LMFDB - Embedded newform 2672.2.a.m.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2672,2,Mod(1,2672)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2672, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2672.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2672 = 2^{4} \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2672.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:	$21.3360274201$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 13x^{7} + 8x^{6} + 56x^{5} - 15x^{4} - 81x^{3} + 2x^{2} + 13x + 1 $ x^9 - x^8 - 13x^7 + 8x^6 + 56x^5 - 15x^4 - 81x^3 + 2x^2 + 13*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1336)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$2.70492$ of defining polynomial
Character	$\chi$	$=$	2672.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.70492 q^{3} -4.24392 q^{5} -1.27743 q^{7} +4.31658 q^{9} +O(q^{10})$	$q+2.70492 q^{3} -4.24392 q^{5} -1.27743 q^{7} +4.31658 q^{9} +0.403841 q^{11} +1.48621 q^{13} -11.4794 q^{15} +3.32654 q^{17} -6.12675 q^{19} -3.45535 q^{21} +4.20028 q^{23} +13.0108 q^{25} +3.56123 q^{27} -9.76774 q^{29} -3.32050 q^{31} +1.09236 q^{33} +5.42131 q^{35} -5.09030 q^{37} +4.02007 q^{39} -4.16778 q^{41} -9.20912 q^{43} -18.3192 q^{45} -1.54157 q^{47} -5.36817 q^{49} +8.99803 q^{51} -0.932016 q^{53} -1.71387 q^{55} -16.5723 q^{57} +1.70002 q^{59} -9.98119 q^{61} -5.51413 q^{63} -6.30735 q^{65} +4.92232 q^{67} +11.3614 q^{69} -6.72094 q^{71} +7.05470 q^{73} +35.1932 q^{75} -0.515879 q^{77} +0.843450 q^{79} -3.31690 q^{81} -4.04828 q^{83} -14.1176 q^{85} -26.4209 q^{87} +1.42451 q^{89} -1.89853 q^{91} -8.98167 q^{93} +26.0014 q^{95} -14.5450 q^{97} +1.74321 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + q^{3} - 8 q^{5} - 2 q^{7}+O(q^{10}) $ 9 * q + q^3 - 8 * q^5 - 2 * q^7	$ 9 q + q^{3} - 8 q^{5} - 2 q^{7} + 10 q^{11} - 13 q^{13} - 2 q^{15} - 8 q^{17} + q^{19} - 19 q^{21} + 3 q^{23} + 3 q^{25} + 10 q^{27} - 25 q^{29} + q^{31} - 12 q^{33} + 17 q^{35} - 35 q^{37} + 4 q^{39} - 16 q^{41} - 9 q^{43} - 24 q^{45} + q^{47} - q^{49} + 10 q^{51} - 29 q^{53} - 9 q^{55} - 17 q^{57} + 14 q^{59} - 28 q^{61} - 4 q^{63} - 31 q^{65} - 19 q^{67} - 19 q^{69} + 9 q^{71} + 7 q^{75} - 33 q^{77} + 18 q^{79} - 27 q^{81} + 13 q^{83} - 36 q^{85} - 18 q^{87} - 21 q^{89} - 20 q^{91} - 35 q^{93} + 12 q^{95} + 2 q^{97} + 3 q^{99}+O(q^{100}) $ 9 * q + q^3 - 8 * q^5 - 2 * q^7 + 10 * q^11 - 13 * q^13 - 2 * q^15 - 8 * q^17 + q^19 - 19 * q^21 + 3 * q^23 + 3 * q^25 + 10 * q^27 - 25 * q^29 + q^31 - 12 * q^33 + 17 * q^35 - 35 * q^37 + 4 * q^39 - 16 * q^41 - 9 * q^43 - 24 * q^45 + q^47 - q^49 + 10 * q^51 - 29 * q^53 - 9 * q^55 - 17 * q^57 + 14 * q^59 - 28 * q^61 - 4 * q^63 - 31 * q^65 - 19 * q^67 - 19 * q^69 + 9 * q^71 + 7 * q^75 - 33 * q^77 + 18 * q^79 - 27 * q^81 + 13 * q^83 - 36 * q^85 - 18 * q^87 - 21 * q^89 - 20 * q^91 - 35 * q^93 + 12 * q^95 + 2 * q^97 + 3 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	2.70492	1.56168	0.780842	−	0.624728i	$-0.214790\pi$
$3$	2.70492	1.56168	0.780842	+	0.624728i	$0.214790\pi$
$4$	0	0
$5$	−4.24392	−1.89794	−0.948969	−	0.315370i	$-0.897871\pi$
$5$	−4.24392	−1.89794	−0.948969	+	0.315370i	$0.897871\pi$
$6$	0	0
$7$	−1.27743	−0.482824	−0.241412	−	0.970423i	$-0.577610\pi$
$7$	−1.27743	−0.482824	−0.241412	+	0.970423i	$0.577610\pi$
$8$	0	0
$9$	4.31658	1.43886
$10$	0	0
$11$	0.403841	0.121763	0.0608813	−	0.998145i	$-0.480609\pi$
$11$	0.403841	0.121763	0.0608813	+	0.998145i	$0.480609\pi$
$12$	0	0
$13$	1.48621	0.412200	0.206100	−	0.978531i	$-0.433923\pi$
$13$	1.48621	0.412200	0.206100	+	0.978531i	$0.433923\pi$
$14$	0	0
$15$	−11.4794	−2.96398
$16$	0	0
$17$	3.32654	0.806805	0.403403	−	0.915023i	$-0.367827\pi$
$17$	3.32654	0.806805	0.403403	+	0.915023i	$0.367827\pi$
$18$	0	0
$19$	−6.12675	−1.40557	−0.702786	−	0.711401i	$-0.748061\pi$
$19$	−6.12675	−1.40557	−0.702786	+	0.711401i	$0.748061\pi$
$20$	0	0
$21$	−3.45535	−0.754019
$22$	0	0
$23$	4.20028	0.875819	0.437910	−	0.899019i	$-0.355719\pi$
$23$	4.20028	0.875819	0.437910	+	0.899019i	$0.355719\pi$
$24$	0	0
$25$	13.0108	2.60217
$26$	0	0
$27$	3.56123	0.685359
$28$	0	0
$29$	−9.76774	−1.81382	−0.906912	−	0.421320i	$-0.861567\pi$
$29$	−9.76774	−1.81382	−0.906912	+	0.421320i	$0.861567\pi$
$30$	0	0
$31$	−3.32050	−0.596379	−0.298189	−	0.954507i	$-0.596383\pi$
$31$	−3.32050	−0.596379	−0.298189	+	0.954507i	$0.596383\pi$
$32$	0	0
$33$	1.09236	0.190155
$34$	0	0
$35$	5.42131	0.916369
$36$	0	0
$37$	−5.09030	−0.836839	−0.418420	−	0.908254i	$-0.637416\pi$
$37$	−5.09030	−0.836839	−0.418420	+	0.908254i	$0.637416\pi$
$38$	0	0
$39$	4.02007	0.643727
$40$	0	0
$41$	−4.16778	−0.650898	−0.325449	−	0.945560i	$-0.605515\pi$
$41$	−4.16778	−0.650898	−0.325449	+	0.945560i	$0.605515\pi$
$42$	0	0
$43$	−9.20912	−1.40438	−0.702189	−	0.711991i	$-0.747794\pi$
