LMFDB - Embedded newform 960.4.f.p.769.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [960,4,Mod(769,960)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("960.769"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(960, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 960 = 2^{6} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	960.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,-6,0,0,0,-36,0,-84] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$56.6418336055$
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\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 15)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			769.4
Root			$3.70156i$ of defining polynomial
Character	$\chi$	$=$	960.769
Dual form			960.4.f.p.769.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000i q^{3} +(8.10469 + 7.70156i) q^{5} +22.2094i q^{7} -9.00000 q^{9} -1.79063 q^{11} -58.2094i q^{13} +(-23.1047 + 24.3141i) q^{15} -18.9844i q^{17} -104.837 q^{19} -66.6281 q^{21} +49.6125i q^{23} +(6.37188 + 124.837i) q^{25} -27.0000i q^{27} -293.466 q^{29} -64.4187 q^{31} -5.37188i q^{33} +(-171.047 + 180.000i) q^{35} -19.8844i q^{37} +174.628 q^{39} -165.581 q^{41} -247.350i q^{43} +(-72.9422 - 69.3141i) q^{45} +384.544i q^{47} -150.256 q^{49} +56.9531 q^{51} -463.528i q^{53} +(-14.5125 - 13.7906i) q^{55} -314.512i q^{57} +73.7906 q^{59} +137.350 q^{61} -199.884i q^{63} +(448.303 - 471.769i) q^{65} -173.906i q^{67} -148.837 q^{69} +594.281 q^{71} +320.231i q^{73} +(-374.512 + 19.1156i) q^{75} -39.7687i q^{77} -770.469 q^{79} +81.0000 q^{81} +173.925i q^{83} +(146.209 - 153.862i) q^{85} -880.397i q^{87} -1019.02 q^{89} +1292.79 q^{91} -193.256i q^{93} +(-849.675 - 807.412i) q^{95} -384.375i q^{97} +16.1156 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{5} - 36 q^{9} - 84 q^{11} - 54 q^{15} - 112 q^{19} - 36 q^{21} + 256 q^{25} - 636 q^{29} - 104 q^{31} - 300 q^{35} + 468 q^{39} - 816 q^{41} + 54 q^{45} - 140 q^{49} + 612 q^{51} + 864 q^{55}+ \cdots + 756 q^{99}+O(q^{100}) $ 4 * q - 6 * q^5 - 36 * q^9 - 84 * q^11 - 54 * q^15 - 112 * q^19 - 36 * q^21 + 256 * q^25 - 636 * q^29 - 104 * q^31 - 300 * q^35 + 468 * q^39 - 816 * q^41 + 54 * q^45 - 140 * q^49 + 612 * q^51 + 864 * q^55 + 372 * q^59 - 680 * q^61 + 948 * q^65 - 288 * q^69 + 72 * q^71 - 576 * q^75 + 760 * q^79 + 324 * q^81 + 508 * q^85 - 2232 * q^89 + 1944 * q^91 - 2784 * q^95 + 756 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/960\mathbb{Z}\right)^\times$.

$n$	$511$	$577$	$641$	$901$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$			3.00000i			0.577350i
$3$			3.00000i			0.577350i
$4$		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
$9$	−9.00000			−0.333333
$10$		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	$-0.786752\pi$
$13$		−	58.2094i		−	1.24188i	0.783860	−	0.620938i	$-0.213248\pi$
$14$		0			0
$15$	−23.1047	+	24.3141i	−0.397707	+	0.418524i
$16$		0			0
$17$		−	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	$0.718148\pi$
$20$		0			0
$21$	−66.6281			−0.692355
$22$		0			0
$23$			49.6125i			0.449779i	0.974384	+	0.224890i	$0.0722021\pi$
$23$			49.6125i			0.449779i	−0.974384	+	0.224890i	$0.927798\pi$
$24$		0			0
$25$	6.37188	+	124.837i	0.0509751	+	0.998700i
$26$		0			0
$27$		−	27.0000i		−	0.192450i
$28$		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
$33$		−	5.37188i		−	0.0283371i
$34$		0			0
$35$	−171.047	+	180.000i	−0.826063	+	0.869302i
$36$		0			0
$37$		−	19.8844i		−	0.0883505i	−0.999024	−	0.0441752i	$-0.985934\pi$
$37$		−	19.8844i		−	0.0883505i	0.999024	−	0.0441752i	$-0.0140660\pi$
$38$		0			0
$39$	174.628			0.716997
$40$		0			0
$41$	−165.581			−0.630718			−0.315359	−	0.948972i	$-0.602125\pi$
$41$	−165.581			−0.630718			−0.315359	+	0.948972i	$0.602125\pi$
$42$		0			0
$43$		−	247.350i		−	0.877221i	−0.898677	−	0.438611i	$-0.855471\pi$
$43$		−	247.350i		−	0.877221i	0.898677	−	0.438611i	$-0.144529\pi$
