LMFDB - Embedded newform 1344.4.c.f.673.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1344,4,Mod(673,1344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1344, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1344.673");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1344 = 2^{6} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1344.c (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:	$79.2985670477$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 2 x^{11} - x^{10} - 861 x^{8} - 2158 x^{7} + 8654 x^{6} + 118244 x^{5} + 707300 x^{4} + \cdots + 43264 $ x^12 - 2x^11 - x^10 - 861x^8 - 2158x^7 + 8654x^6 + 118244x^5 + 707300x^4 + 1646096x^3 + 1391904x^2 + 174720*x + 43264
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{22}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			673.12
Root			$1.59395 - 5.94871i$ of defining polynomial
Character	$\chi$	$=$	1344.673
Dual form			1344.4.c.f.673.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000i q^{3} +17.4720i q^{5} -7.00000 q^{7} -9.00000 q^{9} +O(q^{10})$	$q+3.00000i q^{3} +17.4720i q^{5} -7.00000 q^{7} -9.00000 q^{9} +37.3882i q^{11} +4.32287i q^{13} -52.4161 q^{15} -0.453383 q^{17} -58.2551i q^{19} -21.0000i q^{21} -181.686 q^{23} -180.272 q^{25} -27.0000i q^{27} -255.263i q^{29} -30.0965 q^{31} -112.165 q^{33} -122.304i q^{35} +135.620i q^{37} -12.9686 q^{39} +30.5603 q^{41} +423.547i q^{43} -157.248i q^{45} -28.9804 q^{47} +49.0000 q^{49} -1.36015i q^{51} -117.967i q^{53} -653.247 q^{55} +174.765 q^{57} -447.313i q^{59} +22.4253i q^{61} +63.0000 q^{63} -75.5294 q^{65} +246.131i q^{67} -545.058i q^{69} -56.3796 q^{71} +268.070 q^{73} -540.815i q^{75} -261.717i q^{77} -964.677 q^{79} +81.0000 q^{81} -1289.25i q^{83} -7.92152i q^{85} +765.790 q^{87} -269.033 q^{89} -30.2601i q^{91} -90.2894i q^{93} +1017.83 q^{95} +968.154 q^{97} -336.494i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 84 q^{7} - 108 q^{9}+O(q^{10}) $ 12 * q - 84 * q^7 - 108 * q^9	$ 12 q - 84 q^{7} - 108 q^{9} + 24 q^{15} + 376 q^{17} - 336 q^{23} - 180 q^{25} + 192 q^{31} - 168 q^{33} - 504 q^{39} + 488 q^{41} - 448 q^{47} + 588 q^{49} - 3600 q^{55} - 432 q^{57} + 756 q^{63} + 1408 q^{65} - 5104 q^{71} - 1752 q^{73} - 1632 q^{79} + 972 q^{81} - 336 q^{87} - 3688 q^{89} - 2496 q^{95} - 1944 q^{97}+O(q^{100}) $ 12 * q - 84 * q^7 - 108 * q^9 + 24 * q^15 + 376 * q^17 - 336 * q^23 - 180 * q^25 + 192 * q^31 - 168 * q^33 - 504 * q^39 + 488 * q^41 - 448 * q^47 + 588 * q^49 - 3600 * q^55 - 432 * q^57 + 756 * q^63 + 1408 * q^65 - 5104 * q^71 - 1752 * q^73 - 1632 * q^79 + 972 * q^81 - 336 * q^87 - 3688 * q^89 - 2496 * q^95 - 1944 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times$.

$n$	$127$	$449$	$577$	$1093$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$5$	17.4720i	1.56275i	0.624065	+	0.781373i	$0.285480\pi$
$5$	17.4720i	1.56275i	−0.624065	+	0.781373i	$0.714520\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	−9.00000	−0.333333
$10$	0	0
$11$	37.3882i	1.02481i	0.858743	+	0.512407i	$0.171246\pi$
$11$	37.3882i	1.02481i	−0.858743	+	0.512407i	$0.828754\pi$
$12$	0	0
$13$	4.32287i	0.0922269i	0.998936	+	0.0461134i	$0.0146836\pi$
$13$	4.32287i	0.0922269i	−0.998936	+	0.0461134i	$0.985316\pi$
$14$	0	0
$15$	−52.4161	−0.902251
$16$	0	0
$17$	−0.453383	−0.00646833	−0.00323416	−	0.999995i	$-0.501029\pi$
$17$	−0.453383	−0.00646833	−0.00323416	+	0.999995i	$0.501029\pi$
$18$	0	0
$19$	− 58.2551i	− 0.703402i	−0.936112	−	0.351701i	$-0.885603\pi$
$19$	− 58.2551i	− 0.703402i	0.936112	−	0.351701i	$-0.114397\pi$
$20$	0	0
$21$	− 21.0000i	− 0.218218i
$22$	0	0
$23$	−181.686	−1.64714	−0.823568	−	0.567218i	$-0.808020\pi$
$23$	−181.686	−1.64714	−0.823568	+	0.567218i	$0.808020\pi$
$24$	0	0
$25$	−180.272	−1.44217
$26$	0	0
$27$	− 27.0000i	− 0.192450i
$28$	0	0
$29$	− 255.263i	− 1.63452i	−0.576266	−	0.817262i	$-0.695491\pi$
$29$	− 255.263i	− 1.63452i	0.576266	−	0.817262i	$-0.304509\pi$
$30$	0	0
$31$	−30.0965	−0.174370	−0.0871852	−	0.996192i	$-0.527787\pi$
$31$	−30.0965	−0.174370	−0.0871852	+	0.996192i	$0.527787\pi$
$32$	0	0
$33$	−112.165	−0.591677
$34$	0	0
$35$	− 122.304i	− 0.590662i
$36$	0	0
$37$	135.620i	0.602589i	0.953531	+	0.301294i	$0.0974187\pi$
$37$	135.620i	0.602589i	−0.953531	+	0.301294i	$0.902581\pi$
$38$	0	0
$39$	−12.9686	−0.0532472
$40$	0	0
$41$	30.5603	0.116408	0.0582039	−	0.998305i	$-0.481463\pi$
$41$	30.5603	0.116408	0.0582039	+	0.998305i	$0.481463\pi$
$42$	0	0
$43$	423.547i	1.50210i	0.660245	+	0.751051i	$0.270453\pi$
$43$	423.547i	1.50210i	−0.660245	+	0.751051i	$0.729547\pi$
$44$	0	0
$45$	− 157.248i	− 0.520915i
$46$	0	0
$47$	−28.9804	−0.0899408	−0.0449704	−	0.998988i	$-0.514319\pi$
$47$	−28.9804	−0.0899408	−0.0449704	+	0.998988i	$0.514319\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	− 1.36015i	− 0.00373449i
$52$	0	0
$53$	− 117.967i	− 0.305736i	−0.988247	−	0.152868i	$-0.951149\pi$
