LMFDB - Embedded newform 45.4.b.b.19.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [45,4,Mod(19,45)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(45, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("45.19");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 45 = 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	45.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$2.65508595026$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{41})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 21x^{2} + 100 $ x^4 + 21*x^2 + 100
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 15)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			19.3
Root			$2.70156i$ of defining polynomial
Character	$\chi$	$=$	45.19
Dual form			45.4.b.b.19.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.70156i q^{2} +5.10469 q^{4} +(8.10469 - 7.70156i) q^{5} +22.2094i q^{7} +22.2984i q^{8} +O(q^{10})$	\(q+1.70156i q^{2} +5.10469 q^{4} +(8.10469 - 7.70156i) q^{5} +22.2094i q^{7} +22.2984i q^{8} +(13.1047 + 13.7906i) q^{10} +1.79063 q^{11} -58.2094i q^{13} -37.7906 q^{14} +2.89531 q^{16} -18.9844i q^{17} -104.837 q^{19} +(41.3719 - 39.3141i) q^{20} +3.04686i q^{22} -49.6125i q^{23} +(6.37188 - 124.837i) q^{25} +99.0469 q^{26} +113.372i q^{28} -293.466 q^{29} +64.4187 q^{31} +183.314i q^{32} +32.3031 q^{34} +(171.047 + 180.000i) q^{35} -19.8844i q^{37} -178.388i q^{38} +(171.733 + 180.722i) q^{40} +165.581 q^{41} +247.350i q^{43} +9.14059 q^{44} +84.4187 q^{46} -384.544i q^{47} -150.256 q^{49} +(212.419 + 10.8422i) q^{50} -297.141i q^{52} +463.528i q^{53} +(14.5125 - 13.7906i) q^{55} -495.234 q^{56} -499.350i q^{58} -73.7906 q^{59} -137.350 q^{61} +109.612i q^{62} -288.758 q^{64} +(-448.303 - 471.769i) q^{65} +173.906i q^{67} -96.9093i q^{68} +(-306.281 + 291.047i) q^{70} +594.281 q^{71} -320.231i q^{73} +33.8345 q^{74} -535.163 q^{76} +39.7687i q^{77} +770.469 q^{79} +(23.4656 - 22.2984i) q^{80} +281.747i q^{82} +173.925i q^{83} +(-146.209 - 153.862i) q^{85} -420.881 q^{86} +39.9282i q^{88} +1019.02 q^{89} +1292.79 q^{91} -253.256i q^{92} +654.325 q^{94} +(-849.675 + 807.412i) q^{95} +384.375i q^{97} -255.670i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 18 q^{4} - 6 q^{5}+O(q^{10}) $ 4 * q - 18 * q^4 - 6 * q^5	$ 4 q - 18 q^{4} - 6 q^{5} + 14 q^{10} + 84 q^{11} - 228 q^{14} + 50 q^{16} - 112 q^{19} + 396 q^{20} + 256 q^{25} + 12 q^{26} - 636 q^{29} + 104 q^{31} - 716 q^{34} + 300 q^{35} + 418 q^{40} + 816 q^{41} - 1116 q^{44} + 184 q^{46} - 140 q^{49} + 696 q^{50} - 864 q^{55} - 60 q^{56} - 372 q^{59} + 680 q^{61} + 958 q^{64} - 948 q^{65} + 1080 q^{70} + 72 q^{71} + 3132 q^{74} - 2448 q^{76} - 760 q^{79} - 444 q^{80} - 508 q^{85} - 4296 q^{86} + 2232 q^{89} + 1944 q^{91} + 3232 q^{94} - 2784 q^{95}+O(q^{100}) $ 4 * q - 18 * q^4 - 6 * q^5 + 14 * q^10 + 84 * q^11 - 228 * q^14 + 50 * q^16 - 112 * q^19 + 396 * q^20 + 256 * q^25 + 12 * q^26 - 636 * q^29 + 104 * q^31 - 716 * q^34 + 300 * q^35 + 418 * q^40 + 816 * q^41 - 1116 * q^44 + 184 * q^46 - 140 * q^49 + 696 * q^50 - 864 * q^55 - 60 * q^56 - 372 * q^59 + 680 * q^61 + 958 * q^64 - 948 * q^65 + 1080 * q^70 + 72 * q^71 + 3132 * q^74 - 2448 * q^76 - 760 * q^79 - 444 * q^80 - 508 * q^85 - 4296 * q^86 + 2232 * q^89 + 1944 * q^91 + 3232 * q^94 - 2784 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/45\mathbb{Z}\right)^\times$.

$n$	$11$	$37$
$\chi(n)$	$1$	$-1$

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 $

$2$			1.70156i			0.601593i	0.953688	+	0.300797i	$0.0972525\pi$
$2$			1.70156i			0.601593i	−0.953688	+	0.300797i	$0.902747\pi$
$3$		0			0
$3$		0			0
$4$	5.10469			0.638086
$5$	8.10469	−	7.70156i	0.724905	−	0.688849i
$5$	8.10469	−	7.70156i	0.724905	−	0.688849i
$6$		0			0
$7$			22.2094i			1.19919i	0.800302	+	0.599597i	$0.204672\pi$
$7$			22.2094i			1.19919i	−0.800302	+	0.599597i	$0.795328\pi$
$8$			22.2984i			0.985461i
$9$		0			0
$10$	13.1047	+	13.7906i	0.414407	+	0.436098i
$11$	1.79063			0.0490813			0.0245407	−	0.999699i	$-0.492188\pi$
$11$	1.79063			0.0490813			0.0245407	+	0.999699i	$0.492188\pi$
$12$		0			0
$13$		−	58.2094i		−	1.24188i	−0.783860	−	0.620938i	$-0.786752\pi$
$13$		−	58.2094i		−	1.24188i	0.783860	−	0.620938i	$-0.213248\pi$
$14$	−37.7906			−0.721426
$15$		0			0
$16$	2.89531			0.0452393
$17$		−	18.9844i		−	0.270846i	−0.990788	−	0.135423i	$-0.956761\pi$
$17$		−	18.9844i		−	0.270846i	0.990788	−	0.135423i	$-0.0432394\pi$
$18$		0			0
$19$	−104.837			−1.26586			−0.632931	−	0.774208i	$-0.718148\pi$
$19$	−104.837			−1.26586			−0.632931	+	0.774208i	$0.718148\pi$
$20$	41.3719	−	39.3141i	0.462552	−	0.439545i
$21$		0			0
$22$			3.04686i			0.0295270i
$23$		−	49.6125i		−	0.449779i	−0.974384	−	0.224890i	$-0.927798\pi$
$23$		−	49.6125i		−	0.449779i	0.974384	−	0.224890i	$-0.0722021\pi$
$24$		0			0
$25$	6.37188	−	124.837i	0.0509751	−	0.998700i
$26$	99.0469			0.747103
$27$		0			0
$28$			113.372i			0.765188i
$29$	−293.466			−1.87914			−0.939572	−	0.342350i	$-0.888777\pi$
$29$	−293.466			−1.87914			−0.939572	+	0.342350i	$0.888777\pi$
$30$		0			0
$31$	64.4187			0.373224			0.186612	−	0.982434i	$-0.440249\pi$
$31$	64.4187			0.373224			0.186612	+	0.982434i	$0.440249\pi$
$32$			183.314i			1.01268i
$33$		0			0
$34$	32.3031			0.162939
$35$	171.047	+	180.000i	0.826063	+	0.869302i
$36$		0			0
$37$		−	19.8844i		−	0.0883505i	−0.999024	−	0.0441752i	$-0.985934\pi$
$37$		−	19.8844i		−	0.0883505i	0.999024	−	0.0441752i	$-0.0140660\pi$
$38$		−	178.388i		−	0.761534i
$39$		0			0
$40$	171.733	+	180.722i	0.678834	+	0.714366i
$41$	165.581			0.630718			0.315359	−	0.948972i	$-0.397875\pi$
$41$	165.581			0.630718			0.315359	+	0.948972i	$0.397875\pi$
$42$		0			0
$43$			247.350i			0.877221i	0.898677	+	0.438611i	$0.144529\pi$
$43$			247.350i			0.877221i	−0.898677	+	0.438611i	$0.855471\pi$
$44$	9.14059			0.0313181
$45$		0			0
$46$	84.4187			0.270584
$47$		−	384.544i		−	1.19344i	−0.802451	−	0.596718i	$-0.796471\pi$
$47$		−	384.544i		−	1.19344i	0.802451	−	0.596718i	$-0.203529\pi$
$48$		0			0
$49$	−150.256			−0.438065
$50$	212.419	+	10.8422i	0.600811	+	0.0306662i
$51$		0			0
$52$		−	297.141i		−	0.792423i
$53$			463.528i			1.20133i	0.799501	+	0.600665i	$0.205097\pi$
$53$			463.528i			1.20133i	−0.799501	+	0.600665i	$0.794903\pi$
$54$		0			0
