LMFDB - Embedded newform 4056.2.a.y.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4056,2,Mod(1,4056)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4056, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4056.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4056 = 2^{3} \cdot 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4056.a (trivial)

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:	yes
Analytic conductor:	$32.3873230598$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.837.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 1 $ x^3 - 6*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 312)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.167449$ of defining polynomial
Character	$\chi$	$=$	4056.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -3.80451 q^{5} -5.13941 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -3.80451 q^{5} -5.13941 q^{7} +1.00000 q^{9} -0.334898 q^{11} +3.80451 q^{15} +4.13941 q^{17} +5.94392 q^{19} +5.13941 q^{21} +0.334898 q^{23} +9.47431 q^{25} -1.00000 q^{27} -0.195488 q^{29} -4.80451 q^{31} +0.334898 q^{33} +19.5529 q^{35} +2.13941 q^{37} -3.46961 q^{41} -2.86059 q^{43} -3.80451 q^{45} +3.66510 q^{47} +19.4135 q^{49} -4.13941 q^{51} +9.41353 q^{53} +1.27412 q^{55} -5.94392 q^{57} +6.27882 q^{59} -6.94392 q^{61} -5.13941 q^{63} -13.1394 q^{67} -0.334898 q^{69} -4.33490 q^{71} +10.6090 q^{73} -9.47431 q^{75} +1.72118 q^{77} -0.134715 q^{79} +1.00000 q^{81} +13.2741 q^{83} -15.7484 q^{85} +0.195488 q^{87} -0.390977 q^{89} +4.80451 q^{93} -22.6137 q^{95} -10.8045 q^{97} -0.334898 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} - 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 - 3 * q^7 + 3 * q^9	$ 3 q - 3 q^{3} - 3 q^{7} + 3 q^{9} - 6 q^{19} + 3 q^{21} + 15 q^{25} - 3 q^{27} - 12 q^{29} - 3 q^{31} + 12 q^{35} - 6 q^{37} - 21 q^{43} + 12 q^{47} + 24 q^{49} - 6 q^{53} - 18 q^{55} + 6 q^{57} - 6 q^{59} + 3 q^{61} - 3 q^{63} - 27 q^{67} - 12 q^{71} + 9 q^{73} - 15 q^{75} + 30 q^{77} + 9 q^{79} + 3 q^{81} + 18 q^{83} - 12 q^{85} + 12 q^{87} - 24 q^{89} + 3 q^{93} - 42 q^{95} - 21 q^{97}+O(q^{100}) $ 3 * q - 3 * q^3 - 3 * q^7 + 3 * q^9 - 6 * q^19 + 3 * q^21 + 15 * q^25 - 3 * q^27 - 12 * q^29 - 3 * q^31 + 12 * q^35 - 6 * q^37 - 21 * q^43 + 12 * q^47 + 24 * q^49 - 6 * q^53 - 18 * q^55 + 6 * q^57 - 6 * q^59 + 3 * q^61 - 3 * q^63 - 27 * q^67 - 12 * q^71 + 9 * q^73 - 15 * q^75 + 30 * q^77 + 9 * q^79 + 3 * q^81 + 18 * q^83 - 12 * q^85 + 12 * q^87 - 24 * q^89 + 3 * q^93 - 42 * q^95 - 21 * q^97

Display 100 coefficients 10 coefficients

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−3.80451	−1.70143	−0.850715	−	0.525628i	$-0.823830\pi$
