LMFDB - Embedded newform 475.2.b.e.324.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [475,2,Mod(324,475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(475, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("475.324");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 475 = 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	475.b (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:	$3.79289409601$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.2058981376.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + 2x^{6} + 18x^{4} - 34x^{3} + 32x^{2} - 8x + 1 $ x^8 - 2x^7 + 2x^6 + 18x^4 - 34x^3 + 32x^2 - 8x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 95)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			324.8
Root			$1.52153 + 1.52153i$ of defining polynomial
Character	$\chi$	$=$	475.324
Dual form			475.2.b.e.324.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.63010i q^{2} +3.04306i q^{3} -4.91744 q^{4} -8.00355 q^{6} +0.574672i q^{7} -7.67316i q^{8} -6.26020 q^{9} +O(q^{10})$	\(q+2.63010i q^{2} +3.04306i q^{3} -4.91744 q^{4} -8.00355 q^{6} +0.574672i q^{7} -7.67316i q^{8} -6.26020 q^{9} +2.57467 q^{11} -14.9641i q^{12} -0.468387i q^{13} -1.51145 q^{14} +10.3463 q^{16} +4.08612i q^{17} -16.4650i q^{18} -1.00000 q^{19} -1.74876 q^{21} +6.77165i q^{22} +1.51145i q^{23} +23.3499 q^{24} +1.23191 q^{26} -9.92099i q^{27} -2.82591i q^{28} +4.08612 q^{29} -9.92099 q^{31} +11.8656i q^{32} +7.83488i q^{33} -10.7469 q^{34} +30.7842 q^{36} +8.30326i q^{37} -2.63010i q^{38} +1.42533 q^{39} -1.83488 q^{41} -4.59942i q^{42} -0.574672i q^{43} -12.6608 q^{44} -3.97526 q^{46} -7.09508i q^{47} +31.4845i q^{48} +6.66975 q^{49} -12.4343 q^{51} +2.30326i q^{52} +4.30326i q^{53} +26.0932 q^{54} +4.40955 q^{56} -3.04306i q^{57} +10.7469i q^{58} +2.68553 q^{59} +12.4095 q^{61} -26.0932i q^{62} -3.59756i q^{63} -10.5150 q^{64} -20.6065 q^{66} +2.70570i q^{67} -20.0932i q^{68} -4.59942 q^{69} -7.40058 q^{71} +48.0356i q^{72} +12.0861i q^{73} -21.8384 q^{74} +4.91744 q^{76} +1.47959i q^{77} +3.74876i q^{78} +6.68553 q^{79} +11.4095 q^{81} -4.82591i q^{82} -6.66079i q^{83} +8.59942 q^{84} +1.51145 q^{86} +12.4343i q^{87} -19.7559i q^{88} -14.6065 q^{89} +0.269169 q^{91} -7.43244i q^{92} -30.1902i q^{93} +18.6608 q^{94} -36.1076 q^{96} -17.4526i q^{97} +17.5421i q^{98} -16.1180 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 16 q^{4} - 16 q^{9}+O(q^{10}) $ 8 * q - 16 * q^4 - 16 * q^9	$ 8 q - 16 q^{4} - 16 q^{9} + 8 q^{11} + 16 q^{14} + 8 q^{16} - 8 q^{19} - 8 q^{21} + 48 q^{24} + 8 q^{26} - 8 q^{29} + 8 q^{31} + 8 q^{34} + 80 q^{36} + 24 q^{39} + 32 q^{41} - 48 q^{44} - 40 q^{49} - 72 q^{51} + 40 q^{54} - 24 q^{56} + 40 q^{61} + 8 q^{64} - 56 q^{66} - 56 q^{69} - 40 q^{71} - 64 q^{74} + 16 q^{76} + 32 q^{79} + 32 q^{81} + 88 q^{84} - 16 q^{86} - 8 q^{89} - 72 q^{91} + 96 q^{94} - 104 q^{96} + 8 q^{99}+O(q^{100}) $ 8 * q - 16 * q^4 - 16 * q^9 + 8 * q^11 + 16 * q^14 + 8 * q^16 - 8 * q^19 - 8 * q^21 + 48 * q^24 + 8 * q^26 - 8 * q^29 + 8 * q^31 + 8 * q^34 + 80 * q^36 + 24 * q^39 + 32 * q^41 - 48 * q^44 - 40 * q^49 - 72 * q^51 + 40 * q^54 - 24 * q^56 + 40 * q^61 + 8 * q^64 - 56 * q^66 - 56 * q^69 - 40 * q^71 - 64 * q^74 + 16 * q^76 + 32 * q^79 + 32 * q^81 + 88 * q^84 - 16 * q^86 - 8 * q^89 - 72 * q^91 + 96 * q^94 - 104 * q^96 + 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/475\mathbb{Z}\right)^\times$.

$n$	$77$	$401$
$\chi(n)$	$-1$	$1$

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 $

$2$	2.63010i	1.85976i	0.367859	+	0.929882i	$0.380091\pi$
$2$	2.63010i	1.85976i	−0.367859	+	0.929882i	$0.619909\pi$
$3$	3.04306i	1.75691i	0.477825	+	0.878455i	$0.341425\pi$
$3$	3.04306i	1.75691i	−0.477825	+	0.878455i	$0.658575\pi$
$4$	−4.91744	−2.45872
$5$	0	0
$5$	0	0
$6$	−8.00355	−3.26744
$7$	0.574672i	0.217205i	0.994085	+	0.108603i	$0.0346376\pi$
$7$	0.574672i	0.217205i	−0.994085	+	0.108603i	$0.965362\pi$
$8$	− 7.67316i	− 2.71287i
$9$	−6.26020	−2.08673
$10$	0	0
$11$	2.57467	0.776293	0.388146	−	0.921598i	$-0.373116\pi$
$11$	2.57467	0.776293	0.388146	+	0.921598i	$0.373116\pi$
$12$	− 14.9641i	− 4.31975i
$13$	− 0.468387i	− 0.129907i	−0.997888	−	0.0649536i	$-0.979310\pi$
$13$	− 0.468387i	− 0.129907i	0.997888	−	0.0649536i	$-0.0206899\pi$
$14$	−1.51145	−0.403951
$15$	0	0
$16$	10.3463	2.58658
$17$	4.08612i	0.991029i	0.868600	+	0.495514i	$0.165020\pi$
$17$	4.08612i	0.991029i	−0.868600	+	0.495514i	$0.834980\pi$
$18$	− 16.4650i	− 3.88083i
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−1.74876	−0.381611
$22$	6.77165i	1.44372i
$23$	1.51145i	0.315158i	0.987506	+	0.157579i	$0.0503689\pi$
$23$	1.51145i	0.315158i	−0.987506	+	0.157579i	$0.949631\pi$
$24$	23.3499	4.76627
$25$	0	0
$26$	1.23191	0.241596
$27$	− 9.92099i	− 1.90930i
$28$	− 2.82591i	− 0.534047i
$29$	4.08612	0.758773	0.379386	−	0.925238i	$-0.376135\pi$
$29$	4.08612	0.758773	0.379386	+	0.925238i	$0.376135\pi$
$30$	0	0
$31$	−9.92099	−1.78186	−0.890931	−	0.454138i	$-0.849947\pi$
$31$	−9.92099	−1.78186	−0.890931	+	0.454138i	$0.849947\pi$
$32$	11.8656i	2.09755i
$33$	7.83488i	1.36388i
$34$	−10.7469	−1.84308
$35$	0	0
$36$	30.7842	5.13069
$37$	8.30326i	1.36505i	0.730863	+	0.682524i	$0.239118\pi$
$37$	8.30326i	1.36505i	−0.730863	+	0.682524i	$0.760882\pi$
$38$	− 2.63010i	− 0.426659i
$39$	1.42533	0.228235
$40$	0	0
$41$	−1.83488	−0.286560	−0.143280	−	0.989682i	$-0.545765\pi$
$41$	−1.83488	−0.286560	−0.143280	+	0.989682i	$0.545765\pi$
$42$	− 4.59942i	− 0.709705i
$43$	− 0.574672i	− 0.0876366i	−0.999040	−	0.0438183i	$-0.986048\pi$
$43$	− 0.574672i	− 0.0876366i	0.999040	−	0.0438183i	$-0.0139523\pi$
$44$	−12.6608	−1.90869
$45$	0	0
$46$	−3.97526	−0.586119
$47$	− 7.09508i	− 1.03492i	−0.855706	−	0.517462i	$-0.826877\pi$
