LMFDB - Embedded newform 2151.2.a.k.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2151,2,Mod(1,2151)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2151, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2151.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2151 = 3^{2} \cdot 239 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2151.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:	$17.1758214748$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 4 x^{19} - 21 x^{18} + 96 x^{17} + 164 x^{16} - 936 x^{15} - 540 x^{14} + 4804 x^{13} + \cdots + 1 $ x^20 - 4x^19 - 21x^18 + 96x^17 + 164x^16 - 936x^15 - 540x^14 + 4804x^13 + 229x^12 - 14020x^11 + 3356x^10 + 23404x^9 - 9429x^8 - 21252x^7 + 10479x^6 + 9108x^5 - 4844x^4 - 1184x^3 + 640x^2 - 56*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$0.854685$ of defining polynomial
Character	$\chi$	$=$	2151.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.854685 q^{2} -1.26951 q^{4} -2.82017 q^{5} +3.33105 q^{7} -2.79440 q^{8} +O(q^{10})$	\(q+0.854685 q^{2} -1.26951 q^{4} -2.82017 q^{5} +3.33105 q^{7} -2.79440 q^{8} -2.41036 q^{10} -0.210277 q^{11} -4.81635 q^{13} +2.84700 q^{14} +0.150696 q^{16} +4.49202 q^{17} -4.63640 q^{19} +3.58025 q^{20} -0.179720 q^{22} +7.53504 q^{23} +2.95338 q^{25} -4.11646 q^{26} -4.22882 q^{28} +2.29153 q^{29} -7.14674 q^{31} +5.71760 q^{32} +3.83926 q^{34} -9.39416 q^{35} -1.94848 q^{37} -3.96266 q^{38} +7.88071 q^{40} +8.56569 q^{41} -1.06242 q^{43} +0.266949 q^{44} +6.44008 q^{46} +4.78606 q^{47} +4.09593 q^{49} +2.52421 q^{50} +6.11443 q^{52} +4.11306 q^{53} +0.593017 q^{55} -9.30831 q^{56} +1.95853 q^{58} +7.70829 q^{59} +8.53627 q^{61} -6.10820 q^{62} +4.58536 q^{64} +13.5830 q^{65} +16.0563 q^{67} -5.70268 q^{68} -8.02904 q^{70} -5.59508 q^{71} +11.2368 q^{73} -1.66533 q^{74} +5.88597 q^{76} -0.700443 q^{77} -2.49484 q^{79} -0.424988 q^{80} +7.32097 q^{82} +10.5444 q^{83} -12.6683 q^{85} -0.908034 q^{86} +0.587598 q^{88} +4.26861 q^{89} -16.0435 q^{91} -9.56584 q^{92} +4.09057 q^{94} +13.0755 q^{95} +1.01549 q^{97} +3.50072 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 4 q^{2} + 18 q^{4} + 16 q^{5} - 4 q^{7} + 12 q^{8}+O(q^{10}) $ 20 * q + 4 * q^2 + 18 * q^4 + 16 * q^5 - 4 * q^7 + 12 * q^8	$ 20 q + 4 q^{2} + 18 q^{4} + 16 q^{5} - 4 q^{7} + 12 q^{8} + 4 q^{10} + 12 q^{11} - 4 q^{13} + 20 q^{14} + 22 q^{16} + 24 q^{17} - 4 q^{19} + 40 q^{20} - 6 q^{22} + 12 q^{23} + 22 q^{25} + 30 q^{26} - 12 q^{28} + 24 q^{29} - 4 q^{31} + 28 q^{32} + 8 q^{34} + 20 q^{35} - 10 q^{37} + 26 q^{38} + 6 q^{40} + 66 q^{41} + 8 q^{43} + 36 q^{44} - 12 q^{46} + 28 q^{47} + 18 q^{49} + 28 q^{50} - 18 q^{52} + 28 q^{53} - 4 q^{55} + 60 q^{56} + 54 q^{59} - 4 q^{61} + 20 q^{62} + 22 q^{64} + 42 q^{65} + 12 q^{67} + 12 q^{68} + 20 q^{70} + 36 q^{71} + 14 q^{73} - 50 q^{76} + 8 q^{77} - 12 q^{79} + 88 q^{80} - 8 q^{82} + 20 q^{83} + 4 q^{85} + 18 q^{86} - 10 q^{88} + 130 q^{89} - 6 q^{91} - 46 q^{92} - 26 q^{94} - 2 q^{97} + 12 q^{98}+O(q^{100}) $ 20 * q + 4 * q^2 + 18 * q^4 + 16 * q^5 - 4 * q^7 + 12 * q^8 + 4 * q^10 + 12 * q^11 - 4 * q^13 + 20 * q^14 + 22 * q^16 + 24 * q^17 - 4 * q^19 + 40 * q^20 - 6 * q^22 + 12 * q^23 + 22 * q^25 + 30 * q^26 - 12 * q^28 + 24 * q^29 - 4 * q^31 + 28 * q^32 + 8 * q^34 + 20 * q^35 - 10 * q^37 + 26 * q^38 + 6 * q^40 + 66 * q^41 + 8 * q^43 + 36 * q^44 - 12 * q^46 + 28 * q^47 + 18 * q^49 + 28 * q^50 - 18 * q^52 + 28 * q^53 - 4 * q^55 + 60 * q^56 + 54 * q^59 - 4 * q^61 + 20 * q^62 + 22 * q^64 + 42 * q^65 + 12 * q^67 + 12 * q^68 + 20 * q^70 + 36 * q^71 + 14 * q^73 - 50 * q^76 + 8 * q^77 - 12 * q^79 + 88 * q^80 - 8 * q^82 + 20 * q^83 + 4 * q^85 + 18 * q^86 - 10 * q^88 + 130 * q^89 - 6 * q^91 - 46 * q^92 - 26 * q^94 - 2 * q^97 + 12 * q^98

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 $

$2$	0.854685	0.604353	0.302177	−	0.953252i	$-0.402287\pi$
$2$	0.854685	0.604353	0.302177	+	0.953252i	$0.402287\pi$
$3$	0	0
$3$	0	0
$4$	−1.26951	−0.634757
$5$	−2.82017	−1.26122	−0.630610	−	0.776100i	$-0.717195\pi$
$5$	−2.82017	−1.26122	−0.630610	+	0.776100i	$0.717195\pi$
$6$	0	0
$7$	3.33105	1.25902	0.629510	−	0.776992i	$-0.283256\pi$
$7$	3.33105	1.25902	0.629510	+	0.776992i	$0.283256\pi$
$8$	−2.79440	−0.987971
$9$	0	0
$10$	−2.41036	−0.762223
$11$	−0.210277	−0.0634008	−0.0317004	−	0.999497i	$-0.510092\pi$
$11$	−0.210277	−0.0634008	−0.0317004	+	0.999497i	$0.510092\pi$
$12$	0	0
$13$	−4.81635	−1.33582	−0.667908	−	0.744244i	$-0.732810\pi$
$13$	−4.81635	−1.33582	−0.667908	+	0.744244i	$0.732810\pi$
$14$	2.84700	0.760893
$15$	0	0
$16$	0.150696	0.0376739
$17$	4.49202	1.08947	0.544737	−	0.838607i	$-0.316629\pi$
$17$	4.49202	1.08947	0.544737	+	0.838607i	$0.316629\pi$
$18$	0	0
$19$	−4.63640	−1.06366	−0.531831	−	0.846850i	$-0.678496\pi$
$19$	−4.63640	−1.06366	−0.531831	+	0.846850i	$0.678496\pi$
$20$	3.58025	0.800569
$21$	0	0
$22$	−0.179720	−0.0383165
$23$	7.53504	1.57116	0.785582	−	0.618758i	$-0.212364\pi$
$23$	7.53504	1.57116	0.785582	+	0.618758i	$0.212364\pi$
$24$	0	0
$25$	2.95338	0.590677
$26$	−4.11646	−0.807305
$27$	0	0
$28$	−4.22882	−0.799172
$29$	2.29153	0.425526	0.212763	−	0.977104i	$-0.431754\pi$
$29$	2.29153	0.425526	0.212763	+	0.977104i	$0.431754\pi$
$30$	0	0
$31$	−7.14674	−1.28359	−0.641796	−	0.766876i	$-0.721810\pi$
$31$	−7.14674	−1.28359	−0.641796	+	0.766876i	$0.721810\pi$
