LMFDB - Embedded newform 720.4.f.j.289.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [720,4,Mod(289,720)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(720, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("720.289");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 720 = 2^{4} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	720.f (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:	$42.4813752041$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 15)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			289.4
Root			$3.70156i$ of defining polynomial
Character	$\chi$	$=$	720.289
Dual form			720.4.f.j.289.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(8.10469 + 7.70156i) q^{5} +22.2094i q^{7} -1.79063 q^{11} +58.2094i q^{13} +18.9844i q^{17} +104.837 q^{19} -49.6125i q^{23} +(6.37188 + 124.837i) q^{25} -293.466 q^{29} -64.4187 q^{31} +(-171.047 + 180.000i) q^{35} +19.8844i q^{37} +165.581 q^{41} +247.350i q^{43} -384.544i q^{47} -150.256 q^{49} -463.528i q^{53} +(-14.5125 - 13.7906i) q^{55} +73.7906 q^{59} -137.350 q^{61} +(-448.303 + 471.769i) q^{65} +173.906i q^{67} -594.281 q^{71} +320.231i q^{73} -39.7687i q^{77} -770.469 q^{79} +173.925i q^{83} +(-146.209 + 153.862i) q^{85} +1019.02 q^{89} -1292.79 q^{91} +(849.675 + 807.412i) q^{95} -384.375i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{5} - 84 q^{11} + 112 q^{19} + 256 q^{25} - 636 q^{29} - 104 q^{31} - 300 q^{35} + 816 q^{41} - 140 q^{49} + 864 q^{55} + 372 q^{59} + 680 q^{61} - 948 q^{65} - 72 q^{71} + 760 q^{79} - 508 q^{85}+ \cdots + 2784 q^{95}+O(q^{100}) $ 4 * q - 6 * q^5 - 84 * q^11 + 112 * q^19 + 256 * q^25 - 636 * q^29 - 104 * q^31 - 300 * q^35 + 816 * q^41 - 140 * q^49 + 864 * q^55 + 372 * q^59 + 680 * q^61 - 948 * q^65 - 72 * q^71 + 760 * q^79 - 508 * q^85 + 2232 * q^89 - 1944 * q^91 + 2784 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/720\mathbb{Z}\right)^\times$.

$n$	$181$	$271$	$577$	$641$
$\chi(n)$	$1$	$1$	$-1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	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	$0.795328\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.79063			−0.0490813			−0.0245407	−	0.999699i	$-0.507812\pi$
$11$	−1.79063			−0.0490813			−0.0245407	+	0.999699i	$0.507812\pi$
$12$		0			0
$13$			58.2094i			1.24188i	0.783860	+	0.620938i	$0.213248\pi$
$13$			58.2094i			1.24188i	−0.783860	+	0.620938i	$0.786752\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			18.9844i			0.270846i	0.990788	+	0.135423i	$0.0432394\pi$
$17$			18.9844i			0.270846i	−0.990788	+	0.135423i	$0.956761\pi$
$18$		0			0
$19$	104.837			1.26586			0.632931	−	0.774208i	$-0.281852\pi$
$19$	104.837			1.26586			0.632931	+	0.774208i	$0.281852\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	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
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−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	$-0.559751\pi$
$31$	−64.4187			−0.373224			−0.186612	+	0.982434i	$0.559751\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−171.047	+	180.000i	−0.826063	+	0.869302i
$36$		0			0
$37$			19.8844i			0.0883505i	0.999024	+	0.0441752i	$0.0140660\pi$
$37$			19.8844i			0.0883505i	−0.999024	+	0.0441752i	$0.985934\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	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	$0.855471\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	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
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	463.528i		−	1.20133i	−0.799501	−	0.600665i	$-0.794903\pi$
$53$		−	463.528i		−	1.20133i	0.799501	−	0.600665i	$-0.205097\pi$
$54$		0			0
$55$	−14.5125	−	13.7906i	−0.0355793	−	0.0338096i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	73.7906			0.162826			0.0814129	−	0.996680i	$-0.474057\pi$
$59$	73.7906			0.162826			0.0814129	+	0.996680i	$0.474057\pi$
$60$		0			0
$61$	−137.350			−0.288293			−0.144146	−	0.989556i	$-0.546044\pi$
