LMFDB - Embedded newform 4009.2.a.c.1.54

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4009,2,Mod(1,4009)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4009, 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("4009.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4009 = 19 \cdot 211 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4009.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$32.0120261703$
Analytic rank:	$1$
Dimension:	$71$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.54
Character	$\chi$	$=$	4009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.33446 q^{2} -1.20074 q^{3} -0.219214 q^{4} +3.89007 q^{5} -1.60233 q^{6} -1.22319 q^{7} -2.96145 q^{8} -1.55823 q^{9} +O(q^{10})$	\(q+1.33446 q^{2} -1.20074 q^{3} -0.219214 q^{4} +3.89007 q^{5} -1.60233 q^{6} -1.22319 q^{7} -2.96145 q^{8} -1.55823 q^{9} +5.19115 q^{10} +1.62955 q^{11} +0.263219 q^{12} -5.11688 q^{13} -1.63230 q^{14} -4.67095 q^{15} -3.51352 q^{16} +7.28397 q^{17} -2.07940 q^{18} +1.00000 q^{19} -0.852761 q^{20} +1.46873 q^{21} +2.17458 q^{22} -5.12904 q^{23} +3.55592 q^{24} +10.1327 q^{25} -6.82827 q^{26} +5.47323 q^{27} +0.268142 q^{28} -5.84579 q^{29} -6.23320 q^{30} -1.51703 q^{31} +1.23426 q^{32} -1.95666 q^{33} +9.72017 q^{34} -4.75831 q^{35} +0.341587 q^{36} -7.09052 q^{37} +1.33446 q^{38} +6.14402 q^{39} -11.5203 q^{40} +8.05003 q^{41} +1.95996 q^{42} +1.43962 q^{43} -0.357222 q^{44} -6.06164 q^{45} -6.84450 q^{46} -7.97452 q^{47} +4.21880 q^{48} -5.50380 q^{49} +13.5217 q^{50} -8.74612 q^{51} +1.12169 q^{52} -0.785173 q^{53} +7.30382 q^{54} +6.33909 q^{55} +3.62243 q^{56} -1.20074 q^{57} -7.80097 q^{58} -8.99944 q^{59} +1.02394 q^{60} -2.73849 q^{61} -2.02441 q^{62} +1.90602 q^{63} +8.67410 q^{64} -19.9050 q^{65} -2.61109 q^{66} -12.7085 q^{67} -1.59675 q^{68} +6.15862 q^{69} -6.34978 q^{70} -10.7738 q^{71} +4.61464 q^{72} +7.16088 q^{73} -9.46203 q^{74} -12.1667 q^{75} -0.219214 q^{76} -1.99326 q^{77} +8.19895 q^{78} +1.02335 q^{79} -13.6678 q^{80} -1.89721 q^{81} +10.7424 q^{82} +2.42766 q^{83} -0.321967 q^{84} +28.3352 q^{85} +1.92112 q^{86} +7.01925 q^{87} -4.82585 q^{88} -9.37339 q^{89} -8.08903 q^{90} +6.25893 q^{91} +1.12436 q^{92} +1.82155 q^{93} -10.6417 q^{94} +3.89007 q^{95} -1.48202 q^{96} -13.1838 q^{97} -7.34460 q^{98} -2.53923 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9}+O(q^{10}) $ 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9	\( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9} - 10 q^{10} - 52 q^{11} - 9 q^{12} - 15 q^{13} - 53 q^{14} - 33 q^{15} + 53 q^{16} - 10 q^{17} - 35 q^{18} + 71 q^{19} - 33 q^{20} - 38 q^{21} - 6 q^{22} - 65 q^{23} - 30 q^{24} + 51 q^{25} - 4 q^{26} - 23 q^{27} - 29 q^{28} - 97 q^{29} - 27 q^{30} - 53 q^{31} - 78 q^{32} - 17 q^{33} - 24 q^{34} - 38 q^{35} + 24 q^{36} - 33 q^{37} - 15 q^{38} - 86 q^{39} + 25 q^{40} - 69 q^{41} + 64 q^{42} - 10 q^{43} - 94 q^{44} - 34 q^{45} - 6 q^{46} - 37 q^{47} - q^{48} + 74 q^{49} - 41 q^{50} - 46 q^{51} - 30 q^{52} - 50 q^{53} - 17 q^{54} - 30 q^{55} - 116 q^{56} - 8 q^{57} + 11 q^{58} - 93 q^{59} - 56 q^{60} - 18 q^{61} - q^{62} - 84 q^{63} + 93 q^{64} - 78 q^{65} - 53 q^{66} - 5 q^{67} - 9 q^{68} - 69 q^{69} - 10 q^{70} - 221 q^{71} - 73 q^{72} - 34 q^{73} - 58 q^{74} - 70 q^{75} + 69 q^{76} - 2 q^{77} + 7 q^{78} - 68 q^{79} - 71 q^{80} + 39 q^{81} + 26 q^{82} - 45 q^{83} - 10 q^{84} - 44 q^{85} - 80 q^{86} - 7 q^{87} - 46 q^{88} - 143 q^{89} + 41 q^{90} - 30 q^{91} - 46 q^{92} + 32 q^{93} + 41 q^{94} - 18 q^{95} - 140 q^{96} - 18 q^{97} - 97 q^{98} - 142 q^{99}+O(q^{100}) \) 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9 - 10 * q^10 - 52 * q^11 - 9 * q^12 - 15 * q^13 - 53 * q^14 - 33 * q^15 + 53 * q^16 - 10 * q^17 - 35 * q^18 + 71 * q^19 - 33 * q^20 - 38 * q^21 - 6 * q^22 - 65 * q^23 - 30 * q^24 + 51 * q^25 - 4 * q^26 - 23 * q^27 - 29 * q^28 - 97 * q^29 - 27 * q^30 - 53 * q^31 - 78 * q^32 - 17 * q^33 - 24 * q^34 - 38 * q^35 + 24 * q^36 - 33 * q^37 - 15 * q^38 - 86 * q^39 + 25 * q^40 - 69 * q^41 + 64 * q^42 - 10 * q^43 - 94 * q^44 - 34 * q^45 - 6 * q^46 - 37 * q^47 - q^48 + 74 * q^49 - 41 * q^50 - 46 * q^51 - 30 * q^52 - 50 * q^53 - 17 * q^54 - 30 * q^55 - 116 * q^56 - 8 * q^57 + 11 * q^58 - 93 * q^59 - 56 * q^60 - 18 * q^61 - q^62 - 84 * q^63 + 93 * q^64 - 78 * q^65 - 53 * q^66 - 5 * q^67 - 9 * q^68 - 69 * q^69 - 10 * q^70 - 221 * q^71 - 73 * q^72 - 34 * q^73 - 58 * q^74 - 70 * q^75 + 69 * q^76 - 2 * q^77 + 7 * q^78 - 68 * q^79 - 71 * q^80 + 39 * q^81 + 26 * q^82 - 45 * q^83 - 10 * q^84 - 44 * q^85 - 80 * q^86 - 7 * q^87 - 46 * q^88 - 143 * q^89 + 41 * q^90 - 30 * q^91 - 46 * q^92 + 32 * q^93 + 41 * q^94 - 18 * q^95 - 140 * q^96 - 18 * q^97 - 97 * q^98 - 142 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.33446	0.943606	0.471803	−	0.881704i	$-0.343603\pi$
$2$	1.33446	0.943606	0.471803	+	0.881704i	$0.343603\pi$
$3$	−1.20074	−0.693245	−0.346623	−	0.938005i	$-0.612672\pi$
$3$	−1.20074	−0.693245	−0.346623	+	0.938005i	$0.612672\pi$
$4$	−0.219214	−0.109607
$5$	3.89007	1.73969	0.869847	−	0.493321i	$-0.164217\pi$
$5$	3.89007	1.73969	0.869847	+	0.493321i	$0.164217\pi$
$6$	−1.60233	−0.654150
$7$	−1.22319	−0.462323	−0.231162	−	0.972915i	$-0.574253\pi$
$7$	−1.22319	−0.462323	−0.231162	+	0.972915i	$0.574253\pi$
$8$	−2.96145	−1.04703
$9$	−1.55823	−0.519411
$10$	5.19115	1.64159
$11$	1.62955	0.491329	0.245665	−	0.969355i	$-0.420994\pi$
$11$	1.62955	0.491329	0.245665	+	0.969355i	$0.420994\pi$
$12$	0.263219	0.0759847
$13$	−5.11688	−1.41917	−0.709583	−	0.704622i	$-0.751117\pi$
$13$	−5.11688	−1.41917	−0.709583	+	0.704622i	$0.751117\pi$
$14$	−1.63230	−0.436251
$15$	−4.67095	−1.20603
$16$	−3.51352	−0.878379
$17$	7.28397	1.76662	0.883311	−	0.468787i	$-0.155309\pi$
$17$	7.28397	1.76662	0.883311	+	0.468787i	$0.155309\pi$
$18$	−2.07940	−0.490120
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−0.852761	−0.190683
$21$	1.46873	0.320504
$22$	2.17458	0.463621
$23$	−5.12904	−1.06948	−0.534739	−	0.845017i	$-0.679590\pi$
$23$	−5.12904	−1.06948	−0.534739	+	0.845017i	$0.679590\pi$
$24$	3.55592	0.725850
$25$	10.1327	2.02654
$26$	−6.82827	−1.33913
$27$	5.47323	1.05332
$28$	0.268142	0.0506740