$44$		0			0
$45$	−72.9422	−	69.3141i	−0.241635	−	0.229616i
$46$		0			0
$47$			384.544i			1.19344i	0.802451	+	0.596718i	$0.203529\pi$
$47$			384.544i			1.19344i	−0.802451	+	0.596718i	$0.796471\pi$
$48$		0			0
$49$	−150.256			−0.438065
$50$		0			0
$51$	56.9531			0.156373
$52$		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
$57$		−	314.512i		−	0.730846i
$58$		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	$-0.453956\pi$
$61$	137.350			0.288293			0.144146	+	0.989556i	$0.453956\pi$
$62$		0			0
$63$		−	199.884i		−	0.399731i
$64$		0			0
$65$	448.303	−	471.769i	0.855464	−	0.900242i
$66$		0			0
$67$		−	173.906i		−	0.317105i	−0.987351	−	0.158552i	$-0.949317\pi$
$67$		−	173.906i		−	0.317105i	0.987351	−	0.158552i	$-0.0506827\pi$
$68$		0			0
$69$	−148.837			−0.259680
$70$		0			0
$71$	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	$0.0826398\pi$
$73$			320.231i			0.513428i	−0.966487	+	0.256714i	$0.917360\pi$
$74$		0			0
$75$	−374.512	+	19.1156i	−0.576600	+	0.0294305i
$76$		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
$81$	81.0000			0.111111
$82$		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
$87$		−	880.397i		−	1.08492i
$88$		0			0
$89$	−1019.02			−1.21367			−0.606834	−	0.794829i	$-0.707561\pi$
$89$	−1019.02			−1.21367			−0.606834	+	0.794829i	$0.707561\pi$
$90$		0			0
$91$	1292.79			1.48925
$92$		0			0
$93$		−	193.256i		−	0.215481i
$94$		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
$99$	16.1156			0.0163604
$100$		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
$105$	−540.000	−	513.141i	−0.501891	−	0.476928i
$106$		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	$-0.454939\pi$
$109$	321.119			0.282180			0.141090	+	0.989997i	$0.454939\pi$
$110$		0			0
$111$	59.6531			0.0510092
$112$		0			0
$113$		−	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
$116$		0			0
$117$			523.884i			0.413958i
$118$		0			0
$119$	421.631			0.324797
$120$		0			0
$121$	−1327.79			−0.997591
$122$		0			0
$123$		−	496.744i		−	0.364145i
$124$		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
$129$	742.050			0.506464
$130$		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
$135$	207.942	−	218.827i	0.132569	−	0.139508i
$136$		0			0
$137$		−	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	$0.507504\pi$
$140$		0			0
$141$	−1153.63			−0.689030
$142$		0			0
$143$			104.231i			0.0609529i
$144$		0			0
$145$	−2378.45	−	2260.14i	−1.36220	−	1.29445i
$146$		0			0
$147$		−	450.769i		−	0.252917i
$148$		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
$153$			170.859i			0.0902821i
$154$		0			0
$155$	−522.094	−	496.125i	−0.270552	−	0.257095i
$156$		0			0
$157$		−	1769.05i		−	0.899273i	−0.893212	−	0.449636i	$-0.851554\pi$
$157$		−	1769.05i		−	0.899273i	0.893212	−	0.449636i	$-0.148446\pi$
$158$		0			0
$159$	1390.58			0.693588
$160$		0			0
$161$	−1101.86			−0.539372
$162$		0			0
$163$		−	881.719i		−	0.423690i	−0.977303	−	0.211845i	$-0.932053\pi$
$163$		−	881.719i		−	0.423690i	0.977303	−	0.211845i	$-0.0679473\pi$
$164$		0			0
$165$	41.3719	−	43.5374i	0.0195200	−	0.0205417i
$166$		0			0
$167$		−	216.900i		−	0.100504i	−0.998737	−	0.0502522i	$-0.983997\pi$
$167$		−	216.900i		−	0.100504i	0.998737	−	0.0502522i	$-0.0160025\pi$
$168$		0			0
$169$	−1191.33			−0.542254
$170$		0			0
$171$	943.537			0.421954
$172$		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
$177$			221.372i			0.0940075i
$178$		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	$-0.745471\pi$
$181$	−3394.42			−1.39395			−0.696976	+	0.717095i	$0.745471\pi$
$182$		0			0
$183$			412.050i			0.166446i
$184$		0			0
$185$	153.141	−	161.156i	0.0608601	−	0.0640457i
$186$		0			0
$187$			33.9939i			0.0132935i
$188$		0			0
$189$	599.653			0.230785
$190$		0			0
$191$	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	$0.108466\pi$
$193$			1792.14i			0.668401i	−0.942502	+	0.334200i	$0.891534\pi$
$194$		0			0
$195$	1415.31	+	1344.91i	0.519755	+	0.493902i
