LMFDB - Embedded newform 1682.4.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1682,4,Mod(1,1682)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1682, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1682.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1682 = 2 \cdot 29^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1682.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:	$99.2412126297$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 6 $ x^2 - 6
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 58)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.44949$ of defining polynomial
Character	$\chi$	$=$	1682.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{2} -1.44949 q^{3} +4.00000 q^{4} -19.6969 q^{5} -2.89898 q^{6} +11.5959 q^{7} +8.00000 q^{8} -24.8990 q^{9} +O(q^{10})$	\(q+2.00000 q^{2} -1.44949 q^{3} +4.00000 q^{4} -19.6969 q^{5} -2.89898 q^{6} +11.5959 q^{7} +8.00000 q^{8} -24.8990 q^{9} -39.3939 q^{10} +37.6515 q^{11} -5.79796 q^{12} -44.5959 q^{13} +23.1918 q^{14} +28.5505 q^{15} +16.0000 q^{16} +61.1918 q^{17} -49.7980 q^{18} -63.7980 q^{19} -78.7878 q^{20} -16.8082 q^{21} +75.3031 q^{22} +177.060 q^{23} -11.5959 q^{24} +262.969 q^{25} -89.1918 q^{26} +75.2270 q^{27} +46.3837 q^{28} +57.1010 q^{30} +233.994 q^{31} +32.0000 q^{32} -54.5755 q^{33} +122.384 q^{34} -228.404 q^{35} -99.5959 q^{36} -10.2020 q^{37} -127.596 q^{38} +64.6413 q^{39} -157.576 q^{40} -347.959 q^{41} -33.6163 q^{42} +194.823 q^{43} +150.606 q^{44} +490.434 q^{45} +354.120 q^{46} +14.5005 q^{47} -23.1918 q^{48} -208.535 q^{49} +525.939 q^{50} -88.6969 q^{51} -178.384 q^{52} -606.373 q^{53} +150.454 q^{54} -741.620 q^{55} +92.7673 q^{56} +92.4745 q^{57} -702.372 q^{59} +114.202 q^{60} -543.394 q^{61} +467.989 q^{62} -288.727 q^{63} +64.0000 q^{64} +878.403 q^{65} -109.151 q^{66} -407.010 q^{67} +244.767 q^{68} -256.647 q^{69} -456.808 q^{70} +314.717 q^{71} -199.192 q^{72} +859.110 q^{73} -20.4041 q^{74} -381.171 q^{75} -255.192 q^{76} +436.604 q^{77} +129.283 q^{78} -725.266 q^{79} -315.151 q^{80} +563.232 q^{81} -695.918 q^{82} -820.919 q^{83} -67.2327 q^{84} -1205.29 q^{85} +389.646 q^{86} +301.212 q^{88} +648.363 q^{89} +980.867 q^{90} -517.131 q^{91} +708.241 q^{92} -339.172 q^{93} +29.0010 q^{94} +1256.62 q^{95} -46.3837 q^{96} +60.9490 q^{97} -417.069 q^{98} -937.485 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{2} + 2 q^{3} + 8 q^{4} - 10 q^{5} + 4 q^{6} - 16 q^{7} + 16 q^{8} - 40 q^{9}+O(q^{10}) $ 2 * q + 4 * q^2 + 2 * q^3 + 8 * q^4 - 10 * q^5 + 4 * q^6 - 16 * q^7 + 16 * q^8 - 40 * q^9	\( 2 q + 4 q^{2} + 2 q^{3} + 8 q^{4} - 10 q^{5} + 4 q^{6} - 16 q^{7} + 16 q^{8} - 40 q^{9} - 20 q^{10} + 90 q^{11} + 8 q^{12} - 50 q^{13} - 32 q^{14} + 62 q^{15} + 32 q^{16} + 44 q^{17} - 80 q^{18} - 108 q^{19} - 40 q^{20} - 112 q^{21} + 180 q^{22} - 28 q^{23} + 16 q^{24} + 232 q^{25} - 100 q^{26} - 70 q^{27} - 64 q^{28} + 124 q^{30} - 66 q^{31} + 64 q^{32} + 126 q^{33} + 88 q^{34} - 496 q^{35} - 160 q^{36} - 40 q^{37} - 216 q^{38} + 46 q^{39} - 80 q^{40} - 304 q^{41} - 224 q^{42} + 130 q^{43} + 360 q^{44} + 344 q^{45} - 56 q^{46} + 514 q^{47} + 32 q^{48} + 210 q^{49} + 464 q^{50} - 148 q^{51} - 200 q^{52} - 958 q^{53} - 140 q^{54} - 234 q^{55} - 128 q^{56} - 60 q^{57} - 180 q^{59} + 248 q^{60} - 1028 q^{61} - 132 q^{62} + 128 q^{63} + 128 q^{64} + 826 q^{65} + 252 q^{66} - 912 q^{67} + 176 q^{68} - 964 q^{69} - 992 q^{70} + 796 q^{71} - 320 q^{72} + 856 q^{73} - 80 q^{74} - 488 q^{75} - 432 q^{76} - 1008 q^{77} + 92 q^{78} + 318 q^{79} - 160 q^{80} + 470 q^{81} - 608 q^{82} - 1828 q^{83} - 448 q^{84} - 1372 q^{85} + 260 q^{86} + 720 q^{88} + 944 q^{89} + 688 q^{90} - 368 q^{91} - 112 q^{92} - 1374 q^{93} + 1028 q^{94} + 828 q^{95} + 64 q^{96} - 368 q^{97} + 420 q^{98} - 1728 q^{99}+O(q^{100}) \) 2 * q + 4 * q^2 + 2 * q^3 + 8 * q^4 - 10 * q^5 + 4 * q^6 - 16 * q^7 + 16 * q^8 - 40 * q^9 - 20 * q^10 + 90 * q^11 + 8 * q^12 - 50 * q^13 - 32 * q^14 + 62 * q^15 + 32 * q^16 + 44 * q^17 - 80 * q^18 - 108 * q^19 - 40 * q^20 - 112 * q^21 + 180 * q^22 - 28 * q^23 + 16 * q^24 + 232 * q^25 - 100 * q^26 - 70 * q^27 - 64 * q^28 + 124 * q^30 - 66 * q^31 + 64 * q^32 + 126 * q^33 + 88 * q^34 - 496 * q^35 - 160 * q^36 - 40 * q^37 - 216 * q^38 + 46 * q^39 - 80 * q^40 - 304 * q^41 - 224 * q^42 + 130 * q^43 + 360 * q^44 + 344 * q^45 - 56 * q^46 + 514 * q^47 + 32 * q^48 + 210 * q^49 + 464 * q^50 - 148 * q^51 - 200 * q^52 - 958 * q^53 - 140 * q^54 - 234 * q^55 - 128 * q^56 - 60 * q^57 - 180 * q^59 + 248 * q^60 - 1028 * q^61 - 132 * q^62 + 128 * q^63 + 128 * q^64 + 826 * q^65 + 252 * q^66 - 912 * q^67 + 176 * q^68 - 964 * q^69 - 992 * q^70 + 796 * q^71 - 320 * q^72 + 856 * q^73 - 80 * q^74 - 488 * q^75 - 432 * q^76 - 1008 * q^77 + 92 * q^78 + 318 * q^79 - 160 * q^80 + 470 * q^81 - 608 * q^82 - 1828 * q^83 - 448 * q^84 - 1372 * q^85 + 260 * q^86 + 720 * q^88 + 944 * q^89 + 688 * q^90 - 368 * q^91 - 112 * q^92 - 1374 * q^93 + 1028 * q^94 + 828 * q^95 + 64 * q^96 - 368 * q^97 + 420 * q^98 - 1728 * 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 $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	−1.44949	−0.278954	−0.139477	−	0.990225i	$-0.544542\pi$
$3$	−1.44949	−0.278954	−0.139477	+	0.990225i	$0.544542\pi$
$4$	4.00000	0.500000
$5$	−19.6969	−1.76175	−0.880874	−	0.473351i	$-0.843044\pi$
$5$	−19.6969	−1.76175	−0.880874	+	0.473351i	$0.843044\pi$
$6$	−2.89898	−0.197251
$7$	11.5959	0.626121	0.313060	−	0.949733i	$-0.398646\pi$
$7$	11.5959	0.626121	0.313060	+	0.949733i	$0.398646\pi$
$8$	8.00000	0.353553
$9$	−24.8990	−0.922184
$10$	−39.3939	−1.24574
$11$	37.6515	1.03203	0.516017	−	0.856579i	$-0.327414\pi$
$11$	37.6515	1.03203	0.516017	+	0.856579i	$0.327414\pi$
$12$	−5.79796	−0.139477
$13$	−44.5959	−0.951437	−0.475719	−	0.879598i	$-0.657812\pi$
$13$	−44.5959	−0.951437	−0.475719	+	0.879598i	$0.657812\pi$
$14$	23.1918	0.442734
$15$	28.5505	0.491447
$16$	16.0000	0.250000
$17$	61.1918	0.873012	0.436506	−	0.899701i	$-0.356216\pi$
$17$	61.1918	0.873012	0.436506	+	0.899701i	$0.356216\pi$
$18$	−49.7980	−0.652083
$19$	−63.7980	−0.770329	−0.385165	−	0.922848i	$-0.625855\pi$
$19$	−63.7980	−0.770329	−0.385165	+	0.922848i	$0.625855\pi$
$20$	−78.7878	−0.880874
$21$	−16.8082	−0.174659