$53$	− 117.967i	− 0.305736i	0.988247	−	0.152868i	$-0.0488509\pi$
$54$	0	0
$55$	−653.247	−1.60152
$56$	0	0
$57$	174.765	0.406110
$58$	0	0
$59$	− 447.313i	− 0.987036i	−0.869735	−	0.493518i	$-0.835711\pi$
$59$	− 447.313i	− 0.987036i	0.869735	−	0.493518i	$-0.164289\pi$
$60$	0	0
$61$	22.4253i	0.0470699i	0.999723	+	0.0235350i	$0.00749211\pi$
$61$	22.4253i	0.0470699i	−0.999723	+	0.0235350i	$0.992508\pi$
$62$	0	0
$63$	63.0000	0.125988
$64$	0	0
$65$	−75.5294	−0.144127
$66$	0	0
$67$	246.131i	0.448802i	0.974497	+	0.224401i	$0.0720426\pi$
$67$	246.131i	0.448802i	−0.974497	+	0.224401i	$0.927957\pi$
$68$	0	0
$69$	− 545.058i	− 0.950974i
$70$	0	0
$71$	−56.3796	−0.0942398	−0.0471199	−	0.998889i	$-0.515004\pi$
$71$	−56.3796	−0.0942398	−0.0471199	+	0.998889i	$0.515004\pi$
$72$	0	0
$73$	268.070	0.429797	0.214899	−	0.976636i	$-0.431058\pi$
$73$	268.070	0.429797	0.214899	+	0.976636i	$0.431058\pi$
$74$	0	0
$75$	− 540.815i	− 0.832639i
$76$	0	0
$77$	− 261.717i	− 0.387343i
$78$	0	0
$79$	−964.677	−1.37386	−0.686928	−	0.726725i	$-0.741041\pi$
$79$	−964.677	−1.37386	−0.686928	+	0.726725i	$0.741041\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	− 1289.25i	− 1.70498i	−0.522742	−	0.852491i	$-0.675091\pi$
$83$	− 1289.25i	− 1.70498i	0.522742	−	0.852491i	$-0.324909\pi$
$84$	0	0
$85$	− 7.92152i	− 0.0101083i
$86$	0	0
$87$	765.790	0.943693
$88$	0	0
$89$	−269.033	−0.320420	−0.160210	−	0.987083i	$-0.551217\pi$
$89$	−269.033	−0.320420	−0.160210	+	0.987083i	$0.551217\pi$
$90$	0	0
$91$	− 30.2601i	− 0.0348585i
$92$	0	0
$93$	− 90.2894i	− 0.100673i
$94$	0	0
$95$	1017.83	1.09924
$96$	0	0
$97$	968.154	1.01341	0.506707	−	0.862118i	$-0.330863\pi$
$97$	968.154	1.01341	0.506707	+	0.862118i	$0.330863\pi$
$98$	0	0
$99$	− 336.494i	− 0.341605i
$100$	0	0
$101$	1665.47i	1.64080i	0.571793	+	0.820398i	$0.306248\pi$
$101$	1665.47i	1.64080i	−0.571793	+	0.820398i	$0.693752\pi$
$102$	0	0
$103$	−368.792	−0.352798	−0.176399	−	0.984319i	$-0.556445\pi$
$103$	−368.792	−0.352798	−0.176399	+	0.984319i	$0.556445\pi$
$104$	0	0
$105$	366.912	0.341019
$106$	0	0
$107$	− 476.639i	− 0.430640i	−0.976544	−	0.215320i	$-0.930921\pi$
$107$	− 476.639i	− 0.430640i	0.976544	−	0.215320i	$-0.0690794\pi$
$108$	0	0
$109$	1263.41i	1.11021i	0.831780	+	0.555106i	$0.187322\pi$
$109$	1263.41i	1.11021i	−0.831780	+	0.555106i	$0.812678\pi$
$110$	0	0
$111$	−406.860	−0.347905
$112$	0	0
$113$	−505.593	−0.420905	−0.210452	−	0.977604i	$-0.567494\pi$
$113$	−505.593	−0.420905	−0.210452	+	0.977604i	$0.567494\pi$
$114$	0	0
$115$	− 3174.42i	− 2.57405i
$116$	0	0
$117$	− 38.9059i	− 0.0307423i
$118$	0	0
$119$	3.17368	0.00244480
$120$	0	0
$121$	−66.8758	−0.0502448
$122$	0	0
$123$	91.6809i	0.0672080i
$124$	0	0
$125$	− 965.706i	− 0.691003i
$126$	0	0
$127$	703.892	0.491813	0.245907	−	0.969294i	$-0.420914\pi$
$127$	703.892	0.491813	0.245907	+	0.969294i	$0.420914\pi$
$128$	0	0
$129$	−1270.64	−0.867239
$130$	0	0
$131$	− 1171.40i	− 0.781267i	−0.920546	−	0.390634i	$-0.872256\pi$
$131$	− 1171.40i	− 0.781267i	0.920546	−	0.390634i	$-0.127744\pi$
$132$	0	0
$133$	407.786i	0.265861i
$134$	0	0
$135$	471.745	0.300750
$136$	0	0
$137$	2953.17	1.84165	0.920827	−	0.389972i	$-0.127515\pi$
$137$	2953.17	1.84165	0.920827	+	0.389972i	$0.127515\pi$
$138$	0	0
$139$	2649.44i	1.61671i	0.588698	+	0.808353i	$0.299641\pi$
$139$	2649.44i	1.61671i	−0.588698	+	0.808353i	$0.700359\pi$
$140$	0	0
$141$	− 86.9411i	− 0.0519274i
$142$	0	0
$143$	−161.624	−0.0945155
$144$	0	0
$145$	4459.97	2.55435
$146$	0	0
$147$	147.000i	0.0824786i
$148$	0	0
$149$	− 2667.36i	− 1.46657i	−0.679923	−	0.733284i	$-0.737987\pi$
$149$	− 2667.36i	− 1.46657i	0.679923	−	0.733284i	$-0.262013\pi$
$150$	0	0
$151$	−2149.89	−1.15864	−0.579322	−	0.815099i	$-0.696683\pi$
$151$	−2149.89	−1.15864	−0.579322	+	0.815099i	$0.696683\pi$
$152$	0	0
$153$	4.08045	0.00215611
$154$	0	0
$155$	− 525.846i	− 0.272497i
$156$	0	0
$157$	852.825i	0.433521i	0.976225	+	0.216761i	$0.0695491\pi$
$157$	852.825i	0.433521i	−0.976225	+	0.216761i	$0.930451\pi$
$158$	0	0
$159$	353.901	0.176517
$160$	0	0
$161$	1271.80	0.622559
$162$	0	0
$163$	− 2823.90i	− 1.35696i	−0.734618	−	0.678481i	$-0.762638\pi$
$163$	− 2823.90i	− 1.35696i	0.734618	−	0.678481i	$-0.237362\pi$
$164$	0	0
$165$	− 1959.74i	− 0.924640i
$166$	0	0
$167$	2418.95	1.12086	0.560431	−	0.828201i	$-0.310636\pi$
$167$	2418.95	1.12086	0.560431	+	0.828201i	$0.310636\pi$
$168$	0	0
$169$	2178.31	0.991494
$170$	0	0
$171$	524.296i	0.234467i
$172$	0	0
$173$	1578.81i	0.693844i	0.937894	+	0.346922i	$0.112773\pi$
$173$	1578.81i	0.693844i	−0.937894	+	0.346922i	$0.887227\pi$
$174$	0	0
$175$	1261.90	0.545090
$176$	0	0
$177$	1341.94	0.569866
$178$	0	0
$179$	− 1952.03i	− 0.815093i	−0.913184	−	0.407547i	$-0.866384\pi$
$179$	− 1952.03i	− 0.815093i	0.913184	−	0.407547i	$-0.133616\pi$
$180$	0	0
$181$	− 1098.00i	− 0.450904i	−0.974254	−	0.225452i	$-0.927614\pi$
$181$	− 1098.00i	− 0.450904i	0.974254	−	0.225452i	$-0.0723859\pi$
$182$	0	0
$183$	−67.2759	−0.0271758
$184$	0	0