$196$		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
$201$	521.719			0.183081
$202$		0			0
$203$		−	6517.69i		−	2.25346i
$204$		0			0
$205$	−1341.98	−	1275.23i	−0.457211	−	0.434469i
$206$		0			0
$207$		−	446.512i		−	0.149926i
$208$		0			0
$209$	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	$0.760361\pi$
$212$		0			0
$213$			1782.84i			0.573514i
$214$		0			0
$215$	1904.98	−	2004.69i	0.604273	−	0.635902i
$216$		0			0
$217$		−	1430.70i		−	0.447568i
$218$		0			0
$219$	−960.694			−0.296428
$220$		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
$225$	−57.3469	−	1123.54i	−0.0169917	−	0.332900i
$226$		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	$-0.621516\pi$
$229$	−2582.06			−0.745096			−0.372548	+	0.928013i	$0.621516\pi$
$230$		0			0
$231$	119.306			0.0339817
$232$		0			0
$233$			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
$236$		0			0
$237$		−	2311.41i		−	0.633510i
$238$		0			0
$239$	−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	$0.483731\pi$
$242$		0			0
$243$			243.000i			0.0641500i
$244$		0			0
$245$	−1217.78	−	1157.21i	−0.317555	−	0.301760i
$246$		0			0
$247$			6102.52i			1.57204i
$248$		0			0
$249$	−521.775			−0.132796
$250$		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
$255$	461.587	+	438.628i	0.113356	+	0.107717i
$256$		0			0
$257$			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
$260$		0			0
$261$	2641.19			0.626382
$262$		0			0
$263$			568.312i			0.133246i	0.997778	+	0.0666229i	$0.0212224\pi$
$263$			568.312i			0.133246i	−0.997778	+	0.0666229i	$0.978778\pi$
$264$		0			0
$265$	3569.89	−	3756.75i	0.827534	−	0.870850i
$266$		0			0
$267$		−	3057.07i		−	0.700711i
$268$		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
$273$			3878.38i			0.859818i
$274$		0			0
$275$	−11.4097	−	223.537i	−0.00250192	−	0.0490175i
$276$		0			0
$277$			6823.00i			1.47998i	0.672618	+	0.739990i	$0.265170\pi$
$277$			6823.00i			1.47998i	−0.672618	+	0.739990i	$0.734830\pi$
$278$		0			0
$279$	579.769			0.124408
$280$		0			0
$281$	3315.86			0.703942			0.351971	−	0.936011i	$-0.385512\pi$
$281$	3315.86			0.703942			0.351971	+	0.936011i	$0.385512\pi$
$282$		0			0
$283$			6602.76i			1.38690i	0.720504	+	0.693451i	$0.243910\pi$
$283$			6602.76i			1.38690i	−0.720504	+	0.693451i	$0.756090\pi$
$284$		0			0
$285$	2422.24	−	2549.02i	0.503442	−	0.529794i
$286$		0			0
$287$		−	3677.46i		−	0.756353i
$288$		0			0
$289$	4552.59			0.926642
$290$		0			0
$291$	1153.12			0.232293
$292$		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
$297$			48.3469i			0.00944570i
$298$		0			0
$299$	2887.91			0.558570
$300$		0			0
$301$	5493.49			1.05196
$302$		0			0
$303$		−	103.472i		−	0.0196181i
$304$		0			0
$305$	1113.18	+	1057.81i	0.208985	+	0.198590i
$306$		0			0
$307$			8124.86i			1.51046i	0.655462	+	0.755229i	$0.272474\pi$
$307$			8124.86i			1.51046i	−0.655462	+	0.755229i	$0.727526\pi$
$308$		0			0
$309$	−5268.91			−0.970025
$310$		0			0
$311$	−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	$-0.936268\pi$
$313$		−	2202.66i		−	0.397768i	0.980023	−	0.198884i	$-0.0637318\pi$
$314$		0			0
$315$	1539.42	−	1620.00i	0.275354	−	0.289767i
$316$		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
$321$	−4085.21			−0.710325
$322$		0			0
$323$			1990.27i			0.342854i
$324$		0			0
$325$	7266.71	−	370.903i	1.24026	−	0.0633046i
$326$		0			0
$327$			963.356i			0.162917i
$328$		0			0
$329$	−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	$0.756782\pi$
$332$		0			0
$333$			178.959i			0.0294502i
$334$		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
$339$	4746.74			0.760494
$340$		0			0
$341$	115.350			0.0183183
$342$		0			0
$343$			4280.72i			0.673869i
$344$		0			0
$345$	−1206.28	−	1146.28i	−0.188243	−	0.178880i
$346$		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	$-0.624239\pi$
$349$	−4961.26			−0.760946			−0.380473	+	0.924792i	$0.624239\pi$
$350$		0			0