$29$	−5.84579	−1.08554	−0.542768	−	0.839883i	$-0.682624\pi$
$29$	−5.84579	−1.08554	−0.542768	+	0.839883i	$0.682624\pi$
$30$	−6.23320	−1.13802
$31$	−1.51703	−0.272466	−0.136233	−	0.990677i	$-0.543500\pi$
$31$	−1.51703	−0.272466	−0.136233	+	0.990677i	$0.543500\pi$
$32$	1.23426	0.218188
$33$	−1.95666	−0.340612
$34$	9.72017	1.66700
$35$	−4.75831	−0.804301
$36$	0.341587	0.0569312
$37$	−7.09052	−1.16567	−0.582837	−	0.812589i	$-0.698058\pi$
$37$	−7.09052	−1.16567	−0.582837	+	0.812589i	$0.698058\pi$
$38$	1.33446	0.216478
$39$	6.14402	0.983830
$40$	−11.5203	−1.82152
$41$	8.05003	1.25720	0.628602	−	0.777727i	$-0.283628\pi$
$41$	8.05003	1.25720	0.628602	+	0.777727i	$0.283628\pi$
$42$	1.95996	0.302429
$43$	1.43962	0.219540	0.109770	−	0.993957i	$-0.464989\pi$
$43$	1.43962	0.219540	0.109770	+	0.993957i	$0.464989\pi$
$44$	−0.357222	−0.0538532
$45$	−6.06164	−0.903617
$46$	−6.84450	−1.00917
$47$	−7.97452	−1.16320	−0.581602	−	0.813474i	$-0.697574\pi$
$47$	−7.97452	−1.16320	−0.581602	+	0.813474i	$0.697574\pi$
$48$	4.21880	0.608932
$49$	−5.50380	−0.786257
$50$	13.5217	1.91225
$51$	−8.74612	−1.22470
$52$	1.12169	0.155551
$53$	−0.785173	−0.107852	−0.0539259	−	0.998545i	$-0.517173\pi$
$53$	−0.785173	−0.107852	−0.0539259	+	0.998545i	$0.517173\pi$
$54$	7.30382	0.993924
$55$	6.33909	0.854762
$56$	3.62243	0.484068
$57$	−1.20074	−0.159041
$58$	−7.80097	−1.02432
$59$	−8.99944	−1.17163	−0.585814	−	0.810446i	$-0.699225\pi$
$59$	−8.99944	−1.17163	−0.585814	+	0.810446i	$0.699225\pi$
$60$	1.02394	0.132190
$61$	−2.73849	−0.350627	−0.175314	−	0.984513i	$-0.556094\pi$
$61$	−2.73849	−0.350627	−0.175314	+	0.984513i	$0.556094\pi$
$62$	−2.02441	−0.257100
$63$	1.90602	0.240136
$64$	8.67410	1.08426
$65$	−19.9050	−2.46892
$66$	−2.61109	−0.321403
$67$	−12.7085	−1.55259	−0.776295	−	0.630370i	$-0.782903\pi$
$67$	−12.7085	−1.55259	−0.776295	+	0.630370i	$0.782903\pi$
$68$	−1.59675	−0.193635
$69$	6.15862	0.741411
$70$	−6.34978	−0.758944
$71$	−10.7738	−1.27861	−0.639307	−	0.768951i	$-0.720779\pi$
$71$	−10.7738	−1.27861	−0.639307	+	0.768951i	$0.720779\pi$
$72$	4.61464	0.543840
$73$	7.16088	0.838117	0.419059	−	0.907959i	$-0.362360\pi$
$73$	7.16088	0.838117	0.419059	+	0.907959i	$0.362360\pi$
$74$	−9.46203	−1.09994
$75$	−12.1667	−1.40489
$76$	−0.219214	−0.0251456
$77$	−1.99326	−0.227153
$78$	8.19895	0.928348
$79$	1.02335	0.115135	0.0575677	−	0.998342i	$-0.481665\pi$
$79$	1.02335	0.115135	0.0575677	+	0.998342i	$0.481665\pi$
$80$	−13.6678	−1.52811
$81$	−1.89721	−0.210801
$82$	10.7424	1.18630
$83$	2.42766	0.266471	0.133235	−	0.991084i	$-0.457463\pi$
$83$	2.42766	0.266471	0.133235	+	0.991084i	$0.457463\pi$
$84$	−0.321967	−0.0351295
$85$	28.3352	3.07338
$86$	1.92112	0.207159
$87$	7.01925	0.752542
$88$	−4.82585	−0.514437
$89$	−9.37339	−0.993577	−0.496789	−	0.867872i	$-0.665488\pi$
$89$	−9.37339	−0.993577	−0.496789	+	0.867872i	$0.665488\pi$
$90$	−8.08903	−0.852658
$91$	6.25893	0.656114
$92$	1.12436	0.117223
$93$	1.82155	0.188886
$94$	−10.6417	−1.09761
$95$	3.89007	0.399113
$96$	−1.48202	−0.151258
$97$	−13.1838	−1.33862	−0.669308	−	0.742985i	$-0.733410\pi$
$97$	−13.1838	−1.33862	−0.669308	+	0.742985i	$0.733410\pi$
$98$	−7.34460	−0.741917
$99$	−2.53923	−0.255202
$100$	−2.22123	−0.222123
$101$	−2.91012	−0.289568	−0.144784	−	0.989463i	$-0.546249\pi$
$101$	−2.91012	−0.289568	−0.144784	+	0.989463i	$0.546249\pi$
$102$	−11.6714	−1.15564
$103$	−16.0947	−1.58585	−0.792927	−	0.609316i	$-0.791444\pi$
$103$	−16.0947	−1.58585	−0.792927	+	0.609316i	$0.791444\pi$
$104$	15.1534	1.48591
$105$	5.71348	0.557578
$106$	−1.04778	−0.101770
$107$	3.91397	0.378378	0.189189	−	0.981941i	$-0.439414\pi$
$107$	3.91397	0.378378	0.189189	+	0.981941i	$0.439414\pi$
$108$	−1.19981	−0.115452
$109$	−3.90997	−0.374508	−0.187254	−	0.982312i	$-0.559959\pi$
$109$	−3.90997	−0.374508	−0.187254	+	0.982312i	$0.559959\pi$
$110$	8.45926	0.806559
$111$	8.51385	0.808098
$112$	4.29771	0.406095
$113$	0.715237	0.0672839	0.0336419	−	0.999434i	$-0.489289\pi$
$113$	0.715237	0.0672839	0.0336419	+	0.999434i	$0.489289\pi$
$114$	−1.60233	−0.150072
$115$	−19.9523	−1.86057
$116$	1.28148	0.118983
$117$	7.97329	0.737131
$118$	−12.0094	−1.10556
$119$	−8.90970	−0.816751
$120$	13.8328	1.26276
$121$	−8.34455	−0.758596
$122$	−3.65440	−0.330854
$123$	−9.66596	−0.871550
$124$	0.332554	0.0298642
$125$	19.9665	1.78586
$126$	2.54351	0.226594
$127$	4.67553	0.414886	0.207443	−	0.978247i	$-0.433486\pi$
$127$	4.67553	0.414886	0.207443	+	0.978247i	$0.433486\pi$
$128$	9.10673	0.804929
$129$	−1.72860	−0.152195
$130$	−26.5625	−2.32968
$131$	−10.3054	−0.900390	−0.450195	−	0.892930i	$-0.648646\pi$
$131$	−10.3054	−0.900390	−0.450195	+	0.892930i	$0.648646\pi$
$132$	0.428929	0.0373335
$133$	−1.22319	−0.106064
$134$	−16.9590	−1.46503
$135$	21.2913	1.83246
$136$	−21.5711	−1.84971
$137$	17.0225	1.45433	0.727165	−	0.686463i	$-0.240837\pi$
$137$	17.0225	1.45433	0.727165	+	0.686463i	$0.240837\pi$
$138$	8.21844	0.699600
$139$	4.76663	0.404300	0.202150	−	0.979355i	$-0.435207\pi$
$139$	4.76663	0.404300	0.202150	+	0.979355i	$0.435207\pi$
$140$	1.04309	0.0881573
$141$	9.57529	0.806385
$142$	−14.3772	−1.20651
$143$	−8.33823	−0.697278
$144$	5.47488	0.456240
$145$	−22.7405	−1.88850
$146$	9.55591	0.790853
$147$	6.60861	0.545069
$148$	1.55435	0.127766
$149$	13.3364	1.09256	0.546280	−	0.837603i	$-0.316043\pi$
$149$	13.3364	1.09256	0.546280	+	0.837603i	$0.316043\pi$
$150$	−16.2359	−1.32566
$151$	13.1171	1.06746	0.533729	−	0.845656i	$-0.320790\pi$
$151$	13.1171	1.06746	0.533729	+	0.845656i	$0.320790\pi$
$152$	−2.96145	−0.240206
$153$	−11.3501	−0.917603
$154$	−2.65993	−0.214343
$155$	−5.90134	−0.474007
$156$	−1.34686	−0.107835
$157$	3.59838	0.287182	0.143591	−	0.989637i	$-0.454135\pi$
$157$	3.59838	0.287182	0.143591	+	0.989637i	$0.454135\pi$
$158$	1.36562	0.108643
$159$	0.942785	0.0747677
$160$	4.80136	0.379581
$161$	6.27381	0.494445
$162$	−2.53175	−0.198913
$163$	−10.3577	−0.811282	−0.405641	−	0.914033i	$-0.632952\pi$
$163$	−10.3577	−0.811282	−0.405641	+	0.914033i	$0.632952\pi$
$164$	−1.76468	−0.137799
$165$	−7.61157	−0.592560
$166$	3.23962	0.251443
$167$	17.3739	1.34443	0.672215	−	0.740356i	$-0.265343\pi$
$167$	17.3739	1.34443	0.672215	+	0.740356i	$0.265343\pi$