$22$	75.3031	0.729758
$23$	177.060	1.60520	0.802600	−	0.596517i	$-0.203449\pi$
$23$	177.060	1.60520	0.802600	+	0.596517i	$0.203449\pi$
$24$	−11.5959	−0.0986253
$25$	262.969	2.10376
$26$	−89.1918	−0.672768
$27$	75.2270	0.536202
$28$	46.3837	0.313060
$29$	0	0
$29$	0	0
$30$	57.1010	0.347506
$31$	233.994	1.35570	0.677849	−	0.735201i	$-0.262912\pi$
$31$	233.994	1.35570	0.677849	+	0.735201i	$0.262912\pi$
$32$	32.0000	0.176777
$33$	−54.5755	−0.287890
$34$	122.384	0.617313
$35$	−228.404	−1.10307
$36$	−99.5959	−0.461092
$37$	−10.2020	−0.0453299	−0.0226649	−	0.999743i	$-0.507215\pi$
$37$	−10.2020	−0.0453299	−0.0226649	+	0.999743i	$0.507215\pi$
$38$	−127.596	−0.544705
$39$	64.6413	0.265408
$40$	−157.576	−0.622872
$41$	−347.959	−1.32542	−0.662708	−	0.748877i	$-0.730593\pi$
$41$	−347.959	−1.32542	−0.662708	+	0.748877i	$0.730593\pi$
$42$	−33.6163	−0.123503
$43$	194.823	0.690935	0.345468	−	0.938431i	$-0.387720\pi$
$43$	194.823	0.690935	0.345468	+	0.938431i	$0.387720\pi$
$44$	150.606	0.516017
$45$	490.434	1.62466
$46$	354.120	1.13505
$47$	14.5005	0.0450025	0.0225013	−	0.999747i	$-0.492837\pi$
$47$	14.5005	0.0450025	0.0225013	+	0.999747i	$0.492837\pi$
$48$	−23.1918	−0.0697386
$49$	−208.535	−0.607973
$50$	525.939	1.48758
$51$	−88.6969	−0.243531
$52$	−178.384	−0.475719
$53$	−606.373	−1.57154	−0.785772	−	0.618517i	$-0.787734\pi$
$53$	−606.373	−1.57154	−0.785772	+	0.618517i	$0.787734\pi$
$54$	150.454	0.379152
$55$	−741.620	−1.81818
$56$	92.7673	0.221367
$57$	92.4745	0.214887
$58$	0	0
$59$	−702.372	−1.54985	−0.774925	−	0.632054i	$-0.782212\pi$
$59$	−702.372	−1.54985	−0.774925	+	0.632054i	$0.782212\pi$
$60$	114.202	0.245724
$61$	−543.394	−1.14056	−0.570282	−	0.821449i	$-0.693166\pi$
$61$	−543.394	−1.14056	−0.570282	+	0.821449i	$0.693166\pi$
$62$	467.989	0.958623
$63$	−288.727	−0.577399
$64$	64.0000	0.125000
$65$	878.403	1.67619
$66$	−109.151	−0.203569
$67$	−407.010	−0.742152	−0.371076	−	0.928602i	$-0.621011\pi$
$67$	−407.010	−0.742152	−0.371076	+	0.928602i	$0.621011\pi$
$68$	244.767	0.436506
$69$	−256.647	−0.447778
$70$	−456.808	−0.779986
$71$	314.717	0.526057	0.263029	−	0.964788i	$-0.415279\pi$
$71$	314.717	0.526057	0.263029	+	0.964788i	$0.415279\pi$
$72$	−199.192	−0.326041
$73$	859.110	1.37741	0.688707	−	0.725040i	$-0.258179\pi$
$73$	859.110	1.37741	0.688707	+	0.725040i	$0.258179\pi$
$74$	−20.4041	−0.0320531
$75$	−381.171	−0.586852
$76$	−255.192	−0.385165
$77$	436.604	0.646177
$78$	129.283	0.187672
$79$	−725.266	−1.03290	−0.516448	−	0.856319i	$-0.672746\pi$
$79$	−725.266	−1.03290	−0.516448	+	0.856319i	$0.672746\pi$
$80$	−315.151	−0.440437
$81$	563.232	0.772609
$82$	−695.918	−0.937211
$83$	−820.919	−1.08563	−0.542817	−	0.839851i	$-0.682642\pi$
$83$	−820.919	−1.08563	−0.542817	+	0.839851i	$0.682642\pi$
$84$	−67.2327	−0.0873296
$85$	−1205.29	−1.53803
$86$	389.646	0.488565
$87$	0	0
$88$	301.212	0.364879
$89$	648.363	0.772206	0.386103	−	0.922456i	$-0.373821\pi$
$89$	648.363	0.772206	0.386103	+	0.922456i	$0.373821\pi$
$90$	980.867	1.14881
$91$	−517.131	−0.595714
$92$	708.241	0.802600
$93$	−339.172	−0.378178
$94$	29.0010	0.0318216
$95$	1256.62	1.35713
$96$	−46.3837	−0.0493126
$97$	60.9490	0.0637983	0.0318991	−	0.999491i	$-0.489844\pi$
$97$	60.9490	0.0637983	0.0318991	+	0.999491i	$0.489844\pi$
$98$	−417.069	−0.429902
$99$	−937.485	−0.951725
$100$	1051.88	1.05188
$101$	−1100.16	−1.08386	−0.541930	−	0.840423i	$-0.682306\pi$
$101$	−1100.16	−1.08386	−0.541930	+	0.840423i	$0.682306\pi$
$102$	−177.394	−0.172202
$103$	747.930	0.715492	0.357746	−	0.933819i	$-0.383545\pi$
$103$	747.930	0.715492	0.357746	+	0.933819i	$0.383545\pi$
$104$	−356.767	−0.336384
$105$	331.069	0.307705
$106$	−1212.75	−1.11125
$107$	176.059	0.159068	0.0795340	−	0.996832i	$-0.474657\pi$
$107$	176.059	0.159068	0.0795340	+	0.996832i	$0.474657\pi$
$108$	300.908	0.268101
$109$	173.856	0.152774	0.0763871	−	0.997078i	$-0.475662\pi$
$109$	173.856	0.152774	0.0763871	+	0.997078i	$0.475662\pi$
$110$	−1483.24	−1.28565
$111$	14.7878	0.0126450
$112$	185.535	0.156530
$113$	1707.65	1.42161	0.710807	−	0.703387i	$-0.248330\pi$
$113$	1707.65	1.42161	0.710807	+	0.703387i	$0.248330\pi$
$114$	184.949	0.151948
$115$	−3487.54	−2.82796
$116$	0	0
$117$	1110.39	0.877400
$118$	−1404.74	−1.09591
$119$	709.576	0.546611
$120$	228.404	0.173753
$121$	86.6378	0.0650922
$122$	−1086.79	−0.806501
$123$	504.363	0.369731
$124$	935.978	0.677849
$125$	−2717.57	−1.94454
$126$	−577.453	−0.408283
$127$	881.473	0.615891	0.307945	−	0.951404i	$-0.400359\pi$
$127$	881.473	0.615891	0.307945	+	0.951404i	$0.400359\pi$
$128$	128.000	0.0883883
$129$	−282.394	−0.192739
$130$	1756.81	1.18525
$131$	−2025.49	−1.35090	−0.675451	−	0.737405i	$-0.736051\pi$
$131$	−2025.49	−1.35090	−0.675451	+	0.737405i	$0.736051\pi$
$132$	−218.302	−0.143945
$133$	−739.796	−0.482319
$134$	−814.020	−0.524781
$135$	−1481.74	−0.944652
$136$	489.535	0.308656
$137$	−1594.32	−0.994248	−0.497124	−	0.867679i	$-0.665611\pi$
$137$	−1594.32	−0.994248	−0.497124	+	0.867679i	$0.665611\pi$
$138$	−513.294	−0.316627
$139$	−2855.40	−1.74239	−0.871195	−	0.490938i	$-0.836654\pi$
$139$	−2855.40	−1.74239	−0.871195	+	0.490938i	$0.836654\pi$
$140$	−913.616	−0.551533
$141$	−21.0183	−0.0125536
$142$	629.435	0.371979
$143$	−1679.10	−0.981915
$144$	−398.384	−0.230546
$145$	0	0
$146$	1718.22	0.973979
$147$	302.269	0.169597
$148$	−40.8082	−0.0226649
$149$	−18.3755	−0.0101032	−0.00505162	−	0.999987i	$-0.501608\pi$
$149$	−18.3755	−0.0101032	−0.00505162	+	0.999987i	$0.501608\pi$
$150$	−762.343	−0.414967
$151$	−778.806	−0.419724	−0.209862	−	0.977731i	$-0.567301\pi$
$151$	−778.806	−0.419724	−0.209862	+	0.977731i	$0.567301\pi$
$152$	−510.384	−0.272353
$153$	−1523.61	−0.805078
$154$	873.208	0.456916
$155$	−4608.97	−2.38840
$156$	258.565	0.132704
$157$	511.896	0.260215	0.130107	−	0.991500i	$-0.458468\pi$
$157$	511.896	0.260215	0.130107	+	0.991500i	$0.458468\pi$
$158$	−1450.53	−0.730368
$159$	878.932	0.438389
$160$	−630.302	−0.311436
$161$	2053.18	1.00505
$162$	1126.46	0.546317
$163$	1451.59	0.697529	0.348764	−	0.937210i	$-0.386601\pi$
$163$	1451.59	0.697529	0.348764	+	0.937210i	$0.386601\pi$
$164$	−1391.84	−0.662708
$165$	1074.97	0.507190
$166$	−1641.84	−0.767659
$167$	600.374	0.278194	0.139097	−	0.990279i	$-0.455580\pi$