$43$	−9.20912	−1.40438	−0.702189	+	0.711991i	$0.747794\pi$
$44$	0	0
$45$	−18.3192	−2.73086
$46$	0	0
$47$	−1.54157	−0.224860	−0.112430	−	0.993660i	$-0.535863\pi$
$47$	−1.54157	−0.224860	−0.112430	+	0.993660i	$0.535863\pi$
$48$	0	0
$49$	−5.36817	−0.766881
$50$	0	0
$51$	8.99803	1.25998
$52$	0	0
$53$	−0.932016	−0.128022	−0.0640111	−	0.997949i	$-0.520389\pi$
$53$	−0.932016	−0.128022	−0.0640111	+	0.997949i	$0.520389\pi$
$54$	0	0
$55$	−1.71387	−0.231098
$56$	0	0
$57$	−16.5723	−2.19506
$58$	0	0
$59$	1.70002	0.221324	0.110662	−	0.993858i	$-0.464703\pi$
$59$	1.70002	0.221324	0.110662	+	0.993858i	$0.464703\pi$
$60$	0	0
$61$	−9.98119	−1.27796	−0.638980	−	0.769223i	$-0.720643\pi$
$61$	−9.98119	−1.27796	−0.638980	+	0.769223i	$0.720643\pi$
$62$	0	0
$63$	−5.51413	−0.694715
$64$	0	0
$65$	−6.30735	−0.782330
$66$	0	0
$67$	4.92232	0.601357	0.300679	−	0.953726i	$-0.402787\pi$
$67$	4.92232	0.601357	0.300679	+	0.953726i	$0.402787\pi$
$68$	0	0
$69$	11.3614	1.36775
$70$	0	0
$71$	−6.72094	−0.797629	−0.398814	−	0.917032i	$-0.630578\pi$
$71$	−6.72094	−0.797629	−0.398814	+	0.917032i	$0.630578\pi$
$72$	0	0
$73$	7.05470	0.825690	0.412845	−	0.910801i	$-0.364535\pi$
$73$	7.05470	0.825690	0.412845	+	0.910801i	$0.364535\pi$
$74$	0	0
$75$	35.1932	4.06376
$76$	0	0
$77$	−0.515879	−0.0587899
$78$	0	0
$79$	0.843450	0.0948956	0.0474478	−	0.998874i	$-0.484891\pi$
$79$	0.843450	0.0948956	0.0474478	+	0.998874i	$0.484891\pi$
$80$	0	0
$81$	−3.31690	−0.368545
$82$	0	0
$83$	−4.04828	−0.444357	−0.222178	−	0.975006i	$-0.571317\pi$
$83$	−4.04828	−0.444357	−0.222178	+	0.975006i	$0.571317\pi$
$84$	0	0
$85$	−14.1176	−1.53127
$86$	0	0
$87$	−26.4209	−2.83262
$88$	0	0
$89$	1.42451	0.150997	0.0754986	−	0.997146i	$-0.475945\pi$
$89$	1.42451	0.150997	0.0754986	+	0.997146i	$0.475945\pi$
$90$	0	0
$91$	−1.89853	−0.199020
$92$	0	0
$93$	−8.98167	−0.931356
$94$	0	0
$95$	26.0014	2.66769
$96$	0	0
$97$	−14.5450	−1.47682	−0.738409	−	0.674353i	$-0.764423\pi$
$97$	−14.5450	−1.47682	−0.738409	+	0.674353i	$0.764423\pi$
$98$	0	0
$99$	1.74321	0.175199
$100$	0	0
$101$	15.7304	1.56524	0.782618	−	0.622503i	$-0.213884\pi$
$101$	15.7304	1.56524	0.782618	+	0.622503i	$0.213884\pi$
$102$	0	0
$103$	7.86506	0.774967	0.387484	−	0.921877i	$-0.373344\pi$
$103$	7.86506	0.774967	0.387484	+	0.921877i	$0.373344\pi$
$104$	0	0
$105$	14.6642	1.43108
$106$	0	0
$107$	12.9508	1.25200	0.625999	−	0.779823i	$-0.284691\pi$
$107$	12.9508	1.25200	0.625999	+	0.779823i	$0.284691\pi$
$108$	0	0
$109$	−12.2339	−1.17180	−0.585898	−	0.810385i	$-0.699258\pi$
$109$	−12.2339	−1.17180	−0.585898	+	0.810385i	$0.699258\pi$
$110$	0	0
$111$	−13.7688	−1.30688
$112$	0	0
$113$	−3.89621	−0.366525	−0.183262	−	0.983064i	$-0.558666\pi$
$113$	−3.89621	−0.366525	−0.183262	+	0.983064i	$0.558666\pi$
$114$	0	0
$115$	−17.8256	−1.66225
$116$	0	0
$117$	6.41533	0.593098
$118$	0	0
$119$	−4.24943	−0.389545
$120$	0	0
$121$	−10.8369	−0.985174
$122$	0	0
$123$	−11.2735	−1.01650
$124$	0	0
$125$	−33.9973	−3.04081
$126$	0	0
$127$	3.72218	0.330290	0.165145	−	0.986269i	$-0.447191\pi$
$127$	3.72218	0.330290	0.165145	+	0.986269i	$0.447191\pi$
$128$	0	0
$129$	−24.9099	−2.19320
$130$	0	0
$131$	−17.5303	−1.53163	−0.765815	−	0.643061i	$-0.777664\pi$
$131$	−17.5303	−1.53163	−0.765815	+	0.643061i	$0.777664\pi$
$132$	0	0
$133$	7.82650	0.678644
$134$	0	0
$135$	−15.1136	−1.30077
$136$	0	0
$137$	9.75708	0.833604	0.416802	−	0.908997i	$-0.363151\pi$
$137$	9.75708	0.833604	0.416802	+	0.908997i	$0.363151\pi$
$138$	0	0
$139$	5.10285	0.432818	0.216409	−	0.976303i	$-0.430565\pi$
$139$	5.10285	0.432818	0.216409	+	0.976303i	$0.430565\pi$
$140$	0	0
$141$	−4.16981	−0.351161
$142$	0	0
$143$	0.600192	0.0501905
$144$	0	0
$145$	41.4535	3.44252
$146$	0	0
$147$	−14.5204	−1.19763
$148$	0	0
$149$	−22.9309	−1.87857	−0.939287	−	0.343134i	$-0.888512\pi$
$149$	−22.9309	−1.87857	−0.939287	+	0.343134i	$0.888512\pi$
$150$	0	0
$151$	8.96454	0.729524	0.364762	−	0.931101i	$-0.381150\pi$
$151$	8.96454	0.729524	0.364762	+	0.931101i	$0.381150\pi$
$152$	0	0
$153$	14.3593	1.16088
$154$	0	0
$155$	14.0919	1.13189
$156$	0	0
$157$	3.77912	0.301607	0.150803	−	0.988564i	$-0.451814\pi$
$157$	3.77912	0.301607	0.150803	+	0.988564i	$0.451814\pi$
$158$	0	0
$159$	−2.52103	−0.199930
$160$	0	0
$161$	−5.36557	−0.422866
$162$	0	0
$163$	−14.8808	−1.16555	−0.582776	−	0.812633i	$-0.698034\pi$
$163$	−14.8808	−1.16555	−0.582776	+	0.812633i	$0.698034\pi$
$164$	0	0
$165$	−4.63587	−0.360902
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	−10.7912	−0.830091
$170$	0	0
$171$	−26.4466	−2.02242
$172$	0	0
$173$	23.4547	1.78323	0.891615	−	0.452794i	$-0.149573\pi$
$173$	23.4547	1.78323	0.891615	+	0.452794i	$0.149573\pi$
$174$	0	0
$175$	−16.6204	−1.25639
$176$	0	0
$177$	4.59841	0.345638
$178$	0	0
$179$	9.88961	0.739184	0.369592	−	0.929194i	$-0.379497\pi$
