LMFDB - Embedded newform 722.3.b.b.721.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [722,3,Mod(721,722)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(722, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("722.721");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 722 = 2 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	722.b (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$19.6730750868$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{2} + 4 $ x^4 - 2*x^2 + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 38)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			721.2
Root			$-1.22474 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	722.721
Dual form			722.3.b.b.721.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421i q^{2} +3.14626i q^{3} -2.00000 q^{4} -1.00000 q^{5} +4.44949 q^{6} +6.89898 q^{7} +2.82843i q^{8} -0.898979 q^{9} +O(q^{10})$	\(q-1.41421i q^{2} +3.14626i q^{3} -2.00000 q^{4} -1.00000 q^{5} +4.44949 q^{6} +6.89898 q^{7} +2.82843i q^{8} -0.898979 q^{9} +1.41421i q^{10} -14.8990 q^{11} -6.29253i q^{12} +17.1455i q^{13} -9.75663i q^{14} -3.14626i q^{15} +4.00000 q^{16} -2.10102 q^{17} +1.27135i q^{18} +2.00000 q^{20} +21.7060i q^{21} +21.0703i q^{22} +27.0454 q^{23} -8.89898 q^{24} -24.0000 q^{25} +24.2474 q^{26} +25.4880i q^{27} -13.7980 q^{28} +6.40329i q^{29} -4.44949 q^{30} -31.1769i q^{31} -5.65685i q^{32} -46.8761i q^{33} +2.97129i q^{34} -6.89898 q^{35} +1.79796 q^{36} -28.9199i q^{37} -53.9444 q^{39} -2.82843i q^{40} +64.5931i q^{41} +30.6969 q^{42} -75.3383 q^{43} +29.7980 q^{44} +0.898979 q^{45} -38.2480i q^{46} -11.5403 q^{47} +12.5851i q^{48} -1.40408 q^{49} +33.9411i q^{50} -6.61037i q^{51} -34.2911i q^{52} +80.0066i q^{53} +36.0454 q^{54} +14.8990 q^{55} +19.5133i q^{56} +9.05561 q^{58} +58.7934i q^{59} +6.29253i q^{60} +2.19184 q^{61} -44.0908 q^{62} -6.20204 q^{63} -8.00000 q^{64} -17.1455i q^{65} -66.2929 q^{66} +59.6506i q^{67} +4.20204 q^{68} +85.0920i q^{69} +9.75663i q^{70} +101.063i q^{71} -2.54270i q^{72} -127.384 q^{73} -40.8990 q^{74} -75.5103i q^{75} -102.788 q^{77} +76.2889i q^{78} -6.67458i q^{79} -4.00000 q^{80} -88.2827 q^{81} +91.3485 q^{82} -1.30306 q^{83} -43.4120i q^{84} +2.10102 q^{85} +106.544i q^{86} -20.1464 q^{87} -42.1407i q^{88} -6.75323i q^{89} -1.27135i q^{90} +118.287i q^{91} -54.0908 q^{92} +98.0908 q^{93} +16.3205i q^{94} +17.7980 q^{96} -131.811i q^{97} +1.98567i q^{98} +13.3939 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{4} - 4 q^{5} + 8 q^{6} + 8 q^{7} + 16 q^{9}+O(q^{10}) $ 4 * q - 8 * q^4 - 4 * q^5 + 8 * q^6 + 8 * q^7 + 16 * q^9	$ 4 q - 8 q^{4} - 4 q^{5} + 8 q^{6} + 8 q^{7} + 16 q^{9} - 40 q^{11} + 16 q^{16} - 28 q^{17} + 8 q^{20} + 20 q^{23} - 16 q^{24} - 96 q^{25} + 48 q^{26} - 16 q^{28} - 8 q^{30} - 8 q^{35} - 32 q^{36} - 108 q^{39} + 64 q^{42} - 76 q^{43} + 80 q^{44} - 16 q^{45} + 140 q^{47} - 84 q^{49} + 56 q^{54} + 40 q^{55} + 144 q^{58} - 148 q^{61} - 64 q^{63} - 32 q^{64} - 128 q^{66} + 56 q^{68} - 196 q^{73} - 144 q^{74} - 176 q^{77} - 16 q^{80} - 20 q^{81} + 336 q^{82} - 64 q^{83} + 28 q^{85} - 12 q^{87} - 40 q^{92} + 216 q^{93} + 32 q^{96} - 64 q^{99}+O(q^{100}) $ 4 * q - 8 * q^4 - 4 * q^5 + 8 * q^6 + 8 * q^7 + 16 * q^9 - 40 * q^11 + 16 * q^16 - 28 * q^17 + 8 * q^20 + 20 * q^23 - 16 * q^24 - 96 * q^25 + 48 * q^26 - 16 * q^28 - 8 * q^30 - 8 * q^35 - 32 * q^36 - 108 * q^39 + 64 * q^42 - 76 * q^43 + 80 * q^44 - 16 * q^45 + 140 * q^47 - 84 * q^49 + 56 * q^54 + 40 * q^55 + 144 * q^58 - 148 * q^61 - 64 * q^63 - 32 * q^64 - 128 * q^66 + 56 * q^68 - 196 * q^73 - 144 * q^74 - 176 * q^77 - 16 * q^80 - 20 * q^81 + 336 * q^82 - 64 * q^83 + 28 * q^85 - 12 * q^87 - 40 * q^92 + 216 * q^93 + 32 * q^96 - 64 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/722\mathbb{Z}\right)^\times$.

$n$	$363$
$\chi(n)$	$-1$

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 $

$2$	− 1.41421i	− 0.707107i
$2$	− 1.41421i	− 0.707107i
$3$	3.14626i	1.04875i	0.851486	+	0.524377i	$0.175702\pi$
$3$	3.14626i	1.04875i	−0.851486	+	0.524377i	$0.824298\pi$
$4$	−2.00000	−0.500000
$5$	−1.00000	−0.200000	−0.100000	−	0.994987i	$-0.531884\pi$
$5$	−1.00000	−0.200000	−0.100000	+	0.994987i	$0.531884\pi$
$6$	4.44949	0.741582
$7$	6.89898	0.985568	0.492784	−	0.870152i	$-0.335979\pi$
$7$	6.89898	0.985568	0.492784	+	0.870152i	$0.335979\pi$
$8$	2.82843i	0.353553i
$9$	−0.898979	−0.0998866
$10$	1.41421i	0.141421i
$11$	−14.8990	−1.35445	−0.677226	−	0.735775i	$-0.736818\pi$
$11$	−14.8990	−1.35445	−0.677226	+	0.735775i	$0.736818\pi$
$12$	− 6.29253i	− 0.524377i
$13$	17.1455i	1.31889i	0.751754	+	0.659444i	$0.229208\pi$
$13$	17.1455i	1.31889i	−0.751754	+	0.659444i	$0.770792\pi$
$14$	− 9.75663i	− 0.696902i
$15$	− 3.14626i	− 0.209751i
$16$	4.00000	0.250000
$17$	−2.10102	−0.123589	−0.0617947	−	0.998089i	$-0.519682\pi$
$17$	−2.10102	−0.123589	−0.0617947	+	0.998089i	$0.519682\pi$
$18$	1.27135i	0.0706305i
$19$	0	0
$19$	0	0
$20$	2.00000	0.100000
$21$	21.7060i	1.03362i
$22$	21.0703i	0.957743i
$23$	27.0454	1.17589	0.587944	−	0.808902i	$-0.299938\pi$
$23$	27.0454	1.17589	0.587944	+	0.808902i	$0.299938\pi$
$24$	−8.89898	−0.370791
$25$	−24.0000	−0.960000
$26$	24.2474	0.932594
$27$	25.4880i	0.943998i
$28$	−13.7980	−0.492784
$29$	6.40329i	0.220803i	0.993887	+	0.110401i	$0.0352137\pi$
$29$	6.40329i	0.220803i	−0.993887	+	0.110401i	$0.964786\pi$
$30$	−4.44949	−0.148316
$31$	− 31.1769i	− 1.00571i	−0.864372	−	0.502853i	$-0.832284\pi$
$31$	− 31.1769i	− 1.00571i	0.864372	−	0.502853i	$-0.167716\pi$
$32$	− 5.65685i	− 0.176777i
$33$	− 46.8761i	− 1.42049i
$34$	2.97129i	0.0873909i
$35$	−6.89898	−0.197114
$36$	1.79796	0.0499433
$37$	− 28.9199i	− 0.781620i	−0.920471	−	0.390810i	$-0.872195\pi$
$37$	− 28.9199i	− 0.781620i	0.920471	−	0.390810i	$-0.127805\pi$
$38$	0	0
$39$	−53.9444	−1.38319
$40$	− 2.82843i	− 0.0707107i
$41$	64.5931i	1.57544i	0.616032	+	0.787721i	$0.288739\pi$
$41$	64.5931i	1.57544i	−0.616032	+	0.787721i	$0.711261\pi$
$42$	30.6969	0.730879
$43$	−75.3383	−1.75205	−0.876026	−	0.482263i	$-0.839815\pi$
$43$	−75.3383	−1.75205	−0.876026	+	0.482263i	$0.839815\pi$
$44$	29.7980	0.677226
