LMFDB - Embedded newform 2366.2.d.p.337.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2366,2,Mod(337,2366)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2366, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2366.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2366 = 2 \cdot 7 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2366.d (of order $2$, degree $1$, not 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:	$18.8926051182$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.153664.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 5x^{4} + 6x^{2} + 1 $ x^6 + 5x^4 + 6x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.3
Root			$-1.24698i$ of defining polynomial
Character	$\chi$	$=$	2366.337
Dual form			2366.2.d.p.337.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +2.24698 q^{3} -1.00000 q^{4} -1.69202i q^{5} -2.24698i q^{6} -1.00000i q^{7} +1.00000i q^{8} +2.04892 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} +2.24698 q^{3} -1.00000 q^{4} -1.69202i q^{5} -2.24698i q^{6} -1.00000i q^{7} +1.00000i q^{8} +2.04892 q^{9} -1.69202 q^{10} +0.445042i q^{11} -2.24698 q^{12} -1.00000 q^{14} -3.80194i q^{15} +1.00000 q^{16} +2.15883 q^{17} -2.04892i q^{18} -6.35690i q^{19} +1.69202i q^{20} -2.24698i q^{21} +0.445042 q^{22} +0.911854 q^{23} +2.24698i q^{24} +2.13706 q^{25} -2.13706 q^{27} +1.00000i q^{28} +3.58211 q^{29} -3.80194 q^{30} -8.89977i q^{31} -1.00000i q^{32} +1.00000i q^{33} -2.15883i q^{34} -1.69202 q^{35} -2.04892 q^{36} +10.8019i q^{37} -6.35690 q^{38} +1.69202 q^{40} -2.41789i q^{41} -2.24698 q^{42} -4.63102 q^{43} -0.445042i q^{44} -3.46681i q^{45} -0.911854i q^{46} -9.75063i q^{47} +2.24698 q^{48} -1.00000 q^{49} -2.13706i q^{50} +4.85086 q^{51} -8.74094 q^{53} +2.13706i q^{54} +0.753020 q^{55} +1.00000 q^{56} -14.2838i q^{57} -3.58211i q^{58} -10.1468i q^{59} +3.80194i q^{60} +1.37867 q^{61} -8.89977 q^{62} -2.04892i q^{63} -1.00000 q^{64} +1.00000 q^{66} -6.23490i q^{67} -2.15883 q^{68} +2.04892 q^{69} +1.69202i q^{70} +5.76271i q^{71} +2.04892i q^{72} -9.93661i q^{73} +10.8019 q^{74} +4.80194 q^{75} +6.35690i q^{76} +0.445042 q^{77} -6.30127 q^{79} -1.69202i q^{80} -10.9487 q^{81} -2.41789 q^{82} +2.91185i q^{83} +2.24698i q^{84} -3.65279i q^{85} +4.63102i q^{86} +8.04892 q^{87} -0.445042 q^{88} +18.1075i q^{89} -3.46681 q^{90} -0.911854 q^{92} -19.9976i q^{93} -9.75063 q^{94} -10.7560 q^{95} -2.24698i q^{96} +3.30798i q^{97} +1.00000i q^{98} +0.911854i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{3} - 6 q^{4} - 6 q^{9}+O(q^{10}) $ 6 * q + 4 * q^3 - 6 * q^4 - 6 * q^9	$ 6 q + 4 q^{3} - 6 q^{4} - 6 q^{9} - 4 q^{12} - 6 q^{14} + 6 q^{16} - 4 q^{17} + 2 q^{22} - 2 q^{23} + 2 q^{25} - 2 q^{27} + 10 q^{29} - 14 q^{30} + 6 q^{36} - 30 q^{38} - 4 q^{42} + 2 q^{43} + 4 q^{48} - 6 q^{49} + 2 q^{51} - 24 q^{53} + 14 q^{55} + 6 q^{56} - 6 q^{61} - 8 q^{62} - 6 q^{64} + 6 q^{66} + 4 q^{68} - 6 q^{69} + 56 q^{74} + 20 q^{75} + 2 q^{77} - 4 q^{79} - 2 q^{81} - 26 q^{82} + 30 q^{87} - 2 q^{88} - 14 q^{90} + 2 q^{92} + 14 q^{94} + 14 q^{95}+O(q^{100}) $ 6 * q + 4 * q^3 - 6 * q^4 - 6 * q^9 - 4 * q^12 - 6 * q^14 + 6 * q^16 - 4 * q^17 + 2 * q^22 - 2 * q^23 + 2 * q^25 - 2 * q^27 + 10 * q^29 - 14 * q^30 + 6 * q^36 - 30 * q^38 - 4 * q^42 + 2 * q^43 + 4 * q^48 - 6 * q^49 + 2 * q^51 - 24 * q^53 + 14 * q^55 + 6 * q^56 - 6 * q^61 - 8 * q^62 - 6 * q^64 + 6 * q^66 + 4 * q^68 - 6 * q^69 + 56 * q^74 + 20 * q^75 + 2 * q^77 - 4 * q^79 - 2 * q^81 - 26 * q^82 + 30 * q^87 - 2 * q^88 - 14 * q^90 + 2 * q^92 + 14 * q^94 + 14 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2366\mathbb{Z}\right)^\times$.

$n$	$339$	$2199$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 1.00000i	− 0.707107i
$2$	− 1.00000i	− 0.707107i
$3$	2.24698	1.29729	0.648647	−	0.761089i	$-0.275335\pi$
$3$	2.24698	1.29729	0.648647	+	0.761089i	$0.275335\pi$
$4$	−1.00000	−0.500000
$5$	− 1.69202i	− 0.756695i	−0.925664	−	0.378348i	$-0.876492\pi$
$5$	− 1.69202i	− 0.756695i	0.925664	−	0.378348i	$-0.123508\pi$
$6$	− 2.24698i	− 0.917326i
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	1.00000i	0.353553i
$9$	2.04892	0.682972
$10$	−1.69202	−0.535064
$11$	0.445042i	0.134185i	0.997747	+	0.0670926i	$0.0213723\pi$
$11$	0.445042i	0.134185i	−0.997747	+	0.0670926i	$0.978628\pi$
$12$	−2.24698	−0.648647
$13$	0	0
$13$	0	0
$14$	−1.00000	−0.267261
$15$	− 3.80194i	− 0.981656i
$16$	1.00000	0.250000
$17$	2.15883	0.523594	0.261797	−	0.965123i	$-0.415685\pi$
$17$	2.15883	0.523594	0.261797	+	0.965123i	$0.415685\pi$
$18$	− 2.04892i	− 0.482934i
$19$	− 6.35690i	− 1.45837i	−0.684316	−	0.729186i	$-0.739899\pi$
$19$	− 6.35690i	− 1.45837i	0.684316	−	0.729186i	$-0.260101\pi$
$20$	1.69202i	0.378348i
$21$	− 2.24698i	− 0.490331i
$22$	0.445042	0.0948832
$23$	0.911854	0.190135	0.0950674	−	0.995471i	$-0.469693\pi$
$23$	0.911854	0.190135	0.0950674	+	0.995471i	$0.469693\pi$
$24$	2.24698i	0.458663i
$25$	2.13706	0.427413
$26$	0	0
$27$	−2.13706	−0.411278
$28$	1.00000i	0.188982i
$29$	3.58211	0.665180	0.332590	−	0.943071i	$-0.392077\pi$
$29$	3.58211	0.665180	0.332590	+	0.943071i	$0.392077\pi$
$30$	−3.80194	−0.694136
$31$	− 8.89977i	− 1.59845i	−0.601034	−	0.799223i	$-0.705245\pi$
$31$	− 8.89977i	− 1.59845i	0.601034	−	0.799223i	$-0.294755\pi$
$32$	− 1.00000i	− 0.176777i
$33$	1.00000i	0.174078i
$34$	− 2.15883i	− 0.370237i
$35$	−1.69202	−0.286004
$36$	−2.04892	−0.341486
$37$	10.8019i	1.77583i	0.460010	+	0.887914i	$0.347846\pi$
$37$	10.8019i	1.77583i	−0.460010	+	0.887914i	$0.652154\pi$
$38$	−6.35690	−1.03122
$39$	0	0
$40$	1.69202	0.267532
$41$	− 2.41789i	− 0.377612i	−0.982014	−	0.188806i	$-0.939538\pi$
$41$	− 2.41789i	− 0.377612i	0.982014	−	0.188806i	$-0.0604617\pi$
$42$	−2.24698	−0.346716
$43$	−4.63102	−0.706224	−0.353112	−	0.935581i	$-0.614877\pi$
$43$	−4.63102	−0.706224	−0.353112	+	0.935581i	$0.614877\pi$
$44$	− 0.445042i	− 0.0670926i
$45$	− 3.46681i	− 0.516802i
$46$	− 0.911854i	− 0.134446i
$47$	− 9.75063i	− 1.42228i	−0.703053	−	0.711138i	$-0.748180\pi$
