3164	0.934556	0.467278	+	0.884111i	$0.345235\pi$
$678$	0	0
$679$	2.18517	0.0838592
$680$	0	0
$681$	24.2416	0.928940
$682$	0	0
$683$	4.89320	0.187233	0.0936165	−	0.995608i	$-0.470157\pi$
$683$	4.89320	0.187233	0.0936165	+	0.995608i	$0.470157\pi$
$684$	0	0
$685$	18.9272	0.723173
$686$	0	0
$687$	1.58387	0.0604284
$688$	0	0
$689$	−15.1354	−0.576611
$690$	0	0
$691$	24.0140	0.913536	0.456768	−	0.889586i	$-0.349007\pi$
$691$	24.0140	0.913536	0.456768	+	0.889586i	$0.349007\pi$
$692$	0	0
$693$	14.7417	0.559993
$694$	0	0
$695$	−18.5240	−0.702657
$696$	0	0
$697$	54.2250	2.05392
$698$	0	0
$699$	18.0966	0.684475
$700$	0	0
$701$	19.0044	0.717785	0.358893	−	0.933379i	$-0.383154\pi$
$701$	19.0044	0.717785	0.358893	+	0.933379i	$0.383154\pi$
$702$	0	0
$703$	−25.5718	−0.964457
$704$	0	0
$705$	14.8672	0.559931
$706$	0	0
$707$	−70.5488	−2.65326
$708$	0	0
$709$	33.6743	1.26467	0.632333	−	0.774697i	$-0.282098\pi$
$709$	33.6743	1.26467	0.632333	+	0.774697i	$0.282098\pi$
$710$	0	0
$711$	13.4092	0.502885
$712$	0	0
$713$	−5.39915	−0.202200
$714$	0	0
$715$	−43.9263	−1.64275
$716$	0	0
$717$	14.8153	0.553287
$718$	0	0
$719$	31.8777	1.18884	0.594419	−	0.804155i	$-0.297382\pi$
$719$	31.8777	1.18884	0.594419	+	0.804155i	$0.297382\pi$
$720$	0	0
$721$	37.1307	1.38282
$722$	0	0
$723$	−28.7654	−1.06980
$724$	0	0
$725$	−0.0335065	−0.00124440
$726$	0	0
$727$	14.5765	0.540614	0.270307	−	0.962774i	$-0.412875\pi$
$727$	14.5765	0.540614	0.270307	+	0.962774i	$0.412875\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−53.2770	−1.97052
$732$	0	0
$733$	2.08605	0.0770501	0.0385251	−	0.999258i	$-0.487734\pi$
$733$	2.08605	0.0770501	0.0385251	+	0.999258i	$0.487734\pi$
$734$	0	0
$735$	32.4655	1.19751
$736$	0	0
$737$	−29.7200	−1.09475
$738$	0	0
$739$	17.9185	0.659142	0.329571	−	0.944131i	$-0.393096\pi$
$739$	17.9185	0.659142	0.329571	+	0.944131i	$0.393096\pi$
$740$	0	0
$741$	41.8542	1.53755
$742$	0	0
$743$	41.3924	1.51854	0.759270	−	0.650776i	$-0.225556\pi$
$743$	41.3924	1.51854	0.759270	+	0.650776i	$0.225556\pi$
$744$	0	0
$745$	−28.2534	−1.03512
$746$	0	0
$747$	−3.20203	−0.117156
$748$	0	0
$749$	−13.1879	−0.481875
$750$	0	0
$751$	1.50906	0.0550663	0.0275332	−	0.999621i	$-0.491235\pi$
$751$	1.50906	0.0550663	0.0275332	+	0.999621i	$0.491235\pi$
$752$	0	0
$753$	24.2143	0.882417
$754$	0	0
$755$	34.3567	1.25037
$756$	0	0
$757$	−28.8730	−1.04941	−0.524704	−	0.851285i	$-0.675824\pi$
$757$	−28.8730	−1.04941	−0.524704	+	0.851285i	$0.675824\pi$
$758$	0	0
$759$	−3.60257	−0.130765
$760$	0	0
$761$	−15.2833	−0.554019	−0.277009	−	0.960867i	$-0.589343\pi$
$761$	−15.2833	−0.554019	−0.277009	+	0.960867i	$0.589343\pi$
$762$	0	0
$763$	18.1419	0.656780
$764$	0	0
$765$	14.5962	0.527727
$766$	0	0
$767$	−36.2622	−1.30935
$768$	0	0
$769$	−30.0530	−1.08374	−0.541869	−	0.840463i	$-0.682283\pi$