$167$	600.374	0.278194	0.139097	+	0.990279i	$0.455580\pi$
$168$	−134.465	−0.0617513
$169$	−208.204	−0.0947675
$170$	−2410.58	−1.08755
$171$	1588.50	0.710386
$172$	779.292	0.345468
$173$	−2574.20	−1.13129	−0.565644	−	0.824650i	$-0.691372\pi$
$173$	−2574.20	−1.13129	−0.565644	+	0.824650i	$0.691372\pi$
$174$	0	0
$175$	3049.37	1.31720
$176$	602.424	0.258008
$177$	1018.08	0.432337
$178$	1296.73	0.546032
$179$	3828.72	1.59872	0.799362	−	0.600850i	$-0.205171\pi$
$179$	3828.72	1.59872	0.799362	+	0.600850i	$0.205171\pi$
$180$	1961.73	0.812328
$181$	2075.64	0.852381	0.426190	−	0.904633i	$-0.359855\pi$
$181$	2075.64	0.852381	0.426190	+	0.904633i	$0.359855\pi$
$182$	−1034.26	−0.421234
$183$	787.644	0.318166
$184$	1416.48	0.567524
$185$	200.949	0.0798598
$186$	−678.345	−0.267412
$187$	2303.97	0.900977
$188$	58.0021	0.0225013
$189$	872.327	0.335727
$190$	2513.25	0.959633
$191$	−4070.08	−1.54189	−0.770944	−	0.636902i	$-0.780215\pi$
$191$	−4070.08	−1.54189	−0.770944	+	0.636902i	$0.780215\pi$
$192$	−92.7673	−0.0348693
$193$	−2373.24	−0.885129	−0.442565	−	0.896737i	$-0.645931\pi$
$193$	−2373.24	−0.885129	−0.442565	+	0.896737i	$0.645931\pi$
$194$	121.898	0.0451122
$195$	−1273.24	−0.467581
$196$	−834.139	−0.303986
$197$	−3108.58	−1.12425	−0.562125	−	0.827052i	$-0.690016\pi$
$197$	−3108.58	−1.12425	−0.562125	+	0.827052i	$0.690016\pi$
$198$	−1874.97	−0.672971
$199$	−4048.30	−1.44209	−0.721046	−	0.692887i	$-0.756338\pi$
$199$	−4048.30	−1.44209	−0.721046	+	0.692887i	$0.756338\pi$
$200$	2103.76	0.743790
$201$	589.957	0.207027
$202$	−2200.32	−0.766405
$203$	0	0
$204$	−354.788	−0.121765
$205$	6853.73	2.33505
$206$	1495.86	0.505929
$207$	−4408.62	−1.48029
$208$	−713.535	−0.237859
$209$	−2402.09	−0.795005
$210$	662.139	0.217581
$211$	−3591.66	−1.17185	−0.585924	−	0.810366i	$-0.699268\pi$
$211$	−3591.66	−1.17185	−0.585924	+	0.810366i	$0.699268\pi$
$212$	−2425.49	−0.785772
$213$	−456.180	−0.146746
$214$	352.118	0.112478
$215$	−3837.42	−1.21725
$216$	601.816	0.189576
$217$	2713.38	0.848830
$218$	347.712	0.108028
$219$	−1245.27	−0.384236
$220$	−2966.48	−0.909091
$221$	−2728.91	−0.830616
$222$	29.5755	0.00894134
$223$	−772.085	−0.231850	−0.115925	−	0.993258i	$-0.536983\pi$
$223$	−772.085	−0.231850	−0.115925	+	0.993258i	$0.536983\pi$
$224$	371.069	0.110684
$225$	−6547.67	−1.94005
$226$	3415.31	1.00523
$227$	−4435.16	−1.29679	−0.648397	−	0.761303i	$-0.724560\pi$
$227$	−4435.16	−1.29679	−0.648397	+	0.761303i	$0.724560\pi$
$228$	369.898	0.107443
$229$	−2213.12	−0.638634	−0.319317	−	0.947648i	$-0.603454\pi$
$229$	−2213.12	−0.638634	−0.319317	+	0.947648i	$0.603454\pi$
$230$	−6975.09	−1.99967
$231$	−632.853	−0.180254
$232$	0	0
$233$	−2980.41	−0.837998	−0.418999	−	0.907987i	$-0.637619\pi$
$233$	−2980.41	−0.837998	−0.418999	+	0.907987i	$0.637619\pi$
$234$	2220.79	0.620416
$235$	−285.616	−0.0792831
$236$	−2809.49	−0.774925
$237$	1051.27	0.288131
$238$	1419.15	0.386512
$239$	−557.093	−0.150775	−0.0753877	−	0.997154i	$-0.524019\pi$
$239$	−557.093	−0.150775	−0.0753877	+	0.997154i	$0.524019\pi$
$240$	456.808	0.122862
$241$	−4168.65	−1.11422	−0.557109	−	0.830439i	$-0.688089\pi$
$241$	−4168.65	−1.11422	−0.557109	+	0.830439i	$0.688089\pi$
$242$	173.276	0.0460272
$243$	−2847.53	−0.751724
$244$	−2173.58	−0.570282
$245$	4107.49	1.07109
$246$	1008.73	0.261439
$247$	2845.13	0.732920
$248$	1871.96	0.479312
$249$	1189.91	0.302842
$250$	−5435.15	−1.37500
$251$	7132.56	1.79364	0.896819	−	0.442398i	$-0.145872\pi$
$251$	7132.56	1.79364	0.896819	+	0.442398i	$0.145872\pi$
$252$	−1154.91	−0.288699
$253$	6666.59	1.65662
$254$	1762.95	0.435501
$255$	1747.06	0.429039
$256$	256.000	0.0625000
$257$	5564.76	1.35066	0.675331	−	0.737514i	$-0.264001\pi$
$257$	5564.76	1.35066	0.675331	+	0.737514i	$0.264001\pi$
$258$	−564.788	−0.136287
$259$	−118.302	−0.0283820
$260$	3513.61	0.838096
$261$	0	0
$262$	−4050.98	−0.955231
$263$	−7188.72	−1.68546	−0.842729	−	0.538338i	$-0.819053\pi$
$263$	−7188.72	−1.68546	−0.842729	+	0.538338i	$0.819053\pi$
$264$	−436.604	−0.101785
$265$	11943.7	2.76866
$266$	−1479.59	−0.341051
$267$	−939.796	−0.215410
$268$	−1628.04	−0.371076
$269$	−3184.38	−0.721765	−0.360883	−	0.932611i	$-0.617524\pi$
$269$	−3184.38	−0.721765	−0.360883	+	0.932611i	$0.617524\pi$
$270$	−2963.48	−0.667970
$271$	−1732.34	−0.388311	−0.194155	−	0.980971i	$-0.562197\pi$
$271$	−1732.34	−0.388311	−0.194155	+	0.980971i	$0.562197\pi$
$272$	979.069	0.218253
$273$	749.576	0.166177
$274$	−3188.64	−0.703040
$275$	9901.20	2.17114
$276$	−1026.59	−0.223889
$277$	6061.31	1.31476	0.657381	−	0.753558i	$-0.271664\pi$
$277$	6061.31	1.31476	0.657381	+	0.753558i	$0.271664\pi$
$278$	−5710.81	−1.23206
$279$	−5826.22	−1.25020
$280$	−1827.23	−0.389993
$281$	6183.75	1.31278	0.656391	−	0.754421i	$-0.272082\pi$
$281$	6183.75	1.31278	0.656391	+	0.754421i	$0.272082\pi$
$282$	−42.0367	−0.00887677
$283$	−9076.75	−1.90656	−0.953281	−	0.302086i	$-0.902317\pi$
$283$	−9076.75	−1.90656	−0.953281	+	0.302086i	$0.902317\pi$
$284$	1258.87	0.263029
$285$	−1821.46	−0.378576
$286$	−3358.21	−0.694318
$287$	−4034.91	−0.829871
$288$	−796.767	−0.163021
$289$	−1168.56	−0.237850
$290$	0	0
$291$	−88.3449	−0.0177968
$292$	3436.44	0.688707
$293$	3636.66	0.725105	0.362553	−	0.931963i	$-0.381905\pi$
$293$	3636.66	0.725105	0.362553	+	0.931963i	$0.381905\pi$
$294$	604.538	0.119923
$295$	13834.6	2.73044
$296$	−81.6163	−0.0160265
$297$	2832.41	0.553378
$298$	−36.7511	−0.00714407
$299$	−7896.16	−1.52725
$300$	−1524.69	−0.293426
$301$	2259.15	0.432609
$302$	−1557.61	−0.296790
$303$	1594.67	0.302348
$304$	−1020.77	−0.192582
$305$	10703.2	2.00939
$306$	−3047.23	−0.569276
$307$	4619.06	0.858708	0.429354	−	0.903136i	$-0.358741\pi$
$307$	4619.06	0.858708	0.429354	+	0.903136i	$0.358741\pi$
$308$	1746.42	0.323089
$309$	−1084.12	−0.199590
$310$	−9217.95	−1.68885
$311$	−8094.07	−1.47580	−0.737898	−	0.674912i	$-0.764182\pi$
$311$	−8094.07	−1.47580	−0.737898	+	0.674912i	$0.764182\pi$
$312$	517.131	0.0938358
$313$	6901.79	1.24637	0.623183	−	0.782076i	$-0.285839\pi$
$313$	6901.79	1.24637	0.623183	+	0.782076i	$0.285839\pi$
$314$	1023.79	0.184000
$315$	5687.03	1.01723
$316$	−2901.06	−0.516448
$317$	6119.75	1.08429	0.542144	−	0.840286i	$-0.317613\pi$
$317$	6119.75	1.08429	0.542144	+	0.840286i	$0.317613\pi$
$318$	1757.86	0.309988