$47$	− 9.75063i	− 1.42228i	0.703053	−	0.711138i	$-0.251820\pi$
$48$	2.24698	0.324324
$49$	−1.00000	−0.142857
$50$	− 2.13706i	− 0.302226i
$51$	4.85086	0.679256
$52$	0	0
$53$	−8.74094	−1.20066	−0.600330	−	0.799752i	$-0.704964\pi$
$53$	−8.74094	−1.20066	−0.600330	+	0.799752i	$0.704964\pi$
$54$	2.13706i	0.290817i
$55$	0.753020	0.101537
$56$	1.00000	0.133631
$57$	− 14.2838i	− 1.89194i
$58$	− 3.58211i	− 0.470353i
$59$	− 10.1468i	− 1.32099i	−0.750828	−	0.660497i	$-0.770345\pi$
$59$	− 10.1468i	− 1.32099i	0.750828	−	0.660497i	$-0.229655\pi$
$60$	3.80194i	0.490828i
$61$	1.37867	0.176520	0.0882601	−	0.996097i	$-0.471869\pi$
$61$	1.37867	0.176520	0.0882601	+	0.996097i	$0.471869\pi$
$62$	−8.89977	−1.13027
$63$	− 2.04892i	− 0.258139i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	1.00000	0.123091
$67$	− 6.23490i	− 0.761714i	−0.924634	−	0.380857i	$-0.875629\pi$
$67$	− 6.23490i	− 0.761714i	0.924634	−	0.380857i	$-0.124371\pi$
$68$	−2.15883	−0.261797
$69$	2.04892	0.246661
$70$	1.69202i	0.202235i
$71$	5.76271i	0.683908i	0.939717	+	0.341954i	$0.111089\pi$
$71$	5.76271i	0.683908i	−0.939717	+	0.341954i	$0.888911\pi$
$72$	2.04892i	0.241467i
$73$	− 9.93661i	− 1.16299i	−0.813549	−	0.581496i	$-0.802468\pi$
$73$	− 9.93661i	− 1.16299i	0.813549	−	0.581496i	$-0.197532\pi$
$74$	10.8019	1.25570
$75$	4.80194	0.554480
$76$	6.35690i	0.729186i
$77$	0.445042	0.0507172
$78$	0	0
$79$	−6.30127	−0.708949	−0.354474	−	0.935066i	$-0.615340\pi$
$79$	−6.30127	−0.708949	−0.354474	+	0.935066i	$0.615340\pi$
$80$	− 1.69202i	− 0.189174i
$81$	−10.9487	−1.21652
$82$	−2.41789	−0.267012
$83$	2.91185i	0.319617i	0.987148	+	0.159809i	$0.0510878\pi$
$83$	2.91185i	0.319617i	−0.987148	+	0.159809i	$0.948912\pi$
$84$	2.24698i	0.245166i
$85$	− 3.65279i	− 0.396201i
$86$	4.63102i	0.499376i
$87$	8.04892	0.862935
$88$	−0.445042	−0.0474416
$89$	18.1075i	1.91939i	0.281037	+	0.959697i	$0.409322\pi$
$89$	18.1075i	1.91939i	−0.281037	+	0.959697i	$0.590678\pi$
$90$	−3.46681	−0.365434
$91$	0	0
$92$	−0.911854	−0.0950674
$93$	− 19.9976i	− 2.07366i
$94$	−9.75063	−1.00570
$95$	−10.7560	−1.10354
$96$	− 2.24698i	− 0.229331i
$97$	3.30798i	0.335874i	0.985798	+	0.167937i	$0.0537106\pi$
$97$	3.30798i	0.335874i	−0.985798	+	0.167937i	$0.946289\pi$
$98$	1.00000i	0.101015i
$99$	0.911854i	0.0916448i
$100$	−2.13706	−0.213706
$101$	−2.08815	−0.207778	−0.103889	−	0.994589i	$-0.533129\pi$
$101$	−2.08815	−0.207778	−0.103889	+	0.994589i	$0.533129\pi$
$102$	− 4.85086i	− 0.480306i
$103$	11.3013	1.11355	0.556774	−	0.830664i	$-0.312039\pi$
$103$	11.3013	1.11355	0.556774	+	0.830664i	$0.312039\pi$
$104$	0	0
$105$	−3.80194	−0.371031
$106$	8.74094i	0.848995i
$107$	−7.13467	−0.689735	−0.344867	−	0.938651i	$-0.612076\pi$
$107$	−7.13467	−0.689735	−0.344867	+	0.938651i	$0.612076\pi$
$108$	2.13706	0.205639
$109$	6.57002i	0.629294i	0.949209	+	0.314647i	$0.101886\pi$
$109$	6.57002i	0.629294i	−0.949209	+	0.314647i	$0.898114\pi$
$110$	− 0.753020i	− 0.0717977i
$111$	24.2717i	2.30377i
$112$	− 1.00000i	− 0.0944911i
$113$	10.6015	0.997304	0.498652	−	0.866802i	$-0.333829\pi$
$113$	10.6015	0.997304	0.498652	+	0.866802i	$0.333829\pi$
$114$	−14.2838	−1.33780
$115$	− 1.54288i	− 0.143874i
$116$	−3.58211	−0.332590
$117$	0	0
$118$	−10.1468	−0.934084
$119$	− 2.15883i	− 0.197900i
$120$	3.80194	0.347068
$121$	10.8019	0.981994
$122$	− 1.37867i	− 0.124819i
$123$	− 5.43296i	− 0.489874i
$124$	8.89977i	0.799223i
$125$	− 12.0761i	− 1.08012i
$126$	−2.04892	−0.182532
$127$	21.5646	1.91355	0.956776	−	0.290824i	$-0.0939295\pi$
$127$	21.5646	1.91355	0.956776	+	0.290824i	$0.0939295\pi$
$128$	1.00000i	0.0883883i
$129$	−10.4058	−0.916181
$130$	0	0
$131$	3.14675	0.274933	0.137467	−	0.990506i	$-0.456104\pi$
$131$	3.14675	0.274933	0.137467	+	0.990506i	$0.456104\pi$
$132$	− 1.00000i	− 0.0870388i
$133$	−6.35690	−0.551213
$134$	−6.23490	−0.538613
$135$	3.61596i	0.311212i
$136$	2.15883i	0.185118i
$137$	− 3.93362i	− 0.336072i	−0.985781	−	0.168036i	$-0.946257\pi$
$137$	− 3.93362i	− 0.336072i	0.985781	−	0.168036i	$-0.0537425\pi$
$138$	− 2.04892i	− 0.174415i
$139$	−11.2078	−0.950629	−0.475315	−	0.879816i	$-0.657666\pi$
$139$	−11.2078	−0.950629	−0.475315	+	0.879816i	$0.657666\pi$
$140$	1.69202	0.143002
$141$	− 21.9095i	− 1.84511i
$142$	5.76271	0.483596
$143$	0	0
$144$	2.04892	0.170743
$145$	− 6.06100i	− 0.503339i
$146$	−9.93661	−0.822360
$147$	−2.24698	−0.185328
$148$	− 10.8019i	− 0.887914i
$149$	21.8726i	1.79188i	0.444180	+	0.895938i	$0.353495\pi$
$149$	21.8726i	1.79188i	−0.444180	+	0.895938i	$0.646505\pi$
$150$	− 4.80194i	− 0.392077i
$151$	18.3002i	1.48925i	0.667483	+	0.744625i	$0.267372\pi$
$151$	18.3002i	1.48925i	−0.667483	+	0.744625i	$0.732628\pi$
$152$	6.35690	0.515612
$153$	4.42327	0.357600
$154$	− 0.445042i	− 0.0358625i
$155$	−15.0586	−1.20954
$156$	0	0
$157$	−10.8442	−0.865457	−0.432729	−	0.901524i	$-0.642449\pi$
$157$	−10.8442	−0.865457	−0.432729	+	0.901524i	$0.642449\pi$
$158$	6.30127i	0.501302i
$159$	−19.6407	−1.55761
$160$	−1.69202	−0.133766
$161$	− 0.911854i	− 0.0718642i
$162$	10.9487i	0.860210i
$163$	14.4058i	1.12835i	0.825655	+	0.564175i	$0.190806\pi$
$163$	14.4058i	1.12835i	−0.825655	+	0.564175i	$0.809194\pi$
$164$	2.41789i	0.188806i
$165$	1.69202	0.131724
$166$	2.91185	0.226004
$167$	− 13.6963i	− 1.05985i	−0.848043	−	0.529927i	$-0.822219\pi$
$167$	− 13.6963i	− 1.05985i	0.848043	−	0.529927i	$-0.177781\pi$
$168$	2.24698	0.173358
$169$	0	0
$170$	−3.65279	−0.280156
$171$	− 13.0248i	− 0.996028i
$172$	4.63102	0.353112
$173$	4.47650	0.340342	0.170171	−	0.985415i	$-0.445568\pi$
$173$	4.47650	0.340342	0.170171	+	0.985415i	$0.445568\pi$
$174$	− 8.04892i	− 0.610187i
$175$	− 2.13706i	− 0.161547i
$176$	0.445042i	0.0335463i
$177$	− 22.7995i	− 1.71372i
$178$	18.1075	1.35722
$179$	23.0737	1.72461	0.862304	−	0.506392i	$-0.169021\pi$
$179$	23.0737	1.72461	0.862304	+	0.506392i	$0.169021\pi$
$180$	3.46681i	0.258401i
$181$	−8.46980	−0.629555	−0.314777	−	0.949165i	$-0.601930\pi$
$181$	−8.46980	−0.629555	−0.314777	+	0.949165i	$0.601930\pi$
$182$	0	0
