LMFDB - Embedded newform 2016.3.e.c.1007.13

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2016,3,Mod(1007,2016)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2016, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1, 1])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2016.1007"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 2016 = 2^{5} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2016.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [48,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-336] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)

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

Self dual:	no
Analytic conductor:	$54.9320212950$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	no (minimal twist has level 504)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1007.13
Character	$\chi$	$=$	2016.1007
Dual form			2016.3.e.c.1007.14

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.489356i q^{5} +(5.24868 + 4.63156i) q^{7} +6.94749i q^{11} +16.5922 q^{13} +28.0732 q^{17} -29.3568i q^{19} +7.49845 q^{23} +24.7605 q^{25} -17.9470 q^{29} -30.8259 q^{31} +(2.26648 - 2.56848i) q^{35} -64.8471i q^{37} -50.7873 q^{41} -47.5472 q^{43} +65.0090i q^{47} +(6.09730 + 48.6192i) q^{49} +51.2494 q^{53} +3.39980 q^{55} +27.2645 q^{59} +115.375 q^{61} -8.11952i q^{65} -7.26829 q^{67} +91.6063 q^{71} +27.5052i q^{73} +(-32.1777 + 36.4652i) q^{77} +36.2471i q^{79} -117.776 q^{83} -13.7378i q^{85} +35.2381 q^{89} +(87.0874 + 76.8480i) q^{91} -14.3659 q^{95} +24.7402i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 336 q^{25} + 384 q^{43} + 368 q^{49} + 896 q^{67} + 576 q^{91}+O(q^{100}) $ 48 * q - 336 * q^25 + 384 * q^43 + 368 * q^49 + 896 * q^67 + 576 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2016\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$1765$	$1793$
$\chi(n)$	$-1$	$-1$	$-1$	$-1$

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
$5$		−	0.489356i		−	0.0978713i	−0.998802	−	0.0489356i	$-0.984417\pi$
$5$		−	0.489356i		−	0.0978713i	0.998802	−	0.0489356i	$-0.0155829\pi$
$6$		0			0
$7$	5.24868	+	4.63156i	0.749812	+	0.661651i
$7$	5.24868	+	4.63156i	0.749812	+	0.661651i
$8$		0			0
$9$		0			0
$10$		0			0
$11$			6.94749i			0.631590i	0.948827	+	0.315795i	$0.102271\pi$
$11$			6.94749i			0.631590i	−0.948827	+	0.315795i	$0.897729\pi$
$12$		0			0
$13$	16.5922			1.27633			0.638163	−	0.769901i	$-0.279695\pi$
$13$	16.5922			1.27633			0.638163	+	0.769901i	$0.279695\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	28.0732			1.65137			0.825683	−	0.564135i	$-0.190790\pi$
$17$	28.0732			1.65137			0.825683	+	0.564135i	$0.190790\pi$
$18$		0			0
$19$		−	29.3568i		−	1.54509i	−0.634958	−	0.772547i	$-0.718983\pi$
$19$		−	29.3568i		−	1.54509i	0.634958	−	0.772547i	$-0.281017\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	7.49845			0.326020			0.163010	−	0.986624i	$-0.447880\pi$
$23$	7.49845			0.326020			0.163010	+	0.986624i	$0.447880\pi$
$24$		0			0
$25$	24.7605			0.990421
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−17.9470			−0.618862			−0.309431	−	0.950922i	$-0.600139\pi$
$29$	−17.9470			−0.618862			−0.309431	+	0.950922i	$0.600139\pi$
$30$		0			0
$31$	−30.8259			−0.994385			−0.497193	−	0.867640i	$-0.665636\pi$
$31$	−30.8259			−0.994385			−0.497193	+	0.867640i	$0.665636\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.26648	−	2.56848i	0.0647567	−	0.0733850i
$36$		0			0
$37$		−	64.8471i		−	1.75262i	−0.481744	−	0.876312i	$-0.659996\pi$
$37$		−	64.8471i		−	1.75262i	0.481744	−	0.876312i	$-0.340004\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−50.7873			−1.23871			−0.619357	−	0.785109i	$-0.712607\pi$
$41$	−50.7873			−1.23871			−0.619357	+	0.785109i	$0.712607\pi$
$42$		0			0
$43$	−47.5472			−1.10575			−0.552875	−	0.833264i	$-0.686469\pi$
$43$	−47.5472			−1.10575			−0.552875	+	0.833264i	$0.686469\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			65.0090i			1.38317i	0.722295	+	0.691585i	$0.243087\pi$
$47$			65.0090i			1.38317i	−0.722295	+	0.691585i	$0.756913\pi$
$48$		0			0
$49$	6.09730	+	48.6192i	0.124435	+	0.992228i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	51.2494			0.966969			0.483485	−	0.875353i	$-0.339371\pi$
$53$	51.2494			0.966969			0.483485	+	0.875353i	$0.339371\pi$
$54$		0			0
$55$	3.39980			0.0618145
$56$		0			0
$57$		0			0
$58$		0			0
$59$	27.2645			0.462110			0.231055	−	0.972941i	$-0.425782\pi$
$59$	27.2645			0.462110			0.231055	+	0.972941i	$0.425782\pi$
$60$		0			0
$61$	115.375			1.89140			0.945699	−	0.325043i	$-0.105379\pi$
$61$	115.375			1.89140			0.945699	+	0.325043i	$0.105379\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	8.11952i		−	0.124916i
$66$		0			0
$67$	−7.26829			−0.108482			−0.0542409	−	0.998528i	$-0.517274\pi$
$67$	−7.26829			−0.108482			−0.0542409	+	0.998528i	$0.517274\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	91.6063			1.29023			0.645115	−	0.764085i	$-0.276809\pi$
$71$	91.6063			1.29023			0.645115	+	0.764085i	$0.276809\pi$
$72$		0			0
$73$			27.5052i			0.376783i	0.982094	+	0.188392i	$0.0603274\pi$