$61$	−137.350			−0.288293			−0.144146	+	0.989556i	$0.546044\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−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	$0.949317\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−594.281			−0.993355			−0.496677	−	0.867935i	$-0.665447\pi$
$71$	−594.281			−0.993355			−0.496677	+	0.867935i	$0.665447\pi$
$72$		0			0
$73$			320.231i			0.513428i	0.966487	+	0.256714i	$0.0826398\pi$
$73$			320.231i			0.513428i	−0.966487	+	0.256714i	$0.917360\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	39.7687i		−	0.0588580i
$78$		0			0
$79$	−770.469			−1.09727			−0.548636	−	0.836061i	$-0.684853\pi$
$79$	−770.469			−1.09727			−0.548636	+	0.836061i	$0.684853\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			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
$86$		0			0
$87$		0			0
$88$		0			0
$89$	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
$92$		0			0
$93$		0			0
$94$		0			0
$95$	849.675	+	807.412i	0.917630	+	0.871987i
$96$		0			0
$97$		−	384.375i		−	0.402344i	−0.979556	−	0.201172i	$-0.935525\pi$
$97$		−	384.375i		−	0.402344i	0.979556	−	0.201172i	$-0.0644750\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−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	$0.682516\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			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	$0.545061\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			1582.25i			1.31721i	0.752487	+	0.658607i	$0.228854\pi$
$113$			1582.25i			1.31721i	−0.752487	+	0.658607i	$0.771146\pi$
$114$		0			0
$115$	382.094	−	402.094i	0.309830	−	0.326047i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−421.631			−0.324797
$120$		0			0
$121$	−1327.79			−0.997591
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−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	$-0.137348\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−321.647			−0.214522			−0.107261	−	0.994231i	$-0.534208\pi$
$131$	−321.647			−0.214522			−0.107261	+	0.994231i	$0.534208\pi$
$132$		0			0
$133$			2328.37i			1.51801i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			354.291i			0.220942i	0.993879	+	0.110471i	$0.0352360\pi$
$137$			354.291i			0.220942i	−0.993879	+	0.110471i	$0.964764\pi$
$138$		0			0
$139$	77.2562			0.0471424			0.0235712	−	0.999722i	$-0.492496\pi$
$139$	77.2562			0.0471424			0.0235712	+	0.999722i	$0.492496\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	104.231i		−	0.0609529i
$144$		0			0
$145$	−2378.45	−	2260.14i	−1.36220	−	1.29445i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−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	$-0.565502\pi$
$151$	−758.281			−0.408663			−0.204331	+	0.978902i	$0.565502\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−522.094	−	496.125i	−0.270552	−	0.257095i
$156$		0			0
$157$			1769.05i			0.899273i	0.893212	+	0.449636i	$0.148446\pi$
$157$			1769.05i			0.899273i	−0.893212	+	0.449636i	$0.851554\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	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	$0.932053\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			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
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	4125.91i		−	1.81322i	−0.421970	−	0.906610i	$-0.638661\pi$
$173$		−	4125.91i		−	1.81322i	0.421970	−	0.906610i	$-0.361339\pi$
$174$		0			0
$175$	−2772.56	+	141.515i	−1.19763	+	0.0611289i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	3213.14			1.34168			0.670842	−	0.741600i	$-0.265933\pi$
$179$	3213.14			1.34168			0.670842	+	0.741600i	$0.265933\pi$
$180$		0			0
$181$	3394.42			1.39395			0.696976	−	0.717095i	$-0.254529\pi$
$181$	3394.42			1.39395			0.696976	+	0.717095i	$0.254529\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−153.141	+	161.156i	−0.0608601	+	0.0640457i
$186$		0			0
$187$		−	33.9939i		−	0.0132935i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−3467.49			−1.31361			−0.656804	−	0.754062i	$-0.728092\pi$
$191$	−3467.49			−1.31361			−0.656804	+	0.754062i	$0.728092\pi$
$192$		0			0
$193$			1792.14i			0.668401i	0.942502	+	0.334200i	$0.108466\pi$
$193$			1792.14i			0.668401i	−0.942502	+	0.334200i	$0.891534\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			1678.19i			0.606935i	0.952842	+	0.303467i	$0.0981443\pi$