$185$	−2369.55	−0.941693
$186$	0	0
$187$	− 16.9512i	− 0.00662883i
$188$	0	0
$189$	189.000i	0.0727393i
$190$	0	0
$191$	−2749.65	−1.04166	−0.520831	−	0.853660i	$-0.674378\pi$
$191$	−2749.65	−1.04166	−0.520831	+	0.853660i	$0.674378\pi$
$192$	0	0
$193$	4324.91	1.61302	0.806512	−	0.591218i	$-0.201353\pi$
$193$	4324.91	1.61302	0.806512	+	0.591218i	$0.201353\pi$
$194$	0	0
$195$	− 226.588i	− 0.0832118i
$196$	0	0
$197$	− 4628.81i	− 1.67406i	−0.547161	−	0.837028i	$-0.684291\pi$
$197$	− 4628.81i	− 1.67406i	0.547161	−	0.837028i	$-0.315709\pi$
$198$	0	0
$199$	−5309.40	−1.89132	−0.945661	−	0.325153i	$-0.894584\pi$
$199$	−5309.40	−1.89132	−0.945661	+	0.325153i	$0.894584\pi$
$200$	0	0
$201$	−738.394	−0.259116
$202$	0	0
$203$	1786.84i	0.617792i
$204$	0	0
$205$	533.950i	0.181916i
$206$	0	0
$207$	1635.17	0.549045
$208$	0	0
$209$	2178.05	0.720857
$210$	0	0
$211$	3782.53i	1.23412i	0.786915	+	0.617061i	$0.211677\pi$
$211$	3782.53i	1.23412i	−0.786915	+	0.617061i	$0.788323\pi$
$212$	0	0
$213$	− 169.139i	− 0.0544094i
$214$	0	0
$215$	−7400.23	−2.34740
$216$	0	0
$217$	210.675	0.0659058
$218$	0	0
$219$	804.209i	0.248143i
$220$	0	0
$221$	− 1.95992i	0 0.000596554i
$222$	0	0
$223$	−1866.79	−0.560580	−0.280290	−	0.959915i	$-0.590431\pi$
$223$	−1866.79	−0.560580	−0.280290	+	0.959915i	$0.590431\pi$
$224$	0	0
$225$	1622.44	0.480724
$226$	0	0
$227$	− 4551.03i	− 1.33067i	−0.746545	−	0.665335i	$-0.768289\pi$
$227$	− 4551.03i	− 1.33067i	0.746545	−	0.665335i	$-0.231711\pi$
$228$	0	0
$229$	− 5784.65i	− 1.66926i	−0.550811	−	0.834630i	$-0.685681\pi$
$229$	− 5784.65i	− 1.66926i	0.550811	−	0.834630i	$-0.314319\pi$
$230$	0	0
$231$	785.152	0.223633
$232$	0	0
$233$	−2286.02	−0.642757	−0.321379	−	0.946951i	$-0.604146\pi$
$233$	−2286.02	−0.642757	−0.321379	+	0.946951i	$0.604146\pi$
$234$	0	0
$235$	− 506.345i	− 0.140555i
$236$	0	0
$237$	− 2894.03i	− 0.793196i
$238$	0	0
$239$	−1448.66	−0.392074	−0.196037	−	0.980596i	$-0.562807\pi$
$239$	−1448.66	−0.392074	−0.196037	+	0.980596i	$0.562807\pi$
$240$	0	0
$241$	−778.931	−0.208197	−0.104098	−	0.994567i	$-0.533196\pi$
$241$	−778.931	−0.208197	−0.104098	+	0.994567i	$0.533196\pi$
$242$	0	0
$243$	243.000i	0.0641500i
$244$	0	0
$245$	856.129i	0.223249i
$246$	0	0
$247$	251.830	0.0648726
$248$	0	0
$249$	3867.75	0.984372
$250$	0	0
$251$	− 1351.67i	− 0.339907i	−0.985452	−	0.169954i	$-0.945638\pi$
$251$	− 1351.67i	− 0.339907i	0.985452	−	0.169954i	$-0.0543618\pi$
$252$	0	0
$253$	− 6792.90i	− 1.68801i
$254$	0	0
$255$	23.7646	0.00583606
$256$	0	0
$257$	−5562.00	−1.34999	−0.674996	−	0.737821i	$-0.735855\pi$
$257$	−5562.00	−1.34999	−0.674996	+	0.737821i	$0.735855\pi$
$258$	0	0
$259$	− 949.340i	− 0.227757i
$260$	0	0
$261$	2297.37i	0.544842i
$262$	0	0
$263$	−761.231	−0.178477	−0.0892387	−	0.996010i	$-0.528443\pi$
$263$	−761.231	−0.178477	−0.0892387	+	0.996010i	$0.528443\pi$
$264$	0	0
$265$	2061.12	0.477787
$266$	0	0
$267$	− 807.098i	− 0.184995i
$268$	0	0
$269$	− 1069.55i	− 0.242422i	−0.992627	−	0.121211i	$-0.961322\pi$
$269$	− 1069.55i	− 0.242422i	0.992627	−	0.121211i	$-0.0386778\pi$
$270$	0	0
$271$	132.954	0.0298022	0.0149011	−	0.999889i	$-0.495257\pi$
$271$	132.954	0.0298022	0.0149011	+	0.999889i	$0.495257\pi$
$272$	0	0
$273$	90.7804	0.0201256
$274$	0	0
$275$	− 6740.02i	− 1.47796i
$276$	0	0
$277$	− 4583.31i	− 0.994168i	−0.867703	−	0.497084i	$-0.834404\pi$
$277$	− 4583.31i	− 0.994168i	0.867703	−	0.497084i	$-0.165596\pi$
$278$	0	0
$279$	270.868	0.0581235
$280$	0	0
$281$	7961.29	1.69014	0.845072	−	0.534652i	$-0.179557\pi$
$281$	7961.29	1.69014	0.845072	+	0.534652i	$0.179557\pi$
$282$	0	0
$283$	− 8316.15i	− 1.74680i	−0.487005	−	0.873399i	$-0.661911\pi$
$283$	− 8316.15i	− 1.74680i	0.487005	−	0.873399i	$-0.338089\pi$
$284$	0	0
$285$	3053.50i	0.634646i
$286$	0	0
$287$	−213.922	−0.0439980
$288$	0	0
$289$	−4912.79	−0.999958
$290$	0	0
$291$	2904.46i	0.585095i
$292$	0	0
$293$	− 175.584i	− 0.0350093i	−0.999847	−	0.0175046i	$-0.994428\pi$
$293$	− 175.584i	− 0.0350093i	0.999847	−	0.0175046i	$-0.00557218\pi$
$294$	0	0
$295$	7815.46	1.54249
$296$	0	0
$297$	1009.48	0.197226
$298$	0	0
$299$	− 785.405i	− 0.151910i
$300$	0	0
$301$	− 2964.83i	− 0.567741i
$302$	0	0
$303$	−4996.41	−0.947314
$304$	0	0
$305$	−391.815	−0.0735583
$306$	0	0
$307$	5368.85i	0.998100i	0.866573	+	0.499050i	$0.166318\pi$
$307$	5368.85i	0.998100i	−0.866573	+	0.499050i	$0.833682\pi$
$308$	0	0
$309$	− 1106.38i	− 0.203688i
$310$	0	0
$311$	−4211.61	−0.767906	−0.383953	−	0.923353i	$-0.625438\pi$
$311$	−4211.61	−0.767906	−0.383953	+	0.923353i	$0.625438\pi$
$312$	0	0
$313$	−6473.79	−1.16907	−0.584537	−	0.811367i	$-0.698724\pi$
$313$	−6473.79	−1.16907	−0.584537	+	0.811367i	$0.698724\pi$
$314$	0	0
$315$	1100.74i	0.196887i
$316$	0	0
$317$	8116.09i	1.43800i	0.695011	+	0.718999i	$0.255399\pi$
$317$	8116.09i	1.43800i	−0.695011	+	0.718999i	$0.744601\pi$
$318$	0	0
$319$	9543.83	1.67508
$320$	0	0
$321$	1429.92	0.248630
$322$	0	0
$323$	26.4119i	0.00454984i