$179$	9.88961	0.739184	0.369592	+	0.929194i	$0.379497\pi$
$180$	0	0
$181$	−8.80485	−0.654460	−0.327230	−	0.944945i	$-0.606115\pi$
$181$	−8.80485	−0.654460	−0.327230	+	0.944945i	$0.606115\pi$
$182$	0	0
$183$	−26.9983	−1.99577
$184$	0	0
$185$	21.6028	1.58827
$186$	0	0
$187$	1.34339	0.0982387
$188$	0	0
$189$	−4.54923	−0.330907
$190$	0	0
$191$	−11.9396	−0.863916	−0.431958	−	0.901894i	$-0.642177\pi$
$191$	−11.9396	−0.863916	−0.431958	+	0.901894i	$0.642177\pi$
$192$	0	0
$193$	10.9297	0.786738	0.393369	−	0.919381i	$-0.371310\pi$
$193$	10.9297	0.786738	0.393369	+	0.919381i	$0.371310\pi$
$194$	0	0
$195$	−17.0609	−1.22175
$196$	0	0
$197$	−9.63480	−0.686451	−0.343226	−	0.939253i	$-0.611520\pi$
$197$	−9.63480	−0.686451	−0.343226	+	0.939253i	$0.611520\pi$
$198$	0	0
$199$	−18.1255	−1.28489	−0.642443	−	0.766334i	$-0.722079\pi$
$199$	−18.1255	−1.28489	−0.642443	+	0.766334i	$0.722079\pi$
$200$	0	0
$201$	13.3145	0.939130
$202$	0	0
$203$	12.4776	0.875757
$204$	0	0
$205$	17.6877	1.23536
$206$	0	0
$207$	18.1308	1.26018
$208$	0	0
$209$	−2.47423	−0.171146
$210$	0	0
$211$	17.4655	1.20237	0.601187	−	0.799108i	$-0.294695\pi$
$211$	17.4655	1.20237	0.601187	+	0.799108i	$0.294695\pi$
$212$	0	0
$213$	−18.1796	−1.24564
$214$	0	0
$215$	39.0827	2.66542
$216$	0	0
$217$	4.24171	0.287946
$218$	0	0
$219$	19.0824	1.28947
$220$	0	0
$221$	4.94394	0.332565
$222$	0	0
$223$	0.166086	0.0111219	0.00556096	−	0.999985i	$-0.498230\pi$
$223$	0.166086	0.0111219	0.00556096	+	0.999985i	$0.498230\pi$
$224$	0	0
$225$	56.1622	3.74415
$226$	0	0
$227$	26.7551	1.77580	0.887898	−	0.460041i	$-0.152165\pi$
$227$	26.7551	1.77580	0.887898	+	0.460041i	$0.152165\pi$
$228$	0	0
$229$	26.5428	1.75400	0.877000	−	0.480491i	$-0.159542\pi$
$229$	26.5428	1.75400	0.877000	+	0.480491i	$0.159542\pi$
$230$	0	0
$231$	−1.39541	−0.0918112
$232$	0	0
$233$	27.6384	1.81065	0.905325	−	0.424719i	$-0.139627\pi$
$233$	27.6384	1.81065	0.905325	+	0.424719i	$0.139627\pi$
$234$	0	0
$235$	6.54227	0.426771
$236$	0	0
$237$	2.28146	0.148197
$238$	0	0
$239$	30.3260	1.96163	0.980813	−	0.194951i	$-0.0624549\pi$
$239$	30.3260	1.96163	0.980813	+	0.194951i	$0.0624549\pi$
$240$	0	0
$241$	−24.6851	−1.59011	−0.795054	−	0.606539i	$-0.792558\pi$
$241$	−24.6851	−1.59011	−0.795054	+	0.606539i	$0.792558\pi$
$242$	0	0
$243$	−19.6556	−1.26091
$244$	0	0
$245$	22.7821	1.45549
$246$	0	0
$247$	−9.10563	−0.579377
$248$	0	0
$249$	−10.9503	−0.693945
$250$	0	0
$251$	26.1189	1.64861	0.824304	−	0.566147i	$-0.191566\pi$
$251$	26.1189	1.64861	0.824304	+	0.566147i	$0.191566\pi$
$252$	0	0
$253$	1.69624	0.106642
$254$	0	0
$255$	−38.1869	−2.39135
$256$	0	0
$257$	−3.37604	−0.210592	−0.105296	−	0.994441i	$-0.533579\pi$
$257$	−3.37604	−0.210592	−0.105296	+	0.994441i	$0.533579\pi$
$258$	0	0
$259$	6.50251	0.404046
$260$	0	0
$261$	−42.1632	−2.60984
$262$	0	0
$263$	25.2221	1.55526	0.777631	−	0.628721i	$-0.216421\pi$
$263$	25.2221	1.55526	0.777631	+	0.628721i	$0.216421\pi$
$264$	0	0
$265$	3.95540	0.242978
$266$	0	0
$267$	3.85317	0.235810
$268$	0	0
$269$	−18.0869	−1.10278	−0.551390	−	0.834248i	$-0.685902\pi$
$269$	−18.0869	−1.10278	−0.551390	+	0.834248i	$0.685902\pi$
$270$	0	0
$271$	23.4229	1.42284	0.711421	−	0.702766i	$-0.248052\pi$
$271$	23.4229	1.42284	0.711421	+	0.702766i	$0.248052\pi$
$272$	0	0
$273$	−5.13537	−0.310807
$274$	0	0
$275$	5.25430	0.316846
$276$	0	0
$277$	5.90502	0.354798	0.177399	−	0.984139i	$-0.443232\pi$
$277$	5.90502	0.354798	0.177399	+	0.984139i	$0.443232\pi$
$278$	0	0
$279$	−14.3332	−0.858105
$280$	0	0
$281$	−30.9175	−1.84439	−0.922193	−	0.386730i	$-0.873604\pi$
$281$	−30.9175	−1.84439	−0.922193	+	0.386730i	$0.873604\pi$
$282$	0	0
$283$	3.55188	0.211137	0.105569	−	0.994412i	$-0.466334\pi$
$283$	3.55188	0.211137	0.105569	+	0.994412i	$0.466334\pi$
$284$	0	0
$285$	70.3317	4.16609
$286$	0	0
$287$	5.32405	0.314269
$288$	0	0
$289$	−5.93410	−0.349065
$290$	0	0
$291$	−39.3429	−2.30632
$292$	0	0
$293$	19.2122	1.12239	0.561195	−	0.827683i	$-0.310342\pi$
$293$	19.2122	1.12239	0.561195	+	0.827683i	$0.310342\pi$
$294$	0	0
$295$	−7.21475	−0.420059
$296$	0	0
$297$	1.43817	0.0834510
$298$	0	0
$299$	6.24250	0.361013
$300$	0	0
$301$	11.7640	0.678067
$302$	0	0
$303$	42.5495	2.44440
$304$	0	0
$305$	42.3593	2.42549
$306$	0	0
$307$	1.07774	0.0615096	0.0307548	−	0.999527i	$-0.490209\pi$
$307$	1.07774	0.0615096	0.0307548	+	0.999527i	$0.490209\pi$
$308$	0	0
$309$	21.2743	1.21025
$310$	0	0
$311$	−5.71693	−0.324177	−0.162089	−	0.986776i	$-0.551823\pi$
$311$	−5.71693	−0.324177	−0.162089	+	0.986776i	$0.551823\pi$
$312$	0	0
$313$	−11.6515	−0.658582	−0.329291	−	0.944228i	$-0.606810\pi$
$313$	−11.6515	−0.658582	−0.329291	+	0.944228i	$0.606810\pi$
$314$	0	0
$315$	23.4015	1.31853
$316$	0	0
$317$	−3.79584	−0.213196	−0.106598	−	0.994302i	$-0.533996\pi$
$317$	−3.79584	−0.213196	−0.106598	+	0.994302i	$0.533996\pi$
$318$	0	0