$197$			1678.19i			0.606935i	−0.952842	+	0.303467i	$0.901856\pi$
$198$		0			0
$199$	3108.23			1.10722			0.553610	−	0.832776i	$-0.313250\pi$
$199$	3108.23			1.10722			0.553610	+	0.832776i	$0.313250\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	6517.69i		−	2.25346i
$204$		0			0
$205$	1341.98	+	1275.23i	0.457211	+	0.434469i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−187.725			−0.0621301
$210$		0			0
$211$	4473.27			1.45949			0.729745	−	0.683719i	$-0.239639\pi$
$211$	4473.27			1.45949			0.729745	+	0.683719i	$0.239639\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−1904.98	+	2004.69i	−0.604273	+	0.635902i
$216$		0			0
$217$		−	1430.70i		−	0.447568i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−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	$0.915200\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			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	$0.378484\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	2295.01i		−	0.645284i	−0.946521	−	0.322642i	$-0.895429\pi$
$233$		−	2295.01i		−	0.645284i	0.946521	−	0.322642i	$-0.104571\pi$
$234$		0			0
$235$	2961.59	−	3116.61i	0.822096	−	0.865128i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	2294.01			0.620866			0.310433	−	0.950595i	$-0.399526\pi$
$239$	2294.01			0.620866			0.310433	+	0.950595i	$0.399526\pi$
$240$		0			0
$241$	382.287			0.102180			0.0510898	−	0.998694i	$-0.483731\pi$
$241$	382.287			0.102180			0.0510898	+	0.998694i	$0.483731\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1217.78	−	1157.21i	−0.317555	−	0.301760i
$246$		0			0
$247$			6102.52i			1.57204i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−2259.98			−0.568322			−0.284161	−	0.958777i	$-0.591715\pi$
$251$	−2259.98			−0.568322			−0.284161	+	0.958777i	$0.591715\pi$
$252$		0			0
$253$			88.8375i			0.0220758i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	92.7843i		−	0.0225203i	−0.999937	−	0.0112602i	$-0.996416\pi$
$257$		−	92.7843i		−	0.0225203i	0.999937	−	0.0112602i	$-0.00358430\pi$
$258$		0			0
$259$	−441.619			−0.105949
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	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
$266$		0			0
$267$		0			0
$268$		0			0
$269$	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	$-0.849508\pi$
$271$	−7943.69			−1.78061			−0.890304	+	0.455366i	$0.849508\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−11.4097	−	223.537i	−0.00250192	−	0.0490175i
$276$		0			0
$277$		−	6823.00i		−	1.47998i	−0.672618	−	0.739990i	$-0.734830\pi$
$277$		−	6823.00i		−	1.47998i	0.672618	−	0.739990i	$-0.265170\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−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	$-0.243910\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			3677.46i			0.756353i
$288$		0			0
$289$	4552.59			0.926642
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	5814.14i		−	1.15927i	−0.814877	−	0.579634i	$-0.803195\pi$
$293$		−	5814.14i		−	1.15927i	0.814877	−	0.579634i	$-0.196805\pi$
$294$		0			0
$295$	598.050	+	568.303i	0.118033	+	0.112162i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	2887.91			0.558570
$300$		0			0
$301$	−5493.49			−1.05196
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−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	$-0.272474\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	7336.26			1.33762			0.668812	−	0.743432i	$-0.266803\pi$
$311$	7336.26			1.33762			0.668812	+	0.743432i	$0.266803\pi$
$312$		0			0
$313$		−	2202.66i		−	0.397768i	−0.980023	−	0.198884i	$-0.936268\pi$
$313$		−	2202.66i		−	0.397768i	0.980023	−	0.198884i	$-0.0637318\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			10008.9i			1.77336i	0.462386	+	0.886679i	$0.346993\pi$
$317$			10008.9i			1.77336i	−0.462386	+	0.886679i	$0.653007\pi$
$318$		0			0
$319$	525.488			0.0922309
$320$		0			0
$321$		0			0
$322$		0			0
$323$			1990.27i			0.342854i
$324$		0			0
$325$	−7266.71	+	370.903i	−1.24026	+	0.0633046i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	8540.47			1.43116
$330$		0			0
$331$	8695.94			1.44402			0.722012	−	0.691881i	$-0.243218\pi$