$73$			27.5052i			0.376783i	−0.982094	+	0.188392i	$0.939673\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−32.1777	+	36.4652i	−0.417892	+	0.473574i
$78$		0			0
$79$			36.2471i			0.458824i	0.973329	+	0.229412i	$0.0736803\pi$
$79$			36.2471i			0.458824i	−0.973329	+	0.229412i	$0.926320\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−117.776			−1.41899			−0.709495	−	0.704711i	$-0.751077\pi$
$83$	−117.776			−1.41899			−0.709495	+	0.704711i	$0.751077\pi$
$84$		0			0
$85$		−	13.7378i		−	0.161621i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	35.2381			0.395933			0.197967	−	0.980209i	$-0.436566\pi$
$89$	35.2381			0.395933			0.197967	+	0.980209i	$0.436566\pi$
$90$		0			0
$91$	87.0874	+	76.8480i	0.957004	+	0.844483i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−14.3659			−0.151220
$96$		0			0
$97$			24.7402i			0.255053i	0.991835	+	0.127527i	$0.0407038\pi$
$97$			24.7402i			0.255053i	−0.991835	+	0.127527i	$0.959296\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	54.1213i		−	0.535854i	−0.963439	−	0.267927i	$-0.913661\pi$
$101$		−	54.1213i		−	0.535854i	0.963439	−	0.267927i	$-0.0863387\pi$
$102$		0			0
$103$	118.143			1.14701			0.573507	−	0.819200i	$-0.305582\pi$
$103$	118.143			1.14701			0.573507	+	0.819200i	$0.305582\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			85.2840i			0.797046i	0.917158	+	0.398523i	$0.130477\pi$
$107$			85.2840i			0.797046i	−0.917158	+	0.398523i	$0.869523\pi$
$108$		0			0
$109$		−	88.3506i		−	0.810556i	−0.914193	−	0.405278i	$-0.867175\pi$
$109$		−	88.3506i		−	0.810556i	0.914193	−	0.405278i	$-0.132825\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			113.657i			1.00581i	0.864341	+	0.502907i	$0.167736\pi$
$113$			113.657i			1.00581i	−0.864341	+	0.502907i	$0.832264\pi$
$114$		0			0
$115$		−	3.66942i		−	0.0319080i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	147.347	+	130.023i	1.23821	+	1.09263i
$120$		0			0
$121$	72.7324			0.601094
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	24.3506i		−	0.194805i
$126$		0			0
$127$		−	230.345i		−	1.81374i	−0.421409	−	0.906871i	$-0.638464\pi$
$127$		−	230.345i		−	1.81374i	0.421409	−	0.906871i	$-0.361536\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−65.7427			−0.501853			−0.250926	−	0.968006i	$-0.580735\pi$
$131$	−65.7427			−0.501853			−0.250926	+	0.968006i	$0.580735\pi$
$132$		0			0
$133$	135.968	−	154.084i	1.02231	−	1.15853i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	251.938i		−	1.83896i	−0.393131	−	0.919482i	$-0.628608\pi$
$137$		−	251.938i		−	1.83896i	0.393131	−	0.919482i	$-0.371392\pi$
$138$		0			0
$139$		−	55.1901i		−	0.397051i	−0.980096	−	0.198526i	$-0.936385\pi$
$139$		−	55.1901i		−	0.397051i	0.980096	−	0.198526i	$-0.0636153\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			115.274i			0.806115i
$144$		0			0
$145$			8.78248i			0.0605688i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−18.0084			−0.120862			−0.0604308	−	0.998172i	$-0.519247\pi$
$149$	−18.0084			−0.120862			−0.0604308	+	0.998172i	$0.519247\pi$
$150$		0			0
$151$		−	123.390i		−	0.817155i	−0.912724	−	0.408578i	$-0.866025\pi$
$151$		−	123.390i		−	0.817155i	0.912724	−	0.408578i	$-0.133975\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			15.0849i			0.0973217i
$156$		0			0
$157$	39.5283			0.251773			0.125886	−	0.992045i	$-0.459823\pi$
$157$	39.5283			0.251773			0.125886	+	0.992045i	$0.459823\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	39.3570	+	34.7295i	0.244453	+	0.215711i
$162$		0			0
$163$	5.42728			0.0332962			0.0166481	−	0.999861i	$-0.494700\pi$
$163$	5.42728			0.0332962			0.0166481	+	0.999861i	$0.494700\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			156.148i			0.935019i	0.883988	+	0.467509i	$0.154849\pi$
$167$			156.148i			0.935019i	−0.883988	+	0.467509i	$0.845151\pi$
$168$		0			0
$169$	106.302			0.629009
$170$		0			0
$171$		0			0
$172$		0			0
$173$			148.083i			0.855970i	0.903786	+	0.427985i	$0.140776\pi$
$173$			148.083i			0.855970i	−0.903786	+	0.427985i	$0.859224\pi$
$174$		0			0
$175$	129.960	+	114.680i	0.742629	+	0.655314i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			146.484i			0.818348i	0.912456	+	0.409174i	$0.134183\pi$
$179$			146.484i			0.818348i	−0.912456	+	0.409174i	$0.865817\pi$
$180$		0			0
$181$	150.952			0.833987			0.416993	−	0.908910i	$-0.363084\pi$
$181$	150.952			0.833987			0.416993	+	0.908910i	$0.363084\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−31.7333			−0.171532
$186$		0			0
$187$			195.038i			1.04299i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	243.251			1.27357			0.636783	−	0.771043i	$-0.280265\pi$
$191$	243.251			1.27357			0.636783	+	0.771043i	$0.280265\pi$
$192$		0			0
$193$	−138.025			−0.715158			−0.357579	−	0.933883i	$-0.616398\pi$
$193$	−138.025			−0.715158			−0.357579	+	0.933883i	$0.616398\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−80.7464			−0.409880			−0.204940	−	0.978775i	$-0.565700\pi$
$197$	−80.7464			−0.409880			−0.204940	+	0.978775i	$0.565700\pi$
$198$		0			0
$199$	230.552			1.15855			0.579276	−	0.815131i	$-0.303335\pi$
$199$	230.552			1.15855			0.579276	+	0.815131i	$0.303335\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−94.1981	−	83.1226i	−0.464030	−	0.409471i