$5$	−3.80451	−1.70143	−0.850715	+	0.525628i	$0.823830\pi$
$6$	0	0
$7$	−5.13941	−1.94251	−0.971257	−	0.238033i	$-0.923498\pi$
$7$	−5.13941	−1.94251	−0.971257	+	0.238033i	$0.923498\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.334898	−0.100976	−0.0504878	−	0.998725i	$-0.516078\pi$
$11$	−0.334898	−0.100976	−0.0504878	+	0.998725i	$0.516078\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	3.80451	0.982321
$16$	0	0
$17$	4.13941	1.00395	0.501977	−	0.864881i	$-0.332606\pi$
$17$	4.13941	1.00395	0.501977	+	0.864881i	$0.332606\pi$
$18$	0	0
$19$	5.94392	1.36363	0.681815	−	0.731525i	$-0.261191\pi$
$19$	5.94392	1.36363	0.681815	+	0.731525i	$0.261191\pi$
$20$	0	0
$21$	5.13941	1.12151
$22$	0	0
$23$	0.334898	0.0698311	0.0349156	−	0.999390i	$-0.488884\pi$
$23$	0.334898	0.0698311	0.0349156	+	0.999390i	$0.488884\pi$
$24$	0	0
$25$	9.47431	1.89486
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−0.195488	−0.0363013	−0.0181506	−	0.999835i	$-0.505778\pi$
$29$	−0.195488	−0.0363013	−0.0181506	+	0.999835i	$0.505778\pi$
$30$	0	0
$31$	−4.80451	−0.862916	−0.431458	−	0.902133i	$-0.642001\pi$
$31$	−4.80451	−0.862916	−0.431458	+	0.902133i	$0.642001\pi$
$32$	0	0
$33$	0.334898	0.0582983
$34$	0	0
$35$	19.5529	3.30505
$36$	0	0
$37$	2.13941	0.351717	0.175858	−	0.984415i	$-0.443730\pi$
$37$	2.13941	0.351717	0.175858	+	0.984415i	$0.443730\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−3.46961	−0.541863	−0.270931	−	0.962599i	$-0.587332\pi$
$41$	−3.46961	−0.541863	−0.270931	+	0.962599i	$0.587332\pi$
$42$	0	0
$43$	−2.86059	−0.436236	−0.218118	−	0.975922i	$-0.569992\pi$
$43$	−2.86059	−0.436236	−0.218118	+	0.975922i	$0.569992\pi$
$44$	0	0
$45$	−3.80451	−0.567143
$46$	0	0
$47$	3.66510	0.534610	0.267305	−	0.963612i	$-0.413867\pi$
$47$	3.66510	0.534610	0.267305	+	0.963612i	$0.413867\pi$
$48$	0	0
$49$	19.4135	2.77336
$50$	0	0
$51$	−4.13941	−0.579633
$52$	0	0
$53$	9.41353	1.29305	0.646524	−	0.762893i	$-0.276222\pi$
$53$	9.41353	1.29305	0.646524	+	0.762893i	$0.276222\pi$
$54$	0	0
$55$	1.27412	0.171803
$56$	0	0
$57$	−5.94392	−0.787292
$58$	0	0
$59$	6.27882	0.817433	0.408716	−	0.912661i	$-0.365977\pi$
$59$	6.27882	0.817433	0.408716	+	0.912661i	$0.365977\pi$
$60$	0	0
$61$	−6.94392	−0.889078	−0.444539	−	0.895759i	$-0.646632\pi$
$61$	−6.94392	−0.889078	−0.444539	+	0.895759i	$0.646632\pi$
$62$	0	0
$63$	−5.13941	−0.647505
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−13.1394	−1.60523	−0.802617	−	0.596494i	$-0.796560\pi$
$67$	−13.1394	−1.60523	−0.802617	+	0.596494i	$0.796560\pi$
$68$	0	0
$69$	−0.334898	−0.0403170
$70$	0	0
$71$	−4.33490	−0.514458	−0.257229	−	0.966351i	$-0.582809\pi$
$71$	−4.33490	−0.514458	−0.257229	+	0.966351i	$0.582809\pi$