$168$	−4.34958	−0.335578
$169$	13.1824	1.01403
$170$	37.8122	2.90006
$171$	−1.55823	−0.119161
$172$	−0.315586	−0.0240632
$173$	−2.75369	−0.209359	−0.104680	−	0.994506i	$-0.533382\pi$
$173$	−2.75369	−0.209359	−0.104680	+	0.994506i	$0.533382\pi$
$174$	9.36691	0.710103
$175$	−12.3942	−0.936915
$176$	−5.72547	−0.431573
$177$	10.8060	0.812225
$178$	−12.5084	−0.937546
$179$	4.40200	0.329021	0.164511	−	0.986375i	$-0.447396\pi$
$179$	4.40200	0.329021	0.164511	+	0.986375i	$0.447396\pi$
$180$	1.32880	0.0990429
$181$	−8.80641	−0.654575	−0.327288	−	0.944925i	$-0.606135\pi$
$181$	−8.80641	−0.654575	−0.327288	+	0.944925i	$0.606135\pi$
$182$	8.35229	0.619113
$183$	3.28820	0.243071
$184$	15.1894	1.11978
$185$	−27.5827	−2.02792
$186$	2.43078	0.178234
$187$	11.8696	0.867993
$188$	1.74813	0.127495
$189$	−6.69482	−0.486977
$190$	5.19115	0.376606
$191$	−0.537495	−0.0388918	−0.0194459	−	0.999811i	$-0.506190\pi$
$191$	−0.537495	−0.0388918	−0.0194459	+	0.999811i	$0.506190\pi$
$192$	−10.4153	−0.751660
$193$	−15.5585	−1.11992	−0.559961	−	0.828519i	$-0.689184\pi$
$193$	−15.5585	−1.11992	−0.559961	+	0.828519i	$0.689184\pi$
$194$	−17.5933	−1.26313
$195$	23.9007	1.71156
$196$	1.20651	0.0861795
$197$	7.21972	0.514384	0.257192	−	0.966360i	$-0.417203\pi$
$197$	7.21972	0.514384	0.257192	+	0.966360i	$0.417203\pi$
$198$	−3.38850	−0.240810
$199$	−18.6419	−1.32149	−0.660744	−	0.750611i	$-0.729759\pi$
$199$	−18.6419	−1.32149	−0.660744	+	0.750611i	$0.729759\pi$
$200$	−30.0075	−2.12185
$201$	15.2596	1.07633
$202$	−3.88344	−0.273238
$203$	7.15052	0.501868
$204$	1.91728	0.134236
$205$	31.3152	2.18715
$206$	−21.4777	−1.49642
$207$	7.99224	0.555499
$208$	17.9782	1.24657
$209$	1.62955	0.112719
$210$	7.62441	0.526134
$211$	1.00000	0.0688428
$211$	1.00000	0.0688428
$212$	0.172121	0.0118213
$213$	12.9365	0.886393
$214$	5.22304	0.357040
$215$	5.60023	0.381933
$216$	−16.2087	−1.10286
$217$	1.85561	0.125967
$218$	−5.21771	−0.353388
$219$	−8.59832	−0.581021
$220$	−1.38962	−0.0936882
$221$	−37.2712	−2.50713
$222$	11.3614	0.762527
$223$	−17.7457	−1.18834	−0.594171	−	0.804339i	$-0.702520\pi$
$223$	−17.7457	−1.18834	−0.594171	+	0.804339i	$0.702520\pi$
$224$	−1.50974	−0.100874
$225$	−15.7891	−1.05261
$226$	0.954456	0.0634895
$227$	−0.466640	−0.0309720	−0.0154860	−	0.999880i	$-0.504930\pi$
$227$	−0.466640	−0.0309720	−0.0154860	+	0.999880i	$0.504930\pi$
$228$	0.263219	0.0174321
$229$	−5.23205	−0.345744	−0.172872	−	0.984944i	$-0.555305\pi$
$229$	−5.23205	−0.345744	−0.172872	+	0.984944i	$0.555305\pi$
$230$	−26.6256	−1.75564
$231$	2.39338	0.157473
$232$	17.3120	1.13659
$233$	11.5759	0.758360	0.379180	−	0.925323i	$-0.376206\pi$
$233$	11.5759	0.758360	0.379180	+	0.925323i	$0.376206\pi$
$234$	10.6400	0.695561
$235$	−31.0215	−2.02362
$236$	1.97281	0.128419
$237$	−1.22877	−0.0798171
$238$	−11.8896	−0.770691
$239$	−13.2982	−0.860187	−0.430093	−	0.902784i	$-0.641519\pi$
$239$	−13.2982	−0.860187	−0.430093	+	0.902784i	$0.641519\pi$
$240$	16.4115	1.05936
$241$	24.0072	1.54644	0.773219	−	0.634139i	$-0.218645\pi$
$241$	24.0072	1.54644	0.773219	+	0.634139i	$0.218645\pi$
$242$	−11.1355	−0.715816
$243$	−14.1417	−0.907188
$244$	0.600316	0.0384313
$245$	−21.4102	−1.36785
$246$	−12.8988	−0.822400
$247$	−5.11688	−0.325579
$248$	4.49260	0.285280
$249$	−2.91498	−0.184729
$250$	26.6445	1.68515
$251$	−2.61993	−0.165369	−0.0826844	−	0.996576i	$-0.526349\pi$
$251$	−2.61993	−0.165369	−0.0826844	+	0.996576i	$0.526349\pi$
$252$	−0.417827	−0.0263206
$253$	−8.35805	−0.525466
$254$	6.23931	0.391489
$255$	−34.0231	−2.13061
$256$	−5.19563	−0.324727
$257$	−9.93812	−0.619923	−0.309961	−	0.950749i	$-0.600316\pi$
$257$	−9.93812	−0.619923	−0.309961	+	0.950749i	$0.600316\pi$
$258$	−2.30675	−0.143612
$259$	8.67308	0.538919
$260$	4.36347	0.270611
$261$	9.10910	0.563839
$262$	−13.7522	−0.849614
$263$	−13.8778	−0.855745	−0.427872	−	0.903839i	$-0.640737\pi$
$263$	−13.8778	−0.855745	−0.427872	+	0.903839i	$0.640737\pi$
$264$	5.79457	0.356631
$265$	−3.05438	−0.187629
$266$	−1.63230	−0.100083
$267$	11.2550	0.688793
$268$	2.78589	0.170175
$269$	27.4102	1.67123	0.835614	−	0.549317i	$-0.185112\pi$
$269$	27.4102	1.67123	0.835614	+	0.549317i	$0.185112\pi$
$270$	28.4124	1.72912
$271$	−30.4785	−1.85144	−0.925720	−	0.378210i	$-0.876540\pi$
$271$	−30.4785	−1.85144	−0.925720	+	0.378210i	$0.876540\pi$
$272$	−25.5923	−1.55176
$273$	−7.51532	−0.454848
$274$	22.7159	1.37232
$275$	16.5118	0.995696
$276$	−1.35006	−0.0812640
$277$	−9.38334	−0.563790	−0.281895	−	0.959445i	$-0.590963\pi$
$277$	−9.38334	−0.563790	−0.281895	+	0.959445i	$0.590963\pi$
$278$	6.36088	0.381500
$279$	2.36388	0.141522
$280$	14.0915	0.842130
$281$	−18.5838	−1.10862	−0.554309	−	0.832311i	$-0.687018\pi$
$281$	−18.5838	−1.10862	−0.554309	+	0.832311i	$0.687018\pi$
$282$	12.7779	0.760910
$283$	−19.8116	−1.17768	−0.588839	−	0.808250i	$-0.700415\pi$
$283$	−19.8116	−1.17768	−0.588839	+	0.808250i	$0.700415\pi$
$284$	2.36177	0.140145
$285$	−4.67095	−0.276683
$286$	−11.1270	−0.657956
$287$	−9.84674	−0.581235
$288$	−1.92327	−0.113329
$289$	36.0562	2.12095
$290$	−30.3464	−1.78200
$291$	15.8303	0.927990
$292$	−1.56977	−0.0918637
$293$	32.7701	1.91445	0.957226	−	0.289342i	$-0.0934363\pi$
$293$	32.7701	1.91445	0.957226	+	0.289342i	$0.0934363\pi$
$294$	8.81893	0.514330
$295$	−35.0085	−2.03827
$296$	20.9983	1.22050
$297$	8.91893	0.517529
$298$	17.7969	1.03095
$299$	26.2447	1.51777
$300$	2.66711	0.153986
$301$	−1.76093	−0.101499
$302$	17.5043	1.00726
$303$	3.49428	0.200741
$304$	−3.51352	−0.201514
$305$	−10.6529	−0.609984
$306$	−15.1463	−0.865856
$307$	28.6545	1.63540	0.817700	−	0.575645i	$-0.195249\pi$
$307$	28.6545	1.63540	0.817700	+	0.575645i	$0.195249\pi$
$308$	0.436951	0.0248976
$309$	19.3254	1.09939
$310$	−7.87511	−0.447276
$311$	−14.8378	−0.841376	−0.420688	−	0.907205i	$-0.638211\pi$
$311$	−14.8378	−0.841376	−0.420688	+	0.907205i	$0.638211\pi$
$312$	−18.1952	−1.03010
$313$	−7.82801	−0.442465	−0.221233	−	0.975221i	$-0.571008\pi$
$313$	−7.82801	−0.442465	−0.221233	+	0.975221i	$0.571008\pi$
$314$	4.80190	0.270987
$315$	7.41456	0.417763
$316$	−0.224332	−0.0126197
$317$	−7.13069	−0.400499	−0.200250	−	0.979745i	$-0.564175\pi$
$317$	−7.13069	−0.400499	−0.200250	+	0.979745i	$0.564175\pi$
