LMFDB - Embedded newform 669.2.a.g.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [669,2,Mod(1,669)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("669.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(669, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 669 = 3 \cdot 223 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	669.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,-1,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$5.34199189522$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.17009$ of defining polynomial
Character	$\chi$	$=$	669.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.17009 q^{2} +1.00000 q^{3} +2.70928 q^{4} +0.170086 q^{5} -2.17009 q^{6} -3.70928 q^{7} -1.53919 q^{8} +1.00000 q^{9} -0.369102 q^{10} +0.630898 q^{11} +2.70928 q^{12} -0.290725 q^{13} +8.04945 q^{14} +0.170086 q^{15} -2.07838 q^{16} +4.78765 q^{17} -2.17009 q^{18} -7.70928 q^{19} +0.460811 q^{20} -3.70928 q^{21} -1.36910 q^{22} -6.04945 q^{23} -1.53919 q^{24} -4.97107 q^{25} +0.630898 q^{26} +1.00000 q^{27} -10.0494 q^{28} +5.07838 q^{29} -0.369102 q^{30} -2.29072 q^{31} +7.58864 q^{32} +0.630898 q^{33} -10.3896 q^{34} -0.630898 q^{35} +2.70928 q^{36} +9.83710 q^{37} +16.7298 q^{38} -0.290725 q^{39} -0.261795 q^{40} -11.6020 q^{41} +8.04945 q^{42} -2.44748 q^{43} +1.70928 q^{44} +0.170086 q^{45} +13.1278 q^{46} -10.2195 q^{47} -2.07838 q^{48} +6.75872 q^{49} +10.7877 q^{50} +4.78765 q^{51} -0.787653 q^{52} +7.89269 q^{53} -2.17009 q^{54} +0.107307 q^{55} +5.70928 q^{56} -7.70928 q^{57} -11.0205 q^{58} -10.1568 q^{59} +0.460811 q^{60} -13.7587 q^{61} +4.97107 q^{62} -3.70928 q^{63} -12.3112 q^{64} -0.0494483 q^{65} -1.36910 q^{66} +4.26180 q^{67} +12.9711 q^{68} -6.04945 q^{69} +1.36910 q^{70} +10.3402 q^{71} -1.53919 q^{72} -12.2618 q^{73} -21.3474 q^{74} -4.97107 q^{75} -20.8865 q^{76} -2.34017 q^{77} +0.630898 q^{78} -5.38962 q^{79} -0.353504 q^{80} +1.00000 q^{81} +25.1773 q^{82} -11.2979 q^{83} -10.0494 q^{84} +0.814315 q^{85} +5.31124 q^{86} +5.07838 q^{87} -0.971071 q^{88} -10.8371 q^{89} -0.369102 q^{90} +1.07838 q^{91} -16.3896 q^{92} -2.29072 q^{93} +22.1773 q^{94} -1.31124 q^{95} +7.58864 q^{96} +4.09890 q^{97} -14.6670 q^{98} +0.630898 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + 3 q^{3} + q^{4} - 5 q^{5} - q^{6} - 4 q^{7} - 3 q^{8} + 3 q^{9} - 5 q^{10} - 2 q^{11} + q^{12} - 8 q^{13} + 6 q^{14} - 5 q^{15} - 3 q^{16} + 4 q^{17} - q^{18} - 16 q^{19} + 3 q^{20} - 4 q^{21}+ \cdots - 2 q^{99}+O(q^{100}) $ 3 * q - q^2 + 3 * q^3 + q^4 - 5 * q^5 - q^6 - 4 * q^7 - 3 * q^8 + 3 * q^9 - 5 * q^10 - 2 * q^11 + q^12 - 8 * q^13 + 6 * q^14 - 5 * q^15 - 3 * q^16 + 4 * q^17 - q^18 - 16 * q^19 + 3 * q^20 - 4 * q^21 - 8 * q^22 - 3 * q^24 - 2 * q^26 + 3 * q^27 - 12 * q^28 + 12 * q^29 - 5 * q^30 - 14 * q^31 + 3 * q^32 - 2 * q^33 - 2 * q^34 + 2 * q^35 + q^36 + q^37 + 10 * q^38 - 8 * q^39 + 7 * q^40 - 16 * q^41 + 6 * q^42 - 8 * q^43 - 2 * q^44 - 5 * q^45 + 18 * q^46 - 7 * q^47 - 3 * q^48 - 5 * q^49 + 22 * q^50 + 4 * q^51 + 8 * q^52 + 12 * q^53 - q^54 + 12 * q^55 + 10 * q^56 - 16 * q^57 - 24 * q^59 + 3 * q^60 - 16 * q^61 - 4 * q^63 - 11 * q^64 + 18 * q^65 - 8 * q^66 + 5 * q^67 + 24 * q^68 + 8 * q^70 + 20 * q^71 - 3 * q^72 - 29 * q^73 - 19 * q^74 - 16 * q^76 + 4 * q^77 - 2 * q^78 + 13 * q^79 + 9 * q^80 + 3 * q^81 + 36 * q^82 - 7 * q^83 - 12 * q^84 - 6 * q^85 - 10 * q^86 + 12 * q^87 + 12 * q^88 - 4 * q^89 - 5 * q^90 - 20 * q^92 - 14 * q^93 + 27 * q^94 + 22 * q^95 + 3 * q^96 - 24 * q^97 - 21 * q^98 - 2 * q^99

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 $

$2$	−2.17009	−1.53448	−0.767241	−	0.641358i	$-0.778371\pi$
$2$	−2.17009	−1.53448	−0.767241	+	0.641358i	$0.778371\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	2.70928	1.35464
$5$	0.170086	0.0760650	0.0380325	−	0.999277i	$-0.487891\pi$
$5$	0.170086	0.0760650	0.0380325	+	0.999277i	$0.487891\pi$
$6$	−2.17009	−0.885934
$7$	−3.70928	−1.40197	−0.700987	−	0.713174i	$-0.747257\pi$
$7$	−3.70928	−1.40197	−0.700987	+	0.713174i	$0.747257\pi$
$8$	−1.53919	−0.544185
$9$	1.00000	0.333333
$10$	−0.369102	−0.116720
$11$	0.630898	0.190223	0.0951114	−	0.995467i	$-0.469679\pi$
$11$	0.630898	0.190223	0.0951114	+	0.995467i	$0.469679\pi$
$12$	2.70928	0.782100
$13$	−0.290725	−0.0806325	−0.0403163	−	0.999187i	$-0.512837\pi$
$13$	−0.290725	−0.0806325	−0.0403163	+	0.999187i	$0.512837\pi$
$14$	8.04945	2.15131
$15$	0.170086	0.0439161
$16$	−2.07838	−0.519594
$17$	4.78765	1.16118	0.580588	−	0.814197i	$-0.302823\pi$
$17$	4.78765	1.16118	0.580588	+	0.814197i	$0.302823\pi$
$18$	−2.17009	−0.511494
$19$	−7.70928	−1.76863	−0.884315	−	0.466892i	$-0.845374\pi$
$19$	−7.70928	−1.76863	−0.884315	+	0.466892i	$0.845374\pi$
$20$	0.460811	0.103041
$21$	−3.70928	−0.809430
$22$	−1.36910	−0.291894
$23$	−6.04945	−1.26140	−0.630699	−	0.776028i	$-0.717232\pi$
$23$	−6.04945	−1.26140	−0.630699	+	0.776028i	$0.717232\pi$
$24$	−1.53919	−0.314186
$25$	−4.97107	−0.994214
$26$	0.630898	0.123729
$27$	1.00000	0.192450
$28$	−10.0494	−1.89917
$29$	5.07838	0.943031	0.471516	−	0.881858i	$-0.343707\pi$
$29$	5.07838	0.943031	0.471516	+	0.881858i	$0.343707\pi$
$30$	−0.369102	−0.0673886
$31$	−2.29072	−0.411426	−0.205713	−	0.978612i	$-0.565951\pi$
$31$	−2.29072	−0.411426	−0.205713	+	0.978612i	$0.565951\pi$
$32$	7.58864	1.34149
$33$	0.630898	0.109825
$34$	−10.3896	−1.78181
$35$	−0.630898	−0.106641
$36$	2.70928	0.451546
$37$	9.83710	1.61721	0.808605	−	0.588352i	$-0.200223\pi$
$37$	9.83710	1.61721	0.808605	+	0.588352i	$0.200223\pi$
$38$	16.7298	2.71393
$39$	−0.290725	−0.0465532
$40$	−0.261795	−0.0413935
$41$	−11.6020	−1.81192	−0.905962	−	0.423360i	$-0.860851\pi$
$41$	−11.6020	−1.81192	−0.905962	+	0.423360i	$0.860851\pi$
$42$	8.04945	1.24206
$43$	−2.44748	−0.373237	−0.186619	−	0.982432i	$-0.559753\pi$
$43$	−2.44748	−0.373237	−0.186619	+	0.982432i	$0.559753\pi$
$44$	1.70928	0.257683
$45$	0.170086	0.0253550
$46$	13.1278	1.93559