$319$	0	0
$320$	−1260.60	−0.220218
$321$	−255.196	−0.0443727
$322$	4106.35	0.710677
$323$	−3903.91	−0.672506
$324$	2252.93	0.386304
$325$	−11727.4	−2.00159
$326$	2903.18	0.493227
$327$	−252.003	−0.0426171
$328$	−2783.67	−0.468606
$329$	168.147	0.0281770
$330$	2149.94	0.358637
$331$	−198.924	−0.0330328	−0.0165164	−	0.999864i	$-0.505258\pi$
$331$	−198.924	−0.0330328	−0.0165164	+	0.999864i	$0.505258\pi$
$332$	−3283.68	−0.542817
$333$	254.020	0.0418025
$334$	1200.75	0.196713
$335$	8016.85	1.30749
$336$	−268.931	−0.0436648
$337$	−2102.43	−0.339841	−0.169921	−	0.985458i	$-0.554351\pi$
$337$	−2102.43	−0.339841	−0.169921	+	0.985458i	$0.554351\pi$
$338$	−416.408	−0.0670107
$339$	−2475.23	−0.396566
$340$	−4821.17	−0.769013
$341$	8810.25	1.39912
$342$	3177.01	0.502318
$343$	−6395.55	−1.00679
$344$	1558.58	0.244283
$345$	5055.16	0.788871
$346$	−5148.40	−0.799941
$347$	6543.83	1.01237	0.506183	−	0.862426i	$-0.331056\pi$
$347$	6543.83	1.01237	0.506183	+	0.862426i	$0.331056\pi$
$348$	0	0
$349$	793.427	0.121694	0.0608469	−	0.998147i	$-0.480620\pi$
$349$	793.427	0.121694	0.0608469	+	0.998147i	$0.480620\pi$
$350$	6098.74	0.931404
$351$	−3354.82	−0.510162
$352$	1204.85	0.182439
$353$	7378.84	1.11257	0.556284	−	0.830992i	$-0.312227\pi$
$353$	7378.84	1.11257	0.556284	+	0.830992i	$0.312227\pi$
$354$	2036.16	0.305709
$355$	−6198.97	−0.926780
$356$	2593.45	0.386103
$357$	−1028.52	−0.152479
$358$	7657.43	1.13047
$359$	7142.97	1.05012	0.525058	−	0.851066i	$-0.324044\pi$
$359$	7142.97	1.05012	0.525058	+	0.851066i	$0.324044\pi$
$360$	3923.47	0.574403
$361$	−2788.82	−0.406593
$362$	4151.28	0.602724
$363$	−125.581	−0.0181578
$364$	−2068.52	−0.297857
$365$	−16921.8	−2.42666
$366$	1575.29	0.224977
$367$	1806.47	0.256940	0.128470	−	0.991713i	$-0.458993\pi$
$367$	1806.47	0.256940	0.128470	+	0.991713i	$0.458993\pi$
$368$	2832.96	0.401300
$369$	8663.83	1.22228
$370$	401.898	0.0564694
$371$	−7031.46	−0.983976
$372$	−1356.69	−0.189089
$373$	9396.05	1.30431	0.652157	−	0.758084i	$-0.273864\pi$
$373$	9396.05	1.30431	0.652157	+	0.758084i	$0.273864\pi$
$374$	4607.93	0.637087
$375$	3939.10	0.542437
$376$	116.004	0.0159108
$377$	0	0
$378$	1744.65	0.237395
$379$	421.482	0.0571242	0.0285621	−	0.999592i	$-0.490907\pi$
$379$	421.482	0.0571242	0.0285621	+	0.999592i	$0.490907\pi$
$380$	5026.50	0.678563
$381$	−1277.69	−0.171805
$382$	−8140.16	−1.09028
$383$	3189.84	0.425570	0.212785	−	0.977099i	$-0.431747\pi$
$383$	3189.84	0.425570	0.212785	+	0.977099i	$0.431747\pi$
$384$	−185.535	−0.0246563
$385$	−8599.76	−1.13840
$386$	−4746.49	−0.625881
$387$	−4850.89	−0.637170
$388$	243.796	0.0318991
$389$	4110.96	0.535820	0.267910	−	0.963444i	$-0.413667\pi$
$389$	4110.96	0.535820	0.267910	+	0.963444i	$0.413667\pi$
$390$	−2546.47	−0.330630
$391$	10834.6	1.40136
$392$	−1668.28	−0.214951
$393$	2935.93	0.376840
$394$	−6217.17	−0.794965
$395$	14285.5	1.81970
$396$	−3749.94	−0.475862
$397$	−827.505	−0.104613	−0.0523064	−	0.998631i	$-0.516657\pi$
$397$	−827.505	−0.104613	−0.0523064	+	0.998631i	$0.516657\pi$
$398$	−8096.60	−1.01971
$399$	1072.33	0.134545
$400$	4207.51	0.525939
$401$	675.145	0.0840776	0.0420388	−	0.999116i	$-0.486615\pi$
$401$	675.145	0.0840776	0.0420388	+	0.999116i	$0.486615\pi$
$402$	1179.91	0.146390
$403$	−10435.2	−1.28986
$404$	−4400.64	−0.541930
$405$	−11093.9	−1.36114
$406$	0	0
$407$	−384.122	−0.0467819
$408$	−709.576	−0.0861010
$409$	−764.847	−0.0924676	−0.0462338	−	0.998931i	$-0.514722\pi$
$409$	−764.847	−0.0924676	−0.0462338	+	0.998931i	$0.514722\pi$
$410$	13707.5	1.65113
$411$	2310.95	0.277350
$412$	2991.72	0.357746
$413$	−8144.65	−0.970393
$414$	−8817.24	−1.04672
$415$	16169.6	1.91261
$416$	−1427.07	−0.168192
$417$	4138.88	0.486047
$418$	−4804.18	−0.562154
$419$	−2573.33	−0.300037	−0.150019	−	0.988683i	$-0.547933\pi$
$419$	−2573.33	−0.300037	−0.150019	+	0.988683i	$0.547933\pi$
$420$	1324.28	0.153853
$421$	−14287.4	−1.65398	−0.826992	−	0.562213i	$-0.809950\pi$
$421$	−14287.4	−1.65398	−0.826992	+	0.562213i	$0.809950\pi$
$422$	−7183.32	−0.828622
$423$	−361.048	−0.0415006
$424$	−4850.99	−0.555625
$425$	16091.6	1.83660
$426$	−912.359	−0.103765
$427$	−6301.15	−0.714131
$428$	704.237	0.0795340
$429$	2433.84	0.273909
$430$	−7674.83	−0.860728
$431$	−11283.2	−1.26100	−0.630500	−	0.776189i	$-0.717150\pi$
$431$	−11283.2	−1.26100	−0.630500	+	0.776189i	$0.717150\pi$
$432$	1203.63	0.134050
$433$	−8069.64	−0.895617	−0.447809	−	0.894129i	$-0.647795\pi$
$433$	−8069.64	−0.895617	−0.447809	+	0.894129i	$0.647795\pi$
$434$	5426.76	0.600214
$435$	0	0
$436$	695.424	0.0763871
$437$	−11296.1	−1.23653
$438$	−2490.54	−0.271696
$439$	−7819.42	−0.850116	−0.425058	−	0.905166i	$-0.639746\pi$
$439$	−7819.42	−0.850116	−0.425058	+	0.905166i	$0.639746\pi$
$440$	−5932.96	−0.642824
$441$	5192.30	0.560663
$442$	−5457.81	−0.587334
$443$	10153.4	1.08894	0.544472	−	0.838779i	$-0.316730\pi$
$443$	10153.4	1.08894	0.544472	+	0.838779i	$0.316730\pi$
$444$	59.1510	0.00632248
$445$	−12770.8	−1.36043
$446$	−1544.17	−0.163943
$447$	26.6351	0.00281834
$448$	742.139	0.0782651
$449$	−13608.3	−1.43033	−0.715163	−	0.698958i	$-0.753647\pi$
$449$	−13608.3	−1.43033	−0.715163	+	0.698958i	$0.753647\pi$
$450$	−13095.3	−1.37182
$451$	−13101.2	−1.36787
$452$	6830.61	0.710807
$453$	1128.87	0.117084
$454$	−8870.32	−0.916971
$455$	10185.9	1.04950
$456$	739.796	0.0759739
$457$	−1882.55	−0.192696	−0.0963479	−	0.995348i	$-0.530716\pi$
$457$	−1882.55	−0.192696	−0.0963479	+	0.995348i	$0.530716\pi$
$458$	−4426.24	−0.451583
$459$	4603.28	0.468111
$460$	−13950.2	−1.41398
$461$	13947.2	1.40908	0.704542	−	0.709662i	$-0.251153\pi$
$461$	13947.2	1.40908	0.704542	+	0.709662i	$0.251153\pi$
$462$	−1265.71	−0.127459
$463$	−12454.8	−1.25016	−0.625079	−	0.780562i	$-0.714933\pi$
$463$	−12454.8	−1.25016	−0.625079	+	0.780562i	$0.714933\pi$
$464$	0	0
$465$	6680.66	0.666254
$466$	−5960.83	−0.592554
$467$	6561.59	0.650180	0.325090	−	0.945683i	$-0.394605\pi$
$467$	6561.59	0.650180	0.325090	+	0.945683i	$0.394605\pi$
$468$	4441.57	0.438700
$469$	−4719.66	−0.464677
$470$	−571.232	−0.0560616
$471$	−741.988	−0.0725881
$472$	−5618.98	−0.547954
$473$	7335.38	0.713068
$474$	2102.53	0.203739
$475$	−16776.9	−1.62058
$476$	2838.30	0.273305
$477$	15098.1	1.44925
$478$	−1114.19	−0.106614