$55$	14.5125	−	13.7906i	0.0355793	−	0.0338096i
$56$	−495.234			−1.18176
$57$		0			0
$58$		−	499.350i		−	1.13048i
$59$	−73.7906			−0.162826			−0.0814129	−	0.996680i	$-0.525943\pi$
$59$	−73.7906			−0.162826			−0.0814129	+	0.996680i	$0.525943\pi$
$60$		0			0
$61$	−137.350			−0.288293			−0.144146	−	0.989556i	$-0.546044\pi$
$61$	−137.350			−0.288293			−0.144146	+	0.989556i	$0.546044\pi$
$62$			109.612i			0.224529i
$63$		0			0
$64$	−288.758			−0.563980
$65$	−448.303	−	471.769i	−0.855464	−	0.900242i
$66$		0			0
$67$			173.906i			0.317105i	0.987351	+	0.158552i	$0.0506827\pi$
$67$			173.906i			0.317105i	−0.987351	+	0.158552i	$0.949317\pi$
$68$		−	96.9093i		−	0.172823i
$69$		0			0
$70$	−306.281	+	291.047i	−0.522966	+	0.496954i
$71$	594.281			0.993355			0.496677	−	0.867935i	$-0.334553\pi$
$71$	594.281			0.993355			0.496677	+	0.867935i	$0.334553\pi$
$72$		0			0
$73$		−	320.231i		−	0.513428i	−0.966487	−	0.256714i	$-0.917360\pi$
$73$		−	320.231i		−	0.513428i	0.966487	−	0.256714i	$-0.0826398\pi$
$74$	33.8345			0.0531510
$75$		0			0
$76$	−535.163			−0.807728
$77$			39.7687i			0.0588580i
$78$		0			0
$79$	770.469			1.09727			0.548636	−	0.836061i	$-0.315147\pi$
$79$	770.469			1.09727			0.548636	+	0.836061i	$0.315147\pi$
$80$	23.4656	−	22.2984i	0.0327942	−	0.0311630i
$81$		0			0
$82$			281.747i			0.379436i
$83$			173.925i			0.230009i	0.993365	+	0.115004i	$0.0366882\pi$
$83$			173.925i			0.230009i	−0.993365	+	0.115004i	$0.963312\pi$
$84$		0			0
$85$	−146.209	−	153.862i	−0.186572	−	0.196338i
$86$	−420.881			−0.527730
$87$		0			0
$88$			39.9282i			0.0483677i
$89$	1019.02			1.21367			0.606834	−	0.794829i	$-0.292439\pi$
$89$	1019.02			1.21367			0.606834	+	0.794829i	$0.292439\pi$
$90$		0			0
$91$	1292.79			1.48925
$92$		−	253.256i		−	0.286998i
$93$		0			0
$94$	654.325			0.717962
$95$	−849.675	+	807.412i	−0.917630	+	0.871987i
$96$		0			0
$97$			384.375i			0.402344i	0.979556	+	0.201172i	$0.0644750\pi$
$97$			384.375i			0.402344i	−0.979556	+	0.201172i	$0.935525\pi$
$98$		−	255.670i		−	0.263537i
$99$		0			0
$100$	32.5265	−	637.256i	0.0325265	−	0.637256i
$101$	−34.4906			−0.0339796			−0.0169898	−	0.999856i	$-0.505408\pi$
$101$	−34.4906			−0.0339796			−0.0169898	+	0.999856i	$0.505408\pi$
$102$		0			0
$103$			1756.30i			1.68013i	0.542484	+	0.840066i	$0.317484\pi$
$103$			1756.30i			1.68013i	−0.542484	+	0.840066i	$0.682516\pi$
$104$	1297.98			1.22382
$105$		0			0
$106$	−788.722			−0.722712
$107$			1361.74i			1.23032i	0.788403	+	0.615159i	$0.210908\pi$
$107$			1361.74i			1.23032i	−0.788403	+	0.615159i	$0.789092\pi$
$108$		0			0
$109$	−321.119			−0.282180			−0.141090	−	0.989997i	$-0.545061\pi$
$109$	−321.119			−0.282180			−0.141090	+	0.989997i	$0.545061\pi$
$110$	23.4656	+	24.6939i	0.0203396	+	0.0214043i
$111$		0			0
$112$			64.3031i			0.0542506i
$113$		−	1582.25i		−	1.31721i	−0.752487	−	0.658607i	$-0.771146\pi$
$113$		−	1582.25i		−	1.31721i	0.752487	−	0.658607i	$-0.228854\pi$
$114$		0			0
$115$	−382.094	−	402.094i	−0.309830	−	0.326047i
$116$	−1498.05			−1.19906
$117$		0			0
$118$		−	125.559i		−	0.0979549i
$119$	421.631			0.324797
$120$		0			0
$121$	−1327.79			−0.997591
$122$		−	233.709i		−	0.173435i
$123$		0			0
$124$	328.837			0.238149
$125$	−909.802	−	1060.84i	−0.651001	−	0.759077i
$126$		0			0
$127$		−	1197.14i		−	0.836449i	−0.908344	−	0.418225i	$-0.862652\pi$
$127$		−	1197.14i		−	0.836449i	0.908344	−	0.418225i	$-0.137348\pi$
$128$			975.173i			0.673390i
$129$		0			0
$130$	802.744	−	762.816i	0.541579	−	0.514641i
$131$	321.647			0.214522			0.107261	−	0.994231i	$-0.465792\pi$
$131$	321.647			0.214522			0.107261	+	0.994231i	$0.465792\pi$
$132$		0			0
$133$		−	2328.37i		−	1.51801i
$134$	−295.912			−0.190768
$135$		0			0
$136$	423.322			0.266909
$137$		−	354.291i		−	0.220942i	−0.993879	−	0.110471i	$-0.964764\pi$
$137$		−	354.291i		−	0.220942i	0.993879	−	0.110471i	$-0.0352360\pi$
$138$		0			0
$139$	−77.2562			−0.0471424			−0.0235712	−	0.999722i	$-0.507504\pi$
$139$	−77.2562			−0.0471424			−0.0235712	+	0.999722i	$0.507504\pi$
$140$	873.141	+	918.844i	0.527099	+	0.554689i
$141$		0			0
$142$			1011.21i			0.597595i
$143$		−	104.231i		−	0.0609529i
$144$		0			0
$145$	−2378.45	+	2260.14i	−1.36220	+	1.29445i
$146$	544.893			0.308875
$147$		0			0
$148$		−	101.503i		−	0.0563752i
$149$	−1705.38			−0.937651			−0.468826	−	0.883291i	$-0.655323\pi$
$149$	−1705.38			−0.937651			−0.468826	+	0.883291i	$0.655323\pi$
$150$		0			0
$151$	758.281			0.408663			0.204331	−	0.978902i	$-0.434498\pi$
$151$	758.281			0.408663			0.204331	+	0.978902i	$0.434498\pi$
$152$		−	2337.71i		−	1.24746i
$153$		0			0
$154$	−67.6689			−0.0354086
$155$	522.094	−	496.125i	0.270552	−	0.257095i
$156$		0			0
$157$		−	1769.05i		−	0.899273i	−0.893212	−	0.449636i	$-0.851554\pi$
$157$		−	1769.05i		−	0.899273i	0.893212	−	0.449636i	$-0.148446\pi$
$158$			1311.00i			0.660111i
$159$		0			0
$160$	1411.80	+	1485.70i	0.697581	+	0.734095i
$161$	1101.86			0.539372
$162$		0			0
$163$			881.719i			0.423690i	0.977303	+	0.211845i	$0.0679473\pi$
$163$			881.719i			0.423690i	−0.977303	+	0.211845i	$0.932053\pi$
$164$	845.240			0.402452
$165$		0			0
$166$	−295.944			−0.138372
$167$			216.900i			0.100504i	0.998737	+	0.0502522i	$0.0160025\pi$
$167$			216.900i			0.100504i	−0.998737	+	0.0502522i	$0.983997\pi$
$168$		0			0
$169$	−1191.33			−0.542254
$170$	261.806	−	248.784i	0.118116	−	0.112241i
$171$		0			0
$172$			1262.64i			0.559742i
$173$			4125.91i			1.81322i	0.421970	+	0.906610i	$0.361339\pi$
$173$			4125.91i			1.81322i	−0.421970	+	0.906610i	$0.638661\pi$
$174$		0			0
$175$	2772.56	+	141.515i	1.19763	+	0.0611289i
$176$	5.18443			0.00222040
$177$		0			0
$178$			1733.93i			0.730134i
$179$	−3213.14			−1.34168			−0.670842	−	0.741600i	$-0.734067\pi$
$179$	−3213.14			−1.34168			−0.670842	+	0.741600i	$0.734067\pi$
$180$		0			0
$181$	3394.42			1.39395			0.696976	−	0.717095i	$-0.254529\pi$
$181$	3394.42			1.39395			0.696976	+	0.717095i	$0.254529\pi$
$182$			2199.77i			0.895921i
$183$		0			0
$184$	1106.28			0.443240
$185$	−153.141	−	161.156i	−0.0608601	−	0.0640457i
$186$		0			0
$187$		−	33.9939i		−	0.0132935i
$188$		−	1962.98i		−	0.761514i
$189$		0			0
$190$	−1373.86	−	1445.77i	−0.524581	−	0.552040i
$191$	3467.49			1.31361			0.656804	−	0.754062i	$-0.271908\pi$
$191$	3467.49			1.31361			0.656804	+	0.754062i	$0.271908\pi$
$192$		0			0