$47$	−10.2195	−1.49067	−0.745336	−	0.666689i	$-0.767711\pi$
$47$	−10.2195	−1.49067	−0.745336	+	0.666689i	$0.767711\pi$
$48$	−2.07838	−0.299988
$49$	6.75872	0.965532
$50$	10.7877	1.52560
$51$	4.78765	0.670406
$52$	−0.787653	−0.109228
$53$	7.89269	1.08414	0.542072	−	0.840332i	$-0.317640\pi$
$53$	7.89269	1.08414	0.542072	+	0.840332i	$0.317640\pi$
$54$	−2.17009	−0.295311
$55$	0.107307	0.0144693
$56$	5.70928	0.762934
$57$	−7.70928	−1.02112
$58$	−11.0205	−1.44707
$59$	−10.1568	−1.32230	−0.661148	−	0.750255i	$-0.729931\pi$
$59$	−10.1568	−1.32230	−0.661148	+	0.750255i	$0.729931\pi$
$60$	0.460811	0.0594905
$61$	−13.7587	−1.76162	−0.880812	−	0.473466i	$-0.843003\pi$
$61$	−13.7587	−1.76162	−0.880812	+	0.473466i	$0.843003\pi$
$62$	4.97107	0.631327
$63$	−3.70928	−0.467325
$64$	−12.3112	−1.53891
$65$	−0.0494483	−0.00613331
$66$	−1.36910	−0.168525
$67$	4.26180	0.520661	0.260331	−	0.965520i	$-0.416168\pi$
$67$	4.26180	0.520661	0.260331	+	0.965520i	$0.416168\pi$
$68$	12.9711	1.57297
$69$	−6.04945	−0.728268
$70$	1.36910	0.163639
$71$	10.3402	1.22715	0.613576	−	0.789635i	$-0.289730\pi$
$71$	10.3402	1.22715	0.613576	+	0.789635i	$0.289730\pi$
$72$	−1.53919	−0.181395
$73$	−12.2618	−1.43513	−0.717567	−	0.696489i	$-0.754744\pi$
$73$	−12.2618	−1.43513	−0.717567	+	0.696489i	$0.754744\pi$
$74$	−21.3474	−2.48158
$75$	−4.97107	−0.574010
$76$	−20.8865	−2.39585
$77$	−2.34017	−0.266687
$78$	0.630898	0.0714351
$79$	−5.38962	−0.606380	−0.303190	−	0.952930i	$-0.598052\pi$
$79$	−5.38962	−0.606380	−0.303190	+	0.952930i	$0.598052\pi$
$80$	−0.353504	−0.0395229
$81$	1.00000	0.111111
$82$	25.1773	2.78036
$83$	−11.2979	−1.24011	−0.620054	−	0.784560i	$-0.712889\pi$
$83$	−11.2979	−1.24011	−0.620054	+	0.784560i	$0.712889\pi$
$84$	−10.0494	−1.09648
$85$	0.814315	0.0883249
$86$	5.31124	0.572726
$87$	5.07838	0.544459
$88$	−0.971071	−0.103516
$89$	−10.8371	−1.14873	−0.574365	−	0.818599i	$-0.694751\pi$
$89$	−10.8371	−1.14873	−0.574365	+	0.818599i	$0.694751\pi$
$90$	−0.369102	−0.0389068
$91$	1.07838	0.113045
$92$	−16.3896	−1.70874
$93$	−2.29072	−0.237537
$94$	22.1773	2.28741
$95$	−1.31124	−0.134531
$96$	7.58864	0.774512
$97$	4.09890	0.416180	0.208090	−	0.978110i	$-0.433275\pi$
$97$	4.09890	0.416180	0.208090	+	0.978110i	$0.433275\pi$
$98$	−14.6670	−1.48159
$99$	0.630898	0.0634076
$100$	−13.4680	−1.34680
$101$	0.107307	0.0106775	0.00533873	−	0.999986i	$-0.498301\pi$
$101$	0.107307	0.0106775	0.00533873	+	0.999986i	$0.498301\pi$
$102$	−10.3896	−1.02873
$103$	−5.83710	−0.575147	−0.287573	−	0.957759i	$-0.592849\pi$
$103$	−5.83710	−0.575147	−0.287573	+	0.957759i	$0.592849\pi$
$104$	0.447480	0.0438790
$105$	−0.630898	−0.0615693
$106$	−17.1278	−1.66360
$107$	13.1773	1.27390	0.636948	−	0.770907i	$-0.280197\pi$
$107$	13.1773	1.27390	0.636948	+	0.770907i	$0.280197\pi$
$108$	2.70928	0.260700
$109$	−5.04945	−0.483649	−0.241825	−	0.970320i	$-0.577746\pi$
$109$	−5.04945	−0.483649	−0.241825	+	0.970320i	$0.577746\pi$
$110$	−0.232866	−0.0222029
$111$	9.83710	0.933696
$112$	7.70928	0.728458
$113$	20.5730	1.93535	0.967674	−	0.252203i	$-0.0811551\pi$
$113$	20.5730	1.93535	0.967674	+	0.252203i	$0.0811551\pi$
$114$	16.7298	1.56689
$115$	−1.02893	−0.0959482
$116$	13.7587	1.27747
$117$	−0.290725	−0.0268775
$118$	22.0410	2.02904
$119$	−17.7587	−1.62794
$120$	−0.261795	−0.0238985
$121$	−10.6020	−0.963815
$122$	29.8576	2.70318
$123$	−11.6020	−1.04611
$124$	−6.20620	−0.557334
$125$	−1.69594	−0.151690
$126$	8.04945	0.717102
$127$	−12.8371	−1.13911	−0.569554	−	0.821954i	$-0.692884\pi$
$127$	−12.8371	−1.13911	−0.569554	+	0.821954i	$0.692884\pi$
$128$	11.5392	1.01993
$129$	−2.44748	−0.215489
$130$	0.107307	0.00941146
$131$	−9.35577	−0.817418	−0.408709	−	0.912665i	$-0.634021\pi$
$131$	−9.35577	−0.817418	−0.408709	+	0.912665i	$0.634021\pi$
$132$	1.70928	0.148773
$133$	28.5958	2.47957
$134$	−9.24846	−0.798946
$135$	0.170086	0.0146387
$136$	−7.36910	−0.631895
$137$	−6.61757	−0.565377	−0.282688	−	0.959212i	$-0.591226\pi$
$137$	−6.61757	−0.565377	−0.282688	+	0.959212i	$0.591226\pi$
$138$	13.1278	1.11751
$139$	−0.317387	−0.0269204	−0.0134602	−	0.999909i	$-0.504285\pi$
$139$	−0.317387	−0.0269204	−0.0134602	+	0.999909i	$0.504285\pi$
$140$	−1.70928	−0.144460
$141$	−10.2195	−0.860640
$142$	−22.4391	−1.88304
$143$	−0.183417	−0.0153381
$144$	−2.07838	−0.173198
$145$	0.863763	0.0717316
$146$	26.6092	2.20219
$147$	6.75872	0.557450
$148$	26.6514	2.19073
$149$	2.63090	0.215532	0.107766	−	0.994176i	$-0.465630\pi$
$149$	2.63090	0.215532	0.107766	+	0.994176i	$0.465630\pi$
$150$	10.7877	0.880808
$151$	23.7298	1.93110	0.965552	−	0.260212i	$-0.0837923\pi$
$151$	23.7298	1.93110	0.965552	+	0.260212i	$0.0837923\pi$
$152$	11.8660	0.962462
$153$	4.78765	0.387059
$154$	5.07838	0.409227
$155$	−0.389621	−0.0312951
$156$	−0.787653	−0.0630627
$157$	14.7298	1.17557	0.587783	−	0.809019i	$-0.300001\pi$
$157$	14.7298	1.17557	0.587783	+	0.809019i	$0.300001\pi$
$158$	11.6959	0.930479
$159$	7.89269	0.625931
$160$	1.29072	0.102041
$161$	22.4391	1.76845
$162$	−2.17009	−0.170498
$163$	10.4680	0.819917	0.409958	−	0.912104i	$-0.365543\pi$
$163$	10.4680	0.819917	0.409958	+	0.912104i	$0.365543\pi$
$164$	−31.4329	−2.45450
$165$	0.107307	0.00835385
$166$	24.5174	1.90292
$167$	9.60197	0.743023	0.371511	−	0.928428i	$-0.378840\pi$
$167$	9.60197	0.743023	0.371511	+	0.928428i	$0.378840\pi$
$168$	5.70928	0.440480
$169$	−12.9155	−0.993498
$170$	−1.76713	−0.135533
$171$	−7.70928	−0.589543
$172$	−6.63090	−0.505601
$173$	6.77432	0.515042	0.257521	−	0.966273i	$-0.417094\pi$
$173$	6.77432	0.515042	0.257521	+	0.966273i	$0.417094\pi$
$174$	−11.0205	−0.835463
$175$	18.4391	1.39386
$176$	−1.31124	−0.0988387
$177$	−10.1568	−0.763428
$178$	23.5174	1.76271
$179$	−12.6092	−0.942453	−0.471226	−	0.882012i	$-0.656188\pi$
$179$	−12.6092	−0.942453	−0.471226	+	0.882012i	$0.656188\pi$
$180$	0.460811	0.0343468
$181$	10.6514	0.791714	0.395857	−	0.918312i	$-0.370448\pi$