$72$	0	0
$73$	10.6090	1.24169	0.620846	−	0.783932i	$-0.286789\pi$
$73$	10.6090	1.24169	0.620846	+	0.783932i	$0.286789\pi$
$74$	0	0
$75$	−9.47431	−1.09400
$76$	0	0
$77$	1.72118	0.196147
$78$	0	0
$79$	−0.134715	−0.0151566	−0.00757830	−	0.999971i	$-0.502412\pi$
$79$	−0.134715	−0.0151566	−0.00757830	+	0.999971i	$0.502412\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	13.2741	1.45702	0.728512	−	0.685033i	$-0.240212\pi$
$83$	13.2741	1.45702	0.728512	+	0.685033i	$0.240212\pi$
$84$	0	0
$85$	−15.7484	−1.70816
$86$	0	0
$87$	0.195488	0.0209586
$88$	0	0
$89$	−0.390977	−0.0414435	−0.0207217	−	0.999785i	$-0.506596\pi$
$89$	−0.390977	−0.0414435	−0.0207217	+	0.999785i	$0.506596\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	4.80451	0.498205
$94$	0	0
$95$	−22.6137	−2.32012
$96$	0	0
$97$	−10.8045	−1.09703	−0.548516	−	0.836140i	$-0.684807\pi$
$97$	−10.8045	−1.09703	−0.548516	+	0.836140i	$0.684807\pi$
$98$	0	0
$99$	−0.334898	−0.0336586
$100$	0	0
$101$	−16.0833	−1.60035	−0.800176	−	0.599766i	$-0.795260\pi$
$101$	−16.0833	−1.60035	−0.800176	+	0.599766i	$0.795260\pi$
$102$	0	0
$103$	4.46961	0.440404	0.220202	−	0.975454i	$-0.429328\pi$
$103$	4.46961	0.440404	0.220202	+	0.975454i	$0.429328\pi$
$104$	0	0
$105$	−19.5529	−1.90817
$106$	0	0
$107$	−3.66510	−0.354319	−0.177159	−	0.984182i	$-0.556691\pi$
$107$	−3.66510	−0.354319	−0.177159	+	0.984182i	$0.556691\pi$
$108$	0	0
$109$	9.08333	0.870025	0.435013	−	0.900424i	$-0.356744\pi$
$109$	9.08333	0.870025	0.435013	+	0.900424i	$0.356744\pi$
$110$	0	0
$111$	−2.13941	−0.203064
$112$	0	0
$113$	9.86059	0.927606	0.463803	−	0.885938i	$-0.346484\pi$
$113$	9.86059	0.927606	0.463803	+	0.885938i	$0.346484\pi$
$114$	0	0
$115$	−1.27412	−0.118813
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−21.2741	−1.95020
$120$	0	0
$121$	−10.8878	−0.989804
$122$	0	0
$123$	3.46961	0.312844
$124$	0	0
$125$	−17.0226	−1.52254
$126$	0	0
$127$	−0.134715	−0.0119540	−0.00597700	−	0.999982i	$-0.501903\pi$
$127$	−0.134715	−0.0119540	−0.00597700	+	0.999982i	$0.501903\pi$
$128$	0	0
$129$	2.86059	0.251861
$130$	0	0
$131$	−18.9486	−1.65555	−0.827774	−	0.561061i	$-0.810393\pi$
$131$	−18.9486	−1.65555	−0.827774	+	0.561061i	$0.810393\pi$
$132$	0	0
$133$	−30.5482	−2.64887
$134$	0	0
$135$	3.80451	0.327440
$136$	0	0
$137$	15.0786	1.28825	0.644127	−	0.764918i	$-0.277221\pi$
$137$	15.0786	1.28825	0.644127	+	0.764918i	$0.277221\pi$
$138$	0	0
$139$	−14.4135	−1.22254	−0.611270	−	0.791422i	$-0.709341\pi$
$139$	−14.4135	−1.22254	−0.611270	+	0.791422i	$0.709341\pi$
$140$	0	0
$141$	−3.66510	−0.308657
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0.743738	0.0617641
$146$	0	0
$147$	−19.4135	−1.60120
$148$	0	0