$204$		0			0
$205$			24.8531i			0.121235i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	203.956			0.975866
$210$		0			0
$211$	10.9675			0.0519789			0.0259894	−	0.999662i	$-0.491726\pi$
$211$	10.9675			0.0519789			0.0259894	+	0.999662i	$0.491726\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			23.2675i			0.108221i
$216$		0			0
$217$	−161.796	−	142.772i	−0.745602	−	0.657936i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	465.797			2.10768
$222$		0			0
$223$	150.911			0.676730			0.338365	−	0.941015i	$-0.390126\pi$
$223$	150.911			0.676730			0.338365	+	0.941015i	$0.390126\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−179.331			−0.790002			−0.395001	−	0.918681i	$-0.629256\pi$
$227$	−179.331			−0.790002			−0.395001	+	0.918681i	$0.629256\pi$
$228$		0			0
$229$	−12.7967			−0.0558808			−0.0279404	−	0.999610i	$-0.508895\pi$
$229$	−12.7967			−0.0558808			−0.0279404	+	0.999610i	$0.508895\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			88.7566i			0.380930i	0.981694	+	0.190465i	$0.0609995\pi$
$233$			88.7566i			0.380930i	−0.981694	+	0.190465i	$0.939000\pi$
$234$		0			0
$235$	31.8126			0.135373
$236$		0			0
$237$		0			0
$238$		0			0
$239$	144.716			0.605506			0.302753	−	0.953069i	$-0.402094\pi$
$239$	144.716			0.605506			0.302753	+	0.953069i	$0.402094\pi$
$240$		0			0
$241$		−	40.5889i		−	0.168419i	−0.996448	−	0.0842093i	$-0.973164\pi$
$241$		−	40.5889i		−	0.168419i	0.996448	−	0.0842093i	$-0.0268364\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	23.7921	−	2.98375i	0.0971106	−	0.0121786i
$246$		0			0
$247$		−	487.095i		−	1.97204i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	446.676			1.77959			0.889793	−	0.456365i	$-0.150849\pi$
$251$	446.676			1.77959			0.889793	+	0.456365i	$0.150849\pi$
$252$		0			0
$253$			52.0954i			0.205911i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−311.679			−1.21276			−0.606379	−	0.795176i	$-0.707378\pi$
$257$	−311.679			−1.21276			−0.606379	+	0.795176i	$0.707378\pi$
$258$		0			0
$259$	300.343	−	340.362i	1.15963	−	1.31414i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−254.263			−0.966781			−0.483390	−	0.875405i	$-0.660595\pi$
$263$	−254.263			−0.966781			−0.483390	+	0.875405i	$0.660595\pi$
$264$		0			0
$265$		−	25.0792i		−	0.0946385i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			381.909i			1.41974i	0.704335	+	0.709868i	$0.251245\pi$
$269$			381.909i			1.41974i	−0.704335	+	0.709868i	$0.748755\pi$
$270$		0			0
$271$	108.365			0.399870			0.199935	−	0.979809i	$-0.435927\pi$
$271$	108.365			0.399870			0.199935	+	0.979809i	$0.435927\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			172.024i			0.625540i
$276$		0			0
$277$		−	179.094i		−	0.646547i	−0.946306	−	0.323274i	$-0.895217\pi$
$277$		−	179.094i		−	0.646547i	0.946306	−	0.323274i	$-0.104783\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			347.268i			1.23583i	0.786245	+	0.617915i	$0.212023\pi$
$281$			347.268i			1.23583i	−0.786245	+	0.617915i	$0.787977\pi$
$282$		0			0
$283$			155.999i			0.551233i	0.961268	+	0.275616i	$0.0888819\pi$
$283$			155.999i			0.551233i	−0.961268	+	0.275616i	$0.911118\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−266.566	−	235.224i	−0.928802	−	0.819597i
$288$		0			0
$289$	499.105			1.72701
$290$		0			0
$291$		0			0
$292$		0			0
$293$			479.808i			1.63757i	0.574101	+	0.818785i	$0.305352\pi$
$293$			479.808i			1.63757i	−0.574101	+	0.818785i	$0.694648\pi$
$294$		0			0
$295$		−	13.3420i		−	0.0452272i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	124.416			0.416108
$300$		0			0
$301$	−249.560	−	220.218i	−0.829104	−	0.731621i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	56.4596i		−	0.185114i
$306$		0			0
$307$			409.804i			1.33487i	0.744670	+	0.667433i	$0.232607\pi$
$307$			409.804i			1.33487i	−0.744670	+	0.667433i	$0.767393\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	229.581i		−	0.738202i	−0.929389	−	0.369101i	$-0.879666\pi$
$311$		−	229.581i		−	0.738202i	0.929389	−	0.369101i	$-0.120334\pi$
$312$		0			0
$313$			383.358i			1.22479i	0.790553	+	0.612393i	$0.209793\pi$
$313$			383.358i			1.22479i	−0.790553	+	0.612393i	$0.790207\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	286.747			0.904566			0.452283	−	0.891874i	$-0.350610\pi$
$317$	286.747			0.904566			0.452283	+	0.891874i	$0.350610\pi$
$318$		0			0
$319$		−	124.687i		−	0.390867i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	824.139i		−	2.55151i
$324$		0			0
$325$	410.833			1.26410
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−301.093	+	341.212i	−0.915177	+	1.03712i
$330$		0			0
$331$	−145.600			−0.439880			−0.219940	−	0.975513i	$-0.570586\pi$
$331$	−145.600			−0.439880			−0.219940	+	0.975513i	$0.570586\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			3.55678i			0.0106173i
$336$		0			0
$337$	−115.312			−0.342172			−0.171086	−	0.985256i	$-0.554728\pi$
$337$	−115.312			−0.342172			−0.171086	+	0.985256i	$0.554728\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	214.163i		−	0.628044i
$342$		0			0
