LMFDB - Embedded newform 136.4.s.a.3.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [136,4,Mod(3,136)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(136, base_ring=CyclotomicField(16))

chi = DirichletCharacter(H, H._module([8, 8, 1]))

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

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

chi := DirichletCharacter("136.3");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 136 = 2^{3} \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	136.s (of order $16$, degree $8$, 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:	$8.02425976078$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{16})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 1 $ x^8 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{16}]$

Embedding invariants

Embedding label			3.1
Root			$0.923880 + 0.382683i$ of defining polynomial
Character	$\chi$	$=$	136.3
Dual form			136.4.s.a.91.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.61313 - 1.08239i) q^{2} +(6.61374 + 1.31555i) q^{3} +(5.65685 - 5.65685i) q^{4} +(18.7065 - 3.72095i) q^{6} +(8.65914 - 20.9050i) q^{8} +(17.0661 + 7.06900i) q^{9} +O(q^{10})$	\(q+(2.61313 - 1.08239i) q^{2} +(6.61374 + 1.31555i) q^{3} +(5.65685 - 5.65685i) q^{4} +(18.7065 - 3.72095i) q^{6} +(8.65914 - 20.9050i) q^{8} +(17.0661 + 7.06900i) q^{9} +(0.752886 + 3.78501i) q^{11} +(44.8548 - 29.9710i) q^{12} -64.0000i q^{16} +(-21.0091 + 66.8701i) q^{17} +52.2472 q^{18} +(-13.9013 + 5.75812i) q^{19} +(6.06425 + 9.07580i) q^{22} +(84.7709 - 126.869i) q^{24} +(-47.8354 + 115.485i) q^{25} +(-47.8143 - 31.9485i) q^{27} +(-69.2731 - 167.240i) q^{32} +26.0235i q^{33} +(17.4802 + 197.480i) q^{34} +(136.529 - 56.5520i) q^{36} +(-30.0934 + 30.0934i) q^{38} +(87.4016 - 130.806i) q^{41} +(-76.5305 - 31.7000i) q^{43} +(25.6702 + 17.1523i) q^{44} +(84.1954 - 423.279i) q^{48} +(-131.260 - 316.891i) q^{49} +353.553i q^{50} +(-226.920 + 414.623i) q^{51} +(-159.525 - 31.7316i) q^{54} +(-99.5148 + 19.7947i) q^{57} +(-251.472 + 607.106i) q^{59} +(-362.039 - 362.039i) q^{64} +(28.1677 + 68.0028i) q^{66} +984.492i q^{67} +(259.429 + 497.120i) q^{68} +(295.555 - 295.555i) q^{72} +(-107.388 - 160.717i) q^{73} +(-468.298 + 700.857i) q^{75} +(-46.0649 + 111.211i) q^{76} +(-626.870 - 626.870i) q^{81} +(86.8083 - 436.415i) q^{82} +(43.2971 + 104.528i) q^{83} -234.296 q^{86} +(85.6451 + 17.0359i) q^{88} +(1029.99 - 1029.99i) q^{89} +(-238.141 - 1197.21i) q^{96} +(917.502 - 613.055i) q^{97} +(-686.000 - 686.000i) q^{98} +(-13.9074 + 69.9174i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{3} + 80 q^{6} - 80 q^{9}+O(q^{10}) $ 8 * q - 8 * q^3 + 80 * q^6 - 80 * q^9	$ 8 q - 8 q^{3} + 80 q^{6} - 80 q^{9} + 320 q^{12} + 144 q^{22} - 256 q^{24} + 496 q^{27} - 720 q^{34} - 640 q^{36} - 1440 q^{38} + 160 q^{41} + 208 q^{43} - 1600 q^{44} + 512 q^{48} + 1880 q^{51} - 176 q^{54} + 2840 q^{57} - 920 q^{59} + 5232 q^{66} + 3456 q^{72} - 3312 q^{73} - 2632 q^{81} - 3472 q^{83} - 5120 q^{96} - 5488 q^{98} - 1536 q^{99}+O(q^{100}) $ 8 * q - 8 * q^3 + 80 * q^6 - 80 * q^9 + 320 * q^12 + 144 * q^22 - 256 * q^24 + 496 * q^27 - 720 * q^34 - 640 * q^36 - 1440 * q^38 + 160 * q^41 + 208 * q^43 - 1600 * q^44 + 512 * q^48 + 1880 * q^51 - 176 * q^54 + 2840 * q^57 - 920 * q^59 + 5232 * q^66 + 3456 * q^72 - 3312 * q^73 - 2632 * q^81 - 3472 * q^83 - 5120 * q^96 - 5488 * q^98 - 1536 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/136\mathbb{Z}\right)^\times$.

$n$	$69$	$103$	$105$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{1}{16}\right)$

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 $

$2$	2.61313	−	1.08239i	0.923880	−	0.382683i
$2$	2.61313	−	1.08239i	0.923880	−	0.382683i
$3$	6.61374	+	1.31555i	1.27281	+	0.253178i	0.784850	−	0.619685i	$-0.212740\pi$
$3$	6.61374	+	1.31555i	1.27281	+	0.253178i	0.487964	+	0.872864i	$0.337740\pi$
$4$	5.65685	−	5.65685i	0.707107	−	0.707107i
$5$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$5$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$6$	18.7065	−	3.72095i	1.27281	−	0.253178i
$7$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$7$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$8$	8.65914	−	20.9050i	0.382683	−	0.923880i
$9$	17.0661	+	7.06900i	0.632077	+	0.261815i
$10$		0			0
$11$	0.752886	+	3.78501i	0.0206367	+	0.103748i	0.989731	−	0.142943i	$-0.0456567\pi$
$11$	0.752886	+	3.78501i	0.0206367	+	0.103748i	−0.969094	+	0.246691i	$0.920657\pi$
$12$	44.8548	−	29.9710i	1.07904	−	0.720991i
$13$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$13$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$14$		0			0
$15$		0			0
$16$		−	64.0000i		−	1.00000i
$17$	−21.0091	+	66.8701i	−0.299733	+	0.954023i
$17$	−21.0091	+	66.8701i	−0.299733	+	0.954023i
$18$	52.2472			0.684155
$19$	−13.9013	+	5.75812i	−0.167852	+	0.0695264i	−0.465027	−	0.885296i	$-0.653955\pi$
$19$	−13.9013	+	5.75812i	−0.167852	+	0.0695264i	0.297175	+	0.954823i	$0.403955\pi$
$20$		0			0
$21$		0			0
$22$	6.06425	+	9.07580i	0.0587683	+	0.0879530i
$23$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$23$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$24$	84.7709	−	126.869i	0.720991	−	1.07904i
$25$	−47.8354	+	115.485i	−0.382683	+	0.923880i
$26$		0			0
$27$	−47.8143	−	31.9485i	−0.340810	−	0.227722i
$28$		0			0
$29$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$29$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$30$		0			0
$31$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$31$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$32$	−69.2731	−	167.240i	−0.382683	−	0.923880i
$33$			26.0235i			0.137276i
$34$	17.4802	+	197.480i	0.0881716	+	0.996105i
$35$		0			0
$36$	136.529	−	56.5520i	0.632077	−	0.261815i
$37$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$37$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$38$	−30.0934	+	30.0934i	−0.128468	+	0.128468i
$39$		0			0
$40$		0			0