$319$	−3.94461	−0.220856
$320$	0	0
$321$	35.0308	1.95523
$322$	0	0
$323$	−20.3809	−1.13402
$324$	0	0
$325$	19.3368	1.07261
$326$	0	0
$327$	−33.0917	−1.82998
$328$	0	0
$329$	1.96924	0.108568
$330$	0	0
$331$	−34.3425	−1.88764	−0.943818	−	0.330466i	$-0.892794\pi$
$331$	−34.3425	−1.88764	−0.943818	+	0.330466i	$0.892794\pi$
$332$	0	0
$333$	−21.9726	−1.20409
$334$	0	0
$335$	−20.8899	−1.14134
$336$	0	0
$337$	−5.31445	−0.289497	−0.144748	−	0.989469i	$-0.546237\pi$
$337$	−5.31445	−0.289497	−0.144748	+	0.989469i	$0.546237\pi$
$338$	0	0
$339$	−10.5389	−0.572396
$340$	0	0
$341$	−1.34095	−0.0726166
$342$	0	0
$343$	15.7995	0.853092
$344$	0	0
$345$	−48.2169	−2.59591
$346$	0	0
$347$	−14.5429	−0.780704	−0.390352	−	0.920666i	$-0.627647\pi$
$347$	−14.5429	−0.780704	−0.390352	+	0.920666i	$0.627647\pi$
$348$	0	0
$349$	−11.9375	−0.638999	−0.319499	−	0.947587i	$-0.603515\pi$
$349$	−11.9375	−0.638999	−0.319499	+	0.947587i	$0.603515\pi$
$350$	0	0
$351$	5.29273	0.282505
$352$	0	0
$353$	31.5607	1.67980	0.839902	−	0.542738i	$-0.182612\pi$
$353$	31.5607	1.67980	0.839902	+	0.542738i	$0.182612\pi$
$354$	0	0
$355$	28.5231	1.51385
$356$	0	0
$357$	−11.4944	−0.608346
$358$	0	0
$359$	−16.8047	−0.886916	−0.443458	−	0.896295i	$-0.646248\pi$
$359$	−16.8047	−0.886916	−0.443458	+	0.896295i	$0.646248\pi$
$360$	0	0
$361$	18.5371	0.975634
$362$	0	0
$363$	−29.3129	−1.53853
$364$	0	0
$365$	−29.9396	−1.56711
$366$	0	0
$367$	9.86034	0.514705	0.257353	−	0.966318i	$-0.417150\pi$
$367$	9.86034	0.514705	0.257353	+	0.966318i	$0.417150\pi$
$368$	0	0
$369$	−17.9905	−0.936550
$370$	0	0
$371$	1.19059	0.0618122
$372$	0	0
$373$	−4.70042	−0.243379	−0.121689	−	0.992568i	$-0.538831\pi$
$373$	−4.70042	−0.243379	−0.121689	+	0.992568i	$0.538831\pi$
$374$	0	0
$375$	−91.9599	−4.74879
$376$	0	0
$377$	−14.5169	−0.747659
$378$	0	0
$379$	−14.2996	−0.734523	−0.367261	−	0.930118i	$-0.619705\pi$
$379$	−14.2996	−0.734523	−0.367261	+	0.930118i	$0.619705\pi$
$380$	0	0
$381$	10.0682	0.515808
$382$	0	0
$383$	13.0374	0.666179	0.333089	−	0.942895i	$-0.391909\pi$
$383$	13.0374	0.666179	0.333089	+	0.942895i	$0.391909\pi$
$384$	0	0
$385$	2.18935	0.111579
$386$	0	0
$387$	−39.7519	−2.02070
$388$	0	0
$389$	10.2817	0.521305	0.260652	−	0.965433i	$-0.416062\pi$
$389$	10.2817	0.521305	0.260652	+	0.965433i	$0.416062\pi$
$390$	0	0
$391$	13.9724	0.706616
$392$	0	0
$393$	−47.4180	−2.39192
$394$	0	0
$395$	−3.57953	−0.180106
$396$	0	0
$397$	−18.1577	−0.911311	−0.455656	−	0.890156i	$-0.650595\pi$
$397$	−18.1577	−0.911311	−0.455656	+	0.890156i	$0.650595\pi$
$398$	0	0
$399$	21.1700	1.05983
$400$	0	0
$401$	−8.93888	−0.446386	−0.223193	−	0.974774i	$-0.571648\pi$
$401$	−8.93888	−0.446386	−0.223193	+	0.974774i	$0.571648\pi$
$402$	0	0
$403$	−4.93495	−0.245828
$404$	0	0
$405$	14.0767	0.699475
$406$	0	0
$407$	−2.05567	−0.101896
$408$	0	0
$409$	−12.3707	−0.611694	−0.305847	−	0.952081i	$-0.598940\pi$
$409$	−12.3707	−0.611694	−0.305847	+	0.952081i	$0.598940\pi$
$410$	0	0
$411$	26.3921	1.30183
$412$	0	0
$413$	−2.17166	−0.106860
$414$	0	0
$415$	17.1806	0.843362
$416$	0	0
$417$	13.8028	0.675926
$418$	0	0
$419$	29.2192	1.42745	0.713726	−	0.700425i	$-0.247006\pi$
$419$	29.2192	1.42745	0.713726	+	0.700425i	$0.247006\pi$
$420$	0	0
$421$	4.49591	0.219117	0.109559	−	0.993980i	$-0.465056\pi$
$421$	4.49591	0.219117	0.109559	+	0.993980i	$0.465056\pi$
$422$	0	0
$423$	−6.65428	−0.323542
$424$	0	0
$425$	43.2811	2.09944
$426$	0	0
$427$	12.7503	0.617030
$428$	0	0
$429$	1.62347	0.0783818
$430$	0	0
$431$	−1.46354	−0.0704962	−0.0352481	−	0.999379i	$-0.511222\pi$
$431$	−1.46354	−0.0704962	−0.0352481	+	0.999379i	$0.511222\pi$
$432$	0	0
$433$	−17.4259	−0.837434	−0.418717	−	0.908117i	$-0.637520\pi$
$433$	−17.4259	−0.837434	−0.418717	+	0.908117i	$0.637520\pi$
$434$	0	0
$435$	112.128	5.37614
$436$	0	0
$437$	−25.7341	−1.23103
$438$	0	0
$439$	−3.55586	−0.169712	−0.0848559	−	0.996393i	$-0.527043\pi$
$439$	−3.55586	−0.169712	−0.0848559	+	0.996393i	$0.527043\pi$
$440$	0	0
$441$	−23.1721	−1.10343
$442$	0	0
$443$	28.4618	1.35226	0.676131	−	0.736781i	$-0.263655\pi$
$443$	28.4618	1.35226	0.676131	+	0.736781i	$0.263655\pi$
$444$	0	0
$445$	−6.04548	−0.286583
$446$	0	0
$447$	−62.0262	−2.93374
$448$	0	0
$449$	−9.59661	−0.452892	−0.226446	−	0.974024i	$-0.572711\pi$
$449$	−9.59661	−0.452892	−0.226446	+	0.974024i	$0.572711\pi$
$450$	0	0
$451$	−1.68312	−0.0792550
$452$	0	0
$453$	24.2483	1.13929
$454$	0	0
$455$	8.05721	0.377728
$456$	0	0
$457$	−21.7289	−1.01644	−0.508218	−	0.861228i	$-0.669696\pi$
$457$	−21.7289	−1.01644	−0.508218	+	0.861228i	$0.669696\pi$
$458$	0	0
$459$	11.8466	0.552951
$460$	0	0
$461$	28.8932	1.34569	0.672845	−	0.739783i	$-0.265072\pi$
$461$	28.8932	1.34569	0.672845	+	0.739783i	$0.265072\pi$
$462$	0	0
$463$	23.4238	1.08860	0.544299	−	0.838891i	$-0.316796\pi$
$463$	23.4238	1.08860	0.544299	+	0.838891i	$0.316796\pi$