$45$	0.898979	0.0199773
$46$	− 38.2480i	− 0.831478i
$47$	−11.5403	−0.245538	−0.122769	−	0.992435i	$-0.539177\pi$
$47$	−11.5403	−0.245538	−0.122769	+	0.992435i	$0.539177\pi$
$48$	12.5851i	0.262189i
$49$	−1.40408	−0.0286547
$50$	33.9411i	0.678823i
$51$	− 6.61037i	− 0.129615i
$52$	− 34.2911i	− 0.659444i
$53$	80.0066i	1.50956i	0.655979	+	0.754779i	$0.272256\pi$
$53$	80.0066i	1.50956i	−0.655979	+	0.754779i	$0.727744\pi$
$54$	36.0454	0.667508
$55$	14.8990	0.270891
$56$	19.5133i	0.348451i
$57$	0	0
$58$	9.05561	0.156131
$59$	58.7934i	0.996498i	0.867034	+	0.498249i	$0.166024\pi$
$59$	58.7934i	0.996498i	−0.867034	+	0.498249i	$0.833976\pi$
$60$	6.29253i	0.104875i
$61$	2.19184	0.0359317	0.0179659	−	0.999839i	$-0.494281\pi$
$61$	2.19184	0.0359317	0.0179659	+	0.999839i	$0.494281\pi$
$62$	−44.0908	−0.711142
$63$	−6.20204	−0.0984451
$64$	−8.00000	−0.125000
$65$	− 17.1455i	− 0.263777i
$66$	−66.2929	−1.00444
$67$	59.6506i	0.890307i	0.895454	+	0.445154i	$0.146851\pi$
$67$	59.6506i	0.890307i	−0.895454	+	0.445154i	$0.853149\pi$
$68$	4.20204	0.0617947
$69$	85.0920i	1.23322i
$70$	9.75663i	0.139380i
$71$	101.063i	1.42342i	0.702476	+	0.711708i	$0.252078\pi$
$71$	101.063i	1.42342i	−0.702476	+	0.711708i	$0.747922\pi$
$72$	− 2.54270i	− 0.0353152i
$73$	−127.384	−1.74498	−0.872491	−	0.488630i	$-0.837497\pi$
$73$	−127.384	−1.74498	−0.872491	+	0.488630i	$0.837497\pi$
$74$	−40.8990	−0.552689
$75$	− 75.5103i	− 1.00680i
$76$	0	0
$77$	−102.788	−1.33491
$78$	76.2889i	0.978063i
$79$	− 6.67458i	− 0.0844884i	−0.999107	−	0.0422442i	$-0.986549\pi$
$79$	− 6.67458i	− 0.0844884i	0.999107	−	0.0422442i	$-0.0134507\pi$
$80$	−4.00000	−0.0500000
$81$	−88.2827	−1.08991
$82$	91.3485	1.11401
$83$	−1.30306	−0.0156995	−0.00784977	−	0.999969i	$-0.502499\pi$
$83$	−1.30306	−0.0156995	−0.00784977	+	0.999969i	$0.502499\pi$
$84$	− 43.4120i	− 0.516810i
$85$	2.10102	0.0247179
$86$	106.544i	1.23889i
$87$	−20.1464	−0.231568
$88$	− 42.1407i	− 0.478871i
$89$	− 6.75323i	− 0.0758790i	−0.999280	−	0.0379395i	$-0.987921\pi$
$89$	− 6.75323i	− 0.0758790i	0.999280	−	0.0379395i	$-0.0120794\pi$
$90$	− 1.27135i	− 0.0141261i
$91$	118.287i	1.29985i
$92$	−54.0908	−0.587944
$93$	98.0908	1.05474
$94$	16.3205i	0.173622i
$95$	0	0
$96$	17.7980	0.185395
$97$	− 131.811i	− 1.35887i	−0.733734	−	0.679437i	$-0.762224\pi$
$97$	− 131.811i	− 1.35887i	0.733734	−	0.679437i	$-0.237776\pi$
$98$	1.98567i	0.0202620i
$99$	13.3939	0.135292
$100$	48.0000	0.480000
$101$	6.48469	0.0642049	0.0321024	−	0.999485i	$-0.489780\pi$
$101$	6.48469	0.0642049	0.0321024	+	0.999485i	$0.489780\pi$
$102$	−9.34847	−0.0916517
$103$	80.0243i	0.776935i	0.921462	+	0.388467i	$0.126995\pi$
$103$	80.0243i	0.776935i	−0.921462	+	0.388467i	$0.873005\pi$
$104$	−48.4949	−0.466297
$105$	− 21.7060i	− 0.206724i
$106$	113.146	1.06742
$107$	80.0243i	0.747890i	0.927451	+	0.373945i	$0.121995\pi$
$107$	80.0243i	0.747890i	−0.927451	+	0.373945i	$0.878005\pi$
$108$	− 50.9759i	− 0.471999i
$109$	− 16.4456i	− 0.150877i	−0.997150	−	0.0754387i	$-0.975964\pi$
$109$	− 16.4456i	− 0.150877i	0.997150	−	0.0754387i	$-0.0240357\pi$
$110$	− 21.0703i	− 0.191549i
$111$	90.9898	0.819728
$112$	27.5959	0.246392
$113$	− 132.843i	− 1.17560i	−0.809006	−	0.587801i	$-0.799994\pi$
$113$	− 132.843i	− 1.17560i	0.809006	−	0.587801i	$-0.200006\pi$
$114$	0	0
$115$	−27.0454	−0.235177
$116$	− 12.8066i	− 0.110401i
$117$	− 15.4135i	− 0.131739i
$118$	83.1464	0.704631
$119$	−14.4949	−0.121806
$120$	8.89898	0.0741582
$121$	100.980	0.834542
$122$	− 3.09972i	− 0.0254076i
$123$	−203.227	−1.65225
$124$	62.3538i	0.502853i
$125$	49.0000	0.392000
$126$	8.77101i	0.0696112i
$127$	− 125.661i	− 0.989458i	−0.869047	−	0.494729i	$-0.835267\pi$
$127$	− 125.661i	− 0.989458i	0.869047	−	0.494729i	$-0.164733\pi$
$128$	11.3137i	0.0883883i
$129$	− 237.034i	− 1.83747i
$130$	−24.2474	−0.186519
$131$	107.247	0.818683	0.409341	−	0.912381i	$-0.365759\pi$
$131$	107.247	0.818683	0.409341	+	0.912381i	$0.365759\pi$
$132$	93.7523i	0.710244i
$133$	0	0
$134$	84.3587	0.629542
$135$	− 25.4880i	− 0.188800i
$136$	− 5.94258i	− 0.0436955i
$137$	−2.10102	−0.0153359	−0.00766796	−	0.999971i	$-0.502441\pi$
$137$	−2.10102	−0.0153359	−0.00766796	+	0.999971i	$0.502441\pi$
$138$	120.338	0.872016
$139$	−31.7423	−0.228362	−0.114181	−	0.993460i	$-0.536424\pi$
$139$	−31.7423	−0.228362	−0.114181	+	0.993460i	$0.536424\pi$
$140$	13.7980	0.0985568
$141$	− 36.3089i	− 0.257510i
$142$	142.924	1.00651
$143$	− 255.451i	− 1.78637i
$144$	−3.59592	−0.0249717
$145$	− 6.40329i	− 0.0441606i
$146$	180.148i	1.23389i
$147$	− 4.41761i	− 0.0300518i
$148$	57.8399i	0.390810i
$149$	189.677	1.27300	0.636498	−	0.771278i	$-0.280382\pi$
$149$	189.677	1.27300	0.636498	+	0.771278i	$0.280382\pi$
$150$	−106.788	−0.711918
$151$	16.2707i	0.107753i	0.998548	+	0.0538764i	$0.0171577\pi$
$151$	16.2707i	0.107753i	−0.998548	+	0.0538764i	$0.982842\pi$
$152$	0	0
$153$	1.88877	0.0123449
$154$	145.364i	0.943921i
$155$	31.1769i	0.201141i
$156$	107.889	0.691595
$157$	270.151	1.72071	0.860354	−	0.509698i	$-0.170243\pi$
$157$	270.151	1.72071	0.860354	+	0.509698i	$0.170243\pi$
$158$	−9.43928	−0.0597423
$159$	−251.722	−1.58316
$160$	5.65685i	0.0353553i
$161$	186.586	1.15892
$162$	124.851i	0.770682i
$163$	−156.697	−0.961331	−0.480665	−	0.876904i	$-0.659605\pi$
$163$	−156.697	−0.961331	−0.480665	+	0.876904i	$0.659605\pi$
$164$	− 129.186i	− 0.787721i
$165$	46.8761i	0.284098i
$166$	1.84281i	0.0111012i
$167$	90.1630i	0.539898i	0.962875	+	0.269949i	$0.0870068\pi$
$167$	90.1630i	0.539898i	−0.962875	+	0.269949i	$0.912993\pi$
$168$	−61.3939	−0.365440
$169$	−124.969	−0.739464
$170$	− 2.97129i	− 0.0174782i
$171$	0	0
$172$	150.677	0.876026
$173$	− 304.123i	− 1.75794i	−0.476878	−	0.878969i	$-0.658232\pi$
$173$	− 304.123i	− 1.75794i	0.476878	−	0.878969i	$-0.341768\pi$
$174$	28.4914i	0.163743i
$175$	−165.576	−0.946146
$176$	−59.5959	−0.338613
$177$	−184.980	−1.04508
$178$	−9.55051	−0.0536546
$179$	315.198i	1.76088i	0.474156	+	0.880441i	$0.342753\pi$
$179$	315.198i	1.76088i	−0.474156	+	0.880441i	$0.657247\pi$
$180$	−1.79796	−0.00998866
$181$	32.2091i	0.177951i	0.996034	+	0.0889753i	$0.0283592\pi$
$181$	32.2091i	0.177951i	−0.996034	+	0.0889753i	$0.971641\pi$