$181$	10.6514	0.791714	0.395857	+	0.918312i	$0.370448\pi$
$182$	−2.34017	−0.173465
$183$	−13.7587	−1.01707
$184$	9.31124	0.686434
$185$	1.67316	0.123013
$186$	4.97107	0.364497
$187$	3.02052	0.220882
$188$	−27.6875	−2.01932
$189$	−3.70928	−0.269810
$190$	2.84551	0.206435
$191$	21.1545	1.53069	0.765343	−	0.643623i	$-0.222570\pi$
$191$	21.1545	1.53069	0.765343	+	0.643623i	$0.222570\pi$
$192$	−12.3112	−0.888487
$193$	1.47641	0.106274	0.0531371	−	0.998587i	$-0.483078\pi$
$193$	1.47641	0.106274	0.0531371	+	0.998587i	$0.483078\pi$
$194$	−8.89496	−0.638621
$195$	−0.0494483	−0.00354107
$196$	18.3112	1.30795
$197$	−1.05559	−0.0752078	−0.0376039	−	0.999293i	$-0.511973\pi$
$197$	−1.05559	−0.0752078	−0.0376039	+	0.999293i	$0.511973\pi$
$198$	−1.36910	−0.0972979
$199$	25.3340	1.79588	0.897941	−	0.440116i	$-0.145063\pi$
$199$	25.3340	1.79588	0.897941	+	0.440116i	$0.145063\pi$
$200$	7.65142	0.541037
$201$	4.26180	0.300604
$202$	−0.232866	−0.0163844
$203$	−18.8371	−1.32211
$204$	12.9711	0.908157
$205$	−1.97334	−0.137824
$206$	12.6670	0.882553
$207$	−6.04945	−0.420466
$208$	0.604236	0.0418962
$209$	−4.86376	−0.336434
$210$	1.36910	0.0944770
$211$	9.49466	0.653639	0.326820	−	0.945087i	$-0.394023\pi$
$211$	9.49466	0.653639	0.326820	+	0.945087i	$0.394023\pi$
$212$	21.3835	1.46862
$213$	10.3402	0.708497
$214$	−28.5958	−1.95477
$215$	−0.416283	−0.0283903
$216$	−1.53919	−0.104729
$217$	8.49693	0.576809
$218$	10.9577	0.742152
$219$	−12.2618	−0.828575
$220$	0.290725	0.0196007
$221$	−1.39189	−0.0936286
$222$	−21.3474	−1.43274
$223$	1.00000	0.0669650
$223$	1.00000	0.0669650
$224$	−28.1483	−1.88074
$225$	−4.97107	−0.331405
$226$	−44.6453	−2.96976
$227$	2.88163	0.191260	0.0956302	−	0.995417i	$-0.469513\pi$
$227$	2.88163	0.191260	0.0956302	+	0.995417i	$0.469513\pi$
$228$	−20.8865	−1.38325
$229$	−7.99159	−0.528099	−0.264050	−	0.964509i	$-0.585058\pi$
$229$	−7.99159	−0.528099	−0.264050	+	0.964509i	$0.585058\pi$
$230$	2.23287	0.147231
$231$	−2.34017	−0.153972
$232$	−7.81658	−0.513184
$233$	−3.12783	−0.204911	−0.102455	−	0.994738i	$-0.532670\pi$
$233$	−3.12783	−0.204911	−0.102455	+	0.994738i	$0.532670\pi$
$234$	0.630898	0.0412431
$235$	−1.73820	−0.113388
$236$	−27.5174	−1.79123
$237$	−5.38962	−0.350094
$238$	38.5380	2.49805
$239$	11.7948	0.762945	0.381472	−	0.924380i	$-0.375417\pi$
$239$	11.7948	0.762945	0.381472	+	0.924380i	$0.375417\pi$
$240$	−0.353504	−0.0228186
$241$	−6.39189	−0.411738	−0.205869	−	0.978580i	$-0.566002\pi$
$241$	−6.39189	−0.411738	−0.205869	+	0.978580i	$0.566002\pi$
$242$	23.0072	1.47896
$243$	1.00000	0.0641500
$244$	−37.2762	−2.38636
$245$	1.14957	0.0734432
$246$	25.1773	1.60524
$247$	2.24128	0.142609
$248$	3.52586	0.223892
$249$	−11.2979	−0.715976
$250$	3.68035	0.232766
$251$	12.0277	0.759182	0.379591	−	0.925154i	$-0.376065\pi$
$251$	12.0277	0.759182	0.379591	+	0.925154i	$0.376065\pi$
$252$	−10.0494	−0.633056
$253$	−3.81658	−0.239946
$254$	27.8576	1.74794
$255$	0.814315	0.0509944
$256$	−0.418551	−0.0261594
$257$	−16.9854	−1.05952	−0.529762	−	0.848147i	$-0.677719\pi$
$257$	−16.9854	−1.05952	−0.529762	+	0.848147i	$0.677719\pi$
$258$	5.31124	0.330664
$259$	−36.4885	−2.26729
$260$	−0.133969	−0.00830841
$261$	5.07838	0.314344
$262$	20.3028	1.25431
$263$	−24.8287	−1.53100	−0.765501	−	0.643434i	$-0.777509\pi$
$263$	−24.8287	−1.53100	−0.765501	+	0.643434i	$0.777509\pi$
$264$	−0.971071	−0.0597653
$265$	1.34244	0.0824655
$266$	−62.0554	−3.80486
$267$	−10.8371	−0.663220
$268$	11.5464	0.705307
$269$	9.73698	0.593674	0.296837	−	0.954928i	$-0.404068\pi$
$269$	9.73698	0.593674	0.296837	+	0.954928i	$0.404068\pi$
$270$	−0.369102	−0.0224629
$271$	−24.9939	−1.51827	−0.759134	−	0.650934i	$-0.774377\pi$
$271$	−24.9939	−1.51827	−0.759134	+	0.650934i	$0.774377\pi$
$272$	−9.95055	−0.603341
$273$	1.07838	0.0652664
$274$	14.3607	0.867561
$275$	−3.13624	−0.189122
$276$	−16.3896	−0.986539
$277$	3.60197	0.216421	0.108211	−	0.994128i	$-0.465488\pi$
$277$	3.60197	0.216421	0.108211	+	0.994128i	$0.465488\pi$
$278$	0.688756	0.0413089
$279$	−2.29072	−0.137142
$280$	0.971071	0.0580326
$281$	8.94441	0.533579	0.266789	−	0.963755i	$-0.414037\pi$
$281$	8.94441	0.533579	0.266789	+	0.963755i	$0.414037\pi$
$282$	22.1773	1.32064
$283$	−7.49920	−0.445781	−0.222890	−	0.974843i	$-0.571549\pi$
$283$	−7.49920	−0.445781	−0.222890	+	0.974843i	$0.571549\pi$
$284$	28.0144	1.66235
$285$	−1.31124	−0.0776714
$286$	0.398032	0.0235361
$287$	43.0349	2.54027
$288$	7.58864	0.447165
$289$	5.92162	0.348331
$290$	−1.87444	−0.110071
$291$	4.09890	0.240282
$292$	−33.2206	−1.94409
$293$	25.1689	1.47038	0.735190	−	0.677861i	$-0.237093\pi$
$293$	25.1689	1.47038	0.735190	+	0.677861i	$0.237093\pi$
$294$	−14.6670	−0.855398
$295$	−1.72753	−0.100581
$296$	−15.1412	−0.880062
$297$	0.630898	0.0366084
$298$	−5.70928	−0.330729
$299$	1.75872	0.101710
$300$	−13.4680	−0.777575
$301$	9.07838	0.523269
$302$	−51.4957	−2.96324
$303$	0.107307	0.00616464
$304$	16.0228	0.918970
$305$	−2.34017	−0.133998
$306$	−10.3896	−0.593935
$307$	15.3630	0.876810	0.438405	−	0.898777i	$-0.355543\pi$
$307$	15.3630	0.876810	0.438405	+	0.898777i	$0.355543\pi$
$308$	−6.34017	−0.361265
$309$	−5.83710	−0.332061
$310$	0.845512	0.0480219
$311$	18.9854	1.07657	0.538283	−	0.842764i	$-0.319073\pi$
$311$	18.9854	1.07657	0.538283	+	0.842764i	$0.319073\pi$
$312$	0.447480	0.0253336
$313$	−21.8843	−1.23697	−0.618486	−	0.785796i	$-0.712254\pi$
$313$	−21.8843	−1.23697	−0.618486	+	0.785796i	$0.712254\pi$
$314$	−31.9649	−1.80389
$315$	−0.630898	−0.0355471
$316$	−14.6020	−0.821425
$317$	2.99386	0.168152	0.0840759	−	0.996459i	$-0.473206\pi$
$317$	2.99386	0.168152	0.0840759	+	0.996459i	$0.473206\pi$
$318$	−17.1278	−0.960481
$319$	3.20394	0.179386
$320$	−2.09398	−0.117057
$321$	13.1773	0.735484
$322$	−48.6947	−2.71365
$323$	−36.9093	−2.05369
$324$	2.70928	0.150515
$325$	1.44521	0.0801660
$326$	−22.7165	−1.25815
$327$	−5.04945	−0.279235
$328$	17.8576	0.986022
$329$	37.9071	2.08988
$330$	−0.232866	−0.0128188