$41$	87.4016	−	130.806i	0.332923	−	0.498254i	−0.626809	−	0.779173i	$-0.715639\pi$
$41$	87.4016	−	130.806i	0.332923	−	0.498254i	0.959731	+	0.280919i	$0.0906392\pi$
$42$		0			0
$43$	−76.5305	−	31.7000i	−0.271414	−	0.112423i	0.242826	−	0.970070i	$-0.421926\pi$
$43$	−76.5305	−	31.7000i	−0.271414	−	0.112423i	−0.514239	+	0.857647i	$0.671926\pi$
$44$	25.6702	+	17.1523i	0.0879530	+	0.0587683i
$45$		0			0
$46$		0			0
$47$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$47$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$48$	84.1954	−	423.279i	0.253178	−	1.27281i
$49$	−131.260	−	316.891i	−0.382683	−	0.923880i
$50$			353.553i			1.00000i
$51$	−226.920	+	414.623i	−0.623042	+	1.13841i
$52$		0			0
$53$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$53$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$54$	−159.525	−	31.7316i	−0.402012	−	0.0799652i
$55$		0			0
$56$		0			0
$57$	−99.5148	+	19.7947i	−0.231247	+	0.0459978i
$58$		0			0
$59$	−251.472	+	607.106i	−0.554895	+	1.33964i	0.358868	+	0.933388i	$0.383163\pi$
$59$	−251.472	+	607.106i	−0.554895	+	1.33964i	−0.913763	+	0.406247i	$0.866837\pi$
$60$		0			0
$61$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$61$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$62$		0			0
$63$		0			0
$64$	−362.039	−	362.039i	−0.707107	−	0.707107i
$65$		0			0
$66$	28.1677	+	68.0028i	0.0525334	+	0.126827i
$67$			984.492i			1.79515i	0.440865	+	0.897573i	$0.354672\pi$
$67$			984.492i			1.79515i	−0.440865	+	0.897573i	$0.645328\pi$
$68$	259.429	+	497.120i	0.462653	+	0.886539i
$69$		0			0
$70$		0			0
$71$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$71$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$72$	295.555	−	295.555i	0.483770	−	0.483770i
$73$	−107.388	−	160.717i	−0.172175	−	0.257678i	0.735339	−	0.677699i	$-0.237023\pi$
$73$	−107.388	−	160.717i	−0.172175	−	0.257678i	−0.907515	+	0.420021i	$0.862023\pi$
$74$		0			0
$75$	−468.298	+	700.857i	−0.720991	+	1.07904i
$76$	−46.0649	+	111.211i	−0.0695264	+	0.167852i
$77$		0			0
$78$		0			0
$79$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$79$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$80$		0			0
$81$	−626.870	−	626.870i	−0.859903	−	0.859903i
$82$	86.8083	−	436.415i	0.116907	−	0.587731i
$83$	43.2971	+	104.528i	0.0572587	+	0.138235i	0.949920	−	0.312494i	$-0.101164\pi$
$83$	43.2971	+	104.528i	0.0572587	+	0.138235i	−0.892661	+	0.450728i	$0.851164\pi$
$84$		0			0
$85$		0			0
$86$	−234.296			−0.293776
$87$		0			0
$88$	85.6451	+	17.0359i	0.103748	+	0.0206367i
$89$	1029.99	−	1029.99i	1.22673	−	1.22673i	0.261531	−	0.965195i	$-0.415773\pi$
$89$	1029.99	−	1029.99i	1.22673	−	1.22673i	0.965195	−	0.261531i	$-0.0842275\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$	−238.141	−	1197.21i	−0.253178	−	1.27281i
$97$	917.502	−	613.055i	0.960394	−	0.641715i	0.0266459	−	0.999645i	$-0.491517\pi$
$97$	917.502	−	613.055i	0.960394	−	0.641715i	0.933748	+	0.357930i	$0.116517\pi$
$98$	−686.000	−	686.000i	−0.707107	−	0.707107i
$99$	−13.9074	+	69.9174i	−0.0141187	+	0.0709795i
$100$	382.683	+	923.880i	0.382683	+	0.923880i
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$	−144.186	+	1329.08i	−0.139966	+	1.29018i
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	121.556	+	181.922i	0.109825	+	0.164365i	0.882311	−	0.470667i	$-0.155987\pi$
$107$	121.556	+	181.922i	0.109825	+	0.164365i	−0.772486	+	0.635032i	$0.780987\pi$
$108$	−451.206	+	89.7505i	−0.402012	+	0.0799652i
$109$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$109$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−416.495	−	2093.86i	−0.346730	−	1.74313i	−0.623157	−	0.782097i	$-0.714150\pi$
$113$	−416.495	−	2093.86i	−0.346730	−	1.74313i	0.276427	−	0.961035i	$-0.410850\pi$
$114$	−238.619	+	159.440i	−0.196041	+	0.130991i
$115$		0			0
$116$		0			0
$117$		0			0
$118$			1858.64i			1.45001i
$119$		0			0
$120$		0			0
$121$	1215.92	−	503.652i	0.913542	−	0.378401i
$122$		0			0
$123$	750.133	−	750.133i	0.549896	−	0.549896i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$127$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$128$	−1337.92	−	554.185i	−0.923880	−	0.382683i
$129$	−464.449	−	310.335i	−0.316996	−	0.211810i
$130$		0			0
$131$	2427.30	−	1621.87i	1.61889	−	1.08171i	0.682414	−	0.730966i	$-0.260930\pi$
$131$	2427.30	−	1621.87i	1.61889	−	1.08171i	0.936473	−	0.350740i	$-0.114070\pi$
$132$	147.211	+	147.211i	0.0970690	+	0.0970690i
$133$		0			0
$134$	1065.61	+	2572.60i	0.686973	+	1.65850i
$135$		0			0
$136$	1216.00	+	1018.23i	0.766700	+	0.642006i
$137$	3206.99			1.99994			0.999970	−	0.00779827i	$-0.00248229\pi$
$137$	3206.99			1.99994			0.999970	+	0.00779827i	$0.00248229\pi$
$138$		0			0
$139$	1584.12	+	315.101i	0.966641	+	0.192277i	0.653078	−	0.757291i	$-0.273477\pi$
$139$	1584.12	+	315.101i	0.966641	+	0.192277i	0.313563	+	0.949567i	$0.398477\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	452.416	−	1092.23i	0.261815	−	0.632077i
$145$		0			0
$146$	−454.577	−	303.739i	−0.257678	−	0.172175i
$147$	−451.235	−	2268.51i	−0.253178	−	1.27281i
$148$		0			0
$149$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$149$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$150$	−465.118	+	2338.31i	−0.253178	+	1.27281i
$151$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$151$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$152$			340.468i			0.181681i
$153$	−831.248	+	992.697i	−0.439232	+	0.524541i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$157$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	−2316.61	−	959.570i	−1.12352	−	0.465376i
$163$	−3246.27	−	2169.09i	−1.55992	−	1.04231i	−0.972463	−	0.233056i	$-0.925127\pi$
$163$	−3246.27	−	2169.09i	−1.55992	−	1.04231i	−0.587461	−	0.809252i	$-0.699873\pi$
$164$	−245.531	−	1234.37i	−0.116907	−	0.587731i
$165$		0			0
$166$	226.281	+	226.281i	0.105800	+	0.105800i
$167$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$167$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$168$		0			0