$32$	5.71760	1.01074
$33$	0	0
$34$	3.83926	0.658427
$35$	−9.39416	−1.58790
$36$	0	0
$37$	−1.94848	−0.320327	−0.160164	−	0.987090i	$-0.551202\pi$
$37$	−1.94848	−0.320327	−0.160164	+	0.987090i	$0.551202\pi$
$38$	−3.96266	−0.642828
$39$	0	0
$40$	7.88071	1.24605
$41$	8.56569	1.33774	0.668868	−	0.743381i	$-0.266779\pi$
$41$	8.56569	1.33774	0.668868	+	0.743381i	$0.266779\pi$
$42$	0	0
$43$	−1.06242	−0.162018	−0.0810088	−	0.996713i	$-0.525814\pi$
$43$	−1.06242	−0.162018	−0.0810088	+	0.996713i	$0.525814\pi$
$44$	0.266949	0.0402441
$45$	0	0
$46$	6.44008	0.949538
$47$	4.78606	0.698119	0.349060	−	0.937101i	$-0.386501\pi$
$47$	4.78606	0.698119	0.349060	+	0.937101i	$0.386501\pi$
$48$	0	0
$49$	4.09593	0.585132
$50$	2.52421	0.356977
$51$	0	0
$52$	6.11443	0.847919
$53$	4.11306	0.564973	0.282486	−	0.959271i	$-0.408841\pi$
$53$	4.11306	0.564973	0.282486	+	0.959271i	$0.408841\pi$
$54$	0	0
$55$	0.593017	0.0799624
$56$	−9.30831	−1.24388
$57$	0	0
$58$	1.95853	0.257168
$59$	7.70829	1.00353	0.501767	−	0.865003i	$-0.332684\pi$
$59$	7.70829	1.00353	0.501767	+	0.865003i	$0.332684\pi$
$60$	0	0
$61$	8.53627	1.09296	0.546479	−	0.837473i	$-0.315968\pi$
$61$	8.53627	1.09296	0.546479	+	0.837473i	$0.315968\pi$
$62$	−6.10820	−0.775743
$63$	0	0
$64$	4.58536	0.573170
$65$	13.5830	1.68476
$66$	0	0
$67$	16.0563	1.96159	0.980793	−	0.195053i	$-0.0624879\pi$
$67$	16.0563	1.96159	0.980793	+	0.195053i	$0.0624879\pi$
$68$	−5.70268	−0.691552
$69$	0	0
$70$	−8.02904	−0.959654
$71$	−5.59508	−0.664014	−0.332007	−	0.943277i	$-0.607726\pi$
$71$	−5.59508	−0.664014	−0.332007	+	0.943277i	$0.607726\pi$
$72$	0	0
$73$	11.2368	1.31517	0.657584	−	0.753381i	$-0.271578\pi$
$73$	11.2368	1.31517	0.657584	+	0.753381i	$0.271578\pi$
$74$	−1.66533	−0.193591
$75$	0	0
$76$	5.88597	0.675168
$77$	−0.700443	−0.0798229
$78$	0	0
$79$	−2.49484	−0.280691	−0.140345	−	0.990103i	$-0.544821\pi$
$79$	−2.49484	−0.280691	−0.140345	+	0.990103i	$0.544821\pi$
$80$	−0.424988	−0.0475151
$81$	0	0
$82$	7.32097	0.808465
$83$	10.5444	1.15740	0.578698	−	0.815542i	$-0.303561\pi$
$83$	10.5444	1.15740	0.578698	+	0.815542i	$0.303561\pi$
$84$	0	0
$85$	−12.6683	−1.37407
$86$	−0.908034	−0.0979158
$87$	0	0
$88$	0.587598	0.0626381
$89$	4.26861	0.452472	0.226236	−	0.974073i	$-0.427358\pi$
$89$	4.26861	0.452472	0.226236	+	0.974073i	$0.427358\pi$
$90$	0	0
$91$	−16.0435	−1.68182
$92$	−9.56584	−0.997308
$93$	0	0
$94$	4.09057	0.421911
$95$	13.0755	1.34151
$96$	0	0
$97$	1.01549	0.103107	0.0515536	−	0.998670i	$-0.483583\pi$
$97$	1.01549	0.103107	0.0515536	+	0.998670i	$0.483583\pi$
$98$	3.50072	0.353627
$99$	0	0
$100$	−3.74936	−0.374936
$101$	−15.6127	−1.55352	−0.776760	−	0.629797i	$-0.783138\pi$
$101$	−15.6127	−1.55352	−0.776760	+	0.629797i	$0.783138\pi$
$102$	0	0
$103$	13.4344	1.32373	0.661864	−	0.749624i	$-0.269766\pi$
$103$	13.4344	1.32373	0.661864	+	0.749624i	$0.269766\pi$
$104$	13.4588	1.31975
$105$	0	0
$106$	3.51537	0.341443
$107$	−0.322242	−0.0311523	−0.0155762	−	0.999879i	$-0.504958\pi$
$107$	−0.322242	−0.0311523	−0.0155762	+	0.999879i	$0.504958\pi$
$108$	0	0
$109$	−10.4286	−0.998881	−0.499440	−	0.866348i	$-0.666461\pi$
$109$	−10.4286	−0.998881	−0.499440	+	0.866348i	$0.666461\pi$
$110$	0.506842	0.0483255
$111$	0	0
$112$	0.501975	0.0474322
$113$	−3.87587	−0.364611	−0.182306	−	0.983242i	$-0.558356\pi$
$113$	−3.87587	−0.364611	−0.182306	+	0.983242i	$0.558356\pi$
$114$	0	0
$115$	−21.2501	−1.98158
$116$	−2.90913	−0.270106
$117$	0	0
$118$	6.58816	0.606489
$119$	14.9632	1.37167
$120$	0	0
$121$	−10.9558	−0.995980
$122$	7.29582	0.660532
$123$	0	0
$124$	9.07288	0.814769
$125$	5.77181	0.516247
$126$	0	0
$127$	10.9530	0.971923	0.485962	−	0.873980i	$-0.338469\pi$
$127$	10.9530	0.971923	0.485962	+	0.873980i	$0.338469\pi$
$128$	−7.51617	−0.664342
$129$	0	0
$130$	11.6091	1.01819
$131$	6.21012	0.542581	0.271290	−	0.962498i	$-0.412550\pi$
$131$	6.21012	0.542581	0.271290	+	0.962498i	$0.412550\pi$
$132$	0	0
$133$	−15.4441	−1.33917
$134$	13.7230	1.18549
$135$	0	0
$136$	−12.5525	−1.07637
$137$	−5.68963	−0.486098	−0.243049	−	0.970014i	$-0.578148\pi$
$137$	−5.68963	−0.486098	−0.243049	+	0.970014i	$0.578148\pi$
$138$	0	0
$139$	21.7100	1.84142	0.920708	−	0.390253i	$-0.127612\pi$
$139$	21.7100	1.84142	0.920708	+	0.390253i	$0.127612\pi$
$140$	11.9260	1.00793
$141$	0	0
$142$	−4.78203	−0.401299
$143$	1.01277	0.0846918
$144$	0	0
$145$	−6.46251	−0.536683
$146$	9.60392	0.794826
$147$	0	0
$148$	2.47362	0.203330
$149$	−1.75560	−0.143825	−0.0719123	−	0.997411i	$-0.522910\pi$
$149$	−1.75560	−0.143825	−0.0719123	+	0.997411i	$0.522910\pi$
$150$	0	0
$151$	−3.96839	−0.322943	−0.161471	−	0.986877i	$-0.551624\pi$
$151$	−3.96839	−0.322943	−0.161471	+	0.986877i	$0.551624\pi$
$152$	12.9560	1.05087
$153$	0	0
$154$	−0.598658	−0.0482412
$155$	20.1550	1.61889
$156$	0	0
$157$	−0.806844	−0.0643932	−0.0321966	−	0.999482i	$-0.510250\pi$
$157$	−0.806844	−0.0643932	−0.0321966	+	0.999482i	$0.510250\pi$
$158$	−2.13230	−0.169636
$159$	0	0
$160$	−16.1246	−1.27476
$161$	25.0996	1.97813
$162$	0	0
$163$	−22.8673	−1.79111	−0.895554	−	0.444953i	$-0.853220\pi$
$163$	−22.8673	−1.79111	−0.895554	+	0.444953i	$0.853220\pi$
$164$	−10.8743	−0.849138
$165$	0	0
$166$	9.01212	0.699476
$167$	11.2810	0.872951	0.436475	−	0.899716i	$-0.356227\pi$
$167$	11.2810	0.872951	0.436475	+	0.899716i	$0.356227\pi$
$168$	0	0