$182$	167.283	0.919135
$183$	6.89610i	0.0376836i
$184$	76.4960i	0.415739i
$185$	28.9199i	0.156324i
$186$	− 138.721i	− 0.745814i
$187$	31.3031	0.167396
$188$	23.0806	0.122769
$189$	175.841i	0.930375i
$190$	0	0
$191$	280.252	1.46729	0.733644	−	0.679534i	$-0.237818\pi$
$191$	280.252	1.46729	0.733644	+	0.679534i	$0.237818\pi$
$192$	− 25.1701i	− 0.131094i
$193$	48.5151i	0.251374i	0.992070	+	0.125687i	$0.0401134\pi$
$193$	48.5151i	0.251374i	−0.992070	+	0.125687i	$0.959887\pi$
$194$	−186.409	−0.960869
$195$	53.9444	0.276638
$196$	2.80816	0.0143274
$197$	251.909	1.27873	0.639363	−	0.768905i	$-0.279198\pi$
$197$	251.909	1.27873	0.639363	+	0.768905i	$0.279198\pi$
$198$	− 18.9418i	− 0.0956657i
$199$	259.388	1.30346	0.651729	−	0.758452i	$-0.274044\pi$
$199$	259.388	1.30346	0.651729	+	0.758452i	$0.274044\pi$
$200$	− 67.8823i	− 0.339411i
$201$	−187.677	−0.933714
$202$	− 9.17074i	− 0.0453997i
$203$	44.1761i	0.217616i
$204$	13.2207i	0.0648075i
$205$	− 64.5931i	− 0.315088i
$206$	113.171	0.549376
$207$	−24.3133	−0.117455
$208$	68.5821i	0.329722i
$209$	0	0
$210$	−30.6969	−0.146176
$211$	69.5710i	0.329720i	0.986317	+	0.164860i	$0.0527173\pi$
$211$	69.5710i	0.329720i	−0.986317	+	0.164860i	$0.947283\pi$
$212$	− 160.013i	− 0.754779i
$213$	−317.969	−1.49281
$214$	113.171	0.528838
$215$	75.3383	0.350411
$216$	−72.0908	−0.333754
$217$	− 215.089i	− 0.991193i
$218$	−23.2577	−0.106686
$219$	− 400.783i	− 1.83006i
$220$	−29.7980	−0.135445
$221$	− 36.0231i	− 0.163001i
$222$	− 128.679i	− 0.579635i
$223$	− 6.32464i	− 0.0283616i	−0.999899	−	0.0141808i	$-0.995486\pi$
$223$	− 6.32464i	− 0.0283616i	0.999899	−	0.0141808i	$-0.00451404\pi$
$224$	− 39.0265i	− 0.174226i
$225$	21.5755	0.0958911
$226$	−187.868	−0.831276
$227$	55.7402i	0.245552i	0.992434	+	0.122776i	$0.0391796\pi$
$227$	55.7402i	0.245552i	−0.992434	+	0.122776i	$0.960820\pi$
$228$	0	0
$229$	−132.313	−0.577787	−0.288894	−	0.957361i	$-0.593287\pi$
$229$	−132.313	−0.577787	−0.288894	+	0.957361i	$0.593287\pi$
$230$	38.2480i	0.166296i
$231$	− 323.397i	− 1.39999i
$232$	−18.1112	−0.0780656
$233$	−223.717	−0.960160	−0.480080	−	0.877225i	$-0.659392\pi$
$233$	−223.717	−0.960160	−0.480080	+	0.877225i	$0.659392\pi$
$234$	−21.7980	−0.0931537
$235$	11.5403	0.0491077
$236$	− 117.587i	− 0.498249i
$237$	21.0000	0.0886076
$238$	20.4989i	0.0861298i
$239$	−71.3235	−0.298425	−0.149212	−	0.988805i	$-0.547674\pi$
$239$	−71.3235	−0.298425	−0.149212	+	0.988805i	$0.547674\pi$
$240$	− 12.5851i	− 0.0524377i
$241$	− 139.404i	− 0.578438i	−0.957263	−	0.289219i	$-0.906604\pi$
$241$	− 139.404i	− 0.578438i	0.957263	−	0.289219i	$-0.0933956\pi$
$242$	− 142.807i	− 0.590110i
$243$	− 48.3690i	− 0.199049i
$244$	−4.38367	−0.0179659
$245$	1.40408	0.00573095
$246$	287.406i	1.16832i
$247$	0	0
$248$	88.1816	0.355571
$249$	− 4.09978i	− 0.0164650i
$250$	− 69.2965i	− 0.277186i
$251$	240.076	0.956478	0.478239	−	0.878230i	$-0.341275\pi$
$251$	240.076	0.956478	0.478239	+	0.878230i	$0.341275\pi$
$252$	12.4041	0.0492225
$253$	−402.949	−1.59268
$254$	−177.712	−0.699652
$255$	6.61037i	0.0259230i
$256$	16.0000	0.0625000
$257$	− 9.67474i	− 0.0376449i	−0.999823	−	0.0188224i	$-0.994008\pi$
$257$	− 9.67474i	− 0.0376449i	0.999823	−	0.0188224i	$-0.00599173\pi$
$258$	−335.217	−1.29929
$259$	− 199.518i	− 0.770340i
$260$	34.2911i	0.131889i
$261$	− 5.75642i	− 0.0220553i
$262$	− 151.671i	− 0.578896i
$263$	−113.924	−0.433171	−0.216586	−	0.976264i	$-0.569492\pi$
$263$	−113.924	−0.433171	−0.216586	+	0.976264i	$0.569492\pi$
$264$	132.586	0.502219
$265$	− 80.0066i	− 0.301912i
$266$	0	0
$267$	21.2474	0.0795785
$268$	− 119.301i	− 0.445154i
$269$	− 401.940i	− 1.49420i	−0.664711	−	0.747100i	$-0.731445\pi$
$269$	− 401.940i	− 1.49420i	0.664711	−	0.747100i	$-0.268555\pi$
$270$	−36.0454	−0.133502
$271$	398.984	1.47227	0.736133	−	0.676837i	$-0.236650\pi$
$271$	398.984	1.47227	0.736133	+	0.676837i	$0.236650\pi$
$272$	−8.40408	−0.0308974
$273$	−372.161	−1.36323
$274$	2.97129i	0.0108441i
$275$	357.576	1.30027
$276$	− 170.184i	− 0.616609i
$277$	529.444	1.91135	0.955675	−	0.294424i	$-0.0951278\pi$
$277$	529.444	1.91135	0.955675	+	0.294424i	$0.0951278\pi$
$278$	44.8905i	0.161476i
$279$	28.0274i	0.100457i
$280$	− 19.5133i	− 0.0696902i
$281$	− 54.7434i	− 0.194816i	−0.995245	−	0.0974082i	$-0.968945\pi$
$281$	− 54.7434i	− 0.194816i	0.995245	−	0.0974082i	$-0.0310552\pi$
$282$	−51.3485	−0.182087
$283$	−148.934	−0.526269	−0.263135	−	0.964759i	$-0.584756\pi$
$283$	−148.934	−0.526269	−0.263135	+	0.964759i	$0.584756\pi$
$284$	− 202.125i	− 0.711708i
$285$	0	0
$286$	−361.262	−1.26315
$287$	445.627i	1.55271i
$288$	5.08540i	0.0176576i
$289$	−284.586	−0.984726
$290$	−9.05561	−0.0312263
$291$	414.712	1.42513
$292$	254.767	0.872491
$293$	− 145.685i	− 0.497218i	−0.968604	−	0.248609i	$-0.920027\pi$
$293$	− 145.685i	− 0.497218i	0.968604	−	0.248609i	$-0.0799734\pi$
$294$	−6.24745	−0.0212498
$295$	− 58.7934i	− 0.199300i
$296$	81.7980	0.276344
$297$	− 379.744i	− 1.27860i
$298$	− 268.243i	− 0.900145i
$299$	463.708i	1.55086i
$300$	151.021i	0.503402i
$301$	−519.757	−1.72677
$302$	23.0102	0.0761927
$303$	20.4026i	0.0673352i
$304$	0	0
$305$	−2.19184	−0.00718635
$306$	− 2.67113i	− 0.00872918i
$307$	113.205i	0.368744i	0.982856	+	0.184372i	$0.0590252\pi$
$307$	113.205i	0.368744i	−0.982856	+	0.184372i	$0.940975\pi$
$308$	205.576	0.667453
$309$	−251.778	−0.814814
$310$	44.0908	0.142228
$311$	−105.666	−0.339763	−0.169882	−	0.985464i	$-0.554339\pi$
$311$	−105.666	−0.339763	−0.169882	+	0.985464i	$0.554339\pi$
$312$	− 152.578i	− 0.489031i
$313$	248.242	0.793105	0.396552	−	0.918012i	$-0.370206\pi$
$313$	248.242	0.793105	0.396552	+	0.918012i	$0.370206\pi$
$314$	− 382.051i	− 1.21672i
$315$	6.20204	0.0196890
$316$	13.3492i	0.0422442i
$317$	226.391i	0.714168i	0.934072	+	0.357084i	$0.116229\pi$
$317$	226.391i	0.714168i	−0.934072	+	0.357084i	$0.883771\pi$
$318$	355.989i	1.11946i
$319$	− 95.4024i	− 0.299067i
$320$	8.00000	0.0250000
$321$	−251.778	−0.784354
$322$	− 263.872i	− 0.819478i
$323$	0	0
$324$	176.565	0.544955
$325$	− 411.493i	− 1.26613i
$326$	221.603i	0.679764i
$327$	51.7423	0.158233
$328$	−182.697	−0.557003
$329$	−79.6163	−0.241995
$330$	66.2929	0.200887