$47$	− 7.09508i	− 1.03492i	0.855706	−	0.517462i	$-0.173123\pi$
$48$	31.4845i	4.54439i
$49$	6.66975	0.952822
$50$	0	0
$51$	−12.4343	−1.74115
$52$	2.30326i	0.319405i
$53$	4.30326i	0.591099i	0.955327	+	0.295549i	$0.0955027\pi$
$53$	4.30326i	0.591099i	−0.955327	+	0.295549i	$0.904497\pi$
$54$	26.0932	3.55084
$55$	0	0
$56$	4.40955	0.589251
$57$	− 3.04306i	− 0.403063i
$58$	10.7469i	1.41114i
$59$	2.68553	0.349627	0.174813	−	0.984602i	$-0.444068\pi$
$59$	2.68553	0.349627	0.174813	+	0.984602i	$0.444068\pi$
$60$	0	0
$61$	12.4095	1.58888	0.794440	−	0.607343i	$-0.207765\pi$
$61$	12.4095	1.58888	0.794440	+	0.607343i	$0.207765\pi$
$62$	− 26.0932i	− 3.31384i
$63$	− 3.59756i	− 0.453250i
$64$	−10.5150	−1.31438
$65$	0	0
$66$	−20.6065	−2.53649
$67$	2.70570i	0.330554i	0.986247	+	0.165277i	$0.0528518\pi$
$67$	2.70570i	0.330554i	−0.986247	+	0.165277i	$0.947148\pi$
$68$	− 20.0932i	− 2.43666i
$69$	−4.59942	−0.553705
$70$	0	0
$71$	−7.40058	−0.878288	−0.439144	−	0.898417i	$-0.644718\pi$
$71$	−7.40058	−0.878288	−0.439144	+	0.898417i	$0.644718\pi$
$72$	48.0356i	5.66104i
$73$	12.0861i	1.41457i	0.706927	+	0.707286i	$0.250081\pi$
$73$	12.0861i	1.41457i	−0.706927	+	0.707286i	$0.749919\pi$
$74$	−21.8384	−2.53867
$75$	0	0
$76$	4.91744	0.564069
$77$	1.47959i	0.168615i
$78$	3.74876i	0.424463i
$79$	6.68553	0.752181	0.376091	−	0.926583i	$-0.377268\pi$
$79$	6.68553	0.752181	0.376091	+	0.926583i	$0.377268\pi$
$80$	0	0
$81$	11.4095	1.26773
$82$	− 4.82591i	− 0.532933i
$83$	− 6.66079i	− 0.731117i	−0.930788	−	0.365558i	$-0.880878\pi$
$83$	− 6.66079i	− 0.731117i	0.930788	−	0.365558i	$-0.119122\pi$
$84$	8.59942	0.938273
$85$	0	0
$86$	1.51145	0.162983
$87$	12.4343i	1.33310i
$88$	− 19.7559i	− 2.10598i
$89$	−14.6065	−1.54829	−0.774144	−	0.633009i	$-0.781820\pi$
$89$	−14.6065	−1.54829	−0.774144	+	0.633009i	$0.781820\pi$
$90$	0	0
$91$	0.269169	0.0282165
$92$	− 7.43244i	− 0.774885i
$93$	− 30.1902i	− 3.13057i
$94$	18.6608	1.92471
$95$	0	0
$96$	−36.1076	−3.68522
$97$	− 17.4526i	− 1.77204i	−0.463643	−	0.886022i	$-0.653458\pi$
$97$	− 17.4526i	− 1.77204i	0.463643	−	0.886022i	$-0.346542\pi$
$98$	17.5421i	1.77202i
$99$	−16.1180	−1.61992
$100$	0	0
$101$	14.2831	1.42122	0.710611	−	0.703586i	$-0.248419\pi$
$101$	14.2831	1.42122	0.710611	+	0.703586i	$0.248419\pi$
$102$	− 32.7035i	− 3.23813i
$103$	4.79182i	0.472152i	0.971735	+	0.236076i	$0.0758614\pi$
$103$	4.79182i	0.472152i	−0.971735	+	0.236076i	$0.924139\pi$
$104$	−3.59401	−0.352421
$105$	0	0
$106$	−11.3180	−1.09930
$107$	− 9.22611i	− 0.891922i	−0.895052	−	0.445961i	$-0.852862\pi$
$107$	− 9.22611i	− 0.891922i	0.895052	−	0.445961i	$-0.147138\pi$
$108$	48.7859i	4.69442i
$109$	−4.89810	−0.469153	−0.234577	−	0.972098i	$-0.575370\pi$
$109$	−4.89810	−0.469153	−0.234577	+	0.972098i	$0.575370\pi$
$110$	0	0
$111$	−25.2673	−2.39827
$112$	5.94574i	0.561819i
$113$	1.61773i	0.152183i	0.997101	+	0.0760916i	$0.0242441\pi$
$113$	1.61773i	0.152183i	−0.997101	+	0.0760916i	$0.975756\pi$
$114$	8.00355	0.749602
$115$	0	0
$116$	−20.0932	−1.86561
$117$	2.93220i	0.271082i
$118$	7.06323i	0.650223i
$119$	−2.34818	−0.215257
$120$	0	0
$121$	−4.37107	−0.397370
$122$	32.6384i	2.95494i
$123$	− 5.58364i	− 0.503459i
$124$	48.7859	4.38110
$125$	0	0
$126$	9.46196	0.842938
$127$	13.1292i	1.16503i	0.812821	+	0.582513i	$0.197930\pi$
$127$	13.1292i	1.16503i	−0.812821	+	0.582513i	$0.802070\pi$
$128$	− 3.92440i	− 0.346871i
$129$	1.74876	0.153970
$130$	0	0
$131$	−8.17223	−0.714011	−0.357006	−	0.934102i	$-0.616202\pi$
$131$	−8.17223	−0.714011	−0.357006	+	0.934102i	$0.616202\pi$
$132$	− 38.5275i	− 3.35339i
$133$	− 0.574672i	− 0.0498304i
$134$	−7.11627	−0.614752
$135$	0	0
$136$	31.3534	2.68853
$137$	14.6065i	1.24792i	0.781456	+	0.623960i	$0.214477\pi$
$137$	14.6065i	1.24792i	−0.781456	+	0.623960i	$0.785523\pi$
$138$	− 12.0969i	− 1.02976i
$139$	2.07219	0.175761	0.0878804	−	0.996131i	$-0.471991\pi$
$139$	2.07219	0.175761	0.0878804	+	0.996131i	$0.471991\pi$
$140$	0	0
$141$	21.5907	1.81827
$142$	− 19.4643i	− 1.63341i
$143$	− 1.20594i	− 0.100846i
$144$	−64.7701	−5.39751
$145$	0	0
$146$	−31.7877	−2.63077
$147$	20.2964i	1.67402i
$148$	− 40.8308i	− 3.35627i
$149$	−8.91203	−0.730102	−0.365051	−	0.930988i	$-0.618948\pi$
$149$	−8.91203	−0.730102	−0.365051	+	0.930988i	$0.618948\pi$
$150$	0	0
$151$	11.4572	0.932372	0.466186	−	0.884687i	$-0.345628\pi$
$151$	11.4572	0.932372	0.466186	+	0.884687i	$0.345628\pi$
$152$	7.67316i	0.622376i
$153$	− 25.5799i	− 2.06801i
$154$	−3.89148	−0.313584
$155$	0	0
$156$	−7.00896	−0.561166
$157$	6.60653i	0.527258i	0.964624	+	0.263629i	$0.0849195\pi$
$157$	6.60653i	0.527258i	−0.964624	+	0.263629i	$0.915081\pi$
$158$	17.5836i	1.39888i
$159$	−13.0951	−1.03851
$160$	0	0
$161$	−0.868585	−0.0684541
$162$	30.0083i	2.35767i
$163$	20.2444i	1.58567i	0.609439	+	0.792833i	$0.291395\pi$
$163$	20.2444i	1.58567i	−0.609439	+	0.792833i	$0.708605\pi$
$164$	9.02289	0.704569
$165$	0	0
$166$	17.5186	1.35970
$167$	5.89372i	0.456069i	0.973653	+	0.228035i	$0.0732300\pi$
$167$	5.89372i	0.456069i	−0.973653	+	0.228035i	$0.926770\pi$
$168$	13.4185i	1.03526i
$169$	12.7806	0.983124
$170$	0	0
$171$	6.26020	0.478730
$172$	2.82591i	0.215474i
$173$	− 3.53161i	− 0.268504i	−0.990947	−	0.134252i	$-0.957137\pi$
$173$	− 3.53161i	− 0.268504i	0.990947	−	0.134252i	$-0.0428631\pi$
$174$	−32.7035	−2.47924
$175$	0	0
$176$	26.6384	2.00794
$177$	8.17223i	0.614263i
$178$	− 38.4167i	− 2.87945i
$179$	−7.18801	−0.537257	−0.268629	−	0.963244i	$-0.586570\pi$
$179$	−7.18801	−0.537257	−0.268629	+	0.963244i	$0.586570\pi$
$180$	0	0
$181$	15.5433	1.15532	0.577662	−	0.816276i	$-0.303965\pi$
$181$	15.5433	1.15532	0.577662	+	0.816276i	$0.303965\pi$
$182$	0.707941i	0.0524761i
$183$	37.7630i	2.79152i
$184$	11.5976	0.854984
$185$	0	0
$186$	79.4032	5.82213