$331$	8695.94			1.44402			0.722012	+	0.691881i	$0.243218\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−1339.35	+	1409.46i	−0.218437	+	0.229871i
$336$		0			0
$337$			7400.61i			1.19625i	0.801402	+	0.598126i	$0.204088\pi$
$337$			7400.61i			1.19625i	−0.801402	+	0.598126i	$0.795912\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	115.350			0.0183183
$342$		0			0
$343$			4280.72i			0.673869i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	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	$0.375761\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			12163.0i			1.83392i	0.398981	+	0.916959i	$0.369364\pi$
$353$			12163.0i			1.83392i	−0.398981	+	0.916959i	$0.630636\pi$
$354$		0			0
$355$	−4816.46	−	4576.89i	−0.720088	−	0.684271i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−5193.79			−0.763559			−0.381779	−	0.924253i	$-0.624689\pi$
$359$	−5193.79			−0.763559			−0.381779	+	0.924253i	$0.624689\pi$
$360$		0			0
$361$	4131.90			0.602406
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−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	$-0.142482\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	10294.7			1.44063
$372$		0			0
$373$		−	10581.9i		−	1.46893i	−0.678646	−	0.734466i	$-0.737433\pi$
$373$		−	10581.9i		−	1.46893i	0.678646	−	0.734466i	$-0.262567\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	17082.4i		−	2.33366i
$378$		0			0
$379$	11655.2			1.57964			0.789822	−	0.613336i	$-0.210173\pi$
$379$	11655.2			1.57964			0.789822	+	0.613336i	$0.210173\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			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
$386$		0			0
$387$		0			0
$388$		0			0
$389$	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
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−6244.41	−	5933.81i	−0.795418	−	0.755854i
$396$		0			0
$397$		−	9746.46i		−	1.23214i	−0.787690	−	0.616072i	$-0.788723\pi$
$397$		−	9746.46i		−	1.23214i	0.787690	−	0.616072i	$-0.211277\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	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
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	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	$0.424702\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			1638.84i			0.195260i
$414$		0			0
$415$	−1339.49	+	1409.61i	−0.158441	+	0.166735i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	16022.5			1.86814			0.934071	−	0.357088i	$-0.116230\pi$
$419$	16022.5			1.86814			0.934071	+	0.357088i	$0.116230\pi$
$420$		0			0
$421$	−8119.73			−0.939980			−0.469990	−	0.882672i	$-0.655742\pi$
$421$	−8119.73			−0.939980			−0.469990	+	0.882672i	$0.655742\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−2369.96	+	120.966i	−0.270494	+	0.0138064i
$426$		0			0
$427$		−	3050.46i		−	0.345719i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−5713.99			−0.638592			−0.319296	−	0.947655i	$-0.603446\pi$
$431$	−5713.99			−0.638592			−0.319296	+	0.947655i	$0.603446\pi$
$432$		0			0
$433$			6251.34i			0.693811i	0.937900	+	0.346906i	$0.112768\pi$
$433$			6251.34i			0.693811i	−0.937900	+	0.346906i	$0.887232\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	5201.25i		−	0.569358i
$438$		0			0
$439$	−4230.97			−0.459984			−0.229992	−	0.973192i	$-0.573870\pi$
$439$	−4230.97			−0.459984			−0.229992	+	0.973192i	$0.573870\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			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
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−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
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−10477.7	−	9956.53i	−1.07956	−	1.02587i
$456$		0			0
$457$			9547.46i			0.977268i	0.872489	+	0.488634i	$0.162505\pi$
$457$			9547.46i			0.977268i	−0.872489	+	0.488634i	$0.837495\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−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	$-0.105268\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	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
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	442.912i		−	0.0430552i