$331$	− 293.328i	− 0.886188i	−0.896475	−	0.443094i	$-0.853881\pi$
$331$	− 293.328i	− 0.886188i	0.896475	−	0.443094i	$-0.146119\pi$
$332$	2.60612	0.00784977
$333$	25.9984i	0.0780734i
$334$	127.510	0.381766
$335$	− 59.6506i	− 0.178061i
$336$	86.8241i	0.258405i
$337$	− 461.180i	− 1.36849i	−0.729254	−	0.684243i	$-0.760133\pi$
$337$	− 461.180i	− 1.36849i	0.729254	−	0.684243i	$-0.239867\pi$
$338$	176.733i	0.522880i
$339$	417.959	1.23292
$340$	−4.20204	−0.0123589
$341$	464.504i	1.36218i
$342$	0	0
$343$	−347.737	−1.01381
$344$	− 213.089i	− 0.619444i
$345$	− 85.0920i	− 0.246643i
$346$	−430.095	−1.24305
$347$	304.258	0.876823	0.438412	−	0.898774i	$-0.355541\pi$
$347$	304.258	0.876823	0.438412	+	0.898774i	$0.355541\pi$
$348$	40.2929	0.115784
$349$	50.4337	0.144509	0.0722545	−	0.997386i	$-0.476981\pi$
$349$	50.4337	0.144509	0.0722545	+	0.997386i	$0.476981\pi$
$350$	234.159i	0.669026i
$351$	−437.005	−1.24503
$352$	84.2814i	0.239436i
$353$	−316.817	−0.897500	−0.448750	−	0.893657i	$-0.648131\pi$
$353$	−316.817	−0.897500	−0.448750	+	0.893657i	$0.648131\pi$
$354$	261.601i	0.738985i
$355$	− 101.063i	− 0.284683i
$356$	13.5065i	0.0379395i
$357$	− 45.6048i	− 0.127744i
$358$	445.757	1.24513
$359$	149.409	0.416180	0.208090	−	0.978110i	$-0.433275\pi$
$359$	149.409	0.416180	0.208090	+	0.978110i	$0.433275\pi$
$360$	2.54270i	0.00706305i
$361$	0	0
$362$	45.5505	0.125830
$363$	317.708i	0.875230i
$364$	− 236.573i	− 0.649927i
$365$	127.384	0.348996
$366$	9.75255	0.0266463
$367$	234.076	0.637809	0.318905	−	0.947787i	$-0.396685\pi$
$367$	234.076	0.637809	0.318905	+	0.947787i	$0.396685\pi$
$368$	108.182	0.293972
$369$	− 58.0679i	− 0.157366i
$370$	40.8990	0.110538
$371$	551.964i	1.48777i
$372$	−196.182	−0.527370
$373$	639.738i	1.71512i	0.514387	+	0.857558i	$0.328019\pi$
$373$	639.738i	1.71512i	−0.514387	+	0.857558i	$0.671981\pi$
$374$	− 44.2692i	− 0.118367i
$375$	154.167i	0.411112i
$376$	− 32.6409i	− 0.0868109i
$377$	−109.788	−0.291214
$378$	248.677	0.657874
$379$	− 108.366i	− 0.285927i	−0.989728	−	0.142963i	$-0.954337\pi$
$379$	− 108.366i	− 0.285927i	0.989728	−	0.142963i	$-0.0456631\pi$
$380$	0	0
$381$	395.363	1.03770
$382$	− 396.336i	− 1.03753i
$383$	248.898i	0.649865i	0.945737	+	0.324933i	$0.105342\pi$
$383$	248.898i	0.649865i	−0.945737	+	0.324933i	$0.894658\pi$
$384$	−35.5959	−0.0926977
$385$	102.788	0.266981
$386$	68.6107	0.177748
$387$	67.7276	0.175007
$388$	263.622i	0.679437i
$389$	−671.120	−1.72525	−0.862623	−	0.505848i	$-0.831180\pi$
$389$	−671.120	−1.72525	−0.862623	+	0.505848i	$0.831180\pi$
$390$	− 76.2889i	− 0.195613i
$391$	−56.8230	−0.145327
$392$	− 3.97134i	− 0.0101310i
$393$	337.429i	0.858598i
$394$	− 356.253i	− 0.904196i
$395$	6.67458i	0.0168977i
$396$	−26.7878	−0.0676458
$397$	−93.3633	−0.235172	−0.117586	−	0.993063i	$-0.537516\pi$
$397$	−93.3633	−0.235172	−0.117586	+	0.993063i	$0.537516\pi$
$398$	− 366.830i	− 0.921684i
$399$	0	0
$400$	−96.0000	−0.240000
$401$	279.997i	0.698246i	0.937077	+	0.349123i	$0.113520\pi$
$401$	279.997i	0.698246i	−0.937077	+	0.349123i	$0.886480\pi$
$402$	265.415i	0.660236i
$403$	534.545	1.32641
$404$	−12.9694	−0.0321024
$405$	88.2827	0.217982
$406$	62.4745	0.153878
$407$	430.878i	1.05867i
$408$	18.6969	0.0458258
$409$	641.820i	1.56924i	0.619975	+	0.784621i	$0.287143\pi$
$409$	641.820i	1.56924i	−0.619975	+	0.784621i	$0.712857\pi$
$410$	−91.3485	−0.222801
$411$	− 6.61037i	− 0.0160836i
$412$	− 160.049i	− 0.388467i
$413$	405.614i	0.982117i
$414$	34.3842i	0.0830535i
$415$	1.30306	0.00313991
$416$	96.9898	0.233149
$417$	− 99.8698i	− 0.239496i
$418$	0	0
$419$	668.879	1.59637	0.798184	−	0.602413i	$-0.205794\pi$
$419$	668.879	1.59637	0.798184	+	0.602413i	$0.205794\pi$
$420$	43.4120i	0.103362i
$421$	500.807i	1.18956i	0.803887	+	0.594782i	$0.202762\pi$
$421$	500.807i	1.18956i	−0.803887	+	0.594782i	$0.797238\pi$
$422$	98.3883	0.233148
$423$	10.3745	0.0245260
$424$	−226.293	−0.533710
$425$	50.4245	0.118646
$426$	449.677i	1.05558i
$427$	15.1214	0.0354132
$428$	− 160.049i	− 0.373945i
$429$	803.716	1.87346
$430$	− 106.544i	− 0.247778i
$431$	− 139.203i	− 0.322977i	−0.986875	−	0.161488i	$-0.948371\pi$
$431$	− 139.203i	− 0.322977i	0.986875	−	0.161488i	$-0.0516294\pi$
$432$	101.952i	0.236000i
$433$	− 208.249i	− 0.480945i	−0.970656	−	0.240472i	$-0.922698\pi$
$433$	− 208.249i	− 0.480945i	0.970656	−	0.240472i	$-0.0773023\pi$
$434$	−304.182	−0.700879
$435$	20.1464	0.0463136
$436$	32.8913i	0.0754387i
$437$	0	0
$438$	−566.792	−1.29405
$439$	75.8347i	0.172744i	0.996263	+	0.0863721i	$0.0275274\pi$
$439$	75.8347i	0.172744i	−0.996263	+	0.0863721i	$0.972473\pi$
$440$	42.1407i	0.0957743i
$441$	1.26224	0.00286222
$442$	−50.9444	−0.115259
$443$	−455.257	−1.02767	−0.513834	−	0.857890i	$-0.671775\pi$
$443$	−455.257	−1.02767	−0.513834	+	0.857890i	$0.671775\pi$
$444$	−181.980	−0.409864
$445$	6.75323i	0.0151758i
$446$	−8.94439	−0.0200547
$447$	596.773i	1.33506i
$448$	−55.1918	−0.123196
$449$	− 8.44993i	− 0.0188194i	−0.999956	−	0.00940972i	$-0.997005\pi$
$449$	− 8.44993i	− 0.0188194i	0.999956	−	0.00940972i	$-0.00299525\pi$
$450$	− 30.5124i	− 0.0678053i
$451$	− 962.372i	− 2.13386i
$452$	265.686i	0.587801i
$453$	−51.1918	−0.113006
$454$	78.8286	0.173631
$455$	− 118.287i	− 0.259971i
$456$	0	0
$457$	208.424	0.456071	0.228036	−	0.973653i	$-0.426770\pi$
$457$	208.424	0.456071	0.228036	+	0.973653i	$0.426770\pi$
$458$	187.119i	0.408557i
$459$	− 53.5507i	− 0.116668i
$460$	54.0908	0.117589
$461$	−617.615	−1.33973	−0.669865	−	0.742483i	$-0.733648\pi$
$461$	−617.615	−1.33973	−0.669865	+	0.742483i	$0.733648\pi$
$462$	−457.353	−0.989942
$463$	640.958	1.38436	0.692179	−	0.721725i	$-0.256651\pi$
$463$	640.958	1.38436	0.692179	+	0.721725i	$0.256651\pi$
$464$	25.6131i	0.0552007i
$465$	−98.0908	−0.210948
$466$	316.384i	0.678936i
$467$	164.111	0.351416	0.175708	−	0.984442i	$-0.443779\pi$
$467$	164.111	0.351416	0.175708	+	0.984442i	$0.443779\pi$
$468$	30.8270i	0.0658696i
$469$	411.528i	0.877459i
$470$	− 16.3205i	− 0.0347244i
$471$	849.967i	1.80460i
$472$	−166.293	−0.352315
$473$	1122.46	2.37307
$474$	− 29.6985i	− 0.0626550i
$475$	0	0
$476$	28.9898	0.0609029
$477$	− 71.9243i	− 0.150785i
$478$	100.867i	0.211018i