$324$	0	0
$325$	− 779.291i	− 0.133007i
$326$	0	0
$327$	−3790.24	−0.640981
$328$	0	0
$329$	202.862	0.0339944
$330$	0	0
$331$	2109.02i	0.350218i	0.984549	+	0.175109i	$0.0560278\pi$
$331$	2109.02i	0.350218i	−0.984549	+	0.175109i	$0.943972\pi$
$332$	0	0
$333$	− 1220.58i	− 0.200863i
$334$	0	0
$335$	−4300.41	−0.701363
$336$	0	0
$337$	4799.81	0.775852	0.387926	−	0.921690i	$-0.373192\pi$
$337$	4799.81	0.775852	0.387926	+	0.921690i	$0.373192\pi$
$338$	0	0
$339$	− 1516.78i	− 0.243009i
$340$	0	0
$341$	− 1125.25i	− 0.178697i
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	9523.26	1.48613
$346$	0	0
$347$	6198.46i	0.958937i	0.877559	+	0.479468i	$0.159170\pi$
$347$	6198.46i	0.958937i	−0.877559	+	0.479468i	$0.840830\pi$
$348$	0	0
$349$	8792.41i	1.34856i	0.738476	+	0.674279i	$0.235546\pi$
$349$	8792.41i	1.34856i	−0.738476	+	0.674279i	$0.764454\pi$
$350$	0	0
$351$	116.718	0.0177491
$352$	0	0
$353$	−10149.1	−1.53027	−0.765133	−	0.643872i	$-0.777327\pi$
$353$	−10149.1	−1.53027	−0.765133	+	0.643872i	$0.777327\pi$
$354$	0	0
$355$	− 985.066i	− 0.147273i
$356$	0	0
$357$	9.52104i	0.00141150i
$358$	0	0
$359$	−1114.46	−0.163842	−0.0819208	−	0.996639i	$-0.526105\pi$
$359$	−1114.46	−0.163842	−0.0819208	+	0.996639i	$0.526105\pi$
$360$	0	0
$361$	3465.34	0.505225
$362$	0	0
$363$	− 200.627i	− 0.0290088i
$364$	0	0
$365$	4683.72i	0.671663i
$366$	0	0
$367$	4455.22	0.633680	0.316840	−	0.948479i	$-0.397378\pi$
$367$	4455.22	0.633680	0.316840	+	0.948479i	$0.397378\pi$
$368$	0	0
$369$	−275.043	−0.0388026
$370$	0	0
$371$	825.768i	0.115557i
$372$	0	0
$373$	12462.7i	1.73001i	0.501762	+	0.865006i	$0.332685\pi$
$373$	12462.7i	1.73001i	−0.501762	+	0.865006i	$0.667315\pi$
$374$	0	0
$375$	2897.12	0.398951
$376$	0	0
$377$	1103.47	0.150747
$378$	0	0
$379$	− 8572.04i	− 1.16178i	−0.813981	−	0.580892i	$-0.802704\pi$
$379$	− 8572.04i	− 1.16178i	0.813981	−	0.580892i	$-0.197296\pi$
$380$	0	0
$381$	2111.67i	0.283948i
$382$	0	0
$383$	−1512.38	−0.201772	−0.100886	−	0.994898i	$-0.532168\pi$
$383$	−1512.38	−0.201772	−0.100886	+	0.994898i	$0.532168\pi$
$384$	0	0
$385$	4572.73	0.605319
$386$	0	0
$387$	− 3811.93i	− 0.500700i
$388$	0	0
$389$	7534.31i	0.982017i	0.871155	+	0.491008i	$0.163372\pi$
$389$	7534.31i	0.982017i	−0.871155	+	0.491008i	$0.836628\pi$
$390$	0	0
$391$	82.3733	0.0106542
$392$	0	0
$393$	3514.21	0.451065
$394$	0	0
$395$	− 16854.9i	− 2.14699i
$396$	0	0
$397$	− 1732.14i	− 0.218977i	−0.993988	−	0.109488i	$-0.965079\pi$
$397$	− 1732.14i	− 0.218977i	0.993988	−	0.109488i	$-0.0349213\pi$
$398$	0	0
$399$	−1223.36	−0.153495
$400$	0	0
$401$	4360.84	0.543068	0.271534	−	0.962429i	$-0.412469\pi$
$401$	4360.84	0.543068	0.271534	+	0.962429i	$0.412469\pi$
$402$	0	0
$403$	− 130.103i	− 0.0160816i
$404$	0	0
$405$	1415.23i	0.173638i
$406$	0	0
$407$	−5070.58	−0.617542
$408$	0	0
$409$	−4793.15	−0.579476	−0.289738	−	0.957106i	$-0.593568\pi$
$409$	−4793.15	−0.579476	−0.289738	+	0.957106i	$0.593568\pi$
$410$	0	0
$411$	8859.52i	1.06328i
$412$	0	0
$413$	3131.19i	0.373065i
$414$	0	0
$415$	22525.8	2.66445
$416$	0	0
$417$	−7948.31	−0.933406
$418$	0	0
$419$	− 5001.59i	− 0.583159i	−0.956547	−	0.291580i	$-0.905819\pi$
$419$	− 5001.59i	− 0.583159i	0.956547	−	0.291580i	$-0.0941808\pi$
$420$	0	0
$421$	− 15021.2i	− 1.73892i	−0.494001	−	0.869461i	$-0.664466\pi$
$421$	− 15021.2i	− 1.73892i	0.494001	−	0.869461i	$-0.335534\pi$
$422$	0	0
$423$	260.823	0.0299803
$424$	0	0
$425$	81.7321	0.00932844
$426$	0	0
$427$	− 156.977i	− 0.0177908i
$428$	0	0
$429$	− 484.873i	− 0.0545685i
$430$	0	0
$431$	8850.80	0.989160	0.494580	−	0.869132i	$-0.335322\pi$
$431$	8850.80	0.989160	0.494580	+	0.869132i	$0.335322\pi$
$432$	0	0
$433$	9712.52	1.07795	0.538977	−	0.842321i	$-0.318811\pi$
$433$	9712.52	1.07795	0.538977	+	0.842321i	$0.318811\pi$
$434$	0	0
$435$	13379.9i	1.47475i
$436$	0	0
$437$	10584.1i	1.15860i
$438$	0	0
$439$	−9005.22	−0.979033	−0.489517	−	0.871994i	$-0.662827\pi$
$439$	−9005.22	−0.979033	−0.489517	+	0.871994i	$0.662827\pi$
$440$	0	0
$441$	−441.000	−0.0476190
$442$	0	0
$443$	− 14131.1i	− 1.51555i	−0.652519	−	0.757773i	$-0.726288\pi$
$443$	− 14131.1i	− 1.51555i	0.652519	−	0.757773i	$-0.273712\pi$
$444$	0	0
$445$	− 4700.54i	− 0.500735i
$446$	0	0
$447$	8002.08	0.846723
$448$	0	0
$449$	−17279.4	−1.81619	−0.908093	−	0.418769i	$-0.862462\pi$
$449$	−17279.4	−1.81619	−0.908093	+	0.418769i	$0.862462\pi$
$450$	0	0
$451$	1142.59i	0.119296i
$452$	0	0
$453$	− 6449.66i	− 0.668944i
$454$	0	0
$455$	528.706	0.0544749
$456$	0	0
$457$	−11926.0	−1.22073	−0.610365	−	0.792120i	$-0.708977\pi$
$457$	−11926.0	−1.22073	−0.610365	+	0.792120i	$0.708977\pi$
$458$	0	0
$459$	12.2413i	0.00124483i
$460$	0	0
$461$	− 771.769i	− 0.0779715i	−0.999240	−	0.0389858i	$-0.987587\pi$
$461$	− 771.769i	− 0.0779715i	0.999240	−	0.0389858i	$-0.0124127\pi$
$462$	0	0
$463$	3467.01	0.348003	0.174002	−	0.984745i	$-0.444330\pi$
$463$	3467.01	0.348003	0.174002	+	0.984745i	$0.444330\pi$
$464$	0	0