$331$	21.8888	1.20312	0.601559	−	0.798828i	$-0.294546\pi$
$331$	21.8888	1.20312	0.601559	+	0.798828i	$0.294546\pi$
$332$	−30.6092	−1.67990
$333$	9.83710	0.539070
$334$	−20.8371	−1.14016
$335$	0.724874	0.0396041
$336$	7.70928	0.420575
$337$	−5.27021	−0.287086	−0.143543	−	0.989644i	$-0.545850\pi$
$337$	−5.27021	−0.287086	−0.143543	+	0.989644i	$0.545850\pi$
$338$	28.0277	1.52451
$339$	20.5730	1.11737
$340$	2.20620	0.119648
$341$	−1.44521	−0.0782627
$342$	16.7298	0.904644
$343$	0.894960	0.0483233
$344$	3.76713	0.203110
$345$	−1.02893	−0.0553957
$346$	−14.7009	−0.790323
$347$	0.760991	0.0408521	0.0204261	−	0.999791i	$-0.493498\pi$
$347$	0.760991	0.0408521	0.0204261	+	0.999791i	$0.493498\pi$
$348$	13.7587	0.737545
$349$	−30.3028	−1.62207	−0.811037	−	0.584995i	$-0.801096\pi$
$349$	−30.3028	−1.62207	−0.811037	+	0.584995i	$0.801096\pi$
$350$	−40.0144	−2.13886
$351$	−0.290725	−0.0155177
$352$	4.78765	0.255183
$353$	9.67808	0.515112	0.257556	−	0.966263i	$-0.417083\pi$
$353$	9.67808	0.515112	0.257556	+	0.966263i	$0.417083\pi$
$354$	22.0410	1.17147
$355$	1.75872	0.0933434
$356$	−29.3607	−1.55611
$357$	−17.7587	−0.939891
$358$	27.3630	1.44618
$359$	−14.8865	−0.785682	−0.392841	−	0.919606i	$-0.628508\pi$
$359$	−14.8865	−0.785682	−0.392841	+	0.919606i	$0.628508\pi$
$360$	−0.261795	−0.0137978
$361$	40.4329	2.12805
$362$	−23.1145	−1.21487
$363$	−10.6020	−0.556459
$364$	2.92162	0.153135
$365$	−2.08557	−0.109163
$366$	29.8576	1.56068
$367$	20.1256	1.05055	0.525273	−	0.850934i	$-0.323963\pi$
$367$	20.1256	1.05055	0.525273	+	0.850934i	$0.323963\pi$
$368$	12.5730	0.655415
$369$	−11.6020	−0.603974
$370$	−3.63090	−0.188761
$371$	−29.2762	−1.51994
$372$	−6.20620	−0.321777
$373$	−24.5113	−1.26915	−0.634574	−	0.772862i	$-0.718824\pi$
$373$	−24.5113	−1.26915	−0.634574	+	0.772862i	$0.718824\pi$
$374$	−6.55479	−0.338940
$375$	−1.69594	−0.0875782
$376$	15.7298	0.811202
$377$	−1.47641	−0.0760390
$378$	8.04945	0.414019
$379$	−28.9132	−1.48517	−0.742586	−	0.669751i	$-0.766401\pi$
$379$	−28.9132	−1.48517	−0.742586	+	0.669751i	$0.766401\pi$
$380$	−3.55252	−0.182240
$381$	−12.8371	−0.657665
$382$	−45.9071	−2.34881
$383$	4.25953	0.217652	0.108826	−	0.994061i	$-0.465291\pi$
$383$	4.25953	0.217652	0.108826	+	0.994061i	$0.465291\pi$
$384$	11.5392	0.588857
$385$	−0.398032	−0.0202856
$386$	−3.20394	−0.163076
$387$	−2.44748	−0.124412
$388$	11.1050	0.563773
$389$	29.1734	1.47915	0.739575	−	0.673074i	$-0.235027\pi$
$389$	29.1734	1.47915	0.739575	+	0.673074i	$0.235027\pi$
$390$	0.107307	0.00543371
$391$	−28.9627	−1.46470
$392$	−10.4030	−0.525428
$393$	−9.35577	−0.471936
$394$	2.29072	0.115405
$395$	−0.916702	−0.0461243
$396$	1.70928	0.0858943
$397$	22.5997	1.13425	0.567123	−	0.823633i	$-0.308056\pi$
$397$	22.5997	1.13425	0.567123	+	0.823633i	$0.308056\pi$
$398$	−54.9770	−2.75575
$399$	28.5958	1.43158
$400$	10.3318	0.516588
$401$	9.78992	0.488885	0.244443	−	0.969664i	$-0.421395\pi$
$401$	9.78992	0.488885	0.244443	+	0.969664i	$0.421395\pi$
$402$	−9.24846	−0.461271
$403$	0.665970	0.0331743
$404$	0.290725	0.0144641
$405$	0.170086	0.00845167
$406$	40.8781	2.02875
$407$	6.20620	0.307630
$408$	−7.36910	−0.364825
$409$	−13.4908	−0.667076	−0.333538	−	0.942737i	$-0.608243\pi$
$409$	−13.4908	−0.667076	−0.333538	+	0.942737i	$0.608243\pi$
$410$	4.28231	0.211488
$411$	−6.61757	−0.326420
$412$	−15.8143	−0.779115
$413$	37.6742	1.85383
$414$	13.1278	0.645197
$415$	−1.92162	−0.0943287
$416$	−2.20620	−0.108168
$417$	−0.317387	−0.0155425
$418$	10.5548	0.516252
$419$	−23.8660	−1.16593	−0.582966	−	0.812497i	$-0.698108\pi$
$419$	−23.8660	−1.16593	−0.582966	+	0.812497i	$0.698108\pi$
$420$	−1.70928	−0.0834041
$421$	0.107307	0.00522983	0.00261492	−	0.999997i	$-0.499168\pi$
$421$	0.107307	0.00522983	0.00261492	+	0.999997i	$0.499168\pi$
$422$	−20.6042	−1.00300
$423$	−10.2195	−0.496891
$424$	−12.1483	−0.589976
$425$	−23.7998	−1.15446
$426$	−22.4391	−1.08718
$427$	51.0349	2.46975
$428$	35.7009	1.72567
$429$	−0.183417	−0.00885548
$430$	0.903371	0.0435644
$431$	−22.1795	−1.06835	−0.534175	−	0.845374i	$-0.679378\pi$
$431$	−22.1795	−1.06835	−0.534175	+	0.845374i	$0.679378\pi$
$432$	−2.07838	−0.0999960
$433$	−13.2595	−0.637212	−0.318606	−	0.947887i	$-0.603215\pi$
$433$	−13.2595	−0.637212	−0.318606	+	0.947887i	$0.603215\pi$
$434$	−18.4391	−0.885104
$435$	0.863763	0.0414143
$436$	−13.6803	−0.655170
$437$	46.6369	2.23094
$438$	26.6092	1.27143
$439$	−1.00000	−0.0477274	−0.0238637	−	0.999715i	$-0.507597\pi$
$439$	−1.00000	−0.0477274	−0.0238637	+	0.999715i	$0.507597\pi$
$440$	−0.165166	−0.00787398
$441$	6.75872	0.321844
$442$	3.02052	0.143671
$443$	20.7392	0.985351	0.492676	−	0.870213i	$-0.336019\pi$
$443$	20.7392	0.985351	0.492676	+	0.870213i	$0.336019\pi$
$444$	26.6514	1.26482
$445$	−1.84324	−0.0873782
$446$	−2.17009	−0.102757
$447$	2.63090	0.124437
$448$	45.6658	2.15751
$449$	−27.4101	−1.29356	−0.646782	−	0.762675i	$-0.723886\pi$
$449$	−27.4101	−1.29356	−0.646782	+	0.762675i	$0.723886\pi$
$450$	10.7877	0.508535
$451$	−7.31965	−0.344669
$452$	55.7380	2.62170
$453$	23.7298	1.11492
$454$	−6.25338	−0.293486
$455$	0.183417	0.00859874
$456$	11.8660	0.555678
$457$	−7.86603	−0.367957	−0.183979	−	0.982930i	$-0.558898\pi$
$457$	−7.86603	−0.367957	−0.183979	+	0.982930i	$0.558898\pi$
$458$	17.3424	0.810359
$459$	4.78765	0.223469
$460$	−2.78765	−0.129975
$461$	−30.0638	−1.40021	−0.700106	−	0.714039i	$-0.746864\pi$
$461$	−30.0638	−1.40021	−0.700106	+	0.714039i	$0.746864\pi$
$462$	5.07838	0.236268
$463$	−29.0349	−1.34937	−0.674683	−	0.738108i	$-0.735720\pi$
$463$	−29.0349	−1.34937	−0.674683	+	0.738108i	$0.735720\pi$
$464$	−10.5548	−0.489994
$465$	−0.389621	−0.0180683
$466$	6.78765	0.314432
$467$	12.4969	0.578289	0.289144	−	0.957285i	$-0.406629\pi$
$467$	12.4969	0.578289	0.289144	+	0.957285i	$0.406629\pi$
$468$	−0.787653	−0.0364093
$469$	−15.8082	−0.729954
$470$	3.77205	0.173992
$471$	14.7298	0.678713
$472$	15.6332	0.719575