$187$	10.5204i	0.769329i
$188$	34.8896i	2.54459i
$189$	5.70131	0.414710
$190$	0	0
$191$	13.3216	0.963915	0.481958	−	0.876194i	$-0.339926\pi$
$191$	13.3216	0.963915	0.481958	+	0.876194i	$0.339926\pi$
$192$	− 31.9978i	− 2.30924i
$193$	18.9959i	1.36736i	0.729784	+	0.683678i	$0.239621\pi$
$193$	18.9959i	1.36736i	−0.729784	+	0.683678i	$0.760379\pi$
$194$	45.9021	3.29558
$195$	0	0
$196$	−32.7981	−2.34272
$197$	− 2.17223i	− 0.154765i	−0.997001	−	0.0773826i	$-0.975344\pi$
$197$	− 2.17223i	− 0.154765i	0.997001	−	0.0773826i	$-0.0246563\pi$
$198$	− 42.3919i	− 3.01266i
$199$	1.87355	0.132812	0.0664061	−	0.997793i	$-0.478847\pi$
$199$	1.87355	0.132812	0.0664061	+	0.997793i	$0.478847\pi$
$200$	0	0
$201$	−8.23361	−0.580754
$202$	37.5660i	2.64313i
$203$	2.34818i	0.164810i
$204$	61.1449	4.28100
$205$	0	0
$206$	−12.6030	−0.878091
$207$	− 9.46196i	− 0.657651i
$208$	− 4.84608i	− 0.336015i
$209$	−2.57467	−0.178094
$210$	0	0
$211$	−17.1090	−1.17783	−0.588916	−	0.808194i	$-0.700445\pi$
$211$	−17.1090	−1.17783	−0.588916	+	0.808194i	$0.700445\pi$
$212$	− 21.1610i	− 1.45335i
$213$	− 22.5204i	− 1.54307i
$214$	24.2656	1.65876
$215$	0	0
$216$	−76.1254	−5.17968
$217$	− 5.70131i	− 0.387030i
$218$	− 12.8825i	− 0.872514i
$219$	−36.7788	−2.48528
$220$	0	0
$221$	1.91388	0.128742
$222$	− 66.4556i	− 4.46021i
$223$	5.12918i	0.343475i	0.985143	+	0.171737i	$0.0549381\pi$
$223$	5.12918i	0.343475i	−0.985143	+	0.171737i	$0.945062\pi$
$224$	−6.81880	−0.455600
$225$	0	0
$226$	−4.25480	−0.283025
$227$	7.31223i	0.485330i	0.970110	+	0.242665i	$0.0780215\pi$
$227$	7.31223i	0.485330i	−0.970110	+	0.242665i	$0.921978\pi$
$228$	14.9641i	0.991019i
$229$	8.40955	0.555719	0.277859	−	0.960622i	$-0.410375\pi$
$229$	8.40955	0.555719	0.277859	+	0.960622i	$0.410375\pi$
$230$	0	0
$231$	−4.50248	−0.296242
$232$	− 31.3534i	− 2.05845i
$233$	− 14.1722i	− 0.928454i	−0.885716	−	0.464227i	$-0.846332\pi$
$233$	− 14.1722i	− 0.928454i	0.885716	−	0.464227i	$-0.153668\pi$
$234$	−7.71198	−0.504148
$235$	0	0
$236$	−13.2059	−0.859634
$237$	20.3445i	1.32152i
$238$	− 6.17594i	− 0.400327i
$239$	14.1902	0.917885	0.458943	−	0.888466i	$-0.348228\pi$
$239$	14.1902	0.917885	0.458943	+	0.888466i	$0.348228\pi$
$240$	0	0
$241$	27.8807	1.79595	0.897976	−	0.440045i	$-0.145038\pi$
$241$	27.8807	1.79595	0.897976	+	0.440045i	$0.145038\pi$
$242$	− 11.4964i	− 0.739013i
$243$	4.95694i	0.317988i
$244$	−61.0232	−3.90661
$245$	0	0
$246$	14.6855	0.936315
$247$	0.468387i	0.0298027i
$248$	76.1254i	4.83397i
$249$	20.2692	1.28451
$250$	0	0
$251$	−26.1902	−1.65311	−0.826554	−	0.562857i	$-0.809702\pi$
$251$	−26.1902	−1.65311	−0.826554	+	0.562857i	$0.809702\pi$
$252$	17.6908i	1.11441i
$253$	3.89148i	0.244655i
$254$	−34.5311	−2.16667
$255$	0	0
$256$	−10.7084	−0.669276
$257$	− 9.01831i	− 0.562547i	−0.959628	−	0.281273i	$-0.909243\pi$
$257$	− 9.01831i	− 0.562547i	0.959628	−	0.281273i	$-0.0907568\pi$
$258$	4.59942i	0.286347i
$259$	−4.77165	−0.296496
$260$	0	0
$261$	−25.5799	−1.58336
$262$	− 21.4938i	− 1.32789i
$263$	− 9.00896i	− 0.555517i	−0.960651	−	0.277758i	$-0.910409\pi$
$263$	− 9.00896i	− 0.555517i	0.960651	−	0.277758i	$-0.0895914\pi$
$264$	60.1183	3.70002
$265$	0	0
$266$	1.51145	0.0926727
$267$	− 44.4485i	− 2.72020i
$268$	− 13.3051i	− 0.812739i
$269$	30.6136	1.86655	0.933273	−	0.359167i	$-0.116939\pi$
$269$	30.6136	1.86655	0.933273	+	0.359167i	$0.116939\pi$
$270$	0	0
$271$	24.1180	1.46506	0.732531	−	0.680733i	$-0.238339\pi$
$271$	24.1180	1.46506	0.732531	+	0.680733i	$0.238339\pi$
$272$	42.2763i	2.56338i
$273$	0.819096i	0.0495739i
$274$	−38.4167	−2.32084
$275$	0	0
$276$	22.6173	1.36140
$277$	− 4.56075i	− 0.274029i	−0.990569	−	0.137014i	$-0.956249\pi$
$277$	− 4.56075i	− 0.274029i	0.990569	−	0.137014i	$-0.0437506\pi$
$278$	5.45007i	0.326874i
$279$	62.1074	3.71828
$280$	0	0
$281$	−6.11563	−0.364828	−0.182414	−	0.983222i	$-0.558391\pi$
$281$	−6.11563	−0.364828	−0.182414	+	0.983222i	$0.558391\pi$
$282$	56.7859i	3.38155i
$283$	− 19.0547i	− 1.13269i	−0.824169	−	0.566344i	$-0.808358\pi$
$283$	− 19.0547i	− 1.13269i	0.824169	−	0.566344i	$-0.191642\pi$
$284$	36.3919	2.15946
$285$	0	0
$286$	3.17175	0.187550
$287$	− 1.05445i	− 0.0622423i
$288$	− 74.2808i	− 4.37704i
$289$	0.303649	0.0178617
$290$	0	0
$291$	53.1093	3.11332
$292$	− 59.4327i	− 3.47804i
$293$	− 6.43887i	− 0.376163i	−0.982153	−	0.188081i	$-0.939773\pi$
$293$	− 6.43887i	− 0.376163i	0.982153	−	0.188081i	$-0.0602269\pi$
$294$	−53.3817	−3.11329
$295$	0	0
$296$	63.7123	3.70320
$297$	− 25.5433i	− 1.48217i
$298$	− 23.4395i	− 1.35782i
$299$	0.707941	0.0409413
$300$	0	0
$301$	0.330247	0.0190351
$302$	30.1336i	1.73399i
$303$	43.4643i	2.49696i
$304$	−10.3463	−0.593402
$305$	0	0
$306$	67.2778	3.84602
$307$	6.77389i	0.386606i	0.981139	+	0.193303i	$0.0619201\pi$
$307$	6.77389i	0.386606i	−0.981139	+	0.193303i	$0.938080\pi$
$308$	− 7.27580i	− 0.414577i
$309$	−14.5818	−0.829529
$310$	0	0
$311$	20.6205	1.16928	0.584639	−	0.811293i	$-0.301236\pi$
$311$	20.6205	1.16928	0.584639	+	0.811293i	$0.301236\pi$
$312$	− 10.9368i	− 0.619173i
$313$	19.3711i	1.09492i	0.836833	+	0.547459i	$0.184405\pi$
$313$	19.3711i	1.09492i	−0.836833	+	0.547459i	$0.815595\pi$
$314$	−17.3758	−0.980575
$315$	0	0
$316$	−32.8757	−1.84940
$317$	− 3.37360i	− 0.189480i	−0.995502	−	0.0947401i	$-0.969798\pi$
$317$	− 3.37360i	− 0.189480i	0.995502	−	0.0947401i	$-0.0302020\pi$
$318$	− 34.4414i	− 1.93138i
$319$	10.5204	0.589030
$320$	0	0
$321$	28.0756	1.56703
$322$	− 2.28447i	− 0.127308i
$323$	− 4.08612i	− 0.227358i
$324$	−56.1057	−3.11699
$325$	0	0
$326$	−53.2449	−2.94896
$327$	− 14.9052i	− 0.824260i
$328$	14.0793i	0.777399i
$329$	4.07734	0.224791
$330$	0	0
$331$	−32.7788	−1.80168	−0.900842	−	0.434148i	$-0.857050\pi$
$331$	−32.7788	−1.80168	−0.900842	+	0.434148i	$0.857050\pi$