$318$	1.25811	0.0705513
$319$	−9.52603	−0.533355
$320$	33.7429	1.88629
$321$	−4.69964	−0.262309
$322$	8.37215	0.466562
$323$	7.28397	0.405291
$324$	0.415895	0.0231053
$325$	−51.8477	−2.87599
$326$	−13.8220	−0.765530
$327$	4.69485	0.259626
$328$	−23.8398	−1.31633
$329$	9.75438	0.537776
$330$	−10.1573	−0.559143
$331$	−2.28437	−0.125560	−0.0627800	−	0.998027i	$-0.519997\pi$
$331$	−2.28437	−0.125560	−0.0627800	+	0.998027i	$0.519997\pi$
$332$	−0.532179	−0.0292071
$333$	11.0487	0.605465
$334$	23.1847	1.26861
$335$	−49.4370	−2.70103
$336$	−5.16041	−0.281524
$337$	28.1704	1.53454	0.767271	−	0.641323i	$-0.221614\pi$
$337$	28.1704	1.53454	0.767271	+	0.641323i	$0.221614\pi$
$338$	17.5914	0.956847
$339$	−0.858811	−0.0466442
$340$	−6.21148	−0.336865
$341$	−2.47207	−0.133870
$342$	−2.07940	−0.112441
$343$	15.2946	0.825829
$344$	−4.26337	−0.229866
$345$	23.9575	1.28983
$346$	−3.67469	−0.197552
$347$	10.0607	0.540089	0.270044	−	0.962848i	$-0.412962\pi$
$347$	10.0607	0.540089	0.270044	+	0.962848i	$0.412962\pi$
$348$	−1.53872	−0.0824841
$349$	−27.4359	−1.46861	−0.734306	−	0.678818i	$-0.762492\pi$
$349$	−27.4359	−1.46861	−0.734306	+	0.678818i	$0.762492\pi$
$350$	−16.5396	−0.884079
$351$	−28.0059	−1.49484
$352$	2.01129	0.107202
$353$	−3.19256	−0.169923	−0.0849614	−	0.996384i	$-0.527077\pi$
$353$	−3.19256	−0.169923	−0.0849614	+	0.996384i	$0.527077\pi$
$354$	14.4201	0.766421
$355$	−41.9109	−2.22440
$356$	2.05478	0.108903
$357$	10.6982	0.566209
$358$	5.87430	0.310467
$359$	2.93791	0.155057	0.0775284	−	0.996990i	$-0.475297\pi$
$359$	2.93791	0.155057	0.0775284	+	0.996990i	$0.475297\pi$
$360$	17.9513	0.946116
$361$	1.00000	0.0526316
$362$	−11.7518	−0.617661
$363$	10.0196	0.525893
$364$	−1.37205	−0.0719148
$365$	27.8563	1.45807
$366$	4.38797	0.229363
$367$	9.86537	0.514968	0.257484	−	0.966283i	$-0.417106\pi$
$367$	9.86537	0.514968	0.257484	+	0.966283i	$0.417106\pi$
$368$	18.0210	0.939408
$369$	−12.5438	−0.653005
$370$	−36.8080	−1.91356
$371$	0.960418	0.0498624
$372$	−0.399309	−0.0207032
$373$	−27.7711	−1.43793	−0.718967	−	0.695045i	$-0.755385\pi$
$373$	−27.7711	−1.43793	−0.718967	+	0.695045i	$0.755385\pi$
$374$	15.8395	0.819044
$375$	−23.9745	−1.23804
$376$	23.6162	1.21791
$377$	29.9122	1.54055
$378$	−8.93398	−0.459514
$379$	−9.84866	−0.505892	−0.252946	−	0.967480i	$-0.581399\pi$
$379$	−9.84866	−0.505892	−0.252946	+	0.967480i	$0.581399\pi$
$380$	−0.852761	−0.0437457
$381$	−5.61407	−0.287618
$382$	−0.717266	−0.0366985
$383$	29.8947	1.52755	0.763774	−	0.645484i	$-0.223344\pi$
$383$	29.8947	1.52755	0.763774	+	0.645484i	$0.223344\pi$
$384$	−10.9348	−0.558013
$385$	−7.75393	−0.395177
$386$	−20.7621	−1.05677
$387$	−2.24327	−0.114032
$388$	2.89009	0.146722
$389$	34.5351	1.75100	0.875501	−	0.483217i	$-0.160532\pi$
$389$	34.5351	1.75100	0.875501	+	0.483217i	$0.160532\pi$
$390$	31.8945	1.61504
$391$	−37.3598	−1.88937
$392$	16.2993	0.823237
$393$	12.3741	0.624191
$394$	9.63444	0.485376
$395$	3.98089	0.200300
$396$	0.556635	0.0279720
$397$	−2.72700	−0.136864	−0.0684322	−	0.997656i	$-0.521800\pi$
$397$	−2.72700	−0.136864	−0.0684322	+	0.997656i	$0.521800\pi$
$398$	−24.8769	−1.24696
$399$	1.46873	0.0735285
$400$	−35.6013	−1.78007
$401$	−7.32591	−0.365839	−0.182919	−	0.983128i	$-0.558555\pi$
$401$	−7.32591	−0.365839	−0.182919	+	0.983128i	$0.558555\pi$
$402$	20.3633	1.01563
$403$	7.76243	0.386674
$404$	0.637940	0.0317387
$405$	−7.38028	−0.366729
$406$	9.54209	0.473566
$407$	−11.5544	−0.572730
$408$	25.9012	1.28230
$409$	33.0061	1.63205	0.816024	−	0.578018i	$-0.196174\pi$
$409$	33.0061	1.63205	0.816024	+	0.578018i	$0.196174\pi$
$410$	41.7889	2.06381
$411$	−20.4395	−1.00821
$412$	3.52818	0.173821
$413$	11.0081	0.541671
$414$	10.6653	0.524173
$415$	9.44379	0.463577
$416$	−6.31556	−0.309646
$417$	−5.72346	−0.280279
$418$	2.17458	0.106362
$419$	29.0440	1.41889	0.709445	−	0.704761i	$-0.248946\pi$
$419$	29.0440	1.41889	0.709445	+	0.704761i	$0.248946\pi$
$420$	−1.25248	−0.0611146
$421$	3.67996	0.179350	0.0896751	−	0.995971i	$-0.471417\pi$
$421$	3.67996	0.179350	0.0896751	+	0.995971i	$0.471417\pi$
$422$	1.33446	0.0649605
$423$	12.4262	0.604181
$424$	2.32525	0.112924
$425$	73.8061	3.58012
$426$	17.2632	0.836406
$427$	3.34970	0.162103
$428$	−0.857999	−0.0414729
$429$	10.0120	0.483384
$430$	7.47329	0.360394
$431$	−11.9805	−0.577083	−0.288541	−	0.957467i	$-0.593170\pi$
$431$	−11.9805	−0.577083	−0.288541	+	0.957467i	$0.593170\pi$
$432$	−19.2303	−0.925218
$433$	−39.7468	−1.91011	−0.955054	−	0.296433i	$-0.904203\pi$
$433$	−39.7468	−1.91011	−0.955054	+	0.296433i	$0.904203\pi$
$434$	2.47624	0.118864
$435$	27.3054	1.30919
$436$	0.857123	0.0410488
$437$	−5.12904	−0.245355
$438$	−11.4741	−0.548255
$439$	−29.0457	−1.38628	−0.693139	−	0.720804i	$-0.743773\pi$
$439$	−29.0457	−1.38628	−0.693139	+	0.720804i	$0.743773\pi$
$440$	−18.7729	−0.894964
$441$	8.57620	0.408391
$442$	−49.7369	−2.36574
$443$	19.2231	0.913318	0.456659	−	0.889642i	$-0.349046\pi$
$443$	19.2231	0.913318	0.456659	+	0.889642i	$0.349046\pi$
$444$	−1.86636	−0.0885734
$445$	−36.4632	−1.72852
$446$	−23.6810	−1.12133
$447$	−16.0135	−0.757412
$448$	−10.6101	−0.501280
$449$	9.11527	0.430176	0.215088	−	0.976595i	$-0.430996\pi$
$449$	9.11527	0.430176	0.215088	+	0.976595i	$0.430996\pi$
$450$	−21.0699	−0.993245
$451$	13.1180	0.617701
$452$	−0.156790	−0.00737480
$453$	−15.7502	−0.740010
$454$	−0.622713	−0.0292254
$455$	24.3477	1.14144
$456$	3.55592	0.166521
$457$	28.3975	1.32838	0.664190	−	0.747564i	$-0.268777\pi$
$457$	28.3975	1.32838	0.664190	+	0.747564i	$0.268777\pi$
$458$	−6.98197	−0.326246
$459$	39.8669	1.86083
$460$	4.37384	0.203932
$461$	32.5641	1.51666	0.758332	−	0.651869i	$-0.226015\pi$
$461$	32.5641	1.51666	0.758332	+	0.651869i	$0.226015\pi$
$462$	3.19387	0.148592
$463$	20.8504	0.969000	0.484500	−	0.874791i	$-0.339002\pi$
$463$	20.8504	0.969000	0.484500	+	0.874791i	$0.339002\pi$
$464$	20.5393	0.953511
$465$	7.08595	0.328603
$466$	15.4475	0.715594
$467$	16.1733	0.748413	0.374206	−	0.927345i	$-0.377915\pi$
$467$	16.1733	0.748413	0.374206	+	0.927345i	$0.377915\pi$
$468$	−1.74786	−0.0807949
$469$	15.5449	0.717799
$470$	−41.3969	−1.90950
$471$	−4.32070	−0.199087
$472$	26.6514	1.22673