$169$			2197.00i			1.00000i
$170$		0			0
$171$	−277.945			−0.124298
$172$	−612.244	+	253.600i	−0.271414	+	0.112423i
$173$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$173$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$174$		0			0
$175$		0			0
$176$	242.241	−	48.1847i	0.103748	−	0.0206367i
$177$	−2461.85	+	3684.42i	−1.04545	+	1.56462i
$178$	1576.64	−	3806.34i	0.663899	−	1.60280i
$179$	629.206	+	260.626i	0.262732	+	0.108827i	0.510161	−	0.860079i	$-0.329586\pi$
$179$	629.206	+	260.626i	0.262732	+	0.108827i	−0.247429	+	0.968906i	$0.579586\pi$
$180$		0			0
$181$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$181$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−268.922	−	29.1742i	−0.105163	−	0.0114087i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$191$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$192$	−1918.15	−	2870.71i	−0.720991	−	1.07904i
$193$	−4395.31	+	874.281i	−1.63928	+	0.326073i	−0.926782	−	0.375600i	$-0.877437\pi$
$193$	−4395.31	+	874.281i	−1.63928	+	0.326073i	−0.712499	+	0.701673i	$0.752437\pi$
$194$	1733.98	−	2595.09i	0.641715	−	0.960394i
$195$		0			0
$196$	−2535.13	−	1050.08i	−0.923880	−	0.382683i
$197$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$197$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$198$	39.3362	+	197.756i	0.0141187	+	0.0709795i
$199$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$199$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$200$	2000.00	+	2000.00i	0.707107	+	0.707107i
$201$	−1295.15	+	6511.17i	−0.454492	+	2.28489i
$202$		0			0
$203$		0			0
$204$	1061.81	+	3629.11i	0.364418	+	1.24553i
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−32.2607	−	48.2815i	−0.0106771	−	0.0159794i
$210$		0			0
$211$	−3375.05	+	5051.13i	−1.10118	+	1.64803i	−0.439490	+	0.898248i	$0.644841\pi$
$211$	−3375.05	+	5051.13i	−1.10118	+	1.64803i	−0.661687	+	0.749780i	$0.730159\pi$
$212$		0			0
$213$		0			0
$214$	514.552	+	343.813i	0.164365	+	0.109825i
$215$		0			0
$216$	−1081.91	+	722.911i	−0.340810	+	0.227722i
$217$		0			0
$218$		0			0
$219$	−498.803	−	1204.22i	−0.153908	−	0.371568i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$223$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$224$		0			0
$225$	−1632.73	+	1632.73i	−0.483770	+	0.483770i
$226$	−3354.73	−	5020.71i	−0.987405	−	1.47776i
$227$	−2027.64	+	403.323i	−0.592861	+	0.117927i	−0.482397	−	0.875953i	$-0.660234\pi$
$227$	−2027.64	+	403.323i	−0.592861	+	0.117927i	−0.110464	+	0.993880i	$0.535234\pi$
$228$	−450.965	+	674.916i	−0.130991	+	0.196041i
$229$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$229$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	2425.84	−	1620.89i	0.682068	−	0.455743i	−0.165654	−	0.986184i	$-0.552974\pi$
$233$	2425.84	−	1620.89i	0.682068	−	0.455743i	0.847722	+	0.530441i	$0.177974\pi$
$234$		0			0
$235$		0			0
$236$	2011.77	+	4856.85i	0.554895	+	1.33964i
$237$		0			0
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	−6867.41	−	1366.01i	−1.83555	−	0.365114i	−0.848980	−	0.528426i	$-0.822783\pi$
$241$	−6867.41	−	1366.01i	−1.83555	−	0.365114i	−0.986575	+	0.163311i	$0.947783\pi$
$242$	2632.21	−	2632.21i	0.699195	−	0.699195i
$243$	−2458.66	−	3679.65i	−0.649066	−	0.971397i
$244$		0			0
$245$		0			0
$246$	1148.25	−	2772.13i	0.297602	−	0.718474i
$247$		0			0
$248$		0			0
$249$	148.843	+	748.283i	0.0378816	+	0.190444i
$250$		0			0
$251$	5534.06	+	5534.06i	1.39166	+	1.39166i	0.821607	+	0.570054i	$0.193078\pi$
$251$	5534.06	+	5534.06i	1.39166	+	1.39166i	0.570054	+	0.821607i	$0.306922\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−4096.00			−1.00000
$257$	7577.61	−	3138.75i	1.83922	−	0.761828i	0.882849	−	0.469658i	$-0.155623\pi$
$257$	7577.61	−	3138.75i	1.83922	−	0.761828i	0.956366	−	0.292170i	$-0.0943774\pi$
$258$	−1549.57	−	308.228i	−0.373922	−	0.0743778i
$259$		0			0
$260$		0			0
$261$		0			0
$262$	4587.34	−	6865.44i	1.08171	−	1.61889i
$263$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$263$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$264$	544.022	+	225.341i	0.126827	+	0.0525334i
$265$		0			0
$266$		0			0
$267$	8167.08	−	5457.07i	1.87198	−	1.25081i
$268$	5569.13	+	5569.13i	1.26936	+	1.26936i
$269$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$269$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$	4279.69	+	1344.58i	0.954023	+	0.299733i
$273$		0			0
$274$	8380.27	−	3471.22i	1.84770	−	0.765344i
$275$	−473.127	−	94.1107i	−0.103748	−	0.0206367i
$276$		0			0
$277$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$277$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$278$	4480.56	−	891.239i	0.966641	−	0.192277i
$279$		0			0
$280$		0			0
$281$	6897.49	+	2857.03i	1.46431	+	0.606535i	0.965552	−	0.260209i	$-0.0837914\pi$
$281$	6897.49	+	2857.03i	1.46431	+	0.606535i	0.498753	+	0.866744i	$0.333791\pi$
$282$		0			0
$283$	−673.400	−	3385.41i	−0.141447	−	0.711102i	−0.984793	−	0.173731i	$-0.944418\pi$
$283$	−673.400	−	3385.41i	−0.141447	−	0.711102i	0.843346	−	0.537371i	$-0.180582\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		−	3343.82i		−	0.684155i
$289$	−4030.23	−	2809.77i	−0.820320	−	0.571905i
$290$		0			0
$291$	6874.63	−	2847.56i	1.38487	−	0.573633i
$292$	−1516.63	−	301.677i	−0.303953	−	0.0604599i
$293$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$293$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$294$	−3634.55	−	5439.49i	−0.720991	−	1.07904i
$295$		0			0
$296$		0			0
$297$	84.9267	−	205.031i	0.0165924	−	0.0400576i
$298$		0			0
$299$		0			0
$300$	1315.55	+	6613.74i	0.253178	+	1.27281i
$301$		0			0
$302$		0			0
$303$		0			0
$304$	368.519	+	889.685i	0.0695264	+	0.167852i
$305$		0			0
$306$	−1097.67	+	3493.78i	−0.205064	+	0.652699i
$307$	−7204.00			−1.33926			−0.669632	−	0.742693i	$-0.733548\pi$
$307$	−7204.00			−1.33926			−0.669632	+	0.742693i	$0.733548\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$311$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$312$		0			0