$193$		−	1792.14i		−	0.668401i	−0.942502	−	0.334200i	$-0.891534\pi$
$193$		−	1792.14i		−	0.668401i	0.942502	−	0.334200i	$-0.108466\pi$
$194$	−654.038			−0.242047
$195$		0			0
$196$	−767.011			−0.279523
$197$		−	1678.19i		−	0.606935i	−0.952842	−	0.303467i	$-0.901856\pi$
$197$		−	1678.19i		−	0.606935i	0.952842	−	0.303467i	$-0.0981443\pi$
$198$		0			0
$199$	−3108.23			−1.10722			−0.553610	−	0.832776i	$-0.686750\pi$
$199$	−3108.23			−1.10722			−0.553610	+	0.832776i	$0.686750\pi$
$200$	2783.68	+	142.083i	0.984180	+	0.0502339i
$201$		0			0
$202$		−	58.6878i		−	0.0204419i
$203$		−	6517.69i		−	2.25346i
$204$		0			0
$205$	1341.98	−	1275.23i	0.457211	−	0.434469i
$206$	−2988.46			−1.01076
$207$		0			0
$208$		−	168.534i		−	0.0561815i
$209$	−187.725			−0.0621301
$210$		0			0
$211$	−4473.27			−1.45949			−0.729745	−	0.683719i	$-0.760361\pi$
$211$	−4473.27			−1.45949			−0.729745	+	0.683719i	$0.760361\pi$
$212$			2366.17i			0.766551i
$213$		0			0
$214$	−2317.08			−0.740151
$215$	1904.98	+	2004.69i	0.604273	+	0.635902i
$216$		0			0
$217$			1430.70i			0.447568i
$218$		−	546.403i		−	0.169757i
$219$		0			0
$220$	74.0816	−	70.3968i	0.0227026	−	0.0215734i
$221$	−1105.07			−0.336357
$222$		0			0
$223$			1753.42i			0.526535i	0.964723	+	0.263268i	$0.0848003\pi$
$223$			1753.42i			0.526535i	−0.964723	+	0.263268i	$0.915200\pi$
$224$	−4071.29			−1.21440
$225$		0			0
$226$	2692.29			0.792427
$227$			936.900i			0.273939i	0.990575	+	0.136970i	$0.0437363\pi$
$227$			936.900i			0.273939i	−0.990575	+	0.136970i	$0.956264\pi$
$228$		0			0
$229$	2582.06			0.745096			0.372548	−	0.928013i	$-0.378484\pi$
$229$	2582.06			0.745096			0.372548	+	0.928013i	$0.378484\pi$
$230$	684.187	−	650.156i	0.196148	−	0.186391i
$231$		0			0
$232$		−	6543.82i		−	1.85182i
$233$			2295.01i			0.645284i	0.946521	+	0.322642i	$0.104571\pi$
$233$			2295.01i			0.645284i	−0.946521	+	0.322642i	$0.895429\pi$
$234$		0			0
$235$	−2961.59	−	3116.61i	−0.822096	−	0.865128i
$236$	−376.678			−0.103897
$237$		0			0
$238$			717.432i			0.195396i
$239$	−2294.01			−0.620866			−0.310433	−	0.950595i	$-0.600474\pi$
$239$	−2294.01			−0.620866			−0.310433	+	0.950595i	$0.600474\pi$
$240$		0			0
$241$	382.287			0.102180			0.0510898	−	0.998694i	$-0.483731\pi$
$241$	382.287			0.102180			0.0510898	+	0.998694i	$0.483731\pi$
$242$		−	2259.32i		−	0.600144i
$243$		0			0
$244$	−701.128			−0.183956
$245$	−1217.78	+	1157.21i	−0.317555	+	0.301760i
$246$		0			0
$247$			6102.52i			1.57204i
$248$			1436.44i			0.367798i
$249$		0			0
$250$	1805.09	−	1548.08i	0.456655	−	0.391638i
$251$	2259.98			0.568322			0.284161	−	0.958777i	$-0.408285\pi$
$251$	2259.98			0.568322			0.284161	+	0.958777i	$0.408285\pi$
$252$		0			0
$253$		−	88.8375i		−	0.0220758i
$254$	2037.01			0.503202
$255$		0			0
$256$	−3969.38			−0.969087
$257$			92.7843i			0.0225203i	0.999937	+	0.0112602i	$0.00358430\pi$
$257$			92.7843i			0.0225203i	−0.999937	+	0.0112602i	$0.996416\pi$
$258$		0			0
$259$	441.619			0.105949
$260$	−2288.45	−	2408.23i	−0.545859	−	0.574431i
$261$		0			0
$262$			547.302i			0.129055i
$263$		−	568.312i		−	0.133246i	−0.997778	−	0.0666229i	$-0.978778\pi$
$263$		−	568.312i		−	0.133246i	0.997778	−	0.0666229i	$-0.0212224\pi$
$264$		0			0
$265$	3569.89	+	3756.75i	0.827534	+	0.870850i
$266$	3961.87			0.913226
$267$		0			0
$268$			887.737i			0.202340i
$269$	7582.41			1.71862			0.859309	−	0.511458i	$-0.170894\pi$
$269$	7582.41			1.71862			0.859309	+	0.511458i	$0.170894\pi$
$270$		0			0
$271$	7943.69			1.78061			0.890304	−	0.455366i	$-0.150492\pi$
$271$	7943.69			1.78061			0.890304	+	0.455366i	$0.150492\pi$
$272$		−	54.9657i		−	0.0122529i
$273$		0			0
$274$	602.847			0.132917
$275$	11.4097	−	223.537i	0.00250192	−	0.0490175i
$276$		0			0
$277$			6823.00i			1.47998i	0.672618	+	0.739990i	$0.265170\pi$
$277$			6823.00i			1.47998i	−0.672618	+	0.739990i	$0.734830\pi$
$278$		−	131.456i		−	0.0283605i
$279$		0			0
$280$	−4013.72	+	3814.08i	−0.856663	+	0.814053i
$281$	−3315.86			−0.703942			−0.351971	−	0.936011i	$-0.614488\pi$
$281$	−3315.86			−0.703942			−0.351971	+	0.936011i	$0.614488\pi$
$282$		0			0
$283$		−	6602.76i		−	1.38690i	−0.720504	−	0.693451i	$-0.756090\pi$
$283$		−	6602.76i		−	1.38690i	0.720504	−	0.693451i	$-0.243910\pi$
$284$	3033.62			0.633846
$285$		0			0
$286$	177.356			0.0366688
$287$			3677.46i			0.756353i
$288$		0			0
$289$	4552.59			0.926642
$290$	−3845.77	−	4047.07i	−0.778730	−	0.819491i
$291$		0			0
$292$		−	1634.68i		−	0.327611i
$293$			5814.14i			1.15927i	0.814877	+	0.579634i	$0.196805\pi$
$293$			5814.14i			1.15927i	−0.814877	+	0.579634i	$0.803195\pi$
$294$		0			0
$295$	−598.050	+	568.303i	−0.118033	+	0.112162i
$296$	443.390			0.0870660
$297$		0			0
$298$		−	2901.81i		−	0.564084i
$299$	−2887.91			−0.558570
$300$		0			0
$301$	−5493.49			−1.05196
$302$			1290.26i			0.245849i
$303$		0			0
$304$	−303.537			−0.0572667
$305$	−1113.18	+	1057.81i	−0.208985	+	0.198590i
$306$		0			0
$307$		−	8124.86i		−	1.51046i	−0.655462	−	0.755229i	$-0.727526\pi$
$307$		−	8124.86i		−	1.51046i	0.655462	−	0.755229i	$-0.272474\pi$
$308$			203.007i			0.0375564i
$309$		0			0
$310$	844.187	+	888.375i	0.154667	+	0.162762i
$311$	−7336.26			−1.33762			−0.668812	−	0.743432i	$-0.733197\pi$
$311$	−7336.26			−1.33762			−0.668812	+	0.743432i	$0.733197\pi$
$312$		0			0
$313$			2202.66i			0.397768i	0.980023	+	0.198884i	$0.0637318\pi$
$313$			2202.66i			0.397768i	−0.980023	+	0.198884i	$0.936268\pi$
$314$	3010.15			0.540996
$315$		0			0
$316$	3933.00			0.700154
$317$		−	10008.9i		−	1.77336i	−0.462386	−	0.886679i	$-0.653007\pi$
$317$		−	10008.9i		−	1.77336i	0.462386	−	0.886679i	$-0.346993\pi$
$318$		0			0
$319$	−525.488			−0.0922309
$320$	−2340.29	+	2223.89i	−0.408832	+	0.388497i
$321$		0			0
$322$			1874.89i			0.324483i
$323$			1990.27i			0.342854i
$324$		0			0
$325$	−7266.71	−	370.903i	−1.24026	−	0.0633046i
$326$	−1500.30			−0.254889
$327$		0			0
$328$			3692.20i			0.621548i
$329$	8540.47			1.43116
$330$		0			0
$331$	−8695.94			−1.44402			−0.722012	−	0.691881i	$-0.756782\pi$
$331$	−8695.94			−1.44402			−0.722012	+	0.691881i	$0.756782\pi$
$332$			887.832i			0.146765i
$333$		0			0
$334$	−369.069			−0.0604627
$335$	1339.35	+	1409.46i	0.218437	+	0.229871i
$336$		0			0
$337$		−	7400.61i		−	1.19625i	−0.801402	−	0.598126i	$-0.795912\pi$