$769$	−30.0530	−1.08374	−0.541869	+	0.840463i	$0.682283\pi$
$770$	0	0
$771$	−0.609749	−0.0219596
$772$	0	0
$773$	−36.7691	−1.32249	−0.661247	−	0.750168i	$-0.729972\pi$
$773$	−36.7691	−1.32249	−0.661247	+	0.750168i	$0.729972\pi$
$774$	0	0
$775$	1.68699	0.0605983
$776$	0	0
$777$	16.5566	0.593964
$778$	0	0
$779$	−60.8716	−2.18095
$780$	0	0
$781$	−25.5613	−0.914654
$782$	0	0
$783$	−0.0956753	−0.00341916
$784$	0	0
$785$	−33.2869	−1.18806
$786$	0	0
$787$	5.24409	0.186932	0.0934658	−	0.995622i	$-0.470205\pi$
$787$	5.24409	0.186932	0.0934658	+	0.995622i	$0.470205\pi$
$788$	0	0
$789$	−15.6789	−0.558183
$790$	0	0
$791$	−58.4573	−2.07850
$792$	0	0
$793$	33.4778	1.18883
$794$	0	0
$795$	5.92528	0.210148
$796$	0	0
$797$	−20.4374	−0.723928	−0.361964	−	0.932192i	$-0.617894\pi$
$797$	−20.4374	−0.723928	−0.361964	+	0.932192i	$0.617894\pi$
$798$	0	0
$799$	40.5600	1.43491
$800$	0	0
$801$	−16.0148	−0.565853
$802$	0	0
$803$	48.1084	1.69771
$804$	0	0
$805$	−11.8907	−0.419092
$806$	0	0
$807$	−21.4228	−0.754120
$808$	0	0
$809$	−39.2523	−1.38004	−0.690019	−	0.723791i	$-0.742398\pi$
$809$	−39.2523	−1.38004	−0.690019	+	0.723791i	$0.742398\pi$
$810$	0	0
$811$	−3.47145	−0.121899	−0.0609495	−	0.998141i	$-0.519413\pi$
$811$	−3.47145	−0.121899	−0.0609495	+	0.998141i	$0.519413\pi$
$812$	0	0
$813$	−12.2126	−0.428315
$814$	0	0
$815$	22.7704	0.797612
$816$	0	0
$817$	59.8074	2.09240
$818$	0	0
$819$	−27.0987	−0.946906
$820$	0	0
$821$	−41.2347	−1.43910	−0.719551	−	0.694440i	$-0.755652\pi$
$821$	−41.2347	−1.43910	−0.719551	+	0.694440i	$0.755652\pi$
$822$	0	0
$823$	27.6439	0.963607	0.481803	−	0.876279i	$-0.339982\pi$
$823$	27.6439	0.963607	0.481803	+	0.876279i	$0.339982\pi$
$824$	0	0
$825$	1.12564	0.0391896
$826$	0	0
$827$	−46.1297	−1.60409	−0.802043	−	0.597266i	$-0.796254\pi$
$827$	−46.1297	−1.60409	−0.802043	+	0.597266i	$0.796254\pi$
$828$	0	0
$829$	−45.2938	−1.57312	−0.786559	−	0.617515i	$-0.788139\pi$
$829$	−45.2938	−1.57312	−0.786559	+	0.617515i	$0.788139\pi$
$830$	0	0
$831$	14.4046	0.499690
$832$	0	0
$833$	88.5710	3.06880
$834$	0	0
$835$	2.31305	0.0800465
$836$	0	0
$837$	4.81707	0.166502
$838$	0	0
$839$	52.7011	1.81945	0.909723	−	0.415216i	$-0.136294\pi$
$839$	52.7011	1.81945	0.909723	+	0.415216i	$0.136294\pi$
$840$	0	0
$841$	−28.9908	−0.999684
$842$	0	0
$843$	4.73061	0.162931
$844$	0	0
$845$	50.6769	1.74334
$846$	0	0
$847$	−3.06871	−0.105442
$848$	0	0
$849$	−18.9197	−0.649321
$850$	0	0
$851$	−4.04607	−0.138697
$852$	0	0
$853$	34.3615	1.17651	0.588257	−	0.808674i	$-0.299814\pi$
$853$	34.3615	1.17651	0.588257	+	0.808674i	$0.299814\pi$
$854$	0	0
$855$	−16.3853	−0.560367
$856$	0	0
$857$	9.28567	0.317192	0.158596	−	0.987344i	$-0.449303\pi$
$857$	9.28567	0.317192	0.158596	+	0.987344i	$0.449303\pi$
$858$	0	0
$859$	17.8965	0.610622	0.305311	−	0.952253i	$-0.401240\pi$
$859$	17.8965	0.610622	0.305311	+	0.952253i	$0.401240\pi$