$351$	−1571.65			−0.238999
$352$		0			0
$353$		−	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
$356$		0			0
$357$			1264.89i			0.187522i
$358$		0			0
$359$	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
$362$		0			0
$363$		−	3983.38i		−	0.575959i
$364$		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
$369$	1490.23			0.210239
$370$		0			0
$371$	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	$0.737433\pi$
$374$		0			0
$375$	−3182.53	−	2729.40i	−0.438253	−	0.375856i
$376$		0			0
$377$			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	$0.789827\pi$
$380$		0			0
$381$	3591.42			0.482924
$382$		0			0
$383$		−	6364.97i		−	0.849177i	−0.905387	−	0.424588i	$-0.860419\pi$
$383$		−	6364.97i		−	0.849177i	0.905387	−	0.424588i	$-0.139581\pi$
$384$		0			0
$385$	306.281	−	322.313i	0.0405442	−	0.0426665i
$386$		0			0
$387$			2226.15i			0.292407i
$388$		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
$393$		−	964.941i		−	0.123855i
$394$		0			0
$395$	−6244.41	−	5933.81i	−0.795418	−	0.755854i
$396$		0			0
$397$			9746.46i			1.23214i	0.787690	+	0.616072i	$0.211277\pi$
$397$			9746.46i			1.23214i	−0.787690	+	0.616072i	$0.788723\pi$
$398$		0			0
$399$	6985.12			0.876425
$400$		0			0
$401$	−1306.44			−0.162695			−0.0813474	−	0.996686i	$-0.525922\pi$
$401$	−1306.44			−0.162695			−0.0813474	+	0.996686i	$0.525922\pi$
$402$		0			0
$403$			3749.77i			0.463498i
$404$		0			0
$405$	656.480	+	623.827i	0.0805450	+	0.0765387i
$406$		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
$411$	1062.87			0.127561
$412$		0			0
$413$			1638.84i			0.195260i
$414$		0			0
$415$	−1339.49	+	1409.61i	−0.158441	+	0.166735i
$416$		0			0
$417$		−	231.769i		−	0.0272177i
$418$		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	$-0.344258\pi$
$421$	8119.73			0.939980			0.469990	+	0.882672i	$0.344258\pi$
$422$		0			0
$423$		−	3460.89i		−	0.397812i
$424$		0			0
$425$	2369.96	−	120.966i	0.270494	−	0.0138064i
$426$		0			0
$427$			3050.46i			0.345719i
$428$		0			0
$429$	−312.694			−0.0351911
$430$		0			0
$431$	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	$0.112768\pi$
$433$			6251.34i			0.693811i	−0.937900	+	0.346906i	$0.887232\pi$
$434$		0			0
$435$	6780.43	−	7135.34i	0.747349	−	0.786468i
$436$		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
$441$	1352.31			0.146022
$442$		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
$447$		−	5116.13i		−	0.541353i
$448$		0			0
$449$	9349.71			0.982717			0.491358	−	0.870957i	$-0.336501\pi$
$449$	9349.71			0.982717			0.491358	+	0.870957i	$0.336501\pi$
$450$		0			0
$451$	296.494			0.0309565
$452$		0			0
$453$		−	2274.84i		−	0.235941i
$454$		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
$459$	−512.578			−0.0521244
$460$		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
$465$	1488.37	−	1566.28i	0.148434	−	0.156203i
$466$		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
$471$	5307.16			0.519195
$472$		0			0
$473$			442.912i			0.0430552i
$474$		0			0
$475$	−668.012	−	13087.6i	−0.0645274	−	1.26422i
$476$		0			0
$477$			4171.75i			0.400443i
$478$		0			0
$479$	−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
$482$		0			0
$483$		−	3305.59i		−	0.311407i
$484$		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
$489$	2645.16			0.244618
$490$		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
$495$	130.612	+	124.116i	0.0118598	+	0.0112699i
$496$		0			0
$497$			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	$0.670396\pi$
$500$		0			0
$501$	650.700			0.0580262
$502$		0			0
$503$			5587.37i			0.495285i	0.968851	+	0.247643i	$0.0796559\pi$
$503$			5587.37i			0.495285i	−0.968851	+	0.247643i	$0.920344\pi$
$504$		0			0
$505$	−279.535	−	265.631i	−0.0246320	−	0.0234068i
$506$		0			0
$507$		−	3573.99i		−	0.313070i
$508$		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
$513$			2830.61i			0.243615i
$514$		0			0
$515$	−13526.3	+	14234.3i	−1.15736	+	1.21794i