$169$	10.1973	0.784405
$170$	−10.8274	−0.830422
$171$	0	0
$172$	1.34876	0.102842
$173$	12.4886	0.949491	0.474745	−	0.880123i	$-0.342540\pi$
$173$	12.4886	0.949491	0.474745	+	0.880123i	$0.342540\pi$
$174$	0	0
$175$	9.83789	0.743674
$176$	−0.0316878	−0.00238856
$177$	0	0
$178$	3.64831	0.273453
$179$	2.96249	0.221427	0.110714	−	0.993852i	$-0.464686\pi$
$179$	2.96249	0.221427	0.110714	+	0.993852i	$0.464686\pi$
$180$	0	0
$181$	−15.5039	−1.15239	−0.576197	−	0.817311i	$-0.695464\pi$
$181$	−15.5039	−1.15239	−0.576197	+	0.817311i	$0.695464\pi$
$182$	−13.7122	−1.01641
$183$	0	0
$184$	−21.0559	−1.55226
$185$	5.49504	0.404003
$186$	0	0
$187$	−0.944566	−0.0690735
$188$	−6.07598	−0.443136
$189$	0	0
$190$	11.1754	0.810748
$191$	−23.3311	−1.68818	−0.844088	−	0.536205i	$-0.819858\pi$
$191$	−23.3311	−1.68818	−0.844088	+	0.536205i	$0.819858\pi$
$192$	0	0
$193$	−6.69691	−0.482054	−0.241027	−	0.970518i	$-0.577484\pi$
$193$	−6.69691	−0.482054	−0.241027	+	0.970518i	$0.577484\pi$
$194$	0.867921	0.0623131
$195$	0	0
$196$	−5.19984	−0.371417
$197$	−3.29364	−0.234662	−0.117331	−	0.993093i	$-0.537434\pi$
$197$	−3.29364	−0.234662	−0.117331	+	0.993093i	$0.537434\pi$
$198$	0	0
$199$	21.4825	1.52285	0.761427	−	0.648251i	$-0.224499\pi$
$199$	21.4825	1.52285	0.761427	+	0.648251i	$0.224499\pi$
$200$	−8.25295	−0.583572
$201$	0	0
$202$	−13.3439	−0.938874
$203$	7.63321	0.535746
$204$	0	0
$205$	−24.1568	−1.68718
$206$	11.4821	0.799999
$207$	0	0
$208$	−0.725804	−0.0503254
$209$	0.974926	0.0674371
$210$	0	0
$211$	16.6132	1.14370	0.571850	−	0.820358i	$-0.306226\pi$
$211$	16.6132	1.14370	0.571850	+	0.820358i	$0.306226\pi$
$212$	−5.22159	−0.358621
$213$	0	0
$214$	−0.275415	−0.0188270
$215$	2.99621	0.204340
$216$	0	0
$217$	−23.8062	−1.61607
$218$	−8.91318	−0.603677
$219$	0	0
$220$	−0.752844	−0.0507567
$221$	−21.6351	−1.45534
$222$	0	0
$223$	7.93092	0.531094	0.265547	−	0.964098i	$-0.414448\pi$
$223$	7.93092	0.531094	0.265547	+	0.964098i	$0.414448\pi$
$224$	19.0457	1.27254
$225$	0	0
$226$	−3.31265	−0.220354
$227$	28.7427	1.90772	0.953859	−	0.300254i	$-0.0970715\pi$
$227$	28.7427	1.90772	0.953859	+	0.300254i	$0.0970715\pi$
$228$	0	0
$229$	2.04148	0.134905	0.0674523	−	0.997723i	$-0.478513\pi$
$229$	2.04148	0.134905	0.0674523	+	0.997723i	$0.478513\pi$
$230$	−18.1621	−1.19758
$231$	0	0
$232$	−6.40346	−0.420408
$233$	−5.14041	−0.336760	−0.168380	−	0.985722i	$-0.553854\pi$
$233$	−5.14041	−0.336760	−0.168380	+	0.985722i	$0.553854\pi$
$234$	0	0
$235$	−13.4975	−0.880482
$236$	−9.78578	−0.637000
$237$	0	0
$238$	12.7888	0.828973
$239$	1.00000	0.0646846
$239$	1.00000	0.0646846
$240$	0	0
$241$	20.7729	1.33810	0.669049	−	0.743219i	$-0.266702\pi$
$241$	20.7729	1.33810	0.669049	+	0.743219i	$0.266702\pi$
$242$	−9.36374	−0.601924
$243$	0	0
$244$	−10.8369	−0.693763
$245$	−11.5512	−0.737981
$246$	0	0
$247$	22.3305	1.42086
$248$	19.9709	1.26815
$249$	0	0
$250$	4.93308	0.311995
$251$	−20.7731	−1.31119	−0.655593	−	0.755114i	$-0.727581\pi$
$251$	−20.7731	−1.31119	−0.655593	+	0.755114i	$0.727581\pi$
$252$	0	0
$253$	−1.58444	−0.0996131
$254$	9.36138	0.587385
$255$	0	0
$256$	−15.5947	−0.974667
$257$	−10.7431	−0.670134	−0.335067	−	0.942194i	$-0.608759\pi$
$257$	−10.7431	−0.670134	−0.335067	+	0.942194i	$0.608759\pi$
$258$	0	0
$259$	−6.49048	−0.403299
$260$	−17.2438	−1.06941
$261$	0	0
$262$	5.30769	0.327910
$263$	−9.69482	−0.597808	−0.298904	−	0.954283i	$-0.596621\pi$
$263$	−9.69482	−0.597808	−0.298904	+	0.954283i	$0.596621\pi$
$264$	0	0
$265$	−11.5996	−0.712555
$266$	−13.1998	−0.809333
$267$	0	0
$268$	−20.3837	−1.24513
$269$	−18.8231	−1.14766	−0.573832	−	0.818973i	$-0.694544\pi$
$269$	−18.8231	−1.14766	−0.573832	+	0.818973i	$0.694544\pi$
$270$	0	0
$271$	−16.2771	−0.988762	−0.494381	−	0.869245i	$-0.664605\pi$
$271$	−16.2771	−0.988762	−0.494381	+	0.869245i	$0.664605\pi$
$272$	0.676927	0.0410447
$273$	0	0
$274$	−4.86284	−0.293775
$275$	−0.621028	−0.0374494
$276$	0	0
$277$	31.7697	1.90886	0.954429	−	0.298438i	$-0.0964655\pi$
$277$	31.7697	1.90886	0.954429	+	0.298438i	$0.0964655\pi$
$278$	18.5552	1.11287
$279$	0	0
$280$	26.2511	1.56880
$281$	−26.8446	−1.60142	−0.800709	−	0.599054i	$-0.795543\pi$
$281$	−26.8446	−1.60142	−0.800709	+	0.599054i	$0.795543\pi$
$282$	0	0
$283$	−24.2791	−1.44324	−0.721620	−	0.692289i	$-0.756602\pi$
$283$	−24.2791	−1.44324	−0.721620	+	0.692289i	$0.756602\pi$
$284$	7.10303	0.421488
$285$	0	0
$286$	0.865596	0.0511838
$287$	28.5328	1.68424
$288$	0	0
$289$	3.17821	0.186954
$290$	−5.52341	−0.324346
$291$	0	0
$292$	−14.2653	−0.834813
$293$	6.67176	0.389768	0.194884	−	0.980826i	$-0.437567\pi$
$293$	6.67176	0.389768	0.194884	+	0.980826i	$0.437567\pi$
$294$	0	0
$295$	−21.7387	−1.26568
$296$	5.44483	0.316474
$297$	0	0
$298$	−1.50049	−0.0869208
$299$	−36.2914	−2.09879
$300$	0	0
$301$	−3.53898	−0.203983
$302$	−3.39172	−0.195172
$303$	0	0
$304$	−0.698685	−0.0400723
$305$	−24.0738	−1.37846
$306$	0	0
$307$	−6.54889	−0.373765	−0.186882	−	0.982382i	$-0.559838\pi$
$307$	−6.54889	−0.373765	−0.186882	+	0.982382i	$0.559838\pi$
$308$	0.889223	0.0506682
$309$	0	0
$310$	17.2262	0.978383
$311$	1.24119	0.0703815	0.0351907	−	0.999381i	$-0.488796\pi$
$311$	1.24119	0.0703815	0.0351907	+	0.999381i	$0.488796\pi$
$312$	0	0
$313$	−2.96978	−0.167862	−0.0839311	−	0.996472i	$-0.526748\pi$
$313$	−2.96978	−0.167862	−0.0839311	+	0.996472i	$0.526748\pi$