$473$	2.34594	0.107866
$474$	−1.63974	−0.0753159
$475$	10.1327	0.464919
$476$	1.95314	0.0895218
$477$	1.22348	0.0560194
$478$	−17.7459	−0.811678
$479$	17.2203	0.786816	0.393408	−	0.919364i	$-0.371296\pi$
$479$	17.2203	0.786816	0.393408	+	0.919364i	$0.371296\pi$
$480$	−5.76517	−0.263143
$481$	36.2813	1.65429
$482$	32.0366	1.45923
$483$	−7.53318	−0.342772
$484$	1.82925	0.0831476
$485$	−51.2862	−2.32878
$486$	−18.8715	−0.856028
$487$	−14.0790	−0.637980	−0.318990	−	0.947758i	$-0.603344\pi$
$487$	−14.0790	−0.637980	−0.318990	+	0.947758i	$0.603344\pi$
$488$	8.10990	0.367118
$489$	12.4369	0.562417
$490$	−28.5711	−1.29071
$491$	−2.51351	−0.113433	−0.0567164	−	0.998390i	$-0.518063\pi$
$491$	−2.51351	−0.113433	−0.0567164	+	0.998390i	$0.518063\pi$
$492$	2.11892	0.0955282
$493$	−42.5805	−1.91773
$494$	−6.82827	−0.307218
$495$	−9.87778	−0.443973
$496$	5.33009	0.239328
$497$	13.1784	0.591133
$498$	−3.88993	−0.174312
$499$	−28.0685	−1.25652	−0.628260	−	0.778004i	$-0.716232\pi$
$499$	−28.0685	−1.25652	−0.628260	+	0.778004i	$0.716232\pi$
$500$	−4.37695	−0.195743
$501$	−20.8614	−0.932020
$502$	−3.49620	−0.156043
$503$	−0.867746	−0.0386909	−0.0193454	−	0.999813i	$-0.506158\pi$
$503$	−0.867746	−0.0386909	−0.0193454	+	0.999813i	$0.506158\pi$
$504$	−5.64459	−0.251430
$505$	−11.3206	−0.503759
$506$	−11.1535	−0.495833
$507$	−15.8286	−0.702973
$508$	−1.02494	−0.0454745
$509$	−29.8378	−1.32254	−0.661269	−	0.750149i	$-0.729982\pi$
$509$	−29.8378	−1.32254	−0.661269	+	0.750149i	$0.729982\pi$
$510$	−45.4025	−2.01045
$511$	−8.75913	−0.387481
$512$	−25.1468	−1.11134
$513$	5.47323	0.241649
$514$	−13.2620	−0.584963
$515$	−62.6095	−2.75890
$516$	0.378935	0.0166817
$517$	−12.9949	−0.571516
$518$	11.5739	0.508527
$519$	3.30645	0.145137
$520$	58.9478	2.58503
$521$	−4.01060	−0.175707	−0.0878537	−	0.996133i	$-0.528001\pi$
$521$	−4.01060	−0.175707	−0.0878537	+	0.996133i	$0.528001\pi$
$522$	12.1557	0.532042
$523$	−17.3129	−0.757039	−0.378520	−	0.925593i	$-0.623567\pi$
$523$	−17.3129	−0.757039	−0.378520	+	0.925593i	$0.623567\pi$
$524$	2.25910	0.0986893
$525$	14.8822	0.649512
$526$	−18.5194	−0.807486
$527$	−11.0500	−0.481344
$528$	6.87477	0.299186
$529$	3.30705	0.143785
$530$	−4.07595	−0.177048
$531$	14.0232	0.608557
$532$	0.268142	0.0116254
$533$	−41.1910	−1.78418
$534$	15.0193	0.649949
$535$	15.2256	0.658262
$536$	37.6356	1.62561
$537$	−5.28564	−0.228092
$538$	36.5778	1.57698
$539$	−8.96874	−0.386311
$540$	−4.66736	−0.200851
$541$	−10.9833	−0.472207	−0.236104	−	0.971728i	$-0.575871\pi$
$541$	−10.9833	−0.472207	−0.236104	+	0.971728i	$0.575871\pi$
$542$	−40.6724	−1.74703
$543$	10.5742	0.453781
$544$	8.99031	0.385456
$545$	−15.2101	−0.651529
$546$	−10.0289	−0.429197
$547$	8.34209	0.356682	0.178341	−	0.983969i	$-0.442927\pi$
$547$	8.34209	0.356682	0.178341	+	0.983969i	$0.442927\pi$
$548$	−3.73158	−0.159405
$549$	4.26720	0.182120
$550$	22.0343	0.939545
$551$	−5.84579	−0.249039
$552$	−18.2385	−0.776281
$553$	−1.25175	−0.0532298
$554$	−12.5217	−0.531996
$555$	33.1195	1.40584
$556$	−1.04491	−0.0443142
$557$	33.9255	1.43747	0.718735	−	0.695284i	$-0.244721\pi$
$557$	33.9255	1.43747	0.718735	+	0.695284i	$0.244721\pi$
$558$	3.15450	0.133541
$559$	−7.36636	−0.311564
$560$	16.7184	0.706482
$561$	−14.2523	−0.601732
$562$	−24.7994	−1.04610
$563$	10.5103	0.442956	0.221478	−	0.975165i	$-0.428912\pi$
$563$	10.5103	0.442956	0.221478	+	0.975165i	$0.428912\pi$
$564$	−2.09904	−0.0883856
$565$	2.78233	0.117053
$566$	−26.4378	−1.11126
$567$	2.32065	0.0974582
$568$	31.9061	1.33875
$569$	23.5021	0.985259	0.492629	−	0.870239i	$-0.336036\pi$
$569$	23.5021	0.985259	0.492629	+	0.870239i	$0.336036\pi$
$570$	−6.23320	−0.261080
$571$	−8.50015	−0.355720	−0.177860	−	0.984056i	$-0.556917\pi$
$571$	−8.50015	−0.355720	−0.177860	+	0.984056i	$0.556917\pi$
$572$	1.82786	0.0764267
$573$	0.645390	0.0269615
$574$	−13.1401	−0.548457
$575$	−51.9709	−2.16734
$576$	−13.5163	−0.563178
$577$	−12.1558	−0.506052	−0.253026	−	0.967460i	$-0.581426\pi$
$577$	−12.1558	−0.506052	−0.253026	+	0.967460i	$0.581426\pi$
$578$	48.1156	2.00135
$579$	18.6816	0.776380
$580$	4.98506	0.206993
$581$	−2.96950	−0.123196
$582$	21.1249	0.875657
$583$	−1.27948	−0.0529907
$584$	−21.2066	−0.877536
$585$	31.0167	1.28238
$586$	43.7305	1.80649
$587$	−3.07487	−0.126914	−0.0634568	−	0.997985i	$-0.520213\pi$
$587$	−3.07487	−0.126914	−0.0634568	+	0.997985i	$0.520213\pi$
$588$	−1.44870	−0.0597435
$589$	−1.51703	−0.0625079
$590$	−46.7175	−1.92333
$591$	−8.66898	−0.356594
$592$	24.9127	1.02390
$593$	28.2416	1.15974	0.579872	−	0.814708i	$-0.303103\pi$
$593$	28.2416	1.15974	0.579872	+	0.814708i	$0.303103\pi$
$594$	11.9020	0.488344
$595$	−34.6594	−1.42090
$596$	−2.92353	−0.119752
$597$	22.3840	0.916115
$598$	35.0225	1.43218
$599$	−1.47967	−0.0604575	−0.0302288	−	0.999543i	$-0.509624\pi$
$599$	−1.47967	−0.0604575	−0.0302288	+	0.999543i	$0.509624\pi$
$600$	36.0311	1.47096
$601$	−0.639556	−0.0260880	−0.0130440	−	0.999915i	$-0.504152\pi$
$601$	−0.639556	−0.0260880	−0.0130440	+	0.999915i	$0.504152\pi$
$602$	−2.34990	−0.0957747
$603$	19.8028	0.806433
$604$	−2.87547	−0.117001
$605$	−32.4609	−1.31972
$606$	4.66298	0.189421
$607$	−42.9551	−1.74349	−0.871747	−	0.489957i	$-0.837013\pi$
$607$	−42.9551	−1.74349	−0.871747	+	0.489957i	$0.837013\pi$
$608$	1.23426	0.0500559
$609$	−8.58589	−0.347918
$610$	−14.2159	−0.575585
$611$	40.8046	1.65078
$612$	2.48811	0.100576
$613$	17.6085	0.711199	0.355599	−	0.934638i	$-0.384277\pi$
$613$	17.6085	0.711199	0.355599	+	0.934638i	$0.384277\pi$
$614$	38.2383	1.54317
$615$	−37.6013	−1.51623
$616$	5.90295	0.237837
$617$	19.2645	0.775560	0.387780	−	0.921752i	$-0.373242\pi$
$617$	19.2645	0.775560	0.387780	+	0.921752i	$0.373242\pi$
$618$	25.7890	1.03739
$619$	1.30408	0.0524154	0.0262077	−	0.999657i	$-0.491657\pi$
$619$	1.30408	0.0524154	0.0262077	+	0.999657i	$0.491657\pi$
$620$	1.29366	0.0519546
$621$	−28.0724	−1.12651
$622$	−19.8005	−0.793927
$623$	11.4655	0.459354
$624$	−21.5871	−0.864176
$625$	27.0078	1.08031
$626$	−10.4462	−0.417513
$627$	−1.95666	−0.0781416
$628$	−0.788817	−0.0314772
$629$	−51.6472	−2.05931
$630$	9.89444	0.394204
$631$	31.6313	1.25922	0.629612	−	0.776910i	$-0.283214\pi$