$516$		0			0
$517$		−	688.574i		−	0.0585754i
$518$		0			0
$519$	12377.7			1.04686
$520$		0			0
$521$	19748.4			1.66064			0.830320	−	0.557286i	$-0.188157\pi$
$521$	19748.4			1.66064			0.830320	+	0.557286i	$0.188157\pi$
$522$		0			0
$523$		−	7843.44i		−	0.655774i	−0.944717	−	0.327887i	$-0.893663\pi$
$523$		−	7843.44i		−	0.655774i	0.944717	−	0.327887i	$-0.106337\pi$
$524$		0			0
$525$	−424.546	−	8317.69i	−0.0352928	−	0.691455i
$526$		0			0
$527$			1222.95i			0.101086i
$528$		0			0
$529$	9705.60			0.797699
$530$		0			0
$531$	−664.116			−0.0542753
$532$		0			0
$533$			9638.38i			0.783273i
$534$		0			0
$535$	−10487.5	+	11036.5i	−0.847504	+	0.891865i
$536$		0			0
$537$			9639.42i			0.774622i
$538$		0			0
$539$	269.053			0.0215008
$540$		0			0
$541$	−7383.29			−0.586751			−0.293376	−	0.955997i	$-0.594779\pi$
$541$	−7383.29			−0.586751			−0.293376	+	0.955997i	$0.594779\pi$
$542$		0			0
$543$		−	10183.3i		−	0.804798i
$544$		0			0
$545$	2602.57	+	2473.12i	0.204554	+	0.194379i
$546$		0			0
$547$			3354.90i			0.262240i	0.991367	+	0.131120i	$0.0418573\pi$
$547$			3354.90i			0.262240i	−0.991367	+	0.131120i	$0.958143\pi$
$548$		0			0
$549$	−1236.15			−0.0960976
$550$		0			0
$551$	30766.2			2.37874
$552$		0			0
$553$		−	17111.6i		−	1.31584i
$554$		0			0
$555$	483.469	+	459.422i	0.0369768	+	0.0351376i
$556$		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
$561$	−101.982			−0.00767500
$562$		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
$567$			1798.96i			0.133244i
$568$		0			0
$569$	−11549.5			−0.850931			−0.425466	−	0.904975i	$-0.639890\pi$
$569$	−11549.5			−0.850931			−0.425466	+	0.904975i	$0.639890\pi$
$570$		0			0
$571$	1482.54			0.108655			0.0543277	−	0.998523i	$-0.482698\pi$
$571$	1482.54			0.108655			0.0543277	+	0.998523i	$0.482698\pi$
$572$		0			0
$573$			10402.5i			0.758412i
$574$		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
$579$	−5376.43			−0.385901
$580$		0			0
$581$	−3862.76			−0.275825
$582$		0			0
$583$			830.006i			0.0589628i
$584$		0			0
$585$	−4034.73	+	4245.92i	−0.285155	+	0.300081i
$586$		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
$591$	−5034.57			−0.350414
$592$		0			0
$593$		−	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
$596$		0			0
$597$			9324.69i			0.639253i
$598$		0			0
$599$	−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	$0.533895\pi$
$602$		0			0
$603$			1565.16i			0.105702i
$604$		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
$609$	19553.1			1.30103
$610$		0			0
$611$	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	$-0.401468\pi$
$614$		0			0
$615$	3825.70	−	4025.95i	0.250841	−	0.263971i
$616$		0			0
$617$			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	$0.543022\pi$
$620$		0			0
$621$	1339.54			0.0865600
$622$		0			0
$623$		−	22631.9i		−	1.45542i
$624$		0			0
$625$	−15543.8	+	1590.90i	−0.994803	+	0.101818i
$626$		0			0
$627$			563.175i			0.0358709i
$628$		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
$633$		−	13419.8i		−	0.842637i
$634$		0			0
$635$	9219.85	−	9702.45i	0.576187	−	0.606346i
$636$		0			0
$637$			8746.32i			0.544022i
$638$		0			0
$639$	−5348.53			−0.331118
$640$		0			0
$641$	3909.35			0.240890			0.120445	−	0.992720i	$-0.461568\pi$
$641$	3909.35			0.240890			0.120445	+	0.992720i	$0.461568\pi$
$642$		0			0
$643$			30539.5i			1.87303i	0.350624	+	0.936516i	$0.385969\pi$
$643$			30539.5i			1.87303i	−0.350624	+	0.936516i	$0.614031\pi$
$644$		0			0
$645$	6014.08	+	5714.94i	0.367138	+	0.348877i
$646$		0			0
$647$		−	12707.7i		−	0.772167i	−0.922464	−	0.386083i	$-0.873828\pi$
$647$		−	12707.7i		−	0.772167i	0.922464	−	0.386083i	$-0.126172\pi$
$648$		0			0
$649$	−132.132			−0.00799170
$650$		0			0
$651$	4292.10			0.258403
$652$		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
$657$		−	2882.08i		−	0.171143i
$658$		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	$-0.459042\pi$