$183$	3.09783	0.228999
$184$	0.911854i	0.0672228i
$185$	18.2771	1.34376
$186$	−19.9976	−1.46630
$187$	0.960771i	0.0702586i
$188$	9.75063i	0.711138i
$189$	2.13706i	0.155448i
$190$	10.7560i	0.780323i
$191$	19.7235	1.42714	0.713570	−	0.700583i	$-0.247077\pi$
$191$	19.7235	1.42714	0.713570	+	0.700583i	$0.247077\pi$
$192$	−2.24698	−0.162162
$193$	− 9.15883i	− 0.659267i	−0.944109	−	0.329634i	$-0.893075\pi$
$193$	− 9.15883i	− 0.659267i	0.944109	−	0.329634i	$-0.106925\pi$
$194$	3.30798	0.237499
$195$	0	0
$196$	1.00000	0.0714286
$197$	− 3.73019i	− 0.265765i	−0.991132	−	0.132882i	$-0.957577\pi$
$197$	− 3.73019i	− 0.265765i	0.991132	−	0.132882i	$-0.0424232\pi$
$198$	0.911854	0.0648026
$199$	−19.5918	−1.38883	−0.694413	−	0.719577i	$-0.744336\pi$
$199$	−19.5918	−1.38883	−0.694413	+	0.719577i	$0.744336\pi$
$200$	2.13706i	0.151113i
$201$	− 14.0097i	− 0.988167i
$202$	2.08815i	0.146921i
$203$	− 3.58211i	− 0.251414i
$204$	−4.85086	−0.339628
$205$	−4.09113	−0.285737
$206$	− 11.3013i	− 0.787397i
$207$	1.86831	0.129857
$208$	0	0
$209$	2.82908	0.195692
$210$	3.80194i	0.262359i
$211$	23.3860	1.60996	0.804978	−	0.593305i	$-0.202177\pi$
$211$	23.3860	1.60996	0.804978	+	0.593305i	$0.202177\pi$
$212$	8.74094	0.600330
$213$	12.9487i	0.887230i
$214$	7.13467i	0.487716i
$215$	7.83579i	0.534396i
$216$	− 2.13706i	− 0.145409i
$217$	−8.89977	−0.604156
$218$	6.57002	0.444978
$219$	− 22.3274i	− 1.50874i
$220$	−0.753020	−0.0507686
$221$	0	0
$222$	24.2717	1.62901
$223$	21.7482i	1.45637i	0.685381	+	0.728185i	$0.259636\pi$
$223$	21.7482i	1.45637i	−0.685381	+	0.728185i	$0.740364\pi$
$224$	−1.00000	−0.0668153
$225$	4.37867	0.291911
$226$	− 10.6015i	− 0.705200i
$227$	− 6.55927i	− 0.435354i	−0.976021	−	0.217677i	$-0.930152\pi$
$227$	− 6.55927i	− 0.435354i	0.976021	−	0.217677i	$-0.0698479\pi$
$228$	14.2838i	0.945969i
$229$	− 0.948690i	− 0.0626912i	−0.999509	−	0.0313456i	$-0.990021\pi$
$229$	− 0.948690i	− 0.0626912i	0.999509	−	0.0313456i	$-0.00997925\pi$
$230$	−1.54288	−0.101734
$231$	1.00000	0.0657952
$232$	3.58211i	0.235177i
$233$	15.4765	1.01390	0.506950	−	0.861976i	$-0.330773\pi$
$233$	15.4765	1.01390	0.506950	+	0.861976i	$0.330773\pi$
$234$	0	0
$235$	−16.4983	−1.07623
$236$	10.1468i	0.660497i
$237$	−14.1588	−0.919715
$238$	−2.15883	−0.139936
$239$	− 25.3448i	− 1.63942i	−0.572779	−	0.819710i	$-0.694135\pi$
$239$	− 25.3448i	− 1.63942i	0.572779	−	0.819710i	$-0.305865\pi$
$240$	− 3.80194i	− 0.245414i
$241$	− 28.2664i	− 1.82080i	−0.413733	−	0.910398i	$-0.635775\pi$
$241$	− 28.2664i	− 1.82080i	0.413733	−	0.910398i	$-0.364225\pi$
$242$	− 10.8019i	− 0.694375i
$243$	−18.1903	−1.16691
$244$	−1.37867	−0.0882601
$245$	1.69202i	0.108099i
$246$	−5.43296	−0.346393
$247$	0	0
$248$	8.89977	0.565136
$249$	6.54288i	0.414638i
$250$	−12.0761	−0.763757
$251$	13.7235	0.866218	0.433109	−	0.901341i	$-0.357416\pi$
$251$	13.7235	0.866218	0.433109	+	0.901341i	$0.357416\pi$
$252$	2.04892i	0.129070i
$253$	0.405813i	0.0255133i
$254$	− 21.5646i	− 1.35309i
$255$	− 8.20775i	− 0.513989i
$256$	1.00000	0.0625000
$257$	13.3502	0.832762	0.416381	−	0.909190i	$-0.363298\pi$
$257$	13.3502	0.832762	0.416381	+	0.909190i	$0.363298\pi$
$258$	10.4058i	0.647838i
$259$	10.8019	0.671200
$260$	0	0
$261$	7.33944	0.454300
$262$	− 3.14675i	− 0.194407i
$263$	19.3153	1.19103	0.595515	−	0.803344i	$-0.296948\pi$
$263$	19.3153	1.19103	0.595515	+	0.803344i	$0.296948\pi$
$264$	−1.00000	−0.0615457
$265$	14.7899i	0.908534i
$266$	6.35690i	0.389766i
$267$	40.6872i	2.49002i
$268$	6.23490i	0.380857i
$269$	24.2935	1.48120	0.740601	−	0.671946i	$-0.234541\pi$
$269$	24.2935	1.48120	0.740601	+	0.671946i	$0.234541\pi$
$270$	3.61596	0.220060
$271$	29.0368i	1.76386i	0.471378	+	0.881931i	$0.343757\pi$
$271$	29.0368i	1.76386i	−0.471378	+	0.881931i	$0.656243\pi$
$272$	2.15883	0.130899
$273$	0	0
$274$	−3.93362	−0.237639
$275$	0.951083i	0.0573524i
$276$	−2.04892	−0.123330
$277$	−1.19029	−0.0715177	−0.0357589	−	0.999360i	$-0.511385\pi$
$277$	−1.19029	−0.0715177	−0.0357589	+	0.999360i	$0.511385\pi$
$278$	11.2078i	0.672196i
$279$	− 18.2349i	− 1.09169i
$280$	− 1.69202i	− 0.101118i
$281$	21.2325i	1.26663i	0.773896	+	0.633313i	$0.218305\pi$
$281$	21.2325i	1.26663i	−0.773896	+	0.633313i	$0.781695\pi$
$282$	−21.9095	−1.30469
$283$	−11.6756	−0.694044	−0.347022	−	0.937857i	$-0.612807\pi$
$283$	−11.6756	−0.694044	−0.347022	+	0.937857i	$0.612807\pi$
$284$	− 5.76271i	− 0.341954i
$285$	−24.1685	−1.43162
$286$	0	0
$287$	−2.41789	−0.142724
$288$	− 2.04892i	− 0.120734i
$289$	−12.3394	−0.725849
$290$	−6.06100	−0.355914
$291$	7.43296i	0.435728i
$292$	9.93661i	0.581496i
$293$	14.4101i	0.841848i	0.907096	+	0.420924i	$0.138294\pi$
$293$	14.4101i	0.841848i	−0.907096	+	0.420924i	$0.861706\pi$
$294$	2.24698i	0.131047i
$295$	−17.1685	−0.999590
$296$	−10.8019	−0.627850
$297$	− 0.951083i	− 0.0551874i
$298$	21.8726	1.26705
$299$	0	0
$300$	−4.80194	−0.277240
$301$	4.63102i	0.266928i
$302$	18.3002	1.05306
$303$	−4.69202	−0.269550
$304$	− 6.35690i	− 0.364593i
$305$	− 2.33273i	− 0.133572i
$306$	− 4.42327i	− 0.252862i
$307$	2.55257i	0.145683i	0.997344	+	0.0728413i	$0.0232067\pi$
$307$	2.55257i	0.145683i	−0.997344	+	0.0728413i	$0.976793\pi$
$308$	−0.445042	−0.0253586
$309$	25.3937	1.44460
$310$	15.0586i	0.855271i
$311$	1.63533	0.0927313	0.0463657	−	0.998925i	$-0.485236\pi$
$311$	1.63533	0.0927313	0.0463657	+	0.998925i	$0.485236\pi$
$312$	0	0
$313$	−12.7724	−0.721939	−0.360969	−	0.932578i	$-0.617554\pi$
$313$	−12.7724	−0.721939	−0.360969	+	0.932578i	$0.617554\pi$
$314$	10.8442i	0.611971i
$315$	−3.46681	−0.195333
$316$	6.30127	0.354474
$317$	10.9041i	0.612434i	0.951962	+	0.306217i	$0.0990634\pi$
$317$	10.9041i	0.612434i	−0.951962	+	0.306217i	$0.900937\pi$
$318$	19.6407i	1.10140i
$319$	1.59419i	0.0892573i
$320$	1.69202i	0.0945869i
$321$	−16.0315	−0.894789
$322$	−0.911854	−0.0508156
$323$	− 13.7235i	− 0.763595i
$324$	10.9487	0.608261
$325$	0	0
$326$	14.4058	0.797864
$327$	14.7627i	0.816380i
$328$	2.41789	0.133506
$329$	−9.75063	−0.537569
$330$	− 1.69202i	− 0.0931427i