$332$	32.7540i	1.79761i
$333$	− 51.9801i	− 2.84849i
$334$	−15.5011	−0.848181
$335$	0	0
$336$	−18.0932	−0.987066
$337$	− 8.74915i	− 0.476596i	−0.971192	−	0.238298i	$-0.923410\pi$
$337$	− 8.74915i	− 0.476596i	0.971192	−	0.238298i	$-0.0765895\pi$
$338$	33.6143i	1.82838i
$339$	−4.92285	−0.267372
$340$	0	0
$341$	−25.5433	−1.38325
$342$	16.4650i	0.890324i
$343$	7.85562i	0.424164i
$344$	−4.40955	−0.237747
$345$	0	0
$346$	9.28850	0.499353
$347$	18.5028i	0.993281i	0.867956	+	0.496640i	$0.165433\pi$
$347$	18.5028i	0.993281i	−0.867956	+	0.496640i	$0.834567\pi$
$348$	− 61.1449i	− 3.27771i
$349$	−3.54330	−0.189668	−0.0948342	−	0.995493i	$-0.530232\pi$
$349$	−3.54330	−0.189668	−0.0948342	+	0.995493i	$0.530232\pi$
$350$	0	0
$351$	−4.64686	−0.248031
$352$	30.5499i	1.62832i
$353$	− 3.41140i	− 0.181571i	−0.995870	−	0.0907853i	$-0.971062\pi$
$353$	− 3.41140i	− 0.181571i	0.995870	−	0.0907853i	$-0.0289377\pi$
$354$	−21.4938	−1.14238
$355$	0	0
$356$	71.8267	3.80681
$357$	− 7.14564i	− 0.378187i
$358$	− 18.9052i	− 0.999172i
$359$	1.70609	0.0900438	0.0450219	−	0.998986i	$-0.485664\pi$
$359$	1.70609	0.0900438	0.0450219	+	0.998986i	$0.485664\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	40.8805i	2.14863i
$363$	− 13.3014i	− 0.698143i
$364$	−1.32362	−0.0693765
$365$	0	0
$366$	−99.3205	−5.19157
$367$	− 37.6155i	− 1.96351i	−0.190144	−	0.981756i	$-0.560895\pi$
$367$	− 37.6155i	− 1.96351i	0.190144	−	0.981756i	$-0.439105\pi$
$368$	15.6379i	0.815182i
$369$	11.4867	0.597974
$370$	0	0
$371$	−2.47296	−0.128390
$372$	148.458i	7.69720i
$373$	13.4031i	0.693987i	0.937868	+	0.346994i	$0.112797\pi$
$373$	13.4031i	0.693987i	−0.937868	+	0.346994i	$0.887203\pi$
$374$	−27.6698	−1.43077
$375$	0	0
$376$	−54.4417	−2.80762
$377$	− 1.91388i	− 0.0985700i
$378$	14.9950i	0.771262i
$379$	9.37107	0.481359	0.240680	−	0.970605i	$-0.422630\pi$
$379$	9.37107	0.481359	0.240680	+	0.970605i	$0.422630\pi$
$380$	0	0
$381$	−39.9528	−2.04685
$382$	35.0371i	1.79265i
$383$	− 2.09917i	− 0.107263i	−0.998561	−	0.0536314i	$-0.982920\pi$
$383$	− 2.09917i	− 0.107263i	0.998561	−	0.0536314i	$-0.0170796\pi$
$384$	11.9422	0.609422
$385$	0	0
$386$	−49.9612	−2.54296
$387$	3.59756i	0.182874i
$388$	85.8221i	4.35696i
$389$	−1.07238	−0.0543718	−0.0271859	−	0.999630i	$-0.508655\pi$
$389$	−1.07238	−0.0543718	−0.0271859	+	0.999630i	$0.508655\pi$
$390$	0	0
$391$	−6.17594	−0.312331
$392$	− 51.1781i	− 2.58488i
$393$	− 24.8686i	− 1.25445i
$394$	5.71320	0.287827
$395$	0	0
$396$	79.2591	3.98292
$397$	23.5341i	1.18114i	0.806985	+	0.590572i	$0.201098\pi$
$397$	23.5341i	1.18114i	−0.806985	+	0.590572i	$0.798902\pi$
$398$	4.92762i	0.246999i
$399$	1.74876	0.0875475
$400$	0	0
$401$	18.6469	0.931180	0.465590	−	0.885001i	$-0.345842\pi$
$401$	18.6469	0.931180	0.465590	+	0.885001i	$0.345842\pi$
$402$	− 21.6552i	− 1.08006i
$403$	4.64686i	0.231477i
$404$	−70.2362	−3.49438
$405$	0	0
$406$	−6.17594	−0.306507
$407$	21.3782i	1.05968i
$408$	95.4103i	4.72351i
$409$	6.88017	0.340203	0.170101	−	0.985427i	$-0.445590\pi$
$409$	6.88017	0.340203	0.170101	+	0.985427i	$0.445590\pi$
$410$	0	0
$411$	−44.4485	−2.19248
$412$	− 23.5635i	− 1.16089i
$413$	1.54330i	0.0759408i
$414$	24.8859	1.22308
$415$	0	0
$416$	5.55767	0.272487
$417$	6.30580i	0.308796i
$418$	− 6.77165i	− 0.331212i
$419$	−13.4796	−0.658521	−0.329261	−	0.944239i	$-0.606799\pi$
$419$	−13.4796	−0.658521	−0.329261	+	0.944239i	$0.606799\pi$
$420$	0	0
$421$	−5.83488	−0.284374	−0.142187	−	0.989840i	$-0.545414\pi$
$421$	−5.83488	−0.284374	−0.142187	+	0.989840i	$0.545414\pi$
$422$	− 44.9984i	− 2.19049i
$423$	44.4167i	2.15961i
$424$	33.0196	1.60357
$425$	0	0
$426$	59.2310	2.86975
$427$	7.13142i	0.345113i
$428$	45.3688i	2.19298i
$429$	3.66975	0.177177
$430$	0	0
$431$	29.2039	1.40670	0.703351	−	0.710843i	$-0.251686\pi$
$431$	29.2039	1.40670	0.703351	+	0.710843i	$0.251686\pi$
$432$	− 102.646i	− 4.93855i
$433$	− 12.5229i	− 0.601814i	−0.953653	−	0.300907i	$-0.902711\pi$
$433$	− 12.5229i	− 0.601814i	0.953653	−	0.300907i	$-0.0972894\pi$
$434$	14.9950	0.719785
$435$	0	0
$436$	24.0861	1.15352
$437$	− 1.51145i	− 0.0723022i
$438$	− 96.7319i	− 4.62203i
$439$	−15.6769	−0.748216	−0.374108	−	0.927385i	$-0.622051\pi$
$439$	−15.6769	−0.748216	−0.374108	+	0.927385i	$0.622051\pi$
$440$	0	0
$441$	−41.7540	−1.98829
$442$	5.03371i	0.239429i
$443$	− 29.8281i	− 1.41717i	−0.705624	−	0.708587i	$-0.749333\pi$
$443$	− 29.8281i	− 1.41717i	0.705624	−	0.708587i	$-0.250667\pi$
$444$	124.250	5.89667
$445$	0	0
$446$	−13.4903	−0.638782
$447$	− 27.1198i	− 1.28272i
$448$	− 6.04267i	− 0.285489i
$449$	−29.9668	−1.41422	−0.707110	−	0.707104i	$-0.750001\pi$
$449$	−29.9668	−1.41422	−0.707110	+	0.707104i	$0.750001\pi$
$450$	0	0
$451$	−4.72420	−0.222454
$452$	− 7.95509i	− 0.374176i
$453$	34.8649i	1.63809i
$454$	−19.2319	−0.902598
$455$	0	0
$456$	−23.3499	−1.09346
$457$	− 17.6698i	− 0.826556i	−0.910605	−	0.413278i	$-0.864384\pi$
$457$	− 17.6698i	− 0.826556i	0.910605	−	0.413278i	$-0.135616\pi$
$458$	22.1180i	1.03350i
$459$	40.5383	1.89217
$460$	0	0
$461$	22.3445	1.04069	0.520343	−	0.853957i	$-0.325804\pi$
$461$	22.3445	1.04069	0.520343	+	0.853957i	$0.325804\pi$
$462$	− 11.8420i	− 0.550939i
$463$	− 6.83302i	− 0.317557i	−0.987314	−	0.158779i	$-0.949244\pi$
$463$	− 6.83302i	− 0.317557i	0.987314	−	0.158779i	$-0.0507556\pi$
$464$	42.2763	1.96263
$465$	0	0
$466$	37.2744	1.72670
$467$	− 9.00896i	− 0.416885i	−0.978035	−	0.208443i	$-0.933161\pi$
$467$	− 9.00896i	− 0.416885i	0.978035	−	0.208443i	$-0.0668394\pi$
$468$	− 14.4189i	− 0.666514i
$469$	−1.55489	−0.0717981
$470$	0	0
$471$	−20.1040	−0.926345
$472$	− 20.6065i	− 0.948492i
$473$	− 1.47959i	− 0.0680317i
$474$	−53.5080	−2.45771
$475$	0	0
$476$	11.5470	0.529256
$477$	− 26.9393i	− 1.23347i
$478$	37.3216i	1.70705i