$473$	−1.54411	−0.0709982
$474$	11.6959	0.537213
$475$	38.3234	1.75840
$476$	−48.1133	−2.20527
$477$	7.89269	0.361382
$478$	−25.5958	−1.17073
$479$	−33.7093	−1.54022	−0.770108	−	0.637913i	$-0.779798\pi$
$479$	−33.7093	−1.54022	−0.770108	+	0.637913i	$0.779798\pi$
$480$	1.29072	0.0589133
$481$	−2.85989	−0.130400
$482$	13.8710	0.631805
$483$	22.4391	1.02101
$484$	−28.7237	−1.30562
$485$	0.697167	0.0316567
$486$	−2.17009	−0.0984371
$487$	−9.02893	−0.409140	−0.204570	−	0.978852i	$-0.565580\pi$
$487$	−9.02893	−0.409140	−0.204570	+	0.978852i	$0.565580\pi$
$488$	21.1773	0.958650
$489$	10.4680	0.473379
$490$	−2.49466	−0.112697
$491$	18.2472	0.823486	0.411743	−	0.911300i	$-0.364920\pi$
$491$	18.2472	0.823486	0.411743	+	0.911300i	$0.364920\pi$
$492$	−31.4329	−1.41711
$493$	24.3135	1.09503
$494$	−4.86376	−0.218831
$495$	0.107307	0.00482310
$496$	4.76099	0.213775
$497$	−38.3545	−1.72044
$498$	24.5174	1.09865
$499$	4.24128	0.189866	0.0949328	−	0.995484i	$-0.469736\pi$
$499$	4.24128	0.189866	0.0949328	+	0.995484i	$0.469736\pi$
$500$	−4.59478	−0.205485
$501$	9.60197	0.428984
$502$	−26.1012	−1.16495
$503$	−8.62702	−0.384660	−0.192330	−	0.981330i	$-0.561604\pi$
$503$	−8.62702	−0.384660	−0.192330	+	0.981330i	$0.561604\pi$
$504$	5.70928	0.254311
$505$	0.0182515	0.000812181 0
$506$	8.28231	0.368194
$507$	−12.9155	−0.573597
$508$	−34.7792	−1.54308
$509$	11.3958	0.505108	0.252554	−	0.967583i	$-0.418729\pi$
$509$	11.3958	0.505108	0.252554	+	0.967583i	$0.418729\pi$
$510$	−1.76713	−0.0782500
$511$	45.4824	2.01202
$512$	−22.1701	−0.979789
$513$	−7.70928	−0.340373
$514$	36.8599	1.62582
$515$	−0.992812	−0.0437485
$516$	−6.63090	−0.291909
$517$	−6.44748	−0.283560
$518$	79.1832	3.47911
$519$	6.77432	0.297360
$520$	0.0761103	0.00333766
$521$	−5.08783	−0.222902	−0.111451	−	0.993770i	$-0.535550\pi$
$521$	−5.08783	−0.222902	−0.111451	+	0.993770i	$0.535550\pi$
$522$	−11.0205	−0.482355
$523$	−7.36523	−0.322059	−0.161029	−	0.986950i	$-0.551481\pi$
$523$	−7.36523	−0.322059	−0.161029	+	0.986950i	$0.551481\pi$
$524$	−25.3474	−1.10730
$525$	18.4391	0.804747
$526$	53.8804	2.34930
$527$	−10.9672	−0.477739
$528$	−1.31124	−0.0570646
$529$	13.5958	0.591123
$530$	−2.91321	−0.126542
$531$	−10.1568	−0.440766
$532$	77.4740	3.35892
$533$	3.37298	0.146100
$534$	23.5174	1.01770
$535$	2.24128	0.0968988
$536$	−6.55971	−0.283336
$537$	−12.6092	−0.544125
$538$	−21.1301	−0.910983
$539$	4.26406	0.183666
$540$	0.460811	0.0198302
$541$	−13.5031	−0.580542	−0.290271	−	0.956944i	$-0.593746\pi$
$541$	−13.5031	−0.580542	−0.290271	+	0.956944i	$0.593746\pi$
$542$	54.2388	2.32976
$543$	10.6514	0.457096
$544$	36.3318	1.55771
$545$	−0.858843	−0.0367888
$546$	−2.34017	−0.100150
$547$	0.290725	0.0124305	0.00621524	−	0.999981i	$-0.498022\pi$
$547$	0.290725	0.0124305	0.00621524	+	0.999981i	$0.498022\pi$
$548$	−17.9288	−0.765881
$549$	−13.7587	−0.587208
$550$	6.80590	0.290205
$551$	−39.1506	−1.66787
$552$	9.31124	0.396313
$553$	19.9916	0.850129
$554$	−7.81658	−0.332095
$555$	1.67316	0.0710216
$556$	−0.859888	−0.0364674
$557$	16.2195	0.687244	0.343622	−	0.939108i	$-0.388346\pi$
$557$	16.2195	0.687244	0.343622	+	0.939108i	$0.388346\pi$
$558$	4.97107	0.210442
$559$	0.711543	0.0300951
$560$	1.31124	0.0554102
$561$	3.02052	0.127526
$562$	−19.4101	−0.818767
$563$	16.9588	0.714728	0.357364	−	0.933965i	$-0.383676\pi$
$563$	16.9588	0.714728	0.357364	+	0.933965i	$0.383676\pi$
$564$	−27.6875	−1.16586
$565$	3.49920	0.147212
$566$	16.2739	0.684043
$567$	−3.70928	−0.155775
$568$	−15.9155	−0.667799
$569$	−21.2895	−0.892502	−0.446251	−	0.894908i	$-0.647241\pi$
$569$	−21.2895	−0.892502	−0.446251	+	0.894908i	$0.647241\pi$
$570$	2.84551	0.119185
$571$	−44.5357	−1.86376	−0.931880	−	0.362765i	$-0.881833\pi$
$571$	−44.5357	−1.86376	−0.931880	+	0.362765i	$0.881833\pi$
$572$	−0.496928	−0.0207776
$573$	21.1545	0.883741
$574$	−93.3894	−3.89800
$575$	30.0722	1.25410
$576$	−12.3112	−0.512968
$577$	−18.6781	−0.777579	−0.388789	−	0.921327i	$-0.627107\pi$
$577$	−18.6781	−0.777579	−0.388789	+	0.921327i	$0.627107\pi$
$578$	−12.8504	−0.534508
$579$	1.47641	0.0613575
$580$	2.34017	0.0971704
$581$	41.9071	1.73860
$582$	−8.89496	−0.368708
$583$	4.97948	0.206229
$584$	18.8732	0.780979
$585$	−0.0494483	−0.00204444
$586$	−54.6186	−2.25627
$587$	−10.6719	−0.440478	−0.220239	−	0.975446i	$-0.570684\pi$
$587$	−10.6719	−0.440478	−0.220239	+	0.975446i	$0.570684\pi$
$588$	18.3112	0.755143
$589$	17.6598	0.727660
$590$	3.74888	0.154339
$591$	−1.05559	−0.0434212
$592$	−20.4452	−0.840293
$593$	33.6153	1.38041	0.690207	−	0.723612i	$-0.257519\pi$
$593$	33.6153	1.38041	0.690207	+	0.723612i	$0.257519\pi$
$594$	−1.36910	−0.0561750
$595$	−3.02052	−0.123829
$596$	7.12783	0.291967
$597$	25.3340	1.03685
$598$	−3.81658	−0.156072
$599$	−12.2329	−0.499821	−0.249911	−	0.968269i	$-0.580401\pi$
$599$	−12.2329	−0.499821	−0.249911	+	0.968269i	$0.580401\pi$
$600$	7.65142	0.312368
$601$	16.4969	0.672924	0.336462	−	0.941697i	$-0.390770\pi$
$601$	16.4969	0.672924	0.336462	+	0.941697i	$0.390770\pi$
$602$	−19.7009	−0.802947
$603$	4.26180	0.173554
$604$	64.2905	2.61595
$605$	−1.80325	−0.0733126
$606$	−0.232866	−0.00945953
$607$	34.0577	1.38236	0.691179	−	0.722683i	$-0.257091\pi$
$607$	34.0577	1.38236	0.691179	+	0.722683i	$0.257091\pi$
$608$	−58.5029	−2.37261
$609$	−18.8371	−0.763318
$610$	5.07838	0.205618
$611$	2.97107	0.120197
$612$	12.9711	0.524324
$613$	−25.4101	−1.02631	−0.513153	−	0.858297i	$-0.671523\pi$
$613$	−25.4101	−1.02631	−0.513153	+	0.858297i	$0.671523\pi$
$614$	−33.3390	−1.34545
$615$	−1.97334	−0.0795727
$616$	3.60197	0.145127
$617$	39.5174	1.59091	0.795456	−	0.606011i	$-0.207231\pi$
$617$	39.5174	1.59091	0.795456	+	0.606011i	$0.207231\pi$
$618$	12.6670	0.509542
$619$	23.9854	0.964056	0.482028	−	0.876156i	$-0.339900\pi$
$619$	23.9854	0.964056	0.482028	+	0.876156i	$0.339900\pi$
$620$	−1.05559	−0.0423936
$621$	−6.04945	−0.242756
$622$	−41.2001	−1.65197
$623$	40.1978	1.61049