$149$	−18.0833	−1.48144	−0.740722	−	0.671812i	$-0.765516\pi$
$149$	−18.0833	−1.48144	−0.740722	+	0.671812i	$0.765516\pi$
$150$	0	0
$151$	−5.27412	−0.429202	−0.214601	−	0.976702i	$-0.568845\pi$
$151$	−5.27412	−0.429202	−0.214601	+	0.976702i	$0.568845\pi$
$152$	0	0
$153$	4.13941	0.334651
$154$	0	0
$155$	18.2788	1.46819
$156$	0	0
$157$	4.27412	0.341112	0.170556	−	0.985348i	$-0.445444\pi$
$157$	4.27412	0.341112	0.170556	+	0.985348i	$0.445444\pi$
$158$	0	0
$159$	−9.41353	−0.746542
$160$	0	0
$161$	−1.72118	−0.135648
$162$	0	0
$163$	−3.19549	−0.250290	−0.125145	−	0.992138i	$-0.539940\pi$
$163$	−3.19549	−0.250290	−0.125145	+	0.992138i	$0.539940\pi$
$164$	0	0
$165$	−1.27412	−0.0991905
$166$	0	0
$167$	25.2274	1.95216	0.976079	−	0.217417i	$-0.0697631\pi$
$167$	25.2274	1.95216	0.976079	+	0.217417i	$0.0697631\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	5.94392	0.454543
$172$	0	0
$173$	21.8878	1.66410	0.832051	−	0.554699i	$-0.187167\pi$
$173$	21.8878	1.66410	0.832051	+	0.554699i	$0.187167\pi$
$174$	0	0
$175$	−48.6924	−3.68080
$176$	0	0
$177$	−6.27882	−0.471945
$178$	0	0
$179$	−13.9439	−1.04222	−0.521109	−	0.853490i	$-0.674481\pi$
$179$	−13.9439	−1.04222	−0.521109	+	0.853490i	$0.674481\pi$
$180$	0	0
$181$	9.86059	0.732932	0.366466	−	0.930431i	$-0.380568\pi$
$181$	9.86059	0.732932	0.366466	+	0.930431i	$0.380568\pi$
$182$	0	0
$183$	6.94392	0.513309
$184$	0	0
$185$	−8.13941	−0.598421
$186$	0	0
$187$	−1.38628	−0.101375
$188$	0	0
$189$	5.13941	0.373837
$190$	0	0
$191$	−10.2788	−0.743749	−0.371875	−	0.928283i	$-0.621285\pi$
$191$	−10.2788	−0.743749	−0.371875	+	0.928283i	$0.621285\pi$
$192$	0	0
$193$	−3.66980	−0.264158	−0.132079	−	0.991239i	$-0.542165\pi$
$193$	−3.66980	−0.264158	−0.132079	+	0.991239i	$0.542165\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−20.1667	−1.43682	−0.718408	−	0.695622i	$-0.755129\pi$
$197$	−20.1667	−1.43682	−0.718408	+	0.695622i	$0.755129\pi$
$198$	0	0
$199$	2.74843	0.194831	0.0974156	−	0.995244i	$-0.468942\pi$
$199$	2.74843	0.194831	0.0974156	+	0.995244i	$0.468942\pi$
$200$	0	0
$201$	13.1394	0.926783
$202$	0	0
$203$	1.00470	0.0705158
$204$	0	0
$205$	13.2002	0.921941
$206$	0	0
$207$	0.334898	0.0232770
$208$	0	0
$209$	−1.99061	−0.137693
$210$	0	0
$211$	−12.1347	−0.835388	−0.417694	−	0.908588i	$-0.637162\pi$
$211$	−12.1347	−0.835388	−0.417694	+	0.908588i	$0.637162\pi$
$212$	0	0
$213$	4.33490	0.297022
$214$	0	0
$215$	10.8831	0.742225
$216$	0	0
$217$	24.6924	1.67623
$218$	0	0
$219$	−10.6090	−0.716891
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−16.0000	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	0	0
$225$	9.47431	0.631621
$226$	0	0
$227$	11.2835	0.748913	0.374457	−	0.927244i	$-0.377829\pi$