$479$	−9.26731	−0.423434	−0.211717	−	0.977331i	$-0.567906\pi$
$479$	−9.26731	−0.423434	−0.211717	+	0.977331i	$0.567906\pi$
$480$	0	0
$481$	3.88914	0.177329
$482$	73.3290i	3.34004i
$483$	− 2.64315i	− 0.120268i
$484$	21.4944	0.977020
$485$	0	0
$486$	−13.0373	−0.591382
$487$	38.7694i	1.75681i	0.477917	+	0.878405i	$0.341392\pi$
$487$	38.7694i	1.75681i	−0.477917	+	0.878405i	$0.658608\pi$
$488$	− 95.2205i	− 4.31043i
$489$	−61.6050	−2.78587
$490$	0	0
$491$	21.6877	0.978751	0.489376	−	0.872073i	$-0.337225\pi$
$491$	21.6877	0.978751	0.489376	+	0.872073i	$0.337225\pi$
$492$	27.4572i	1.23787i
$493$	16.6964i	0.751966i
$494$	−1.23191	−0.0554260
$495$	0	0
$496$	−102.646	−4.60893
$497$	− 4.25291i	− 0.190769i
$498$	53.3100i	2.38888i
$499$	27.8372	1.24616	0.623082	−	0.782156i	$-0.285880\pi$
$499$	27.8372	1.24616	0.623082	+	0.782156i	$0.285880\pi$
$500$	0	0
$501$	−17.9349	−0.801273
$502$	− 68.8828i	− 3.07439i
$503$	− 41.2449i	− 1.83902i	−0.393067	−	0.919510i	$-0.628586\pi$
$503$	− 41.2449i	− 1.83902i	0.393067	−	0.919510i	$-0.371414\pi$
$504$	−27.6047	−1.22961
$505$	0	0
$506$	−10.2350	−0.455000
$507$	38.8922i	1.72726i
$508$	− 64.5619i	− 2.86447i
$509$	−0.416364	−0.0184550	−0.00922751	−	0.999957i	$-0.502937\pi$
$509$	−0.416364	−0.0184550	−0.00922751	+	0.999957i	$0.502937\pi$
$510$	0	0
$511$	−6.94555	−0.307253
$512$	− 36.0131i	− 1.59157i
$513$	9.92099i	0.438023i
$514$	23.7191	1.04620
$515$	0	0
$516$	−8.59942	−0.378568
$517$	− 18.2675i	− 0.803404i
$518$	− 12.5499i	− 0.551412i
$519$	10.7469	0.471737
$520$	0	0
$521$	24.0458	1.05346	0.526732	−	0.850031i	$-0.323417\pi$
$521$	24.0458	1.05346	0.526732	+	0.850031i	$0.323417\pi$
$522$	− 67.2778i	− 2.94467i
$523$	− 5.19736i	− 0.227265i	−0.993523	−	0.113632i	$-0.963751\pi$
$523$	− 5.19736i	− 0.227265i	0.993523	−	0.113632i	$-0.0362486\pi$
$524$	40.1865	1.75555
$525$	0	0
$526$	23.6945	1.03313
$527$	− 40.5383i	− 1.76588i
$528$	81.0621i	3.52778i
$529$	20.7155	0.900675
$530$	0	0
$531$	−16.8120	−0.729578
$532$	2.82591i	0.122519i
$533$	0.859432i	0.0372261i
$534$	116.904	5.05894
$535$	0	0
$536$	20.7613	0.896751
$537$	− 21.8735i	− 0.943913i
$538$	80.5170i	3.47133i
$539$	17.1724	0.739669
$540$	0	0
$541$	−1.93863	−0.0833481	−0.0416741	−	0.999131i	$-0.513269\pi$
$541$	−1.93863	−0.0833481	−0.0416741	+	0.999131i	$0.513269\pi$
$542$	63.4327i	2.72467i
$543$	47.2992i	2.02980i
$544$	−48.4841	−2.07874
$545$	0	0
$546$	−2.15431	−0.0921958
$547$	12.9075i	0.551883i	0.961174	+	0.275941i	$0.0889896\pi$
$547$	12.9075i	0.551883i	−0.961174	+	0.275941i	$0.911010\pi$
$548$	− 71.8267i	− 3.06828i
$549$	−77.6863	−3.31557
$550$	0	0
$551$	−4.08612	−0.174074
$552$	35.2921i	1.50213i
$553$	3.84199i	0.163378i
$554$	11.9952	0.509628
$555$	0	0
$556$	−10.1899	−0.432147
$557$	34.1040i	1.44503i	0.691353	+	0.722517i	$0.257015\pi$
$557$	34.1040i	1.44503i	−0.691353	+	0.722517i	$0.742985\pi$
$558$	163.349i	6.91511i
$559$	−0.269169	−0.0113846
$560$	0	0
$561$	−32.0142	−1.35164
$562$	− 16.0847i	− 0.678494i
$563$	− 14.4911i	− 0.610727i	−0.952236	−	0.305363i	$-0.901222\pi$
$563$	− 14.4911i	− 0.610727i	0.952236	−	0.305363i	$-0.0987779\pi$
$564$	−106.171	−4.47061
$565$	0	0
$566$	50.1159	2.10653
$567$	6.55674i	0.275357i
$568$	56.7859i	2.38268i
$569$	39.2110	1.64381	0.821906	−	0.569624i	$-0.192911\pi$
$569$	39.2110	1.64381	0.821906	+	0.569624i	$0.192911\pi$
$570$	0	0
$571$	21.9915	0.920316	0.460158	−	0.887837i	$-0.347793\pi$
$571$	21.9915	0.920316	0.460158	+	0.887837i	$0.347793\pi$
$572$	5.93015i	0.247952i
$573$	40.5383i	1.69351i
$574$	2.77331	0.115756
$575$	0	0
$576$	65.8261	2.74275
$577$	− 23.8735i	− 0.993869i	−0.867788	−	0.496934i	$-0.834459\pi$
$577$	− 23.8735i	− 0.993869i	0.867788	−	0.496934i	$-0.165541\pi$
$578$	0.798628i	0.0332185i
$579$	−57.8057	−2.40232
$580$	0	0
$581$	3.82777	0.158803
$582$	139.683i	5.79004i
$583$	11.0795i	0.458866i
$584$	92.7387	3.83756
$585$	0	0
$586$	16.9349	0.699574
$587$	− 17.8281i	− 0.735843i	−0.929857	−	0.367921i	$-0.880070\pi$
$587$	− 17.8281i	− 0.735843i	0.929857	−	0.367921i	$-0.119930\pi$
$588$	− 99.8065i	− 4.11595i
$589$	9.92099	0.408787
$590$	0	0
$591$	6.61023	0.271909
$592$	85.9082i	3.53081i
$593$	− 21.2446i	− 0.872412i	−0.899847	−	0.436206i	$-0.856322\pi$
$593$	− 21.2446i	− 0.872412i	0.899847	−	0.436206i	$-0.143678\pi$
$594$	67.1815	2.75649
$595$	0	0
$596$	43.8244	1.79512
$597$	5.70131i	0.233339i
$598$	1.86196i	0.0761411i
$599$	−24.3374	−0.994397	−0.497199	−	0.867637i	$-0.665638\pi$
$599$	−24.3374	−0.994397	−0.497199	+	0.867637i	$0.665638\pi$
$600$	0	0
$601$	15.7891	0.644051	0.322025	−	0.946731i	$-0.395636\pi$
$601$	15.7891	0.644051	0.322025	+	0.946731i	$0.395636\pi$
$602$	0.868585i	0.0354009i
$603$	− 16.9382i	− 0.689779i
$604$	−56.3400	−2.29244
$605$	0	0
$606$	−114.316	−4.64375
$607$	− 7.81471i	− 0.317189i	−0.987344	−	0.158595i	$-0.949304\pi$
$607$	− 7.81471i	− 0.317189i	0.987344	−	0.158595i	$-0.0506963\pi$
$608$	− 11.8656i	− 0.481212i
$609$	−7.14564	−0.289556
$610$	0	0
$611$	−3.32324	−0.134444
$612$	125.788i	5.08467i
$613$	12.9547i	0.523235i	0.965172	+	0.261618i	$0.0842560\pi$
$613$	12.9547i	0.523235i	−0.965172	+	0.261618i	$0.915744\pi$
$614$	−17.8160	−0.718996
$615$	0	0
$616$	11.3531	0.457431
$617$	− 18.8873i	− 0.760373i	−0.924910	−	0.380187i	$-0.875860\pi$
$617$	− 18.8873i	− 0.760373i	0.924910	−	0.380187i	$-0.124140\pi$
$618$	− 38.3516i	− 1.54273i
$619$	−29.2673	−1.17635	−0.588176	−	0.808733i	$-0.700154\pi$
$619$	−29.2673	−1.17635	−0.588176	+	0.808733i	$0.700154\pi$
$620$	0	0
$621$	14.9950	0.601730
$622$	54.2339i	2.17458i
$623$	− 8.39396i	− 0.336297i
$624$	14.7469	0.590349
$625$	0	0
$626$	−50.9479	−2.03629
$627$	− 7.83488i	− 0.312895i
$628$	− 32.4872i	− 1.29638i
$629$	−33.9281	−1.35280
$630$	0	0
$631$	−5.25309	−0.209122	−0.104561	−	0.994518i	$-0.533344\pi$