$314$	−0.689597	−0.0389162
$315$	0	0
$316$	3.16723	0.178171
$317$	4.94899	0.277963	0.138982	−	0.990295i	$-0.455617\pi$
$317$	4.94899	0.277963	0.138982	+	0.990295i	$0.455617\pi$
$318$	0	0
$319$	−0.481855	−0.0269787
$320$	−12.9315	−0.722893
$321$	0	0
$322$	21.4523	1.19549
$323$	−20.8268	−1.15883
$324$	0	0
$325$	−14.2245	−0.789036
$326$	−19.5444	−1.08246
$327$	0	0
$328$	−23.9360	−1.32164
$329$	15.9426	0.878946
$330$	0	0
$331$	11.8841	0.653211	0.326605	−	0.945161i	$-0.394095\pi$
$331$	11.8841	0.653211	0.326605	+	0.945161i	$0.394095\pi$
$332$	−13.3862	−0.734665
$333$	0	0
$334$	9.64170	0.527570
$335$	−45.2815	−2.47399
$336$	0	0
$337$	23.8172	1.29741	0.648703	−	0.761042i	$-0.275312\pi$
$337$	23.8172	1.29741	0.648703	+	0.761042i	$0.275312\pi$
$338$	8.71545	0.474058
$339$	0	0
$340$	16.0826	0.872199
$341$	1.50279	0.0813807
$342$	0	0
$343$	−9.67363	−0.522327
$344$	2.96883	0.160069
$345$	0	0
$346$	10.6738	0.573828
$347$	−22.0441	−1.18339	−0.591696	−	0.806162i	$-0.701541\pi$
$347$	−22.0441	−1.18339	−0.591696	+	0.806162i	$0.701541\pi$
$348$	0	0
$349$	−13.3807	−0.716254	−0.358127	−	0.933673i	$-0.616585\pi$
$349$	−13.3807	−0.716254	−0.358127	+	0.933673i	$0.616585\pi$
$350$	8.40829	0.449442
$351$	0	0
$352$	−1.20228	−0.0640817
$353$	32.7460	1.74289	0.871447	−	0.490490i	$-0.163182\pi$
$353$	32.7460	1.74289	0.871447	+	0.490490i	$0.163182\pi$
$354$	0	0
$355$	15.7791	0.837468
$356$	−5.41906	−0.287210
$357$	0	0
$358$	2.53200	0.133820
$359$	5.72619	0.302217	0.151108	−	0.988517i	$-0.451716\pi$
$359$	5.72619	0.302217	0.151108	+	0.988517i	$0.451716\pi$
$360$	0	0
$361$	2.49619	0.131378
$362$	−13.2509	−0.696453
$363$	0	0
$364$	20.3675	1.06755
$365$	−31.6898	−1.65872
$366$	0	0
$367$	5.40110	0.281935	0.140968	−	0.990014i	$-0.454979\pi$
$367$	5.40110	0.281935	0.140968	+	0.990014i	$0.454979\pi$
$368$	1.13550	0.0591919
$369$	0	0
$370$	4.69653	0.244161
$371$	13.7008	0.711312
$372$	0	0
$373$	13.1487	0.680813	0.340407	−	0.940278i	$-0.389435\pi$
$373$	13.1487	0.680813	0.340407	+	0.940278i	$0.389435\pi$
$374$	−0.807306	−0.0417448
$375$	0	0
$376$	−13.3742	−0.689721
$377$	−11.0368	−0.568425
$378$	0	0
$379$	−22.2643	−1.14364	−0.571821	−	0.820379i	$-0.693763\pi$
$379$	−22.2643	−1.14364	−0.571821	+	0.820379i	$0.693763\pi$
$380$	−16.5995	−0.851535
$381$	0	0
$382$	−19.9407	−1.02025
$383$	35.4884	1.81337	0.906686	−	0.421806i	$-0.138604\pi$
$383$	35.4884	1.81337	0.906686	+	0.421806i	$0.138604\pi$
$384$	0	0
$385$	1.97537	0.100674
$386$	−5.72374	−0.291331
$387$	0	0
$388$	−1.28918	−0.0654480
$389$	12.1136	0.614183	0.307091	−	0.951680i	$-0.400644\pi$
$389$	12.1136	0.614183	0.307091	+	0.951680i	$0.400644\pi$
$390$	0	0
$391$	33.8475	1.71174
$392$	−11.4457	−0.578093
$393$	0	0
$394$	−2.81502	−0.141819
$395$	7.03587	0.354013
$396$	0	0
$397$	−13.8215	−0.693681	−0.346840	−	0.937924i	$-0.612745\pi$
$397$	−13.8215	−0.693681	−0.346840	+	0.937924i	$0.612745\pi$
$398$	18.3608	0.920341
$399$	0	0
$400$	0.445062	0.0222531
$401$	10.0053	0.499640	0.249820	−	0.968292i	$-0.419629\pi$
$401$	10.0053	0.499640	0.249820	+	0.968292i	$0.419629\pi$
$402$	0	0
$403$	34.4212	1.71464
$404$	19.8205	0.986107
$405$	0	0
$406$	6.52399	0.323780
$407$	0.409719	0.0203090
$408$	0	0
$409$	−3.05615	−0.151117	−0.0755585	−	0.997141i	$-0.524074\pi$
$409$	−3.05615	−0.151117	−0.0755585	+	0.997141i	$0.524074\pi$
$410$	−20.6464	−1.01965
$411$	0	0
$412$	−17.0551	−0.840246
$413$	25.6767	1.26347
$414$	0	0
$415$	−29.7370	−1.45973
$416$	−27.5380	−1.35016
$417$	0	0
$418$	0.833255	0.0407558
$419$	−12.7158	−0.621209	−0.310605	−	0.950539i	$-0.600531\pi$
$419$	−12.7158	−0.621209	−0.310605	+	0.950539i	$0.600531\pi$
$420$	0	0
$421$	−38.6710	−1.88471	−0.942355	−	0.334616i	$-0.891393\pi$
$421$	−38.6710	−1.88471	−0.942355	+	0.334616i	$0.891393\pi$
$422$	14.1990	0.691199
$423$	0	0
$424$	−11.4936	−0.558177
$425$	13.2667	0.643527
$426$	0	0
$427$	28.4348	1.37606
$428$	0.409091	0.0197742
$429$	0	0
$430$	2.56081	0.123493
$431$	15.2077	0.732531	0.366266	−	0.930510i	$-0.380636\pi$
$431$	15.2077	0.732531	0.366266	+	0.930510i	$0.380636\pi$
$432$	0	0
$433$	−6.68681	−0.321348	−0.160674	−	0.987008i	$-0.551367\pi$
$433$	−6.68681	−0.321348	−0.160674	+	0.987008i	$0.551367\pi$
$434$	−20.3468	−0.976676
$435$	0	0
$436$	13.2393	0.634047
$437$	−34.9354	−1.67119
$438$	0	0
$439$	14.6690	0.700113	0.350057	−	0.936729i	$-0.386162\pi$
$439$	14.6690	0.700113	0.350057	+	0.936729i	$0.386162\pi$
$440$	−1.65713	−0.0790005
$441$	0	0
$442$	−18.4912	−0.879538
$443$	−16.9264	−0.804198	−0.402099	−	0.915596i	$-0.631719\pi$
$443$	−16.9264	−0.804198	−0.402099	+	0.915596i	$0.631719\pi$
$444$	0	0
$445$	−12.0382	−0.570667
$446$	6.77843	0.320968
$447$	0	0
$448$	15.2741	0.721632
$449$	28.4725	1.34370	0.671850	−	0.740687i	$-0.265500\pi$
$449$	28.4725	1.34370	0.671850	+	0.740687i	$0.265500\pi$
$450$	0	0
$451$	−1.80117	−0.0848136
$452$	4.92047	0.231440
$453$	0	0
$454$	24.5659	1.15294
$455$	45.2456	2.12115
$456$	0	0
$457$	−7.17324	−0.335550	−0.167775	−	0.985825i	$-0.553658\pi$
$457$	−7.17324	−0.335550	−0.167775	+	0.985825i	$0.553658\pi$
$458$	1.74482	0.0815300
$459$	0	0
$460$	26.9773	1.25782
$461$	17.1051	0.796666	0.398333	−	0.917241i	$-0.369589\pi$
$461$	17.1051	0.796666	0.398333	+	0.917241i	$0.369589\pi$
$462$	0	0
$463$	10.2474	0.476235	0.238117	−	0.971236i	$-0.423470\pi$
$463$	10.2474	0.476235	0.238117	+	0.971236i	$0.423470\pi$