$474$		0			0
$475$	668.012	+	13087.6i	0.0645274	+	1.26422i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	10851.8			1.03514			0.517571	−	0.855640i	$-0.326836\pi$
$479$	10851.8			1.03514			0.517571	+	0.855640i	$0.326836\pi$
$480$		0			0
$481$	−1157.46			−0.109720
$482$		0			0
$483$		0			0
$484$		0			0
$485$	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	$0.797742\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−7016.52			−0.644911			−0.322455	−	0.946585i	$-0.604508\pi$
$491$	−7016.52			−0.644911			−0.322455	+	0.946585i	$0.604508\pi$
$492$		0			0
$493$		−	5571.26i		−	0.508960i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	13198.6i		−	1.19122i
$498$		0			0
$499$	11372.3			1.02023			0.510113	−	0.860107i	$-0.329604\pi$
$499$	11372.3			1.02023			0.510113	+	0.860107i	$0.329604\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	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
$506$		0			0
$507$		0			0
$508$		0			0
$509$	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
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−13526.3	+	14234.3i	−1.15736	+	1.21794i
$516$		0			0
$517$			688.574i			0.0585754i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−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	$0.893663\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	1222.95i		−	0.101086i
$528$		0			0
$529$	9705.60			0.797699
$530$		0			0
$531$		0			0
$532$		0			0
$533$			9638.38i			0.783273i
$534$		0			0
$535$	−10487.5	+	11036.5i	−0.847504	+	0.891865i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	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	$0.405221\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−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	$-0.0418573\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−30766.2			−2.37874
$552$		0			0
$553$		−	17111.6i		−	1.31584i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			20771.8i			1.58012i	0.613028	+	0.790061i	$0.289951\pi$
$557$			20771.8i			1.58012i	−0.613028	+	0.790061i	$0.710049\pi$
$558$		0			0
$559$	−14398.1			−1.08940
$560$		0			0
$561$		0			0
$562$		0			0
$563$			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
$566$		0			0
$567$		0			0
$568$		0			0
$569$	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	$-0.517302\pi$
$571$	−1482.54			−0.108655			−0.0543277	+	0.998523i	$0.517302\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	6193.50	−	316.125i	0.449194	−	0.0229275i
$576$		0			0
$577$			15264.0i			1.10130i	0.834737	+	0.550649i	$0.185620\pi$
$577$			15264.0i			1.10130i	−0.834737	+	0.550649i	$0.814380\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−3862.76			−0.275825
$582$		0			0
$583$			830.006i			0.0589628i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	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
$590$		0			0
$591$		0			0
$592$		0			0
$593$			11764.8i			0.814707i	0.913271	+	0.407353i	$0.133548\pi$
$593$			11764.8i			0.814707i	−0.913271	+	0.407353i	$0.866452\pi$
$594$		0			0
$595$	−3417.19	−	3247.22i	−0.235447	−	0.223736i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	9451.99			0.644737			0.322369	−	0.946614i	$-0.395521\pi$
$599$	9451.99			0.644737			0.322369	+	0.946614i	$0.395521\pi$
$600$		0			0
$601$	−3131.93			−0.212569			−0.106285	−	0.994336i	$-0.533895\pi$
$601$	−3131.93			−0.212569			−0.106285	+	0.994336i	$0.533895\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−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	$0.725700\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	22384.0			1.48210
$612$		0			0
$613$			28911.6i			1.90494i	0.304629	+	0.952471i	$0.401468\pi$
$613$			28911.6i			1.90494i	−0.304629	+	0.952471i	$0.598532\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	5566.87i		−	0.363231i	−0.983370	−	0.181616i	$-0.941867\pi$
$617$		−	5566.87i		−	0.363231i	0.983370	−	0.181616i	$-0.0581326\pi$
$618$		0			0
$619$	4150.32			0.269492			0.134746	−	0.990880i	$-0.456978\pi$
$619$	4150.32			0.269492			0.134746	+	0.990880i	$0.456978\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			22631.9i			1.45542i
$624$		0			0