$624$	0.604236	0.0241888
$625$	24.5669	0.982676
$626$	47.4908	1.89811
$627$	−4.86376	−0.194240
$628$	39.9071	1.59247
$629$	47.0966	1.87787
$630$	1.36910	0.0545463
$631$	−46.4596	−1.84953	−0.924764	−	0.380542i	$-0.875737\pi$
$631$	−46.4596	−1.84953	−0.924764	+	0.380542i	$0.875737\pi$
$632$	8.29565	0.329983
$633$	9.49466	0.377379
$634$	−6.49693	−0.258026
$635$	−2.18342	−0.0866463
$636$	21.3835	0.847910
$637$	−1.96493	−0.0778533
$638$	−6.95282	−0.275265
$639$	10.3402	0.409051
$640$	1.96266	0.0775810
$641$	−9.92881	−0.392164	−0.196082	−	0.980587i	$-0.562822\pi$
$641$	−9.92881	−0.392164	−0.196082	+	0.980587i	$0.562822\pi$
$642$	−28.5958	−1.12859
$643$	−38.7480	−1.52807	−0.764037	−	0.645173i	$-0.776785\pi$
$643$	−38.7480	−1.52807	−0.764037	+	0.645173i	$0.776785\pi$
$644$	60.7936	2.39560
$645$	−0.416283	−0.0163911
$646$	80.0965	3.15135
$647$	−32.6453	−1.28342	−0.641709	−	0.766948i	$-0.721774\pi$
$647$	−32.6453	−1.28342	−0.641709	+	0.766948i	$0.721774\pi$
$648$	−1.53919	−0.0604650
$649$	−6.40787	−0.251531
$650$	−3.13624	−0.123013
$651$	8.49693	0.333021
$652$	28.3607	1.11069
$653$	−12.8554	−0.503069	−0.251534	−	0.967848i	$-0.580935\pi$
$653$	−12.8554	−0.503069	−0.251534	+	0.967848i	$0.580935\pi$
$654$	10.9577	0.428482
$655$	−1.59129	−0.0621769
$656$	24.1133	0.941465
$657$	−12.2618	−0.478378
$658$	−82.2616	−3.20689
$659$	34.2651	1.33478	0.667390	−	0.744709i	$-0.267412\pi$
$659$	34.2651	1.33478	0.667390	+	0.744709i	$0.267412\pi$
$660$	0.290725	0.0113164
$661$	20.4619	0.795874	0.397937	−	0.917413i	$-0.369726\pi$
$661$	20.4619	0.795874	0.397937	+	0.917413i	$0.369726\pi$
$662$	−47.5006	−1.84616
$663$	−1.39189	−0.0540565
$664$	17.3896	0.674848
$665$	4.86376	0.188609
$666$	−21.3474	−0.827193
$667$	−30.7214	−1.18954
$668$	26.0144	1.00653
$669$	1.00000	0.0386622
$670$	−1.57304	−0.0607718
$671$	−8.68035	−0.335101
$672$	−28.1483	−1.08585
$673$	3.84324	0.148146	0.0740731	−	0.997253i	$-0.476400\pi$
$673$	3.84324	0.148146	0.0740731	+	0.997253i	$0.476400\pi$
$674$	11.4368	0.440529
$675$	−4.97107	−0.191337
$676$	−34.9916	−1.34583
$677$	11.7177	0.450347	0.225174	−	0.974319i	$-0.427705\pi$
$677$	11.7177	0.450347	0.225174	+	0.974319i	$0.427705\pi$
$678$	−44.6453	−1.71459
$679$	−15.2039	−0.583474
$680$	−1.25338	−0.0480651
$681$	2.88163	0.110424
$682$	3.13624	0.120093
$683$	−24.2690	−0.928627	−0.464313	−	0.885671i	$-0.653699\pi$
$683$	−24.2690	−0.928627	−0.464313	+	0.885671i	$0.653699\pi$
$684$	−20.8865	−0.798617
$685$	−1.12556	−0.0430054
$686$	−1.94214	−0.0741513
$687$	−7.99159	−0.304898
$688$	5.08679	0.193932
$689$	−2.29460	−0.0874173
$690$	2.23287	0.0850037
$691$	7.12556	0.271069	0.135535	−	0.990773i	$-0.456725\pi$
$691$	7.12556	0.271069	0.135535	+	0.990773i	$0.456725\pi$
$692$	18.3535	0.697696
$693$	−2.34017	−0.0888958
$694$	−1.65142	−0.0626869
$695$	−0.0539832	−0.00204770
$696$	−7.81658	−0.296287
$697$	−55.5462	−2.10396
$698$	65.7598	2.48904
$699$	−3.12783	−0.118305
$700$	49.9565	1.88818
$701$	−22.4307	−0.847194	−0.423597	−	0.905851i	$-0.639233\pi$
$701$	−22.4307	−0.847194	−0.423597	+	0.905851i	$0.639233\pi$
$702$	0.630898	0.0238117
$703$	−75.8369	−2.86024
$704$	−7.76713	−0.292735
$705$	−1.73820	−0.0654646
$706$	−21.0023	−0.790431
$707$	−0.398032	−0.0149695
$708$	−27.5174	−1.03417
$709$	17.1461	0.643934	0.321967	−	0.946751i	$-0.395656\pi$
$709$	17.1461	0.643934	0.321967	+	0.946751i	$0.395656\pi$
$710$	−3.81658	−0.143234
$711$	−5.38962	−0.202127
$712$	16.6803	0.625122
$713$	13.8576	0.518972
$714$	38.5380	1.44225
$715$	−0.0311968	−0.00116670
$716$	−34.1617	−1.27668
$717$	11.7948	0.440486
$718$	32.3051	1.20562
$719$	4.28580	0.159834	0.0799168	−	0.996802i	$-0.474535\pi$
$719$	4.28580	0.159834	0.0799168	+	0.996802i	$0.474535\pi$
$720$	−0.353504	−0.0131743
$721$	21.6514	0.806341
$722$	−87.7429	−3.26545
$723$	−6.39189	−0.237717
$724$	28.8576	1.07249
$725$	−25.2450	−0.937575
$726$	23.0072	0.853877
$727$	−29.6886	−1.10109	−0.550544	−	0.834806i	$-0.685580\pi$
$727$	−29.6886	−1.10109	−0.550544	+	0.834806i	$0.685580\pi$
$728$	−1.65983	−0.0615173
$729$	1.00000	0.0370370
$730$	4.52586	0.167510
$731$	−11.7177	−0.433394
$732$	−37.2762	−1.37777
$733$	17.9155	0.661723	0.330862	−	0.943679i	$-0.392661\pi$
$733$	17.9155	0.661723	0.330862	+	0.943679i	$0.392661\pi$
$734$	−43.6742	−1.61204
$735$	1.14957	0.0424024
$736$	−45.9071	−1.69216
$737$	2.68876	0.0990416
$738$	25.1773	0.926788
$739$	−4.50307	−0.165648	−0.0828241	−	0.996564i	$-0.526394\pi$
$739$	−4.50307	−0.165648	−0.0828241	+	0.996564i	$0.526394\pi$
$740$	4.53305	0.166638
$741$	2.24128	0.0823353
$742$	63.5318	2.33233
$743$	10.6235	0.389740	0.194870	−	0.980829i	$-0.437572\pi$
$743$	10.6235	0.389740	0.194870	+	0.980829i	$0.437572\pi$
$744$	3.52586	0.129264
$745$	0.447480	0.0163944
$746$	53.1917	1.94748
$747$	−11.2979	−0.413369
$748$	8.18342	0.299215
$749$	−48.8781	−1.78597
$750$	3.68035	0.134387
$751$	33.1194	1.20854	0.604272	−	0.796778i	$-0.293464\pi$
$751$	33.1194	1.20854	0.604272	+	0.796778i	$0.293464\pi$
$752$	21.2401	0.774545
$753$	12.0277	0.438314
$754$	3.20394	0.116680
$755$	4.03612	0.146889
$756$	−10.0494	−0.365495
$757$	−13.0289	−0.473544	−0.236772	−	0.971565i	$-0.576090\pi$
$757$	−13.0289	−0.473544	−0.236772	+	0.971565i	$0.576090\pi$
$758$	62.7442	2.27897
$759$	−3.81658	−0.138533
$760$	2.01825	0.0732097
$761$	−15.5392	−0.563295	−0.281648	−	0.959518i	$-0.590881\pi$
$761$	−15.5392	−0.563295	−0.281648	+	0.959518i	$0.590881\pi$
$762$	27.8576	1.00918
$763$	18.7298	0.678064
$764$	57.3133	2.07352
$765$	0.814315	0.0294416
$766$	−9.24354	−0.333983
$767$	2.95282	0.106620
$768$	−0.418551	−0.0151031
$769$	3.62863	0.130852	0.0654259	−	0.997857i	$-0.479159\pi$
$769$	3.62863	0.130852	0.0654259	+	0.997857i	$0.479159\pi$
$770$	0.863763	0.0311279
$771$	−16.9854	−0.611716
$772$	4.00000	0.143963
$773$	−8.57304	−0.308351	−0.154175	−	0.988043i	$-0.549272\pi$
$773$	−8.57304	−0.308351	−0.154175	+	0.988043i	$0.549272\pi$
$774$	5.31124	0.190909