$227$	11.2835	0.748913	0.374457	+	0.927244i	$0.377829\pi$
$228$	0	0
$229$	−2.66980	−0.176425	−0.0882126	−	0.996102i	$-0.528115\pi$
$229$	−2.66980	−0.176425	−0.0882126	+	0.996102i	$0.528115\pi$
$230$	0	0
$231$	−1.72118	−0.113245
$232$	0	0
$233$	21.2180	1.39004	0.695020	−	0.718990i	$-0.255395\pi$
$233$	21.2180	1.39004	0.695020	+	0.718990i	$0.255395\pi$
$234$	0	0
$235$	−13.9439	−0.909601
$236$	0	0
$237$	0.134715	0.00875067
$238$	0	0
$239$	13.2741	0.858632	0.429316	−	0.903154i	$-0.358755\pi$
$239$	13.2741	0.858632	0.429316	+	0.903154i	$0.358755\pi$
$240$	0	0
$241$	−0.586465	−0.0377775	−0.0188888	−	0.999822i	$-0.506013\pi$
$241$	−0.586465	−0.0377775	−0.0188888	+	0.999822i	$0.506013\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−73.8590	−4.71868
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−13.2741	−0.841213
$250$	0	0
$251$	0.669797	0.0422772	0.0211386	−	0.999777i	$-0.493271\pi$
$251$	0.669797	0.0422772	0.0211386	+	0.999777i	$0.493271\pi$
$252$	0	0
$253$	−0.112157	−0.00705125
$254$	0	0
$255$	15.7484	0.986205
$256$	0	0
$257$	−0.530387	−0.0330846	−0.0165423	−	0.999863i	$-0.505266\pi$
$257$	−0.530387	−0.0330846	−0.0165423	+	0.999863i	$0.505266\pi$
$258$	0	0
$259$	−10.9953	−0.683215
$260$	0	0
$261$	−0.195488	−0.0121004
$262$	0	0
$263$	−2.61372	−0.161169	−0.0805844	−	0.996748i	$-0.525679\pi$
$263$	−2.61372	−0.161169	−0.0805844	+	0.996748i	$0.525679\pi$
$264$	0	0
$265$	−35.8139	−2.20003
$266$	0	0
$267$	0.390977	0.0239274
$268$	0	0
$269$	12.2788	0.748653	0.374326	−	0.927297i	$-0.377874\pi$
$269$	12.2788	0.748653	0.374326	+	0.927297i	$0.377874\pi$
$270$	0	0
$271$	−2.14411	−0.130245	−0.0651226	−	0.997877i	$-0.520744\pi$
$271$	−2.14411	−0.130245	−0.0651226	+	0.997877i	$0.520744\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−3.17293	−0.191335
$276$	0	0
$277$	6.53039	0.392373	0.196186	−	0.980567i	$-0.437144\pi$
$277$	6.53039	0.392373	0.196186	+	0.980567i	$0.437144\pi$
$278$	0	0
$279$	−4.80451	−0.287639
$280$	0	0
$281$	29.3575	1.75132	0.875660	−	0.482929i	$-0.160427\pi$
$281$	29.3575	1.75132	0.875660	+	0.482929i	$0.160427\pi$
$282$	0	0
$283$	−12.0880	−0.718559	−0.359279	−	0.933230i	$-0.616977\pi$
$283$	−12.0880	−0.718559	−0.359279	+	0.933230i	$0.616977\pi$
$284$	0	0
$285$	22.6137	1.33952
$286$	0	0
$287$	17.8318	1.05258
$288$	0	0
$289$	0.134715	0.00792440
$290$	0	0
$291$	10.8045	0.633372
$292$	0	0
$293$	−3.69235	−0.215710	−0.107855	−	0.994167i	$-0.534398\pi$
$293$	−3.69235	−0.215710	−0.107855	+	0.994167i	$0.534398\pi$
$294$	0	0
$295$	−23.8878	−1.39080
$296$	0	0
$297$	0.334898	0.0194328
$298$	0	0
$299$	0	0
$300$	0	0
$301$	14.7017	0.847394
$302$	0	0
$303$	16.0833	0.923963
$304$	0	0
$305$	26.4182	1.51270
$306$	0	0