$313$	−1644.24	+	2460.78i	−0.296927	+	0.444382i	−0.949695	−	0.313175i	$-0.898607\pi$
$313$	−1644.24	+	2460.78i	−0.296927	+	0.444382i	0.652769	+	0.757557i	$0.273607\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$317$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	564.612	+	1363.09i	0.0981732	+	0.237011i
$322$		0			0
$323$	−92.9914	−	1050.56i	−0.0160191	−	0.180974i
$324$	−7092.22			−1.21609
$325$		0			0
$326$	−10830.7	−	2154.37i	−1.84006	−	0.366010i
$327$		0			0
$328$	−1977.67	−	2959.80i	−0.332923	−	0.498254i
$329$		0			0
$330$		0			0
$331$	−4312.01	+	10410.1i	−0.716041	+	1.72868i	−0.0317191	+	0.999497i	$0.510098\pi$
$331$	−4312.01	+	10410.1i	−0.716041	+	1.72868i	−0.684322	+	0.729180i	$0.739902\pi$
$332$	836.227	+	346.377i	0.138235	+	0.0572587i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−2012.98	+	10120.0i	−0.325384	+	1.63581i	0.378574	+	0.925571i	$0.376415\pi$
$337$	−2012.98	+	10120.0i	−0.325384	+	1.63581i	−0.703957	+	0.710242i	$0.748585\pi$
$338$	2378.02	+	5741.04i	0.382683	+	0.923880i
$339$		−	14396.2i		−	2.30647i
$340$		0			0
$341$		0			0
$342$	−726.305	+	300.845i	−0.114837	+	0.0475668i
$343$		0			0
$344$	−1325.38	+	1325.38i	−0.207731	+	0.207731i
$345$		0			0
$346$		0			0
$347$	−5577.49	+	8347.30i	−0.862868	+	1.29137i	0.0924274	+	0.995719i	$0.470537\pi$
$347$	−5577.49	+	8347.30i	−0.862868	+	1.29137i	−0.955295	+	0.295654i	$0.904463\pi$
$348$		0			0
$349$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$349$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$350$		0			0
$351$		0			0
$352$	580.851	−	388.112i	0.0879530	−	0.0587683i
$353$	−6465.40	−	6465.40i	−0.974840	−	0.974840i	0.0248514	−	0.999691i	$-0.492089\pi$
$353$	−6465.40	−	6465.40i	−0.974840	−	0.974840i	−0.999691	+	0.0248514i	$0.992089\pi$
$354$	−2445.14	+	12292.5i	−0.367112	+	1.84559i
$355$		0			0
$356$		−	11653.0i		−	1.73485i
$357$		0			0
$358$	1926.29			0.284379
$359$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$359$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$360$		0			0
$361$	−4689.95	+	4689.95i	−0.683767	+	0.683767i
$362$		0			0
$363$	8704.38	−	1731.41i	1.25857	−	0.250346i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$367$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$368$		0			0
$369$	2416.27	−	1614.50i	0.340883	−	0.227771i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$	−734.305	+	214.843i	−0.101524	+	0.0297039i
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−3659.69	−	5477.11i	−0.496004	−	0.742322i	0.496028	−	0.868306i	$-0.334791\pi$
$379$	−3659.69	−	5477.11i	−0.496004	−	0.742322i	−0.992032	+	0.125984i	$0.959791\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$383$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$384$	−8119.59	−	5425.34i	−1.07904	−	0.720991i
$385$		0			0
$386$	−10539.2	+	7042.06i	−1.38972	+	0.928578i
$387$	−1081.99	−	1081.99i	−0.142120	−	0.142120i
$388$	1722.21	−	8658.14i	0.225340	−	1.13286i
$389$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$389$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$390$		0			0
$391$		0			0
$392$	−7761.20			−1.00000
$393$	18187.2	−	7533.38i	2.33441	−	0.966943i
$394$		0			0
$395$		0			0
$396$	316.840	+	474.185i	0.0402066	+	0.0601735i
$397$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$397$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$398$		0			0
$399$		0			0
$400$	7391.04	+	3061.47i	0.923880	+	0.382683i
$401$	−1835.76	−	1226.62i	−0.228612	−	0.152754i	0.435989	−	0.899952i	$-0.356399\pi$
$401$	−1835.76	−	1226.62i	−0.228612	−	0.152754i	−0.664601	+	0.747198i	$0.731399\pi$
$402$	3663.24	+	18416.4i	0.454492	+	2.28489i
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$	6702.76	+	8334.04i	0.813324	+	1.01127i
$409$	−3903.53			−0.471925			−0.235962	−	0.971762i	$-0.575824\pi$
$409$	−3903.53			−0.471925			−0.235962	+	0.971762i	$0.575824\pi$
$410$		0			0
$411$	21210.2	+	4218.97i	2.54555	+	0.506341i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	10062.4	+	4167.98i	1.18167	+	0.489465i
$418$	−136.561	−	91.2469i	−0.0159794	−	0.0106771i
$419$	−2386.86	−	11999.6i	−0.278296	−	1.39909i	−0.826591	−	0.562803i	$-0.809723\pi$
$419$	−2386.86	−	11999.6i	−0.278296	−	1.39909i	0.548296	−	0.836285i	$-0.315277\pi$
$420$		0			0
$421$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$421$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$422$	−3352.14	+	16852.4i	−0.386682	+	1.94398i
$423$		0			0
$424$		0			0
$425$	−6717.51	−	5625.00i	−0.766700	−	0.642006i
$426$		0			0
$427$		0			0
$428$	1716.73	+	341.479i	0.193881	+	0.0385654i
$429$		0			0
$430$		0			0
$431$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$431$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$432$	−2044.70	+	3060.11i	−0.227722	+	0.340810i
$433$	3151.72	−	7608.93i	0.349797	−	0.844484i	−0.646847	−	0.762620i	$-0.723913\pi$
$433$	3151.72	−	7608.93i	0.349797	−	0.844484i	0.996644	−	0.0818641i	$-0.0260873\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	−2606.87	−	2606.87i	−0.284386	−	0.284386i
$439$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$439$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$440$		0			0
$441$		−	6335.96i		−	0.684155i
$442$		0			0
$443$	−15449.4			−1.65694			−0.828471	−	0.560032i	$-0.810789\pi$
$443$	−15449.4			−1.65694			−0.828471	+	0.560032i	$0.810789\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	8842.10	−	13233.1i	0.929364	−	1.39089i	0.00894477	−	0.999960i	$-0.497153\pi$
$449$	8842.10	−	13233.1i	0.929364	−	1.39089i	0.920420	−	0.390932i	$-0.127847\pi$
$450$	−2499.27	+	6033.77i	−0.261815	+	0.632077i
$451$	560.905	+	232.334i	0.0585632	+	0.0242577i
$452$	−14200.7	−	9488.62i	−1.47776	−	0.987405i
$453$		0			0
$454$	−4861.93	+	3248.64i	−0.502603	+	0.335829i
$455$		0			0
$456$	−447.903	+	2251.76i	−0.0459978	+	0.231247i
$457$	2878.32	+	6948.89i	0.294622	+	0.711280i	0.999997	+	0.00244611i	$0.000778620\pi$