$337$		−	7400.61i		−	1.19625i	0.801402	−	0.598126i	$-0.204088\pi$
$338$		−	2027.12i		−	0.326216i
$339$		0			0
$340$	−746.353	−	785.419i	−0.119049	−	0.125280i
$341$	115.350			0.0183183
$342$		0			0
$343$			4280.72i			0.673869i
$344$	−5515.52			−0.864467
$345$		0			0
$346$	−7020.49			−1.09082
$347$		−	7841.44i		−	1.21311i	−0.795040	−	0.606557i	$-0.792550\pi$
$347$		−	7841.44i		−	1.21311i	0.795040	−	0.606557i	$-0.207450\pi$
$348$		0			0
$349$	4961.26			0.760946			0.380473	−	0.924792i	$-0.375761\pi$
$349$	4961.26			0.760946			0.380473	+	0.924792i	$0.375761\pi$
$350$	−240.797	+	4717.69i	−0.0367748	+	0.720489i
$351$		0			0
$352$			328.247i			0.0497035i
$353$		−	12163.0i		−	1.83392i	−0.398981	−	0.916959i	$-0.630636\pi$
$353$		−	12163.0i		−	1.83392i	0.398981	−	0.916959i	$-0.369364\pi$
$354$		0			0
$355$	4816.46	−	4576.89i	0.720088	−	0.684271i
$356$	5201.80			0.774424
$357$		0			0
$358$		−	5467.36i		−	0.807148i
$359$	5193.79			0.763559			0.381779	−	0.924253i	$-0.375311\pi$
$359$	5193.79			0.763559			0.381779	+	0.924253i	$0.375311\pi$
$360$		0			0
$361$	4131.90			0.602406
$362$			5775.81i			0.838591i
$363$		0			0
$364$	6599.31			0.950268
$365$	−2466.28	−	2595.37i	−0.353674	−	0.372187i
$366$		0			0
$367$		−	6086.09i		−	0.865644i	−0.901479	−	0.432822i	$-0.857518\pi$
$367$		−	6086.09i		−	0.865644i	0.901479	−	0.432822i	$-0.142482\pi$
$368$		−	143.644i		−	0.0203477i
$369$		0			0
$370$	274.218	−	260.578i	0.0385295	−	0.0366130i
$371$	−10294.7			−1.44063
$372$		0			0
$373$			10581.9i			1.46893i	0.678646	+	0.734466i	$0.262567\pi$
$373$			10581.9i			1.46893i	−0.678646	+	0.734466i	$0.737433\pi$
$374$	57.8428			0.00799727
$375$		0			0
$376$	8574.72			1.17608
$377$			17082.4i			2.33366i
$378$		0			0
$379$	−11655.2			−1.57964			−0.789822	−	0.613336i	$-0.789827\pi$
$379$	−11655.2			−1.57964			−0.789822	+	0.613336i	$0.789827\pi$
$380$	−4337.32	+	4121.59i	−0.585526	+	0.556403i
$381$		0			0
$382$			5900.16i			0.790257i
$383$			6364.97i			0.849177i	0.905387	+	0.424588i	$0.139581\pi$
$383$			6364.97i			0.849177i	−0.905387	+	0.424588i	$0.860419\pi$
$384$		0			0
$385$	306.281	+	322.313i	0.0405442	+	0.0426665i
$386$	3049.44			0.402105
$387$		0			0
$388$			1962.11i			0.256730i
$389$	6134.33			0.799545			0.399773	−	0.916614i	$-0.369089\pi$
$389$	6134.33			0.799545			0.399773	+	0.916614i	$0.369089\pi$
$390$		0			0
$391$	−941.862			−0.121821
$392$		−	3350.48i		−	0.431696i
$393$		0			0
$394$	2855.55			0.365128
$395$	6244.41	−	5933.81i	0.795418	−	0.755854i
$396$		0			0
$397$			9746.46i			1.23214i	0.787690	+	0.616072i	$0.211277\pi$
$397$			9746.46i			1.23214i	−0.787690	+	0.616072i	$0.788723\pi$
$398$		−	5288.85i		−	0.666096i
$399$		0			0
$400$	18.4486	−	361.444i	0.00230607	−	0.0451805i
$401$	1306.44			0.162695			0.0813474	−	0.996686i	$-0.474078\pi$
$401$	1306.44			0.162695			0.0813474	+	0.996686i	$0.474078\pi$
$402$		0			0
$403$		−	3749.77i		−	0.463498i
$404$	−176.063			−0.0216819
$405$		0			0
$406$	11090.2			1.35566
$407$		−	35.6055i		−	0.00433636i
$408$		0			0
$409$	3876.93			0.468709			0.234354	−	0.972151i	$-0.424702\pi$
$409$	3876.93			0.468709			0.234354	+	0.972151i	$0.424702\pi$
$410$	2169.89	+	2283.47i	0.261374	+	0.275055i
$411$		0			0
$412$			8965.38i			1.07207i
$413$		−	1638.84i		−	0.195260i
$414$		0			0
$415$	1339.49	+	1409.61i	0.158441	+	0.166735i
$416$	10670.6			1.25762
$417$		0			0
$418$		−	319.426i		−	0.0373771i
$419$	−16022.5			−1.86814			−0.934071	−	0.357088i	$-0.883770\pi$
$419$	−16022.5			−1.86814			−0.934071	+	0.357088i	$0.883770\pi$
$420$		0			0
$421$	−8119.73			−0.939980			−0.469990	−	0.882672i	$-0.655742\pi$
$421$	−8119.73			−0.939980			−0.469990	+	0.882672i	$0.655742\pi$
$422$		−	7611.54i		−	0.878019i
$423$		0			0
$424$	−10336.0			−1.18386
$425$	−2369.96	−	120.966i	−0.270494	−	0.0138064i
$426$		0			0
$427$		−	3050.46i		−	0.345719i
$428$			6951.24i			0.785049i
$429$		0			0
$430$	−3411.11	+	3241.44i	−0.382554	+	0.363526i
$431$	5713.99			0.638592			0.319296	−	0.947655i	$-0.396554\pi$
$431$	5713.99			0.638592			0.319296	+	0.947655i	$0.396554\pi$
$432$		0			0
$433$		−	6251.34i		−	0.693811i	−0.937900	−	0.346906i	$-0.887232\pi$
$433$		−	6251.34i		−	0.693811i	0.937900	−	0.346906i	$-0.112768\pi$
$434$	−2434.42			−0.269254
$435$		0			0
$436$	−1639.21			−0.180055
$437$			5201.25i			0.569358i
$438$		0			0
$439$	4230.97			0.459984			0.229992	−	0.973192i	$-0.426130\pi$
$439$	4230.97			0.459984			0.229992	+	0.973192i	$0.426130\pi$
$440$	307.509	+	323.605i	0.0333180	+	0.0350620i
$441$		0			0
$442$		−	1880.34i		−	0.202350i
$443$			6314.29i			0.677203i	0.940930	+	0.338601i	$0.109954\pi$
$443$			6314.29i			0.677203i	−0.940930	+	0.338601i	$0.890046\pi$
$444$		0			0
$445$	8258.88	−	7848.08i	0.879794	−	0.836033i
$446$	−2983.55			−0.316760
$447$		0			0
$448$		−	6413.13i		−	0.676321i
$449$	−9349.71			−0.982717			−0.491358	−	0.870957i	$-0.663499\pi$
$449$	−9349.71			−0.982717			−0.491358	+	0.870957i	$0.663499\pi$
$450$		0			0
$451$	296.494			0.0309565
$452$		−	8076.87i		−	0.840496i
$453$		0			0
$454$	−1594.19			−0.164800
$455$	10477.7	−	9956.53i	1.07956	−	1.02587i
$456$		0			0
$457$		−	9547.46i		−	0.977268i	−0.872489	−	0.488634i	$-0.837495\pi$
$457$		−	9547.46i		−	0.977268i	0.872489	−	0.488634i	$-0.162505\pi$
$458$			4393.53i			0.448245i
$459$		0			0
$460$	−1950.47	−	2052.56i	−0.197698	−	0.208046i
$461$	−6237.23			−0.630145			−0.315073	−	0.949068i	$-0.602029\pi$
$461$	−6237.23			−0.630145			−0.315073	+	0.949068i	$0.602029\pi$
$462$		0			0
$463$		−	6469.98i		−	0.649428i	−0.945812	−	0.324714i	$-0.894732\pi$
$463$		−	6469.98i		−	0.649428i	0.945812	−	0.324714i	$-0.105268\pi$
$464$	−849.675			−0.0850111
$465$		0			0
$466$	−3905.10			−0.388198
$467$		−	7206.64i		−	0.714097i	−0.934086	−	0.357049i	$-0.883783\pi$
$467$		−	7206.64i		−	0.714097i	0.934086	−	0.357049i	$-0.116217\pi$
$468$		0			0
$469$	−3862.35			−0.380270
$470$	5303.10	−	5039.32i	0.520455	−	0.494567i
$471$		0			0
$472$		−	1645.42i		−	0.160458i
$473$			442.912i			0.0430552i
$474$		0			0
$475$	−668.012	+	13087.6i	−0.0645274	+	1.26422i
$476$	2152.29			0.207248
$477$		0			0
$478$		−	3903.39i		−	0.373509i
$479$	−10851.8			−1.03514			−0.517571	−	0.855640i	$-0.673164\pi$
$479$	−10851.8			−1.03514			−0.517571	+	0.855640i	$0.673164\pi$
$480$		0			0
$481$	−1157.46			−0.109720
$482$			650.485i			0.0614705i