$465$	1577.54	0.157326
$466$	0	0
$467$	7236.61i	0.717067i	0.933517	+	0.358533i	$0.116723\pi$
$467$	7236.61i	0.717067i	−0.933517	+	0.358533i	$0.883277\pi$
$468$	0	0
$469$	− 1722.92i	− 0.169631i
$470$	0	0
$471$	−2558.47	−0.250294
$472$	0	0
$473$	−15835.7	−1.53938
$474$	0	0
$475$	10501.7i	1.01443i
$476$	0	0
$477$	1061.70i	0.101912i
$478$	0	0
$479$	−6144.44	−0.586110	−0.293055	−	0.956096i	$-0.594672\pi$
$479$	−6144.44	−0.586110	−0.293055	+	0.956096i	$0.594672\pi$
$480$	0	0
$481$	−586.268	−0.0555749
$482$	0	0
$483$	3815.40i	0.359434i
$484$	0	0
$485$	16915.6i	1.58371i
$486$	0	0
$487$	−12917.5	−1.20195	−0.600974	−	0.799268i	$-0.705221\pi$
$487$	−12917.5	−1.20195	−0.600974	+	0.799268i	$0.705221\pi$
$488$	0	0
$489$	8471.70	0.783443
$490$	0	0
$491$	− 7964.15i	− 0.732011i	−0.930613	−	0.366005i	$-0.880725\pi$
$491$	− 7964.15i	− 0.732011i	0.930613	−	0.366005i	$-0.119275\pi$
$492$	0	0
$493$	115.732i	0.0105726i
$494$	0	0
$495$	5879.22	0.533841
$496$	0	0
$497$	394.657	0.0356193
$498$	0	0
$499$	17549.3i	1.57437i	0.616714	+	0.787187i	$0.288463\pi$
$499$	17549.3i	1.57437i	−0.616714	+	0.787187i	$0.711537\pi$
$500$	0	0
$501$	7256.85i	0.647130i
$502$	0	0
$503$	−12194.7	−1.08098	−0.540492	−	0.841349i	$-0.681762\pi$
$503$	−12194.7	−1.08098	−0.540492	+	0.841349i	$0.681762\pi$
$504$	0	0
$505$	−29099.1	−2.56415
$506$	0	0
$507$	6534.94i	0.572439i
$508$	0	0
$509$	10089.8i	0.878633i	0.898332	+	0.439316i	$0.144779\pi$
$509$	10089.8i	0.878633i	−0.898332	+	0.439316i	$0.855221\pi$
$510$	0	0
$511$	−1876.49	−0.162448
$512$	0	0
$513$	−1572.89	−0.135370
$514$	0	0
$515$	− 6443.55i	− 0.551333i
$516$	0	0
$517$	− 1083.52i	− 0.0921727i
$518$	0	0
$519$	−4736.44	−0.400591
$520$	0	0
$521$	−17003.4	−1.42982	−0.714908	−	0.699218i	$-0.753532\pi$
$521$	−17003.4	−1.42982	−0.714908	+	0.699218i	$0.753532\pi$
$522$	0	0
$523$	− 3626.50i	− 0.303204i	−0.988442	−	0.151602i	$-0.951557\pi$
$523$	− 3626.50i	− 0.303204i	0.988442	−	0.151602i	$-0.0484432\pi$
$524$	0	0
$525$	3785.70i	0.314708i
$526$	0	0
$527$	13.6452	0.00112789
$528$	0	0
$529$	20842.8	1.71306
$530$	0	0
$531$	4025.81i	0.329012i
$532$	0	0
$533$	132.108i	0.0107359i
$534$	0	0
$535$	8327.85	0.672980
$536$	0	0
$537$	5856.09	0.470594
$538$	0	0
$539$	1832.02i	0.146402i
$540$	0	0
$541$	− 13102.4i	− 1.04125i	−0.853786	−	0.520624i	$-0.825699\pi$
$541$	− 13102.4i	− 1.04125i	0.853786	−	0.520624i	$-0.174301\pi$
$542$	0	0
$543$	3294.00	0.260330
$544$	0	0
$545$	−22074.4	−1.73498
$546$	0	0
$547$	19947.4i	1.55921i	0.626269	+	0.779607i	$0.284581\pi$
$547$	19947.4i	1.55921i	−0.626269	+	0.779607i	$0.715419\pi$
$548$	0	0
$549$	− 201.828i	− 0.0156900i
$550$	0	0
$551$	−14870.4	−1.14973
$552$	0	0
$553$	6752.74	0.519269
$554$	0	0
$555$	− 7108.66i	− 0.543686i
$556$	0	0
$557$	− 17285.5i	− 1.31492i	−0.753489	−	0.657461i	$-0.771630\pi$
$557$	− 17285.5i	− 1.31492i	0.753489	−	0.657461i	$-0.228370\pi$
$558$	0	0
$559$	−1830.94	−0.138534
$560$	0	0
$561$	50.8535	0.00382716
$562$	0	0
$563$	26089.4i	1.95300i	0.215519	+	0.976500i	$0.430856\pi$
$563$	26089.4i	1.95300i	−0.215519	+	0.976500i	$0.569144\pi$
$564$	0	0
$565$	− 8833.74i	− 0.657767i
$566$	0	0
$567$	−567.000	−0.0419961
$568$	0	0
$569$	10919.5	0.804513	0.402257	−	0.915527i	$-0.368226\pi$
$569$	10919.5	0.804513	0.402257	+	0.915527i	$0.368226\pi$
$570$	0	0
$571$	− 1828.86i	− 0.134037i	−0.997752	−	0.0670186i	$-0.978651\pi$
$571$	− 1828.86i	− 0.134037i	0.997752	−	0.0670186i	$-0.0213487\pi$
$572$	0	0
$573$	− 8248.94i	− 0.601404i
$574$	0	0
$575$	32752.8	2.37545
$576$	0	0
$577$	16607.3	1.19822	0.599110	−	0.800667i	$-0.295521\pi$
$577$	16607.3	1.19822	0.599110	+	0.800667i	$0.295521\pi$
$578$	0	0
$579$	12974.7i	0.931279i
$580$	0	0
$581$	9024.75i	0.644423i
$582$	0	0
$583$	4410.57	0.313322
$584$	0	0
$585$	679.764	0.0480424
$586$	0	0
$587$	14955.1i	1.05156i	0.850621	+	0.525779i	$0.176226\pi$
$587$	14955.1i	1.05156i	−0.850621	+	0.525779i	$0.823774\pi$
$588$	0	0
$589$	1753.27i	0.122653i
$590$	0	0
$591$	13886.4	0.966516
$592$	0	0
$593$	−11985.7	−0.830003	−0.415002	−	0.909821i	$-0.636219\pi$
$593$	−11985.7	−0.830003	−0.415002	+	0.909821i	$0.636219\pi$
$594$	0	0
$595$	55.4506i	0.00382060i
$596$	0	0
$597$	− 15928.2i	− 1.09196i
$598$	0	0
$599$	4441.72	0.302978	0.151489	−	0.988459i	$-0.451593\pi$
$599$	4441.72	0.302978	0.151489	+	0.988459i	$0.451593\pi$
$600$	0	0
$601$	−16036.0	−1.08839	−0.544194	−	0.838959i	$-0.683164\pi$
$601$	−16036.0	−1.08839	−0.544194	+	0.838959i	$0.683164\pi$
$602$	0	0
$603$	− 2215.18i	− 0.149601i
$604$	0	0
$605$	− 1168.46i	− 0.0785198i
$606$	0	0
$607$	−27300.6	−1.82553	−0.912765	−	0.408484i	$-0.866057\pi$
$607$	−27300.6	−1.82553	−0.912765	+	0.408484i	$0.866057\pi$
$608$	0	0
$609$	−5360.53	−0.356683
$610$	0	0
$611$	− 125.278i	− 0.00829496i
$612$	0	0
$613$	9028.45i	0.594871i	0.954742	+	0.297435i	$0.0961312\pi$
$613$	9028.45i	0.594871i	−0.954742	+	0.297435i	$0.903869\pi$
$614$	0	0
$615$	−1601.85	−0.105029
$616$	0	0
$617$	−13495.7	−0.880580	−0.440290	−	0.897856i	$-0.645124\pi$