$331$	− 9.94869i	− 0.546829i	−0.961896	−	0.273415i	$-0.911847\pi$
$331$	− 9.94869i	− 0.546829i	0.961896	−	0.273415i	$-0.0881531\pi$
$332$	− 2.91185i	− 0.159809i
$333$	22.1323i	1.21284i
$334$	−13.6963	−0.749430
$335$	−10.5496	−0.576385
$336$	− 2.24698i	− 0.122583i
$337$	−8.06829	−0.439508	−0.219754	−	0.975555i	$-0.570525\pi$
$337$	−8.06829	−0.439508	−0.219754	+	0.975555i	$0.570525\pi$
$338$	0	0
$339$	23.8213	1.29380
$340$	3.65279i	0.198101i
$341$	3.96077	0.214488
$342$	−13.0248	−0.704298
$343$	1.00000i	0.0539949i
$344$	− 4.63102i	− 0.249688i
$345$	− 3.46681i	− 0.186647i
$346$	− 4.47650i	− 0.240658i
$347$	−33.1769	−1.78103	−0.890514	−	0.454955i	$-0.849655\pi$
$347$	−33.1769	−1.78103	−0.890514	+	0.454955i	$0.849655\pi$
$348$	−8.04892	−0.431467
$349$	24.6872i	1.32148i	0.750616	+	0.660739i	$0.229757\pi$
$349$	24.6872i	1.32148i	−0.750616	+	0.660739i	$0.770243\pi$
$350$	−2.13706	−0.114231
$351$	0	0
$352$	0.445042	0.0237208
$353$	− 22.5754i	− 1.20157i	−0.799412	−	0.600784i	$-0.794855\pi$
$353$	− 22.5754i	− 1.20157i	0.799412	−	0.600784i	$-0.205145\pi$
$354$	−22.7995	−1.21178
$355$	9.75063	0.517510
$356$	− 18.1075i	− 0.959697i
$357$	− 4.85086i	− 0.256734i
$358$	− 23.0737i	− 1.21948i
$359$	2.30798i	0.121810i	0.998144	+	0.0609052i	$0.0193987\pi$
$359$	2.30798i	0.121810i	−0.998144	+	0.0609052i	$0.980601\pi$
$360$	3.46681	0.182717
$361$	−21.4101	−1.12685
$362$	8.46980i	0.445163i
$363$	24.2717	1.27394
$364$	0	0
$365$	−16.8130	−0.880030
$366$	− 3.09783i	− 0.161926i
$367$	25.1685	1.31379	0.656893	−	0.753984i	$-0.271870\pi$
$367$	25.1685	1.31379	0.656893	+	0.753984i	$0.271870\pi$
$368$	0.911854	0.0475337
$369$	− 4.95407i	− 0.257898i
$370$	− 18.2771i	− 0.950182i
$371$	8.74094i	0.453807i
$372$	19.9976i	1.03683i
$373$	11.6407	0.602733	0.301367	−	0.953508i	$-0.402557\pi$
$373$	11.6407	0.602733	0.301367	+	0.953508i	$0.402557\pi$
$374$	0.960771	0.0496803
$375$	− 27.1347i	− 1.40123i
$376$	9.75063	0.502850
$377$	0	0
$378$	2.13706	0.109919
$379$	16.5623i	0.850746i	0.905018	+	0.425373i	$0.139857\pi$
$379$	16.5623i	0.850746i	−0.905018	+	0.425373i	$0.860143\pi$
$380$	10.7560	0.551771
$381$	48.4553	2.48244
$382$	− 19.7235i	− 1.00914i
$383$	7.39506i	0.377870i	0.981990	+	0.188935i	$0.0605035\pi$
$383$	7.39506i	0.377870i	−0.981990	+	0.188935i	$0.939496\pi$
$384$	2.24698i	0.114666i
$385$	− 0.753020i	− 0.0383775i
$386$	−9.15883	−0.466172
$387$	−9.48858	−0.482332
$388$	− 3.30798i	− 0.167937i
$389$	−23.4674	−1.18984	−0.594922	−	0.803783i	$-0.702817\pi$
$389$	−23.4674	−1.18984	−0.594922	+	0.803783i	$0.702817\pi$
$390$	0	0
$391$	1.96854	0.0995534
$392$	− 1.00000i	− 0.0505076i
$393$	7.07069	0.356669
$394$	−3.73019	−0.187924
$395$	10.6619i	0.536458i
$396$	− 0.911854i	− 0.0458224i
$397$	− 0.0499823i	− 0.00250854i	−0.999999	−	0.00125427i	$-0.999601\pi$
$397$	− 0.0499823i	− 0.00250854i	0.999999	−	0.00125427i	$-0.000399246\pi$
$398$	19.5918i	0.982048i
$399$	−14.2838	−0.715085
$400$	2.13706	0.106853
$401$	4.84010i	0.241703i	0.992671	+	0.120852i	$0.0385625\pi$
$401$	4.84010i	0.241703i	−0.992671	+	0.120852i	$0.961437\pi$
$402$	−14.0097	−0.698740
$403$	0	0
$404$	2.08815	0.103889
$405$	18.5254i	0.920535i
$406$	−3.58211	−0.177777
$407$	−4.80731	−0.238290
$408$	4.85086i	0.240153i
$409$	− 29.8866i	− 1.47780i	−0.673816	−	0.738899i	$-0.735346\pi$
$409$	− 29.8866i	− 1.47780i	0.673816	−	0.738899i	$-0.264654\pi$
$410$	4.09113i	0.202047i
$411$	− 8.83877i	− 0.435985i
$412$	−11.3013	−0.556774
$413$	−10.1468	−0.499289
$414$	− 1.86831i	− 0.0918226i
$415$	4.92692	0.241853
$416$	0	0
$417$	−25.1836	−1.23325
$418$	− 2.82908i	− 0.138375i
$419$	8.13898	0.397615	0.198808	−	0.980039i	$-0.436293\pi$
$419$	8.13898	0.397615	0.198808	+	0.980039i	$0.436293\pi$
$420$	3.80194	0.185516
$421$	17.0398i	0.830470i	0.909714	+	0.415235i	$0.136301\pi$
$421$	17.0398i	0.830470i	−0.909714	+	0.415235i	$0.863699\pi$
$422$	− 23.3860i	− 1.13841i
$423$	− 19.9782i	− 0.971375i
$424$	− 8.74094i	− 0.424498i
$425$	4.61356	0.223791
$426$	12.9487	0.627366
$427$	− 1.37867i	− 0.0667183i
$428$	7.13467	0.344867
$429$	0	0
$430$	7.83579	0.377875
$431$	6.95779i	0.335145i	0.985860	+	0.167572i	$0.0535928\pi$
$431$	6.95779i	0.335145i	−0.985860	+	0.167572i	$0.946407\pi$
$432$	−2.13706	−0.102820
$433$	6.80731	0.327139	0.163569	−	0.986532i	$-0.447699\pi$
$433$	6.80731	0.327139	0.163569	+	0.986532i	$0.447699\pi$
$434$	8.89977i	0.427203i
$435$	− 13.6189i	− 0.652978i
$436$	− 6.57002i	− 0.314647i
$437$	− 5.79656i	− 0.277287i
$438$	−22.3274	−1.06684
$439$	−26.3351	−1.25691	−0.628453	−	0.777847i	$-0.716312\pi$
$439$	−26.3351	−1.25691	−0.628453	+	0.777847i	$0.716312\pi$
$440$	0.753020i	0.0358988i
$441$	−2.04892	−0.0975675
$442$	0	0
$443$	7.32437	0.347991	0.173996	−	0.984746i	$-0.444332\pi$
$443$	7.32437	0.347991	0.173996	+	0.984746i	$0.444332\pi$
$444$	− 24.2717i	− 1.15189i
$445$	30.6383	1.45240
$446$	21.7482	1.02981
$447$	49.1473i	2.32459i
$448$	1.00000i	0.0472456i
$449$	− 41.4228i	− 1.95486i	−0.211253	−	0.977431i	$-0.567755\pi$
$449$	− 41.4228i	− 1.95486i	0.211253	−	0.977431i	$-0.432245\pi$
$450$	− 4.37867i	− 0.206412i
$451$	1.07606	0.0506699
$452$	−10.6015	−0.498652
$453$	41.1202i	1.93200i
$454$	−6.55927	−0.307842
$455$	0	0
$456$	14.2838	0.668901
$457$	− 11.7125i	− 0.547886i	−0.961746	−	0.273943i	$-0.911672\pi$
$457$	− 11.7125i	− 0.547886i	0.961746	−	0.273943i	$-0.0883280\pi$
$458$	−0.948690	−0.0443294
$459$	−4.61356	−0.215343
$460$	1.54288i	0.0719370i
$461$	20.6461i	0.961584i	0.876835	+	0.480792i	$0.159651\pi$
$461$	20.6461i	0.961584i	−0.876835	+	0.480792i	$0.840349\pi$
$462$	− 1.00000i	− 0.0465242i
$463$	10.9312i	0.508017i	0.967202	+	0.254009i	$0.0817492\pi$
$463$	10.9312i	0.508017i	−0.967202	+	0.254009i	$0.918251\pi$
$464$	3.58211	0.166295
$465$	−33.8364	−1.56912
$466$	− 15.4765i	− 0.716935i
$467$	−15.4450	−0.714711	−0.357356	−	0.933968i	$-0.616322\pi$
$467$	−15.4450	−0.714711	−0.357356	+	0.933968i	$0.616322\pi$
$468$	0	0
$469$	−6.23490	−0.287901
$470$	16.4983i	0.761008i
$471$	−24.3666	−1.12275
$472$	10.1468	0.467042
$473$	− 2.06100i	− 0.0947648i