$479$	802.803	1.67600	0.837998	−	0.545673i	$-0.183726\pi$
$479$	802.803	1.67600	0.837998	+	0.545673i	$0.183726\pi$
$480$	−17.7980	−0.0370791
$481$	495.848	1.03087
$482$	−197.146	−0.409017
$483$	587.048i	1.21542i
$484$	−201.959	−0.417271
$485$	131.811i	0.271775i
$486$	−68.4041	−0.140749
$487$	454.391i	0.933041i	0.884510	+	0.466521i	$0.154493\pi$
$487$	454.391i	0.933041i	−0.884510	+	0.466521i	$0.845507\pi$
$488$	6.19945i	0.0127038i
$489$	− 493.010i	− 1.00820i
$490$	− 1.98567i	− 0.00405239i
$491$	519.652	1.05835	0.529177	−	0.848512i	$-0.322501\pi$
$491$	519.652	1.05835	0.529177	+	0.848512i	$0.322501\pi$
$492$	406.454	0.826126
$493$	− 13.4534i	− 0.0272889i
$494$	0	0
$495$	−13.3939	−0.0270583
$496$	− 124.708i	− 0.251427i
$497$	697.228i	1.40287i
$498$	−5.79796	−0.0116425
$499$	443.086	0.887948	0.443974	−	0.896040i	$-0.353568\pi$
$499$	443.086	0.887948	0.443974	+	0.896040i	$0.353568\pi$
$500$	−98.0000	−0.196000
$501$	−283.677	−0.566221
$502$	− 339.519i	− 0.676332i
$503$	−241.681	−0.480479	−0.240240	−	0.970714i	$-0.577226\pi$
$503$	−241.681	−0.480479	−0.240240	+	0.970714i	$0.577226\pi$
$504$	− 17.5420i	− 0.0348056i
$505$	−6.48469	−0.0128410
$506$	569.856i	1.12620i
$507$	− 393.187i	− 0.775516i
$508$	251.322i	0.494729i
$509$	− 412.788i	− 0.810979i	−0.914100	−	0.405490i	$-0.867101\pi$
$509$	− 412.788i	− 0.810979i	0.914100	−	0.405490i	$-0.132899\pi$
$510$	9.34847	0.0183303
$511$	−878.817	−1.71980
$512$	− 22.6274i	− 0.0441942i
$513$	0	0
$514$	−13.6821	−0.0266190
$515$	− 80.0243i	− 0.155387i
$516$	474.068i	0.918737i
$517$	171.939	0.332570
$518$	−282.161	−0.544713
$519$	956.853	1.84365
$520$	48.4949	0.0932594
$521$	− 888.839i	− 1.70602i	−0.521891	−	0.853012i	$-0.674773\pi$
$521$	− 888.839i	− 1.70602i	0.521891	−	0.853012i	$-0.325227\pi$
$522$	−8.14081	−0.0155954
$523$	− 153.409i	− 0.293326i	−0.989187	−	0.146663i	$-0.953147\pi$
$523$	− 153.409i	− 0.293326i	0.989187	−	0.146663i	$-0.0468532\pi$
$524$	−214.495	−0.409341
$525$	− 520.944i	− 0.992275i
$526$	161.113i	0.306298i
$527$	65.5033i	0.124295i
$528$	− 187.505i	− 0.355122i
$529$	202.454	0.382711
$530$	−113.146	−0.213484
$531$	− 52.8541i	− 0.0995368i
$532$	0	0
$533$	−1107.48	−2.07783
$534$	− 30.0484i	− 0.0562705i
$535$	− 80.0243i	− 0.149578i
$536$	−168.717	−0.314771
$537$	−991.696	−1.84673
$538$	−568.429	−1.05656
$539$	20.9194	0.0388115
$540$	50.9759i	0.0943998i
$541$	−892.848	−1.65037	−0.825183	−	0.564866i	$-0.808928\pi$
$541$	−892.848	−1.65037	−0.825183	+	0.564866i	$0.808928\pi$
$542$	− 564.249i	− 1.04105i
$543$	−101.338	−0.186627
$544$	11.8852i	0.0218477i
$545$	16.4456i	0.0301755i
$546$	526.315i	0.963948i
$547$	864.687i	1.58078i	0.612604	+	0.790390i	$0.290122\pi$
$547$	864.687i	1.58078i	−0.612604	+	0.790390i	$0.709878\pi$
$548$	4.20204	0.00766796
$549$	−1.97042	−0.00358910
$550$	− 505.688i	− 0.919433i
$551$	0	0
$552$	−240.677	−0.436008
$553$	− 46.0478i	− 0.0832691i
$554$	− 748.747i	− 1.35153i
$555$	−90.9898	−0.163946
$556$	63.4847	0.114181
$557$	749.706	1.34597	0.672986	−	0.739655i	$-0.265012\pi$
$557$	749.706	1.34597	0.672986	+	0.739655i	$0.265012\pi$
$558$	39.6367	0.0710336
$559$	− 1291.71i	− 2.31076i
$560$	−27.5959	−0.0492784
$561$	98.4877i	0.175557i
$562$	−77.4189	−0.137756
$563$	162.777i	0.289125i	0.989496	+	0.144563i	$0.0461775\pi$
$563$	162.777i	0.289125i	−0.989496	+	0.144563i	$0.953823\pi$
$564$	72.6177i	0.128755i
$565$	132.843i	0.235120i
$566$	210.625i	0.372129i
$567$	−609.060	−1.07418
$568$	−285.848	−0.503253
$569$	296.076i	0.520345i	0.965562	+	0.260173i	$0.0837795\pi$
$569$	296.076i	0.520345i	−0.965562	+	0.260173i	$0.916221\pi$
$570$	0	0
$571$	274.758	0.481188	0.240594	−	0.970626i	$-0.422658\pi$
$571$	274.758	0.481188	0.240594	+	0.970626i	$0.422658\pi$
$572$	510.902i	0.893185i
$573$	881.747i	1.53883i
$574$	630.211	1.09793
$575$	−649.090	−1.12885
$576$	7.19184	0.0124858
$577$	524.595	0.909177	0.454588	−	0.890702i	$-0.349786\pi$
$577$	524.595	0.909177	0.454588	+	0.890702i	$0.349786\pi$
$578$	402.465i	0.696306i
$579$	−152.641	−0.263629
$580$	12.8066i	0.0220803i
$581$	−8.98979	−0.0154730
$582$	− 586.491i	− 1.00772i
$583$	− 1192.02i	− 2.04463i
$584$	− 360.295i	− 0.616944i
$585$	15.4135i	0.0263478i
$586$	−206.030	−0.351586
$587$	−751.136	−1.27962	−0.639809	−	0.768534i	$-0.720987\pi$
$587$	−751.136	−1.27962	−0.639809	+	0.768534i	$0.720987\pi$
$588$	8.83523i	0.0150259i
$589$	0	0
$590$	−83.1464	−0.140926
$591$	792.573i	1.34107i
$592$	− 115.680i	− 0.195405i
$593$	−570.060	−0.961316	−0.480658	−	0.876908i	$-0.659602\pi$
$593$	−570.060	−0.961316	−0.480658	+	0.876908i	$0.659602\pi$
$594$	−537.040	−0.904107
$595$	14.4949	0.0243612
$596$	−379.353	−0.636498
$597$	816.104i	1.36701i
$598$	655.782	1.09663
$599$	394.269i	0.658211i	0.944293	+	0.329106i	$0.106747\pi$
$599$	394.269i	0.658211i	−0.944293	+	0.329106i	$0.893253\pi$
$600$	213.576	0.355959
$601$	− 241.665i	− 0.402105i	−0.979580	−	0.201053i	$-0.935564\pi$
$601$	− 241.665i	− 0.402105i	0.979580	−	0.201053i	$-0.0644362\pi$
$602$	735.048i	1.22101i
$603$	− 53.6247i	− 0.0889298i
$604$	− 32.5413i	− 0.0538764i
$605$	−100.980	−0.166908
$606$	28.8536	0.0476132
$607$	− 369.224i	− 0.608276i	−0.952628	−	0.304138i	$-0.901632\pi$
$607$	− 369.224i	− 0.608276i	0.952628	−	0.304138i	$-0.0983685\pi$
$608$	0	0
$609$	−138.990	−0.228226
$610$	3.09972i	0.00508151i
$611$	− 197.865i	− 0.323837i
$612$	−3.77755	−0.00617247
$613$	380.342	0.620460	0.310230	−	0.950662i	$-0.399594\pi$
$613$	380.342	0.620460	0.310230	+	0.950662i	$0.399594\pi$
$614$	160.095	0.260742
$615$	203.227	0.330450
$616$	− 290.728i	− 0.471961i
$617$	555.738	0.900709	0.450355	−	0.892850i	$-0.351298\pi$
$617$	555.738	0.900709	0.450355	+	0.892850i	$0.351298\pi$
$618$	356.067i	0.576161i
$619$	−624.152	−1.00832	−0.504162	−	0.863609i	$-0.668198\pi$
$619$	−624.152	−1.00832	−0.504162	+	0.863609i	$0.668198\pi$
$620$	− 62.3538i	− 0.100571i
$621$	689.332i	1.11004i
$622$	149.435i	0.240249i
$623$	− 46.5904i	− 0.0747839i
$624$	−215.778	−0.345797
$625$	551.000	0.881600
$626$	− 351.067i	− 0.560810i
$627$	0	0
$628$	−540.302	−0.860354
$629$	60.7614i	0.0966000i
$630$	− 8.77101i	− 0.0139222i
$631$	85.4495	0.135419	0.0677096	−	0.997705i	$-0.478431\pi$