$617$	−13495.7	−0.880580	−0.440290	+	0.897856i	$0.645124\pi$
$618$	0	0
$619$	20208.8i	1.31222i	0.754667	+	0.656108i	$0.227798\pi$
$619$	20208.8i	1.31222i	−0.754667	+	0.656108i	$0.772202\pi$
$620$	0	0
$621$	4905.52i	0.316991i
$622$	0	0
$623$	1883.23	0.121107
$624$	0	0
$625$	−5661.11	−0.362311
$626$	0	0
$627$	6534.16i	0.416187i
$628$	0	0
$629$	− 61.4878i	− 0.00389774i
$630$	0	0
$631$	11819.1	0.745657	0.372828	−	0.927900i	$-0.378388\pi$
$631$	11819.1	0.745657	0.372828	+	0.927900i	$0.378388\pi$
$632$	0	0
$633$	−11347.6	−0.712521
$634$	0	0
$635$	12298.4i	0.768579i
$636$	0	0
$637$	211.821i	0.0131753i
$638$	0	0
$639$	507.416	0.0314133
$640$	0	0
$641$	−21519.7	−1.32602	−0.663009	−	0.748611i	$-0.730721\pi$
$641$	−21519.7	−1.32602	−0.663009	+	0.748611i	$0.730721\pi$
$642$	0	0
$643$	12695.3i	0.778621i	0.921107	+	0.389311i	$0.127287\pi$
$643$	12695.3i	0.778621i	−0.921107	+	0.389311i	$0.872713\pi$
$644$	0	0
$645$	− 22200.7i	− 1.35527i
$646$	0	0
$647$	19246.1	1.16946	0.584732	−	0.811227i	$-0.301200\pi$
$647$	19246.1	1.16946	0.584732	+	0.811227i	$0.301200\pi$
$648$	0	0
$649$	16724.2	1.01153
$650$	0	0
$651$	632.026i	0.0380508i
$652$	0	0
$653$	− 11147.0i	− 0.668020i	−0.942569	−	0.334010i	$-0.891598\pi$
$653$	− 11147.0i	− 0.668020i	0.942569	−	0.334010i	$-0.108402\pi$
$654$	0	0
$655$	20466.8	1.22092
$656$	0	0
$657$	−2412.63	−0.143266
$658$	0	0
$659$	− 5236.88i	− 0.309560i	−0.987949	−	0.154780i	$-0.950533\pi$
$659$	− 5236.88i	− 0.309560i	0.987949	−	0.154780i	$-0.0494669\pi$
$660$	0	0
$661$	− 5099.28i	− 0.300059i	−0.988682	−	0.150030i	$-0.952063\pi$
$661$	− 5099.28i	− 0.300059i	0.988682	−	0.150030i	$-0.0479369\pi$
$662$	0	0
$663$	5.87975	0.000344420 0
$664$	0	0
$665$	−7124.84	−0.415473
$666$	0	0
$667$	46377.8i	2.69228i
$668$	0	0
$669$	− 5600.37i	− 0.323651i
$670$	0	0
$671$	−838.441	−0.0482380
$672$	0	0
$673$	653.573	0.0374344	0.0187172	−	0.999825i	$-0.494042\pi$
$673$	653.573	0.0374344	0.0187172	+	0.999825i	$0.494042\pi$
$674$	0	0
$675$	4867.33i	0.277546i
$676$	0	0
$677$	− 1139.25i	− 0.0646751i	−0.999477	−	0.0323375i	$-0.989705\pi$
$677$	− 1139.25i	− 0.0646751i	0.999477	−	0.0323375i	$-0.0102951\pi$
$678$	0	0
$679$	−6777.08	−0.383034
$680$	0	0
$681$	13653.1	0.768263
$682$	0	0
$683$	− 29280.6i	− 1.64040i	−0.572079	−	0.820199i	$-0.693863\pi$
$683$	− 29280.6i	− 1.64040i	0.572079	−	0.820199i	$-0.306137\pi$
$684$	0	0
$685$	51597.9i	2.87804i
$686$	0	0
$687$	17354.0	0.963748
$688$	0	0
$689$	509.956	0.0281971
$690$	0	0
$691$	− 6960.69i	− 0.383208i	−0.981472	−	0.191604i	$-0.938631\pi$
$691$	− 6960.69i	− 0.383208i	0.981472	−	0.191604i	$-0.0613690\pi$
$692$	0	0
$693$	2355.46i	0.129114i
$694$	0	0
$695$	−46291.0	−2.52650
$696$	0	0
$697$	−13.8555	−0.000752963 0
$698$	0	0
$699$	− 6858.07i	− 0.371096i
$700$	0	0
$701$	− 8817.36i	− 0.475074i	−0.971378	−	0.237537i	$-0.923660\pi$
$701$	− 8817.36i	− 0.475074i	0.971378	−	0.237537i	$-0.0763401\pi$
$702$	0	0
$703$	7900.56	0.423862
$704$	0	0
$705$	1519.04	0.0811492
$706$	0	0
$707$	− 11658.3i	− 0.620163i
$708$	0	0
$709$	27251.2i	1.44350i	0.692155	+	0.721749i	$0.256661\pi$
$709$	27251.2i	1.44350i	−0.692155	+	0.721749i	$0.743339\pi$
$710$	0	0
$711$	8682.09	0.457952
$712$	0	0
$713$	5468.10	0.287212
$714$	0	0
$715$	− 2823.91i	− 0.147704i
$716$	0	0
$717$	− 4345.97i	− 0.226364i
$718$	0	0
$719$	19977.3	1.03620	0.518099	−	0.855321i	$-0.326640\pi$
$719$	19977.3	1.03620	0.518099	+	0.855321i	$0.326640\pi$
$720$	0	0
$721$	2581.55	0.133345
$722$	0	0
$723$	− 2336.79i	− 0.120202i
$724$	0	0
$725$	46016.7i	2.35727i
$726$	0	0
$727$	−20613.8	−1.05161	−0.525807	−	0.850604i	$-0.676237\pi$
$727$	−20613.8	−1.05161	−0.525807	+	0.850604i	$0.676237\pi$
$728$	0	0
$729$	−729.000	−0.0370370
$730$	0	0
$731$	− 192.029i	− 0.00971608i
$732$	0	0
$733$	20567.6i	1.03640i	0.855259	+	0.518200i	$0.173398\pi$
$733$	20567.6i	1.03640i	−0.855259	+	0.518200i	$0.826602\pi$
$734$	0	0
$735$	−2568.39	−0.128893
$736$	0	0
$737$	−9202.41	−0.459939
$738$	0	0
$739$	12830.6i	0.638673i	0.947641	+	0.319337i	$0.103460\pi$
$739$	12830.6i	0.638673i	−0.947641	+	0.319337i	$0.896540\pi$
$740$	0	0
$741$	755.489i	0.0374542i
$742$	0	0
$743$	−11598.4	−0.572686	−0.286343	−	0.958127i	$-0.592440\pi$
$743$	−11598.4	−0.572686	−0.286343	+	0.958127i	$0.592440\pi$
$744$	0	0
$745$	46604.2	2.29187
$746$	0	0
$747$	11603.2i	0.568327i
$748$	0	0
$749$	3336.48i	0.162767i
$750$	0	0
$751$	20829.8	1.01210	0.506052	−	0.862503i	$-0.331105\pi$
$751$	20829.8	1.01210	0.506052	+	0.862503i	$0.331105\pi$
$752$	0	0
$753$	4055.01	0.196245
$754$	0	0
$755$	− 37562.9i	− 1.81067i
$756$	0	0
$757$	9455.66i	0.453992i	0.973896	+	0.226996i	$0.0728904\pi$
$757$	9455.66i	0.453992i	−0.973896	+	0.226996i	$0.927110\pi$
$758$	0	0
$759$	20378.7	0.974572
$760$	0	0
$761$	−18076.5	−0.861070	−0.430535	−	0.902574i	$-0.641675\pi$
$761$	−18076.5	−0.861070	−0.430535	+	0.902574i	$0.641675\pi$
$762$	0	0
$763$	− 8843.89i	− 0.419620i
$764$	0	0
$765$	71.2937i	0.00336945i