$631$	31.6313	1.25922	0.629612	+	0.776910i	$0.283214\pi$
$632$	−3.03059	−0.120551
$633$	−1.20074	−0.0477250
$634$	−9.51562	−0.377914
$635$	18.1881	0.721775
$636$	−0.206672	−0.00819508
$637$	28.1623	1.11583
$638$	−12.7121	−0.503277
$639$	16.7881	0.664127
$640$	35.4259	1.40033
$641$	−26.7587	−1.05690	−0.528452	−	0.848963i	$-0.677228\pi$
$641$	−26.7587	−1.05690	−0.528452	+	0.848963i	$0.677228\pi$
$642$	−6.27149	−0.247516
$643$	−18.9549	−0.747507	−0.373753	−	0.927528i	$-0.621929\pi$
$643$	−18.9549	−0.747507	−0.373753	+	0.927528i	$0.621929\pi$
$644$	−1.37531	−0.0541948
$645$	−6.72440	−0.264773
$646$	9.72017	0.382435
$647$	−3.99984	−0.157250	−0.0786249	−	0.996904i	$-0.525053\pi$
$647$	−3.99984	−0.157250	−0.0786249	+	0.996904i	$0.525053\pi$
$648$	5.61850	0.220715
$649$	−14.6651	−0.575655
$650$	−69.1887	−2.71380
$651$	−2.22810	−0.0873262
$652$	2.27057	0.0889223
$653$	−23.7552	−0.929612	−0.464806	−	0.885412i	$-0.653876\pi$
$653$	−23.7552	−0.929612	−0.464806	+	0.885412i	$0.653876\pi$
$654$	6.26509	0.244984
$655$	−40.0889	−1.56640
$656$	−28.2839	−1.10430
$657$	−11.1583	−0.435327
$658$	13.0168	0.507449
$659$	39.7873	1.54989	0.774946	−	0.632028i	$-0.217777\pi$
$659$	39.7873	1.54989	0.774946	+	0.632028i	$0.217777\pi$
$660$	1.66857	0.0649489
$661$	−11.5122	−0.447771	−0.223886	−	0.974615i	$-0.571874\pi$
$661$	−11.5122	−0.447771	−0.223886	+	0.974615i	$0.571874\pi$
$662$	−3.04840	−0.118479
$663$	44.7528	1.73806
$664$	−7.18941	−0.279003
$665$	−4.75831	−0.184519
$666$	14.7440	0.571320
$667$	29.9833	1.16096
$668$	−3.80860	−0.147359
$669$	21.3079	0.823812
$670$	−65.9718	−2.54871
$671$	−4.46251	−0.172273
$672$	1.81280	0.0699301
$673$	17.4148	0.671290	0.335645	−	0.941989i	$-0.391046\pi$
$673$	17.4148	0.671290	0.335645	+	0.941989i	$0.391046\pi$
$674$	37.5924	1.44800
$675$	55.4585	2.13460
$676$	−2.88978	−0.111145
$677$	39.1476	1.50456	0.752282	−	0.658842i	$-0.228953\pi$
$677$	39.1476	1.50456	0.752282	+	0.658842i	$0.228953\pi$
$678$	−1.14605	−0.0440138
$679$	16.1264	0.618874
$680$	−83.9134	−3.21793
$681$	0.560312	0.0214712
$682$	−3.29889	−0.126321
$683$	44.3243	1.69602	0.848011	−	0.529979i	$-0.177800\pi$
$683$	44.3243	1.69602	0.848011	+	0.529979i	$0.177800\pi$
$684$	0.341587	0.0130609
$685$	66.2188	2.53009
$686$	20.4100	0.779257
$687$	6.28231	0.239685
$688$	−5.05813	−0.192839
$689$	4.01763	0.153060
$690$	31.9703	1.21709
$691$	30.4646	1.15893	0.579464	−	0.814998i	$-0.303262\pi$
$691$	30.4646	1.15893	0.579464	+	0.814998i	$0.303262\pi$
$692$	0.603648	0.0229473
$693$	3.10596	0.117986
$694$	13.4257	0.509631
$695$	18.5425	0.703359
$696$	−20.7872	−0.787936
$697$	58.6362	2.22100
$698$	−36.6122	−1.38579
$699$	−13.8996	−0.525730
$700$	2.71699	0.102693
$701$	13.3794	0.505333	0.252667	−	0.967553i	$-0.418692\pi$
$701$	13.3794	0.505333	0.252667	+	0.967553i	$0.418692\pi$
$702$	−37.3727	−1.41054
$703$	−7.09052	−0.267424
$704$	14.1349	0.532730
$705$	37.2486	1.40286
$706$	−4.26035	−0.160340
$707$	3.55964	0.133874
$708$	−2.36882	−0.0890258
$709$	31.4220	1.18008	0.590039	−	0.807375i	$-0.299112\pi$
$709$	31.4220	1.18008	0.590039	+	0.807375i	$0.299112\pi$
$710$	−55.9284	−2.09896
$711$	−1.59461	−0.0598026
$712$	27.7589	1.04031
$713$	7.78088	0.291396
$714$	14.2763	0.534278
$715$	−32.4363	−1.21305
$716$	−0.964983	−0.0360631
$717$	15.9676	0.596320
$718$	3.92052	0.146312
$719$	0.603625	0.0225114	0.0112557	−	0.999937i	$-0.496417\pi$
$719$	0.603625	0.0225114	0.0112557	+	0.999937i	$0.496417\pi$
$720$	21.2977	0.793718
$721$	19.6869	0.733178
$722$	1.33446	0.0496635
$723$	−28.8263	−1.07206
$724$	1.93049	0.0717462
$725$	−59.2335	−2.19988
$726$	13.3708	0.496236
$727$	2.98097	0.110558	0.0552790	−	0.998471i	$-0.482395\pi$
$727$	2.98097	0.110558	0.0552790	+	0.998471i	$0.482395\pi$
$728$	−18.5355	−0.686972
$729$	22.6720	0.839704
$730$	37.1732	1.37584
$731$	10.4862	0.387844
$732$	−0.720820	−0.0266423
$733$	16.6928	0.616562	0.308281	−	0.951295i	$-0.400246\pi$
$733$	16.6928	0.616562	0.308281	+	0.951295i	$0.400246\pi$
$734$	13.1650	0.485927
$735$	25.7080	0.948253
$736$	−6.33057	−0.233348
$737$	−20.7092	−0.762833
$738$	−16.7392	−0.616180
$739$	9.15487	0.336767	0.168384	−	0.985722i	$-0.446145\pi$
$739$	9.15487	0.336767	0.168384	+	0.985722i	$0.446145\pi$
$740$	6.04652	0.222274
$741$	6.14402	0.225706
$742$	1.28164	0.0470505
$743$	−35.1364	−1.28903	−0.644515	−	0.764591i	$-0.722941\pi$
$743$	−35.1364	−1.28903	−0.644515	+	0.764591i	$0.722941\pi$
$744$	−5.39443	−0.197769
$745$	51.8796	1.90072
$746$	−37.0595	−1.35684
$747$	−3.78287	−0.138408
$748$	−2.60199	−0.0951383
$749$	−4.78754	−0.174933
$750$	−31.9930	−1.16822
$751$	31.0248	1.13211	0.566056	−	0.824367i	$-0.308468\pi$
$751$	31.0248	1.13211	0.566056	+	0.824367i	$0.308468\pi$
$752$	28.0186	1.02173
$753$	3.14585	0.114641
$754$	39.9166	1.45368
$755$	51.0266	1.85705
$756$	1.46760	0.0533762
$757$	30.7401	1.11727	0.558634	−	0.829414i	$-0.311326\pi$
$757$	30.7401	1.11727	0.558634	+	0.829414i	$0.311326\pi$
$758$	−13.1426	−0.477362
$759$	10.0358	0.364277
$760$	−11.5203	−0.417884
$761$	−37.8349	−1.37151	−0.685756	−	0.727831i	$-0.740528\pi$
$761$	−37.8349	−1.37151	−0.685756	+	0.727831i	$0.740528\pi$
$762$	−7.49176	−0.271398
$763$	4.78265	0.173144
$764$	0.117827	0.00426282
$765$	−44.1528	−1.59635
$766$	39.8933	1.44140
$767$	46.0490	1.66273
$768$	6.23858	0.225115
$769$	−16.7590	−0.604344	−0.302172	−	0.953253i	$-0.597712\pi$
$769$	−16.7590	−0.604344	−0.302172	+	0.953253i	$0.597712\pi$
$770$	−10.3473	−0.372891
$771$	11.9331	0.429758
$772$	3.41064	0.122752
$773$	36.6044	1.31657	0.658285	−	0.752769i	$-0.271282\pi$
$773$	36.6044	1.31657	0.658285	+	0.752769i	$0.271282\pi$
$774$	−2.99355	−0.107601
$775$	−15.3715	−0.552162
$776$	39.0434	1.40158
$777$	−10.4141	−0.373603
$778$	46.0858	1.65226
$779$	8.05003	0.288422
$780$	−5.23938	−0.187600
$781$	−17.5565	−0.628220
$782$	−49.8552	−1.78282
$783$	−31.9954	−1.14342
$784$	19.3377	0.690632
$785$	13.9980	0.499609
$786$	16.5128	0.588991
$787$	−11.7496	−0.418827	−0.209413	−	0.977827i	$-0.567155\pi$
$787$	−11.7496	−0.418827	−0.209413	+	0.977827i	$0.567155\pi$
$788$	−1.58267	−0.0563802
$789$	16.6636	0.593241
$790$	5.31235	0.189005
$791$	−0.874873	−0.0311069
$792$	7.51980	0.267205
$793$	14.0125	0.497598