$474$	14.1588i	0.650337i
$475$	− 13.5851i	− 0.623327i
$476$	2.15883i	0.0989500i
$477$	−17.9095	−0.820018
$478$	−25.3448	−1.15924
$479$	25.2828i	1.15520i	0.816321	+	0.577599i	$0.196010\pi$
$479$	25.2828i	1.15520i	−0.816321	+	0.577599i	$0.803990\pi$
$480$	−3.80194	−0.173534
$481$	0	0
$482$	−28.2664	−1.28750
$483$	− 2.04892i	− 0.0932290i
$484$	−10.8019	−0.490997
$485$	5.59717	0.254154
$486$	18.1903i	0.825128i
$487$	− 1.01938i	− 0.0461924i	−0.999733	−	0.0230962i	$-0.992648\pi$
$487$	− 1.01938i	− 0.0461924i	0.999733	−	0.0230962i	$-0.00735240\pi$
$488$	1.37867i	0.0624093i
$489$	32.3696i	1.46380i
$490$	1.69202	0.0764377
$491$	36.7198	1.65714	0.828570	−	0.559886i	$-0.189155\pi$
$491$	36.7198	1.65714	0.828570	+	0.559886i	$0.189155\pi$
$492$	5.43296i	0.244937i
$493$	7.73317	0.348284
$494$	0	0
$495$	1.54288	0.0693471
$496$	− 8.89977i	− 0.399612i
$497$	5.76271	0.258493
$498$	6.54288	0.293193
$499$	41.4685i	1.85638i	0.372102	+	0.928192i	$0.378637\pi$
$499$	41.4685i	1.85638i	−0.372102	+	0.928192i	$0.621363\pi$
$500$	12.0761i	0.540058i
$501$	− 30.7754i	− 1.37494i
$502$	− 13.7235i	− 0.612509i
$503$	−2.15346	−0.0960179	−0.0480089	−	0.998847i	$-0.515288\pi$
$503$	−2.15346	−0.0960179	−0.0480089	+	0.998847i	$0.515288\pi$
$504$	2.04892	0.0912660
$505$	3.53319i	0.157225i
$506$	0.405813	0.0180406
$507$	0	0
$508$	−21.5646	−0.956776
$509$	7.62565i	0.338001i	0.985616	+	0.169000i	$0.0540539\pi$
$509$	7.62565i	0.338001i	−0.985616	+	0.169000i	$0.945946\pi$
$510$	−8.20775	−0.363445
$511$	−9.93661	−0.439570
$512$	− 1.00000i	− 0.0441942i
$513$	13.5851i	0.599796i
$514$	− 13.3502i	− 0.588852i
$515$	− 19.1220i	− 0.842616i
$516$	10.4058	0.458090
$517$	4.33944	0.190848
$518$	− 10.8019i	− 0.474610i
$519$	10.0586	0.441524
$520$	0	0
$521$	20.8377	0.912917	0.456458	−	0.889745i	$-0.349118\pi$
$521$	20.8377	0.912917	0.456458	+	0.889745i	$0.349118\pi$
$522$	− 7.33944i	− 0.321238i
$523$	38.3491	1.67689	0.838445	−	0.544986i	$-0.183465\pi$
$523$	38.3491	1.67689	0.838445	+	0.544986i	$0.183465\pi$
$524$	−3.14675	−0.137467
$525$	− 4.80194i	− 0.209574i
$526$	− 19.3153i	− 0.842186i
$527$	− 19.2131i	− 0.836937i
$528$	1.00000i	0.0435194i
$529$	−22.1685	−0.963849
$530$	14.7899	0.642430
$531$	− 20.7899i	− 0.902203i
$532$	6.35690	0.275606
$533$	0	0
$534$	40.6872	1.76071
$535$	12.0720i	0.521919i
$536$	6.23490	0.269307
$537$	51.8461	2.23732
$538$	− 24.2935i	− 1.04737i
$539$	− 0.445042i	− 0.0191693i
$540$	− 3.61596i	− 0.155606i
$541$	− 5.46740i	− 0.235062i	−0.993069	−	0.117531i	$-0.962502\pi$
$541$	− 5.46740i	− 0.235062i	0.993069	−	0.117531i	$-0.0374980\pi$
$542$	29.0368	1.24724
$543$	−19.0315	−0.816718
$544$	− 2.15883i	− 0.0925592i
$545$	11.1166	0.476184
$546$	0	0
$547$	−6.26636	−0.267930	−0.133965	−	0.990986i	$-0.542771\pi$
$547$	−6.26636	−0.267930	−0.133965	+	0.990986i	$0.542771\pi$
$548$	3.93362i	0.168036i
$549$	2.82477	0.120558
$550$	0.951083	0.0405543
$551$	− 22.7711i	− 0.970080i
$552$	2.04892i	0.0872077i
$553$	6.30127i	0.267957i
$554$	1.19029i	0.0505707i
$555$	41.0683	1.74325
$556$	11.2078	0.475315
$557$	− 23.1371i	− 0.980349i	−0.871624	−	0.490174i	$-0.836933\pi$
$557$	− 23.1371i	− 0.980349i	0.871624	−	0.490174i	$-0.163067\pi$
$558$	−18.2349	−0.771945
$559$	0	0
$560$	−1.69202	−0.0715010
$561$	2.15883i	0.0911460i
$562$	21.2325	0.895639
$563$	−1.21685	−0.0512841	−0.0256420	−	0.999671i	$-0.508163\pi$
$563$	−1.21685	−0.0512841	−0.0256420	+	0.999671i	$0.508163\pi$
$564$	21.9095i	0.922555i
$565$	− 17.9379i	− 0.754655i
$566$	11.6756i	0.490763i
$567$	10.9487i	0.459802i
$568$	−5.76271	−0.241798
$569$	28.5816	1.19820	0.599102	−	0.800673i	$-0.295524\pi$
$569$	28.5816	1.19820	0.599102	+	0.800673i	$0.295524\pi$
$570$	24.1685i	1.01231i
$571$	32.1825	1.34680	0.673398	−	0.739280i	$-0.264834\pi$
$571$	32.1825	1.34680	0.673398	+	0.739280i	$0.264834\pi$
$572$	0	0
$573$	44.3183	1.85142
$574$	2.41789i	0.100921i
$575$	1.94869	0.0812660
$576$	−2.04892	−0.0853716
$577$	2.94810i	0.122731i	0.998115	+	0.0613655i	$0.0195455\pi$
$577$	2.94810i	0.122731i	−0.998115	+	0.0613655i	$0.980454\pi$
$578$	12.3394i	0.513253i
$579$	− 20.5797i	− 0.855264i
$580$	6.06100i	0.251669i
$581$	2.91185	0.120804
$582$	7.43296	0.308106
$583$	− 3.89008i	− 0.161111i
$584$	9.93661	0.411180
$585$	0	0
$586$	14.4101	0.595277
$587$	− 2.77538i	− 0.114552i	−0.998358	−	0.0572761i	$-0.981758\pi$
$587$	− 2.77538i	− 0.114552i	0.998358	−	0.0572761i	$-0.0182415\pi$
$588$	2.24698	0.0926639
$589$	−56.5749	−2.33113
$590$	17.1685i	0.706817i
$591$	− 8.38165i	− 0.344775i
$592$	10.8019i	0.443957i
$593$	21.5241i	0.883888i	0.897043	+	0.441944i	$0.145711\pi$
$593$	21.5241i	0.883888i	−0.897043	+	0.441944i	$0.854289\pi$
$594$	−0.951083	−0.0390234
$595$	−3.65279	−0.149750
$596$	− 21.8726i	− 0.895938i
$597$	−44.0224	−1.80172
$598$	0	0
$599$	23.3230	0.952954	0.476477	−	0.879187i	$-0.341914\pi$
$599$	23.3230	0.952954	0.476477	+	0.879187i	$0.341914\pi$
$600$	4.80194i	0.196038i
$601$	15.7429	0.642165	0.321082	−	0.947051i	$-0.395953\pi$
$601$	15.7429	0.642165	0.321082	+	0.947051i	$0.395953\pi$
$602$	4.63102	0.188746
$603$	− 12.7748i	− 0.520230i
$604$	− 18.3002i	− 0.744625i
$605$	− 18.2771i	− 0.743070i
$606$	4.69202i	0.190600i
$607$	−19.7845	−0.803027	−0.401514	−	0.915853i	$-0.631516\pi$
$607$	−19.7845	−0.803027	−0.401514	+	0.915853i	$0.631516\pi$
$608$	−6.35690	−0.257806
$609$	− 8.04892i	− 0.326159i
$610$	−2.33273	−0.0944496
$611$	0	0
$612$	−4.42327	−0.178800
$613$	7.24996i	0.292823i	0.989224	+	0.146412i	$0.0467724\pi$
$613$	7.24996i	0.292823i	−0.989224	+	0.146412i	$0.953228\pi$
$614$	2.55257	0.103013
$615$	−9.19269	−0.370685
$616$	0.445042i	0.0179312i
$617$	36.7415i	1.47916i	0.673070	+	0.739579i	$0.264975\pi$
$617$	36.7415i	1.47916i	−0.673070	+	0.739579i	$0.735025\pi$
$618$	− 25.3937i	− 1.02149i
$619$	− 17.4959i	− 0.703219i	−0.936147	−	0.351609i	$-0.885635\pi$
$619$	− 17.4959i	− 0.703219i	0.936147	−	0.351609i	$-0.114365\pi$
$620$	15.0586	0.604768
$621$	−1.94869	−0.0781982
$622$	− 1.63533i	− 0.0655709i
$623$	18.1075	0.725463
$624$	0	0
$625$	−9.74764	−0.389906