$631$	−5.25309	−0.209122	−0.104561	+	0.994518i	$0.533344\pi$
$632$	− 51.2992i	− 2.04057i
$633$	− 52.0637i	− 2.06935i
$634$	8.87291	0.352388
$635$	0	0
$636$	64.3942	2.55340
$637$	− 3.12402i	− 0.123778i
$638$	27.6698i	1.09546i
$639$	46.3292	1.83275
$640$	0	0
$641$	−13.0021	−0.513554	−0.256777	−	0.966471i	$-0.582661\pi$
$641$	−13.0021	−0.513554	−0.256777	+	0.966471i	$0.582661\pi$
$642$	73.8417i	2.91430i
$643$	− 17.3534i	− 0.684353i	−0.939636	−	0.342176i	$-0.888836\pi$
$643$	− 17.3534i	− 0.684353i	0.939636	−	0.342176i	$-0.111164\pi$
$644$	4.27121	0.168309
$645$	0	0
$646$	10.7469	0.422831
$647$	− 12.4848i	− 0.490830i	−0.969418	−	0.245415i	$-0.921076\pi$
$647$	− 12.4848i	− 0.490830i	0.969418	−	0.245415i	$-0.0789242\pi$
$648$	− 87.5473i	− 3.43918i
$649$	6.91437	0.271413
$650$	0	0
$651$	17.3494	0.679978
$652$	− 99.5507i	− 3.89871i
$653$	− 28.3532i	− 1.10955i	−0.832001	−	0.554774i	$-0.812805\pi$
$653$	− 28.3532i	− 1.10955i	0.832001	−	0.554774i	$-0.187195\pi$
$654$	39.2022	1.53293
$655$	0	0
$656$	−18.9842	−0.741209
$657$	− 75.6616i	− 2.95184i
$658$	10.7238i	0.418058i
$659$	−45.2202	−1.76153	−0.880764	−	0.473556i	$-0.842970\pi$
$659$	−45.2202	−1.76153	−0.880764	+	0.473556i	$0.842970\pi$
$660$	0	0
$661$	−14.6086	−0.568207	−0.284104	−	0.958794i	$-0.591696\pi$
$661$	−14.6086	−0.568207	−0.284104	+	0.958794i	$0.591696\pi$
$662$	− 86.2115i	− 3.35070i
$663$	5.82406i	0.226188i
$664$	−51.1093	−1.98343
$665$	0	0
$666$	136.713	5.29752
$667$	6.17594i	0.239133i
$668$	− 28.9820i	− 1.12135i
$669$	−15.6084	−0.603455
$670$	0	0
$671$	31.9505	1.23344
$672$	− 20.7500i	− 0.800449i
$673$	− 0.440534i	− 0.0169813i	−0.999964	−	0.00849067i	$-0.997297\pi$
$673$	− 0.440534i	− 0.0169813i	0.999964	−	0.00849067i	$-0.00270270\pi$
$674$	23.0111	0.886356
$675$	0	0
$676$	−62.8479	−2.41723
$677$	− 32.8057i	− 1.26083i	−0.776259	−	0.630414i	$-0.782885\pi$
$677$	− 32.8057i	− 1.26083i	0.776259	−	0.630414i	$-0.217115\pi$
$678$	− 12.9476i	− 0.497249i
$679$	10.0295	0.384898
$680$	0	0
$681$	−22.2515	−0.852681
$682$	− 67.1815i	− 2.57251i
$683$	− 39.6092i	− 1.51561i	−0.652484	−	0.757803i	$-0.726273\pi$
$683$	− 39.6092i	− 1.51561i	0.652484	−	0.757803i	$-0.273727\pi$
$684$	−30.7842	−1.17706
$685$	0	0
$686$	−20.6611	−0.788844
$687$	25.5907i	0.976348i
$688$	− 5.94574i	− 0.226679i
$689$	2.01559	0.0767879
$690$	0	0
$691$	−19.8962	−0.756889	−0.378444	−	0.925624i	$-0.623541\pi$
$691$	−19.8962	−0.756889	−0.378444	+	0.925624i	$0.623541\pi$
$692$	17.3665i	0.660175i
$693$	− 9.26254i	− 0.351855i
$694$	−48.6642	−1.84727
$695$	0	0
$696$	95.4103	3.61652
$697$	− 7.49752i	− 0.283989i
$698$	− 9.31924i	− 0.352738i
$699$	43.1269	1.63121
$700$	0	0
$701$	−14.1251	−0.533497	−0.266748	−	0.963766i	$-0.585949\pi$
$701$	−14.1251	−0.533497	−0.266748	+	0.963766i	$0.585949\pi$
$702$	− 12.2217i	− 0.461279i
$703$	− 8.30326i	− 0.313163i
$704$	−27.0727	−1.02034
$705$	0	0
$706$	8.97234	0.337678
$707$	8.20809i	0.308697i
$708$	− 40.1865i	− 1.51030i
$709$	41.1815	1.54660	0.773302	−	0.634038i	$-0.218604\pi$
$709$	41.1815	1.54660	0.773302	+	0.634038i	$0.218604\pi$
$710$	0	0
$711$	−41.8528	−1.56960
$712$	112.078i	4.20031i
$713$	− 14.9950i	− 0.561569i
$714$	18.7938	0.703338
$715$	0	0
$716$	35.3466	1.32097
$717$	43.1815i	1.61264i
$718$	4.48718i	0.167460i
$719$	18.0227	0.672133	0.336067	−	0.941838i	$-0.390903\pi$
$719$	18.0227	0.672133	0.336067	+	0.941838i	$0.390903\pi$
$720$	0	0
$721$	−2.75372	−0.102554
$722$	2.63010i	0.0978823i
$723$	84.8425i	3.15533i
$724$	−76.4332	−2.84062
$725$	0	0
$726$	34.9841	1.29838
$727$	21.1266i	0.783544i	0.920062	+	0.391772i	$0.128138\pi$
$727$	21.1266i	0.783544i	−0.920062	+	0.391772i	$0.871862\pi$
$728$	− 2.06537i	− 0.0765479i
$729$	19.1444	0.709051
$730$	0	0
$731$	2.34818	0.0868504
$732$	− 185.697i	− 6.86356i
$733$	− 27.6660i	− 1.02187i	−0.859620	−	0.510934i	$-0.829300\pi$
$733$	− 27.6660i	− 1.02187i	0.859620	−	0.510934i	$-0.170700\pi$
$734$	98.9326	3.65167
$735$	0	0
$736$	−17.9341	−0.661061
$737$	6.96629i	0.256607i
$738$	30.2112i	1.11209i
$739$	14.2987	0.525986	0.262993	−	0.964798i	$-0.415290\pi$
$739$	14.2987	0.525986	0.262993	+	0.964798i	$0.415290\pi$
$740$	0	0
$741$	−1.42533	−0.0523607
$742$	− 6.50415i	− 0.238775i
$743$	49.9438i	1.83226i	0.400881	+	0.916130i	$0.368704\pi$
$743$	49.9438i	1.83226i	−0.400881	+	0.916130i	$0.631296\pi$
$744$	−231.654	−8.49285
$745$	0	0
$746$	−35.2516	−1.29065
$747$	41.6979i	1.52565i
$748$	− 51.7335i	− 1.89156i
$749$	5.30198	0.193730
$750$	0	0
$751$	9.69216	0.353672	0.176836	−	0.984240i	$-0.443414\pi$
$751$	9.69216	0.353672	0.176836	+	0.984240i	$0.443414\pi$
$752$	− 73.4080i	− 2.67691i
$753$	− 79.6982i	− 2.90436i
$754$	5.03371	0.183317
$755$	0	0
$756$	−28.0359	−1.01965
$757$	− 46.6889i	− 1.69694i	−0.529245	−	0.848469i	$-0.677525\pi$
$757$	− 46.6889i	− 1.69694i	0.529245	−	0.848469i	$-0.322475\pi$
$758$	24.6469i	0.895214i
$759$	−11.8420	−0.429837
$760$	0	0
$761$	−38.7361	−1.40418	−0.702091	−	0.712087i	$-0.747750\pi$
$761$	−38.7361	−1.40418	−0.702091	+	0.712087i	$0.747750\pi$
$762$	− 105.080i	− 3.80665i
$763$	− 2.81480i	− 0.101903i
$764$	−65.5080	−2.37000
$765$	0	0
$766$	5.52104	0.199483
$767$	− 1.25787i	− 0.0454190i
$768$	− 32.5864i	− 1.17586i
$769$	5.38666	0.194248	0.0971239	−	0.995272i	$-0.469036\pi$
$769$	5.38666	0.194248	0.0971239	+	0.995272i	$0.469036\pi$
$770$	0	0
$771$	27.4433	0.988345
$772$	− 93.4112i	− 3.36194i
$773$	7.20137i	0.259015i	0.991578	+	0.129508i	$0.0413397\pi$
$773$	7.20137i	0.259015i	−0.991578	+	0.129508i	$0.958660\pi$
$774$	−9.46196	−0.340103
$775$	0	0
$776$	−133.917	−4.80733
$777$	− 14.5204i	− 0.520917i
$778$	− 2.82047i	− 0.101119i
$779$	1.83488	0.0657413
$780$	0	0
$781$	−19.0541	−0.681808
$782$	− 16.2434i	− 0.580861i
$783$	− 40.5383i	− 1.44872i