$794$	−3.63908	−0.129146
$795$	3.66750	0.130073
$796$	4.08657	0.144845
$797$	−37.3665	−1.32359	−0.661794	−	0.749685i	$-0.730205\pi$
$797$	−37.3665	−1.32359	−0.661794	+	0.749685i	$0.730205\pi$
$798$	1.95996	0.0693820
$799$	−58.0862	−2.05494
$800$	12.5064	0.442167
$801$	14.6059	0.516075
$802$	−9.77614	−0.345208
$803$	11.6690	0.411791
$804$	−3.34511	−0.117973
$805$	24.4056	0.860183
$806$	10.3587	0.364868
$807$	−32.9124	−1.15857
$808$	8.61818	0.303187
$809$	13.7433	0.483187	0.241594	−	0.970378i	$-0.422330\pi$
$809$	13.7433	0.483187	0.241594	+	0.970378i	$0.422330\pi$
$810$	−9.84870	−0.346048
$811$	−10.2843	−0.361131	−0.180566	−	0.983563i	$-0.557793\pi$
$811$	−10.2843	−0.361131	−0.180566	+	0.983563i	$0.557793\pi$
$812$	−1.56750	−0.0550084
$813$	36.5967	1.28350
$814$	−15.4189	−0.540432
$815$	−40.2924	−1.41138
$816$	30.7296	1.07575
$817$	1.43962	0.0503660
$818$	44.0454	1.54001
$819$	−9.75287	−0.340793
$820$	−6.86475	−0.239727
$821$	48.0125	1.67565	0.837823	−	0.545942i	$-0.183828\pi$
$821$	48.0125	1.67565	0.837823	+	0.545942i	$0.183828\pi$
$822$	−27.2757	−0.951351
$823$	−53.8144	−1.87585	−0.937925	−	0.346837i	$-0.887256\pi$
$823$	−53.8144	−1.87585	−0.937925	+	0.346837i	$0.887256\pi$
$824$	47.6636	1.66044
$825$	−19.8263	−0.690262
$826$	14.6898	0.511124
$827$	−49.5358	−1.72253	−0.861265	−	0.508156i	$-0.830327\pi$
$827$	−49.5358	−1.72253	−0.861265	+	0.508156i	$0.830327\pi$
$828$	−1.75202	−0.0608867
$829$	13.9816	0.485601	0.242801	−	0.970076i	$-0.421934\pi$
$829$	13.9816	0.485601	0.242801	+	0.970076i	$0.421934\pi$
$830$	12.6024	0.437434
$831$	11.2669	0.390845
$832$	−44.3843	−1.53875
$833$	−40.0895	−1.38902
$834$	−7.63774	−0.264473
$835$	67.5856	2.33890
$836$	−0.357222	−0.0123548
$837$	−8.30303	−0.286995
$838$	38.7580	1.33887
$839$	19.2895	0.665947	0.332973	−	0.942936i	$-0.391948\pi$
$839$	19.2895	0.665947	0.332973	+	0.942936i	$0.391948\pi$
$840$	−16.9202	−0.583802
$841$	5.17322	0.178387
$842$	4.91076	0.169236
$843$	22.3143	0.768544
$844$	−0.219214	−0.00754567
$845$	51.2806	1.76411
$846$	16.5822	0.570109
$847$	10.2070	0.350717
$848$	2.75872	0.0947347
$849$	23.7885	0.816420
$850$	98.4914	3.37823
$851$	36.3676	1.24666
$852$	−2.83586	−0.0971551
$853$	23.3158	0.798319	0.399159	−	0.916882i	$-0.369302\pi$
$853$	23.3158	0.798319	0.399159	+	0.916882i	$0.369302\pi$
$854$	4.47004	0.152962
$855$	−6.06164	−0.207304
$856$	−11.5910	−0.396174
$857$	14.6480	0.500366	0.250183	−	0.968199i	$-0.419509\pi$
$857$	14.6480	0.500366	0.250183	+	0.968199i	$0.419509\pi$
$858$	13.3606	0.456124
$859$	−41.8699	−1.42858	−0.714291	−	0.699848i	$-0.753251\pi$
$859$	−41.8699	−1.42858	−0.714291	+	0.699848i	$0.753251\pi$
$860$	−1.22765	−0.0418626
$861$	11.8233	0.402938
$862$	−15.9876	−0.544539
$863$	−26.5586	−0.904066	−0.452033	−	0.892001i	$-0.649301\pi$
$863$	−26.5586	−0.904066	−0.452033	+	0.892001i	$0.649301\pi$
$864$	6.75539	0.229823
$865$	−10.7121	−0.364221
$866$	−53.0405	−1.80239
$867$	−43.2940	−1.47034
$868$	−0.406778	−0.0138069
$869$	1.66760	0.0565694
$870$	36.4380	1.23536
$871$	65.0278	2.20338
$872$	11.5792	0.392122
$873$	20.5435	0.695293
$874$	−6.84450	−0.231519
$875$	−24.4229	−0.825645
$876$	1.88488	0.0636841
$877$	21.8758	0.738692	0.369346	−	0.929292i	$-0.379582\pi$
$877$	21.8758	0.738692	0.369346	+	0.929292i	$0.379582\pi$
$878$	−38.7604	−1.30810
$879$	−39.3483	−1.32718
$880$	−22.2725	−0.750805
$881$	−10.6259	−0.357997	−0.178999	−	0.983849i	$-0.557286\pi$
$881$	−10.6259	−0.357997	−0.178999	+	0.983849i	$0.557286\pi$
$882$	11.4446	0.385360
$883$	9.87480	0.332313	0.166157	−	0.986099i	$-0.446864\pi$
$883$	9.87480	0.332313	0.166157	+	0.986099i	$0.446864\pi$
$884$	8.17038	0.274800
$885$	42.0360	1.41302
$886$	25.6525	0.861813
$887$	38.5010	1.29274	0.646369	−	0.763025i	$-0.276287\pi$
$887$	38.5010	1.29274	0.646369	+	0.763025i	$0.276287\pi$
$888$	−25.2134	−0.846105
$889$	−5.71907	−0.191811
$890$	−48.6587	−1.63104
$891$	−3.09160	−0.103573
$892$	3.89012	0.130251
$893$	−7.97452	−0.266857
$894$	−21.3694	−0.714699
$895$	17.1241	0.572396
$896$	−11.1393	−0.372138
$897$	−31.5129	−1.05219
$898$	12.1640	0.405917
$899$	8.86821	0.295771
$900$	3.46120	0.115373
$901$	−5.71917	−0.190533
$902$	17.5054	0.582866
$903$	2.11442	0.0703634
$904$	−2.11814	−0.0704484
$905$	−34.2576	−1.13876
$906$	−21.0180	−0.698278
$907$	−7.88809	−0.261920	−0.130960	−	0.991388i	$-0.541806\pi$
$907$	−7.88809	−0.261920	−0.130960	+	0.991388i	$0.541806\pi$
$908$	0.102294	0.00339476
$909$	4.53464	0.150405
$910$	32.4910	1.07707
$911$	2.12373	0.0703624	0.0351812	−	0.999381i	$-0.488799\pi$
$911$	2.12373	0.0703624	0.0351812	+	0.999381i	$0.488799\pi$
$912$	4.21880	0.139699
$913$	3.95601	0.130925
$914$	37.8954	1.25347
$915$	12.7913	0.422868
$916$	1.14694	0.0378960
$917$	12.6055	0.416271
$918$	53.2008	1.75589
$919$	−33.3971	−1.10167	−0.550835	−	0.834614i	$-0.685691\pi$
$919$	−33.3971	−1.10167	−0.550835	+	0.834614i	$0.685691\pi$
$920$	59.0880	1.94807
$921$	−34.4065	−1.13373
$922$	43.4556	1.43113
$923$	55.1282	1.81457
$924$	−0.524663	−0.0172601
$925$	−71.8460	−2.36228
$926$	27.8240	0.914354
$927$	25.0793	0.823711
$928$	−7.21522	−0.236851
$929$	7.74693	0.254168	0.127084	−	0.991892i	$-0.459438\pi$
$929$	7.74693	0.254168	0.127084	+	0.991892i	$0.459438\pi$
$930$	9.45592	0.310072
$931$	−5.50380	−0.180380
$932$	−2.53760	−0.0831218
$933$	17.8163	0.583280
$934$	21.5827	0.706207
$935$	46.1737	1.51004
$936$	−23.6125	−0.771800
$937$	27.2881	0.891462	0.445731	−	0.895167i	$-0.352944\pi$
$937$	27.2881	0.891462	0.445731	+	0.895167i	$0.352944\pi$
$938$	20.7441	0.677320
$939$	9.39937	0.306737
$940$	6.80036	0.221803
$941$	24.7621	0.807221	0.403610	−	0.914931i	$-0.367755\pi$
$941$	24.7621	0.807221	0.403610	+	0.914931i	$0.367755\pi$
$942$	−5.76581	−0.187860
$943$	−41.2889	−1.34455
$944$	31.6197	1.02913
$945$	−26.0434	−0.847190
$946$	3.13057	0.101783
$947$	22.3198	0.725296	0.362648	−	0.931926i	$-0.381873\pi$
$947$	22.3198	0.725296	0.362648	+	0.931926i	$0.381873\pi$
$948$	0.269364	0.00874853
$949$	−36.6413	−1.18943
$950$	13.5217	0.438701
$951$	8.56207	0.277644
$952$	26.3857	0.855165
$953$	52.3570	1.69601	0.848004	−	0.529989i	$-0.177804\pi$
$953$	52.3570	1.69601	0.848004	+	0.529989i	$0.177804\pi$
$954$	1.63269	0.0528603