$626$	12.7724i	0.510488i
$627$	6.35690	0.253870
$628$	10.8442	0.432729
$629$	23.3196i	0.929813i
$630$	3.46681i	0.138121i
$631$	31.4644i	1.25258i	0.779591	+	0.626289i	$0.215427\pi$
$631$	31.4644i	1.25258i	−0.779591	+	0.626289i	$0.784573\pi$
$632$	− 6.30127i	− 0.250651i
$633$	52.5478	2.08859
$634$	10.9041	0.433057
$635$	− 36.4878i	− 1.44798i
$636$	19.6407	0.778805
$637$	0	0
$638$	1.59419	0.0631145
$639$	11.8073i	0.467090i
$640$	1.69202	0.0668830
$641$	24.6799	0.974799	0.487400	−	0.873179i	$-0.337946\pi$
$641$	24.6799	0.974799	0.487400	+	0.873179i	$0.337946\pi$
$642$	16.0315i	0.632711i
$643$	11.1588i	0.440061i	0.975493	+	0.220031i	$0.0706158\pi$
$643$	11.1588i	0.440061i	−0.975493	+	0.220031i	$0.929384\pi$
$644$	0.911854i	0.0359321i
$645$	17.6069i	0.693269i
$646$	−13.7235	−0.539943
$647$	−14.8791	−0.584956	−0.292478	−	0.956272i	$-0.594480\pi$
$647$	−14.8791	−0.584956	−0.292478	+	0.956272i	$0.594480\pi$
$648$	− 10.9487i	− 0.430105i
$649$	4.51573	0.177258
$650$	0	0
$651$	−19.9976	−0.783768
$652$	− 14.4058i	− 0.564175i
$653$	−5.13600	−0.200987	−0.100494	−	0.994938i	$-0.532042\pi$
$653$	−5.13600	−0.200987	−0.100494	+	0.994938i	$0.532042\pi$
$654$	14.7627	0.577268
$655$	− 5.32437i	− 0.208040i
$656$	− 2.41789i	− 0.0944029i
$657$	− 20.3593i	− 0.794292i
$658$	9.75063i	0.380119i
$659$	8.42998	0.328385	0.164193	−	0.986428i	$-0.447498\pi$
$659$	8.42998	0.328385	0.164193	+	0.986428i	$0.447498\pi$
$660$	−1.69202	−0.0658618
$661$	20.3793i	0.792661i	0.918108	+	0.396331i	$0.129717\pi$
$661$	20.3793i	0.792661i	−0.918108	+	0.396331i	$0.870283\pi$
$662$	−9.94869	−0.386667
$663$	0	0
$664$	−2.91185	−0.113002
$665$	10.7560i	0.417100i
$666$	22.1323	0.857608
$667$	3.26636	0.126474
$668$	13.6963i	0.529927i
$669$	48.8678i	1.88934i
$670$	10.5496i	0.407566i
$671$	0.613564i	0.0236864i
$672$	−2.24698	−0.0866791
$673$	−21.6383	−0.834096	−0.417048	−	0.908884i	$-0.636935\pi$
$673$	−21.6383	−0.834096	−0.417048	+	0.908884i	$0.636935\pi$
$674$	8.06829i	0.310779i
$675$	−4.56704	−0.175785
$676$	0	0
$677$	40.5351	1.55789	0.778945	−	0.627092i	$-0.215755\pi$
$677$	40.5351	1.55789	0.778945	+	0.627092i	$0.215755\pi$
$678$	− 23.8213i	− 0.914852i
$679$	3.30798	0.126949
$680$	3.65279	0.140078
$681$	− 14.7385i	− 0.564782i
$682$	− 3.96077i	− 0.151666i
$683$	21.6969i	0.830210i	0.909774	+	0.415105i	$0.136255\pi$
$683$	21.6969i	0.830210i	−0.909774	+	0.415105i	$0.863745\pi$
$684$	13.0248i	0.498014i
$685$	−6.65578	−0.254304
$686$	1.00000	0.0381802
$687$	− 2.13169i	− 0.0813289i
$688$	−4.63102	−0.176556
$689$	0	0
$690$	−3.46681	−0.131979
$691$	− 28.0610i	− 1.06749i	−0.845645	−	0.533745i	$-0.820784\pi$
$691$	− 28.0610i	− 1.06749i	0.845645	−	0.533745i	$-0.179216\pi$
$692$	−4.47650	−0.170171
$693$	0.911854	0.0346385
$694$	33.1769i	1.25938i
$695$	18.9638i	0.719336i
$696$	8.04892i	0.305093i
$697$	− 5.21983i	− 0.197715i
$698$	24.6872	0.934426
$699$	34.7754	1.31533
$700$	2.13706i	0.0807734i
$701$	13.7380	0.518875	0.259438	−	0.965760i	$-0.416463\pi$
$701$	13.7380	0.518875	0.259438	+	0.965760i	$0.416463\pi$
$702$	0	0
$703$	68.6668	2.58982
$704$	− 0.445042i	− 0.0167731i
$705$	−37.0713	−1.39619
$706$	−22.5754	−0.849636
$707$	2.08815i	0.0785328i
$708$	22.7995i	0.856859i
$709$	− 23.0586i	− 0.865984i	−0.901398	−	0.432992i	$-0.857458\pi$
$709$	− 23.0586i	− 0.865984i	0.901398	−	0.432992i	$-0.142542\pi$
$710$	− 9.75063i	− 0.365935i
$711$	−12.9108	−0.484192
$712$	−18.1075	−0.678608
$713$	− 8.11529i	− 0.303920i
$714$	−4.85086	−0.181539
$715$	0	0
$716$	−23.0737	−0.862304
$717$	− 56.9493i	− 2.12681i
$718$	2.30798	0.0861330
$719$	−22.8750	−0.853094	−0.426547	−	0.904465i	$-0.640270\pi$
$719$	−22.8750	−0.853094	−0.426547	+	0.904465i	$0.640270\pi$
$720$	− 3.46681i	− 0.129200i
$721$	− 11.3013i	− 0.420881i
$722$	21.4101i	0.796802i
$723$	− 63.5139i	− 2.36211i
$724$	8.46980	0.314777
$725$	7.65519	0.284306
$726$	− 24.2717i	− 0.900809i
$727$	−9.76377	−0.362118	−0.181059	−	0.983472i	$-0.557953\pi$
$727$	−9.76377	−0.362118	−0.181059	+	0.983472i	$0.557953\pi$
$728$	0	0
$729$	−8.02715	−0.297302
$730$	16.8130i	0.622275i
$731$	−9.99761	−0.369775
$732$	−3.09783	−0.114499
$733$	− 2.65519i	− 0.0980715i	−0.998797	−	0.0490358i	$-0.984385\pi$
$733$	− 2.65519i	− 0.0980715i	0.998797	−	0.0490358i	$-0.0156148\pi$
$734$	− 25.1685i	− 0.928987i
$735$	3.80194i	0.140237i
$736$	− 0.911854i	− 0.0336114i
$737$	2.77479	0.102211
$738$	−4.95407	−0.182362
$739$	0.135144i	0.00497137i	0.999997	+	0.00248568i	$0.000791218\pi$
$739$	0.135144i	0.00497137i	−0.999997	+	0.00248568i	$0.999209\pi$
$740$	−18.2771	−0.671880
$741$	0	0
$742$	8.74094	0.320890
$743$	38.1943i	1.40121i	0.713547	+	0.700607i	$0.247087\pi$
$743$	38.1943i	1.40121i	−0.713547	+	0.700607i	$0.752913\pi$
$744$	19.9976	0.733148
$745$	37.0090	1.35590
$746$	− 11.6407i	− 0.426197i
$747$	5.96615i	0.218290i
$748$	− 0.960771i	− 0.0351293i
$749$	7.13467i	0.260695i
$750$	−27.1347	−0.990818
$751$	−31.9614	−1.16629	−0.583143	−	0.812369i	$-0.698177\pi$
$751$	−31.9614	−1.16629	−0.583143	+	0.812369i	$0.698177\pi$
$752$	− 9.75063i	− 0.355569i
$753$	30.8364	1.12374
$754$	0	0
$755$	30.9643	1.12691
$756$	− 2.13706i	− 0.0777242i
$757$	−51.4258	−1.86910	−0.934551	−	0.355829i	$-0.884198\pi$
$757$	−51.4258	−1.86910	−0.934551	+	0.355829i	$0.884198\pi$
$758$	16.5623	0.601568
$759$	0.911854i	0.0330982i
$760$	− 10.7560i	− 0.390161i
$761$	29.4782i	1.06858i	0.845301	+	0.534291i	$0.179421\pi$
$761$	29.4782i	1.06858i	−0.845301	+	0.534291i	$0.820579\pi$
$762$	− 48.4553i	− 1.75535i
$763$	6.57002	0.237851
$764$	−19.7235	−0.713570
$765$	− 7.48427i	− 0.270594i
$766$	7.39506	0.267194
$767$	0	0
$768$	2.24698	0.0810809
$769$	24.7114i	0.891116i	0.895253	+	0.445558i	$0.146995\pi$
$769$	24.7114i	0.891116i	−0.895253	+	0.445558i	$0.853005\pi$
$770$	−0.753020	−0.0271370
$771$	29.9976	1.08034
$772$	9.15883i	0.329634i
$773$	22.3032i	0.802190i	0.916036	+	0.401095i	$0.131370\pi$
$773$	22.3032i	0.802190i	−0.916036	+	0.401095i	$0.868630\pi$
$774$	9.48858i	0.341060i
$775$	− 19.0194i	− 0.683196i
$776$	−3.30798	−0.118750