$661$	4361.31			0.256634			0.128317	+	0.991733i	$0.459042\pi$
$662$		0			0
$663$		−	3315.21i		−	0.194196i
$664$		0			0
$665$	17932.1	−	18870.7i	1.04568	−	1.10042i
$666$		0			0
$667$		−	14559.6i		−	0.845200i
$668$		0			0
$669$	−5260.25			−0.303995
$670$		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
$675$	3370.61	−	172.041i	0.192200	−	0.00981015i
$676$		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
$681$	−2810.70			−0.158159
$682$		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
$687$		−	7746.17i		−	0.430182i
$688$		0			0
$689$	−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	$0.383841\pi$
$692$		0			0
$693$			357.918i			0.0196193i
$694$		0			0
$695$	−626.138	−	594.994i	−0.0341737	−	0.0324740i
$696$		0			0
$697$			3143.46i			0.170828i
$698$		0			0
$699$	−6885.03			−0.372555
$700$		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
$705$	−9349.82	−	8884.76i	−0.499482	−	0.474638i
$706$		0			0
$707$		−	766.014i		−	0.0407481i
$708$		0			0
$709$	−14244.4			−0.754529			−0.377265	−	0.926105i	$-0.623135\pi$
$709$	−14244.4			−0.754529			−0.377265	+	0.926105i	$0.623135\pi$
$710$		0			0
$711$	6934.22			0.365757
$712$		0			0
$713$		−	3195.97i		−	0.167868i
$714$		0			0
$715$	−802.744	+	844.762i	−0.0419873	+	0.0441850i
$716$		0			0
$717$		−	6882.02i		−	0.358457i
$718$		0			0
$719$	−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
$722$		0			0
$723$			1146.86i			0.0589934i
$724$		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
$729$	−729.000			−0.0370370
$730$		0			0
$731$	−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	$0.932118\pi$
$734$		0			0
$735$	3471.62	−	3653.34i	0.174221	−	0.183341i
$736$		0			0
$737$			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	$0.336983\pi$
$740$		0			0
$741$	−18307.6			−0.907619
$742$		0			0
$743$			22526.6i			1.11227i	0.831091	+	0.556137i	$0.187717\pi$
$743$			22526.6i			1.11227i	−0.831091	+	0.556137i	$0.812283\pi$
$744$		0			0
$745$	−13821.6	−	13134.1i	−0.679708	−	0.645900i
$746$		0			0
$747$		−	1565.32i		−	0.0766696i
$748$		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
$753$		−	6779.95i		−	0.328121i
$754$		0			0
$755$	−6145.63	−	5839.95i	−0.296242	−	0.281507i
$756$		0			0
$757$			6619.98i			0.317843i	0.987291	+	0.158922i	$0.0508017\pi$
$757$			6619.98i			0.317843i	−0.987291	+	0.158922i	$0.949198\pi$
$758$		0			0
$759$	266.512			0.0127454
$760$		0			0
$761$	−29368.7			−1.39897			−0.699483	−	0.714649i	$-0.746586\pi$
$761$	−29368.7			−1.39897			−0.699483	+	0.714649i	$0.746586\pi$
$762$		0			0
$763$			7131.84i			0.338388i
$764$		0			0
$765$	−1315.88	+	1384.76i	−0.0621907	+	0.0654460i
$766$		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
$771$	−278.353			−0.0130021
$772$		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
$777$			1324.86i			0.0611699i
$778$		0			0
$779$	17359.1			0.798402
$780$		0			0
$781$	−1064.14			−0.0487552
$782$		0			0
$783$			7923.57i			0.361642i
$784$		0			0
$785$	13624.5	−	14337.6i	0.619463	−	0.651887i
$786$		0			0
$787$			22172.1i			1.00426i	0.864793	+	0.502128i	$0.167449\pi$
$787$			22172.1i			1.00426i	−0.864793	+	0.502128i	$0.832551\pi$
$788$		0			0
$789$	−1704.94			−0.0769295
$790$		0			0
$791$	35140.7			1.57960
$792$		0			0
$793$		−	7995.06i		−	0.358024i
$794$		0			0
$795$	11270.2	+	10709.7i	0.502786	+	0.477777i
$796$		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
$801$	9171.22			0.404556
$802$		0			0
$803$		−	573.415i		−	0.0251997i
$804$		0			0
$805$	−8930.25	−	8486.06i	−0.390994	−	0.371546i
$806$		0			0
$807$			22747.2i			0.992244i
$808$		0			0
$809$	−15304.2			−0.665102			−0.332551	−	0.943085i	$-0.607909\pi$
$809$	−15304.2			−0.665102			−0.332551	+	0.943085i	$0.607909\pi$
$810$		0			0
$811$	−27002.2			−1.16914			−0.584572	−	0.811342i	$-0.698738\pi$
$811$	−27002.2			−1.16914			−0.584572	+	0.811342i	$0.698738\pi$
$812$		0			0
$813$		−	23831.1i		−	1.02803i
$814$		0			0
$815$	6790.61	−	7146.05i	0.291859	−	0.307135i