$784$	69.0074	2.46455
$785$	0	0
$786$	65.4069	2.33299
$787$	33.5231i	1.19497i	0.801880	+	0.597485i	$0.203833\pi$
$787$	33.5231i	1.19497i	−0.801880	+	0.597485i	$0.796167\pi$
$788$	10.6818i	0.380524i
$789$	27.4148	0.975993
$790$	0	0
$791$	−0.929664	−0.0330550
$792$	123.676i	4.39463i
$793$	− 5.81247i	− 0.206407i
$794$	−61.8972	−2.19665
$795$	0	0
$796$	−9.21305	−0.326548
$797$	10.4979i	0.371855i	0.982563	+	0.185927i	$0.0595289\pi$
$797$	10.4979i	0.371855i	−0.982563	+	0.185927i	$0.940471\pi$
$798$	4.59942i	0.162818i
$799$	28.9913	1.02564
$800$	0	0
$801$	91.4398	3.23087
$802$	49.0432i	1.73177i
$803$	31.1178i	1.09812i
$804$	40.4882	1.42791
$805$	0	0
$806$	−12.2217	−0.430492
$807$	93.1591i	3.27936i
$808$	− 109.596i	− 3.85559i
$809$	13.0724	0.459600	0.229800	−	0.973238i	$-0.426193\pi$
$809$	13.0724	0.459600	0.229800	+	0.973238i	$0.426193\pi$
$810$	0	0
$811$	−10.0790	−0.353922	−0.176961	−	0.984218i	$-0.556627\pi$
$811$	−10.0790	−0.353922	−0.176961	+	0.984218i	$0.556627\pi$
$812$	− 11.5470i	− 0.405221i
$813$	73.3924i	2.57398i
$814$	−56.2268	−1.97075
$815$	0	0
$816$	−128.649	−4.50362
$817$	0.574672i	0.0201052i
$818$	18.0956i	0.632697i
$819$	−1.68505	−0.0588804
$820$	0	0
$821$	40.2987	1.40643	0.703217	−	0.710975i	$-0.251746\pi$
$821$	40.2987	1.40643	0.703217	+	0.710975i	$0.251746\pi$
$822$	− 116.904i	− 4.07750i
$823$	− 5.07715i	− 0.176978i	−0.996077	−	0.0884892i	$-0.971796\pi$
$823$	− 5.07715i	− 0.176978i	0.996077	−	0.0884892i	$-0.0282039\pi$
$824$	36.7684	1.28089
$825$	0	0
$826$	−4.05904	−0.141232
$827$	42.8077i	1.48857i	0.667862	+	0.744285i	$0.267209\pi$
$827$	42.8077i	1.48857i	−0.667862	+	0.744285i	$0.732791\pi$
$828$	46.5286i	1.61698i
$829$	−24.2018	−0.840562	−0.420281	−	0.907394i	$-0.638068\pi$
$829$	−24.2018	−0.840562	−0.420281	+	0.907394i	$0.638068\pi$
$830$	0	0
$831$	13.8786	0.481444
$832$	4.92509i	0.170747i
$833$	27.2534i	0.944274i
$834$	−16.5849	−0.574288
$835$	0	0
$836$	12.6608	0.437883
$837$	98.4261i	3.40210i
$838$	− 35.4527i	− 1.22469i
$839$	8.63975	0.298277	0.149139	−	0.988816i	$-0.452350\pi$
$839$	8.63975	0.298277	0.149139	+	0.988816i	$0.452350\pi$
$840$	0	0
$841$	−12.3036	−0.424264
$842$	− 15.3463i	− 0.528869i
$843$	− 18.6102i	− 0.640971i
$844$	84.1325	2.89596
$845$	0	0
$846$	−116.820	−4.01637
$847$	− 2.51193i	− 0.0863109i
$848$	44.5229i	1.52892i
$849$	57.9847	1.99003
$850$	0	0
$851$	−12.5499	−0.430206
$852$	110.743i	3.79398i
$853$	− 13.0229i	− 0.445895i	−0.974830	−	0.222948i	$-0.928432\pi$
$853$	− 13.0229i	− 0.445895i	0.974830	−	0.222948i	$-0.0715679\pi$
$854$	−18.7564	−0.641829
$855$	0	0
$856$	−70.7934	−2.41967
$857$	2.82367i	0.0964548i	0.998836	+	0.0482274i	$0.0153572\pi$
$857$	2.82367i	0.0964548i	−0.998836	+	0.0482274i	$0.984643\pi$
$858$	9.65182i	0.329508i
$859$	−24.1227	−0.823057	−0.411529	−	0.911397i	$-0.635005\pi$
$859$	−24.1227	−0.823057	−0.411529	+	0.911397i	$0.635005\pi$
$860$	0	0
$861$	3.20876	0.109354
$862$	76.8092i	2.61613i
$863$	32.7307i	1.11417i	0.830456	+	0.557084i	$0.188080\pi$
$863$	32.7307i	1.11417i	−0.830456	+	0.557084i	$0.811920\pi$
$864$	117.718	4.00485
$865$	0	0
$866$	32.9366	1.11923
$867$	0.924022i	0.0313814i
$868$	28.0359i	0.951599i
$869$	17.2131	0.583913
$870$	0	0
$871$	1.26731	0.0429413
$872$	37.5839i	1.27275i
$873$	109.257i	3.69779i
$874$	3.97526	0.134465
$875$	0	0
$876$	180.857	6.11060
$877$	49.5072i	1.67174i	0.548929	+	0.835869i	$0.315036\pi$
$877$	49.5072i	1.67174i	−0.548929	+	0.835869i	$0.684964\pi$
$878$	− 41.2318i	− 1.39150i
$879$	19.5939	0.660884
$880$	0	0
$881$	40.3152	1.35826	0.679128	−	0.734020i	$-0.262358\pi$
$881$	40.3152	1.35826	0.679128	+	0.734020i	$0.262358\pi$
$882$	− 109.817i	− 3.69774i
$883$	− 29.3347i	− 0.987192i	−0.869691	−	0.493596i	$-0.835682\pi$
$883$	− 29.3347i	− 0.987192i	0.869691	−	0.493596i	$-0.164318\pi$
$884$	−9.41140	−0.316540
$885$	0	0
$886$	78.4508	2.63561
$887$	− 43.3214i	− 1.45459i	−0.686325	−	0.727295i	$-0.740777\pi$
$887$	− 43.3214i	− 1.45459i	0.686325	−	0.727295i	$-0.259223\pi$
$888$	193.880i	6.50619i
$889$	−7.54496	−0.253050
$890$	0	0
$891$	29.3758	0.984128
$892$	− 25.2224i	− 0.844508i
$893$	7.09508i	0.237428i
$894$	71.3279	2.38556
$895$	0	0
$896$	2.25524	0.0753424
$897$	2.15431i	0.0719302i
$898$	− 78.8157i	− 2.63011i
$899$	−40.5383	−1.35203
$900$	0	0
$901$	−17.5836	−0.585796
$902$	− 12.4251i	− 0.413712i
$903$	1.00496i	0.0334431i
$904$	12.4131	0.412854
$905$	0	0
$906$	−91.6982	−3.04647
$907$	− 14.0731i	− 0.467288i	−0.972322	−	0.233644i	$-0.924935\pi$
$907$	− 14.0731i	− 0.467288i	0.972322	−	0.233644i	$-0.0750651\pi$
$908$	− 35.9574i	− 1.19329i
$909$	−89.4151	−2.96571
$910$	0	0
$911$	30.0725	0.996346	0.498173	−	0.867078i	$-0.334004\pi$
$911$	30.0725	0.996346	0.498173	+	0.867078i	$0.334004\pi$
$912$	− 31.4845i	− 1.04255i
$913$	− 17.1493i	− 0.567560i
$914$	46.4733	1.53720
$915$	0	0
$916$	−41.3534	−1.36636
$917$	− 4.69635i	− 0.155087i
$918$	106.620i	3.51898i
$919$	−29.6518	−0.978123	−0.489062	−	0.872249i	$-0.662661\pi$
$919$	−29.6518	−0.978123	−0.489062	+	0.872249i	$0.662661\pi$
$920$	0	0
$921$	−20.6133	−0.679233
$922$	58.7682i	1.93543i
$923$	3.46634i	0.114096i
$924$	22.1407	0.728375
$925$	0	0
$926$	17.9715	0.590582
$927$	− 29.9978i	− 0.985256i
$928$	48.4841i	1.59157i
$929$	−58.3803	−1.91540	−0.957698	−	0.287775i	$-0.907085\pi$
$929$	−58.3803	−1.91540	−0.957698	+	0.287775i	$0.907085\pi$
$930$	0	0
$931$	−6.66975	−0.218592
$932$	69.6911i	2.28281i
$933$	62.7492i	2.05432i
$934$	23.6945	0.775308
$935$	0	0
$936$	22.4992	0.735410
$937$	3.15850i	0.103184i	0.998668	+	0.0515918i	$0.0164295\pi$
$937$	3.15850i	0.103184i	−0.998668	+	0.0515918i	$0.983571\pi$
$938$	− 4.08952i	− 0.133528i
$939$	−58.9473	−1.92367
$940$	0	0
$941$	−9.66975	−0.315225	−0.157612	−	0.987501i	$-0.550380\pi$
$941$	−9.66975	−0.315225	−0.157612	+	0.987501i	$0.550380\pi$