$479$	17407.5	1.66048	0.830240	−	0.557406i	$-0.188203\pi$
$479$	17407.5	1.66048	0.830240	+	0.557406i	$0.188203\pi$
$480$	913.616	0.0868764
$481$	454.969	0.0431285
$482$	−8337.31	−0.787871
$483$	−2976.06	−0.280363
$484$	346.551	0.0325461
$485$	−1200.51	−0.112396
$486$	−5695.06	−0.531549
$487$	−2544.79	−0.236787	−0.118393	−	0.992967i	$-0.537774\pi$
$487$	−2544.79	−0.236787	−0.118393	+	0.992967i	$0.537774\pi$
$488$	−4347.15	−0.403251
$489$	−2104.06	−0.194579
$490$	8214.99	0.757378
$491$	−2712.89	−0.249351	−0.124675	−	0.992198i	$-0.539789\pi$
$491$	−2712.89	−0.249351	−0.124675	+	0.992198i	$0.539789\pi$
$492$	2017.45	0.184865
$493$	0	0
$494$	5690.26	0.518253
$495$	18465.6	1.67670
$496$	3743.91	0.338924
$497$	3649.44	0.329375
$498$	2379.83	0.214142
$499$	4583.79	0.411220	0.205610	−	0.978634i	$-0.434082\pi$
$499$	4583.79	0.411220	0.205610	+	0.978634i	$0.434082\pi$
$500$	−10870.3	−0.972269
$501$	−870.237	−0.0776034
$502$	14265.1	1.26829
$503$	1950.19	0.172872	0.0864359	−	0.996257i	$-0.472452\pi$
$503$	1950.19	0.172872	0.0864359	+	0.996257i	$0.472452\pi$
$504$	−2309.81	−0.204141
$505$	21669.8	1.90949
$506$	13333.2	1.17141
$507$	301.790	0.0264358
$508$	3525.89	0.307945
$509$	−8853.80	−0.770997	−0.385499	−	0.922708i	$-0.625971\pi$
$509$	−8853.80	−0.770997	−0.385499	+	0.922708i	$0.625971\pi$
$510$	3494.12	0.303377
$511$	9962.17	0.862428
$512$	512.000	0.0441942
$513$	−4799.33	−0.413052
$514$	11129.5	0.955063
$515$	−14731.9	−1.26052
$516$	−1129.58	−0.0963697
$517$	545.967	0.0464441
$518$	−236.604	−0.0200691
$519$	3731.28	0.315578
$520$	7027.22	0.592623
$521$	−19939.5	−1.67671	−0.838355	−	0.545125i	$-0.816482\pi$
$521$	−19939.5	−1.67671	−0.838355	+	0.545125i	$0.816482\pi$
$522$	0	0
$523$	−12185.8	−1.01883	−0.509416	−	0.860520i	$-0.670138\pi$
$523$	−12185.8	−1.01883	−0.509416	+	0.860520i	$0.670138\pi$
$524$	−8101.97	−0.675451
$525$	−4420.03	−0.367440
$526$	−14377.4	−1.19180
$527$	14318.5	1.18354
$528$	−873.208	−0.0719725
$529$	19183.3	1.57667
$530$	23887.4	1.95774
$531$	17488.4	1.42925
$532$	−2959.18	−0.241160
$533$	15517.6	1.26105
$534$	−1879.59	−0.152318
$535$	−3467.83	−0.280238
$536$	−3256.08	−0.262390
$537$	−5549.68	−0.445971
$538$	−6368.75	−0.510365
$539$	−7851.65	−0.627448
$540$	−5926.97	−0.472326
$541$	−700.188	−0.0556440	−0.0278220	−	0.999613i	$-0.508857\pi$
$541$	−700.188	−0.0556440	−0.0278220	+	0.999613i	$0.508857\pi$
$542$	−3464.68	−0.274577
$543$	−3008.62	−0.237775
$544$	1958.14	0.154328
$545$	−3424.43	−0.269150
$546$	1499.15	0.117505
$547$	10664.2	0.833576	0.416788	−	0.909004i	$-0.363156\pi$
$547$	10664.2	0.833576	0.416788	+	0.909004i	$0.363156\pi$
$548$	−6377.28	−0.497124
$549$	13530.0	1.05181
$550$	19802.4	1.53523
$551$	0	0
$552$	−2053.18	−0.158313
$553$	−8410.12	−0.646718
$554$	12122.6	0.929677
$555$	−291.273	−0.0222772
$556$	−11421.6	−0.871195
$557$	8410.60	0.639800	0.319900	−	0.947451i	$-0.396351\pi$
$557$	8410.60	0.639800	0.319900	+	0.947451i	$0.396351\pi$
$558$	−11652.4	−0.884027
$559$	−8688.31	−0.657382
$560$	−3654.47	−0.275767
$561$	−3339.58	−0.251332
$562$	12367.5	0.928277
$563$	−8770.15	−0.656515	−0.328257	−	0.944588i	$-0.606461\pi$
$563$	−8770.15	−0.656515	−0.328257	+	0.944588i	$0.606461\pi$
$564$	−84.0734	−0.00627682
$565$	−33635.5	−2.50453
$566$	−18153.5	−1.34814
$567$	6531.19	0.483746
$568$	2517.74	0.185989
$569$	10493.4	0.773125	0.386562	−	0.922263i	$-0.373662\pi$
$569$	10493.4	0.773125	0.386562	+	0.922263i	$0.373662\pi$
$570$	−3642.93	−0.267694
$571$	−2517.50	−0.184508	−0.0922542	−	0.995735i	$-0.529407\pi$
$571$	−2517.50	−0.184508	−0.0922542	+	0.995735i	$0.529407\pi$
$572$	−6716.42	−0.490957
$573$	5899.54	0.430117
$574$	−8069.81	−0.586807
$575$	46561.4	3.37695
$576$	−1593.53	−0.115273
$577$	−1394.88	−0.100641	−0.0503203	−	0.998733i	$-0.516024\pi$
$577$	−1394.88	−0.100641	−0.0503203	+	0.998733i	$0.516024\pi$
$578$	−2337.12	−0.168186
$579$	3439.99	0.246911
$580$	0	0
$581$	−9519.31	−0.679738
$582$	−176.690	−0.0125842
$583$	−22830.9	−1.62188
$584$	6872.88	0.486989
$585$	−21871.3	−1.54576
$586$	7273.31	0.512727
$587$	24391.0	1.71504	0.857518	−	0.514455i	$-0.172006\pi$
$587$	24391.0	1.71504	0.857518	+	0.514455i	$0.172006\pi$
$588$	1209.08	0.0847984
$589$	−14928.4	−1.04433
$590$	27669.2	1.93071
$591$	4505.86	0.313615
$592$	−163.233	−0.0113325
$593$	−16089.4	−1.11418	−0.557092	−	0.830451i	$-0.688083\pi$
$593$	−16089.4	−1.11418	−0.557092	+	0.830451i	$0.688083\pi$
$594$	5664.83	0.391297
$595$	−13976.5	−0.962990
$596$	−73.5021	−0.00505162
$597$	5867.97	0.402278
$598$	−15792.3	−1.07993
$599$	19228.2	1.31159	0.655796	−	0.754938i	$-0.272333\pi$
$599$	19228.2	1.31159	0.655796	+	0.754938i	$0.272333\pi$
$600$	−3049.37	−0.207483
$601$	2345.94	0.159223	0.0796115	−	0.996826i	$-0.474632\pi$
$601$	2345.94	0.159223	0.0796115	+	0.996826i	$0.474632\pi$
$602$	4518.30	0.305901
$603$	10134.1	0.684401
$604$	−3115.22	−0.209862
$605$	−1706.50	−0.114676
$606$	3189.34	0.213792
$607$	4621.11	0.309004	0.154502	−	0.987993i	$-0.450623\pi$
$607$	4621.11	0.309004	0.154502	+	0.987993i	$0.450623\pi$
$608$	−2041.53	−0.136176
$609$	0	0
$610$	21406.4	1.42085
$611$	−646.664	−0.0428170
$612$	−6094.46	−0.402539
$613$	−10464.8	−0.689511	−0.344755	−	0.938693i	$-0.612038\pi$
$613$	−10464.8	−0.689511	−0.344755	+	0.938693i	$0.612038\pi$
$614$	9238.11	0.607198
$615$	−9934.41	−0.651373
$616$	3492.83	0.228458
$617$	11873.3	0.774718	0.387359	−	0.921929i	$-0.373387\pi$
$617$	11873.3	0.774718	0.387359	+	0.921929i	$0.373387\pi$
$618$	−2168.23	−0.141131
$619$	−13102.2	−0.850762	−0.425381	−	0.905014i	$-0.639860\pi$
$619$	−13102.2	−0.850762	−0.425381	+	0.905014i	$0.639860\pi$
$620$	−18435.9	−1.19420
$621$	13319.7	0.860711
$622$	−16188.1	−1.04355
$623$	7518.37	0.483494
$624$	1034.26	0.0663519
$625$	20656.7	1.32203
$626$	13803.6	0.881313
$627$	3481.81	0.221770
$628$	2047.58	0.130107
$629$	−624.282	−0.0395735
$630$	11374.1	0.719291
$631$	12279.7	0.774719	0.387359	−	0.921929i	$-0.373387\pi$
$631$	12279.7	0.774719	0.387359	+	0.921929i	$0.373387\pi$
$632$	−5802.13	−0.365184
$633$	5206.07	0.326892
$634$	12239.5	0.766707
$635$	−17362.3	−1.08504
$636$	3515.73	0.219195
$637$	9299.80	0.578448
$638$	0	0
$639$	−7836.14	−0.485122
$640$	−2521.21	−0.155718
$641$	−17054.4	−1.05087	−0.525435	−	0.850834i	$-0.676097\pi$