$775$	11.3874	0.409046
$776$	−6.30898	−0.226479
$777$	−36.4885	−1.30902
$778$	−63.3088	−2.26973
$779$	89.4428	3.20462
$780$	−0.133969	−0.00479687
$781$	6.52359	0.233432
$782$	62.8515	2.24756
$783$	5.07838	0.181486
$784$	−14.0472	−0.501685
$785$	2.50534	0.0894194
$786$	20.3028	0.724178
$787$	−47.5692	−1.69566	−0.847829	−	0.530270i	$-0.822091\pi$
$787$	−47.5692	−1.69566	−0.847829	+	0.530270i	$0.822091\pi$
$788$	−2.85989	−0.101879
$789$	−24.8287	−0.883925
$790$	1.98932	0.0707769
$791$	−76.3111	−2.71331
$792$	−0.971071	−0.0345055
$793$	4.00000	0.142044
$794$	−49.0433	−1.74048
$795$	1.34244	0.0476115
$796$	68.6369	2.43277
$797$	−41.8492	−1.48238	−0.741188	−	0.671298i	$-0.765737\pi$
$797$	−41.8492	−1.48238	−0.741188	+	0.671298i	$0.765737\pi$
$798$	−62.0554	−2.19674
$799$	−48.9276	−1.73093
$800$	−37.7237	−1.33373
$801$	−10.8371	−0.382910
$802$	−21.2450	−0.750186
$803$	−7.73594	−0.272995
$804$	11.5464	0.407209
$805$	3.81658	0.134517
$806$	−1.44521	−0.0509054
$807$	9.73698	0.342758
$808$	−0.165166	−0.00581052
$809$	49.4196	1.73750	0.868750	−	0.495250i	$-0.164924\pi$
$809$	49.4196	1.73750	0.868750	+	0.495250i	$0.164924\pi$
$810$	−0.369102	−0.0129689
$811$	−14.3402	−0.503552	−0.251776	−	0.967786i	$-0.581015\pi$
$811$	−14.3402	−0.503552	−0.251776	+	0.967786i	$0.581015\pi$
$812$	−51.0349	−1.79097
$813$	−24.9939	−0.876573
$814$	−13.4680	−0.472053
$815$	1.78047	0.0623670
$816$	−9.95055	−0.348339
$817$	18.8683	0.660118
$818$	29.2762	1.02362
$819$	1.07838	0.0376816
$820$	−5.34632	−0.186701
$821$	−46.5608	−1.62498	−0.812491	−	0.582974i	$-0.801889\pi$
$821$	−46.5608	−1.62498	−0.812491	+	0.582974i	$0.801889\pi$
$822$	14.3607	0.500887
$823$	−16.2313	−0.565786	−0.282893	−	0.959151i	$-0.591294\pi$
$823$	−16.2313	−0.565786	−0.282893	+	0.959151i	$0.591294\pi$
$824$	8.98440	0.312986
$825$	−3.13624	−0.109190
$826$	−81.7563	−2.84466
$827$	5.39576	0.187629	0.0938146	−	0.995590i	$-0.470094\pi$
$827$	5.39576	0.187629	0.0938146	+	0.995590i	$0.470094\pi$
$828$	−16.3896	−0.569579
$829$	0.451356	0.0156762	0.00783812	−	0.999969i	$-0.497505\pi$
$829$	0.451356	0.0156762	0.00783812	+	0.999969i	$0.497505\pi$
$830$	4.17009	0.144746
$831$	3.60197	0.124951
$832$	3.57918	0.124086
$833$	32.3584	1.12115
$834$	0.688756	0.0238497
$835$	1.63317	0.0565180
$836$	−13.1773	−0.455746
$837$	−2.29072	−0.0791790
$838$	51.7914	1.78910
$839$	−4.05786	−0.140093	−0.0700464	−	0.997544i	$-0.522315\pi$
$839$	−4.05786	−0.140093	−0.0700464	+	0.997544i	$0.522315\pi$
$840$	0.971071	0.0335051
$841$	−3.21008	−0.110692
$842$	−0.232866	−0.00802509
$843$	8.94441	0.308062
$844$	25.7237	0.885445
$845$	−2.19675	−0.0755704
$846$	22.1773	0.762470
$847$	39.3256	1.35124
$848$	−16.4040	−0.563316
$849$	−7.49920	−0.257372
$850$	51.6475	1.77150
$851$	−59.5090	−2.03994
$852$	28.0144	0.959757
$853$	−8.73367	−0.299035	−0.149518	−	0.988759i	$-0.547772\pi$
$853$	−8.73367	−0.299035	−0.149518	+	0.988759i	$0.547772\pi$
$854$	−110.750	−3.78979
$855$	−1.31124	−0.0448436
$856$	−20.2823	−0.693235
$857$	30.0183	1.02540	0.512702	−	0.858567i	$-0.328645\pi$
$857$	30.0183	1.02540	0.512702	+	0.858567i	$0.328645\pi$
$858$	0.398032	0.0135886
$859$	−50.5669	−1.72532	−0.862660	−	0.505784i	$-0.831203\pi$
$859$	−50.5669	−1.72532	−0.862660	+	0.505784i	$0.831203\pi$
$860$	−1.12783	−0.0384586
$861$	43.0349	1.46663
$862$	48.1315	1.63937
$863$	6.96719	0.237166	0.118583	−	0.992944i	$-0.462165\pi$
$863$	6.96719	0.237166	0.118583	+	0.992944i	$0.462165\pi$
$864$	7.58864	0.258171
$865$	1.15222	0.0391767
$866$	28.7743	0.977791
$867$	5.92162	0.201109
$868$	23.0205	0.781367
$869$	−3.40030	−0.115347
$870$	−1.87444	−0.0635495
$871$	−1.23901	−0.0419822
$872$	7.77205	0.263195
$873$	4.09890	0.138727
$874$	−101.206	−3.42334
$875$	6.29072	0.212665
$876$	−33.2206	−1.12242
$877$	42.0372	1.41949	0.709747	−	0.704457i	$-0.248809\pi$
$877$	42.0372	1.41949	0.709747	+	0.704457i	$0.248809\pi$
$878$	2.17009	0.0732369
$879$	25.1689	0.848925
$880$	−0.223025	−0.00751816
$881$	−10.8911	−0.366930	−0.183465	−	0.983026i	$-0.558731\pi$
$881$	−10.8911	−0.366930	−0.183465	+	0.983026i	$0.558731\pi$
$882$	−14.6670	−0.493864
$883$	−12.9071	−0.434357	−0.217179	−	0.976132i	$-0.569685\pi$
$883$	−12.9071	−0.434357	−0.217179	+	0.976132i	$0.569685\pi$
$884$	−3.77101	−0.126833
$885$	−1.72753	−0.0580702
$886$	−45.0060	−1.51200
$887$	−35.7910	−1.20174	−0.600872	−	0.799346i	$-0.705180\pi$
$887$	−35.7910	−1.20174	−0.600872	+	0.799346i	$0.705180\pi$
$888$	−15.1412	−0.508104
$889$	47.6163	1.59700
$890$	4.00000	0.134080
$891$	0.630898	0.0211359
$892$	2.70928	0.0907132
$893$	78.7852	2.63645
$894$	−5.70928	−0.190947
$895$	−2.14465	−0.0716876
$896$	−42.8020	−1.42992
$897$	1.75872	0.0587221
$898$	59.4824	1.98495
$899$	−11.6332	−0.387988
$900$	−13.4680	−0.448933
$901$	37.7875	1.25888
$902$	15.8843	0.528889
$903$	9.07838	0.302109
$904$	−31.6658	−1.05319
$905$	1.81166	0.0602217
$906$	−51.4957	−1.71083
$907$	−55.0843	−1.82905	−0.914523	−	0.404534i	$-0.867434\pi$
$907$	−55.0843	−1.82905	−0.914523	+	0.404534i	$0.867434\pi$
$908$	7.80713	0.259089
$909$	0.107307	0.00355915
$910$	−0.398032	−0.0131946
$911$	−15.3414	−0.508283	−0.254142	−	0.967167i	$-0.581793\pi$
$911$	−15.3414	−0.508283	−0.254142	+	0.967167i	$0.581793\pi$
$912$	16.0228	0.530567
$913$	−7.12783	−0.235897
$914$	17.0700	0.564624
$915$	−2.34017	−0.0773637
$916$	−21.6514	−0.715383
$917$	34.7031	1.14600
$918$	−10.3896	−0.342909
$919$	5.29299	0.174600	0.0872998	−	0.996182i	$-0.472176\pi$
$919$	5.29299	0.174600	0.0872998	+	0.996182i	$0.472176\pi$
$920$	1.58372	0.0522136
$921$	15.3630	0.506227
$922$	65.2411	2.14860
$923$	−3.00614	−0.0989484
$924$	−6.34017	−0.208576
$925$	−48.9009	−1.60785
$926$	63.0082	2.07058
$927$	−5.83710	−0.191716
$928$	38.5380	1.26507
$929$	7.89269	0.258951	0.129475	−	0.991583i	$-0.458671\pi$
$929$	7.89269	0.258951	0.129475	+	0.991583i	$0.458671\pi$
$930$	0.845512	0.0277254
$931$	−52.1049	−1.70767
$932$	−8.47414	−0.277580