$942$	− 52.8757i	− 1.72278i
$943$	− 2.77331i	− 0.0903116i
$944$	27.7854	0.904337
$945$	0	0
$946$	3.89148	0.126523
$947$	− 31.3714i	− 1.01943i	−0.860343	−	0.509716i	$-0.829750\pi$
$947$	− 31.3714i	− 1.01943i	0.860343	−	0.509716i	$-0.170250\pi$
$948$	− 100.043i	− 3.24923i
$949$	5.66098	0.183763
$950$	0	0
$951$	10.2661	0.332900
$952$	18.0179i	0.583964i
$953$	− 13.1224i	− 0.425075i	−0.977153	−	0.212537i	$-0.931827\pi$
$953$	− 13.1224i	− 0.425075i	0.977153	−	0.212537i	$-0.0681727\pi$
$954$	70.8531	2.29395
$955$	0	0
$956$	−69.7792	−2.25682
$957$	32.0142i	1.03487i
$958$	− 24.3740i	− 0.787488i
$959$	−8.39396	−0.271055
$960$	0	0
$961$	67.4261	2.17504
$962$	10.2288i	0.329791i
$963$	57.7573i	1.86120i
$964$	−137.101	−4.41574
$965$	0	0
$966$	6.95177	0.223669
$967$	44.4400i	1.42910i	0.699587	+	0.714548i	$0.253367\pi$
$967$	44.4400i	1.42910i	−0.699587	+	0.714548i	$0.746633\pi$
$968$	33.5399i	1.07801i
$969$	12.4343	0.399447
$970$	0	0
$971$	−5.69927	−0.182898	−0.0914491	−	0.995810i	$-0.529150\pi$
$971$	−5.69927	−0.182898	−0.0914491	+	0.995810i	$0.529150\pi$
$972$	− 24.3755i	− 0.781843i
$973$	1.19083i	0.0381762i
$974$	−101.968	−3.26725
$975$	0	0
$976$	128.393	4.10977
$977$	23.9164i	0.765154i	0.923924	+	0.382577i	$0.124963\pi$
$977$	23.9164i	0.765154i	−0.923924	+	0.382577i	$0.875037\pi$
$978$	− 162.027i	− 5.18106i
$979$	−37.6070	−1.20193
$980$	0	0
$981$	30.6631	0.978998
$982$	57.0408i	1.82025i
$983$	− 45.5302i	− 1.45219i	−0.687595	−	0.726095i	$-0.741333\pi$
$983$	− 45.5302i	− 1.45219i	0.687595	−	0.726095i	$-0.258667\pi$
$984$	−42.8441	−1.36582
$985$	0	0
$986$	−43.9131	−1.39848
$987$	12.4076i	0.394938i
$988$	− 2.30326i	− 0.0732766i
$989$	0.868585	0.0276194
$990$	0	0
$991$	27.5521	0.875220	0.437610	−	0.899165i	$-0.355825\pi$
$991$	27.5521	0.875220	0.437610	+	0.899165i	$0.355825\pi$
$992$	− 117.718i	− 3.73756i
$993$	− 99.7477i	− 3.16540i
$994$	11.1856	0.354785
$995$	0	0
$996$	−99.6724	−3.15824
$997$	− 16.8557i	− 0.533826i	−0.963721	−	0.266913i	$-0.913996\pi$
$997$	− 16.8557i	− 0.533826i	0.963721	−	0.266913i	$-0.0860037\pi$
$998$	73.2147i	2.31757i
$999$	82.3766	2.60628

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.2.b.e.324.8		8
5.2	odd	4		95.2.a.b.1.1	✓	4
5.3	odd	4		475.2.a.i.1.4		4
5.4	even	2	inner	475.2.b.e.324.1		8
15.2	even	4		855.2.a.m.1.4		4
15.8	even	4		4275.2.a.bo.1.1		4
20.3	even	4		7600.2.a.cf.1.4		4
20.7	even	4		1520.2.a.t.1.1		4
35.27	even	4		4655.2.a.y.1.1		4
40.27	even	4		6080.2.a.ch.1.4		4
40.37	odd	4		6080.2.a.cc.1.1		4
95.18	even	4		9025.2.a.bf.1.1		4
95.37	even	4		1805.2.a.p.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.2.a.b.1.1	✓	4	5.2	odd	4
475.2.a.i.1.4		4	5.3	odd	4
475.2.b.e.324.1		8	5.4	even	2	inner
475.2.b.e.324.8		8	1.1	even	1	trivial
855.2.a.m.1.4		4	15.2	even	4
1520.2.a.t.1.1		4	20.7	even	4
1805.2.a.p.1.4		4	95.37	even	4
4275.2.a.bo.1.1		4	15.8	even	4
4655.2.a.y.1.1		4	35.27	even	4
6080.2.a.cc.1.1		4	40.37	odd	4
6080.2.a.ch.1.4		4	40.27	even	4
7600.2.a.cf.1.4		4	20.3	even	4
9025.2.a.bf.1.1		4	95.18	even	4

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:	\( N \)	\(=\)	\( 475 = 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	475.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.63010i q^{2} +3.04306i q^{3} -4.91744 q^{4} -8.00355 q^{6} +0.574672i q^{7} -7.67316i q^{8} -6.26020 q^{9} +O(q^{10})\)	\(q+2.63010i q^{2} +3.04306i q^{3} -4.91744 q^{4} -8.00355 q^{6} +0.574672i q^{7} -7.67316i q^{8} -6.26020 q^{9} +2.57467 q^{11} -14.9641i q^{12} -0.468387i q^{13} -1.51145 q^{14} +10.3463 q^{16} +4.08612i q^{17} -16.4650i q^{18} -1.00000 q^{19} -1.74876 q^{21} +6.77165i q^{22} +1.51145i q^{23} +23.3499 q^{24} +1.23191 q^{26} -9.92099i q^{27} -2.82591i q^{28} +4.08612 q^{29} -9.92099 q^{31} +11.8656i q^{32} +7.83488i q^{33} -10.7469 q^{34} +30.7842 q^{36} +8.30326i q^{37} -2.63010i q^{38} +1.42533 q^{39} -1.83488 q^{41} -4.59942i q^{42} -0.574672i q^{43} -12.6608 q^{44} -3.97526 q^{46} -7.09508i q^{47} +31.4845i q^{48} +6.66975 q^{49} -12.4343 q^{51} +2.30326i q^{52} +4.30326i q^{53} +26.0932 q^{54} +4.40955 q^{56} -3.04306i q^{57} +10.7469i q^{58} +2.68553 q^{59} +12.4095 q^{61} -26.0932i q^{62} -3.59756i q^{63} -10.5150 q^{64} -20.6065 q^{66} +2.70570i q^{67} -20.0932i q^{68} -4.59942 q^{69} -7.40058 q^{71} +48.0356i q^{72} +12.0861i q^{73} -21.8384 q^{74} +4.91744 q^{76} +1.47959i q^{77} +3.74876i q^{78} +6.68553 q^{79} +11.4095 q^{81} -4.82591i q^{82} -6.66079i q^{83} +8.59942 q^{84} +1.51145 q^{86} +12.4343i q^{87} -19.7559i q^{88} -14.6065 q^{89} +0.269169 q^{91} -7.43244i q^{92} -30.1902i q^{93} +18.6608 q^{94} -36.1076 q^{96} -17.4526i q^{97} +17.5421i q^{98} -16.1180 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{4} - 16 q^{9}+O(q^{10}) \) 8 * q - 16 * q^4 - 16 * q^9	\( 8 q - 16 q^{4} - 16 q^{9} + 8 q^{11} + 16 q^{14} + 8 q^{16} - 8 q^{19} - 8 q^{21} + 48 q^{24} + 8 q^{26} - 8 q^{29} + 8 q^{31} + 8 q^{34} + 80 q^{36} + 24 q^{39} + 32 q^{41} - 48 q^{44} - 40 q^{49} - 72 q^{51} + 40 q^{54} - 24 q^{56} + 40 q^{61} + 8 q^{64} - 56 q^{66} - 56 q^{69} - 40 q^{71} - 64 q^{74} + 16 q^{76} + 32 q^{79} + 32 q^{81} + 88 q^{84} - 16 q^{86} - 8 q^{89} - 72 q^{91} + 96 q^{94} - 104 q^{96} + 8 q^{99}+O(q^{100}) \) 8 * q - 16 * q^4 - 16 * q^9 + 8 * q^11 + 16 * q^14 + 8 * q^16 - 8 * q^19 - 8 * q^21 + 48 * q^24 + 8 * q^26 - 8 * q^29 + 8 * q^31 + 8 * q^34 + 80 * q^36 + 24 * q^39 + 32 * q^41 - 48 * q^44 - 40 * q^49 - 72 * q^51 + 40 * q^54 - 24 * q^56 + 40 * q^61 + 8 * q^64 - 56 * q^66 - 56 * q^69 - 40 * q^71 - 64 * q^74 + 16 * q^76 + 32 * q^79 + 32 * q^81 + 88 * q^84 - 16 * q^86 - 8 * q^89 - 72 * q^91 + 96 * q^94 - 104 * q^96 + 8 * q^99

Introduction

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