$343$	−193.180	+	283.426i	−0.563206	+	0.826316i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	166.848i		−	0.480831i	−0.970670	−	0.240416i	$-0.922716\pi$
$347$		−	166.848i		−	0.480831i	0.970670	−	0.240416i	$-0.0772838\pi$
$348$		0			0
$349$	−65.9305			−0.188913			−0.0944563	−	0.995529i	$-0.530111\pi$
$349$	−65.9305			−0.188913			−0.0944563	+	0.995529i	$0.530111\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−418.652			−1.18598			−0.592992	−	0.805209i	$-0.702053\pi$
$353$	−418.652			−1.18598			−0.592992	+	0.805209i	$0.702053\pi$
$354$		0			0
$355$		−	44.8281i		−	0.126276i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−612.879			−1.70718			−0.853592	−	0.520942i	$-0.825581\pi$
$359$	−612.879			−1.70718			−0.853592	+	0.520942i	$0.825581\pi$
$360$		0			0
$361$	−500.820			−1.38731
$362$		0			0
$363$		0			0
$364$		0			0
$365$	13.4598			0.0368763
$366$		0			0
$367$	114.767			0.312716			0.156358	−	0.987700i	$-0.450025\pi$
$367$	114.767			0.312716			0.156358	+	0.987700i	$0.450025\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	268.992	+	237.364i	0.725045	+	0.639796i
$372$		0			0
$373$			546.200i			1.46434i	0.681120	+	0.732171i	$0.261493\pi$
$373$			546.200i			1.46434i	−0.681120	+	0.732171i	$0.738507\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−297.781			−0.789870
$378$		0			0
$379$	−250.576			−0.661150			−0.330575	−	0.943780i	$-0.607243\pi$
$379$	−250.576			−0.661150			−0.330575	+	0.943780i	$0.607243\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	423.702i		−	1.10627i	−0.833092	−	0.553135i	$-0.813431\pi$
$383$		−	423.702i		−	1.10627i	0.833092	−	0.553135i	$-0.186569\pi$
$384$		0			0
$385$	17.8445	+	15.7464i	0.0463492	+	0.0408997i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	27.1194			0.0697157			0.0348578	−	0.999392i	$-0.488902\pi$
$389$	27.1194			0.0697157			0.0348578	+	0.999392i	$0.488902\pi$
$390$		0			0
$391$	210.506			0.538378
$392$		0			0
$393$		0			0
$394$		0			0
$395$	17.7377			0.0449057
$396$		0			0
$397$	−215.658			−0.543220			−0.271610	−	0.962407i	$-0.587556\pi$
$397$	−215.658			−0.543220			−0.271610	+	0.962407i	$0.587556\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	423.953i		−	1.05724i	−0.848859	−	0.528620i	$-0.822710\pi$
$401$		−	423.953i		−	1.05724i	0.848859	−	0.528620i	$-0.177290\pi$
$402$		0			0
$403$	−511.471			−1.26916
$404$		0			0
$405$		0			0
$406$		0			0
$407$	450.525			1.10694
$408$		0			0
$409$			51.8874i			0.126864i	0.997986	+	0.0634320i	$0.0202046\pi$
$409$			51.8874i			0.126864i	−0.997986	+	0.0634320i	$0.979795\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	143.102	+	126.277i	0.346495	+	0.305755i
$414$		0			0
$415$			57.6345i			0.138878i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−418.176			−0.998034			−0.499017	−	0.866592i	$-0.666306\pi$
$419$	−418.176			−0.998034			−0.499017	+	0.866592i	$0.666306\pi$
$420$		0			0
$421$		−	48.3444i		−	0.114832i	−0.998350	−	0.0574161i	$-0.981714\pi$
$421$		−	48.3444i		−	0.114832i	0.998350	−	0.0574161i	$-0.0182862\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	695.107			1.63555
$426$		0			0
$427$	605.568	+	534.368i	1.41819	+	1.25145i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	106.413			0.246898			0.123449	−	0.992351i	$-0.460604\pi$
$431$	106.413			0.246898			0.123449	+	0.992351i	$0.460604\pi$
$432$		0			0
$433$			261.405i			0.603707i	0.953354	+	0.301854i	$0.0976054\pi$
$433$			261.405i			0.603707i	−0.953354	+	0.301854i	$0.902395\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	220.130i		−	0.503731i
$438$		0			0
$439$	−611.541			−1.39303			−0.696517	−	0.717541i	$-0.745268\pi$
$439$	−611.541			−1.39303			−0.696517	+	0.717541i	$0.745268\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			695.830i			1.57072i	0.619038	+	0.785361i	$0.287523\pi$
$443$			695.830i			1.57072i	−0.619038	+	0.785361i	$0.712477\pi$
$444$		0			0
$445$		−	17.2440i		−	0.0387505i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			45.4904i			0.101315i	0.998716	+	0.0506574i	$0.0161317\pi$
$449$			45.4904i			0.101315i	−0.998716	+	0.0506574i	$0.983868\pi$
$450$		0			0
$451$		−	352.844i		−	0.782360i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	37.6060	−	42.6168i	0.0826506	−	0.0936632i
$456$		0			0
$457$	567.174			1.24108			0.620540	−	0.784175i	$-0.286913\pi$
$457$	567.174			1.24108			0.620540	+	0.784175i	$0.286913\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	166.006i		−	0.360099i	−0.983658	−	0.180050i	$-0.942374\pi$
$461$		−	166.006i		−	0.360099i	0.983658	−	0.180050i	$-0.0576258\pi$
$462$		0			0
$463$		−	194.577i		−	0.420254i	−0.977674	−	0.210127i	$-0.932612\pi$
$463$		−	194.577i		−	0.420254i	0.977674	−	0.210127i	$-0.0673877\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	86.2300			0.184647			0.0923234	−	0.995729i	$-0.470571\pi$
$467$	86.2300			0.184647			0.0923234	+	0.995729i	$0.470571\pi$
$468$		0			0
$469$	−38.1489	−	33.6635i	−0.0813410	−	0.0717772i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	330.334i		−	0.698380i
$474$		0			0
$475$		−	726.889i		−	1.53029i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			574.253i			1.19886i	0.800428	+	0.599429i	$0.204606\pi$