$631$	85.4495	0.135419	0.0677096	+	0.997705i	$0.478431\pi$
$632$	18.8786	0.0298712
$633$	−218.889	−0.345796
$634$	320.166	0.504993
$635$	125.661i	0.197892i
$636$	503.444	0.791578
$637$	− 24.0737i	− 0.0377924i
$638$	−134.919	−0.211472
$639$	− 90.8531i	− 0.142180i
$640$	− 11.3137i	− 0.0176777i
$641$	692.697i	1.08065i	0.841456	+	0.540325i	$0.181699\pi$
$641$	692.697i	1.08065i	−0.841456	+	0.540325i	$0.818301\pi$
$642$	356.067i	0.554622i
$643$	−504.246	−0.784209	−0.392105	−	0.919921i	$-0.628253\pi$
$643$	−504.246	−0.784209	−0.392105	+	0.919921i	$0.628253\pi$
$644$	−373.171	−0.579459
$645$	237.034i	0.367495i
$646$	0	0
$647$	−797.868	−1.23318	−0.616591	−	0.787284i	$-0.711487\pi$
$647$	−797.868	−1.23318	−0.616591	+	0.787284i	$0.711487\pi$
$648$	− 249.701i	− 0.385341i
$649$	− 875.962i	− 1.34971i
$650$	−581.939	−0.895290
$651$	676.727	1.03952
$652$	313.394	0.480665
$653$	−651.383	−0.997523	−0.498762	−	0.866739i	$-0.666212\pi$
$653$	−651.383	−0.997523	−0.498762	+	0.866739i	$0.666212\pi$
$654$	− 73.1747i	− 0.111888i
$655$	−107.247	−0.163737
$656$	258.372i	0.393861i
$657$	114.515	0.174300
$658$	112.594i	0.171116i
$659$	456.202i	0.692264i	0.938186	+	0.346132i	$0.112505\pi$
$659$	456.202i	0.692264i	−0.938186	+	0.346132i	$0.887495\pi$
$660$	− 93.7523i	− 0.142049i
$661$	− 572.381i	− 0.865932i	−0.901410	−	0.432966i	$-0.857467\pi$
$661$	− 572.381i	− 0.865932i	0.901410	−	0.432966i	$-0.142533\pi$
$662$	−414.829	−0.626629
$663$	113.338	0.170948
$664$	− 3.68561i	− 0.00555062i
$665$	0	0
$666$	36.7673	0.0552062
$667$	173.179i	0.259639i
$668$	− 180.326i	− 0.269949i
$669$	19.8990	0.0297444
$670$	−84.3587	−0.125908
$671$	−32.6561	−0.0486678
$672$	122.788	0.182720
$673$	− 914.574i	− 1.35895i	−0.733698	−	0.679476i	$-0.762207\pi$
$673$	− 914.574i	− 1.35895i	0.733698	−	0.679476i	$-0.237793\pi$
$674$	−652.207	−0.967666
$675$	− 611.711i	− 0.906238i
$676$	249.939	0.369732
$677$	− 272.492i	− 0.402500i	−0.979540	−	0.201250i	$-0.935500\pi$
$677$	− 272.492i	− 0.402500i	0.979540	−	0.201250i	$-0.0645003\pi$
$678$	− 591.084i	− 0.871805i
$679$	− 909.360i	− 1.33926i
$680$	5.94258i	0.00873909i
$681$	−175.373	−0.257523
$682$	656.908	0.963208
$683$	164.842i	0.241350i	0.992692	+	0.120675i	$0.0385058\pi$
$683$	164.842i	0.241350i	−0.992692	+	0.120675i	$0.961494\pi$
$684$	0	0
$685$	2.10102	0.00306718
$686$	491.774i	0.716872i
$687$	− 416.293i	− 0.605957i
$688$	−301.353	−0.438013
$689$	−1371.76	−1.99094
$690$	−120.338	−0.174403
$691$	1040.21	1.50537	0.752685	−	0.658380i	$-0.228758\pi$
$691$	1040.21	1.50537	0.752685	+	0.658380i	$0.228758\pi$
$692$	608.247i	0.878969i
$693$	92.4041	0.133339
$694$	− 430.285i	− 0.620008i
$695$	31.7423	0.0456724
$696$	− 56.9827i	− 0.0818717i
$697$	− 135.711i	− 0.194708i
$698$	− 71.3240i	− 0.102183i
$699$	− 703.874i	− 1.00697i
$700$	331.151	0.473073
$701$	−75.6357	−0.107897	−0.0539484	−	0.998544i	$-0.517181\pi$
$701$	−75.6357	−0.107897	−0.0539484	+	0.998544i	$0.517181\pi$
$702$	618.018i	0.880367i
$703$	0	0
$704$	119.192	0.169307
$705$	36.3089i	0.0515019i
$706$	448.047i	0.634628i
$707$	44.7378	0.0632783
$708$	369.959	0.522541
$709$	−902.757	−1.27328	−0.636641	−	0.771160i	$-0.719677\pi$
$709$	−902.757	−1.27328	−0.636641	+	0.771160i	$0.719677\pi$
$710$	−142.924	−0.201301
$711$	6.00031i	0.00843926i
$712$	19.1010	0.0268273
$713$	− 843.192i	− 1.18260i
$714$	−64.4949	−0.0903290
$715$	255.451i	0.357274i
$716$	− 630.396i	− 0.880441i
$717$	− 224.402i	− 0.312974i
$718$	− 211.296i	− 0.294284i
$719$	1173.47	1.63208	0.816042	−	0.577993i	$-0.196164\pi$
$719$	1173.47	1.63208	0.816042	+	0.577993i	$0.196164\pi$
$720$	3.59592	0.00499433
$721$	552.086i	0.765722i
$722$	0	0
$723$	438.601	0.606640
$724$	− 64.4181i	− 0.0889753i
$725$	− 153.679i	− 0.211971i
$726$	449.308	0.618881
$727$	577.105	0.793816	0.396908	−	0.917858i	$-0.370083\pi$
$727$	577.105	0.793816	0.396908	+	0.917858i	$0.370083\pi$
$728$	−334.565	−0.459568
$729$	−642.362	−0.881155
$730$	− 180.148i	− 0.246778i
$731$	158.287	0.216535
$732$	− 13.7922i	− 0.0188418i
$733$	−812.332	−1.10823	−0.554114	−	0.832441i	$-0.686943\pi$
$733$	−812.332	−1.10823	−0.554114	+	0.832441i	$0.686943\pi$
$734$	− 331.033i	− 0.450999i
$735$	4.41761i	0.00601036i
$736$	− 152.992i	− 0.207869i
$737$	− 888.733i	− 1.20588i
$738$	−82.1204	−0.111274
$739$	−761.863	−1.03094	−0.515469	−	0.856908i	$-0.672382\pi$
$739$	−761.863	−1.03094	−0.515469	+	0.856908i	$0.672382\pi$
$740$	− 57.8399i	− 0.0781620i
$741$	0	0
$742$	780.595	1.05201
$743$	− 137.190i	− 0.184643i	−0.995729	−	0.0923216i	$-0.970571\pi$
$743$	− 137.190i	− 0.184643i	0.995729	−	0.0923216i	$-0.0294288\pi$
$744$	277.443i	0.372907i
$745$	−189.677	−0.254599
$746$	904.727	1.21277
$747$	1.17143	0.00156817
$748$	−62.6061	−0.0836980
$749$	552.086i	0.737097i
$750$	218.025	0.290700
$751$	− 109.776i	− 0.146173i	−0.997326	−	0.0730864i	$-0.976715\pi$
$751$	− 109.776i	− 0.146173i	0.997326	−	0.0730864i	$-0.0232849\pi$
$752$	−46.1612	−0.0613846
$753$	755.343i	1.00311i
$754$	155.263i	0.205920i
$755$	− 16.2707i	− 0.0215506i
$756$	− 351.682i	− 0.465187i
$757$	266.514	0.352066	0.176033	−	0.984384i	$-0.443673\pi$
$757$	266.514	0.352066	0.176033	+	0.984384i	$0.443673\pi$
$758$	−153.253	−0.202181
$759$	− 1267.78i	− 1.67033i
$760$	0	0
$761$	1227.85	1.61346	0.806732	−	0.590918i	$-0.201234\pi$
$761$	1227.85	1.61346	0.806732	+	0.590918i	$0.201234\pi$
$762$	− 559.128i	− 0.733764i
$763$	− 113.458i	− 0.148700i
$764$	−560.504	−0.733644
$765$	−1.88877	−0.00246899
$766$	351.995	0.459524
$767$	−1008.04	−1.31427
$768$	50.3402i	0.0655472i
$769$	668.231	0.868960	0.434480	−	0.900681i	$-0.356932\pi$
$769$	668.231	0.868960	0.434480	+	0.900681i	$0.356932\pi$
$770$	− 145.364i	− 0.188784i
$771$	30.4393	0.0394803
$772$	− 97.0302i	− 0.125687i
$773$	568.795i	0.735828i	0.929860	+	0.367914i	$0.119928\pi$
$773$	568.795i	0.735828i	−0.929860	+	0.367914i	$0.880072\pi$
$774$	− 95.7812i	− 0.123748i
$775$	748.246i	0.965479i
$776$	372.817	0.480435
$777$	627.737	0.807898
$778$	949.108i	1.21993i
$779$	0	0
$780$	−107.889	−0.138319
$781$	− 1505.73i	− 1.92795i
$782$	80.3598i	0.102762i
$783$	−163.207	−0.208438
$784$	−5.61633	−0.00716368
$785$	−270.151	−0.344141