$464$	0.345323	0.0160312
$465$	0	0
$466$	−4.39343	−0.203522
$467$	31.6949	1.46666	0.733332	−	0.679871i	$-0.237964\pi$
$467$	31.6949	1.46666	0.733332	+	0.679871i	$0.237964\pi$
$468$	0	0
$469$	53.4843	2.46968
$470$	−11.5361	−0.532122
$471$	0	0
$472$	−21.5401	−0.991462
$473$	0.223402	0.0102720
$474$	0	0
$475$	−13.6931	−0.628281
$476$	−18.9959	−0.870677
$477$	0	0
$478$	0.854685	0.0390924
$479$	23.7240	1.08397	0.541987	−	0.840387i	$-0.317672\pi$
$479$	23.7240	1.08397	0.541987	+	0.840387i	$0.317672\pi$
$480$	0	0
$481$	9.38455	0.427899
$482$	17.7542	0.808683
$483$	0	0
$484$	13.9085	0.632206
$485$	−2.86385	−0.130041
$486$	0	0
$487$	18.1842	0.824006	0.412003	−	0.911182i	$-0.364829\pi$
$487$	18.1842	0.824006	0.412003	+	0.911182i	$0.364829\pi$
$488$	−23.8538	−1.07981
$489$	0	0
$490$	−9.87265	−0.446001
$491$	−27.2627	−1.23035	−0.615174	−	0.788392i	$-0.710914\pi$
$491$	−27.2627	−1.23035	−0.615174	+	0.788392i	$0.710914\pi$
$492$	0	0
$493$	10.2936	0.463600
$494$	19.0856	0.858700
$495$	0	0
$496$	−1.07698	−0.0483579
$497$	−18.6375	−0.836007
$498$	0	0
$499$	−8.57191	−0.383732	−0.191866	−	0.981421i	$-0.561454\pi$
$499$	−8.57191	−0.383732	−0.191866	+	0.981421i	$0.561454\pi$
$500$	−7.32740	−0.327691
$501$	0	0
$502$	−17.7544	−0.792420
$503$	−23.6499	−1.05450	−0.527249	−	0.849711i	$-0.676776\pi$
$503$	−23.6499	−1.05450	−0.527249	+	0.849711i	$0.676776\pi$
$504$	0	0
$505$	44.0305	1.95933
$506$	−1.35420	−0.0602015
$507$	0	0
$508$	−13.9050	−0.616935
$509$	25.3730	1.12464	0.562319	−	0.826920i	$-0.309909\pi$
$509$	25.3730	1.12464	0.562319	+	0.826920i	$0.309909\pi$
$510$	0	0
$511$	37.4304	1.65582
$512$	1.70383	0.0752992
$513$	0	0
$514$	−9.18193	−0.404997
$515$	−37.8873	−1.66951
$516$	0	0
$517$	−1.00640	−0.0442613
$518$	−5.54731	−0.243735
$519$	0	0
$520$	−37.9563	−1.66449
$521$	15.8794	0.695688	0.347844	−	0.937553i	$-0.386914\pi$
$521$	15.8794	0.695688	0.347844	+	0.937553i	$0.386914\pi$
$522$	0	0
$523$	16.5175	0.722259	0.361129	−	0.932516i	$-0.382391\pi$
$523$	16.5175	0.722259	0.361129	+	0.932516i	$0.382391\pi$
$524$	−7.88384	−0.344407
$525$	0	0
$526$	−8.28601	−0.361287
$527$	−32.1033	−1.39844
$528$	0	0
$529$	33.7768	1.46856
$530$	−9.91396	−0.430635
$531$	0	0
$532$	19.6065	0.850050
$533$	−41.2554	−1.78697
$534$	0	0
$535$	0.908779	0.0392899
$536$	−44.8677	−1.93799
$537$	0	0
$538$	−16.0878	−0.693594
$539$	−0.861278	−0.0370979
$540$	0	0
$541$	−43.3424	−1.86344	−0.931719	−	0.363181i	$-0.881691\pi$
$541$	−43.3424	−1.86344	−0.931719	+	0.363181i	$0.881691\pi$
$542$	−13.9118	−0.597562
$543$	0	0
$544$	25.6836	1.10117
$545$	29.4105	1.25981
$546$	0	0
$547$	18.6320	0.796648	0.398324	−	0.917245i	$-0.369592\pi$
$547$	18.6320	0.796648	0.398324	+	0.917245i	$0.369592\pi$
$548$	7.22307	0.308554
$549$	0	0
$550$	−0.530783	−0.0226327
$551$	−10.6244	−0.452617
$552$	0	0
$553$	−8.31043	−0.353396
$554$	27.1531	1.15362
$555$	0	0
$556$	−27.5611	−1.16885
$557$	−17.0235	−0.721309	−0.360654	−	0.932699i	$-0.617447\pi$
$557$	−17.0235	−0.721309	−0.360654	+	0.932699i	$0.617447\pi$
$558$	0	0
$559$	5.11699	0.216426
$560$	−1.41566	−0.0598225
$561$	0	0
$562$	−22.9437	−0.967822
$563$	−19.5189	−0.822622	−0.411311	−	0.911495i	$-0.634929\pi$
$563$	−19.5189	−0.822622	−0.411311	+	0.911495i	$0.634929\pi$
$564$	0	0
$565$	10.9306	0.459855
$566$	−20.7509	−0.872227
$567$	0	0
$568$	15.6349	0.656026
$569$	40.5460	1.69978	0.849888	−	0.526964i	$-0.176670\pi$
$569$	40.5460	1.69978	0.849888	+	0.526964i	$0.176670\pi$
$570$	0	0
$571$	−7.45404	−0.311942	−0.155971	−	0.987762i	$-0.549851\pi$
$571$	−7.45404	−0.311942	−0.155971	+	0.987762i	$0.549851\pi$
$572$	−1.28572	−0.0537588
$573$	0	0
$574$	24.3865	1.01787
$575$	22.2539	0.928050
$576$	0	0
$577$	3.39347	0.141272	0.0706360	−	0.997502i	$-0.477497\pi$
$577$	3.39347	0.141272	0.0706360	+	0.997502i	$0.477497\pi$
$578$	2.71637	0.112986
$579$	0	0
$580$	8.20425	0.340663
$581$	35.1239	1.45719
$582$	0	0
$583$	−0.864882	−0.0358197
$584$	−31.4002	−1.29935
$585$	0	0
$586$	5.70225	0.235558
$587$	8.97090	0.370269	0.185134	−	0.982713i	$-0.440728\pi$
$587$	8.97090	0.370269	0.185134	+	0.982713i	$0.440728\pi$
$588$	0	0
$589$	33.1351	1.36531
$590$	−18.5797	−0.764916
$591$	0	0
$592$	−0.293627	−0.0120680
$593$	29.9761	1.23097	0.615486	−	0.788148i	$-0.288960\pi$
$593$	29.9761	1.23097	0.615486	+	0.788148i	$0.288960\pi$
$594$	0	0
$595$	−42.1987	−1.72998
$596$	2.22876	0.0912937
$597$	0	0
$598$	−31.0177	−1.26841
$599$	−33.9162	−1.38578	−0.692890	−	0.721044i	$-0.743663\pi$
$599$	−33.9162	−1.38578	−0.692890	+	0.721044i	$0.743663\pi$
$600$	0	0
$601$	−28.1595	−1.14865	−0.574325	−	0.818628i	$-0.694735\pi$
$601$	−28.1595	−1.14865	−0.574325	+	0.818628i	$0.694735\pi$
$602$	−3.02471	−0.123278
$603$	0	0
$604$	5.03793	0.204990
$605$	30.8972	1.25615
$606$	0	0
$607$	−14.1651	−0.574942	−0.287471	−	0.957789i	$-0.592814\pi$
$607$	−14.1651	−0.574942	−0.287471	+	0.957789i	$0.592814\pi$
$608$	−26.5091	−1.07509
$609$	0	0
$610$	−20.5755	−0.833077
$611$	−23.0514	−0.932559
$612$	0	0
$613$	21.2591	0.858648	0.429324	−	0.903151i	$-0.358752\pi$
$613$	21.2591	0.858648	0.429324	+	0.903151i	$0.358752\pi$
$614$	−5.59724	−0.225886
$615$	0	0
$616$	1.95732	0.0788627
$617$	1.39712	0.0562458	0.0281229	−	0.999604i	$-0.491047\pi$
$617$	1.39712	0.0562458	0.0281229	+	0.999604i	$0.491047\pi$
$618$	0	0
$619$	−7.38678	−0.296900	−0.148450	−	0.988920i	$-0.547428\pi$