$483$		0			0
$484$	−6777.97			−0.636549
$485$	2960.29	+	3115.24i	0.277154	+	0.291661i
$486$		0			0
$487$			12757.1i			1.18702i	0.804827	+	0.593510i	$0.202258\pi$
$487$			12757.1i			1.18702i	−0.804827	+	0.593510i	$0.797742\pi$
$488$		−	3062.69i		−	0.284101i
$489$		0			0
$490$	−1969.06	−	2072.13i	−0.181537	−	0.191039i
$491$	7016.52			0.644911			0.322455	−	0.946585i	$-0.395492\pi$
$491$	7016.52			0.644911			0.322455	+	0.946585i	$0.395492\pi$
$492$		0			0
$493$			5571.26i			0.508960i
$494$	−10383.8			−0.945729
$495$		0			0
$496$	186.512			0.0168844
$497$			13198.6i			1.19122i
$498$		0			0
$499$	−11372.3			−1.02023			−0.510113	−	0.860107i	$-0.670396\pi$
$499$	−11372.3			−1.02023			−0.510113	+	0.860107i	$0.670396\pi$
$500$	−4644.25	−	5415.27i	−0.415395	−	0.484356i
$501$		0			0
$502$			3845.50i			0.341899i
$503$		−	5587.37i		−	0.495285i	−0.968851	−	0.247643i	$-0.920344\pi$
$503$		−	5587.37i		−	0.495285i	0.968851	−	0.247643i	$-0.0796559\pi$
$504$		0			0
$505$	−279.535	+	265.631i	−0.0246320	+	0.0234068i
$506$	151.163			0.0132806
$507$		0			0
$508$		−	6111.03i		−	0.533726i
$509$	16256.7			1.41565			0.707825	−	0.706388i	$-0.249676\pi$
$509$	16256.7			1.41565			0.707825	+	0.706388i	$0.249676\pi$
$510$		0			0
$511$	7112.14			0.615699
$512$			1047.24i			0.0903943i
$513$		0			0
$514$	−157.878			−0.0135481
$515$	13526.3	+	14234.3i	1.15736	+	1.21794i
$516$		0			0
$517$		−	688.574i		−	0.0585754i
$518$			751.442i			0.0637384i
$519$		0			0
$520$	10519.7	−	9996.46i	0.887153	−	0.843026i
$521$	−19748.4			−1.66064			−0.830320	−	0.557286i	$-0.811843\pi$
$521$	−19748.4			−1.66064			−0.830320	+	0.557286i	$0.811843\pi$
$522$		0			0
$523$			7843.44i			0.655774i	0.944717	+	0.327887i	$0.106337\pi$
$523$			7843.44i			0.655774i	−0.944717	+	0.327887i	$0.893663\pi$
$524$	1641.91			0.136884
$525$		0			0
$526$	967.019			0.0801597
$527$		−	1222.95i		−	0.101086i
$528$		0			0
$529$	9705.60			0.797699
$530$	−6392.34	+	6074.39i	−0.523897	+	0.497839i
$531$		0			0
$532$		−	11885.6i		−	0.968622i
$533$		−	9638.38i		−	0.783273i
$534$		0			0
$535$	10487.5	+	11036.5i	0.847504	+	0.891865i
$536$	−3877.84			−0.312495
$537$		0			0
$538$			12902.0i			1.03391i
$539$	−269.053			−0.0215008
$540$		0			0
$541$	7383.29			0.586751			0.293376	−	0.955997i	$-0.405221\pi$
$541$	7383.29			0.586751			0.293376	+	0.955997i	$0.405221\pi$
$542$			13516.7i			1.07120i
$543$		0			0
$544$	3480.10			0.274280
$545$	−2602.57	+	2473.12i	−0.204554	+	0.194379i
$546$		0			0
$547$		−	3354.90i		−	0.262240i	−0.991367	−	0.131120i	$-0.958143\pi$
$547$		−	3354.90i		−	0.262240i	0.991367	−	0.131120i	$-0.0418573\pi$
$548$		−	1808.54i		−	0.140980i
$549$		0			0
$550$	380.363	+	19.4143i	0.0294886	+	0.00150514i
$551$	30766.2			2.37874
$552$		0			0
$553$			17111.6i			1.31584i
$554$	−11609.8			−0.890346
$555$		0			0
$556$	−394.369			−0.0300809
$557$		−	20771.8i		−	1.58012i	−0.613028	−	0.790061i	$-0.710049\pi$
$557$		−	20771.8i		−	1.58012i	0.613028	−	0.790061i	$-0.289951\pi$
$558$		0			0
$559$	14398.1			1.08940
$560$	495.234	+	521.156i	0.0373705	+	0.0393266i
$561$		0			0
$562$		−	5642.15i		−	0.423487i
$563$			7194.86i			0.538592i	0.963057	+	0.269296i	$0.0867910\pi$
$563$			7194.86i			0.538592i	−0.963057	+	0.269296i	$0.913209\pi$
$564$		0			0
$565$	−12185.8	−	12823.6i	−0.907361	−	0.954856i
$566$	11235.0			0.834350
$567$		0			0
$568$			13251.5i			0.978913i
$569$	11549.5			0.850931			0.425466	−	0.904975i	$-0.360110\pi$
$569$	11549.5			0.850931			0.425466	+	0.904975i	$0.360110\pi$
$570$		0			0
$571$	1482.54			0.108655			0.0543277	−	0.998523i	$-0.482698\pi$
$571$	1482.54			0.108655			0.0543277	+	0.998523i	$0.482698\pi$
$572$		−	532.068i		−	0.0388932i
$573$		0			0
$574$	−6257.42			−0.455017
$575$	−6193.50	−	316.125i	−0.449194	−	0.0229275i
$576$		0			0
$577$		−	15264.0i		−	1.10130i	−0.834737	−	0.550649i	$-0.814380\pi$
$577$		−	15264.0i		−	1.10130i	0.834737	−	0.550649i	$-0.185620\pi$
$578$			7746.52i			0.557462i
$579$		0			0
$580$	−12141.2	+	11537.3i	−0.869202	+	0.825968i
$581$	−3862.76			−0.275825
$582$		0			0
$583$			830.006i			0.0589628i
$584$	7140.66			0.505963
$585$		0			0
$586$	−9893.12			−0.697408
$587$		−	1736.89i		−	0.122128i	−0.998134	−	0.0610639i	$-0.980551\pi$
$587$		−	1736.89i		−	0.122128i	0.998134	−	0.0610639i	$-0.0194493\pi$
$588$		0			0
$589$	−6753.50			−0.472450
$590$	−967.003	−	1017.62i	−0.0674761	−	0.0710080i
$591$		0			0
$592$		−	57.5714i		−	0.00399691i
$593$		−	11764.8i		−	0.814707i	−0.913271	−	0.407353i	$-0.866452\pi$
$593$		−	11764.8i		−	0.814707i	0.913271	−	0.407353i	$-0.133548\pi$
$594$		0			0
$595$	3417.19	−	3247.22i	0.235447	−	0.223736i
$596$	−8705.42			−0.598302
$597$		0			0
$598$		−	4913.96i		−	0.336032i
$599$	−9451.99			−0.644737			−0.322369	−	0.946614i	$-0.604479\pi$
$599$	−9451.99			−0.644737			−0.322369	+	0.946614i	$0.604479\pi$
$600$		0			0
$601$	−3131.93			−0.212569			−0.106285	−	0.994336i	$-0.533895\pi$
$601$	−3131.93			−0.212569			−0.106285	+	0.994336i	$0.533895\pi$
$602$		−	9347.51i		−	0.632851i
$603$		0			0
$604$	3870.79			0.260762
$605$	−10761.4	+	10226.1i	−0.723159	+	0.687189i
$606$		0			0
$607$			22700.8i			1.51795i	0.651120	+	0.758975i	$0.274300\pi$
$607$			22700.8i			1.51795i	−0.651120	+	0.758975i	$0.725700\pi$
$608$		−	19218.2i		−	1.28191i
$609$		0			0
$610$	−1799.93	−	1894.14i	−0.119470	−	0.125724i
$611$	−22384.0			−1.48210
$612$		0			0
$613$		−	28911.6i		−	1.90494i	−0.304629	−	0.952471i	$-0.598532\pi$
$613$		−	28911.6i		−	1.90494i	0.304629	−	0.952471i	$-0.401468\pi$
$614$	13825.0			0.908681
$615$		0			0
$616$	−886.780			−0.0580023
$617$			5566.87i			0.363231i	0.983370	+	0.181616i	$0.0581326\pi$
$617$			5566.87i			0.363231i	−0.983370	+	0.181616i	$0.941867\pi$
$618$		0			0
$619$	−4150.32			−0.269492			−0.134746	−	0.990880i	$-0.543022\pi$
$619$	−4150.32			−0.269492			−0.134746	+	0.990880i	$0.543022\pi$
$620$	2665.12	−	2532.56i	0.172635	−	0.164049i
$621$		0			0
$622$		−	12483.1i		−	0.804705i
$623$			22631.9i			1.45542i
$624$		0			0
$625$	−15543.8	−	1590.90i	−0.994803	−	0.101818i
$626$	−3747.96			−0.239295
$627$		0			0
$628$		−	9030.46i		−	0.573813i
$629$	−377.492			−0.0239294
$630$		0			0
$631$	−4090.09			−0.258041			−0.129021	−	0.991642i	$-0.541183\pi$
$631$	−4090.09			−0.258041			−0.129021	+	0.991642i	$0.541183\pi$
$632$			17180.2i			1.08132i
$633$		0			0