$777$	24.2717	0.870744
$778$	23.4674i	0.841347i
$779$	−15.3703	−0.550698
$780$	0	0
$781$	−2.56465	−0.0917703
$782$	− 1.96854i	− 0.0703949i
$783$	−7.65519	−0.273574
$784$	−1.00000	−0.0357143
$785$	18.3485i	0.654887i
$786$	− 7.07069i	− 0.252203i
$787$	− 16.4896i	− 0.587792i	−0.955837	−	0.293896i	$-0.905048\pi$
$787$	− 16.4896i	− 0.587792i	0.955837	−	0.293896i	$-0.0949520\pi$
$788$	3.73019i	0.132882i
$789$	43.4010	1.54512
$790$	10.6619	0.379333
$791$	− 10.6015i	− 0.376945i
$792$	−0.911854	−0.0324013
$793$	0	0
$794$	−0.0499823	−0.00177380
$795$	33.2325i	1.17864i
$796$	19.5918	0.694413
$797$	1.00106	0.0354595	0.0177298	−	0.999843i	$-0.494356\pi$
$797$	1.00106	0.0354595	0.0177298	+	0.999843i	$0.494356\pi$
$798$	14.2838i	0.505642i
$799$	− 21.0500i	− 0.744695i
$800$	− 2.13706i	− 0.0755566i
$801$	37.1008i	1.31089i
$802$	4.84010	0.170910
$803$	4.42221	0.156056
$804$	14.0097i	0.494084i
$805$	−1.54288	−0.0543793
$806$	0	0
$807$	54.5870	1.92155
$808$	− 2.08815i	− 0.0734607i
$809$	−49.9172	−1.75500	−0.877498	−	0.479580i	$-0.840789\pi$
$809$	−49.9172	−1.75500	−0.877498	+	0.479580i	$0.840789\pi$
$810$	18.5254	0.650917
$811$	− 32.9554i	− 1.15722i	−0.815604	−	0.578610i	$-0.803595\pi$
$811$	− 32.9554i	− 1.15722i	0.815604	−	0.578610i	$-0.196405\pi$
$812$	3.58211i	0.125707i
$813$	65.2452i	2.28825i
$814$	4.80731i	0.168496i
$815$	24.3749	0.853817
$816$	4.85086	0.169814
$817$	29.4389i	1.02994i
$818$	−29.8866	−1.04496
$819$	0	0
$820$	4.09113	0.142868
$821$	− 29.3129i	− 1.02303i	−0.859275	−	0.511513i	$-0.829085\pi$
$821$	− 29.3129i	− 1.02303i	0.859275	−	0.511513i	$-0.170915\pi$
$822$	−8.83877	−0.308288
$823$	11.3013	0.393938	0.196969	−	0.980410i	$-0.436890\pi$
$823$	11.3013	0.393938	0.196969	+	0.980410i	$0.436890\pi$
$824$	11.3013i	0.393699i
$825$	2.13706i	0.0744030i
$826$	10.1468i	0.353051i
$827$	− 32.8568i	− 1.14254i	−0.820761	−	0.571272i	$-0.806450\pi$
$827$	− 32.8568i	− 1.14254i	0.820761	−	0.571272i	$-0.193550\pi$
$828$	−1.86831	−0.0649284
$829$	−36.6853	−1.27413	−0.637067	−	0.770809i	$-0.719852\pi$
$829$	−36.6853	−1.27413	−0.637067	+	0.770809i	$0.719852\pi$
$830$	− 4.92692i	− 0.171016i
$831$	−2.67456	−0.0927796
$832$	0	0
$833$	−2.15883	−0.0747992
$834$	25.1836i	0.872036i
$835$	−23.1745	−0.801986
$836$	−2.82908	−0.0978459
$837$	19.0194i	0.657406i
$838$	− 8.13898i	− 0.281156i
$839$	− 23.7687i	− 0.820586i	−0.911954	−	0.410293i	$-0.865426\pi$
$839$	− 23.7687i	− 0.820586i	0.911954	−	0.410293i	$-0.134574\pi$
$840$	− 3.80194i	− 0.131179i
$841$	−16.1685	−0.557535
$842$	17.0398	0.587231
$843$	47.7090i	1.64319i
$844$	−23.3860	−0.804978
$845$	0	0
$846$	−19.9782	−0.686866
$847$	− 10.8019i	− 0.371159i
$848$	−8.74094	−0.300165
$849$	−26.2349	−0.900379
$850$	− 4.61356i	− 0.158244i
$851$	9.84979i	0.337646i
$852$	− 12.9487i	− 0.443615i
$853$	33.6437i	1.15194i	0.817471	+	0.575969i	$0.195375\pi$
$853$	33.6437i	1.15194i	−0.817471	+	0.575969i	$0.804625\pi$
$854$	−1.37867	−0.0471770
$855$	−22.0382	−0.753689
$856$	− 7.13467i	− 0.243858i
$857$	35.2646	1.20461	0.602307	−	0.798264i	$-0.294248\pi$
$857$	35.2646	1.20461	0.602307	+	0.798264i	$0.294248\pi$
$858$	0	0
$859$	9.80433	0.334519	0.167260	−	0.985913i	$-0.446508\pi$
$859$	9.80433	0.334519	0.167260	+	0.985913i	$0.446508\pi$
$860$	− 7.83579i	− 0.267198i
$861$	−5.43296	−0.185155
$862$	6.95779	0.236983
$863$	− 34.2586i	− 1.16618i	−0.812409	−	0.583088i	$-0.801844\pi$
$863$	− 34.2586i	− 1.16618i	0.812409	−	0.583088i	$-0.198156\pi$
$864$	2.13706i	0.0727044i
$865$	− 7.57434i	− 0.257535i
$866$	− 6.80731i	− 0.231322i
$867$	−27.7265	−0.941640
$868$	8.89977	0.302078
$869$	− 2.80433i	− 0.0951304i
$870$	−13.6189	−0.461725
$871$	0	0
$872$	−6.57002	−0.222489
$873$	6.77777i	0.229393i
$874$	−5.79656	−0.196072
$875$	−12.0761	−0.408245
$876$	22.3274i	0.754371i
$877$	28.5174i	0.962964i	0.876456	+	0.481482i	$0.159901\pi$
$877$	28.5174i	0.962964i	−0.876456	+	0.481482i	$0.840099\pi$
$878$	26.3351i	0.888767i
$879$	32.3793i	1.09213i
$880$	0.753020	0.0253843
$881$	55.3193	1.86376	0.931878	−	0.362773i	$-0.118170\pi$
$881$	55.3193	1.86376	0.931878	+	0.362773i	$0.118170\pi$
$882$	2.04892i	0.0689906i
$883$	−0.815938	−0.0274585	−0.0137293	−	0.999906i	$-0.504370\pi$
$883$	−0.815938	−0.0274585	−0.0137293	+	0.999906i	$0.504370\pi$
$884$	0	0
$885$	−38.5773	−1.29676
$886$	− 7.32437i	− 0.246067i
$887$	39.8713	1.33875	0.669374	−	0.742926i	$-0.266563\pi$
$887$	39.8713	1.33875	0.669374	+	0.742926i	$0.266563\pi$
$888$	−24.2717	−0.814506
$889$	− 21.5646i	− 0.723255i
$890$	− 30.6383i	− 1.02700i
$891$	− 4.87263i	− 0.163239i
$892$	− 21.7482i	− 0.728185i
$893$	−61.9837	−2.07421
$894$	49.1473	1.64373
$895$	− 39.0411i	− 1.30500i
$896$	1.00000	0.0334077
$897$	0	0
$898$	−41.4228	−1.38230
$899$	− 31.8799i	− 1.06325i
$900$	−4.37867	−0.145956
$901$	−18.8702	−0.628659
$902$	− 1.07606i	− 0.0358290i
$903$	10.4058i	0.346284i
$904$	10.6015i	0.352600i
$905$	14.3311i	0.476381i
$906$	41.1202	1.36613
$907$	−5.54719	−0.184191	−0.0920957	−	0.995750i	$-0.529357\pi$
$907$	−5.54719	−0.184191	−0.0920957	+	0.995750i	$0.529357\pi$
$908$	6.55927i	0.217677i
$909$	−4.27844	−0.141907
$910$	0	0
$911$	12.3357	0.408701	0.204350	−	0.978898i	$-0.434492\pi$
$911$	12.3357	0.408701	0.204350	+	0.978898i	$0.434492\pi$
$912$	− 14.2838i	− 0.472984i
$913$	−1.29590	−0.0428879
$914$	−11.7125	−0.387414
$915$	− 5.24160i	− 0.173282i
$916$	0.948690i	0.0313456i
$917$	− 3.14675i	− 0.103915i
$918$	4.61356i	0.152270i
$919$	5.22223	0.172265	0.0861327	−	0.996284i	$-0.472549\pi$
$919$	5.22223	0.172265	0.0861327	+	0.996284i	$0.472549\pi$
$920$	1.54288	0.0508671
$921$	5.73556i	0.188993i
$922$	20.6461	0.679943
$923$	0	0
$924$	−1.00000	−0.0328976
$925$	23.0844i	0.759011i
$926$	10.9312	0.359223
$927$	23.1554	0.760522
$928$	− 3.58211i	− 0.117588i
$929$	− 10.9957i	− 0.360757i	−0.983597	−	0.180378i	$-0.942268\pi$
$929$	− 10.9957i	− 0.360757i	0.983597	−	0.180378i	$-0.0577322\pi$
$930$	33.8364i	1.10954i
$931$	6.35690i	0.208339i
$932$	−15.4765	−0.506950
$933$	3.67456	0.120300
$934$	15.4450i	0.505377i