$619$	−7.38678	−0.296900	−0.148450	+	0.988920i	$0.547428\pi$
$620$	−25.5871	−1.02760
$621$	0	0
$622$	1.06083	0.0425353
$623$	14.2190	0.569671
$624$	0	0
$625$	−31.0444	−1.24178
$626$	−2.53823	−0.101448
$627$	0	0
$628$	1.02430	0.0408740
$629$	−8.75258	−0.348988
$630$	0	0
$631$	19.0951	0.760166	0.380083	−	0.924952i	$-0.375895\pi$
$631$	19.0951	0.760166	0.380083	+	0.924952i	$0.375895\pi$
$632$	6.97158	0.277314
$633$	0	0
$634$	4.22983	0.167988
$635$	−30.8894	−1.22581
$636$	0	0
$637$	−19.7274	−0.781629
$638$	−0.411834	−0.0163047
$639$	0	0
$640$	21.1969	0.837882
$641$	41.5685	1.64186	0.820929	−	0.571030i	$-0.193456\pi$
$641$	41.5685	1.64186	0.820929	+	0.571030i	$0.193456\pi$
$642$	0	0
$643$	11.4830	0.452845	0.226423	−	0.974029i	$-0.427297\pi$
$643$	11.4830	0.452845	0.226423	+	0.974029i	$0.427297\pi$
$644$	−31.8643	−1.25563
$645$	0	0
$646$	−17.8003	−0.700344
$647$	39.2841	1.54442	0.772209	−	0.635369i	$-0.219152\pi$
$647$	39.2841	1.54442	0.772209	+	0.635369i	$0.219152\pi$
$648$	0	0
$649$	−1.62087	−0.0636249
$650$	−12.1575	−0.476856
$651$	0	0
$652$	29.0304	1.13692
$653$	42.2701	1.65416	0.827079	−	0.562086i	$-0.190001\pi$
$653$	42.2701	1.65416	0.827079	+	0.562086i	$0.190001\pi$
$654$	0	0
$655$	−17.5136	−0.684314
$656$	1.29081	0.0503978
$657$	0	0
$658$	13.6259	0.531194
$659$	−27.8321	−1.08419	−0.542093	−	0.840319i	$-0.682368\pi$
$659$	−27.8321	−1.08419	−0.542093	+	0.840319i	$0.682368\pi$
$660$	0	0
$661$	−38.8101	−1.50954	−0.754769	−	0.655990i	$-0.772251\pi$
$661$	−38.8101	−1.50954	−0.754769	+	0.655990i	$0.772251\pi$
$662$	10.1572	0.394770
$663$	0	0
$664$	−29.4652	−1.14347
$665$	43.5550	1.68899
$666$	0	0
$667$	17.2668	0.668572
$668$	−14.3214	−0.554112
$669$	0	0
$670$	−38.7014	−1.49516
$671$	−1.79498	−0.0692944
$672$	0	0
$673$	−25.1330	−0.968806	−0.484403	−	0.874845i	$-0.660963\pi$
$673$	−25.1330	−0.968806	−0.484403	+	0.874845i	$0.660963\pi$
$674$	20.3562	0.784091
$675$	0	0
$676$	−12.9456	−0.497907
$677$	−40.8531	−1.57011	−0.785055	−	0.619425i	$-0.787366\pi$
$677$	−40.8531	−1.57011	−0.785055	+	0.619425i	$0.787366\pi$
$678$	0	0
$679$	3.38264	0.129814
$680$	35.4003	1.35754
$681$	0	0
$682$	1.28441	0.0491827
$683$	4.28639	0.164014	0.0820070	−	0.996632i	$-0.473867\pi$
$683$	4.28639	0.164014	0.0820070	+	0.996632i	$0.473867\pi$
$684$	0	0
$685$	16.0458	0.613077
$686$	−8.26790	−0.315670
$687$	0	0
$688$	−0.160102	−0.00610383
$689$	−19.8100	−0.754700
$690$	0	0
$691$	44.6869	1.69997	0.849985	−	0.526807i	$-0.176611\pi$
$691$	44.6869	1.69997	0.849985	+	0.526807i	$0.176611\pi$
$692$	−15.8545	−0.602696
$693$	0	0
$694$	−18.8408	−0.715186
$695$	−61.2259	−2.32243
$696$	0	0
$697$	38.4772	1.45743
$698$	−11.4363	−0.432870
$699$	0	0
$700$	−12.4893	−0.472053
$701$	42.8739	1.61932	0.809662	−	0.586896i	$-0.199650\pi$
$701$	42.8739	1.61932	0.809662	+	0.586896i	$0.199650\pi$
$702$	0	0
$703$	9.03391	0.340720
$704$	−0.964194	−0.0363394
$705$	0	0
$706$	27.9875	1.05332
$707$	−52.0067	−1.95591
$708$	0	0
$709$	30.5886	1.14878	0.574389	−	0.818583i	$-0.305240\pi$
$709$	30.5886	1.14878	0.574389	+	0.818583i	$0.305240\pi$
$710$	13.4862	0.506126
$711$	0	0
$712$	−11.9282	−0.447029
$713$	−53.8509	−2.01673
$714$	0	0
$715$	−2.85618	−0.106815
$716$	−3.76093	−0.140553
$717$	0	0
$718$	4.89408	0.182646
$719$	−7.37273	−0.274956	−0.137478	−	0.990505i	$-0.543900\pi$
$719$	−7.37273	−0.274956	−0.137478	+	0.990505i	$0.543900\pi$
$720$	0	0
$721$	44.7506	1.66660
$722$	2.13345	0.0793989
$723$	0	0
$724$	19.6824	0.731490
$725$	6.76777	0.251349
$726$	0	0
$727$	16.3311	0.605687	0.302843	−	0.953040i	$-0.402064\pi$
$727$	16.3311	0.605687	0.302843	+	0.953040i	$0.402064\pi$
$728$	44.8321	1.66159
$729$	0	0
$730$	−27.0847	−1.00245
$731$	−4.77241	−0.176514
$732$	0	0
$733$	−24.3359	−0.898866	−0.449433	−	0.893314i	$-0.648374\pi$
$733$	−24.3359	−0.898866	−0.449433	+	0.893314i	$0.648374\pi$
$734$	4.61624	0.170389
$735$	0	0
$736$	43.0824	1.58804
$737$	−3.37626	−0.124366
$738$	0	0
$739$	−1.89263	−0.0696215	−0.0348107	−	0.999394i	$-0.511083\pi$
$739$	−1.89263	−0.0696215	−0.0348107	+	0.999394i	$0.511083\pi$
$740$	−6.97603	−0.256444
$741$	0	0
$742$	11.7099	0.429884
$743$	51.3232	1.88286	0.941432	−	0.337203i	$-0.109481\pi$
$743$	51.3232	1.88286	0.941432	+	0.337203i	$0.109481\pi$
$744$	0	0
$745$	4.95110	0.181394
$746$	11.2380	0.411452
$747$	0	0
$748$	1.19914	0.0438449
$749$	−1.07341	−0.0392214
$750$	0	0
$751$	−24.9517	−0.910499	−0.455249	−	0.890364i	$-0.650450\pi$
$751$	−24.9517	−0.910499	−0.455249	+	0.890364i	$0.650450\pi$
$752$	0.721239	0.0263009
$753$	0	0
$754$	−9.43300	−0.343530
$755$	11.1915	0.407302
$756$	0	0
$757$	−28.0172	−1.01830	−0.509151	−	0.860677i	$-0.670041\pi$
$757$	−28.0172	−1.01830	−0.509151	+	0.860677i	$0.670041\pi$
$758$	−19.0290	−0.691163
$759$	0	0
$760$	−36.5381	−1.32538
$761$	−18.8832	−0.684514	−0.342257	−	0.939606i	$-0.611191\pi$
$761$	−18.8832	−0.684514	−0.342257	+	0.939606i	$0.611191\pi$
$762$	0	0
$763$	−34.7383	−1.25761
$764$	29.6191	1.07158
$765$	0	0
$766$	30.3314	1.09592
$767$	−37.1259	−1.34054
$768$	0	0
$769$	28.6249	1.03224	0.516121	−	0.856516i	$-0.327376\pi$
$769$	28.6249	1.03224	0.516121	+	0.856516i	$0.327376\pi$
$770$	1.68832	0.0608428
$771$	0	0
$772$	8.50182	0.305987
$773$	−23.0387	−0.828643	−0.414322	−	0.910131i	$-0.635981\pi$
$773$	−23.0387	−0.828643	−0.414322	+	0.910131i	$0.635981\pi$
$774$	0	0