$457$	2878.32	+	6948.89i	0.294622	+	0.711280i	−0.705375	+	0.708834i	$0.749221\pi$
$458$		0			0
$459$	3140.93	−	2526.14i	0.319404	−	0.256884i
$460$		0			0
$461$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$461$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$462$		0			0
$463$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$463$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$464$		0			0
$465$		0			0
$466$	4584.57	−	6861.30i	0.455743	−	0.682068i
$467$	−7712.53	+	18619.7i	−0.764226	+	1.84500i	−0.331679	+	0.943392i	$0.607615\pi$
$467$	−7712.53	+	18619.7i	−0.764226	+	1.84500i	−0.432546	+	0.901612i	$0.642385\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	10514.0	+	10514.0i	1.02531	+	1.02531i
$473$	62.3661	−	313.535i	0.00606257	−	0.0304786i
$474$		0			0
$475$		−	1880.84i		−	0.181681i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$479$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$480$		0			0
$481$		0			0
$482$	−19424.0	+	3863.67i	−1.83555	+	0.365114i
$483$		0			0
$484$	4029.22	−	9727.39i	0.378401	−	0.913542i
$485$		0			0
$486$	−10407.6	−	6954.14i	−0.971397	−	0.649066i
$487$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$487$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$488$		0			0
$489$	−18616.4	−	18616.4i	−1.72160	−	1.72160i
$490$		0			0
$491$	8178.81	+	19745.4i	0.751740	+	1.81486i	0.549429	+	0.835540i	$0.314845\pi$
$491$	8178.81	+	19745.4i	0.751740	+	1.81486i	0.202311	+	0.979321i	$0.435155\pi$
$492$		−	8486.79i		−	0.777671i
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$	1198.88	+	1794.25i	0.107878	+	0.161450i
$499$	−15851.0	+	3152.97i	−1.42202	+	0.282858i	−0.845392	−	0.534147i	$-0.820633\pi$
$499$	−15851.0	+	3152.97i	−1.42202	+	0.282858i	−0.576631	+	0.817005i	$0.695633\pi$
$500$		0			0
$501$		0			0
$502$	20451.2	+	8471.18i	1.81829	+	0.753162i
$503$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$503$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−2890.27	+	14530.4i	−0.253178	+	1.27281i
$508$		0			0
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$		0			0
$512$	−10703.4	+	4433.48i	−0.923880	+	0.382683i
$513$	848.644	+	168.806i	0.0730381	+	0.0145282i
$514$	16403.9	−	16403.9i	1.40767	−	1.40767i
$515$		0			0
$516$	−4382.84	+	871.802i	−0.373922	+	0.0743778i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−2588.44	−	13013.0i	−0.217662	−	1.09426i	−0.922827	−	0.385215i	$-0.874127\pi$
$521$	−2588.44	−	13013.0i	−0.217662	−	1.09426i	0.705165	−	0.709043i	$-0.250873\pi$
$522$		0			0
$523$	9447.21	+	9447.21i	0.789862	+	0.789862i	0.981471	−	0.191609i	$-0.0613707\pi$
$523$	9447.21	+	9447.21i	0.789862	+	0.789862i	−0.191609	+	0.981471i	$0.561371\pi$
$524$	4556.20	−	22905.6i	0.379845	−	1.90961i
$525$		0			0
$526$		0			0
$527$		0			0
$528$	1665.51			0.137276
$529$	11240.8	−	4656.11i	0.923880	−	0.382683i
$530$		0			0
$531$	−8583.26	+	8583.26i	−0.701472	+	0.701472i
$532$		0			0
$533$		0			0
$534$	15434.9	−	23100.0i	1.25081	−	1.87198i
$535$		0			0
$536$	20580.8	+	8524.85i	1.65850	+	0.686973i
$537$	3818.53	+	2551.46i	0.306856	+	0.205035i
$538$		0			0
$539$	1100.61	−	735.405i	0.0879530	−	0.0587683i
$540$		0			0
$541$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$541$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$542$		0			0
$543$		0			0
$544$	12638.7	−	1118.73i	0.996105	−	0.0881716i
$545$		0			0
$546$		0			0
$547$	22762.9	+	4527.83i	1.77929	+	0.353923i	0.971773	−	0.235917i	$-0.0758092\pi$
$547$	22762.9	+	4527.83i	1.77929	+	0.353923i	0.807520	+	0.589840i	$0.200809\pi$
$548$	18141.5	−	18141.5i	1.41417	−	1.41417i
$549$		0			0
$550$	−1338.20	+	266.185i	−0.103748	+	0.0206367i
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	10743.6	−	7178.64i	0.819479	−	0.547558i
$557$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$557$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	−1740.20	−	546.732i	−0.130965	−	0.0411462i
$562$	21116.4			1.58495
$563$	−18945.3	+	7847.39i	−1.41820	+	0.587439i	−0.954408	−	0.298504i	$-0.903512\pi$
$563$	−18945.3	+	7847.39i	−1.41820	+	0.587439i	−0.463795	+	0.885943i	$0.653512\pi$
$564$		0			0
$565$		0			0
$566$	−5424.02	−	8117.62i	−0.402807	−	0.602843i
$567$		0			0
$568$		0			0
$569$	9209.41	−	22233.5i	0.678521	−	1.63809i	−0.0881913	−	0.996104i	$-0.528109\pi$
$569$	9209.41	−	22233.5i	0.678521	−	1.63809i	0.766712	−	0.641991i	$-0.221891\pi$
$570$		0			0
$571$	21450.6	+	14332.8i	1.57212	+	1.05045i	0.967151	+	0.254202i	$0.0818128\pi$
$571$	21450.6	+	14332.8i	1.57212	+	1.05045i	0.604965	+	0.796252i	$0.293187\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−3619.33	−	8737.83i	−0.261815	−	0.632077i
$577$			19651.9i			1.41789i	0.705266	+	0.708943i	$0.250827\pi$
$577$			19651.9i			1.41789i	−0.705266	+	0.708943i	$0.749173\pi$
$578$	−13572.8	−	2979.98i	−0.976735	−	0.214448i
$579$	−30219.6			−2.16905
$580$		0			0
$581$		0			0
$582$	14882.1	−	14882.1i	1.05993	−	1.05993i
$583$		0			0
$584$	−4289.68	+	853.271i	−0.303953	+	0.0604599i
$585$		0			0
$586$		0			0
$587$	−9562.15	−	3960.77i	−0.672355	−	0.278498i	0.0202721	−	0.999795i	$-0.493547\pi$
$587$	−9562.15	−	3960.77i	−0.672355	−	0.278498i	−0.692627	+	0.721296i	$0.743547\pi$
$588$	−15385.2	−	10280.1i	−1.07904	−	0.720991i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	8139.44	+	19650.3i	0.563654	+	1.36078i	0.906825	+	0.421507i	$0.138499\pi$
$593$	8139.44	+	19650.3i	0.563654	+	1.36078i	−0.343171	+	0.939273i	$0.611501\pi$
$594$		−	627.696i		−	0.0433581i
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$599$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$600$	10596.4	+	15858.6i	0.720991	+	1.07904i
$601$	−25750.9	+	5122.18i	−1.74776	+	0.347651i	−0.962451	−	0.271455i	$-0.912495\pi$
$601$	−25750.9	+	5122.18i	−1.74776	+	0.347651i	−0.785307	+	0.619106i	$0.787495\pi$
$602$		0			0
$603$	−6959.37	+	16801.4i	−0.469996	+	1.13467i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$607$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$608$	1925.98	+	1925.98i	0.128468	+	0.128468i