$786$	477.196	0.607120
$787$	460.132i	0.584665i	0.956317	+	0.292333i	$0.0944315\pi$
$787$	460.132i	0.584665i	−0.956317	+	0.292333i	$0.905569\pi$
$788$	−503.818	−0.639363
$789$	− 358.435i	− 0.454290i
$790$	9.43928	0.0119485
$791$	− 916.481i	− 1.15864i
$792$	37.8836i	0.0478328i
$793$	37.5802i	0.0473899i
$794$	132.036i	0.166292i
$795$	251.722	0.316631
$796$	−518.777	−0.651729
$797$	1211.02i	1.51947i	0.650234	+	0.759734i	$0.274671\pi$
$797$	1211.02i	1.51947i	−0.650234	+	0.759734i	$0.725329\pi$
$798$	0	0
$799$	24.2464	0.0303460
$800$	135.765i	0.169706i
$801$	6.07102i	0.00757930i
$802$	395.975	0.493734
$803$	1897.89	2.36350
$804$	375.353	0.466857
$805$	−186.586	−0.231783
$806$	− 755.961i	− 0.937916i
$807$	1264.61	1.56705
$808$	18.3415i	0.0226999i
$809$	914.089	1.12990	0.564950	−	0.825125i	$-0.308896\pi$
$809$	914.089	1.12990	0.564950	+	0.825125i	$0.308896\pi$
$810$	− 124.851i	− 0.154136i
$811$	465.615i	0.574125i	0.957912	+	0.287062i	$0.0926787\pi$
$811$	465.615i	0.574125i	−0.957912	+	0.287062i	$0.907321\pi$
$812$	− 88.3523i	− 0.108808i
$813$	1255.31i	1.54405i
$814$	609.353	0.748591
$815$	156.697	0.192266
$816$	− 26.4415i	− 0.0324038i
$817$	0	0
$818$	907.671	1.10962
$819$	− 106.337i	− 0.129838i
$820$	129.186i	0.157544i
$821$	−97.1112	−0.118284	−0.0591420	−	0.998250i	$-0.518836\pi$
$821$	−97.1112	−0.118284	−0.0591420	+	0.998250i	$0.518836\pi$
$822$	−9.34847	−0.0113728
$823$	−1124.10	−1.36586	−0.682930	−	0.730483i	$-0.739295\pi$
$823$	−1124.10	−1.36586	−0.682930	+	0.730483i	$0.739295\pi$
$824$	−226.343	−0.274688
$825$	1125.03i	1.36367i
$826$	573.626	0.694462
$827$	65.6704i	0.0794079i	0.999211	+	0.0397040i	$0.0126415\pi$
$827$	65.6704i	0.0794079i	−0.999211	+	0.0397040i	$0.987359\pi$
$828$	48.6265	0.0587277
$829$	− 276.987i	− 0.334121i	−0.985947	−	0.167061i	$-0.946572\pi$
$829$	− 276.987i	− 0.334121i	0.985947	−	0.167061i	$-0.0534276\pi$
$830$	− 1.84281i	− 0.00222025i
$831$	1665.77i	2.00454i
$832$	− 137.164i	− 0.164861i
$833$	2.95001	0.00354142
$834$	−141.237	−0.169349
$835$	− 90.1630i	− 0.107980i
$836$	0	0
$837$	794.636	0.949386
$838$	− 945.937i	− 1.12880i
$839$	− 184.288i	− 0.219651i	−0.993951	−	0.109826i	$-0.964971\pi$
$839$	− 184.288i	− 0.219651i	0.993951	−	0.109826i	$-0.0350293\pi$
$840$	61.3939	0.0730879
$841$	799.998	0.951246
$842$	708.247	0.841149
$843$	172.237	0.204315
$844$	− 139.142i	− 0.164860i
$845$	124.969	0.147893
$846$	− 14.6718i	− 0.0173425i
$847$	696.656	0.822498
$848$	320.026i	0.377390i
$849$	− 468.586i	− 0.551927i
$850$	− 71.3110i	− 0.0838953i
$851$	− 782.152i	− 0.919097i
$852$	635.939	0.746407
$853$	−1281.24	−1.50204	−0.751020	−	0.660279i	$-0.770438\pi$
$853$	−1281.24	−1.50204	−0.751020	+	0.660279i	$0.770438\pi$
$854$	− 21.3849i	− 0.0250409i
$855$	0	0
$856$	−226.343	−0.264419
$857$	104.185i	0.121569i	0.998151	+	0.0607845i	$0.0193602\pi$
$857$	104.185i	0.121569i	−0.998151	+	0.0607845i	$0.980640\pi$
$858$	− 1136.63i	− 1.32474i
$859$	−748.367	−0.871207	−0.435604	−	0.900139i	$-0.643465\pi$
$859$	−748.367	−0.871207	−0.435604	+	0.900139i	$0.643465\pi$
$860$	−150.677	−0.175205
$861$	−1402.06	−1.62841
$862$	−196.863	−0.228379
$863$	19.3495i	0.0224212i	0.999937	+	0.0112106i	$0.00356852\pi$
$863$	19.3495i	0.0224212i	−0.999937	+	0.0112106i	$0.996431\pi$
$864$	144.182	0.166877
$865$	304.123i	0.351588i
$866$	−294.509	−0.340079
$867$	− 895.382i	− 1.03274i
$868$	430.178i	0.495597i
$869$	99.4445i	0.114436i
$870$	− 28.4914i	− 0.0327487i
$871$	−1022.74	−1.17422
$872$	46.5153	0.0533432
$873$	118.495i	0.135733i
$874$	0	0
$875$	338.050	0.386343
$876$	801.565i	0.915029i
$877$	− 349.943i	− 0.399023i	−0.979895	−	0.199512i	$-0.936064\pi$
$877$	− 349.943i	− 0.399023i	0.979895	−	0.199512i	$-0.0639355\pi$
$878$	107.246	0.122149
$879$	458.363	0.521460
$880$	59.5959	0.0677226
$881$	−449.707	−0.510451	−0.255225	−	0.966882i	$-0.582150\pi$
$881$	−449.707	−0.510451	−0.255225	+	0.966882i	$0.582150\pi$
$882$	− 1.78508i	− 0.00202390i
$883$	1115.06	1.26281	0.631406	−	0.775452i	$-0.282478\pi$
$883$	1115.06	1.26281	0.631406	+	0.775452i	$0.282478\pi$
$884$	72.0462i	0.0815003i
$885$	184.980	0.209016
$886$	643.830i	0.726671i
$887$	− 945.324i	− 1.06575i	−0.846193	−	0.532877i	$-0.821111\pi$
$887$	− 945.324i	− 1.06575i	0.846193	−	0.532877i	$-0.178889\pi$
$888$	257.358i	0.289818i
$889$	− 866.934i	− 0.975179i
$890$	9.55051	0.0107309
$891$	1315.32	1.47623
$892$	12.6493i	0.0141808i
$893$	0	0
$894$	843.964	0.944031
$895$	− 315.198i	− 0.352176i
$896$	78.0530i	0.0871128i
$897$	−1458.95	−1.62647
$898$	−11.9500	−0.0133074
$899$	199.635	0.222063
$900$	−43.1510	−0.0479456
$901$	− 168.096i	− 0.186566i
$902$	−1361.00	−1.50887
$903$	− 1635.29i	− 1.81096i
$904$	375.737	0.415638
$905$	− 32.2091i	− 0.0355901i
$906$	72.3962i	0.0799075i
$907$	713.473i	0.786630i	0.919404	+	0.393315i	$0.128672\pi$
$907$	713.473i	0.786630i	−0.919404	+	0.393315i	$0.871328\pi$
$908$	− 111.480i	− 0.122776i
$909$	−5.82961	−0.00641321
$910$	−167.283	−0.183827
$911$	1490.37i	1.63597i	0.575242	+	0.817984i	$0.304908\pi$
$911$	1490.37i	1.63597i	−0.575242	+	0.817984i	$0.695092\pi$
$912$	0	0
$913$	19.4143	0.0212643
$914$	− 294.757i	− 0.322491i
$915$	− 6.89610i	− 0.00753672i
$916$	264.627	0.288894
$917$	739.898	0.806868
$918$	−75.7321	−0.0824969
$919$	−543.666	−0.591585	−0.295792	−	0.955252i	$-0.595584\pi$
$919$	−543.666	−0.591585	−0.295792	+	0.955252i	$0.595584\pi$
$920$	− 76.4960i	− 0.0831478i
$921$	−356.171	−0.386723
$922$	873.440i	0.947332i
$923$	−1732.77	−1.87732
$924$	646.795i	0.699994i
$925$	694.079i	0.750355i
$926$	− 906.452i	− 0.978890i
$927$	− 71.9402i	− 0.0776054i
$928$	36.2225	0.0390328
$929$	112.516	0.121116	0.0605578	−	0.998165i	$-0.480712\pi$
$929$	112.516	0.121116	0.0605578	+	0.998165i	$0.480712\pi$
$930$	138.721i	0.149163i
$931$	0	0
$932$	447.435	0.480080
$933$	− 332.454i	− 0.356328i
$934$	− 232.088i	− 0.248489i
$935$	−31.3031	−0.0334792
$936$	43.5959	0.0465768
$937$	−251.788	−0.268717	−0.134358	−	0.990933i	$-0.542897\pi$
$937$	−251.788	−0.268717	−0.134358	+	0.990933i	$0.542897\pi$
$938$	581.989	0.620457
$939$	781.034i	0.831773i
$940$	−23.0806	−0.0245538
$941$	21.5375i	0.0228879i	0.999935	+	0.0114440i	$0.00364280\pi$