$860$	0	0
$861$	39.4116	1.34314
$862$	0	0
$863$	−29.3429	−0.998845	−0.499422	−	0.866359i	$-0.666454\pi$
$863$	−29.3429	−0.998845	−0.499422	+	0.866359i	$0.666454\pi$
$864$	0	0
$865$	−49.7108	−1.69022
$866$	0	0
$867$	22.8207	0.775032
$868$	0	0
$869$	43.0996	1.46205
$870$	0	0
$871$	54.6321	1.85114
$872$	0	0
$873$	0.476438	0.0161250
$874$	0	0
$875$	−49.3285	−1.66761
$876$	0	0
$877$	−43.6823	−1.47505	−0.737523	−	0.675322i	$-0.764005\pi$
$877$	−43.6823	−1.47505	−0.737523	+	0.675322i	$0.764005\pi$
$878$	0	0
$879$	4.27544	0.144207
$880$	0	0
$881$	−22.1789	−0.747227	−0.373614	−	0.927584i	$-0.621881\pi$
$881$	−22.1789	−0.747227	−0.373614	+	0.927584i	$0.621881\pi$
$882$	0	0
$883$	7.72940	0.260115	0.130057	−	0.991506i	$-0.458484\pi$
$883$	7.72940	0.260115	0.130057	+	0.991506i	$0.458484\pi$
$884$	0	0
$885$	14.1961	0.477198
$886$	0	0
$887$	−24.0092	−0.806150	−0.403075	−	0.915167i	$-0.632059\pi$
$887$	−24.0092	−0.806150	−0.403075	+	0.915167i	$0.632059\pi$
$888$	0	0
$889$	29.5996	0.992740
$890$	0	0
$891$	3.21418	0.107679
$892$	0	0
$893$	−45.5317	−1.52366
$894$	0	0
$895$	−29.4670	−0.984972
$896$	0	0
$897$	6.62234	0.221114
$898$	0	0
$899$	−0.460875	−0.0153710
$900$	0	0
$901$	16.1651	0.538537
$902$	0	0
$903$	−38.7226	−1.28861
$904$	0	0
$905$	37.2984	1.23984
$906$	0	0
$907$	17.6999	0.587715	0.293857	−	0.955849i	$-0.405061\pi$
$907$	17.6999	0.587715	0.293857	+	0.955849i	$0.405061\pi$
$908$	0	0
$909$	−15.3819	−0.510186
$910$	0	0
$911$	−9.84469	−0.326169	−0.163085	−	0.986612i	$-0.552144\pi$
$911$	−9.84469	−0.326169	−0.163085	+	0.986612i	$0.552144\pi$
$912$	0	0
$913$	−10.2919	−0.340612
$914$	0	0
$915$	−13.1061	−0.433274
$916$	0	0
$917$	39.8691	1.31659
$918$	0	0
$919$	−16.2682	−0.536638	−0.268319	−	0.963330i	$-0.586468\pi$
$919$	−16.2682	−0.536638	−0.268319	+	0.963330i	$0.586468\pi$
$920$	0	0
$921$	20.2247	0.666428
$922$	0	0
$923$	46.9875	1.54661
$924$	0	0
$925$	1.26421	0.0415670
$926$	0	0
$927$	8.09569	0.265897
$928$	0	0
$929$	−1.38263	−0.0453625	−0.0226812	−	0.999743i	$-0.507220\pi$
$929$	−1.38263	−0.0453625	−0.0226812	+	0.999743i	$0.507220\pi$
$930$	0	0
$931$	−99.4276	−3.25861
$932$	0	0
$933$	−19.1435	−0.626729
$934$	0	0
$935$	46.9148	1.53428
$936$	0	0
$937$	47.9877	1.56769	0.783845	−	0.620956i	$-0.213256\pi$
$937$	47.9877	1.56769	0.783845	+	0.620956i	$0.213256\pi$
$938$	0	0
$939$	−27.2504	−0.889282
$940$	0	0
$941$	38.3869	1.25138	0.625688	−	0.780073i	$-0.284818\pi$
$941$	38.3869	1.25138	0.625688	+	0.780073i	$0.284818\pi$
$942$	0	0
$943$	−9.63135	−0.313640
$944$	0	0
$945$	10.6088	0.345103
$946$	0	0
$947$	−46.9162	−1.52457	−0.762285	−	0.647241i	$-0.775923\pi$
$947$	−46.9162	−1.52457	−0.762285	+	0.647241i	$0.775923\pi$
$948$	0	0
$949$	−88.4342	−2.87070
$950$	0	0
$951$	5.80299	0.188175
$952$	0	0
$953$	21.6138	0.700139	0.350070	−	0.936724i	$-0.386158\pi$