$609$		0			0
$610$		0			0
$611$		0			0
$612$	913.293	+	10317.8i	0.0603230	+	0.681490i
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$	−18825.0	+	7797.56i	−1.23732	+	0.512514i
$615$		0			0
$616$		0			0
$617$	9198.48	+	13766.5i	0.600190	+	0.898247i	0.999830	−	0.0184247i	$-0.00586511\pi$
$617$	9198.48	+	13766.5i	0.600190	+	0.898247i	−0.399641	+	0.916672i	$0.630865\pi$
$618$		0			0
$619$	8360.55	−	12512.4i	0.542874	−	0.812468i	−0.454039	−	0.890982i	$-0.650017\pi$
$619$	8360.55	−	12512.4i	0.542874	−	0.812468i	0.996913	+	0.0785136i	$0.0250174\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−11048.5	−	11048.5i	−0.707107	−	0.707107i
$626$	−1633.08	+	8210.05i	−0.104267	+	0.524185i
$627$	−149.847	−	361.762i	−0.00954433	−	0.0230420i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$631$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$632$		0			0
$633$	−28966.7	+	28966.7i	−1.81884	+	1.81884i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−293.496	+	196.107i	−0.0180848	+	0.0120839i	−0.564580	−	0.825378i	$-0.690962\pi$
$641$	−293.496	+	196.107i	−0.0180848	+	0.0120839i	0.546495	+	0.837462i	$0.315962\pi$
$642$	2950.81	+	2950.81i	0.181400	+	0.181400i
$643$	6062.53	−	30478.4i	0.371824	−	1.86928i	−0.111381	−	0.993778i	$-0.535528\pi$
$643$	6062.53	−	30478.4i	0.371824	−	1.86928i	0.483205	−	0.875507i	$-0.339472\pi$
$644$		0			0
$645$		0			0
$646$	−1380.11	−	2644.58i	−0.0840554	−	0.161068i
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$	−18532.9	+	7676.56i	−1.12352	+	0.465376i
$649$	−2487.23	−	494.742i	−0.150435	−	0.0299234i
$650$		0			0
$651$		0			0
$652$	−30633.9	+	6093.47i	−1.84006	+	0.366010i
$653$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$653$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$654$		0			0
$655$		0			0
$656$	−8371.57	−	5593.70i	−0.498254	−	0.332923i
$657$	−696.578	−	3501.94i	−0.0413639	−	0.207951i
$658$		0			0
$659$	−21039.3	−	21039.3i	−1.24366	−	1.24366i	−0.958471	−	0.285191i	$-0.907943\pi$
$659$	−21039.3	−	21039.3i	−1.24366	−	1.24366i	−0.285191	−	0.958471i	$-0.592057\pi$
$660$		0			0
$661$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$661$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$662$			31870.2i			1.87111i
$663$		0			0
$664$	2560.08			0.149624
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	26902.7	+	17975.8i	1.54090	+	1.02959i	0.979355	+	0.202147i	$0.0647919\pi$
$673$	26902.7	+	17975.8i	1.54090	+	1.02959i	0.561542	+	0.827448i	$0.310208\pi$
$674$	5693.58	+	28623.6i	0.325384	+	1.63581i
$675$	5976.78	−	3993.56i	0.340810	−	0.227722i
$676$	12428.1	+	12428.1i	0.707107	+	0.707107i
$677$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$677$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$678$	−15582.3	−	37619.0i	−0.882647	−	2.13090i
$679$		0			0
$680$		0			0
$681$	−13940.9			−0.784459
$682$		0			0
$683$	30062.4	+	5979.78i	1.68420	+	0.335007i	0.942112	−	0.335300i	$-0.108838\pi$
$683$	30062.4	+	5979.78i	1.68420	+	0.335007i	0.742084	+	0.670307i	$0.233838\pi$
$684$	−1572.29	+	1572.29i	−0.0878921	+	0.0878921i
$685$		0			0
$686$		0			0
$687$		0			0
$688$	−2028.80	+	4897.95i	−0.112423	+	0.271414i
$689$		0			0
$690$		0			0
$691$	−6865.56	−	34515.5i	−0.377971	−	1.90019i	−0.432419	−	0.901673i	$-0.642340\pi$
$691$	−6865.56	−	34515.5i	−0.377971	−	1.90019i	0.0544477	−	0.998517i	$-0.482660\pi$
$692$		0			0
$693$		0			0
$694$	−5539.62	+	27849.6i	−0.302999	+	1.52328i
$695$		0			0
$696$		0			0
$697$	6910.77	+	8592.67i	0.375558	+	0.466959i
$698$		0			0
$699$	18176.2	−	7528.83i	0.983530	−	0.407391i
$700$		0			0
$701$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$701$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$702$		0			0
$703$		0			0
$704$	1097.75	−	1642.90i	0.0587683	−	0.0879530i
$705$		0			0
$706$	−23893.0	−	9896.80i	−1.27369	−	0.527579i
$707$		0			0
$708$	6915.89	+	34768.5i	0.367112	+	1.84559i
$709$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$709$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$710$		0			0
$711$		0			0
$712$	−12613.1	−	30450.8i	−0.663899	−	1.60280i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	5033.65	−	2085.00i	0.262732	−	0.108827i
$717$		0			0
$718$		0			0
$719$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$719$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$720$		0			0
$721$		0			0
$722$	−7179.07	+	17331.8i	−0.370052	+	0.893384i
$723$	−43622.2	−	18068.9i	−2.24388	−	0.929446i
$724$		0			0
$725$		0			0
$726$	20871.6	−	13945.9i	1.06697	−	0.712924i
$727$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$727$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$728$		0			0
$729$	−2260.16	−	5456.50i	−0.114828	−	0.277219i
$730$		0			0
$731$	3727.62	−	4451.62i	0.188606	−	0.225238i
$732$		0			0
$733$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$733$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−3726.31	+	741.210i	−0.186242	+	0.0370459i
$738$	4566.49	−	6834.24i	0.227771	−	0.340883i
$739$	6770.36	−	16345.1i	0.337012	−	0.813619i	−0.660988	−	0.750397i	$-0.729862\pi$
$739$	6770.36	−	16345.1i	0.337012	−	0.813619i	0.997999	−	0.0632220i	$-0.0201376\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$743$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$			2089.96i			0.102366i
$748$	−1686.29	+	1356.22i	−0.0824288	+	0.0662944i
$749$		0			0
$750$		0			0
$751$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$751$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$752$		0			0
$753$	29320.5	+	43881.2i	1.41899	+	2.12366i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$757$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$758$	−15491.6	−	10351.2i	−0.742322	−	0.496004i
$759$		0			0
$760$		0			0
$761$	−29281.5	−	29281.5i	−1.39482	−	1.39482i	−0.814124	−	0.580691i	$-0.802782\pi$