$641$	−17054.4	−1.05087	−0.525435	+	0.850834i	$0.676097\pi$
$642$	−510.392	−0.0313763
$643$	−20697.3	−1.26940	−0.634698	−	0.772761i	$-0.718875\pi$
$643$	−20697.3	−1.26940	−0.634698	+	0.772761i	$0.718875\pi$
$644$	8212.70	0.502525
$645$	5562.29	0.339558
$646$	−7807.83	−0.475534
$647$	−10613.6	−0.644924	−0.322462	−	0.946582i	$-0.604510\pi$
$647$	−10613.6	−0.644924	−0.322462	+	0.946582i	$0.604510\pi$
$648$	4505.85	0.273158
$649$	−26445.4	−1.59950
$650$	−23454.7	−1.41534
$651$	−3933.02	−0.236785
$652$	5806.35	0.348764
$653$	10019.4	0.600441	0.300220	−	0.953870i	$-0.402940\pi$
$653$	10019.4	0.600441	0.300220	+	0.953870i	$0.402940\pi$
$654$	−504.005	−0.0301348
$655$	39896.0	2.37995
$656$	−5567.35	−0.331354
$657$	−21391.0	−1.27023
$658$	336.294	0.0199241
$659$	−2400.70	−0.141909	−0.0709545	−	0.997480i	$-0.522605\pi$
$659$	−2400.70	−0.141909	−0.0709545	+	0.997480i	$0.522605\pi$
$660$	4299.88	0.253595
$661$	−5037.05	−0.296397	−0.148199	−	0.988958i	$-0.547347\pi$
$661$	−5037.05	−0.296397	−0.148199	+	0.988958i	$0.547347\pi$
$662$	−397.848	−0.0233577
$663$	3955.52	0.231704
$664$	−6567.36	−0.383830
$665$	14571.7	0.849725
$666$	508.041	0.0295588
$667$	0	0
$668$	2401.50	0.139097
$669$	1119.13	0.0646757
$670$	16033.7	0.924532
$671$	−20459.6	−1.17710
$672$	−537.861	−0.0308757
$673$	−20842.1	−1.19377	−0.596884	−	0.802328i	$-0.703595\pi$
$673$	−20842.1	−1.19377	−0.596884	+	0.802328i	$0.703595\pi$
$674$	−4204.85	−0.240304
$675$	19782.4	1.12804
$676$	−832.816	−0.0473837
$677$	23394.0	1.32807	0.664037	−	0.747700i	$-0.268842\pi$
$677$	23394.0	1.32807	0.664037	+	0.747700i	$0.268842\pi$
$678$	−4950.45	−0.280414
$679$	706.759	0.0399454
$680$	−9642.33	−0.543774
$681$	6428.72	0.361746
$682$	17620.5	0.989331
$683$	−8567.11	−0.479958	−0.239979	−	0.970778i	$-0.577141\pi$
$683$	−8567.11	−0.479958	−0.239979	+	0.970778i	$0.577141\pi$
$684$	6354.02	0.355193
$685$	31403.2	1.75161
$686$	−12791.1	−0.711905
$687$	3207.90	0.178150
$688$	3117.17	0.172734
$689$	27041.8	1.49522
$690$	10110.3	0.557816
$691$	−11987.6	−0.659959	−0.329979	−	0.943988i	$-0.607042\pi$
$691$	−11987.6	−0.659959	−0.329979	+	0.943988i	$0.607042\pi$
$692$	−10296.8	−0.565644
$693$	−10871.0	−0.595895
$694$	13087.7	0.715851
$695$	56242.7	3.06965
$696$	0	0
$697$	−21292.3	−1.15710
$698$	1586.85	0.0860505
$699$	4320.08	0.233763
$700$	12197.5	0.658602
$701$	19429.6	1.04686	0.523428	−	0.852070i	$-0.324653\pi$
$701$	19429.6	1.04686	0.523428	+	0.852070i	$0.324653\pi$
$702$	−6709.64	−0.360739
$703$	650.869	0.0349189
$704$	2409.70	0.129004
$705$	413.997	0.0221164
$706$	14757.7	0.786704
$707$	−12757.4	−0.678628
$708$	4072.33	0.216169
$709$	−804.014	−0.0425887	−0.0212944	−	0.999773i	$-0.506779\pi$
$709$	−804.014	−0.0425887	−0.0212944	+	0.999773i	$0.506779\pi$
$710$	−12397.9	−0.655333
$711$	18058.4	0.952521
$712$	5186.91	0.273016
$713$	41431.1	2.17617
$714$	−2057.04	−0.107819
$715$	33073.2	1.72989
$716$	15314.9	0.799362
$717$	807.500	0.0420595
$718$	14285.9	0.742544
$719$	4975.94	0.258096	0.129048	−	0.991638i	$-0.458808\pi$
$719$	4975.94	0.258096	0.129048	+	0.991638i	$0.458808\pi$
$720$	7846.94	0.406164
$721$	8672.93	0.447984
$722$	−5577.64	−0.287505
$723$	6042.42	0.310816
$724$	8302.55	0.426190
$725$	0	0
$726$	−251.161	−0.0128395
$727$	3761.32	0.191884	0.0959419	−	0.995387i	$-0.469414\pi$
$727$	3761.32	0.191884	0.0959419	+	0.995387i	$0.469414\pi$
$728$	−4137.04	−0.210617
$729$	−11079.8	−0.562912
$730$	−33843.7	−1.71591
$731$	11921.6	0.603195
$732$	3150.58	0.159083
$733$	−21439.2	−1.08032	−0.540161	−	0.841562i	$-0.681637\pi$
$733$	−21439.2	−1.08032	−0.540161	+	0.841562i	$0.681637\pi$
$734$	3612.94	0.181684
$735$	−5953.77	−0.298787
$736$	5665.93	0.283762
$737$	−15324.6	−0.765926
$738$	17327.7	0.864282
$739$	−22000.3	−1.09512	−0.547561	−	0.836766i	$-0.684444\pi$
$739$	−22000.3	−1.09512	−0.547561	+	0.836766i	$0.684444\pi$
$740$	803.796	0.0399299
$741$	−4123.98	−0.204451
$742$	−14062.9	−0.695776
$743$	13371.8	0.660246	0.330123	−	0.943938i	$-0.392910\pi$
$743$	13371.8	0.660246	0.330123	+	0.943938i	$0.392910\pi$
$744$	−2713.38	−0.133706
$745$	361.942	0.0177994
$746$	18792.1	0.922289
$747$	20440.1	1.00115
$748$	9215.87	0.450489
$749$	2041.57	0.0995958
$750$	7878.19	0.383561
$751$	17764.8	0.863178	0.431589	−	0.902070i	$-0.357953\pi$
$751$	17764.8	0.863178	0.431589	+	0.902070i	$0.357953\pi$
$752$	232.008	0.0112506
$753$	−10338.6	−0.500343
$754$	0	0
$755$	15340.1	0.739448
$756$	3489.31	0.167864
$757$	−33266.1	−1.59719	−0.798597	−	0.601867i	$-0.794424\pi$
$757$	−33266.1	−1.59719	−0.798597	+	0.601867i	$0.794424\pi$
$758$	842.963	0.0403929
$759$	−9663.15	−0.462121
$760$	10053.0	0.479816
$761$	−25327.3	−1.20646	−0.603228	−	0.797569i	$-0.706119\pi$
$761$	−25327.3	−1.20646	−0.603228	+	0.797569i	$0.706119\pi$
$762$	−2555.37	−0.121485
$763$	2016.02	0.0956551
$764$	−16280.3	−0.770944
$765$	30010.5	1.41834
$766$	6379.69	0.300924
$767$	31322.9	1.47458
$768$	−371.069	−0.0174347
$769$	9587.28	0.449579	0.224789	−	0.974407i	$-0.427831\pi$
$769$	9587.28	0.449579	0.224789	+	0.974407i	$0.427831\pi$
$770$	−17199.5	−0.804971
$771$	−8066.07	−0.376773
$772$	−9492.98	−0.442565
$773$	−2825.54	−0.131472	−0.0657359	−	0.997837i	$-0.520939\pi$
$773$	−2825.54	−0.131472	−0.0657359	+	0.997837i	$0.520939\pi$
$774$	−9701.79	−0.450547
$775$	61533.4	2.85206
$776$	487.592	0.0225561
$777$	171.478	0.00791728
$778$	8221.92	0.378882
$779$	22199.1	1.02101
$780$	−5092.94	−0.233791
$781$	11849.6	0.542909
$782$	21669.3	0.990910
$783$	0	0
$784$	−3336.55	−0.151993
$785$	−10082.8	−0.458433
$786$	5871.86	0.266466
$787$	2132.09	0.0965702	0.0482851	−	0.998834i	$-0.484624\pi$
$787$	2132.09	0.0965702	0.0482851	+	0.998834i	$0.484624\pi$
$788$	−12434.3	−0.562125
$789$	10420.0	0.470166
$790$	28571.0	1.28672
$791$	19801.8	0.890103
$792$	−7499.88	−0.336486
$793$	24233.1	1.08518
$794$	−1655.01	−0.0739724
$795$	−17312.3	−0.772331
$796$	−16193.2	−0.721046
$797$	−5443.62	−0.241936	−0.120968	−	0.992656i	$-0.538600\pi$
$797$	−5443.62	−0.241936	−0.120968	+	0.992656i	$0.538600\pi$
$798$	2144.65	0.0951377
$799$	887.313	0.0392877
$800$	8415.02	0.371895
$801$	−16143.6	−0.712117
$802$	1350.29	0.0594519
$803$	32346.8	1.42154
$804$	2359.83	0.103513
$805$	−40441.3	−1.77064
$806$	−20870.4	−0.912070
$807$	4615.72	0.201340
$808$	−8801.27	−0.383203