$479$			574.253i			1.19886i	−0.800428	+	0.599429i	$0.795394\pi$
$480$		0			0
$481$		−	1075.96i		−	2.23692i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	12.1068			0.0249624
$486$		0			0
$487$			595.755i			1.22332i	0.791122	+	0.611658i	$0.209497\pi$
$487$			595.755i			1.22332i	−0.791122	+	0.611658i	$0.790503\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	571.410i		−	1.16377i	−0.813272	−	0.581883i	$-0.802316\pi$
$491$		−	571.410i		−	1.16377i	0.813272	−	0.581883i	$-0.197684\pi$
$492$		0			0
$493$	−503.830			−1.02197
$494$		0			0
$495$		0			0
$496$		0			0
$497$	480.812	+	424.280i	0.967429	+	0.853683i
$498$		0			0
$499$	22.8477			0.0457870			0.0228935	−	0.999738i	$-0.492712\pi$
$499$	22.8477			0.0457870			0.0228935	+	0.999738i	$0.492712\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	688.951i		−	1.36968i	−0.728692	−	0.684842i	$-0.759871\pi$
$503$		−	688.951i		−	1.36968i	0.728692	−	0.684842i	$-0.240129\pi$
$504$		0			0
$505$	−26.4846			−0.0524447
$506$		0			0
$507$		0			0
$508$		0			0
$509$			62.9269i			0.123628i	0.998088	+	0.0618142i	$0.0196886\pi$
$509$			62.9269i			0.123628i	−0.998088	+	0.0618142i	$0.980311\pi$
$510$		0			0
$511$	−127.392	+	144.366i	−0.249299	+	0.282516i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	57.8138i		−	0.112260i
$516$		0			0
$517$	−451.650			−0.873597
$518$		0			0
$519$		0			0
$520$		0			0
$521$	612.508			1.17564			0.587820	−	0.808992i	$-0.299986\pi$
$521$	612.508			1.17564			0.587820	+	0.808992i	$0.299986\pi$
$522$		0			0
$523$		−	692.859i		−	1.32478i	−0.749160	−	0.662389i	$-0.769542\pi$
$523$		−	692.859i		−	1.32478i	0.749160	−	0.662389i	$-0.230458\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−865.383			−1.64209
$528$		0			0
$529$	−472.773			−0.893711
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−842.675			−1.58100
$534$		0			0
$535$	41.7343			0.0780079
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−337.781	+	42.3609i	−0.626681	+	0.0785917i
$540$		0			0
$541$			634.171i			1.17222i	0.810232	+	0.586110i	$0.199341\pi$
$541$			634.171i			1.17222i	−0.810232	+	0.586110i	$0.800659\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−43.2349			−0.0793301
$546$		0			0
$547$	361.551			0.660971			0.330486	−	0.943811i	$-0.392787\pi$
$547$	361.551			0.660971			0.330486	+	0.943811i	$0.392787\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			526.866i			0.956200i
$552$		0			0
$553$	−167.881	+	190.249i	−0.303581	+	0.344031i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−687.553			−1.23439			−0.617193	−	0.786812i	$-0.711730\pi$
$557$	−687.553			−1.23439			−0.617193	+	0.786812i	$0.711730\pi$
$558$		0			0
$559$	−788.915			−1.41130
$560$		0			0
$561$		0			0
$562$		0			0
$563$	960.105			1.70534			0.852669	−	0.522452i	$-0.174983\pi$
$563$	960.105			1.70534			0.852669	+	0.522452i	$0.174983\pi$
$564$		0			0
$565$	55.6187			0.0984403
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	569.590i		−	1.00104i	−0.865726	−	0.500518i	$-0.833143\pi$
$569$		−	569.590i		−	1.00104i	0.865726	−	0.500518i	$-0.166857\pi$
$570$		0			0
$571$	−413.155			−0.723564			−0.361782	−	0.932263i	$-0.617832\pi$
$571$	−413.155			−0.723564			−0.361782	+	0.932263i	$0.617832\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	185.666			0.322897
$576$		0			0
$577$			21.1560i			0.0366654i	0.999832	+	0.0183327i	$0.00583581\pi$
$577$			21.1560i			0.0366654i	−0.999832	+	0.0183327i	$0.994164\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−618.169	−	545.487i	−1.06397	−	0.938876i
$582$		0			0
$583$			356.054i			0.610728i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−395.721			−0.674141			−0.337071	−	0.941479i	$-0.609436\pi$
$587$	−395.721			−0.674141			−0.337071	+	0.941479i	$0.609436\pi$
$588$		0			0
$589$			904.950i			1.53642i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−488.061			−0.823037			−0.411519	−	0.911401i	$-0.635001\pi$
$593$	−488.061			−0.823037			−0.411519	+	0.911401i	$0.635001\pi$
$594$		0			0
$595$	63.6275	−	72.1053i	0.106937	−	0.121185i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	215.241			0.359334			0.179667	−	0.983727i	$-0.442498\pi$
$599$	215.241			0.359334			0.179667	+	0.983727i	$0.442498\pi$
$600$		0			0
$601$		−	810.565i		−	1.34869i	−0.738415	−	0.674347i	$-0.764425\pi$
$601$		−	810.565i		−	1.34869i	0.738415	−	0.674347i	$-0.235575\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		−	35.5920i		−	0.0588298i
$606$		0			0
$607$	621.412			1.02374			0.511872	−	0.859062i	$-0.328952\pi$
$607$	621.412			1.02374			0.511872	+	0.859062i	$0.328952\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			1078.65i			1.76538i
$612$		0			0
$613$		−	748.837i		−	1.22159i	−0.791787	−	0.610797i	$-0.790849\pi$
$613$		−	748.837i		−	1.22159i	0.791787	−	0.610797i	$-0.209151\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			439.740i			0.712707i	0.934351	+	0.356353i	$0.115980\pi$
$617$			439.740i			0.712707i	−0.934351	+	0.356353i	$0.884020\pi$
$618$		0			0
$619$			598.114i			0.966259i	0.875549	+	0.483129i	$0.160500\pi$
$619$			598.114i			0.966259i	−0.875549	+	0.483129i	$0.839500\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	184.953	+	163.207i	0.296875	+	0.261970i