$307$	6.07864	0.346926	0.173463	−	0.984840i	$-0.444504\pi$
$307$	6.07864	0.346926	0.173463	+	0.984840i	$0.444504\pi$
$308$	0	0
$309$	−4.46961	−0.254267
$310$	0	0
$311$	10.0561	0.570228	0.285114	−	0.958494i	$-0.407969\pi$
$311$	10.0561	0.570228	0.285114	+	0.958494i	$0.407969\pi$
$312$	0	0
$313$	−11.3622	−0.642227	−0.321113	−	0.947041i	$-0.604057\pi$
$313$	−11.3622	−0.642227	−0.321113	+	0.947041i	$0.604057\pi$
$314$	0	0
$315$	19.5529	1.10168
$316$	0	0
$317$	−19.4229	−1.09090	−0.545450	−	0.838143i	$-0.683641\pi$
$317$	−19.4229	−1.09090	−0.545450	+	0.838143i	$0.683641\pi$
$318$	0	0
$319$	0.0654688	0.00366555
$320$	0	0
$321$	3.66510	0.204566
$322$	0	0
$323$	24.6043	1.36902
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−9.08333	−0.502309
$328$	0	0
$329$	−18.8365	−1.03849
$330$	0	0
$331$	−28.8045	−1.58324	−0.791620	−	0.611014i	$-0.790762\pi$
$331$	−28.8045	−1.58324	−0.791620	+	0.611014i	$0.790762\pi$
$332$	0	0
$333$	2.13941	0.117239
$334$	0	0
$335$	49.9890	2.73119
$336$	0	0
$337$	0.0513833	0.00279903	0.00139951	−	0.999999i	$-0.499555\pi$
$337$	0.0513833	0.00279903	0.00139951	+	0.999999i	$0.499555\pi$
$338$	0	0
$339$	−9.86059	−0.535554
$340$	0	0
$341$	1.60902	0.0871335
$342$	0	0
$343$	−63.7982	−3.44478
$344$	0	0
$345$	1.27412	0.0685966
$346$	0	0
$347$	2.72588	0.146333	0.0731663	−	0.997320i	$-0.476690\pi$
$347$	2.72588	0.146333	0.0731663	+	0.997320i	$0.476690\pi$
$348$	0	0
$349$	−27.7437	−1.48509	−0.742544	−	0.669797i	$-0.766381\pi$
$349$	−27.7437	−1.48509	−0.742544	+	0.669797i	$0.766381\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	22.0273	1.17239	0.586196	−	0.810169i	$-0.300625\pi$
$353$	22.0273	1.17239	0.586196	+	0.810169i	$0.300625\pi$
$354$	0	0
$355$	16.4922	0.875314
$356$	0	0
$357$	21.2741	1.12595
$358$	0	0
$359$	1.94392	0.102596	0.0512981	−	0.998683i	$-0.483664\pi$
$359$	1.94392	0.102596	0.0512981	+	0.998683i	$0.483664\pi$
$360$	0	0
$361$	16.3302	0.859484
$362$	0	0
$363$	10.8878	0.571464
$364$	0	0
$365$	−40.3622	−2.11265
$366$	0	0
$367$	−1.13941	−0.0594767	−0.0297384	−	0.999558i	$-0.509467\pi$
$367$	−1.13941	−0.0594767	−0.0297384	+	0.999558i	$0.509467\pi$
$368$	0	0
$369$	−3.46961	−0.180621
$370$	0	0
$371$	−48.3800	−2.51177
$372$	0	0
$373$	4.83176	0.250179	0.125090	−	0.992145i	$-0.460078\pi$
$373$	4.83176	0.250179	0.125090	+	0.992145i	$0.460078\pi$
$374$	0	0
$375$	17.0226	0.879041
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−28.9618	−1.48767	−0.743833	−	0.668365i	$-0.766994\pi$
$379$	−28.9618	−1.48767	−0.743833	+	0.668365i	$0.766994\pi$
$380$	0	0
$381$	0.134715	0.00690165
$382$	0	0
$383$	−9.33959	−0.477231	−0.238615	−	0.971114i	$-0.576694\pi$
$383$	−9.33959	−0.477231	−0.238615	+	0.971114i	$0.576694\pi$
$384$	0	0