$809$	−41193.5	−1.79022	−0.895109	−	0.445847i	$-0.852902\pi$
$809$	−41193.5	−1.79022	−0.895109	+	0.445847i	$0.852902\pi$
$810$	−22187.9	−0.962472
$811$	−18225.9	−0.789145	−0.394572	−	0.918865i	$-0.629107\pi$
$811$	−18225.9	−0.789145	−0.394572	+	0.918865i	$0.629107\pi$
$812$	0	0
$813$	2511.01	0.108321
$814$	−768.245	−0.0330798
$815$	−28591.8	−1.22887
$816$	−1419.15	−0.0608826
$817$	−12429.3	−0.532248
$818$	−1529.69	−0.0653845
$819$	12876.0	0.549359
$820$	27414.9	1.16753
$821$	46061.7	1.95806	0.979029	−	0.203720i	$-0.0653033\pi$
$821$	46061.7	1.95806	0.979029	+	0.203720i	$0.0653033\pi$
$822$	4621.90	0.196116
$823$	−40166.3	−1.70122	−0.850612	−	0.525793i	$-0.823769\pi$
$823$	−40166.3	−1.70122	−0.850612	+	0.525793i	$0.823769\pi$
$824$	5983.44	0.252965
$825$	−14351.7	−0.605650
$826$	−16289.3	−0.686171
$827$	23994.5	1.00891	0.504457	−	0.863437i	$-0.331693\pi$
$827$	23994.5	1.00891	0.504457	+	0.863437i	$0.331693\pi$
$828$	−17634.5	−0.740145
$829$	13549.6	0.567668	0.283834	−	0.958873i	$-0.408394\pi$
$829$	13549.6	0.567668	0.283834	+	0.958873i	$0.408394\pi$
$830$	32339.2	1.35242
$831$	−8785.81	−0.366759
$832$	−2854.14	−0.118930
$833$	−12760.6	−0.530767
$834$	8277.75	0.343687
$835$	−11825.5	−0.490107
$836$	−9608.36	−0.397503
$837$	17602.7	0.726928
$838$	−5146.67	−0.212158
$839$	−34768.5	−1.43068	−0.715341	−	0.698776i	$-0.753729\pi$
$839$	−34768.5	−1.43068	−0.715341	+	0.698776i	$0.753729\pi$
$840$	2648.55	0.108790
$841$	0	0
$842$	−28574.9	−1.16954
$843$	−8963.29	−0.366207
$844$	−14366.6	−0.585924
$845$	4100.98	0.166956
$846$	−722.096	−0.0293454
$847$	1004.64	0.0407556
$848$	−9701.98	−0.392886
$849$	13156.7	0.531844
$850$	32183.2	1.29867
$851$	−1806.38	−0.0727635
$852$	−1824.72	−0.0733730
$853$	−4691.07	−0.188299	−0.0941496	−	0.995558i	$-0.530013\pi$
$853$	−4691.07	−0.188299	−0.0941496	+	0.995558i	$0.530013\pi$
$854$	−12602.3	−0.504967
$855$	−31288.7	−1.25152
$856$	1408.47	0.0562391
$857$	−14362.7	−0.572485	−0.286242	−	0.958157i	$-0.592406\pi$
$857$	−14362.7	−0.572485	−0.286242	+	0.958157i	$0.592406\pi$
$858$	4867.69	0.193683
$859$	−1392.43	−0.0553073	−0.0276537	−	0.999618i	$-0.508804\pi$
$859$	−1392.43	−0.0553073	−0.0276537	+	0.999618i	$0.508804\pi$
$860$	−15349.7	−0.608627
$861$	5848.55	0.231496
$862$	−22566.3	−0.891661
$863$	19162.2	0.755840	0.377920	−	0.925838i	$-0.376639\pi$
$863$	19162.2	0.755840	0.377920	+	0.925838i	$0.376639\pi$
$864$	2407.27	0.0947880
$865$	50703.9	1.99304
$866$	−16139.3	−0.633297
$867$	1693.81	0.0663494
$868$	10853.5	0.424415
$869$	−27307.4	−1.06598
$870$	0	0
$871$	18151.0	0.706111
$872$	1390.85	0.0540139
$873$	−1517.57	−0.0588338
$874$	−22592.2	−0.874361
$875$	−31512.8	−1.21752
$876$	−4981.09	−0.192118
$877$	−6509.75	−0.250649	−0.125324	−	0.992116i	$-0.539997\pi$
$877$	−6509.75	−0.250649	−0.125324	+	0.992116i	$0.539997\pi$
$878$	−15638.8	−0.601122
$879$	−5271.30	−0.202271
$880$	−11865.9	−0.454545
$881$	589.496	0.0225433	0.0112716	−	0.999936i	$-0.496412\pi$
$881$	589.496	0.0225433	0.0112716	+	0.999936i	$0.496412\pi$
$882$	10384.6	0.396449
$883$	−3903.57	−0.148772	−0.0743859	−	0.997230i	$-0.523700\pi$
$883$	−3903.57	−0.148772	−0.0743859	+	0.997230i	$0.523700\pi$
$884$	−10915.6	−0.415308
$885$	−20053.1	−0.761669
$886$	20306.8	0.770000
$887$	24136.6	0.913674	0.456837	−	0.889550i	$-0.348982\pi$
$887$	24136.6	0.913674	0.456837	+	0.889550i	$0.348982\pi$
$888$	118.302	0.00447067
$889$	10221.5	0.385622
$890$	−25541.5	−0.961971
$891$	21206.5	0.797358
$892$	−3088.34	−0.115925
$893$	−925.103	−0.0346667
$894$	53.2703	0.00199287
$895$	−75414.0	−2.81655
$896$	1484.28	0.0553418
$897$	11445.4	0.426032
$898$	−27216.6	−1.01139
$899$	0	0
$900$	−26190.7	−0.970025
$901$	−37105.1	−1.37198
$902$	−26202.4	−0.967233
$903$	−3274.62	−0.120678
$904$	13661.2	0.502617
$905$	−40883.7	−1.50168
$906$	2257.74	0.0827908
$907$	−22174.3	−0.811781	−0.405890	−	0.913922i	$-0.633038\pi$
$907$	−22174.3	−0.811781	−0.405890	+	0.913922i	$0.633038\pi$
$908$	−17740.6	−0.648397
$909$	27392.8	0.999519
$910$	20371.8	0.742108
$911$	46956.1	1.70771	0.853856	−	0.520509i	$-0.174258\pi$
$911$	46956.1	1.70771	0.853856	+	0.520509i	$0.174258\pi$
$912$	1479.59	0.0537217
$913$	−30908.9	−1.12041
$914$	−3765.10	−0.136257
$915$	−15514.2	−0.560528
$916$	−8852.49	−0.319317
$917$	−23487.4	−0.845827
$918$	9206.56	0.331004
$919$	−37166.4	−1.33407	−0.667033	−	0.745028i	$-0.732436\pi$
$919$	−37166.4	−1.33407	−0.667033	+	0.745028i	$0.732436\pi$
$920$	−27900.4	−0.999834
$921$	−6695.27	−0.239540
$922$	27894.5	0.996373
$923$	−14035.1	−0.500511
$924$	−2531.41	−0.0901270
$925$	−2682.82	−0.0953629
$926$	−24909.6	−0.883995
$927$	−18622.7	−0.659816
$928$	0	0
$929$	29571.7	1.04437	0.522183	−	0.852833i	$-0.325118\pi$
$929$	29571.7	1.04437	0.522183	+	0.852833i	$0.325118\pi$
$930$	13361.3	0.471113
$931$	13304.1	0.468339
$932$	−11921.7	−0.418999
$933$	11732.3	0.411680
$934$	13123.2	0.459747
$935$	−45381.1	−1.58729
$936$	8883.14	0.310208
$937$	8691.51	0.303030	0.151515	−	0.988455i	$-0.451585\pi$
$937$	8691.51	0.303030	0.151515	+	0.988455i	$0.451585\pi$
$938$	−9439.31	−0.328576
$939$	−10004.1	−0.347679
$940$	−1142.46	−0.0396415
$941$	−2862.43	−0.0991632	−0.0495816	−	0.998770i	$-0.515789\pi$
$941$	−2862.43	−0.0991632	−0.0495816	+	0.998770i	$0.515789\pi$
$942$	−1483.98	−0.0513275
$943$	−61609.7	−2.12756
$944$	−11238.0	−0.387462
$945$	−17182.2	−0.591466
$946$	14670.8	0.504215
$947$	−36997.8	−1.26955	−0.634777	−	0.772695i	$-0.718908\pi$
$947$	−36997.8	−1.26955	−0.634777	+	0.772695i	$0.718908\pi$
$948$	4205.06	0.144065
$949$	−38312.8	−1.31052
$950$	−33553.8	−1.14593
$951$	−8870.51	−0.302467
$952$	5676.60	0.193256
$953$	20998.8	0.713764	0.356882	−	0.934150i	$-0.383840\pi$
$953$	20998.8	0.713764	0.356882	+	0.934150i	$0.383840\pi$
$954$	30196.2	1.02478
$955$	80168.1	2.71642
$956$	−2228.37	−0.0753877
$957$	0	0
$958$	34815.0	1.17414
$959$	−18487.6	−0.622519
$960$	1827.23	0.0614309
$961$	24962.4	0.837917
$962$	909.939	0.0304965
$963$	−4383.69	−0.146690
$964$	−16674.6	−0.557109
$965$	46745.7	1.55937
$966$	−5952.11	−0.198247
$967$	30803.0	1.02436	0.512181	−	0.858878i	$-0.328838\pi$
$967$	30803.0	1.02436	0.512181	+	0.858878i	$0.328838\pi$
$968$	693.102	0.0230136
$969$	5658.68	0.187599
$970$	−2401.02	−0.0794763