$766$	0	0
$767$	1933.68	0.0910313
$768$	0	0
$769$	−19101.1	−0.895713	−0.447857	−	0.894105i	$-0.647813\pi$
$769$	−19101.1	−0.895713	−0.447857	+	0.894105i	$0.647813\pi$
$770$	0	0
$771$	− 16686.0i	− 0.779419i
$772$	0	0
$773$	− 23720.6i	− 1.10371i	−0.833939	−	0.551857i	$-0.813919\pi$
$773$	− 23720.6i	− 1.10371i	0.833939	−	0.551857i	$-0.186081\pi$
$774$	0	0
$775$	5425.54	0.251472
$776$	0	0
$777$	2848.02	0.131496
$778$	0	0
$779$	− 1780.29i	− 0.0818815i
$780$	0	0
$781$	− 2107.93i	− 0.0965783i
$782$	0	0
$783$	−6892.11	−0.314564
$784$	0	0
$785$	−14900.6	−0.677483
$786$	0	0
$787$	− 25919.3i	− 1.17398i	−0.809593	−	0.586991i	$-0.800312\pi$
$787$	− 25919.3i	− 1.17398i	0.809593	−	0.586991i	$-0.199688\pi$
$788$	0	0
$789$	− 2283.69i	− 0.103044i
$790$	0	0
$791$	3539.15	0.159087
$792$	0	0
$793$	−96.9418	−0.00434112
$794$	0	0
$795$	6183.36i	0.275851i
$796$	0	0
$797$	18011.6i	0.800505i	0.916405	+	0.400253i	$0.131078\pi$
$797$	18011.6i	0.800505i	−0.916405	+	0.400253i	$0.868922\pi$
$798$	0	0
$799$	13.1392	0.000581767 0
$800$	0	0
$801$	2421.29	0.106807
$802$	0	0
$803$	10022.6i	0.440462i
$804$	0	0
$805$	22220.9i	0.972901i
$806$	0	0
$807$	3208.65	0.139963
$808$	0	0
$809$	−8800.52	−0.382459	−0.191230	−	0.981545i	$-0.561248\pi$
$809$	−8800.52	−0.382459	−0.191230	+	0.981545i	$0.561248\pi$
$810$	0	0
$811$	36022.4i	1.55970i	0.625967	+	0.779850i	$0.284704\pi$
$811$	36022.4i	1.55970i	−0.625967	+	0.779850i	$0.715296\pi$
$812$	0	0
$813$	398.862i	0.0172063i
$814$	0	0
$815$	49339.2	2.12059
$816$	0	0
$817$	24673.8	1.05658
$818$	0	0
$819$	272.341i	0.0116195i
$820$	0	0
$821$	17624.7i	0.749216i	0.927183	+	0.374608i	$0.122223\pi$
$821$	17624.7i	0.749216i	−0.927183	+	0.374608i	$0.877777\pi$
$822$	0	0
$823$	3193.42	0.135256	0.0676280	−	0.997711i	$-0.478457\pi$
$823$	3193.42	0.135256	0.0676280	+	0.997711i	$0.478457\pi$
$824$	0	0
$825$	20220.1	0.853300
$826$	0	0
$827$	− 35413.5i	− 1.48906i	−0.667591	−	0.744528i	$-0.732675\pi$
$827$	− 35413.5i	− 1.48906i	0.667591	−	0.744528i	$-0.267325\pi$
$828$	0	0
$829$	− 41344.0i	− 1.73213i	−0.499932	−	0.866065i	$-0.666642\pi$
$829$	− 41344.0i	− 1.73213i	0.499932	−	0.866065i	$-0.333358\pi$
$830$	0	0
$831$	13749.9	0.573983
$832$	0	0
$833$	−22.2158	−0.000924047 0
$834$	0	0
$835$	42263.9i	1.75162i
$836$	0	0
$837$	812.605i	0.0335576i
$838$	0	0
$839$	−39777.5	−1.63679	−0.818397	−	0.574653i	$-0.805137\pi$
$839$	−39777.5	−1.63679	−0.818397	+	0.574653i	$0.805137\pi$
$840$	0	0
$841$	−40770.4	−1.67167
$842$	0	0
$843$	23883.9i	0.975805i
$844$	0	0
$845$	38059.5i	1.54945i
$846$	0	0
$847$	468.131	0.0189907
$848$	0	0
$849$	24948.5	1.00851
$850$	0	0
$851$	− 24640.2i	− 0.992545i
$852$	0	0
$853$	− 27194.4i	− 1.09158i	−0.837922	−	0.545790i	$-0.816230\pi$
$853$	− 27194.4i	− 1.09158i	0.837922	−	0.545790i	$-0.183770\pi$
$854$	0	0
$855$	−9160.51	−0.366413
$856$	0	0
$857$	−18965.3	−0.755943	−0.377972	−	0.925817i	$-0.623378\pi$
$857$	−18965.3	−0.755943	−0.377972	+	0.925817i	$0.623378\pi$
$858$	0	0
$859$	32183.8i	1.27834i	0.769064	+	0.639172i	$0.220723\pi$
$859$	32183.8i	1.27834i	−0.769064	+	0.639172i	$0.779277\pi$
$860$	0	0
$861$	− 641.766i	− 0.0254022i
$862$	0	0
$863$	3324.30	0.131125	0.0655624	−	0.997848i	$-0.479116\pi$
$863$	3324.30	0.131125	0.0655624	+	0.997848i	$0.479116\pi$
$864$	0	0
$865$	−27585.1	−1.08430
$866$	0	0
$867$	− 14738.4i	− 0.577326i
$868$	0	0
$869$	− 36067.5i	− 1.40795i
$870$	0	0
$871$	−1064.00	−0.0413916
$872$	0	0
$873$	−8713.38	−0.337805
$874$	0	0
$875$	6759.94i	0.261174i
$876$	0	0
$877$	22741.2i	0.875618i	0.899068	+	0.437809i	$0.144245\pi$
$877$	22741.2i	0.875618i	−0.899068	+	0.437809i	$0.855755\pi$
$878$	0	0
$879$	526.751	0.0202126
$880$	0	0
$881$	34545.2	1.32107	0.660533	−	0.750797i	$-0.270330\pi$
$881$	34545.2	1.32107	0.660533	+	0.750797i	$0.270330\pi$
$882$	0	0
$883$	− 31171.3i	− 1.18799i	−0.804467	−	0.593997i	$-0.797549\pi$
$883$	− 31171.3i	− 1.18799i	0.804467	−	0.593997i	$-0.202451\pi$
$884$	0	0
$885$	23446.4i	0.890555i
$886$	0	0
$887$	−11015.2	−0.416974	−0.208487	−	0.978025i	$-0.566854\pi$
$887$	−11015.2	−0.416974	−0.208487	+	0.978025i	$0.566854\pi$
$888$	0	0
$889$	−4927.24	−0.185888
$890$	0	0
$891$	3028.44i	0.113868i
$892$	0	0
$893$	1688.25i	0.0632646i
$894$	0	0
$895$	34105.9	1.27378
$896$	0	0
$897$	2356.22	0.0877054
$898$	0	0
$899$	7682.53i	0.285013i
$900$	0	0
$901$	53.4842i	0.00197760i
$902$	0	0
$903$	8894.49	0.327785
$904$	0	0
$905$	19184.3	0.704649
$906$	0	0
$907$	− 28824.5i	− 1.05524i	−0.849481	−	0.527620i	$-0.823085\pi$
$907$	− 28824.5i	− 1.05524i	0.849481	−	0.527620i	$-0.176915\pi$
$908$	0	0
$909$	− 14989.2i	− 0.546932i
$910$	0	0
$911$	−4484.99	−0.163111	−0.0815556	−	0.996669i	$-0.525989\pi$
$911$	−4484.99	−0.163111	−0.0815556	+	0.996669i	$0.525989\pi$
$912$	0	0
$913$	48202.7	1.74729
$914$	0	0
$915$	− 1175.45i	− 0.0424689i
$916$	0	0
$917$	8199.83i	0.295291i
$918$	0	0
$919$	−35988.2	−1.29177	−0.645887	−	0.763433i	$-0.723512\pi$
$919$	−35988.2	−1.29177	−0.645887	+	0.763433i	$0.723512\pi$