$464$	0	0
$465$	38.1175	1.76765
$466$	0	0
$467$	−31.5852	−1.46159	−0.730793	−	0.682599i	$-0.760850\pi$
$467$	−31.5852	−1.46159	−0.730793	+	0.682599i	$0.760850\pi$
$468$	0	0
$469$	−6.28793	−0.290350
$470$	0	0
$471$	10.2222	0.471015
$472$	0	0
$473$	−3.71902	−0.171001
$474$	0	0
$475$	−79.7141	−3.65753
$476$	0	0
$477$	−4.02312	−0.184206
$478$	0	0
$479$	33.7955	1.54416	0.772078	−	0.635527i	$-0.219217\pi$
$479$	33.7955	1.54416	0.772078	+	0.635527i	$0.219217\pi$
$480$	0	0
$481$	−7.56524	−0.344945
$482$	0	0
$483$	−14.5134	−0.660384
$484$	0	0
$485$	61.7277	2.80291
$486$	0	0
$487$	−38.2331	−1.73251	−0.866253	−	0.499605i	$-0.833479\pi$
$487$	−38.2331	−1.73251	−0.866253	+	0.499605i	$0.833479\pi$
$488$	0	0
$489$	−40.2513	−1.82023
$490$	0	0
$491$	−2.32648	−0.104992	−0.0524962	−	0.998621i	$-0.516718\pi$
$491$	−2.32648	−0.104992	−0.0524962	+	0.998621i	$0.516718\pi$
$492$	0	0
$493$	−32.4928	−1.46340
$494$	0	0
$495$	−7.39803	−0.332517
$496$	0	0
$497$	8.58554	0.385114
$498$	0	0
$499$	31.8384	1.42528	0.712642	−	0.701528i	$-0.247498\pi$
$499$	31.8384	1.42528	0.712642	+	0.701528i	$0.247498\pi$
$500$	0	0
$501$	−2.70492	−0.120847
$502$	0	0
$503$	−4.97511	−0.221829	−0.110915	−	0.993830i	$-0.535378\pi$
$503$	−4.97511	−0.221829	−0.110915	+	0.993830i	$0.535378\pi$
$504$	0	0
$505$	−66.7586	−2.97072
$506$	0	0
$507$	−29.1893	−1.29634
$508$	0	0
$509$	−34.2648	−1.51876	−0.759381	−	0.650647i	$-0.774498\pi$
$509$	−34.2648	−1.51876	−0.759381	+	0.650647i	$0.774498\pi$
$510$	0	0
$511$	−9.01190	−0.398663
$512$	0	0
$513$	−21.8187	−0.963321
$514$	0	0
$515$	−33.3787	−1.47084
$516$	0	0
$517$	−0.622547	−0.0273796
$518$	0	0
$519$	63.4431	2.78484
$520$	0	0
$521$	10.8100	0.473596	0.236798	−	0.971559i	$-0.423902\pi$
$521$	10.8100	0.473596	0.236798	+	0.971559i	$0.423902\pi$
$522$	0	0
$523$	15.1590	0.662856	0.331428	−	0.943481i	$-0.392470\pi$
$523$	15.1590	0.662856	0.331428	+	0.943481i	$0.392470\pi$
$524$	0	0
$525$	−44.9569	−1.96208
$526$	0	0
$527$	−11.0458	−0.481162
$528$	0	0
$529$	−5.35764	−0.232941
$530$	0	0
$531$	7.33827	0.318454
$532$	0	0
$533$	−6.19419	−0.268300
$534$	0	0
$535$	−54.9620	−2.37622
$536$	0	0
$537$	26.7506	1.15437
$538$	0	0
$539$	−2.16788	−0.0933774
$540$	0	0
$541$	−7.68358	−0.330343	−0.165171	−	0.986265i	$-0.552818\pi$
$541$	−7.68358	−0.330343	−0.165171	+	0.986265i	$0.552818\pi$
$542$	0	0
$543$	−23.8164	−1.02206
$544$	0	0
$545$	51.9197	2.22400
$546$	0	0
$547$	5.79317	0.247698	0.123849	−	0.992301i	$-0.460476\pi$
$547$	5.79317	0.247698	0.123849	+	0.992301i	$0.460476\pi$
$548$	0	0
$549$	−43.0846	−1.83880
$550$	0	0
$551$	59.8445	2.54946
$552$	0	0
$553$	−1.07745	−0.0458178
$554$	0	0
$555$	58.4338	2.48037
$556$	0	0
$557$	−15.2484	−0.646094	−0.323047	−	0.946383i	$-0.604707\pi$
$557$	−15.2484	−0.646094	−0.323047	+	0.946383i	$0.604707\pi$
$558$	0	0
$559$	−13.6867	−0.578885
$560$	0	0
$561$	3.63377	0.153418
$562$	0	0
$563$	−13.5679	−0.571818	−0.285909	−	0.958257i	$-0.592296\pi$
$563$	−13.5679	−0.571818	−0.285909	+	0.958257i	$0.592296\pi$
$564$	0	0
$565$	16.5352	0.695641
$566$	0	0
$567$	4.23712	0.177942
$568$	0	0
$569$	3.43479	0.143994	0.0719970	−	0.997405i	$-0.477063\pi$
$569$	3.43479	0.143994	0.0719970	+	0.997405i	$0.477063\pi$
$570$	0	0
$571$	−43.1389	−1.80531	−0.902654	−	0.430368i	$-0.858384\pi$
$571$	−43.1389	−1.80531	−0.902654	+	0.430368i	$0.858384\pi$
$572$	0	0
$573$	−32.2955	−1.34916
$574$	0	0
$575$	54.6491	2.27903
$576$	0	0
$577$	17.6883	0.736372	0.368186	−	0.929752i	$-0.379979\pi$
$577$	17.6883	0.736372	0.368186	+	0.929752i	$0.379979\pi$
$578$	0	0
$579$	29.5640	1.22864
$580$	0	0
$581$	5.17141	0.214546
$582$	0	0
$583$	−0.376386	−0.0155883
$584$	0	0
$585$	−27.2261	−1.12566
$586$	0	0
$587$	−32.5290	−1.34261	−0.671307	−	0.741180i	$-0.734267\pi$
$587$	−32.5290	−1.34261	−0.671307	+	0.741180i	$0.734267\pi$
$588$	0	0
$589$	20.3439	0.838254
$590$	0	0
$591$	−26.0613	−1.07202
$592$	0	0
$593$	−1.04886	−0.0430716	−0.0215358	−	0.999768i	$-0.506856\pi$
$593$	−1.04886	−0.0430716	−0.0215358	+	0.999768i	$0.506856\pi$
$594$	0	0
$595$	18.0342	0.739332
$596$	0	0
$597$	−49.0281	−2.00659
$598$	0	0
$599$	42.5697	1.73935	0.869675	−	0.493624i	$-0.164328\pi$
$599$	42.5697	1.73935	0.869675	+	0.493624i	$0.164328\pi$
$600$	0	0
$601$	−9.46471	−0.386073	−0.193037	−	0.981192i	$-0.561834\pi$
$601$	−9.46471	−0.386073	−0.193037	+	0.981192i	$0.561834\pi$
$602$	0	0
$603$	21.2476	0.865268
$604$	0	0
$605$	45.9910	1.86980
$606$	0	0
$607$	34.1161	1.38473	0.692365	−	0.721548i	$-0.256569\pi$
$607$	34.1161	1.38473	0.692365	+	0.721548i	$0.256569\pi$
$608$	0	0
$609$	33.7509	1.36766
$610$	0	0
$611$	−2.29109	−0.0926875
$612$	0	0
$613$	17.8616	0.721425	0.360712	−	0.932677i	$-0.382534\pi$
$613$	17.8616	0.721425	0.360712	+	0.932677i	$0.382534\pi$
$614$	0	0
$615$	47.8438	1.92925
$616$	0	0
$617$	−36.4070	−1.46569	−0.732846	−	0.680395i	$-0.761808\pi$