$971$	755.015	0.0249532	0.0124766	−	0.999922i	$-0.496028\pi$
$971$	755.015	0.0249532	0.0124766	+	0.999922i	$0.496028\pi$
$972$	−11390.1	−0.375862
$973$	−33111.0	−1.09095
$974$	−5089.57	−0.167434
$975$	16998.7	0.558353
$976$	−8694.30	−0.285141
$977$	−44011.7	−1.44121	−0.720603	−	0.693348i	$-0.756135\pi$
$977$	−44011.7	−1.44121	−0.720603	+	0.693348i	$0.756135\pi$
$978$	−4208.12	−0.137588
$979$	24411.9	0.796943
$980$	16430.0	0.535547
$981$	−4328.84	−0.140886
$982$	−5425.79	−0.176318
$983$	36334.8	1.17894	0.589471	−	0.807790i	$-0.299336\pi$
$983$	36334.8	1.17894	0.589471	+	0.807790i	$0.299336\pi$
$984$	4034.91	0.130720
$985$	61229.6	1.98065
$986$	0	0
$987$	−243.727	−0.00786010
$988$	11380.5	0.366460
$989$	34495.4	1.10909
$990$	36931.2	1.18561
$991$	9848.60	0.315692	0.157846	−	0.987464i	$-0.449545\pi$
$991$	9848.60	0.315692	0.157846	+	0.987464i	$0.449545\pi$
$992$	7487.82	0.239656
$993$	288.338	0.00921464
$994$	7298.87	0.232904
$995$	79739.1	2.54060
$996$	4759.66	0.151421
$997$	−3378.02	−0.107305	−0.0536525	−	0.998560i	$-0.517086\pi$
$997$	−3378.02	−0.107305	−0.0536525	+	0.998560i	$0.517086\pi$
$998$	9167.59	0.290777
$999$	−767.469	−0.0243060

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1682.4.a.c.1.1		2
29.28	even	2		58.4.a.c.1.2	✓	2
87.86	odd	2		522.4.a.j.1.2		2
116.115	odd	2		464.4.a.e.1.1		2
145.144	even	2		1450.4.a.g.1.1		2
232.115	odd	2		1856.4.a.i.1.2		2
232.173	even	2		1856.4.a.l.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
58.4.a.c.1.2	✓	2	29.28	even	2
464.4.a.e.1.1		2	116.115	odd	2
522.4.a.j.1.2		2	87.86	odd	2
1450.4.a.g.1.1		2	145.144	even	2
1682.4.a.c.1.1		2	1.1	even	1	trivial
1856.4.a.i.1.2		2	232.115	odd	2
1856.4.a.l.1.1		2	232.173	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1682 = 2 \cdot 29^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1682.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} -1.44949 q^{3} +4.00000 q^{4} -19.6969 q^{5} -2.89898 q^{6} +11.5959 q^{7} +8.00000 q^{8} -24.8990 q^{9} +O(q^{10})\)	\(q+2.00000 q^{2} -1.44949 q^{3} +4.00000 q^{4} -19.6969 q^{5} -2.89898 q^{6} +11.5959 q^{7} +8.00000 q^{8} -24.8990 q^{9} -39.3939 q^{10} +37.6515 q^{11} -5.79796 q^{12} -44.5959 q^{13} +23.1918 q^{14} +28.5505 q^{15} +16.0000 q^{16} +61.1918 q^{17} -49.7980 q^{18} -63.7980 q^{19} -78.7878 q^{20} -16.8082 q^{21} +75.3031 q^{22} +177.060 q^{23} -11.5959 q^{24} +262.969 q^{25} -89.1918 q^{26} +75.2270 q^{27} +46.3837 q^{28} +57.1010 q^{30} +233.994 q^{31} +32.0000 q^{32} -54.5755 q^{33} +122.384 q^{34} -228.404 q^{35} -99.5959 q^{36} -10.2020 q^{37} -127.596 q^{38} +64.6413 q^{39} -157.576 q^{40} -347.959 q^{41} -33.6163 q^{42} +194.823 q^{43} +150.606 q^{44} +490.434 q^{45} +354.120 q^{46} +14.5005 q^{47} -23.1918 q^{48} -208.535 q^{49} +525.939 q^{50} -88.6969 q^{51} -178.384 q^{52} -606.373 q^{53} +150.454 q^{54} -741.620 q^{55} +92.7673 q^{56} +92.4745 q^{57} -702.372 q^{59} +114.202 q^{60} -543.394 q^{61} +467.989 q^{62} -288.727 q^{63} +64.0000 q^{64} +878.403 q^{65} -109.151 q^{66} -407.010 q^{67} +244.767 q^{68} -256.647 q^{69} -456.808 q^{70} +314.717 q^{71} -199.192 q^{72} +859.110 q^{73} -20.4041 q^{74} -381.171 q^{75} -255.192 q^{76} +436.604 q^{77} +129.283 q^{78} -725.266 q^{79} -315.151 q^{80} +563.232 q^{81} -695.918 q^{82} -820.919 q^{83} -67.2327 q^{84} -1205.29 q^{85} +389.646 q^{86} +301.212 q^{88} +648.363 q^{89} +980.867 q^{90} -517.131 q^{91} +708.241 q^{92} -339.172 q^{93} +29.0010 q^{94} +1256.62 q^{95} -46.3837 q^{96} +60.9490 q^{97} -417.069 q^{98} -937.485 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} + 2 q^{3} + 8 q^{4} - 10 q^{5} + 4 q^{6} - 16 q^{7} + 16 q^{8} - 40 q^{9}+O(q^{10}) \) 2 * q + 4 * q^2 + 2 * q^3 + 8 * q^4 - 10 * q^5 + 4 * q^6 - 16 * q^7 + 16 * q^8 - 40 * q^9	\( 2 q + 4 q^{2} + 2 q^{3} + 8 q^{4} - 10 q^{5} + 4 q^{6} - 16 q^{7} + 16 q^{8} - 40 q^{9} - 20 q^{10} + 90 q^{11} + 8 q^{12} - 50 q^{13} - 32 q^{14} + 62 q^{15} + 32 q^{16} + 44 q^{17} - 80 q^{18} - 108 q^{19} - 40 q^{20} - 112 q^{21} + 180 q^{22} - 28 q^{23} + 16 q^{24} + 232 q^{25} - 100 q^{26} - 70 q^{27} - 64 q^{28} + 124 q^{30} - 66 q^{31} + 64 q^{32} + 126 q^{33} + 88 q^{34} - 496 q^{35} - 160 q^{36} - 40 q^{37} - 216 q^{38} + 46 q^{39} - 80 q^{40} - 304 q^{41} - 224 q^{42} + 130 q^{43} + 360 q^{44} + 344 q^{45} - 56 q^{46} + 514 q^{47} + 32 q^{48} + 210 q^{49} + 464 q^{50} - 148 q^{51} - 200 q^{52} - 958 q^{53} - 140 q^{54} - 234 q^{55} - 128 q^{56} - 60 q^{57} - 180 q^{59} + 248 q^{60} - 1028 q^{61} - 132 q^{62} + 128 q^{63} + 128 q^{64} + 826 q^{65} + 252 q^{66} - 912 q^{67} + 176 q^{68} - 964 q^{69} - 992 q^{70} + 796 q^{71} - 320 q^{72} + 856 q^{73} - 80 q^{74} - 488 q^{75} - 432 q^{76} - 1008 q^{77} + 92 q^{78} + 318 q^{79} - 160 q^{80} + 470 q^{81} - 608 q^{82} - 1828 q^{83} - 448 q^{84} - 1372 q^{85} + 260 q^{86} + 720 q^{88} + 944 q^{89} + 688 q^{90} - 368 q^{91} - 112 q^{92} - 1374 q^{93} + 1028 q^{94} + 828 q^{95} + 64 q^{96} - 368 q^{97} + 420 q^{98} - 1728 q^{99}+O(q^{100}) \) 2 * q + 4 * q^2 + 2 * q^3 + 8 * q^4 - 10 * q^5 + 4 * q^6 - 16 * q^7 + 16 * q^8 - 40 * q^9 - 20 * q^10 + 90 * q^11 + 8 * q^12 - 50 * q^13 - 32 * q^14 + 62 * q^15 + 32 * q^16 + 44 * q^17 - 80 * q^18 - 108 * q^19 - 40 * q^20 - 112 * q^21 + 180 * q^22 - 28 * q^23 + 16 * q^24 + 232 * q^25 - 100 * q^26 - 70 * q^27 - 64 * q^28 + 124 * q^30 - 66 * q^31 + 64 * q^32 + 126 * q^33 + 88 * q^34 - 496 * q^35 - 160 * q^36 - 40 * q^37 - 216 * q^38 + 46 * q^39 - 80 * q^40 - 304 * q^41 - 224 * q^42 + 130 * q^43 + 360 * q^44 + 344 * q^45 - 56 * q^46 + 514 * q^47 + 32 * q^48 + 210 * q^49 + 464 * q^50 - 148 * q^51 - 200 * q^52 - 958 * q^53 - 140 * q^54 - 234 * q^55 - 128 * q^56 - 60 * q^57 - 180 * q^59 + 248 * q^60 - 1028 * q^61 - 132 * q^62 + 128 * q^63 + 128 * q^64 + 826 * q^65 + 252 * q^66 - 912 * q^67 + 176 * q^68 - 964 * q^69 - 992 * q^70 + 796 * q^71 - 320 * q^72 + 856 * q^73 - 80 * q^74 - 488 * q^75 - 432 * q^76 - 1008 * q^77 + 92 * q^78 + 318 * q^79 - 160 * q^80 + 470 * q^81 - 608 * q^82 - 1828 * q^83 - 448 * q^84 - 1372 * q^85 + 260 * q^86 + 720 * q^88 + 944 * q^89 + 688 * q^90 - 368 * q^91 - 112 * q^92 - 1374 * q^93 + 1028 * q^94 + 828 * q^95 + 64 * q^96 - 368 * q^97 + 420 * q^98 - 1728 * q^99

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