$920$	0	0
$921$	−16106.6	−0.576253
$922$	0	0
$923$	− 243.722i	− 0.00869145i
$924$	0	0
$925$	− 24448.4i	− 0.869037i
$926$	0	0
$927$	3319.13	0.117599
$928$	0	0
$929$	4954.03	0.174958	0.0874792	−	0.996166i	$-0.472119\pi$
$929$	4954.03	0.174958	0.0874792	+	0.996166i	$0.472119\pi$
$930$	0	0
$931$	− 2854.50i	− 0.100486i
$932$	0	0
$933$	− 12634.8i	− 0.443351i
$934$	0	0
$935$	296.171	0.0103592
$936$	0	0
$937$	−17123.9	−0.597027	−0.298513	−	0.954405i	$-0.596491\pi$
$937$	−17123.9	−0.597027	−0.298513	+	0.954405i	$0.596491\pi$
$938$	0	0
$939$	− 19421.4i	− 0.674965i
$940$	0	0
$941$	− 54586.0i	− 1.89102i	−0.325590	−	0.945511i	$-0.605563\pi$
$941$	− 54586.0i	− 1.89102i	0.325590	−	0.945511i	$-0.394437\pi$
$942$	0	0
$943$	−5552.37	−0.191739
$944$	0	0
$945$	−3302.21	−0.113673
$946$	0	0
$947$	24841.2i	0.852409i	0.904627	+	0.426204i	$0.140150\pi$
$947$	24841.2i	0.852409i	−0.904627	+	0.426204i	$0.859850\pi$
$948$	0	0
$949$	1158.83i	0.0396388i
$950$	0	0
$951$	−24348.3	−0.830228
$952$	0	0
$953$	−20388.7	−0.693028	−0.346514	−	0.938045i	$-0.612635\pi$
$953$	−20388.7	−0.693028	−0.346514	+	0.938045i	$0.612635\pi$
$954$	0	0
$955$	− 48041.9i	− 1.62785i
$956$	0	0
$957$	28631.5i	0.967111i
$958$	0	0
$959$	−20672.2	−0.696080
$960$	0	0
$961$	−28885.2	−0.969595
$962$	0	0
$963$	4289.75i	0.143547i
$964$	0	0
$965$	75564.8i	2.52074i
$966$	0	0
$967$	27543.1	0.915952	0.457976	−	0.888964i	$-0.348575\pi$
$967$	27543.1	0.915952	0.457976	+	0.888964i	$0.348575\pi$
$968$	0	0
$969$	−79.2357	−0.00262685
$970$	0	0
$971$	5698.52i	0.188336i	0.995556	+	0.0941680i	$0.0300191\pi$
$971$	5698.52i	0.188336i	−0.995556	+	0.0941680i	$0.969981\pi$
$972$	0	0
$973$	− 18546.0i	− 0.611057i
$974$	0	0
$975$	2337.87	0.0767917
$976$	0	0
$977$	−25368.3	−0.830711	−0.415355	−	0.909659i	$-0.636343\pi$
$977$	−25368.3	−0.830711	−0.415355	+	0.909659i	$0.636343\pi$
$978$	0	0
$979$	− 10058.6i	− 0.328371i
$980$	0	0
$981$	− 11370.7i	− 0.370070i
$982$	0	0
$983$	−18984.9	−0.615995	−0.307998	−	0.951387i	$-0.599659\pi$
$983$	−18984.9	−0.615995	−0.307998	+	0.951387i	$0.599659\pi$
$984$	0	0
$985$	80874.6	2.61612
$986$	0	0
$987$	608.587i	0.0196267i
$988$	0	0
$989$	− 76952.6i	− 2.47416i
$990$	0	0
$991$	−52094.6	−1.66987	−0.834935	−	0.550349i	$-0.814495\pi$
$991$	−52094.6	−1.66987	−0.834935	+	0.550349i	$0.814495\pi$
$992$	0	0
$993$	−6327.06	−0.202199
$994$	0	0
$995$	− 92765.9i	− 2.95566i
$996$	0	0
$997$	− 37144.6i	− 1.17992i	−0.807432	−	0.589961i	$-0.799143\pi$
$997$	− 37144.6i	− 1.17992i	0.807432	−	0.589961i	$-0.200857\pi$
$998$	0	0
$999$	3661.74	0.115968

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.4.c.f.673.12	yes	12
4.3	odd	2		1344.4.c.g.673.6	yes	12
8.3	odd	2		1344.4.c.g.673.7	yes	12
8.5	even	2	inner	1344.4.c.f.673.1	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1344.4.c.f.673.1	✓	12	8.5	even	2	inner
1344.4.c.f.673.12	yes	12	1.1	even	1	trivial
1344.4.c.g.673.6	yes	12	4.3	odd	2
1344.4.c.g.673.7	yes	12	8.3	odd	2

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:	\( N \)	\(=\)	\( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1344.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.00000i q^{3} +17.4720i q^{5} -7.00000 q^{7} -9.00000 q^{9} +O(q^{10})\)	\(q+3.00000i q^{3} +17.4720i q^{5} -7.00000 q^{7} -9.00000 q^{9} +37.3882i q^{11} +4.32287i q^{13} -52.4161 q^{15} -0.453383 q^{17} -58.2551i q^{19} -21.0000i q^{21} -181.686 q^{23} -180.272 q^{25} -27.0000i q^{27} -255.263i q^{29} -30.0965 q^{31} -112.165 q^{33} -122.304i q^{35} +135.620i q^{37} -12.9686 q^{39} +30.5603 q^{41} +423.547i q^{43} -157.248i q^{45} -28.9804 q^{47} +49.0000 q^{49} -1.36015i q^{51} -117.967i q^{53} -653.247 q^{55} +174.765 q^{57} -447.313i q^{59} +22.4253i q^{61} +63.0000 q^{63} -75.5294 q^{65} +246.131i q^{67} -545.058i q^{69} -56.3796 q^{71} +268.070 q^{73} -540.815i q^{75} -261.717i q^{77} -964.677 q^{79} +81.0000 q^{81} -1289.25i q^{83} -7.92152i q^{85} +765.790 q^{87} -269.033 q^{89} -30.2601i q^{91} -90.2894i q^{93} +1017.83 q^{95} +968.154 q^{97} -336.494i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 84 q^{7} - 108 q^{9}+O(q^{10}) \) 12 * q - 84 * q^7 - 108 * q^9	\( 12 q - 84 q^{7} - 108 q^{9} + 24 q^{15} + 376 q^{17} - 336 q^{23} - 180 q^{25} + 192 q^{31} - 168 q^{33} - 504 q^{39} + 488 q^{41} - 448 q^{47} + 588 q^{49} - 3600 q^{55} - 432 q^{57} + 756 q^{63} + 1408 q^{65} - 5104 q^{71} - 1752 q^{73} - 1632 q^{79} + 972 q^{81} - 336 q^{87} - 3688 q^{89} - 2496 q^{95} - 1944 q^{97}+O(q^{100}) \) 12 * q - 84 * q^7 - 108 * q^9 + 24 * q^15 + 376 * q^17 - 336 * q^23 - 180 * q^25 + 192 * q^31 - 168 * q^33 - 504 * q^39 + 488 * q^41 - 448 * q^47 + 588 * q^49 - 3600 * q^55 - 432 * q^57 + 756 * q^63 + 1408 * q^65 - 5104 * q^71 - 1752 * q^73 - 1632 * q^79 + 972 * q^81 - 336 * q^87 - 3688 * q^89 - 2496 * q^95 - 1944 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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