$634$	17030.7			1.06684
$635$	−9219.85	−	9702.45i	−0.576187	−	0.606346i
$636$		0			0
$637$			8746.32i			0.544022i
$638$		−	894.150i		−	0.0554855i
$639$		0			0
$640$	7510.36	+	7903.47i	0.463864	+	0.488144i
$641$	−3909.35			−0.240890			−0.120445	−	0.992720i	$-0.538432\pi$
$641$	−3909.35			−0.240890			−0.120445	+	0.992720i	$0.538432\pi$
$642$		0			0
$643$		−	30539.5i		−	1.87303i	−0.350624	−	0.936516i	$-0.614031\pi$
$643$		−	30539.5i		−	1.87303i	0.350624	−	0.936516i	$-0.385969\pi$
$644$	5624.66			0.344166
$645$		0			0
$646$	−3386.58			−0.206259
$647$			12707.7i			0.772167i	0.922464	+	0.386083i	$0.126172\pi$
$647$			12707.7i			0.772167i	−0.922464	+	0.386083i	$0.873828\pi$
$648$		0			0
$649$	−132.132			−0.00799170
$650$	631.115	−	12364.8i	0.0380836	−	0.746132i
$651$		0			0
$652$			4500.90i			0.270351i
$653$			12777.6i			0.765737i	0.923803	+	0.382869i	$0.125064\pi$
$653$			12777.6i			0.765737i	−0.923803	+	0.382869i	$0.874936\pi$
$654$		0			0
$655$	2606.85	−	2477.18i	0.155508	−	0.147773i
$656$	479.410			0.0285332
$657$		0			0
$658$			14532.1i			0.860976i
$659$	23563.5			1.39287			0.696435	−	0.717620i	$-0.254768\pi$
$659$	23563.5			1.39287			0.696435	+	0.717620i	$0.254768\pi$
$660$		0			0
$661$	−4361.31			−0.256634			−0.128317	−	0.991733i	$-0.540958\pi$
$661$	−4361.31			−0.256634			−0.128317	+	0.991733i	$0.540958\pi$
$662$		−	14796.7i		−	0.868715i
$663$		0			0
$664$	−3878.25			−0.226665
$665$	−17932.1	−	18870.7i	−1.04568	−	1.10042i
$666$		0			0
$667$			14559.6i			0.845200i
$668$			1107.21i			0.0641304i
$669$		0			0
$670$	−2398.28	+	2278.99i	−0.138289	+	0.131410i
$671$	−245.943			−0.0141498
$672$		0			0
$673$			8203.52i			0.469870i	0.972011	+	0.234935i	$0.0754877\pi$
$673$			8203.52i			0.469870i	−0.972011	+	0.234935i	$0.924512\pi$
$674$	12592.6			0.719657
$675$		0			0
$676$	−6081.37			−0.346004
$677$		−	28057.1i		−	1.59279i	−0.604774	−	0.796397i	$-0.706737\pi$
$677$		−	28057.1i		−	1.59279i	0.604774	−	0.796397i	$-0.293263\pi$
$678$		0			0
$679$	−8536.73			−0.482488
$680$	3430.89	−	3260.24i	0.193483	−	0.183860i
$681$		0			0
$682$			196.275i			0.0110202i
$683$		−	3344.62i		−	0.187377i	−0.995602	−	0.0936885i	$-0.970134\pi$
$683$		−	3344.62i		−	0.187377i	0.995602	−	0.0936885i	$-0.0298658\pi$
$684$		0			0
$685$	−2728.59	−	2871.41i	−0.152196	−	0.160162i
$686$	−7283.91			−0.405395
$687$		0			0
$688$			716.156i			0.0396849i
$689$	26981.7			1.49190
$690$		0			0
$691$	12964.8			0.713757			0.356879	−	0.934151i	$-0.383841\pi$
$691$	12964.8			0.713757			0.356879	+	0.934151i	$0.383841\pi$
$692$			21061.5i			1.15699i
$693$		0			0
$694$	13342.7			0.729801
$695$	−626.138	+	594.994i	−0.0341737	+	0.0324740i
$696$		0			0
$697$		−	3143.46i		−	0.170828i
$698$			8441.90i			0.457780i
$699$		0			0
$700$	14153.1	+	722.392i	0.764193	+	0.0390055i
$701$	16162.1			0.870806			0.435403	−	0.900236i	$-0.356606\pi$
$701$	16162.1			0.870806			0.435403	+	0.900236i	$0.356606\pi$
$702$		0			0
$703$			2084.63i			0.111839i
$704$	−517.058			−0.0276809
$705$		0			0
$706$	20696.2			1.10327
$707$		−	766.014i		−	0.0407481i
$708$		0			0
$709$	14244.4			0.754529			0.377265	−	0.926105i	$-0.376865\pi$
$709$	14244.4			0.754529			0.377265	+	0.926105i	$0.376865\pi$
$710$	7787.87	+	8195.51i	0.411653	+	0.433200i
$711$		0			0
$712$			22722.7i			1.19602i
$713$		−	3195.97i		−	0.167868i
$714$		0			0
$715$	−802.744	−	844.762i	−0.0419873	−	0.0441850i
$716$	−16402.1			−0.856109
$717$		0			0
$718$			8837.55i			0.459352i
$719$	−27638.5			−1.43358			−0.716790	−	0.697289i	$-0.754389\pi$
$719$	−27638.5			−1.43358			−0.716790	+	0.697289i	$0.754389\pi$
$720$		0			0
$721$	−39006.4			−2.01480
$722$			7030.68i			0.362403i
$723$		0			0
$724$	17327.4			0.889460
$725$	−1869.93	+	36635.5i	−0.0957895	+	1.87670i
$726$		0			0
$727$		−	2525.52i		−	0.128840i	−0.997923	−	0.0644199i	$-0.979480\pi$
$727$		−	2525.52i		−	0.128840i	0.997923	−	0.0644199i	$-0.0205197\pi$
$728$			28827.3i			1.46760i
$729$		0			0
$730$	4416.19	−	4196.53i	0.223905	−	0.212768i
$731$	4695.79			0.237592
$732$		0			0
$733$			8400.27i			0.423289i	0.977347	+	0.211645i	$0.0678820\pi$
$733$			8400.27i			0.423289i	−0.977347	+	0.211645i	$0.932118\pi$
$734$	10355.9			0.520765
$735$		0			0
$736$	9094.67			0.455481
$737$			311.401i			0.0155639i
$738$		0			0
$739$	19689.1			0.980074			0.490037	−	0.871702i	$-0.336983\pi$
$739$	19689.1			0.980074			0.490037	+	0.871702i	$0.336983\pi$
$740$	−781.735	−	822.653i	−0.0388340	−	0.0408667i
$741$		0			0
$742$		−	17517.0i		−	0.866671i
$743$		−	22526.6i		−	1.11227i	−0.831091	−	0.556137i	$-0.812283\pi$
$743$		−	22526.6i		−	1.11227i	0.831091	−	0.556137i	$-0.187717\pi$
$744$		0			0
$745$	−13821.6	+	13134.1i	−0.679708	+	0.645900i
$746$	−18005.8			−0.883699
$747$		0			0
$748$		−	173.528i		−	0.00848239i
$749$	−30243.3			−1.47539
$750$		0			0
$751$	34691.1			1.68562			0.842808	−	0.538215i	$-0.180901\pi$
$751$	34691.1			1.68562			0.842808	+	0.538215i	$0.180901\pi$
$752$		−	1113.37i		−	0.0539902i
$753$		0			0
$754$	−29066.8			−1.40392
$755$	6145.63	−	5839.95i	0.296242	−	0.281507i
$756$		0			0
$757$			6619.98i			0.317843i	0.987291	+	0.158922i	$0.0508017\pi$
$757$			6619.98i			0.317843i	−0.987291	+	0.158922i	$0.949198\pi$
$758$		−	19832.0i		−	0.950303i
$759$		0			0
$760$	−18004.0	−	18946.4i	−0.859309	−	0.904288i
$761$	29368.7			1.39897			0.699483	−	0.714649i	$-0.253414\pi$
$761$	29368.7			1.39897			0.699483	+	0.714649i	$0.253414\pi$
$762$		0			0
$763$		−	7131.84i		−	0.338388i
$764$	17700.5			0.838194
$765$		0			0
$766$	−10830.4			−0.510859
$767$			4295.31i			0.202209i
$768$		0			0
$769$	−32677.4			−1.53235			−0.766174	−	0.642633i	$-0.777842\pi$
$769$	−32677.4			−1.53235			−0.766174	+	0.642633i	$0.777842\pi$
$770$	−548.435	+	521.156i	−0.0256678	+	0.0243911i
$771$		0			0
$772$		−	9148.33i		−	0.426497i
$773$		−	28047.5i		−	1.30504i	−0.757770	−	0.652522i	$-0.773711\pi$
$773$		−	28047.5i		−	1.30504i	0.757770	−	0.652522i	$-0.226289\pi$
$774$		0			0
$775$	410.469	−	8041.87i	0.0190251	−	0.372739i
$776$	−8570.96			−0.396494
$777$		0			0
$778$			10438.0i			0.481001i
$779$	−17359.1			−0.798402
$780$		0			0
$781$	1064.14			0.0487552
$782$		−	1602.64i		−	0.0732867i
$783$		0			0
$784$	−435.039			−0.0198177
$785$	−13624.5	−	14337.6i	−0.619463	−	0.651887i
$786$		0			0
$787$		−	22172.1i		−	1.00426i	−0.864793	−	0.502128i	$-0.832551\pi$