$816$		0			0
$817$			25931.5i			1.11044i
$818$		0			0
$819$	−11635.1			−0.496416
$820$		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
$825$	670.612	−	34.2290i	0.0283003	−	0.00144449i
$826$		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	$-0.593936\pi$
$829$	−13884.2			−0.581687			−0.290844	+	0.956771i	$0.593936\pi$
$830$		0			0
$831$	−20469.0			−0.854467
$832$		0			0
$833$			2852.52i			0.118648i
$834$		0			0
$835$	1670.47	−	1757.91i	0.0692323	−	0.0728561i
$836$		0			0
$837$			1739.31i			0.0718270i
$838$		0			0
$839$	−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
$842$		0			0
$843$			9947.59i			0.406421i
$844$		0			0
$845$	−9655.36	−	9175.11i	−0.393082	−	0.373531i
$846$		0			0
$847$		−	29489.5i		−	1.19630i
$848$		0			0
$849$	−19808.3			−0.800728
$850$		0			0
$851$	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	$-0.204094\pi$
$854$		0			0
$855$	7647.07	+	7266.71i	0.305877	+	0.290662i
$856$		0			0
$857$		−	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	$0.732630\pi$
$860$		0			0
$861$	11032.4			0.436681
$862$		0			0
$863$		−	33775.6i		−	1.33226i	−0.745838	−	0.666128i	$-0.767951\pi$
$863$		−	33775.6i		−	1.33226i	0.745838	−	0.666128i	$-0.232049\pi$
$864$		0			0
$865$	31775.9	−	33439.2i	1.24903	−	1.31441i
$866$		0			0
$867$			13657.8i			0.534997i
$868$		0			0
$869$	1379.62			0.0538556
$870$		0			0
$871$	−10123.0			−0.393805
$872$		0			0
$873$			3459.37i			0.134115i
$874$		0			0
$875$	−23560.6	−	20206.1i	−0.910280	−	0.780676i
$876$		0			0
$877$			12637.0i			0.486570i	0.969955	+	0.243285i	$0.0782250\pi$
$877$			12637.0i			0.486570i	−0.969955	+	0.243285i	$0.921775\pi$
$878$		0			0
$879$	17442.4			0.669304
$880$		0			0
$881$	−6579.45			−0.251609			−0.125804	−	0.992055i	$-0.540151\pi$
$881$	−6579.45			−0.251609			−0.125804	+	0.992055i	$0.540151\pi$
$882$		0			0
$883$		−	50442.1i		−	1.92244i	−0.275786	−	0.961219i	$-0.588938\pi$
$883$		−	50442.1i		−	1.92244i	0.275786	−	0.961219i	$-0.411062\pi$
$884$		0			0
$885$	−1704.91	+	1794.15i	−0.0647569	+	0.0681465i
$886$		0			0
$887$			984.823i			0.0372797i	0.999826	+	0.0186399i	$0.00593360\pi$
$887$			984.823i			0.0372797i	−0.999826	+	0.0186399i	$0.994066\pi$
$888$		0			0
$889$	26587.7			1.00306
$890$		0			0
$891$	−145.041			−0.00545348
$892$		0			0
$893$		−	40314.6i		−	1.51072i
$894$		0			0
$895$	26041.5	+	24746.2i	0.972594	+	0.924217i
$896$		0			0
$897$			8663.74i			0.322490i
$898$		0			0
$899$	18904.7			0.701342
$900$		0			0
$901$	−8799.79			−0.325376
$902$		0			0
$903$			16480.5i			0.607348i
$904$		0			0
$905$	−27510.7	−	26142.3i	−1.01048	−	0.960221i
$906$		0			0
$907$		−	43679.9i		−	1.59908i	−0.600612	−	0.799541i	$-0.705076\pi$
$907$		−	43679.9i		−	1.59908i	0.600612	−	0.799541i	$-0.294924\pi$
$908$		0			0
$909$	310.415			0.0113265
$910$		0			0
$911$	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
$914$		0			0
$915$	−3173.43	+	3339.54i	−0.114656	+	0.120658i
$916$		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
$921$	−24374.6			−0.872063
$922$		0			0
$923$		−	34592.7i		−	1.23362i
$924$		0			0
$925$	2482.31	−	126.701i	0.0882356	−	0.00450367i
$926$		0			0
$927$		−	15806.7i		−	0.560044i
$928$		0			0
$929$	27701.8			0.978326			0.489163	−	0.872192i	$-0.337302\pi$
$929$	27701.8			0.978326			0.489163	+	0.872192i	$0.337302\pi$
$930$		0			0
$931$	15752.5			0.554529
$932$		0			0
$933$		−	22008.8i		−	0.772277i
$934$		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
$939$	6607.97			0.229652
$940$		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
$945$	4860.00	+	4618.27i	0.167297	+	0.158976i
$946$		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
$951$	−30026.6			−1.02385
$952$		0			0
$953$			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
$956$		0			0
$957$			1576.46i			0.0532495i
$958$		0			0
$959$	7868.57			0.264952
$960$		0			0
$961$	−25641.2			−0.860704
$962$		0			0
$963$		−	12255.6i		−	0.410106i