$935$	1.62565	0.0531643
$936$	0	0
$937$	−21.5190	−0.702994	−0.351497	−	0.936189i	$-0.614327\pi$
$937$	−21.5190	−0.702994	−0.351497	+	0.936189i	$0.614327\pi$
$938$	6.23490i	0.203577i
$939$	−28.6993	−0.936567
$940$	16.4983	0.538114
$941$	11.2784i	0.367667i	0.982957	+	0.183833i	$0.0588507\pi$
$941$	11.2784i	0.367667i	−0.982957	+	0.183833i	$0.941149\pi$
$942$	24.3666i	0.793906i
$943$	− 2.20477i	− 0.0717971i
$944$	− 10.1468i	− 0.330249i
$945$	3.61596	0.117627
$946$	−2.06100	−0.0670089
$947$	− 1.92154i	− 0.0624417i	−0.999513	−	0.0312209i	$-0.990060\pi$
$947$	− 1.92154i	− 0.0624417i	0.999513	−	0.0312209i	$-0.00993953\pi$
$948$	14.1588	0.459858
$949$	0	0
$950$	−13.5851	−0.440758
$951$	24.5013i	0.794508i
$952$	2.15883	0.0699682
$953$	−32.0538	−1.03833	−0.519163	−	0.854676i	$-0.673756\pi$
$953$	−32.0538	−1.03833	−0.519163	+	0.854676i	$0.673756\pi$
$954$	17.9095i	0.579840i
$955$	− 33.3726i	− 1.07991i
$956$	25.3448i	0.819710i
$957$	3.58211i	0.115793i
$958$	25.2828	0.816849
$959$	−3.93362	−0.127023
$960$	3.80194i	0.122707i
$961$	−48.2059	−1.55503
$962$	0	0
$963$	−14.6183	−0.471070
$964$	28.2664i	0.910398i
$965$	−15.4969	−0.498864
$966$	−2.04892	−0.0659228
$967$	− 47.5080i	− 1.52775i	−0.645362	−	0.763876i	$-0.723294\pi$
$967$	− 47.5080i	− 1.52775i	0.645362	−	0.763876i	$-0.276706\pi$
$968$	10.8019i	0.347187i
$969$	− 30.8364i	− 0.990607i
$970$	− 5.59717i	− 0.179714i
$971$	36.0224	1.15601	0.578006	−	0.816032i	$-0.303831\pi$
$971$	36.0224	1.15601	0.578006	+	0.816032i	$0.303831\pi$
$972$	18.1903	0.583454
$973$	11.2078i	0.359304i
$974$	−1.01938	−0.0326630
$975$	0	0
$976$	1.37867	0.0441300
$977$	28.2620i	0.904183i	0.891972	+	0.452091i	$0.149322\pi$
$977$	28.2620i	0.904183i	−0.891972	+	0.452091i	$0.850678\pi$
$978$	32.3696	1.03506
$979$	−8.05861	−0.257554
$980$	− 1.69202i	− 0.0540496i
$981$	13.4614i	0.429791i
$982$	− 36.7198i	− 1.17177i
$983$	8.60196i	0.274360i	0.990546	+	0.137180i	$0.0438038\pi$
$983$	8.60196i	0.274360i	−0.990546	+	0.137180i	$0.956196\pi$
$984$	5.43296	0.173196
$985$	−6.31155	−0.201103
$986$	− 7.73317i	− 0.246274i
$987$	−21.9095	−0.697386
$988$	0	0
$989$	−4.22282	−0.134278
$990$	− 1.54288i	− 0.0490358i
$991$	−17.8549	−0.567180	−0.283590	−	0.958946i	$-0.591525\pi$
$991$	−17.8549	−0.567180	−0.283590	+	0.958946i	$0.591525\pi$
$992$	−8.89977	−0.282568
$993$	− 22.3545i	− 0.709399i
$994$	− 5.76271i	− 0.182782i
$995$	33.1497i	1.05092i
$996$	− 6.54288i	− 0.207319i
$997$	−36.3247	−1.15041	−0.575207	−	0.818008i	$-0.695079\pi$
$997$	−36.3247	−1.15041	−0.575207	+	0.818008i	$0.695079\pi$
$998$	41.4685	1.31266
$999$	− 23.0844i	− 0.730359i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2366.2.d.p.337.3		6
13.5	odd	4		2366.2.a.y.1.3	✓	3
13.8	odd	4		2366.2.a.bd.1.3	yes	3
13.12	even	2	inner	2366.2.d.p.337.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2366.2.a.y.1.3	✓	3	13.5	odd	4
2366.2.a.bd.1.3	yes	3	13.8	odd	4
2366.2.d.p.337.3		6	1.1	even	1	trivial
2366.2.d.p.337.6		6	13.12	even	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:	\( N \)	\(=\)	\( 2366 = 2 \cdot 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2366.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +2.24698 q^{3} -1.00000 q^{4} -1.69202i q^{5} -2.24698i q^{6} -1.00000i q^{7} +1.00000i q^{8} +2.04892 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} +2.24698 q^{3} -1.00000 q^{4} -1.69202i q^{5} -2.24698i q^{6} -1.00000i q^{7} +1.00000i q^{8} +2.04892 q^{9} -1.69202 q^{10} +0.445042i q^{11} -2.24698 q^{12} -1.00000 q^{14} -3.80194i q^{15} +1.00000 q^{16} +2.15883 q^{17} -2.04892i q^{18} -6.35690i q^{19} +1.69202i q^{20} -2.24698i q^{21} +0.445042 q^{22} +0.911854 q^{23} +2.24698i q^{24} +2.13706 q^{25} -2.13706 q^{27} +1.00000i q^{28} +3.58211 q^{29} -3.80194 q^{30} -8.89977i q^{31} -1.00000i q^{32} +1.00000i q^{33} -2.15883i q^{34} -1.69202 q^{35} -2.04892 q^{36} +10.8019i q^{37} -6.35690 q^{38} +1.69202 q^{40} -2.41789i q^{41} -2.24698 q^{42} -4.63102 q^{43} -0.445042i q^{44} -3.46681i q^{45} -0.911854i q^{46} -9.75063i q^{47} +2.24698 q^{48} -1.00000 q^{49} -2.13706i q^{50} +4.85086 q^{51} -8.74094 q^{53} +2.13706i q^{54} +0.753020 q^{55} +1.00000 q^{56} -14.2838i q^{57} -3.58211i q^{58} -10.1468i q^{59} +3.80194i q^{60} +1.37867 q^{61} -8.89977 q^{62} -2.04892i q^{63} -1.00000 q^{64} +1.00000 q^{66} -6.23490i q^{67} -2.15883 q^{68} +2.04892 q^{69} +1.69202i q^{70} +5.76271i q^{71} +2.04892i q^{72} -9.93661i q^{73} +10.8019 q^{74} +4.80194 q^{75} +6.35690i q^{76} +0.445042 q^{77} -6.30127 q^{79} -1.69202i q^{80} -10.9487 q^{81} -2.41789 q^{82} +2.91185i q^{83} +2.24698i q^{84} -3.65279i q^{85} +4.63102i q^{86} +8.04892 q^{87} -0.445042 q^{88} +18.1075i q^{89} -3.46681 q^{90} -0.911854 q^{92} -19.9976i q^{93} -9.75063 q^{94} -10.7560 q^{95} -2.24698i q^{96} +3.30798i q^{97} +1.00000i q^{98} +0.911854i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{3} - 6 q^{4} - 6 q^{9}+O(q^{10}) \) 6 * q + 4 * q^3 - 6 * q^4 - 6 * q^9	\( 6 q + 4 q^{3} - 6 q^{4} - 6 q^{9} - 4 q^{12} - 6 q^{14} + 6 q^{16} - 4 q^{17} + 2 q^{22} - 2 q^{23} + 2 q^{25} - 2 q^{27} + 10 q^{29} - 14 q^{30} + 6 q^{36} - 30 q^{38} - 4 q^{42} + 2 q^{43} + 4 q^{48} - 6 q^{49} + 2 q^{51} - 24 q^{53} + 14 q^{55} + 6 q^{56} - 6 q^{61} - 8 q^{62} - 6 q^{64} + 6 q^{66} + 4 q^{68} - 6 q^{69} + 56 q^{74} + 20 q^{75} + 2 q^{77} - 4 q^{79} - 2 q^{81} - 26 q^{82} + 30 q^{87} - 2 q^{88} - 14 q^{90} + 2 q^{92} + 14 q^{94} + 14 q^{95}+O(q^{100}) \) 6 * q + 4 * q^3 - 6 * q^4 - 6 * q^9 - 4 * q^12 - 6 * q^14 + 6 * q^16 - 4 * q^17 + 2 * q^22 - 2 * q^23 + 2 * q^25 - 2 * q^27 + 10 * q^29 - 14 * q^30 + 6 * q^36 - 30 * q^38 - 4 * q^42 + 2 * q^43 + 4 * q^48 - 6 * q^49 + 2 * q^51 - 24 * q^53 + 14 * q^55 + 6 * q^56 - 6 * q^61 - 8 * q^62 - 6 * q^64 + 6 * q^66 + 4 * q^68 - 6 * q^69 + 56 * q^74 + 20 * q^75 + 2 * q^77 - 4 * q^79 - 2 * q^81 - 26 * q^82 + 30 * q^87 - 2 * q^88 - 14 * q^90 + 2 * q^92 + 14 * q^94 + 14 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

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