$787$		−	22172.1i		−	1.00426i	0.864793	−	0.502128i	$-0.167449\pi$
$788$		−	8566.64i		−	0.387276i
$789$		0			0
$790$	10096.7	+	10625.2i	0.454717	+	0.478518i
$791$	35140.7			1.57960
$792$		0			0
$793$			7995.06i			0.358024i
$794$	−16584.2			−0.741249
$795$		0			0
$796$	−15866.5			−0.706501
$797$			24170.3i			1.07422i	0.843511	+	0.537112i	$0.180485\pi$
$797$			24170.3i			1.07422i	−0.843511	+	0.537112i	$0.819515\pi$
$798$		0			0
$799$	−7300.32			−0.323238
$800$	22884.5	+	1168.06i	1.01136	+	0.0516212i
$801$		0			0
$802$			2222.99i			0.0978761i
$803$		−	573.415i		−	0.0251997i
$804$		0			0
$805$	8930.25	−	8486.06i	0.390994	−	0.371546i
$806$	6380.47			0.278837
$807$		0			0
$808$		−	769.085i		−	0.0334856i
$809$	15304.2			0.665102			0.332551	−	0.943085i	$-0.392091\pi$
$809$	15304.2			0.665102			0.332551	+	0.943085i	$0.392091\pi$
$810$		0			0
$811$	−27002.2			−1.16914			−0.584572	−	0.811342i	$-0.698738\pi$
$811$	−27002.2			−1.16914			−0.584572	+	0.811342i	$0.698738\pi$
$812$		−	33270.7i		−	1.43790i
$813$		0			0
$814$	60.5849			0.00260872
$815$	6790.61	+	7146.05i	0.291859	+	0.307135i
$816$		0			0
$817$		−	25931.5i		−	1.11044i
$818$			6596.84i			0.281972i
$819$		0			0
$820$	6850.41	−	6509.67i	0.291740	−	0.277229i
$821$	−25061.4			−1.06535			−0.532673	−	0.846321i	$-0.678812\pi$
$821$	−25061.4			−1.06535			−0.532673	+	0.846321i	$0.678812\pi$
$822$		0			0
$823$			24896.4i			1.05448i	0.849718	+	0.527238i	$0.176772\pi$
$823$			24896.4i			1.05448i	−0.849718	+	0.527238i	$0.823228\pi$
$824$	−39162.8			−1.65571
$825$		0			0
$826$	2788.59			0.117467
$827$			20063.2i			0.843612i	0.906686	+	0.421806i	$0.138604\pi$
$827$			20063.2i			0.843612i	−0.906686	+	0.421806i	$0.861396\pi$
$828$		0			0
$829$	13884.2			0.581687			0.290844	−	0.956771i	$-0.406064\pi$
$829$	13884.2			0.581687			0.290844	+	0.956771i	$0.406064\pi$
$830$	−2398.53	+	2279.23i	−0.100306	+	0.0953172i
$831$		0			0
$832$			16808.4i			0.700393i
$833$			2852.52i			0.118648i
$834$		0			0
$835$	1670.47	+	1757.91i	0.0692323	+	0.0728561i
$836$	−958.277			−0.0396444
$837$		0			0
$838$		−	27263.3i		−	1.12386i
$839$	−13678.1			−0.562838			−0.281419	−	0.959585i	$-0.590805\pi$
$839$	−13678.1			−0.562838			−0.281419	+	0.959585i	$0.590805\pi$
$840$		0			0
$841$	61733.1			2.53118
$842$		−	13816.2i		−	0.565485i
$843$		0			0
$844$	−22834.6			−0.931280
$845$	−9655.36	+	9175.11i	−0.393082	+	0.373531i
$846$		0			0
$847$		−	29489.5i		−	1.19630i
$848$			1342.06i			0.0543473i
$849$		0			0
$850$	205.832	−	4032.64i	0.00830584	−	0.162727i
$851$	−986.512			−0.0397382
$852$		0			0
$853$		−	29802.9i		−	1.19629i	−0.801390	−	0.598143i	$-0.795906\pi$
$853$		−	29802.9i		−	1.19629i	0.801390	−	0.598143i	$-0.204094\pi$
$854$	5190.54			0.207982
$855$		0			0
$856$	−30364.6			−1.21243
$857$		−	22045.2i		−	0.878706i	−0.898314	−	0.439353i	$-0.855208\pi$
$857$		−	22045.2i		−	0.878706i	0.898314	−	0.439353i	$-0.144792\pi$
$858$		0			0
$859$	−33609.5			−1.33497			−0.667487	−	0.744622i	$-0.732630\pi$
$859$	−33609.5			−1.33497			−0.667487	+	0.744622i	$0.732630\pi$
$860$	9724.33	+	10233.3i	0.385578	+	0.405760i
$861$		0			0
$862$			9722.70i			0.384172i
$863$			33775.6i			1.33226i	0.745838	+	0.666128i	$0.232049\pi$
$863$			33775.6i			1.33226i	−0.745838	+	0.666128i	$0.767951\pi$
$864$		0			0
$865$	31775.9	+	33439.2i	1.24903	+	1.31441i
$866$	10637.0			0.417392
$867$		0			0
$868$			7303.27i			0.285587i
$869$	1379.62			0.0538556
$870$		0			0
$871$	10123.0			0.393805
$872$		−	7160.44i		−	0.278077i
$873$		0			0
$874$	−8850.25			−0.342522
$875$	23560.6	−	20206.1i	0.910280	−	0.780676i
$876$		0			0
$877$			12637.0i			0.486570i	0.969955	+	0.243285i	$0.0782250\pi$
$877$			12637.0i			0.486570i	−0.969955	+	0.243285i	$0.921775\pi$
$878$			7199.25i			0.276723i
$879$		0			0
$880$	42.0182	−	39.9282i	0.00160958	−	0.00152952i
$881$	6579.45			0.251609			0.125804	−	0.992055i	$-0.459849\pi$
$881$	6579.45			0.251609			0.125804	+	0.992055i	$0.459849\pi$
$882$		0			0
$883$			50442.1i			1.92244i	0.275786	+	0.961219i	$0.411062\pi$
$883$			50442.1i			1.92244i	−0.275786	+	0.961219i	$0.588938\pi$
$884$	−5641.03			−0.214625
$885$		0			0
$886$	−10744.2			−0.407401
$887$		−	984.823i		−	0.0372797i	−0.999826	−	0.0186399i	$-0.994066\pi$
$887$		−	984.823i		−	0.0372797i	0.999826	−	0.0186399i	$-0.00593360\pi$
$888$		0			0
$889$	26587.7			1.00306
$890$	13354.0	+	14053.0i	0.502952	+	0.529278i
$891$		0			0
$892$			8950.64i			0.335975i
$893$			40314.6i			1.51072i
$894$		0			0
$895$	−26041.5	+	24746.2i	−0.972594	+	0.924217i
$896$	−21658.0			−0.807525
$897$		0			0
$898$		−	15909.1i		−	0.591196i
$899$	−18904.7			−0.701342
$900$		0			0
$901$	8799.79			0.325376
$902$			504.503i			0.0186232i
$903$		0			0
$904$	35281.6			1.29806
$905$	27510.7	−	26142.3i	1.01048	−	0.960221i
$906$		0			0
$907$			43679.9i			1.59908i	0.600612	+	0.799541i	$0.294924\pi$
$907$			43679.9i			1.59908i	−0.600612	+	0.799541i	$0.705076\pi$
$908$			4782.58i			0.174797i
$909$		0			0
$910$	16941.7	+	17828.4i	0.617154	+	0.649458i
$911$	10364.3			0.376930			0.188465	−	0.982080i	$-0.439649\pi$
$911$	10364.3			0.376930			0.188465	+	0.982080i	$0.439649\pi$
$912$		0			0
$913$			311.435i			0.0112891i
$914$	16245.6			0.587918
$915$		0			0
$916$	13180.6			0.475435
$917$			7143.58i			0.257254i
$918$		0			0
$919$	−11451.9			−0.411059			−0.205530	−	0.978651i	$-0.565892\pi$
$919$	−11451.9			−0.411059			−0.205530	+	0.978651i	$0.565892\pi$
$920$	8966.06	−	8520.09i	0.321307	−	0.305325i
$921$		0			0
$922$		−	10613.0i		−	0.379091i
$923$		−	34592.7i		−	1.23362i
$924$		0			0
$925$	−2482.31	−	126.701i	−0.0882356	−	0.00450367i
$926$	11009.1			0.390692
$927$		0			0
$928$		−	53796.4i		−	1.90297i
$929$	−27701.8			−0.978326			−0.489163	−	0.872192i	$-0.662698\pi$
$929$	−27701.8			−0.978326			−0.489163	+	0.872192i	$0.662698\pi$
$930$		0			0
$931$	15752.5			0.554529
$932$			11715.3i			0.411746i
$933$		0			0
$934$	12262.5			0.429596
$935$	−261.806	−	275.510i	−0.00915721	−	0.00963652i
$936$		0			0
$937$		−	5878.01i		−	0.204937i	−0.994736	−	0.102469i	$-0.967326\pi$
$937$		−	5878.01i		−	0.204937i	0.994736	−	0.102469i	$-0.0326741\pi$
$938$		−	6572.03i		−	0.228768i
$939$		0			0
$940$	−15118.0	−	15909.3i	−0.524568	−	0.552026i
$941$	28786.0			0.997234			0.498617	−	0.866823i	$-0.333841\pi$