$624$		0			0
$625$	607.097			0.971355
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	1820.47i		−	2.89422i
$630$		0			0
$631$			480.690i			0.761791i	0.924618	+	0.380895i	$0.124384\pi$
$631$			480.690i			0.761791i	−0.924618	+	0.380895i	$0.875616\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−112.721			−0.177513
$636$		0			0
$637$	101.168	+	806.701i	0.158819	+	1.26641i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	890.272i		−	1.38888i	−0.719551	−	0.694440i	$-0.755652\pi$
$641$		−	890.272i		−	1.38888i	0.719551	−	0.694440i	$-0.244348\pi$
$642$		0			0
$643$			748.676i			1.16435i	0.813064	+	0.582174i	$0.197798\pi$
$643$			748.676i			1.16435i	−0.813064	+	0.582174i	$0.802202\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	1011.64i		−	1.56359i	−0.623535	−	0.781795i	$-0.714304\pi$
$647$		−	1011.64i		−	1.56359i	0.623535	−	0.781795i	$-0.285696\pi$
$648$		0			0
$649$			189.420i			0.291864i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	970.047			1.48552			0.742762	−	0.669556i	$-0.233515\pi$
$653$	970.047			1.48552			0.742762	+	0.669556i	$0.233515\pi$
$654$		0			0
$655$			32.1716i			0.0491170i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	122.648i		−	0.186112i	−0.995661	−	0.0930562i	$-0.970336\pi$
$659$		−	122.648i		−	0.186112i	0.995661	−	0.0930562i	$-0.0296636\pi$
$660$		0			0
$661$	−550.866			−0.833382			−0.416691	−	0.909048i	$-0.636810\pi$
$661$	−550.866			−0.833382			−0.416691	+	0.909048i	$0.636810\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−75.4022	−	66.5366i	−0.113387	−	0.100055i
$666$		0			0
$667$	−134.575			−0.201761
$668$		0			0
$669$		0			0
$670$		0			0
$671$			801.569i			1.19459i
$672$		0			0
$673$	−929.335			−1.38088			−0.690442	−	0.723387i	$-0.742584\pi$
$673$	−929.335			−1.38088			−0.690442	+	0.723387i	$0.742584\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	426.954i		−	0.630656i	−0.948983	−	0.315328i	$-0.897885\pi$
$677$		−	426.954i		−	0.630656i	0.948983	−	0.315328i	$-0.102115\pi$
$678$		0			0
$679$	−114.586	+	129.853i	−0.168756	+	0.191242i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	1012.14i		−	1.48190i	−0.671558	−	0.740952i	$-0.734375\pi$
$683$		−	1012.14i		−	1.48190i	0.671558	−	0.740952i	$-0.265625\pi$
$684$		0			0
$685$	−123.288			−0.179982
$686$		0			0
$687$		0			0
$688$		0			0
$689$	850.342			1.23417
$690$		0			0
$691$			637.724i			0.922901i	0.887166	+	0.461450i	$0.152671\pi$
$691$			637.724i			0.922901i	−0.887166	+	0.461450i	$0.847329\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−27.0076			−0.0388599
$696$		0			0
$697$	−1425.76			−2.04557
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−915.416			−1.30587			−0.652936	−	0.757413i	$-0.726463\pi$
$701$	−915.416			−1.30587			−0.652936	+	0.757413i	$0.726463\pi$
$702$		0			0
$703$	−1903.70			−2.70797
$704$		0			0
$705$		0			0
$706$		0			0
$707$	250.666	−	284.065i	0.354549	−	0.401790i
$708$		0			0
$709$		−	26.7631i		−	0.0377477i	−0.999822	−	0.0188739i	$-0.993992\pi$
$709$		−	26.7631i		−	0.0377477i	0.999822	−	0.0188739i	$-0.00600809\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−231.147			−0.324189
$714$		0			0
$715$	56.4103			0.0788955
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	812.958i		−	1.13068i	−0.824859	−	0.565339i	$-0.808745\pi$
$719$		−	812.958i		−	1.13068i	0.824859	−	0.565339i	$-0.191255\pi$
$720$		0			0
$721$	620.092	+	547.184i	0.860045	+	0.758924i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−444.377			−0.612934
$726$		0			0
$727$	−402.292			−0.553359			−0.276679	−	0.960962i	$-0.589234\pi$
$727$	−402.292			−0.553359			−0.276679	+	0.960962i	$0.589234\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−1334.80			−1.82600
$732$		0			0
$733$	−707.087			−0.964648			−0.482324	−	0.875993i	$-0.660207\pi$
$733$	−707.087			−0.964648			−0.482324	+	0.875993i	$0.660207\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	50.4963i		−	0.0685161i
$738$		0			0
$739$	−1178.59			−1.59484			−0.797420	−	0.603425i	$-0.793802\pi$
$739$	−1178.59			−1.59484			−0.797420	+	0.603425i	$0.793802\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	227.166			0.305741			0.152870	−	0.988246i	$-0.451148\pi$
$743$	227.166			0.305741			0.152870	+	0.988246i	$0.451148\pi$
$744$		0			0
$745$			8.81251i			0.0118289i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−394.998	+	447.628i	−0.527367	+	0.597635i
$750$		0			0
$751$		−	19.6849i		−	0.0262116i	−0.999914	−	0.0131058i	$-0.995828\pi$
$751$		−	19.6849i		−	0.0262116i	0.999914	−	0.0131058i	$-0.00417182\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−60.3819			−0.0799760
$756$		0			0
$757$		−	1077.26i		−	1.42307i	−0.702650	−	0.711535i	$-0.748000\pi$
$757$		−	1077.26i		−	1.42307i	0.702650	−	0.711535i	$-0.252000\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	98.7893			0.129815			0.0649075	−	0.997891i	$-0.479325\pi$
$761$	98.7893			0.129815			0.0649075	+	0.997891i	$0.479325\pi$
$762$		0			0
$763$	409.201	−	463.724i	0.536306	−	0.607764i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	452.379			0.589803
$768$		0			0
$769$			332.048i			0.431792i	0.976416	+	0.215896i	$0.0692672\pi$