$761$	−29281.5	−	29281.5i	−1.39482	−	1.39482i	−0.580691	−	0.814124i	$-0.697218\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−27089.9	−	5388.51i	−1.27281	−	0.253178i
$769$	−27307.9	+	27307.9i	−1.28056	+	1.28056i	−0.340207	+	0.940351i	$0.610497\pi$
$769$	−27307.9	+	27307.9i	−1.28056	+	1.28056i	−0.940351	+	0.340207i	$0.889503\pi$
$770$		0			0
$771$	54245.5	−	10790.1i	2.53386	−	0.504015i
$772$	−19917.9	+	29809.3i	−0.928578	+	1.38972i
$773$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$773$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$774$	−3998.50	−	1656.23i	−0.185689	−	0.0769149i
$775$		0			0
$776$	−4871.15	−	24488.9i	−0.225340	−	1.13286i
$777$		0			0
$778$		0			0
$779$	−461.803	+	2321.64i	−0.0212398	+	0.106780i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−20281.0	+	8400.67i	−0.923880	+	0.382683i
$785$		0			0
$786$	39371.3	−	39371.3i	1.78668	−	1.78668i
$787$	−10248.8	−	15338.5i	−0.464207	−	0.694736i	0.523328	−	0.852131i	$-0.324690\pi$
$787$	−10248.8	−	15338.5i	−0.464207	−	0.694736i	−0.987536	+	0.157396i	$0.949690\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$	1341.20	+	896.160i	0.0601735	+	0.0402066i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$797$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$798$		0			0
$799$		0			0
$800$	22627.4			1.00000
$801$	24858.9	−	10296.9i	1.09656	−	0.454210i
$802$	−6124.76	−	1218.29i	−0.269667	−	0.0536400i
$803$	527.466	−	527.466i	0.0231804	−	0.0231804i
$804$	29506.2	+	44159.2i	1.29428	+	1.93703i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	30117.5	+	20123.8i	1.30887	+	0.874557i	0.997137	−	0.0756140i	$-0.0240917\pi$
$809$	30117.5	+	20123.8i	1.30887	+	0.874557i	0.311730	+	0.950171i	$0.399092\pi$
$810$		0			0
$811$	38387.8	−	25649.9i	1.66212	−	1.11059i	0.813614	−	0.581405i	$-0.197497\pi$
$811$	38387.8	−	25649.9i	1.66212	−	1.11059i	0.848505	−	0.529188i	$-0.177503\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$	26535.9	+	14522.9i	1.13841	+	0.623042i
$817$	1246.41			0.0533736
$818$	−10200.4	+	4225.15i	−0.436001	+	0.180598i
$819$		0			0
$820$		0			0
$821$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$821$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$822$	59991.5	−	11933.0i	2.54555	−	0.506341i
$823$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$823$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$824$		0			0
$825$	−3005.33	−	1244.85i	−0.126827	−	0.0525334i
$826$		0			0
$827$	672.518	+	3380.98i	0.0282778	+	0.142162i	0.992345	−	0.123496i	$-0.0394107\pi$
$827$	672.518	+	3380.98i	0.0282778	+	0.142162i	−0.964067	+	0.265658i	$0.914411\pi$
$828$		0			0
$829$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$829$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	23948.2	−	2119.81i	0.996105	−	0.0881716i
$834$	30805.7			1.27903
$835$		0			0
$836$	−455.615	−	90.6275i	−0.0188490	−	0.00374930i
$837$		0			0
$838$	−19225.4	−	28772.9i	−0.792519	−	1.18609i
$839$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$839$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$840$		0			0
$841$	9333.27	−	22532.5i	0.382683	−	0.923880i
$842$		0			0
$843$	41859.6	+	27969.7i	1.71023	+	1.14274i
$844$	9481.29	+	47665.7i	0.386682	+	1.94398i
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$		−	23276.1i		−	0.940911i
$850$	−23642.2	−	7427.85i	−0.954023	−	0.299733i
$851$		0			0
$852$		0			0
$853$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$853$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$854$		0			0
$855$		0			0
$856$	4855.64	−	965.848i	0.193881	−	0.0385654i
$857$	−15201.1	+	22750.1i	−0.605904	+	0.906799i	−0.999924	−	0.0123023i	$-0.996084\pi$
$857$	−15201.1	+	22750.1i	−0.605904	+	0.906799i	0.394020	+	0.919102i	$0.371084\pi$
$858$		0			0
$859$	−31446.3	−	13025.5i	−1.24905	−	0.517373i	−0.342518	−	0.939511i	$-0.611280\pi$
$859$	−31446.3	−	13025.5i	−1.24905	−	0.517373i	−0.906532	+	0.422138i	$0.861280\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$863$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$864$	−2030.82	+	10209.6i	−0.0799652	+	0.402012i
$865$		0			0
$866$		−	23294.5i		−	0.914063i
$867$	−22958.5	−	23885.0i	−0.899321	−	0.935615i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	19991.8	−	3976.62i	0.775053	−	0.154168i
$874$		0			0
$875$		0			0
$876$	−9633.73	−	3990.42i	−0.371568	−	0.153908i
$877$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$877$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	6787.76	−	34124.4i	0.259575	−	1.30497i	−0.602471	−	0.798141i	$-0.705817\pi$
$881$	6787.76	−	34124.4i	0.259575	−	1.30497i	0.862046	−	0.506830i	$-0.169183\pi$
$882$	−6857.99	−	16556.7i	−0.261815	−	0.632077i
$883$			24867.2i			0.947733i	0.880597	+	0.473866i	$0.157142\pi$
$883$			24867.2i			0.947733i	−0.880597	+	0.473866i	$0.842858\pi$
$884$		0			0
$885$		0			0
$886$	−40371.3	+	16722.4i	−1.53081	+	0.634084i
$887$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$887$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	1900.75	−	2844.67i	0.0714674	−	0.106959i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	8782.08	−	44150.5i	0.326349	−	1.64067i
$899$		0			0
$900$			18472.2i			0.684155i
$901$		0			0
$902$	1717.19			0.0633883
$903$		0			0
$904$	−47378.7	−	9424.20i	−1.74313	−	0.346730i
$905$		0			0
$906$		0			0
$907$	34384.1	−	6839.42i	1.25877	−	0.250385i	0.479762	−	0.877399i	$-0.340723\pi$
$907$	34384.1	−	6839.42i	1.25877	−	0.250385i	0.779008	+	0.627013i	$0.215723\pi$
$908$	−9188.54	+	13751.6i	−0.335829	+	0.502603i
$909$		0			0
$910$		0			0
$911$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$911$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$912$	1266.86	+	6368.95i	0.0459978	+	0.231247i
$913$	−363.044	+	242.578i	−0.0131599	+	0.00879316i
$914$	15042.8	+	15042.8i	0.544390	+	0.544390i
$915$		0			0
$916$		0			0
$917$		0			0
$918$	5473.39	−	10000.8i	0.196785	−	0.359561i