$964$		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
$969$	−5970.82			−0.197947
$970$		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
$975$	1112.71	+	21800.1i	0.0365490	+	0.716065i
$976$		0			0
$977$			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
$980$		0			0
$981$	−2890.07			−0.0940599
$982$		0			0
$983$			25297.7i			0.820826i	0.911900	+	0.410413i	$0.134615\pi$
$983$			25297.7i			0.820826i	−0.911900	+	0.410413i	$0.865385\pi$
$984$		0			0
$985$	−12924.7	+	13601.2i	−0.418086	+	0.439970i
$986$		0			0
$987$		−	25621.4i		−	0.826281i
$988$		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
$993$		−	26087.8i		−	0.833708i
$994$		0			0
$995$	25191.2	+	23938.2i	0.802629	+	0.762707i
$996$		0			0
$997$			25465.9i			0.808939i	0.914551	+	0.404470i	$0.132544\pi$
$997$			25465.9i			0.808939i	−0.914551	+	0.404470i	$0.867456\pi$
$998$		0			0
$999$	−536.878			−0.0170031

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	960.4.f.p.769.4		4
4.3	odd	2		960.4.f.q.769.2		4
5.4	even	2	inner	960.4.f.p.769.2		4
8.3	odd	2		15.4.b.a.4.3	yes	4
8.5	even	2		240.4.f.f.49.1		4
20.19	odd	2		960.4.f.q.769.4		4
24.5	odd	2		720.4.f.j.289.4		4
24.11	even	2		45.4.b.b.19.2		4
40.3	even	4		75.4.a.c.1.2		2
40.13	odd	4		1200.4.a.bt.1.2		2
40.19	odd	2		15.4.b.a.4.2	✓	4
40.27	even	4		75.4.a.f.1.1		2
40.29	even	2		240.4.f.f.49.3		4
40.37	odd	4		1200.4.a.bn.1.1		2
120.29	odd	2		720.4.f.j.289.3		4
120.59	even	2		45.4.b.b.19.3		4
120.83	odd	4		225.4.a.o.1.1		2
120.107	odd	4		225.4.a.i.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.4.b.a.4.2	✓	4	40.19	odd	2
15.4.b.a.4.3	yes	4	8.3	odd	2
45.4.b.b.19.2		4	24.11	even	2
45.4.b.b.19.3		4	120.59	even	2
75.4.a.c.1.2		2	40.3	even	4
75.4.a.f.1.1		2	40.27	even	4
225.4.a.i.1.2		2	120.107	odd	4
225.4.a.o.1.1		2	120.83	odd	4
240.4.f.f.49.1		4	8.5	even	2
240.4.f.f.49.3		4	40.29	even	2
720.4.f.j.289.3		4	120.29	odd	2
720.4.f.j.289.4		4	24.5	odd	2
960.4.f.p.769.2		4	5.4	even	2	inner
960.4.f.p.769.4		4	1.1	even	1	trivial
960.4.f.q.769.2		4	4.3	odd	2
960.4.f.q.769.4		4	20.19	odd	2
1200.4.a.bn.1.1		2	40.37	odd	4
1200.4.a.bt.1.2		2	40.13	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	960.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.00000i q^{3} +(8.10469 + 7.70156i) q^{5} +22.2094i q^{7} -9.00000 q^{9} -1.79063 q^{11} -58.2094i q^{13} +(-23.1047 + 24.3141i) q^{15} -18.9844i q^{17} -104.837 q^{19} -66.6281 q^{21} +49.6125i q^{23} +(6.37188 + 124.837i) q^{25} -27.0000i q^{27} -293.466 q^{29} -64.4187 q^{31} -5.37188i q^{33} +(-171.047 + 180.000i) q^{35} -19.8844i q^{37} +174.628 q^{39} -165.581 q^{41} -247.350i q^{43} +(-72.9422 - 69.3141i) q^{45} +384.544i q^{47} -150.256 q^{49} +56.9531 q^{51} -463.528i q^{53} +(-14.5125 - 13.7906i) q^{55} -314.512i q^{57} +73.7906 q^{59} +137.350 q^{61} -199.884i q^{63} +(448.303 - 471.769i) q^{65} -173.906i q^{67} -148.837 q^{69} +594.281 q^{71} +320.231i q^{73} +(-374.512 + 19.1156i) q^{75} -39.7687i q^{77} -770.469 q^{79} +81.0000 q^{81} +173.925i q^{83} +(146.209 - 153.862i) q^{85} -880.397i q^{87} -1019.02 q^{89} +1292.79 q^{91} -193.256i q^{93} +(-849.675 - 807.412i) q^{95} -384.375i q^{97} +16.1156 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{5} - 36 q^{9} - 84 q^{11} - 54 q^{15} - 112 q^{19} - 36 q^{21} + 256 q^{25} - 636 q^{29} - 104 q^{31} - 300 q^{35} + 468 q^{39} - 816 q^{41} + 54 q^{45} - 140 q^{49} + 612 q^{51} + 864 q^{55}+ \cdots + 756 q^{99}+O(q^{100}) \) 4 * q - 6 * q^5 - 36 * q^9 - 84 * q^11 - 54 * q^15 - 112 * q^19 - 36 * q^21 + 256 * q^25 - 636 * q^29 - 104 * q^31 - 300 * q^35 + 468 * q^39 - 816 * q^41 + 54 * q^45 - 140 * q^49 + 612 * q^51 + 864 * q^55 + 372 * q^59 - 680 * q^61 + 948 * q^65 - 288 * q^69 + 72 * q^71 - 576 * q^75 + 760 * q^79 + 324 * q^81 + 508 * q^85 - 2232 * q^89 + 1944 * q^91 - 2784 * q^95 + 756 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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