$941$	21.5375i	0.0228879i	−0.999935	+	0.0114440i	$0.996357\pi$
$942$	1202.03	1.27604
$943$	1746.95i	1.85254i
$944$	235.174i	0.249125i
$945$	− 175.841i	− 0.186075i
$946$	− 1587.40i	− 1.67802i
$947$	629.470	0.664699	0.332349	−	0.943156i	$-0.392159\pi$
$947$	629.470	0.664699	0.332349	+	0.943156i	$0.392159\pi$
$948$	−42.0000	−0.0443038
$949$	− 2184.06i	− 2.30143i
$950$	0	0
$951$	−712.287	−0.748988
$952$	− 40.9978i	− 0.0430649i
$953$	− 485.971i	− 0.509938i	−0.966949	−	0.254969i	$-0.917935\pi$
$953$	− 485.971i	− 0.509938i	0.966949	−	0.254969i	$-0.0820653\pi$
$954$	−101.716	−0.106621
$955$	−280.252	−0.293458
$956$	142.647	0.149212
$957$	300.161	0.313648
$958$	− 1135.33i	− 1.18511i
$959$	−14.4949	−0.0151146
$960$	25.1701i	0.0262189i
$961$	−11.0000	−0.0114464
$962$	− 701.235i	− 0.728934i
$963$	− 71.9402i	− 0.0747042i
$964$	278.807i	0.289219i
$965$	− 48.5151i	− 0.0502747i
$966$	830.211	0.859432
$967$	432.743	0.447511	0.223756	−	0.974645i	$-0.428168\pi$
$967$	432.743	0.447511	0.223756	+	0.974645i	$0.428168\pi$
$968$	285.613i	0.295055i
$969$	0	0
$970$	186.409	0.192174
$971$	− 1222.15i	− 1.25865i	−0.777140	−	0.629327i	$-0.783331\pi$
$971$	− 1222.15i	− 1.25865i	0.777140	−	0.629327i	$-0.216669\pi$
$972$	96.7380i	0.0995247i
$973$	−218.990	−0.225067
$974$	642.606	0.659760
$975$	1294.67	1.32786
$976$	8.76734	0.00898293
$977$	867.932i	0.888365i	0.895936	+	0.444182i	$0.146506\pi$
$977$	867.932i	0.888365i	−0.895936	+	0.444182i	$0.853494\pi$
$978$	−697.221	−0.712905
$979$	100.616i	0.102775i
$980$	−2.80816	−0.00286547
$981$	14.7843i	0.0150706i
$982$	− 734.898i	− 0.748369i
$983$	1074.05i	1.09263i	0.837580	+	0.546315i	$0.183970\pi$
$983$	1074.05i	1.09263i	−0.837580	+	0.546315i	$0.816030\pi$
$984$	− 574.813i	− 0.584159i
$985$	−251.909	−0.255745
$986$	−19.0260	−0.0192962
$987$	− 250.494i	− 0.253793i
$988$	0	0
$989$	−2037.55	−2.06022
$990$	18.9418i	0.0191331i
$991$	− 1763.88i	− 1.77990i	−0.456057	−	0.889951i	$-0.650739\pi$
$991$	− 1763.88i	− 1.77990i	0.456057	−	0.889951i	$-0.349261\pi$
$992$	−176.363	−0.177786
$993$	922.888	0.929393
$994$	986.030	0.991981
$995$	−259.388	−0.260692
$996$	8.19955i	0.00823248i
$997$	−23.2429	−0.0233128	−0.0116564	−	0.999932i	$-0.503710\pi$
$997$	−23.2429	−0.0233128	−0.0116564	+	0.999932i	$0.503710\pi$
$998$	− 626.619i	− 0.627874i
$999$	737.110	0.737848

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	722.3.b.b.721.2		4
19.11	even	3		38.3.d.a.31.1	yes	4
19.12	odd	6		38.3.d.a.27.1	✓	4
19.18	odd	2	inner	722.3.b.b.721.3		4
57.11	odd	6		342.3.m.a.145.2		4
57.50	even	6		342.3.m.a.217.2		4
76.11	odd	6		304.3.r.a.145.1		4
76.31	even	6		304.3.r.a.65.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.3.d.a.27.1	✓	4	19.12	odd	6
38.3.d.a.31.1	yes	4	19.11	even	3
304.3.r.a.65.1		4	76.31	even	6
304.3.r.a.145.1		4	76.11	odd	6
342.3.m.a.145.2		4	57.11	odd	6
342.3.m.a.217.2		4	57.50	even	6
722.3.b.b.721.2		4	1.1	even	1	trivial
722.3.b.b.721.3		4	19.18	odd	2	inner

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:	\( N \)	\(=\)	\( 722 = 2 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	722.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} +3.14626i q^{3} -2.00000 q^{4} -1.00000 q^{5} +4.44949 q^{6} +6.89898 q^{7} +2.82843i q^{8} -0.898979 q^{9} +O(q^{10})\)	\(q-1.41421i q^{2} +3.14626i q^{3} -2.00000 q^{4} -1.00000 q^{5} +4.44949 q^{6} +6.89898 q^{7} +2.82843i q^{8} -0.898979 q^{9} +1.41421i q^{10} -14.8990 q^{11} -6.29253i q^{12} +17.1455i q^{13} -9.75663i q^{14} -3.14626i q^{15} +4.00000 q^{16} -2.10102 q^{17} +1.27135i q^{18} +2.00000 q^{20} +21.7060i q^{21} +21.0703i q^{22} +27.0454 q^{23} -8.89898 q^{24} -24.0000 q^{25} +24.2474 q^{26} +25.4880i q^{27} -13.7980 q^{28} +6.40329i q^{29} -4.44949 q^{30} -31.1769i q^{31} -5.65685i q^{32} -46.8761i q^{33} +2.97129i q^{34} -6.89898 q^{35} +1.79796 q^{36} -28.9199i q^{37} -53.9444 q^{39} -2.82843i q^{40} +64.5931i q^{41} +30.6969 q^{42} -75.3383 q^{43} +29.7980 q^{44} +0.898979 q^{45} -38.2480i q^{46} -11.5403 q^{47} +12.5851i q^{48} -1.40408 q^{49} +33.9411i q^{50} -6.61037i q^{51} -34.2911i q^{52} +80.0066i q^{53} +36.0454 q^{54} +14.8990 q^{55} +19.5133i q^{56} +9.05561 q^{58} +58.7934i q^{59} +6.29253i q^{60} +2.19184 q^{61} -44.0908 q^{62} -6.20204 q^{63} -8.00000 q^{64} -17.1455i q^{65} -66.2929 q^{66} +59.6506i q^{67} +4.20204 q^{68} +85.0920i q^{69} +9.75663i q^{70} +101.063i q^{71} -2.54270i q^{72} -127.384 q^{73} -40.8990 q^{74} -75.5103i q^{75} -102.788 q^{77} +76.2889i q^{78} -6.67458i q^{79} -4.00000 q^{80} -88.2827 q^{81} +91.3485 q^{82} -1.30306 q^{83} -43.4120i q^{84} +2.10102 q^{85} +106.544i q^{86} -20.1464 q^{87} -42.1407i q^{88} -6.75323i q^{89} -1.27135i q^{90} +118.287i q^{91} -54.0908 q^{92} +98.0908 q^{93} +16.3205i q^{94} +17.7980 q^{96} -131.811i q^{97} +1.98567i q^{98} +13.3939 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{4} - 4 q^{5} + 8 q^{6} + 8 q^{7} + 16 q^{9}+O(q^{10}) \) 4 * q - 8 * q^4 - 4 * q^5 + 8 * q^6 + 8 * q^7 + 16 * q^9	\( 4 q - 8 q^{4} - 4 q^{5} + 8 q^{6} + 8 q^{7} + 16 q^{9} - 40 q^{11} + 16 q^{16} - 28 q^{17} + 8 q^{20} + 20 q^{23} - 16 q^{24} - 96 q^{25} + 48 q^{26} - 16 q^{28} - 8 q^{30} - 8 q^{35} - 32 q^{36} - 108 q^{39} + 64 q^{42} - 76 q^{43} + 80 q^{44} - 16 q^{45} + 140 q^{47} - 84 q^{49} + 56 q^{54} + 40 q^{55} + 144 q^{58} - 148 q^{61} - 64 q^{63} - 32 q^{64} - 128 q^{66} + 56 q^{68} - 196 q^{73} - 144 q^{74} - 176 q^{77} - 16 q^{80} - 20 q^{81} + 336 q^{82} - 64 q^{83} + 28 q^{85} - 12 q^{87} - 40 q^{92} + 216 q^{93} + 32 q^{96} - 64 q^{99}+O(q^{100}) \) 4 * q - 8 * q^4 - 4 * q^5 + 8 * q^6 + 8 * q^7 + 16 * q^9 - 40 * q^11 + 16 * q^16 - 28 * q^17 + 8 * q^20 + 20 * q^23 - 16 * q^24 - 96 * q^25 + 48 * q^26 - 16 * q^28 - 8 * q^30 - 8 * q^35 - 32 * q^36 - 108 * q^39 + 64 * q^42 - 76 * q^43 + 80 * q^44 - 16 * q^45 + 140 * q^47 - 84 * q^49 + 56 * q^54 + 40 * q^55 + 144 * q^58 - 148 * q^61 - 64 * q^63 - 32 * q^64 - 128 * q^66 + 56 * q^68 - 196 * q^73 - 144 * q^74 - 176 * q^77 - 16 * q^80 - 20 * q^81 + 336 * q^82 - 64 * q^83 + 28 * q^85 - 12 * q^87 - 40 * q^92 + 216 * q^93 + 32 * q^96 - 64 * q^99

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