$775$	−21.1071	−0.758188
$776$	−2.83768	−0.101867
$777$	0	0
$778$	10.3533	0.371183
$779$	−39.7140	−1.42290
$780$	0	0
$781$	1.17651	0.0420990
$782$	28.9289	1.03450
$783$	0	0
$784$	0.617238	0.0220442
$785$	2.27544	0.0812140
$786$	0	0
$787$	14.1619	0.504816	0.252408	−	0.967621i	$-0.418777\pi$
$787$	14.1619	0.504816	0.252408	+	0.967621i	$0.418777\pi$
$788$	4.18132	0.148953
$789$	0	0
$790$	6.01345	0.213949
$791$	−12.9107	−0.459053
$792$	0	0
$793$	−41.1137	−1.45999
$794$	−11.8130	−0.419228
$795$	0	0
$796$	−27.2723	−0.966642
$797$	−20.7689	−0.735671	−0.367836	−	0.929891i	$-0.619901\pi$
$797$	−20.7689	−0.735671	−0.367836	+	0.929891i	$0.619901\pi$
$798$	0	0
$799$	21.4991	0.760583
$800$	16.8863	0.597020
$801$	0	0
$802$	8.55136	0.301959
$803$	−2.36284	−0.0833828
$804$	0	0
$805$	−70.7853	−2.49485
$806$	29.4193	1.03625
$807$	0	0
$808$	43.6281	1.53483
$809$	−11.5690	−0.406746	−0.203373	−	0.979101i	$-0.565190\pi$
$809$	−11.5690	−0.406746	−0.203373	+	0.979101i	$0.565190\pi$
$810$	0	0
$811$	−33.8449	−1.18846	−0.594228	−	0.804296i	$-0.702542\pi$
$811$	−33.8449	−1.18846	−0.594228	+	0.804296i	$0.702542\pi$
$812$	−9.69047	−0.340069
$813$	0	0
$814$	0.350180	0.0122738
$815$	64.4899	2.25898
$816$	0	0
$817$	4.92580	0.172332
$818$	−2.61205	−0.0913281
$819$	0	0
$820$	30.6673	1.07095
$821$	42.0603	1.46791	0.733957	−	0.679196i	$-0.237671\pi$
$821$	42.0603	1.46791	0.733957	+	0.679196i	$0.237671\pi$
$822$	0	0
$823$	−0.495945	−0.0172876	−0.00864379	−	0.999963i	$-0.502751\pi$
$823$	−0.495945	−0.0172876	−0.00864379	+	0.999963i	$0.502751\pi$
$824$	−37.5410	−1.30780
$825$	0	0
$826$	21.9455	0.763582
$827$	−35.9757	−1.25100	−0.625498	−	0.780226i	$-0.715104\pi$
$827$	−35.9757	−1.25100	−0.625498	+	0.780226i	$0.715104\pi$
$828$	0	0
$829$	−55.6039	−1.93120	−0.965601	−	0.260028i	$-0.916268\pi$
$829$	−55.6039	−1.93120	−0.965601	+	0.260028i	$0.916268\pi$
$830$	−25.4157	−0.882193
$831$	0	0
$832$	−22.0847	−0.765649
$833$	18.3990	0.637486
$834$	0	0
$835$	−31.8144	−1.10098
$836$	−1.23768	−0.0428062
$837$	0	0
$838$	−10.8680	−0.375430
$839$	0.160571	0.00554352	0.00277176	−	0.999996i	$-0.499118\pi$
$839$	0.160571	0.00554352	0.00277176	+	0.999996i	$0.499118\pi$
$840$	0	0
$841$	−23.7489	−0.818927
$842$	−33.0515	−1.13903
$843$	0	0
$844$	−21.0907	−0.725972
$845$	−28.7581	−0.989308
$846$	0	0
$847$	−36.4943	−1.25396
$848$	0.619821	0.0212847
$849$	0	0
$850$	11.3388	0.388918
$851$	−14.6818	−0.503287
$852$	0	0
$853$	25.5042	0.873246	0.436623	−	0.899645i	$-0.356174\pi$
$853$	25.5042	0.873246	0.436623	+	0.899645i	$0.356174\pi$
$854$	24.3028	0.831624
$855$	0	0
$856$	0.900474	0.0307776
$857$	50.9080	1.73899	0.869493	−	0.493946i	$-0.164446\pi$
$857$	50.9080	1.73899	0.869493	+	0.493946i	$0.164446\pi$
$858$	0	0
$859$	18.4338	0.628955	0.314477	−	0.949265i	$-0.398171\pi$
$859$	18.4338	0.628955	0.314477	+	0.949265i	$0.398171\pi$
$860$	−3.80373	−0.129706
$861$	0	0
$862$	12.9978	0.442708
$863$	31.1929	1.06182	0.530909	−	0.847429i	$-0.321851\pi$
$863$	31.1929	1.06182	0.530909	+	0.847429i	$0.321851\pi$
$864$	0	0
$865$	−35.2200	−1.19752
$866$	−5.71511	−0.194207
$867$	0	0
$868$	30.2223	1.02581
$869$	0.524606	0.0177960
$870$	0	0
$871$	−77.3327	−2.62032
$872$	29.1418	0.986865
$873$	0	0
$874$	−29.8588	−1.00999
$875$	19.2262	0.649965
$876$	0	0
$877$	18.1105	0.611548	0.305774	−	0.952104i	$-0.401085\pi$
$877$	18.1105	0.611548	0.305774	+	0.952104i	$0.401085\pi$
$878$	12.5374	0.423116
$879$	0	0
$880$	0.0893651	0.00301250
$881$	25.5190	0.859758	0.429879	−	0.902887i	$-0.358556\pi$
$881$	25.5190	0.859758	0.429879	+	0.902887i	$0.358556\pi$
$882$	0	0
$883$	27.9850	0.941770	0.470885	−	0.882195i	$-0.343935\pi$
$883$	27.9850	0.941770	0.470885	+	0.882195i	$0.343935\pi$
$884$	27.4661	0.923786
$885$	0	0
$886$	−14.4667	−0.486020
$887$	36.2906	1.21852	0.609260	−	0.792971i	$-0.291467\pi$
$887$	36.2906	1.21852	0.609260	+	0.792971i	$0.291467\pi$
$888$	0	0
$889$	36.4851	1.22367
$890$	−10.2889	−0.344884
$891$	0	0
$892$	−10.0684	−0.337115
$893$	−22.1901	−0.742563
$894$	0	0
$895$	−8.35475	−0.279269
$896$	−25.0368	−0.836420
$897$	0	0
$898$	24.3350	0.812069
$899$	−16.3770	−0.546202
$900$	0	0
$901$	18.4760	0.615523
$902$	−1.53943	−0.0512574
$903$	0	0
$904$	10.8307	0.360225
$905$	43.7236	1.45342
$906$	0	0
$907$	47.3395	1.57188	0.785941	−	0.618301i	$-0.212179\pi$
$907$	47.3395	1.57188	0.785941	+	0.618301i	$0.212179\pi$
$908$	−36.4892	−1.21094
$909$	0	0
$910$	38.6707	1.28192
$911$	−19.7190	−0.653319	−0.326660	−	0.945142i	$-0.605923\pi$
$911$	−19.7190	−0.653319	−0.326660	+	0.945142i	$0.605923\pi$
$912$	0	0
$913$	−2.21724	−0.0733798
$914$	−6.13086	−0.202791
$915$	0	0
$916$	−2.59168	−0.0856317
$917$	20.6862	0.683120
$918$	0	0
$919$	−11.8733	−0.391665	−0.195833	−	0.980637i	$-0.562741\pi$
$919$	−11.8733	−0.391665	−0.195833	+	0.980637i	$0.562741\pi$
$920$	59.3814	1.95775
$921$	0	0
$922$	14.6195	0.481468
$923$	26.9479	0.887000
$924$	0	0
$925$	−5.75460	−0.189210
$926$	8.75825	0.287814
$927$	0	0
$928$	13.1021	0.430096
$929$	31.5247	1.03429	0.517146	−	0.855897i	$-0.326994\pi$
$929$	31.5247	1.03429	0.517146	+	0.855897i	$0.326994\pi$
$930$	0	0
$931$	−18.9903	−0.622383
$932$	6.52583	0.213761
$933$	0	0
$934$	27.0891	0.886383
$935$	2.66384	0.0871170
$936$	0	0
$937$	7.82762	0.255717	0.127859	−	0.991792i	$-0.459190\pi$
$937$	7.82762	0.255717	0.127859	+	0.991792i	$0.459190\pi$