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$	−47645.4	−	9477.25i	−1.70463	−	0.339073i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−7221.37	−	36304.3i	−0.255033	−	1.28214i	−0.869791	−	0.493421i	$-0.835746\pi$
$929$	−7221.37	−	36304.3i	−0.255033	−	1.28214i	0.614758	−	0.788716i	$-0.289254\pi$
$930$		0			0
$931$	3649.39	+	3649.39i	0.128468	+	0.128468i
$932$	4553.45	−	22891.7i	0.160036	−	0.804554i
$933$		0			0
$934$			57003.6i			1.99702i
$935$		0			0
$936$		0			0
$937$	−10300.9	+	4266.79i	−0.359143	+	0.148762i	−0.554957	−	0.831879i	$-0.687265\pi$
$937$	−10300.9	+	4266.79i	−0.359143	+	0.148762i	0.195814	+	0.980641i	$0.437265\pi$
$938$		0			0
$939$	−14111.9	+	14111.9i	−0.490440	+	0.490440i
$940$		0			0
$941$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$941$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$942$		0			0
$943$		0			0
$944$	38854.8	+	16094.2i	1.33964	+	0.554895i
$945$		0			0
$946$	−176.398	−	886.812i	−0.00606257	−	0.0304786i
$947$	−39090.1	+	26119.1i	−1.34135	+	0.896260i	−0.999061	−	0.0433353i	$-0.986202\pi$
$947$	−39090.1	+	26119.1i	−1.34135	+	0.896260i	−0.342287	+	0.939595i	$0.611202\pi$
$948$		0			0
$949$		0			0
$950$	−2035.80	−	4914.86i	−0.0695264	−	0.167852i
$951$		0			0
$952$		0			0
$953$	28444.1			0.966836			0.483418	−	0.875390i	$-0.339395\pi$
$953$	28444.1			0.966836			0.483418	+	0.875390i	$0.339395\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−27523.3	−	11400.5i	−0.923880	−	0.382683i
$962$		0			0
$963$	788.482	+	3963.97i	0.0263847	+	0.132645i
$964$	−46575.3	+	31120.6i	−1.55611	+	1.03976i
$965$		0			0
$966$		0			0
$967$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$967$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$968$		−	29780.1i		−	0.988811i
$969$	767.043	−	7070.44i	0.0254293	−	0.234402i
$970$		0			0
$971$	−21256.7	+	8804.82i	−0.702534	+	0.290999i	−0.705211	−	0.708997i	$-0.749148\pi$
$971$	−21256.7	+	8804.82i	−0.702534	+	0.290999i	0.00267705	+	0.999996i	$0.499148\pi$
$972$	−34723.5	−	6906.93i	−1.14584	−	0.227922i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	2433.83	−	5875.78i	0.0796981	−	0.192408i	−0.879008	−	0.476807i	$-0.841794\pi$
$977$	2433.83	−	5875.78i	0.0796981	−	0.192408i	0.958706	+	0.284399i	$0.0917940\pi$
$978$	−68797.4	−	28496.8i	−2.24938	−	0.931725i
$979$	4673.99	+	3123.06i	0.152586	+	0.101954i
$980$		0			0
$981$		0			0
$982$	42744.5	+	42744.5i	1.38904	+	1.38904i
$983$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$983$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$984$	−9186.03	−	22177.0i	−0.297602	−	0.718474i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$991$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$992$		0			0
$993$	−42213.6	+	63177.1i	−1.34905	+	2.01900i
$994$		0			0
$995$		0			0
$996$	5074.91	+	3390.95i	0.161450	+	0.107878i
$997$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$997$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$998$	−38008.0	+	25396.1i	−1.20553	+	0.805511i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	136.4.s.a.3.1	✓	8
8.3	odd	2	CM	136.4.s.a.3.1	✓	8
17.6	odd	16	inner	136.4.s.a.91.1	yes	8
136.91	even	16	inner	136.4.s.a.91.1	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.4.s.a.3.1	✓	8	1.1	even	1	trivial
136.4.s.a.3.1	✓	8	8.3	odd	2	CM
136.4.s.a.91.1	yes	8	17.6	odd	16	inner
136.4.s.a.91.1	yes	8	136.91	even	16	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:	\( N \)	\(=\)	\( 136 = 2^{3} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	136.s (of order \(16\), degree \(8\), minimal)

\(f(q)\)	\(=\)	\(q+(2.61313 - 1.08239i) q^{2} +(6.61374 + 1.31555i) q^{3} +(5.65685 - 5.65685i) q^{4} +(18.7065 - 3.72095i) q^{6} +(8.65914 - 20.9050i) q^{8} +(17.0661 + 7.06900i) q^{9} +O(q^{10})\)	\(q+(2.61313 - 1.08239i) q^{2} +(6.61374 + 1.31555i) q^{3} +(5.65685 - 5.65685i) q^{4} +(18.7065 - 3.72095i) q^{6} +(8.65914 - 20.9050i) q^{8} +(17.0661 + 7.06900i) q^{9} +(0.752886 + 3.78501i) q^{11} +(44.8548 - 29.9710i) q^{12} -64.0000i q^{16} +(-21.0091 + 66.8701i) q^{17} +52.2472 q^{18} +(-13.9013 + 5.75812i) q^{19} +(6.06425 + 9.07580i) q^{22} +(84.7709 - 126.869i) q^{24} +(-47.8354 + 115.485i) q^{25} +(-47.8143 - 31.9485i) q^{27} +(-69.2731 - 167.240i) q^{32} +26.0235i q^{33} +(17.4802 + 197.480i) q^{34} +(136.529 - 56.5520i) q^{36} +(-30.0934 + 30.0934i) q^{38} +(87.4016 - 130.806i) q^{41} +(-76.5305 - 31.7000i) q^{43} +(25.6702 + 17.1523i) q^{44} +(84.1954 - 423.279i) q^{48} +(-131.260 - 316.891i) q^{49} +353.553i q^{50} +(-226.920 + 414.623i) q^{51} +(-159.525 - 31.7316i) q^{54} +(-99.5148 + 19.7947i) q^{57} +(-251.472 + 607.106i) q^{59} +(-362.039 - 362.039i) q^{64} +(28.1677 + 68.0028i) q^{66} +984.492i q^{67} +(259.429 + 497.120i) q^{68} +(295.555 - 295.555i) q^{72} +(-107.388 - 160.717i) q^{73} +(-468.298 + 700.857i) q^{75} +(-46.0649 + 111.211i) q^{76} +(-626.870 - 626.870i) q^{81} +(86.8083 - 436.415i) q^{82} +(43.2971 + 104.528i) q^{83} -234.296 q^{86} +(85.6451 + 17.0359i) q^{88} +(1029.99 - 1029.99i) q^{89} +(-238.141 - 1197.21i) q^{96} +(917.502 - 613.055i) q^{97} +(-686.000 - 686.000i) q^{98} +(-13.9074 + 69.9174i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{3} + 80 q^{6} - 80 q^{9}+O(q^{10}) \) 8 * q - 8 * q^3 + 80 * q^6 - 80 * q^9	\( 8 q - 8 q^{3} + 80 q^{6} - 80 q^{9} + 320 q^{12} + 144 q^{22} - 256 q^{24} + 496 q^{27} - 720 q^{34} - 640 q^{36} - 1440 q^{38} + 160 q^{41} + 208 q^{43} - 1600 q^{44} + 512 q^{48} + 1880 q^{51} - 176 q^{54} + 2840 q^{57} - 920 q^{59} + 5232 q^{66} + 3456 q^{72} - 3312 q^{73} - 2632 q^{81} - 3472 q^{83} - 5120 q^{96} - 5488 q^{98} - 1536 q^{99}+O(q^{100}) \) 8 * q - 8 * q^3 + 80 * q^6 - 80 * q^9 + 320 * q^12 + 144 * q^22 - 256 * q^24 + 496 * q^27 - 720 * q^34 - 640 * q^36 - 1440 * q^38 + 160 * q^41 + 208 * q^43 - 1600 * q^44 + 512 * q^48 + 1880 * q^51 - 176 * q^54 + 2840 * q^57 - 920 * q^59 + 5232 * q^66 + 3456 * q^72 - 3312 * q^73 - 2632 * q^81 - 3472 * q^83 - 5120 * q^96 - 5488 * q^98 - 1536 * q^99

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