$769$			332.048i			0.431792i	−0.976416	+	0.215896i	$0.930733\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	1460.93i		−	1.88995i	−0.327149	−	0.944973i	$-0.606088\pi$
$773$		−	1460.93i		−	1.88995i	0.327149	−	0.944973i	$-0.393912\pi$
$774$		0			0
$775$	−763.267			−0.984860
$776$		0			0
$777$		0			0
$778$		0			0
$779$			1490.95i			1.91393i
$780$		0			0
$781$			636.434i			0.814897i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	19.3434i		−	0.0246413i
$786$		0			0
$787$		−	889.382i		−	1.13009i	−0.825059	−	0.565046i	$-0.808858\pi$
$787$		−	889.382i		−	1.13009i	0.825059	−	0.565046i	$-0.191142\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−526.409	+	596.549i	−0.665498	+	0.754171i
$792$		0			0
$793$	1914.33			2.41404
$794$		0			0
$795$		0			0
$796$		0			0
$797$			1370.49i			1.71956i	0.510662	+	0.859782i	$0.329400\pi$
$797$			1370.49i			1.71956i	−0.510662	+	0.859782i	$0.670600\pi$
$798$		0			0
$799$			1825.01i			2.28412i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−191.092			−0.237973
$804$		0			0
$805$	16.9951	−	19.2596i	0.0211120	−	0.0239250i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	1096.60i		−	1.35550i	−0.735291	−	0.677751i	$-0.762955\pi$
$809$		−	1096.60i		−	1.35550i	0.735291	−	0.677751i	$-0.237045\pi$
$810$		0			0
$811$		−	1357.90i		−	1.67436i	−0.546929	−	0.837179i	$-0.684203\pi$
$811$		−	1357.90i		−	1.67436i	0.546929	−	0.837179i	$-0.315797\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	2.65587i		−	0.00325874i
$816$		0			0
$817$			1395.83i			1.70849i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−936.137			−1.14024			−0.570120	−	0.821561i	$-0.693103\pi$
$821$	−936.137			−1.14024			−0.570120	+	0.821561i	$0.693103\pi$
$822$		0			0
$823$		−	129.080i		−	0.156841i	−0.996920	−	0.0784207i	$-0.975012\pi$
$823$		−	129.080i		−	0.156841i	0.996920	−	0.0784207i	$-0.0249877\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	22.8284i		−	0.0276039i	−0.999905	−	0.0138020i	$-0.995607\pi$
$827$		−	22.8284i		−	0.0276039i	0.999905	−	0.0138020i	$-0.00439344\pi$
$828$		0			0
$829$	−630.043			−0.760004			−0.380002	−	0.924986i	$-0.624077\pi$
$829$	−630.043			−0.760004			−0.380002	+	0.924986i	$0.624077\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	171.171	+	1364.90i	0.205487	+	1.63853i
$834$		0			0
$835$	76.4121			0.0915115
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	265.666i		−	0.316647i	−0.987387	−	0.158323i	$-0.949391\pi$
$839$		−	265.666i		−	0.316647i	0.987387	−	0.158323i	$-0.0506088\pi$
$840$		0			0
$841$	−518.905			−0.617010
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	52.0198i		−	0.0615619i
$846$		0			0
$847$	381.749	+	336.864i	0.450707	+	0.397715i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	486.253i		−	0.571390i
$852$		0			0
$853$	760.591			0.891665			0.445833	−	0.895116i	$-0.352908\pi$
$853$	760.591			0.891665			0.445833	+	0.895116i	$0.352908\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−962.482			−1.12308			−0.561542	−	0.827449i	$-0.689792\pi$
$857$	−962.482			−1.12308			−0.561542	+	0.827449i	$0.689792\pi$
$858$		0			0
$859$		−	160.713i		−	0.187093i	−0.995615	−	0.0935466i	$-0.970180\pi$
$859$		−	160.713i		−	0.187093i	0.995615	−	0.0935466i	$-0.0298204\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−1275.31			−1.47777			−0.738884	−	0.673832i	$-0.764647\pi$
$863$	−1275.31			−1.47777			−0.738884	+	0.673832i	$0.764647\pi$
$864$		0			0
$865$	72.4652			0.0837748
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−251.826			−0.289789
$870$		0			0
$871$	−120.597			−0.138458
$872$		0			0
$873$		0			0
$874$		0			0
$875$	112.781	−	127.809i	0.128893	−	0.146067i
$876$		0			0
$877$			1148.72i			1.30983i	0.755704	+	0.654913i	$0.227295\pi$
$877$			1148.72i			1.30983i	−0.755704	+	0.654913i	$0.772705\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	1589.04			1.80368			0.901840	−	0.432070i	$-0.142217\pi$
$881$	1589.04			1.80368			0.901840	+	0.432070i	$0.142217\pi$
$882$		0			0
$883$	−523.232			−0.592561			−0.296281	−	0.955101i	$-0.595746\pi$
$883$	−523.232			−0.592561			−0.296281	+	0.955101i	$0.595746\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			1190.80i			1.34250i	0.741231	+	0.671250i	$0.234242\pi$
$887$			1190.80i			1.34250i	−0.741231	+	0.671250i	$0.765758\pi$
$888$		0			0
$889$	1066.86	−	1209.01i	1.20006	−	1.35996i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	1908.46			2.13713
$894$		0			0
$895$	71.6831			0.0800928
$896$		0			0
$897$		0			0
$898$		0			0
$899$	553.233			0.615387
$900$		0			0
$901$	1438.73			1.59682
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	73.8691i		−	0.0816233i
$906$		0			0
$907$	−1762.93			−1.94369			−0.971845	−	0.235623i	$-0.924287\pi$
$907$	−1762.93			−1.94369			−0.971845	+	0.235623i	$0.924287\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−49.2079			−0.0540152			−0.0270076	−	0.999635i	$-0.508598\pi$
$911$	−49.2079			−0.0540152			−0.0270076	+	0.999635i	$0.508598\pi$
$912$		0			0
$913$		−	818.248i		−	0.896219i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−345.062	−	304.491i	−0.376295	−	0.332052i
$918$		0			0