$938$	45.7122	1.49256
$939$	0	0
$940$	17.1353	0.558892
$941$	39.5843	1.29041	0.645206	−	0.764009i	$-0.276772\pi$
$941$	39.5843	1.29041	0.645206	+	0.764009i	$0.276772\pi$
$942$	0	0
$943$	64.5428	2.10180
$944$	1.16161	0.0378070
$945$	0	0
$946$	0.190938	0.00620794
$947$	−20.0897	−0.652828	−0.326414	−	0.945227i	$-0.605840\pi$
$947$	−20.0897	−0.652828	−0.326414	+	0.945227i	$0.605840\pi$
$948$	0	0
$949$	−54.1204	−1.75682
$950$	−11.7033	−0.379704
$951$	0	0
$952$	−41.8131	−1.35517
$953$	21.7639	0.705003	0.352501	−	0.935811i	$-0.385331\pi$
$953$	21.7639	0.705003	0.352501	+	0.935811i	$0.385331\pi$
$954$	0	0
$955$	65.7977	2.12916
$956$	−1.26951	−0.0410590
$957$	0	0
$958$	20.2765	0.655104
$959$	−18.9525	−0.612007
$960$	0	0
$961$	20.0758	0.647607
$962$	8.02083	0.258602
$963$	0	0
$964$	−26.3714	−0.849367
$965$	18.8864	0.607976
$966$	0	0
$967$	−6.19414	−0.199190	−0.0995950	−	0.995028i	$-0.531755\pi$
$967$	−6.19414	−0.199190	−0.0995950	+	0.995028i	$0.531755\pi$
$968$	30.6149	0.983999
$969$	0	0
$970$	−2.44769	−0.0785906
$971$	39.2831	1.26065	0.630327	−	0.776330i	$-0.282921\pi$
$971$	39.2831	1.26065	0.630327	+	0.776330i	$0.282921\pi$
$972$	0	0
$973$	72.3171	2.31838
$974$	15.5418	0.497991
$975$	0	0
$976$	1.28638	0.0411760
$977$	23.6188	0.755633	0.377817	−	0.925880i	$-0.376675\pi$
$977$	23.6188	0.755633	0.377817	+	0.925880i	$0.376675\pi$
$978$	0	0
$979$	−0.897589	−0.0286871
$980$	14.6644	0.468439
$981$	0	0
$982$	−23.3010	−0.743564
$983$	−50.3413	−1.60564	−0.802819	−	0.596223i	$-0.796668\pi$
$983$	−50.3413	−1.60564	−0.802819	+	0.596223i	$0.796668\pi$
$984$	0	0
$985$	9.28863	0.295960
$986$	8.79777	0.280178
$987$	0	0
$988$	−28.3489	−0.901900
$989$	−8.00537	−0.254556
$990$	0	0
$991$	17.5107	0.556247	0.278123	−	0.960545i	$-0.410288\pi$
$991$	17.5107	0.556247	0.278123	+	0.960545i	$0.410288\pi$
$992$	−40.8622	−1.29738
$993$	0	0
$994$	−15.9292	−0.505243
$995$	−60.5844	−1.92065
$996$	0	0
$997$	19.0698	0.603945	0.301973	−	0.953317i	$-0.402355\pi$
$997$	19.0698	0.603945	0.301973	+	0.953317i	$0.402355\pi$
$998$	−7.32628	−0.231909
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.2.a.k.1.12	yes	20
3.2	odd	2		2151.2.a.j.1.9	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2151.2.a.j.1.9	✓	20	3.2	odd	2
2151.2.a.k.1.12	yes	20	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:	\( N \)	\(=\)	\( 2151 = 3^{2} \cdot 239 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2151.a (trivial)

\(f(q)\)	\(=\)	\(q+0.854685 q^{2} -1.26951 q^{4} -2.82017 q^{5} +3.33105 q^{7} -2.79440 q^{8} +O(q^{10})\)	\(q+0.854685 q^{2} -1.26951 q^{4} -2.82017 q^{5} +3.33105 q^{7} -2.79440 q^{8} -2.41036 q^{10} -0.210277 q^{11} -4.81635 q^{13} +2.84700 q^{14} +0.150696 q^{16} +4.49202 q^{17} -4.63640 q^{19} +3.58025 q^{20} -0.179720 q^{22} +7.53504 q^{23} +2.95338 q^{25} -4.11646 q^{26} -4.22882 q^{28} +2.29153 q^{29} -7.14674 q^{31} +5.71760 q^{32} +3.83926 q^{34} -9.39416 q^{35} -1.94848 q^{37} -3.96266 q^{38} +7.88071 q^{40} +8.56569 q^{41} -1.06242 q^{43} +0.266949 q^{44} +6.44008 q^{46} +4.78606 q^{47} +4.09593 q^{49} +2.52421 q^{50} +6.11443 q^{52} +4.11306 q^{53} +0.593017 q^{55} -9.30831 q^{56} +1.95853 q^{58} +7.70829 q^{59} +8.53627 q^{61} -6.10820 q^{62} +4.58536 q^{64} +13.5830 q^{65} +16.0563 q^{67} -5.70268 q^{68} -8.02904 q^{70} -5.59508 q^{71} +11.2368 q^{73} -1.66533 q^{74} +5.88597 q^{76} -0.700443 q^{77} -2.49484 q^{79} -0.424988 q^{80} +7.32097 q^{82} +10.5444 q^{83} -12.6683 q^{85} -0.908034 q^{86} +0.587598 q^{88} +4.26861 q^{89} -16.0435 q^{91} -9.56584 q^{92} +4.09057 q^{94} +13.0755 q^{95} +1.01549 q^{97} +3.50072 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 4 q^{2} + 18 q^{4} + 16 q^{5} - 4 q^{7} + 12 q^{8}+O(q^{10}) \) 20 * q + 4 * q^2 + 18 * q^4 + 16 * q^5 - 4 * q^7 + 12 * q^8	\( 20 q + 4 q^{2} + 18 q^{4} + 16 q^{5} - 4 q^{7} + 12 q^{8} + 4 q^{10} + 12 q^{11} - 4 q^{13} + 20 q^{14} + 22 q^{16} + 24 q^{17} - 4 q^{19} + 40 q^{20} - 6 q^{22} + 12 q^{23} + 22 q^{25} + 30 q^{26} - 12 q^{28} + 24 q^{29} - 4 q^{31} + 28 q^{32} + 8 q^{34} + 20 q^{35} - 10 q^{37} + 26 q^{38} + 6 q^{40} + 66 q^{41} + 8 q^{43} + 36 q^{44} - 12 q^{46} + 28 q^{47} + 18 q^{49} + 28 q^{50} - 18 q^{52} + 28 q^{53} - 4 q^{55} + 60 q^{56} + 54 q^{59} - 4 q^{61} + 20 q^{62} + 22 q^{64} + 42 q^{65} + 12 q^{67} + 12 q^{68} + 20 q^{70} + 36 q^{71} + 14 q^{73} - 50 q^{76} + 8 q^{77} - 12 q^{79} + 88 q^{80} - 8 q^{82} + 20 q^{83} + 4 q^{85} + 18 q^{86} - 10 q^{88} + 130 q^{89} - 6 q^{91} - 46 q^{92} - 26 q^{94} - 2 q^{97} + 12 q^{98}+O(q^{100}) \) 20 * q + 4 * q^2 + 18 * q^4 + 16 * q^5 - 4 * q^7 + 12 * q^8 + 4 * q^10 + 12 * q^11 - 4 * q^13 + 20 * q^14 + 22 * q^16 + 24 * q^17 - 4 * q^19 + 40 * q^20 - 6 * q^22 + 12 * q^23 + 22 * q^25 + 30 * q^26 - 12 * q^28 + 24 * q^29 - 4 * q^31 + 28 * q^32 + 8 * q^34 + 20 * q^35 - 10 * q^37 + 26 * q^38 + 6 * q^40 + 66 * q^41 + 8 * q^43 + 36 * q^44 - 12 * q^46 + 28 * q^47 + 18 * q^49 + 28 * q^50 - 18 * q^52 + 28 * q^53 - 4 * q^55 + 60 * q^56 + 54 * q^59 - 4 * q^61 + 20 * q^62 + 22 * q^64 + 42 * q^65 + 12 * q^67 + 12 * q^68 + 20 * q^70 + 36 * q^71 + 14 * q^73 - 50 * q^76 + 8 * q^77 - 12 * q^79 + 88 * q^80 - 8 * q^82 + 20 * q^83 + 4 * q^85 + 18 * q^86 - 10 * q^88 + 130 * q^89 - 6 * q^91 - 46 * q^92 - 26 * q^94 - 2 * q^97 + 12 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more