$617$	−36.4070	−1.46569	−0.732846	+	0.680395i	$0.761808\pi$
$618$	0	0
$619$	−15.8973	−0.638968	−0.319484	−	0.947592i	$-0.603510\pi$
$619$	−15.8973	−0.638968	−0.319484	+	0.947592i	$0.603510\pi$
$620$	0	0
$621$	14.9582	0.600250
$622$	0	0
$623$	−1.81971	−0.0729051
$624$	0	0
$625$	79.2276	3.16910
$626$	0	0
$627$	−6.69259	−0.267276
$628$	0	0
$629$	−16.9331	−0.675167
$630$	0	0
$631$	−17.2964	−0.688557	−0.344279	−	0.938868i	$-0.611876\pi$
$631$	−17.2964	−0.688557	−0.344279	+	0.938868i	$0.611876\pi$
$632$	0	0
$633$	47.2427	1.87773
$634$	0	0
$635$	−15.7966	−0.626869
$636$	0	0
$637$	−7.97822	−0.316109
$638$	0	0
$639$	−29.0114	−1.14767
$640$	0	0
$641$	19.0444	0.752207	0.376103	−	0.926578i	$-0.377264\pi$
$641$	19.0444	0.752207	0.376103	+	0.926578i	$0.377264\pi$
$642$	0	0
$643$	17.0174	0.671100	0.335550	−	0.942022i	$-0.391078\pi$
$643$	17.0174	0.671100	0.335550	+	0.942022i	$0.391078\pi$
$644$	0	0
$645$	105.716	4.16255
$646$	0	0
$647$	−11.0099	−0.432844	−0.216422	−	0.976300i	$-0.569439\pi$
$647$	−11.0099	−0.432844	−0.216422	+	0.976300i	$0.569439\pi$
$648$	0	0
$649$	0.686537	0.0269489
$650$	0	0
$651$	11.4735	0.449681
$652$	0	0
$653$	−19.1819	−0.750644	−0.375322	−	0.926894i	$-0.622468\pi$
$653$	−19.1819	−0.750644	−0.375322	+	0.926894i	$0.622468\pi$
$654$	0	0
$655$	74.3972	2.90694
$656$	0	0
$657$	30.4521	1.18805
$658$	0	0
$659$	−30.3453	−1.18209	−0.591043	−	0.806640i	$-0.701283\pi$
$659$	−30.3453	−1.18209	−0.591043	+	0.806640i	$0.701283\pi$
$660$	0	0
$661$	−14.9037	−0.579685	−0.289842	−	0.957074i	$-0.593603\pi$
$661$	−14.9037	−0.579685	−0.289842	+	0.957074i	$0.593603\pi$
$662$	0	0
$663$	13.3729	0.519362
$664$	0	0
$665$	−33.2150	−1.28802
$666$	0	0
$667$	−41.0273	−1.58858
$668$	0	0
$669$	0.449248	0.0173689
$670$	0	0
$671$	−4.03081	−0.155608
$672$	0	0
$673$	12.6160	0.486313	0.243156	−	0.969987i	$-0.421817\pi$
$673$	12.6160	0.486313	0.243156	+	0.969987i	$0.421817\pi$
$674$	0	0
$675$	46.3345	1.78342
$676$	0	0
$677$	−11.9550	−0.459468	−0.229734	−	0.973253i	$-0.573786\pi$
$677$	−11.9550	−0.459468	−0.229734	+	0.973253i	$0.573786\pi$
$678$	0	0
$679$	18.5802	0.713043
$680$	0	0
$681$	72.3702	2.77323
$682$	0	0
$683$	35.8260	1.37084	0.685421	−	0.728147i	$-0.259618\pi$
$683$	35.8260	1.37084	0.685421	+	0.728147i	$0.259618\pi$
$684$	0	0
$685$	−41.4082	−1.58213
$686$	0	0
$687$	71.7961	2.73919
$688$	0	0
$689$	−1.38517	−0.0527708
$690$	0	0
$691$	−24.0214	−0.913816	−0.456908	−	0.889514i	$-0.651043\pi$
$691$	−24.0214	−0.913816	−0.456908	+	0.889514i	$0.651043\pi$
$692$	0	0
$693$	−2.22683	−0.0845903
$694$	0	0
$695$	−21.6561	−0.821462
$696$	0	0
$697$	−13.8643	−0.525148
$698$	0	0
$699$	74.7595	2.82767
$700$	0	0
$701$	−25.0403	−0.945759	−0.472879	−	0.881127i	$-0.656785\pi$
$701$	−25.0403	−0.945759	−0.472879	+	0.881127i	$0.656785\pi$
$702$	0	0
$703$	31.1870	1.17624
$704$	0	0
$705$	17.6963	0.666482
$706$	0	0
$707$	−20.0945	−0.755733
$708$	0	0
$709$	27.7976	1.04396	0.521980	−	0.852957i	$-0.325193\pi$
$709$	27.7976	1.04396	0.521980	+	0.852957i	$0.325193\pi$
$710$	0	0
$711$	3.64082	0.136541
$712$	0	0
$713$	−13.9470	−0.522320
$714$	0	0
$715$	−2.54716	−0.0952585
$716$	0	0
$717$	82.0293	3.06344
$718$	0	0
$719$	−36.4516	−1.35942	−0.679708	−	0.733483i	$-0.737894\pi$
$719$	−36.4516	−1.35942	−0.679708	+	0.733483i	$0.737894\pi$
$720$	0	0
$721$	−10.0471	−0.374173
$722$	0	0
$723$	−66.7712	−2.48325
$724$	0	0
$725$	−127.086	−4.71987
$726$	0	0
$727$	21.0330	0.780070	0.390035	−	0.920800i	$-0.372463\pi$
$727$	21.0330	0.780070	0.390035	+	0.920800i	$0.372463\pi$
$728$	0	0
$729$	−43.2161	−1.60060
$730$	0	0
$731$	−30.6345	−1.13306
$732$	0	0
$733$	11.5226	0.425596	0.212798	−	0.977096i	$-0.431742\pi$
$733$	11.5226	0.425596	0.212798	+	0.977096i	$0.431742\pi$
$734$	0	0
$735$	61.6236	2.27302
$736$	0	0
$737$	1.98783	0.0732228
$738$	0	0
$739$	10.4455	0.384245	0.192123	−	0.981371i	$-0.438463\pi$
$739$	10.4455	0.384245	0.192123	+	0.981371i	$0.438463\pi$
$740$	0	0
$741$	−24.6300	−0.904805
$742$	0	0
$743$	29.0541	1.06589	0.532946	−	0.846149i	$-0.321085\pi$
$743$	29.0541	1.06589	0.532946	+	0.846149i	$0.321085\pi$
$744$	0	0
$745$	97.3169	3.56541
$746$	0	0
$747$	−17.4747	−0.639367
$748$	0	0
$749$	−16.5437	−0.604495
$750$	0	0
$751$	36.3643	1.32695	0.663476	−	0.748198i	$-0.269080\pi$
$751$	36.3643	1.32695	0.663476	+	0.748198i	$0.269080\pi$
$752$	0	0
$753$	70.6494	2.57461
$754$	0	0
$755$	−38.0448	−1.38459
$756$	0	0
$757$	−36.7704	−1.33644	−0.668221	−	0.743963i	$-0.732944\pi$
$757$	−36.7704	−1.33644	−0.668221	+	0.743963i	$0.732944\pi$
$758$	0	0
$759$	4.58820	0.166541
$760$	0	0
$761$	−28.6610	−1.03896	−0.519481	−	0.854482i	$-0.673875\pi$
$761$	−28.6610	−1.03896	−0.519481	+	0.854482i	$0.673875\pi$
$762$	0	0
$763$	15.6280	0.565771
$764$	0	0
$765$	−60.9396	−2.20328
$766$	0	0
$767$	2.52659	0.0912297
$768$	0	0