$953$	21.6138	0.700139	0.350070	+	0.936724i	$0.386158\pi$
$954$	0	0
$955$	−7.09968	−0.229740
$956$	0	0
$957$	−0.307517	−0.00994062
$958$	0	0
$959$	37.5302	1.21191
$960$	0	0
$961$	−7.79581	−0.251478
$962$	0	0
$963$	−2.87538	−0.0926579
$964$	0	0
$965$	−47.0809	−1.51559
$966$	0	0
$967$	38.9732	1.25330	0.626648	−	0.779303i	$-0.284427\pi$
$967$	38.9732	1.25330	0.626648	+	0.779303i	$0.284427\pi$
$968$	0	0
$969$	−44.7017	−1.43603
$970$	0	0
$971$	−31.0949	−0.997882	−0.498941	−	0.866636i	$-0.666278\pi$
$971$	−31.0949	−0.997882	−0.498941	+	0.866636i	$0.666278\pi$
$972$	0	0
$973$	−36.7308	−1.17753
$974$	0	0
$975$	−2.06918	−0.0662667
$976$	0	0
$977$	−25.3796	−0.811966	−0.405983	−	0.913881i	$-0.633071\pi$
$977$	−25.3796	−0.811966	−0.405983	+	0.913881i	$0.633071\pi$
$978$	0	0
$979$	−51.4742	−1.64512
$980$	0	0
$981$	3.95551	0.126290
$982$	0	0
$983$	−9.01822	−0.287636	−0.143818	−	0.989604i	$-0.545938\pi$
$983$	−9.01822	−0.287636	−0.143818	+	0.989604i	$0.545938\pi$
$984$	0	0
$985$	−14.7005	−0.468397
$986$	0	0
$987$	29.4797	0.938350
$988$	0	0
$989$	9.46297	0.300905
$990$	0	0
$991$	−3.25211	−0.103307	−0.0516534	−	0.998665i	$-0.516449\pi$
$991$	−3.25211	−0.103307	−0.0516534	+	0.998665i	$0.516449\pi$
$992$	0	0
$993$	−10.6624	−0.338360
$994$	0	0
$995$	−29.7685	−0.943725
$996$	0	0
$997$	44.1469	1.39815	0.699073	−	0.715051i	$-0.253596\pi$
$997$	44.1469	1.39815	0.699073	+	0.715051i	$0.253596\pi$
$998$	0	0
$999$	3.60986	0.114211

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4008.2.a.l.1.9	✓	12
4.3	odd	2		8016.2.a.bf.1.9		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4008.2.a.l.1.9	✓	12	1.1	even	1	trivial
8016.2.a.bf.1.9		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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +2.31305 q^{5} +4.58648 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +2.31305 q^{5} +4.58648 q^{7} +1.00000 q^{9} +3.21418 q^{11} -5.90839 q^{13} +2.31305 q^{15} +6.31037 q^{17} -7.08386 q^{19} +4.58648 q^{21} -1.12084 q^{23} +0.350210 q^{25} +1.00000 q^{27} -0.0956753 q^{29} +4.81707 q^{31} +3.21418 q^{33} +10.6088 q^{35} +3.60986 q^{37} -5.90839 q^{39} +8.59300 q^{41} -8.44278 q^{43} +2.31305 q^{45} +6.42753 q^{47} +14.0358 q^{49} +6.31037 q^{51} +2.56167 q^{53} +7.43455 q^{55} -7.08386 q^{57} +6.13741 q^{59} -5.66615 q^{61} +4.58648 q^{63} -13.6664 q^{65} -9.24652 q^{67} -1.12084 q^{69} -7.95267 q^{71} +14.9676 q^{73} +0.350210 q^{75} +14.7417 q^{77} +13.4092 q^{79} +1.00000 q^{81} -3.20203 q^{83} +14.5962 q^{85} -0.0956753 q^{87} -16.0148 q^{89} -27.0987 q^{91} +4.81707 q^{93} -16.3853 q^{95} +0.476438 q^{97} +3.21418 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{3} + 4 q^{5} + 11 q^{7} + 12 q^{9}+O(q^{10}) \) 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

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