$625$	−15543.8	+	1590.90i	−0.994803	+	0.101818i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−377.492			−0.0239294
$630$		0			0
$631$	4090.09			0.258041			0.129021	−	0.991642i	$-0.458817\pi$
$631$	4090.09			0.258041			0.129021	+	0.991642i	$0.458817\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	9219.85	−	9702.45i	0.576187	−	0.606346i
$636$		0			0
$637$		−	8746.32i		−	0.544022i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−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	$-0.385969\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			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
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	12777.6i		−	0.765737i	−0.923803	−	0.382869i	$-0.874936\pi$
$653$		−	12777.6i		−	0.765737i	0.923803	−	0.382869i	$-0.125064\pi$
$654$		0			0
$655$	−2606.85	−	2477.18i	−0.155508	−	0.147773i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−23563.5			−1.39287			−0.696435	−	0.717620i	$-0.745232\pi$
$659$	−23563.5			−1.39287			−0.696435	+	0.717620i	$0.745232\pi$
$660$		0			0
$661$	−4361.31			−0.256634			−0.128317	−	0.991733i	$-0.540958\pi$
$661$	−4361.31			−0.256634			−0.128317	+	0.991733i	$0.540958\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−17932.1	+	18870.7i	−1.04568	+	1.10042i
$666$		0			0
$667$			14559.6i			0.845200i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	245.943			0.0141498
$672$		0			0
$673$		−	8203.52i		−	0.469870i	−0.972011	−	0.234935i	$-0.924512\pi$
$673$		−	8203.52i		−	0.469870i	0.972011	−	0.234935i	$-0.0754877\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			28057.1i			1.59279i	0.604774	+	0.796397i	$0.293263\pi$
$677$			28057.1i			1.59279i	−0.604774	+	0.796397i	$0.706737\pi$
$678$		0			0
$679$	8536.73			0.482488
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	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
$686$		0			0
$687$		0			0
$688$		0			0
$689$	26981.7			1.49190
$690$		0			0
$691$	−12964.8			−0.713757			−0.356879	−	0.934151i	$-0.616159\pi$
$691$	−12964.8			−0.713757			−0.356879	+	0.934151i	$0.616159\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	626.138	+	594.994i	0.0341737	+	0.0324740i
$696$		0			0
$697$			3143.46i			0.170828i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	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
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	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	$0.376865\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			3195.97i			0.167868i
$714$		0			0
$715$	802.744	−	844.762i	0.0419873	−	0.0441850i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	27638.5			1.43358			0.716790	−	0.697289i	$-0.245611\pi$
$719$	27638.5			1.43358			0.716790	+	0.697289i	$0.245611\pi$
$720$		0			0
$721$	−39006.4			−2.01480
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−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	$-0.0205197\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−4695.79			−0.237592
$732$		0			0
$733$		−	8400.27i		−	0.423289i	−0.977347	−	0.211645i	$-0.932118\pi$
$733$		−	8400.27i		−	0.423289i	0.977347	−	0.211645i	$-0.0678820\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	311.401i		−	0.0155639i
$738$		0			0
$739$	−19689.1			−0.980074			−0.490037	−	0.871702i	$-0.663017\pi$
$739$	−19689.1			−0.980074			−0.490037	+	0.871702i	$0.663017\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	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
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−30243.3			−1.47539
$750$		0			0
$751$	−34691.1			−1.68562			−0.842808	−	0.538215i	$-0.819099\pi$
$751$	−34691.1			−1.68562			−0.842808	+	0.538215i	$0.819099\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−6145.63	−	5839.95i	−0.296242	−	0.281507i
$756$		0			0
$757$		−	6619.98i		−	0.317843i	−0.987291	−	0.158922i	$-0.949198\pi$
$757$		−	6619.98i		−	0.317843i	0.987291	−	0.158922i	$-0.0508017\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	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
$764$		0			0
$765$		0			0
$766$		0			0