$769$	−40.6341	−1.46530	−0.732652	−	0.680604i	$-0.761718\pi$
$769$	−40.6341	−1.46530	−0.732652	+	0.680604i	$0.761718\pi$
$770$	0	0
$771$	−9.13191	−0.328878
$772$	0	0
$773$	−12.1122	−0.435646	−0.217823	−	0.975988i	$-0.569895\pi$
$773$	−12.1122	−0.435646	−0.217823	+	0.975988i	$0.569895\pi$
$774$	0	0
$775$	−43.2024	−1.55188
$776$	0	0
$777$	17.5887	0.630992
$778$	0	0
$779$	25.5349	0.914884
$780$	0	0
$781$	−2.71419	−0.0971213
$782$	0	0
$783$	−34.7852	−1.24312
$784$	0	0
$785$	−16.0383	−0.572431
$786$	0	0
$787$	−43.2132	−1.54038	−0.770191	−	0.637813i	$-0.779839\pi$
$787$	−43.2132	−1.54038	−0.770191	+	0.637813i	$0.779839\pi$
$788$	0	0
$789$	68.2237	2.42883
$790$	0	0
$791$	4.97714	0.176967
$792$	0	0
$793$	−14.8341	−0.526776
$794$	0	0
$795$	10.6990	0.379455
$796$	0	0
$797$	32.6861	1.15780	0.578901	−	0.815398i	$-0.303482\pi$
$797$	32.6861	1.15780	0.578901	+	0.815398i	$0.303482\pi$
$798$	0	0
$799$	−5.12808	−0.181419
$800$	0	0
$801$	6.14899	0.217264
$802$	0	0
$803$	2.84897	0.100538
$804$	0	0
$805$	22.7710	0.802574
$806$	0	0
$807$	−48.9237	−1.72219
$808$	0	0
$809$	−10.4760	−0.368316	−0.184158	−	0.982897i	$-0.558956\pi$
$809$	−10.4760	−0.368316	−0.184158	+	0.982897i	$0.558956\pi$
$810$	0	0
$811$	12.9589	0.455049	0.227525	−	0.973772i	$-0.426937\pi$
$811$	12.9589	0.455049	0.227525	+	0.973772i	$0.426937\pi$
$812$	0	0
$813$	63.3571	2.22203
$814$	0	0
$815$	63.1528	2.21215
$816$	0	0
$817$	56.4220	1.97395
$818$	0	0
$819$	−8.19515	−0.286362
$820$	0	0
$821$	−16.4621	−0.574532	−0.287266	−	0.957851i	$-0.592746\pi$
$821$	−16.4621	−0.574532	−0.287266	+	0.957851i	$0.592746\pi$
$822$	0	0
$823$	−42.7238	−1.48926	−0.744630	−	0.667478i	$-0.767374\pi$
$823$	−42.7238	−1.48926	−0.744630	+	0.667478i	$0.767374\pi$
$824$	0	0
$825$	14.2124	0.494814
$826$	0	0
$827$	−33.0454	−1.14910	−0.574550	−	0.818469i	$-0.694823\pi$
$827$	−33.0454	−1.14910	−0.574550	+	0.818469i	$0.694823\pi$
$828$	0	0
$829$	18.0587	0.627206	0.313603	−	0.949554i	$-0.398464\pi$
$829$	18.0587	0.627206	0.313603	+	0.949554i	$0.398464\pi$
$830$	0	0
$831$	15.9726	0.554083
$832$	0	0
$833$	−17.8574	−0.618724
$834$	0	0
$835$	4.24392	0.146867
$836$	0	0
$837$	−11.8250	−0.408733
$838$	0	0
$839$	28.1830	0.972986	0.486493	−	0.873685i	$-0.338276\pi$
$839$	28.1830	0.972986	0.486493	+	0.873685i	$0.338276\pi$
$840$	0	0
$841$	66.4088	2.28996
$842$	0	0
$843$	−83.6294	−2.88035
$844$	0	0
$845$	45.7969	1.57546
$846$	0	0
$847$	13.8434	0.475665
$848$	0	0
$849$	9.60755	0.329730
$850$	0	0
$851$	−21.3807	−0.732920
$852$	0	0
$853$	42.4867	1.45472	0.727359	−	0.686257i	$-0.240747\pi$
$853$	42.4867	1.45472	0.727359	+	0.686257i	$0.240747\pi$
$854$	0	0
$855$	112.237	3.83843
$856$	0	0
$857$	49.9048	1.70471	0.852357	−	0.522960i	$-0.175172\pi$
$857$	49.9048	1.70471	0.852357	+	0.522960i	$0.175172\pi$
$858$	0	0
$859$	−8.47277	−0.289087	−0.144544	−	0.989498i	$-0.546171\pi$
$859$	−8.47277	−0.289087	−0.144544	+	0.989498i	$0.546171\pi$
$860$	0	0
$861$	14.4011	0.490789
$862$	0	0
$863$	−1.46856	−0.0499902	−0.0249951	−	0.999688i	$-0.507957\pi$
$863$	−1.46856	−0.0499902	−0.0249951	+	0.999688i	$0.507957\pi$
$864$	0	0
$865$	−99.5399	−3.38446
$866$	0	0
$867$	−16.0513	−0.545129
$868$	0	0
$869$	0.340619	0.0115547
$870$	0	0
$871$	7.31560	0.247880
$872$	0	0
$873$	−62.7845	−2.12493
$874$	0	0
$875$	43.4292	1.46818
$876$	0	0
$877$	−39.0735	−1.31942	−0.659709	−	0.751521i	$-0.729321\pi$
$877$	−39.0735	−1.31942	−0.659709	+	0.751521i	$0.729321\pi$
$878$	0	0
$879$	51.9675	1.75282
$880$	0	0
$881$	−23.4707	−0.790747	−0.395374	−	0.918520i	$-0.629385\pi$
$881$	−23.4707	−0.790747	−0.395374	+	0.918520i	$0.629385\pi$
$882$	0	0
$883$	−8.49116	−0.285750	−0.142875	−	0.989741i	$-0.545635\pi$
$883$	−8.49116	−0.285750	−0.142875	+	0.989741i	$0.545635\pi$
$884$	0	0
$885$	−19.5153	−0.655999
$886$	0	0
$887$	8.79252	0.295224	0.147612	−	0.989045i	$-0.452841\pi$
$887$	8.79252	0.295224	0.147612	+	0.989045i	$0.452841\pi$
$888$	0	0
$889$	−4.75483	−0.159472
$890$	0	0
$891$	−1.33950	−0.0448749
$892$	0	0
$893$	9.44478	0.316058
$894$	0	0
$895$	−41.9707	−1.40293
$896$	0	0
$897$	16.8854	0.563788
$898$	0	0
$899$	32.4338	1.08173
$900$	0	0
$901$	−3.10039	−0.103289
$902$	0	0
$903$	31.8207	1.05893
$904$	0	0
$905$	37.3671	1.24212
$906$	0	0
$907$	−19.8502	−0.659116	−0.329558	−	0.944135i	$-0.606900\pi$
$907$	−19.8502	−0.659116	−0.329558	+	0.944135i	$0.606900\pi$
$908$	0	0
$909$	67.9016	2.25215
$910$	0	0
$911$	−28.6417	−0.948942	−0.474471	−	0.880271i	$-0.657361\pi$
$911$	−28.6417	−0.948942	−0.474471	+	0.880271i	$0.657361\pi$
$912$	0	0
$913$	−1.63486	−0.0541060
$914$	0	0
$915$	114.579	3.78785
$916$	0	0
$917$	22.3938	0.739508
$918$	0	0
$919$	−2.99821	−0.0989018	−0.0494509	−	0.998777i	$-0.515747\pi$
$919$	−2.99821	−0.0989018	−0.0494509	+	0.998777i	$0.515747\pi$
$920$	0	0
$921$	2.91519	0.0960586
$922$	0	0
$923$	−9.98872	−0.328783
$924$	0	0
$925$	−66.2290	−2.17760