$955$	−2.09090	−0.0676598
$956$	2.91515	0.0942827
$957$	11.4382	0.369746
$958$	22.9798	0.742445
$959$	−20.8218	−0.672371
$960$	−40.5163	−1.30766
$961$	−28.6986	−0.925762
$962$	48.4160	1.56099
$963$	−6.09888	−0.196534
$964$	−5.26272	−0.169501
$965$	−60.5235	−1.94832
$966$	−10.0527	−0.323442
$967$	25.3829	0.816259	0.408129	−	0.912924i	$-0.366181\pi$
$967$	25.3829	0.816259	0.408129	+	0.912924i	$0.366181\pi$
$968$	24.7120	0.794274
$969$	−8.74612	−0.280966
$970$	−68.4394	−2.19746
$971$	5.59153	0.179441	0.0897203	−	0.995967i	$-0.471403\pi$
$971$	5.59153	0.179441	0.0897203	+	0.995967i	$0.471403\pi$
$972$	3.10006	0.0994343
$973$	−5.83051	−0.186918
$974$	−18.7879	−0.602002
$975$	62.2554	1.99377
$976$	9.62171	0.307983
$977$	5.22076	0.167027	0.0835135	−	0.996507i	$-0.473386\pi$
$977$	5.22076	0.167027	0.0835135	+	0.996507i	$0.473386\pi$
$978$	16.5966	0.530700
$979$	−15.2744	−0.488174
$980$	4.69342	0.149926
$981$	6.09265	0.194523
$982$	−3.35417	−0.107036
$983$	29.3194	0.935145	0.467572	−	0.883955i	$-0.345129\pi$
$983$	29.3194	0.935145	0.467572	+	0.883955i	$0.345129\pi$
$984$	28.6253	0.912541
$985$	28.0853	0.894871
$986$	−56.8221	−1.80958
$987$	−11.7124	−0.372811
$988$	1.12169	0.0356858
$989$	−7.38387	−0.234793
$990$	−13.1815	−0.418936
$991$	48.6860	1.54656	0.773280	−	0.634064i	$-0.218615\pi$
$991$	48.6860	1.54656	0.773280	+	0.634064i	$0.218615\pi$
$992$	−1.87240	−0.0594489
$993$	2.74292	0.0870439
$994$	17.5861	0.557797
$995$	−72.5183	−2.29898
$996$	0.639006	0.0202477
$997$	−39.5857	−1.25369	−0.626846	−	0.779143i	$-0.715654\pi$
$997$	−39.5857	−1.25369	−0.626846	+	0.779143i	$0.715654\pi$
$998$	−37.4563	−1.18566
$999$	−38.8081	−1.22783

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4009.2.a.c.1.54	✓	71

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4009.2.a.c.1.54	✓	71	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:	\( N \)	\(=\)	\( 4009 = 19 \cdot 211 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4009.a (trivial)

\(f(q)\)	\(=\)	\(q+1.33446 q^{2} -1.20074 q^{3} -0.219214 q^{4} +3.89007 q^{5} -1.60233 q^{6} -1.22319 q^{7} -2.96145 q^{8} -1.55823 q^{9} +O(q^{10})\)	\(q+1.33446 q^{2} -1.20074 q^{3} -0.219214 q^{4} +3.89007 q^{5} -1.60233 q^{6} -1.22319 q^{7} -2.96145 q^{8} -1.55823 q^{9} +5.19115 q^{10} +1.62955 q^{11} +0.263219 q^{12} -5.11688 q^{13} -1.63230 q^{14} -4.67095 q^{15} -3.51352 q^{16} +7.28397 q^{17} -2.07940 q^{18} +1.00000 q^{19} -0.852761 q^{20} +1.46873 q^{21} +2.17458 q^{22} -5.12904 q^{23} +3.55592 q^{24} +10.1327 q^{25} -6.82827 q^{26} +5.47323 q^{27} +0.268142 q^{28} -5.84579 q^{29} -6.23320 q^{30} -1.51703 q^{31} +1.23426 q^{32} -1.95666 q^{33} +9.72017 q^{34} -4.75831 q^{35} +0.341587 q^{36} -7.09052 q^{37} +1.33446 q^{38} +6.14402 q^{39} -11.5203 q^{40} +8.05003 q^{41} +1.95996 q^{42} +1.43962 q^{43} -0.357222 q^{44} -6.06164 q^{45} -6.84450 q^{46} -7.97452 q^{47} +4.21880 q^{48} -5.50380 q^{49} +13.5217 q^{50} -8.74612 q^{51} +1.12169 q^{52} -0.785173 q^{53} +7.30382 q^{54} +6.33909 q^{55} +3.62243 q^{56} -1.20074 q^{57} -7.80097 q^{58} -8.99944 q^{59} +1.02394 q^{60} -2.73849 q^{61} -2.02441 q^{62} +1.90602 q^{63} +8.67410 q^{64} -19.9050 q^{65} -2.61109 q^{66} -12.7085 q^{67} -1.59675 q^{68} +6.15862 q^{69} -6.34978 q^{70} -10.7738 q^{71} +4.61464 q^{72} +7.16088 q^{73} -9.46203 q^{74} -12.1667 q^{75} -0.219214 q^{76} -1.99326 q^{77} +8.19895 q^{78} +1.02335 q^{79} -13.6678 q^{80} -1.89721 q^{81} +10.7424 q^{82} +2.42766 q^{83} -0.321967 q^{84} +28.3352 q^{85} +1.92112 q^{86} +7.01925 q^{87} -4.82585 q^{88} -9.37339 q^{89} -8.08903 q^{90} +6.25893 q^{91} +1.12436 q^{92} +1.82155 q^{93} -10.6417 q^{94} +3.89007 q^{95} -1.48202 q^{96} -13.1838 q^{97} -7.34460 q^{98} -2.53923 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9}+O(q^{10}) \) 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9	\( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9} - 10 q^{10} - 52 q^{11} - 9 q^{12} - 15 q^{13} - 53 q^{14} - 33 q^{15} + 53 q^{16} - 10 q^{17} - 35 q^{18} + 71 q^{19} - 33 q^{20} - 38 q^{21} - 6 q^{22} - 65 q^{23} - 30 q^{24} + 51 q^{25} - 4 q^{26} - 23 q^{27} - 29 q^{28} - 97 q^{29} - 27 q^{30} - 53 q^{31} - 78 q^{32} - 17 q^{33} - 24 q^{34} - 38 q^{35} + 24 q^{36} - 33 q^{37} - 15 q^{38} - 86 q^{39} + 25 q^{40} - 69 q^{41} + 64 q^{42} - 10 q^{43} - 94 q^{44} - 34 q^{45} - 6 q^{46} - 37 q^{47} - q^{48} + 74 q^{49} - 41 q^{50} - 46 q^{51} - 30 q^{52} - 50 q^{53} - 17 q^{54} - 30 q^{55} - 116 q^{56} - 8 q^{57} + 11 q^{58} - 93 q^{59} - 56 q^{60} - 18 q^{61} - q^{62} - 84 q^{63} + 93 q^{64} - 78 q^{65} - 53 q^{66} - 5 q^{67} - 9 q^{68} - 69 q^{69} - 10 q^{70} - 221 q^{71} - 73 q^{72} - 34 q^{73} - 58 q^{74} - 70 q^{75} + 69 q^{76} - 2 q^{77} + 7 q^{78} - 68 q^{79} - 71 q^{80} + 39 q^{81} + 26 q^{82} - 45 q^{83} - 10 q^{84} - 44 q^{85} - 80 q^{86} - 7 q^{87} - 46 q^{88} - 143 q^{89} + 41 q^{90} - 30 q^{91} - 46 q^{92} + 32 q^{93} + 41 q^{94} - 18 q^{95} - 140 q^{96} - 18 q^{97} - 97 q^{98} - 142 q^{99}+O(q^{100}) \) 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9 - 10 * q^10 - 52 * q^11 - 9 * q^12 - 15 * q^13 - 53 * q^14 - 33 * q^15 + 53 * q^16 - 10 * q^17 - 35 * q^18 + 71 * q^19 - 33 * q^20 - 38 * q^21 - 6 * q^22 - 65 * q^23 - 30 * q^24 + 51 * q^25 - 4 * q^26 - 23 * q^27 - 29 * q^28 - 97 * q^29 - 27 * q^30 - 53 * q^31 - 78 * q^32 - 17 * q^33 - 24 * q^34 - 38 * q^35 + 24 * q^36 - 33 * q^37 - 15 * q^38 - 86 * q^39 + 25 * q^40 - 69 * q^41 + 64 * q^42 - 10 * q^43 - 94 * q^44 - 34 * q^45 - 6 * q^46 - 37 * q^47 - q^48 + 74 * q^49 - 41 * q^50 - 46 * q^51 - 30 * q^52 - 50 * q^53 - 17 * q^54 - 30 * q^55 - 116 * q^56 - 8 * q^57 + 11 * q^58 - 93 * q^59 - 56 * q^60 - 18 * q^61 - q^62 - 84 * q^63 + 93 * q^64 - 78 * q^65 - 53 * q^66 - 5 * q^67 - 9 * q^68 - 69 * q^69 - 10 * q^70 - 221 * q^71 - 73 * q^72 - 34 * q^73 - 58 * q^74 - 70 * q^75 + 69 * q^76 - 2 * q^77 + 7 * q^78 - 68 * q^79 - 71 * q^80 + 39 * q^81 + 26 * q^82 - 45 * q^83 - 10 * q^84 - 44 * q^85 - 80 * q^86 - 7 * q^87 - 46 * q^88 - 143 * q^89 + 41 * q^90 - 30 * q^91 - 46 * q^92 + 32 * q^93 + 41 * q^94 - 18 * q^95 - 140 * q^96 - 18 * q^97 - 97 * q^98 - 142 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more