$767$			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	$0.777842\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			28047.5i			1.30504i	0.757770	+	0.652522i	$0.226289\pi$
$773$			28047.5i			1.30504i	−0.757770	+	0.652522i	$0.773711\pi$
$774$		0			0
$775$	−410.469	−	8041.87i	−0.0190251	−	0.372739i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	17359.1			0.798402
$780$		0			0
$781$	1064.14			0.0487552
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−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	$-0.167449\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−35140.7			−1.57960
$792$		0			0
$793$		−	7995.06i		−	0.358024i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	24170.3i		−	1.07422i	−0.843511	−	0.537112i	$-0.819515\pi$
$797$		−	24170.3i		−	1.07422i	0.843511	−	0.537112i	$-0.180485\pi$
$798$		0			0
$799$	7300.32			0.323238
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	573.415i		−	0.0251997i
$804$		0			0
$805$	8930.25	+	8486.06i	0.390994	+	0.371546i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	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	$-0.301262\pi$
$811$	27002.2			1.16914			0.584572	+	0.811342i	$0.301262\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−6790.61	+	7146.05i	−0.291859	+	0.307135i
$816$		0			0
$817$			25931.5i			1.11044i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−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	$0.823228\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			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	$0.406064\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	2852.52i		−	0.118648i
$834$		0			0
$835$	−1670.47	+	1757.91i	−0.0692323	+	0.0728561i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	13678.1			0.562838			0.281419	−	0.959585i	$-0.409195\pi$
$839$	13678.1			0.562838			0.281419	+	0.959585i	$0.409195\pi$
$840$		0			0
$841$	61733.1			2.53118
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−9655.36	−	9175.11i	−0.393082	−	0.373531i
$846$		0			0
$847$		−	29489.5i		−	1.19630i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	986.512			0.0397382
$852$		0			0
$853$			29802.9i			1.19629i	0.801390	+	0.598143i	$0.204094\pi$
$853$			29802.9i			1.19629i	−0.801390	+	0.598143i	$0.795906\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			22045.2i			0.878706i	0.898314	+	0.439353i	$0.144792\pi$
$857$			22045.2i			0.878706i	−0.898314	+	0.439353i	$0.855208\pi$
$858$		0			0
$859$	33609.5			1.33497			0.667487	−	0.744622i	$-0.267370\pi$
$859$	33609.5			1.33497			0.667487	+	0.744622i	$0.267370\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			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
$866$		0			0
$867$		0			0
$868$		0			0
$869$	1379.62			0.0538556
$870$		0			0
$871$	−10123.0			−0.393805
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−23560.6	−	20206.1i	−0.910280	−	0.780676i
$876$		0			0
$877$		−	12637.0i		−	0.486570i	−0.969955	−	0.243285i	$-0.921775\pi$
$877$		−	12637.0i		−	0.486570i	0.969955	−	0.243285i	$-0.0782250\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	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	$0.588938\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	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
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	40314.6i		−	1.51072i
$894$		0			0
$895$	26041.5	+	24746.2i	0.972594	+	0.924217i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	18904.7			0.701342
$900$		0			0
$901$	8799.79			0.325376
$902$		0			0
$903$		0			0
$904$		0			0
$905$	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	$0.705076\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−10364.3			−0.376930			−0.188465	−	0.982080i	$-0.560351\pi$
$911$	−10364.3			−0.376930			−0.188465	+	0.982080i	$0.560351\pi$
$912$		0			0
$913$		−	311.435i		−	0.0112891i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	7143.58i		−	0.257254i
$918$		0			0
$919$	11451.9			0.411059			0.205530	−	0.978651i	$-0.434108\pi$
$919$	11451.9			0.411059			0.205530	+	0.978651i	$0.434108\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	34592.7i		−	1.23362i
$924$		0			0