$926$	0	0
$927$	33.9501	1.11507
$928$	0	0
$929$	9.34190	0.306498	0.153249	−	0.988188i	$-0.451026\pi$
$929$	9.34190	0.306498	0.153249	+	0.988188i	$0.451026\pi$
$930$	0	0
$931$	32.8894	1.07791
$932$	0	0
$933$	−15.4638	−0.506262
$934$	0	0
$935$	−5.70125	−0.186451
$936$	0	0
$937$	−37.3671	−1.22073	−0.610365	−	0.792121i	$-0.708977\pi$
$937$	−37.3671	−1.22073	−0.610365	+	0.792121i	$0.708977\pi$
$938$	0	0
$939$	−31.5164	−1.02850
$940$	0	0
$941$	−11.7055	−0.381587	−0.190794	−	0.981630i	$-0.561106\pi$
$941$	−11.7055	−0.381587	−0.190794	+	0.981630i	$0.561106\pi$
$942$	0	0
$943$	−17.5058	−0.570069
$944$	0	0
$945$	19.3065	0.628042
$946$	0	0
$947$	33.8526	1.10006	0.550030	−	0.835145i	$-0.314616\pi$
$947$	33.8526	1.10006	0.550030	+	0.835145i	$0.314616\pi$
$948$	0	0
$949$	10.4848	0.340350
$950$	0	0
$951$	−10.2674	−0.332944
$952$	0	0
$953$	21.3850	0.692728	0.346364	−	0.938100i	$-0.387416\pi$
$953$	21.3850	0.692728	0.346364	+	0.938100i	$0.387416\pi$
$954$	0	0
$955$	50.6705	1.63966
$956$	0	0
$957$	−10.6698	−0.344907
$958$	0	0
$959$	−12.4640	−0.402484
$960$	0	0
$961$	−19.9743	−0.644332
$962$	0	0
$963$	55.9030	1.80145
$964$	0	0
$965$	−46.3848	−1.49318
$966$	0	0
$967$	38.4901	1.23776	0.618879	−	0.785486i	$-0.287587\pi$
$967$	38.4901	1.23776	0.618879	+	0.785486i	$0.287587\pi$
$968$	0	0
$969$	−55.1286	−1.77099
$970$	0	0
$971$	11.1981	0.359365	0.179682	−	0.983725i	$-0.442493\pi$
$971$	11.1981	0.359365	0.179682	+	0.983725i	$0.442493\pi$
$972$	0	0
$973$	−6.51855	−0.208975
$974$	0	0
$975$	52.3045	1.67508
$976$	0	0
$977$	45.5724	1.45799	0.728996	−	0.684518i	$-0.239987\pi$
$977$	45.5724	1.45799	0.728996	+	0.684518i	$0.239987\pi$
$978$	0	0
$979$	0.575273	0.0183858
$980$	0	0
$981$	−52.8086	−1.68605
$982$	0	0
$983$	−53.6365	−1.71074	−0.855370	−	0.518018i	$-0.826670\pi$
$983$	−53.6365	−1.71074	−0.855370	+	0.518018i	$0.826670\pi$
$984$	0	0
$985$	40.8893	1.30284
$986$	0	0
$987$	5.32664	0.169549
$988$	0	0
$989$	−38.6809	−1.22998
$990$	0	0
$991$	−27.2527	−0.865710	−0.432855	−	0.901464i	$-0.642494\pi$
$991$	−27.2527	−0.865710	−0.432855	+	0.901464i	$0.642494\pi$
$992$	0	0
$993$	−92.8937	−2.94789
$994$	0	0
$995$	76.9233	2.43863
$996$	0	0
$997$	33.3821	1.05722	0.528610	−	0.848865i	$-0.322713\pi$
$997$	33.3821	1.05722	0.528610	+	0.848865i	$0.322713\pi$
$998$	0	0
$999$	−18.1277	−0.573535

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2672.2.a.m.1.9		9
4.3	odd	2		1336.2.a.c.1.1	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1336.2.a.c.1.1	✓	9	4.3	odd	2
2672.2.a.m.1.9		9	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2672 = 2^{4} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2672.a (trivial)

\(f(q)\)	\(=\)	\(q+2.70492 q^{3} -4.24392 q^{5} -1.27743 q^{7} +4.31658 q^{9} +O(q^{10})\)	\(q+2.70492 q^{3} -4.24392 q^{5} -1.27743 q^{7} +4.31658 q^{9} +0.403841 q^{11} +1.48621 q^{13} -11.4794 q^{15} +3.32654 q^{17} -6.12675 q^{19} -3.45535 q^{21} +4.20028 q^{23} +13.0108 q^{25} +3.56123 q^{27} -9.76774 q^{29} -3.32050 q^{31} +1.09236 q^{33} +5.42131 q^{35} -5.09030 q^{37} +4.02007 q^{39} -4.16778 q^{41} -9.20912 q^{43} -18.3192 q^{45} -1.54157 q^{47} -5.36817 q^{49} +8.99803 q^{51} -0.932016 q^{53} -1.71387 q^{55} -16.5723 q^{57} +1.70002 q^{59} -9.98119 q^{61} -5.51413 q^{63} -6.30735 q^{65} +4.92232 q^{67} +11.3614 q^{69} -6.72094 q^{71} +7.05470 q^{73} +35.1932 q^{75} -0.515879 q^{77} +0.843450 q^{79} -3.31690 q^{81} -4.04828 q^{83} -14.1176 q^{85} -26.4209 q^{87} +1.42451 q^{89} -1.89853 q^{91} -8.98167 q^{93} +26.0014 q^{95} -14.5450 q^{97} +1.74321 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + q^{3} - 8 q^{5} - 2 q^{7}+O(q^{10}) \) 9 * q + q^3 - 8 * q^5 - 2 * q^7	\( 9 q + q^{3} - 8 q^{5} - 2 q^{7} + 10 q^{11} - 13 q^{13} - 2 q^{15} - 8 q^{17} + q^{19} - 19 q^{21} + 3 q^{23} + 3 q^{25} + 10 q^{27} - 25 q^{29} + q^{31} - 12 q^{33} + 17 q^{35} - 35 q^{37} + 4 q^{39} - 16 q^{41} - 9 q^{43} - 24 q^{45} + q^{47} - q^{49} + 10 q^{51} - 29 q^{53} - 9 q^{55} - 17 q^{57} + 14 q^{59} - 28 q^{61} - 4 q^{63} - 31 q^{65} - 19 q^{67} - 19 q^{69} + 9 q^{71} + 7 q^{75} - 33 q^{77} + 18 q^{79} - 27 q^{81} + 13 q^{83} - 36 q^{85} - 18 q^{87} - 21 q^{89} - 20 q^{91} - 35 q^{93} + 12 q^{95} + 2 q^{97} + 3 q^{99}+O(q^{100}) \) 9 * q + q^3 - 8 * q^5 - 2 * q^7 + 10 * q^11 - 13 * q^13 - 2 * q^15 - 8 * q^17 + q^19 - 19 * q^21 + 3 * q^23 + 3 * q^25 + 10 * q^27 - 25 * q^29 + q^31 - 12 * q^33 + 17 * q^35 - 35 * q^37 + 4 * q^39 - 16 * q^41 - 9 * q^43 - 24 * q^45 + q^47 - q^49 + 10 * q^51 - 29 * q^53 - 9 * q^55 - 17 * q^57 + 14 * q^59 - 28 * q^61 - 4 * q^63 - 31 * q^65 - 19 * q^67 - 19 * q^69 + 9 * q^71 + 7 * q^75 - 33 * q^77 + 18 * q^79 - 27 * q^81 + 13 * q^83 - 36 * q^85 - 18 * q^87 - 21 * q^89 - 20 * q^91 - 35 * q^93 + 12 * q^95 + 2 * q^97 + 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more