$919$			685.900i			0.746355i	0.927760	+	0.373177i	$0.121732\pi$
$919$			685.900i			0.746355i	−0.927760	+	0.373177i	$0.878268\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	1519.95			1.64675
$924$		0			0
$925$		−	1605.65i		−	1.73584i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−194.049			−0.208880			−0.104440	−	0.994531i	$-0.533305\pi$
$929$	−194.049			−0.208880			−0.104440	+	0.994531i	$0.533305\pi$
$930$		0			0
$931$	1427.30	−	178.997i	1.53308	−	0.192263i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	95.4432			0.102078
$936$		0			0
$937$			523.728i			0.558942i	0.960154	+	0.279471i	$0.0901590\pi$
$937$			523.728i			0.558942i	−0.960154	+	0.279471i	$0.909841\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			144.646i			0.153716i	0.997042	+	0.0768578i	$0.0244887\pi$
$941$			144.646i			0.153716i	−0.997042	+	0.0768578i	$0.975511\pi$
$942$		0			0
$943$	−380.826			−0.403845
$944$		0			0
$945$		0			0
$946$		0			0
$947$			177.497i			0.187431i	0.995599	+	0.0937155i	$0.0298744\pi$
$947$			177.497i			0.187431i	−0.995599	+	0.0937155i	$0.970126\pi$
$948$		0			0
$949$			456.373i			0.480898i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			59.3443i			0.0622710i	0.999515	+	0.0311355i	$0.00991234\pi$
$953$			59.3443i			0.0622710i	−0.999515	+	0.0311355i	$0.990088\pi$
$954$		0			0
$955$		−	119.036i		−	0.124645i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	1166.87	−	1322.34i	1.21675	−	1.37888i
$960$		0			0
$961$	−10.7613			−0.0111980
$962$		0			0
$963$		0			0
$964$		0			0
$965$			67.5436i			0.0699934i
$966$		0			0
$967$		−	1022.00i		−	1.05687i	−0.848973	−	0.528437i	$-0.822778\pi$
$967$		−	1022.00i		−	1.05687i	0.848973	−	0.528437i	$-0.177222\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−121.308			−0.124931			−0.0624654	−	0.998047i	$-0.519896\pi$
$971$	−121.308			−0.124931			−0.0624654	+	0.998047i	$0.519896\pi$
$972$		0			0
$973$	255.616	−	289.675i	0.262710	−	0.297714i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	485.885i		−	0.497323i	−0.968590	−	0.248662i	$-0.920009\pi$
$977$		−	485.885i		−	0.497323i	0.968590	−	0.248662i	$-0.0799907\pi$
$978$		0			0
$979$			244.816i			0.250068i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	258.044i		−	0.262507i	−0.991349	−	0.131253i	$-0.958100\pi$
$983$		−	258.044i		−	0.262507i	0.991349	−	0.131253i	$-0.0419001\pi$
$984$		0			0
$985$			39.5138i			0.0401155i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−356.531			−0.360496
$990$		0			0
$991$		−	679.844i		−	0.686019i	−0.939332	−	0.343009i	$-0.888554\pi$
$991$		−	679.844i		−	0.686019i	0.939332	−	0.343009i	$-0.111446\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	112.822i		−	0.113389i
$996$		0			0
$997$	982.612			0.985569			0.492784	−	0.870152i	$-0.335979\pi$
$997$	982.612			0.985569			0.492784	+	0.870152i	$0.335979\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2016.3.e.c.1007.13		48
3.2	odd	2	inner	2016.3.e.c.1007.48		48
4.3	odd	2		504.3.e.c.251.27	yes	48
7.6	odd	2	inner	2016.3.e.c.1007.20		48
8.3	odd	2	inner	2016.3.e.c.1007.46		48
8.5	even	2		504.3.e.c.251.23	yes	48
12.11	even	2		504.3.e.c.251.22	yes	48
21.20	even	2	inner	2016.3.e.c.1007.45		48
24.5	odd	2		504.3.e.c.251.26	yes	48
24.11	even	2	inner	2016.3.e.c.1007.19		48
28.27	even	2		504.3.e.c.251.28	yes	48
56.13	odd	2		504.3.e.c.251.24	yes	48
56.27	even	2	inner	2016.3.e.c.1007.47		48
84.83	odd	2		504.3.e.c.251.21	✓	48
168.83	odd	2	inner	2016.3.e.c.1007.14		48
168.125	even	2		504.3.e.c.251.25	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.3.e.c.251.21	✓	48	84.83	odd	2
504.3.e.c.251.22	yes	48	12.11	even	2
504.3.e.c.251.23	yes	48	8.5	even	2
504.3.e.c.251.24	yes	48	56.13	odd	2
504.3.e.c.251.25	yes	48	168.125	even	2
504.3.e.c.251.26	yes	48	24.5	odd	2
504.3.e.c.251.27	yes	48	4.3	odd	2
504.3.e.c.251.28	yes	48	28.27	even	2
2016.3.e.c.1007.13		48	1.1	even	1	trivial
2016.3.e.c.1007.14		48	168.83	odd	2	inner
2016.3.e.c.1007.19		48	24.11	even	2	inner
2016.3.e.c.1007.20		48	7.6	odd	2	inner
2016.3.e.c.1007.45		48	21.20	even	2	inner
2016.3.e.c.1007.46		48	8.3	odd	2	inner
2016.3.e.c.1007.47		48	56.27	even	2	inner
2016.3.e.c.1007.48		48	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2016.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.489356i q^{5} +(5.24868 + 4.63156i) q^{7} +6.94749i q^{11} +16.5922 q^{13} +28.0732 q^{17} -29.3568i q^{19} +7.49845 q^{23} +24.7605 q^{25} -17.9470 q^{29} -30.8259 q^{31} +(2.26648 - 2.56848i) q^{35} -64.8471i q^{37} -50.7873 q^{41} -47.5472 q^{43} +65.0090i q^{47} +(6.09730 + 48.6192i) q^{49} +51.2494 q^{53} +3.39980 q^{55} +27.2645 q^{59} +115.375 q^{61} -8.11952i q^{65} -7.26829 q^{67} +91.6063 q^{71} +27.5052i q^{73} +(-32.1777 + 36.4652i) q^{77} +36.2471i q^{79} -117.776 q^{83} -13.7378i q^{85} +35.2381 q^{89} +(87.0874 + 76.8480i) q^{91} -14.3659 q^{95} +24.7402i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 336 q^{25} + 384 q^{43} + 368 q^{49} + 896 q^{67} + 576 q^{91}+O(q^{100}) \) 48 * q - 336 * q^25 + 384 * q^43 + 368 * q^49 + 896 * q^67 + 576 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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