$925$	−2482.31	+	126.701i	−0.0882356	+	0.00450367i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−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
$932$		0			0
$933$		0			0
$934$		0			0
$935$	261.806	−	275.510i	0.00915721	−	0.00963652i
$936$		0			0
$937$			5878.01i			0.204937i	0.994736	+	0.102469i	$0.0326741\pi$
$937$			5878.01i			0.204937i	−0.994736	+	0.102469i	$0.967326\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	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
$944$		0			0
$945$		0			0
$946$		0			0
$947$			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
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	31929.4i		−	1.08530i	−0.839958	−	0.542651i	$-0.817420\pi$
$953$		−	31929.4i		−	1.08530i	0.839958	−	0.542651i	$-0.182580\pi$
$954$		0			0
$955$	−28102.9	−	26705.1i	−0.952241	−	0.904877i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−7868.57			−0.264952
$960$		0			0
$961$	−25641.2			−0.860704
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−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	$-0.0579957\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	7041.97			0.232737			0.116368	−	0.993206i	$-0.462875\pi$
$971$	7041.97			0.232737			0.116368	+	0.993206i	$0.462875\pi$
$972$		0			0
$973$			1715.81i			0.0565328i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	37607.6i		−	1.23150i	−0.787943	−	0.615749i	$-0.788854\pi$
$977$		−	37607.6i		−	1.23150i	0.787943	−	0.615749i	$-0.211146\pi$
$978$		0			0
$979$	−1824.69			−0.0595684
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	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
$986$		0			0
$987$		0			0
$988$		0			0
$989$	12271.6			0.394556
$990$		0			0
$991$	41686.5			1.33624			0.668120	−	0.744053i	$-0.267099\pi$
$991$	41686.5			1.33624			0.668120	+	0.744053i	$0.267099\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	25191.2	+	23938.2i	0.802629	+	0.762707i
$996$		0			0
$997$		−	25465.9i		−	0.808939i	−0.914551	−	0.404470i	$-0.867456\pi$
$997$		−	25465.9i		−	0.808939i	0.914551	−	0.404470i	$-0.132544\pi$
$998$		0			0
$999$		0			0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.4.b.a.4.2	✓	4	60.59	even	2
15.4.b.a.4.3	yes	4	12.11	even	2
45.4.b.b.19.2		4	4.3	odd	2
45.4.b.b.19.3		4	20.19	odd	2
75.4.a.c.1.2		2	60.23	odd	4
75.4.a.f.1.1		2	60.47	odd	4
225.4.a.i.1.2		2	20.7	even	4
225.4.a.o.1.1		2	20.3	even	4
240.4.f.f.49.1		4	3.2	odd	2
240.4.f.f.49.3		4	15.14	odd	2
720.4.f.j.289.3		4	5.4	even	2	inner
720.4.f.j.289.4		4	1.1	even	1	trivial
960.4.f.p.769.2		4	120.29	odd	2
960.4.f.p.769.4		4	24.5	odd	2
960.4.f.q.769.2		4	24.11	even	2
960.4.f.q.769.4		4	120.59	even	2
1200.4.a.bn.1.1		2	15.2	even	4
1200.4.a.bt.1.2		2	15.8	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:	\( N \)	\(=\)	\( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	720.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(8.10469 + 7.70156i) q^{5} +22.2094i q^{7} -1.79063 q^{11} +58.2094i q^{13} +18.9844i q^{17} +104.837 q^{19} -49.6125i q^{23} +(6.37188 + 124.837i) q^{25} -293.466 q^{29} -64.4187 q^{31} +(-171.047 + 180.000i) q^{35} +19.8844i q^{37} +165.581 q^{41} +247.350i q^{43} -384.544i q^{47} -150.256 q^{49} -463.528i q^{53} +(-14.5125 - 13.7906i) q^{55} +73.7906 q^{59} -137.350 q^{61} +(-448.303 + 471.769i) q^{65} +173.906i q^{67} -594.281 q^{71} +320.231i q^{73} -39.7687i q^{77} -770.469 q^{79} +173.925i q^{83} +(-146.209 + 153.862i) q^{85} +1019.02 q^{89} -1292.79 q^{91} +(849.675 + 807.412i) q^{95} -384.375i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{5} - 84 q^{11} + 112 q^{19} + 256 q^{25} - 636 q^{29} - 104 q^{31} - 300 q^{35} + 816 q^{41} - 140 q^{49} + 864 q^{55} + 372 q^{59} + 680 q^{61} - 948 q^{65} - 72 q^{71} + 760 q^{79} - 508 q^{85}+ \cdots + 2784 q^{95}+O(q^{100}) \) 4 * q - 6 * q^5 - 84 * q^11 + 112 * q^19 + 256 * q^25 - 636 * q^29 - 104 * q^31 - 300 * q^35 + 816 * q^41 - 140 * q^49 + 864 * q^55 + 372 * q^59 + 680 * q^61 - 948 * q^65 - 72 * q^71 + 760 * q^79 - 508 * q^85 + 2232 * q^89 - 1944 * q^91 + 2784 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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