$933$	18.9854	0.621556
$934$	−27.1194	−0.887374
$935$	0.513749	0.0168014
$936$	0.447480	0.0146263
$937$	40.1627	1.31206	0.656029	−	0.754735i	$-0.272235\pi$
$937$	40.1627	1.31206	0.656029	+	0.754735i	$0.272235\pi$
$938$	34.3051	1.12010
$939$	−21.8843	−0.714167
$940$	−4.70928	−0.153600
$941$	−9.20006	−0.299913	−0.149957	−	0.988693i	$-0.547913\pi$
$941$	−9.20006	−0.299913	−0.149957	+	0.988693i	$0.547913\pi$
$942$	−31.9649	−1.04147
$943$	70.1855	2.28555
$944$	21.1096	0.687058
$945$	−0.630898	−0.0205231
$946$	3.35085	0.108946
$947$	34.2907	1.11430	0.557149	−	0.830412i	$-0.311895\pi$
$947$	34.2907	1.11430	0.557149	+	0.830412i	$0.311895\pi$
$948$	−14.6020	−0.474250
$949$	3.56481	0.115718
$950$	−83.1650	−2.69823
$951$	2.99386	0.0970825
$952$	27.3340	0.885901
$953$	29.8937	0.968353	0.484177	−	0.874970i	$-0.339119\pi$
$953$	29.8937	0.968353	0.484177	+	0.874970i	$0.339119\pi$
$954$	−17.1278	−0.554534
$955$	3.59809	0.116432
$956$	31.9555	1.03351
$957$	3.20394	0.103569
$958$	73.1520	2.36344
$959$	24.5464	0.792644
$960$	−2.09398	−0.0675828
$961$	−25.7526	−0.830728
$962$	6.20620	0.200096
$963$	13.1773	0.424632
$964$	−17.3174	−0.557755
$965$	0.251117	0.00808375
$966$	−48.6947	−1.56673
$967$	16.3135	0.524607	0.262304	−	0.964985i	$-0.415518\pi$
$967$	16.3135	0.524607	0.262304	+	0.964985i	$0.415518\pi$
$968$	16.3184	0.524494
$969$	−36.9093	−1.18570
$970$	−1.51291	−0.0485767
$971$	16.6225	0.533441	0.266720	−	0.963774i	$-0.414060\pi$
$971$	16.6225	0.533441	0.266720	+	0.963774i	$0.414060\pi$
$972$	2.70928	0.0869000
$973$	1.17727	0.0377417
$974$	19.5936	0.627818
$975$	1.44521	0.0462838
$976$	28.5958	0.915330
$977$	32.3679	1.03554	0.517770	−	0.855520i	$-0.326762\pi$
$977$	32.3679	1.03554	0.517770	+	0.855520i	$0.326762\pi$
$978$	−22.7165	−0.726392
$979$	−6.83710	−0.218515
$980$	3.11450	0.0994889
$981$	−5.04945	−0.161216
$982$	−39.5981	−1.26363
$983$	40.0456	1.27726	0.638628	−	0.769516i	$-0.279502\pi$
$983$	40.0456	1.27726	0.638628	+	0.769516i	$0.279502\pi$
$984$	17.8576	0.569280
$985$	−0.179542	−0.00572068
$986$	−52.7624	−1.68030
$987$	37.9071	1.20660
$988$	6.07223	0.193184
$989$	14.8059	0.470800
$990$	−0.232866	−0.00740096
$991$	39.4161	1.25209	0.626047	−	0.779785i	$-0.284672\pi$
$991$	39.4161	1.25209	0.626047	+	0.779785i	$0.284672\pi$
$992$	−17.3835	−0.551926
$993$	21.8888	0.694621
$994$	83.2327	2.63998
$995$	4.30898	0.136604
$996$	−30.6092	−0.969888
$997$	48.9337	1.54975	0.774873	−	0.632116i	$-0.217814\pi$
$997$	48.9337	1.54975	0.774873	+	0.632116i	$0.217814\pi$
$998$	−9.20394	−0.291346
$999$	9.83710	0.311232

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	669.2.a.g.1.1	✓	3
3.2	odd	2		2007.2.a.f.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
669.2.a.g.1.1	✓	3	1.1	even	1	trivial
2007.2.a.f.1.3		3	3.2	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 669 = 3 \cdot 223 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	669.a (trivial)

\(f(q)\)	\(=\)	\(q-2.17009 q^{2} +1.00000 q^{3} +2.70928 q^{4} +0.170086 q^{5} -2.17009 q^{6} -3.70928 q^{7} -1.53919 q^{8} +1.00000 q^{9} -0.369102 q^{10} +0.630898 q^{11} +2.70928 q^{12} -0.290725 q^{13} +8.04945 q^{14} +0.170086 q^{15} -2.07838 q^{16} +4.78765 q^{17} -2.17009 q^{18} -7.70928 q^{19} +0.460811 q^{20} -3.70928 q^{21} -1.36910 q^{22} -6.04945 q^{23} -1.53919 q^{24} -4.97107 q^{25} +0.630898 q^{26} +1.00000 q^{27} -10.0494 q^{28} +5.07838 q^{29} -0.369102 q^{30} -2.29072 q^{31} +7.58864 q^{32} +0.630898 q^{33} -10.3896 q^{34} -0.630898 q^{35} +2.70928 q^{36} +9.83710 q^{37} +16.7298 q^{38} -0.290725 q^{39} -0.261795 q^{40} -11.6020 q^{41} +8.04945 q^{42} -2.44748 q^{43} +1.70928 q^{44} +0.170086 q^{45} +13.1278 q^{46} -10.2195 q^{47} -2.07838 q^{48} +6.75872 q^{49} +10.7877 q^{50} +4.78765 q^{51} -0.787653 q^{52} +7.89269 q^{53} -2.17009 q^{54} +0.107307 q^{55} +5.70928 q^{56} -7.70928 q^{57} -11.0205 q^{58} -10.1568 q^{59} +0.460811 q^{60} -13.7587 q^{61} +4.97107 q^{62} -3.70928 q^{63} -12.3112 q^{64} -0.0494483 q^{65} -1.36910 q^{66} +4.26180 q^{67} +12.9711 q^{68} -6.04945 q^{69} +1.36910 q^{70} +10.3402 q^{71} -1.53919 q^{72} -12.2618 q^{73} -21.3474 q^{74} -4.97107 q^{75} -20.8865 q^{76} -2.34017 q^{77} +0.630898 q^{78} -5.38962 q^{79} -0.353504 q^{80} +1.00000 q^{81} +25.1773 q^{82} -11.2979 q^{83} -10.0494 q^{84} +0.814315 q^{85} +5.31124 q^{86} +5.07838 q^{87} -0.971071 q^{88} -10.8371 q^{89} -0.369102 q^{90} +1.07838 q^{91} -16.3896 q^{92} -2.29072 q^{93} +22.1773 q^{94} -1.31124 q^{95} +7.58864 q^{96} +4.09890 q^{97} -14.6670 q^{98} +0.630898 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + 3 q^{3} + q^{4} - 5 q^{5} - q^{6} - 4 q^{7} - 3 q^{8} + 3 q^{9} - 5 q^{10} - 2 q^{11} + q^{12} - 8 q^{13} + 6 q^{14} - 5 q^{15} - 3 q^{16} + 4 q^{17} - q^{18} - 16 q^{19} + 3 q^{20} - 4 q^{21}+ \cdots - 2 q^{99}+O(q^{100}) \) 3 * q - q^2 + 3 * q^3 + q^4 - 5 * q^5 - q^6 - 4 * q^7 - 3 * q^8 + 3 * q^9 - 5 * q^10 - 2 * q^11 + q^12 - 8 * q^13 + 6 * q^14 - 5 * q^15 - 3 * q^16 + 4 * q^17 - q^18 - 16 * q^19 + 3 * q^20 - 4 * q^21 - 8 * q^22 - 3 * q^24 - 2 * q^26 + 3 * q^27 - 12 * q^28 + 12 * q^29 - 5 * q^30 - 14 * q^31 + 3 * q^32 - 2 * q^33 - 2 * q^34 + 2 * q^35 + q^36 + q^37 + 10 * q^38 - 8 * q^39 + 7 * q^40 - 16 * q^41 + 6 * q^42 - 8 * q^43 - 2 * q^44 - 5 * q^45 + 18 * q^46 - 7 * q^47 - 3 * q^48 - 5 * q^49 + 22 * q^50 + 4 * q^51 + 8 * q^52 + 12 * q^53 - q^54 + 12 * q^55 + 10 * q^56 - 16 * q^57 - 24 * q^59 + 3 * q^60 - 16 * q^61 - 4 * q^63 - 11 * q^64 + 18 * q^65 - 8 * q^66 + 5 * q^67 + 24 * q^68 + 8 * q^70 + 20 * q^71 - 3 * q^72 - 29 * q^73 - 19 * q^74 - 16 * q^76 + 4 * q^77 - 2 * q^78 + 13 * q^79 + 9 * q^80 + 3 * q^81 + 36 * q^82 - 7 * q^83 - 12 * q^84 - 6 * q^85 - 10 * q^86 + 12 * q^87 + 12 * q^88 - 4 * q^89 - 5 * q^90 - 20 * q^92 - 14 * q^93 + 27 * q^94 + 22 * q^95 + 3 * q^96 - 24 * q^97 - 21 * q^98 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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