$941$	28786.0			0.997234			0.498617	+	0.866823i	$0.333841\pi$
$942$		0			0
$943$		−	8214.90i		−	0.283684i
$944$	−213.647			−0.00736612
$945$		0			0
$946$	−753.642			−0.0259017
$947$			1695.04i			0.0581641i	0.999577	+	0.0290821i	$0.00925841\pi$
$947$			1695.04i			0.0581641i	−0.999577	+	0.0290821i	$0.990742\pi$
$948$		0			0
$949$	−18640.5			−0.637613
$950$	−22269.4	−	1136.66i	−0.760543	−	0.0388192i
$951$		0			0
$952$			9401.72i			0.320075i
$953$			31929.4i			1.08530i	0.839958	+	0.542651i	$0.182580\pi$
$953$			31929.4i			1.08530i	−0.839958	+	0.542651i	$0.817420\pi$
$954$		0			0
$955$	28102.9	−	26705.1i	0.952241	−	0.904877i
$956$	−11710.2			−0.396166
$957$		0			0
$958$		−	18465.1i		−	0.622735i
$959$	7868.57			0.264952
$960$		0			0
$961$	−25641.2			−0.860704
$962$		−	1969.48i		−	0.0660069i
$963$		0			0
$964$	1951.46			0.0651994
$965$	−13802.3	−	14524.8i	−0.460427	−	0.484527i
$966$		0			0
$967$		−	10897.1i		−	0.362385i	−0.983448	−	0.181193i	$-0.942004\pi$
$967$		−	10897.1i		−	0.362385i	0.983448	−	0.181193i	$-0.0579957\pi$
$968$		−	29607.7i		−	0.983087i
$969$		0			0
$970$	−5300.77	+	5037.11i	−0.175461	+	0.166734i
$971$	−7041.97			−0.232737			−0.116368	−	0.993206i	$-0.537125\pi$
$971$	−7041.97			−0.232737			−0.116368	+	0.993206i	$0.537125\pi$
$972$		0			0
$973$		−	1715.81i		−	0.0565328i
$974$	−21707.0			−0.714103
$975$		0			0
$976$	−397.671			−0.0130422
$977$			37607.6i			1.23150i	0.787943	+	0.615749i	$0.211146\pi$
$977$			37607.6i			1.23150i	−0.787943	+	0.615749i	$0.788854\pi$
$978$		0			0
$979$	1824.69			0.0595684
$980$	−6216.38	+	5907.18i	−0.202628	+	0.192549i
$981$		0			0
$982$			11939.0i			0.387974i
$983$		−	25297.7i		−	0.820826i	−0.911900	−	0.410413i	$-0.865385\pi$
$983$		−	25297.7i		−	0.820826i	0.911900	−	0.410413i	$-0.134615\pi$
$984$		0			0
$985$	−12924.7	−	13601.2i	−0.418086	−	0.439970i
$986$	−9479.85			−0.306187
$987$		0			0
$988$			31151.5i			1.00310i
$989$	12271.6			0.394556
$990$		0			0
$991$	−41686.5			−1.33624			−0.668120	−	0.744053i	$-0.732901\pi$
$991$	−41686.5			−1.33624			−0.668120	+	0.744053i	$0.732901\pi$
$992$			11808.9i			0.377955i
$993$		0			0
$994$	−22458.3			−0.716633
$995$	−25191.2	+	23938.2i	−0.802629	+	0.762707i
$996$		0			0
$997$			25465.9i			0.808939i	0.914551	+	0.404470i	$0.132544\pi$
$997$			25465.9i			0.808939i	−0.914551	+	0.404470i	$0.867456\pi$
$998$		−	19350.6i		−	0.613761i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	45.4.b.b.19.3		4
3.2	odd	2		15.4.b.a.4.2	✓	4
4.3	odd	2		720.4.f.j.289.3		4
5.2	odd	4		225.4.a.o.1.1		2
5.3	odd	4		225.4.a.i.1.2		2
5.4	even	2	inner	45.4.b.b.19.2		4
12.11	even	2		240.4.f.f.49.3		4
15.2	even	4		75.4.a.c.1.2		2
15.8	even	4		75.4.a.f.1.1		2
15.14	odd	2		15.4.b.a.4.3	yes	4
20.19	odd	2		720.4.f.j.289.4		4
24.5	odd	2		960.4.f.q.769.4		4
24.11	even	2		960.4.f.p.769.2		4
60.23	odd	4		1200.4.a.bn.1.1		2
60.47	odd	4		1200.4.a.bt.1.2		2
60.59	even	2		240.4.f.f.49.1		4
120.29	odd	2		960.4.f.q.769.2		4
120.59	even	2		960.4.f.p.769.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.4.b.a.4.2	✓	4	3.2	odd	2
15.4.b.a.4.3	yes	4	15.14	odd	2
45.4.b.b.19.2		4	5.4	even	2	inner
45.4.b.b.19.3		4	1.1	even	1	trivial
75.4.a.c.1.2		2	15.2	even	4
75.4.a.f.1.1		2	15.8	even	4
225.4.a.i.1.2		2	5.3	odd	4
225.4.a.o.1.1		2	5.2	odd	4
240.4.f.f.49.1		4	60.59	even	2
240.4.f.f.49.3		4	12.11	even	2
720.4.f.j.289.3		4	4.3	odd	2
720.4.f.j.289.4		4	20.19	odd	2
960.4.f.p.769.2		4	24.11	even	2
960.4.f.p.769.4		4	120.59	even	2
960.4.f.q.769.2		4	120.29	odd	2
960.4.f.q.769.4		4	24.5	odd	2
1200.4.a.bn.1.1		2	60.23	odd	4
1200.4.a.bt.1.2		2	60.47	odd	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:	\( N \)	\(=\)	\( 45 = 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	45.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.70156i q^{2} +5.10469 q^{4} +(8.10469 - 7.70156i) q^{5} +22.2094i q^{7} +22.2984i q^{8} +O(q^{10})\)	\(q+1.70156i q^{2} +5.10469 q^{4} +(8.10469 - 7.70156i) q^{5} +22.2094i q^{7} +22.2984i q^{8} +(13.1047 + 13.7906i) q^{10} +1.79063 q^{11} -58.2094i q^{13} -37.7906 q^{14} +2.89531 q^{16} -18.9844i q^{17} -104.837 q^{19} +(41.3719 - 39.3141i) q^{20} +3.04686i q^{22} -49.6125i q^{23} +(6.37188 - 124.837i) q^{25} +99.0469 q^{26} +113.372i q^{28} -293.466 q^{29} +64.4187 q^{31} +183.314i q^{32} +32.3031 q^{34} +(171.047 + 180.000i) q^{35} -19.8844i q^{37} -178.388i q^{38} +(171.733 + 180.722i) q^{40} +165.581 q^{41} +247.350i q^{43} +9.14059 q^{44} +84.4187 q^{46} -384.544i q^{47} -150.256 q^{49} +(212.419 + 10.8422i) q^{50} -297.141i q^{52} +463.528i q^{53} +(14.5125 - 13.7906i) q^{55} -495.234 q^{56} -499.350i q^{58} -73.7906 q^{59} -137.350 q^{61} +109.612i q^{62} -288.758 q^{64} +(-448.303 - 471.769i) q^{65} +173.906i q^{67} -96.9093i q^{68} +(-306.281 + 291.047i) q^{70} +594.281 q^{71} -320.231i q^{73} +33.8345 q^{74} -535.163 q^{76} +39.7687i q^{77} +770.469 q^{79} +(23.4656 - 22.2984i) q^{80} +281.747i q^{82} +173.925i q^{83} +(-146.209 - 153.862i) q^{85} -420.881 q^{86} +39.9282i q^{88} +1019.02 q^{89} +1292.79 q^{91} -253.256i q^{92} +654.325 q^{94} +(-849.675 + 807.412i) q^{95} +384.375i q^{97} -255.670i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 18 q^{4} - 6 q^{5}+O(q^{10}) \) 4 * q - 18 * q^4 - 6 * q^5	\( 4 q - 18 q^{4} - 6 q^{5} + 14 q^{10} + 84 q^{11} - 228 q^{14} + 50 q^{16} - 112 q^{19} + 396 q^{20} + 256 q^{25} + 12 q^{26} - 636 q^{29} + 104 q^{31} - 716 q^{34} + 300 q^{35} + 418 q^{40} + 816 q^{41} - 1116 q^{44} + 184 q^{46} - 140 q^{49} + 696 q^{50} - 864 q^{55} - 60 q^{56} - 372 q^{59} + 680 q^{61} + 958 q^{64} - 948 q^{65} + 1080 q^{70} + 72 q^{71} + 3132 q^{74} - 2448 q^{76} - 760 q^{79} - 444 q^{80} - 508 q^{85} - 4296 q^{86} + 2232 q^{89} + 1944 q^{91} + 3232 q^{94} - 2784 q^{95}+O(q^{100}) \) 4 * q - 18 * q^4 - 6 * q^5 + 14 * q^10 + 84 * q^11 - 228 * q^14 + 50 * q^16 - 112 * q^19 + 396 * q^20 + 256 * q^25 + 12 * q^26 - 636 * q^29 + 104 * q^31 - 716 * q^34 + 300 * q^35 + 418 * q^40 + 816 * q^41 - 1116 * q^44 + 184 * q^46 - 140 * q^49 + 696 * q^50 - 864 * q^55 - 60 * q^56 - 372 * q^59 + 680 * q^61 + 958 * q^64 - 948 * q^65 + 1080 * q^70 + 72 * q^71 + 3132 * q^74 - 2448 * q^76 - 760 * q^79 - 444 * q^80 - 508 * q^85 - 4296 * q^86 + 2232 * q^89 + 1944 * q^91 + 3232 * q^94 - 2784 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more