$385$	−6.54825	−0.333730
$386$	0	0
$387$	−2.86059	−0.145412
$388$	0	0
$389$	0.204879	0.0103878	0.00519388	−	0.999987i	$-0.498347\pi$
$389$	0.204879	0.0103878	0.00519388	+	0.999987i	$0.498347\pi$
$390$	0	0
$391$	1.38628	0.0701073
$392$	0	0
$393$	18.9486	0.955831
$394$	0	0
$395$	0.512524	0.0257879
$396$	0	0
$397$	8.81390	0.442357	0.221179	−	0.975233i	$-0.429010\pi$
$397$	8.81390	0.442357	0.221179	+	0.975233i	$0.429010\pi$
$398$	0	0
$399$	30.5482	1.52933
$400$	0	0
$401$	23.6363	1.18034	0.590170	−	0.807279i	$-0.299061\pi$
$401$	23.6363	1.18034	0.590170	+	0.807279i	$0.299061\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−3.80451	−0.189048
$406$	0	0
$407$	−0.716485	−0.0355148
$408$	0	0
$409$	28.4875	1.40862	0.704308	−	0.709895i	$-0.251258\pi$
$409$	28.4875	1.40862	0.704308	+	0.709895i	$0.251258\pi$
$410$	0	0
$411$	−15.0786	−0.743774
$412$	0	0
$413$	−32.2694	−1.58787
$414$	0	0
$415$	−50.5016	−2.47902
$416$	0	0
$417$	14.4135	0.705834
$418$	0	0
$419$	20.5576	1.00431	0.502153	−	0.864779i	$-0.332541\pi$
$419$	20.5576	1.00431	0.502153	+	0.864779i	$0.332541\pi$
$420$	0	0
$421$	8.55294	0.416845	0.208423	−	0.978039i	$-0.433167\pi$
$421$	8.55294	0.416845	0.208423	+	0.978039i	$0.433167\pi$
$422$	0	0
$423$	3.66510	0.178203
$424$	0	0
$425$	39.2180	1.90235
$426$	0	0
$427$	35.6877	1.72705
$428$	0	0
$429$	0	0
$430$	0	0
$431$	25.2741	1.21741	0.608706	−	0.793396i	$-0.291689\pi$
$431$	25.2741	1.21741	0.608706	+	0.793396i	$0.291689\pi$
$432$	0	0
$433$	16.3396	0.785231	0.392615	−	0.919703i	$-0.371570\pi$
$433$	16.3396	0.785231	0.392615	+	0.919703i	$0.371570\pi$
$434$	0	0
$435$	−0.743738	−0.0356595
$436$	0	0
$437$	1.99061	0.0952238
$438$	0	0
$439$	−3.41823	−0.163143	−0.0815716	−	0.996667i	$-0.525994\pi$
$439$	−3.41823	−0.163143	−0.0815716	+	0.996667i	$0.525994\pi$
$440$	0	0
$441$	19.4135	0.924454
$442$	0	0
$443$	0.669797	0.0318230	0.0159115	−	0.999873i	$-0.494935\pi$
$443$	0.669797	0.0318230	0.0159115	+	0.999873i	$0.494935\pi$
$444$	0	0
$445$	1.48748	0.0705131
$446$	0	0
$447$	18.0833	0.855312
$448$	0	0
$449$	−21.2180	−1.00134	−0.500671	−	0.865638i	$-0.666913\pi$
$449$	−21.2180	−1.00134	−0.500671	+	0.865638i	$0.666913\pi$
$450$	0	0
$451$	1.16197	0.0547149
$452$	0	0
$453$	5.27412	0.247800
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−6.61841	−0.309596	−0.154798	−	0.987946i	$-0.549473\pi$
$457$	−6.61841	−0.309596	−0.154798	+	0.987946i	$0.549473\pi$
$458$	0	0
$459$	−4.13941	−0.193211
$460$	0	0
$461$	29.6924	1.38291	0.691455	−	0.722419i	$-0.256970\pi$
$461$	29.6924	1.38291	0.691455	+	0.722419i	$0.256970\pi$
$462$	0	0
$463$	−18.7484	−0.871314	−0.435657	−	0.900113i	$-0.643484\pi$
$463$	−18.7484	−0.871314	−0.435657	+	0.900113i	$0.643484\pi$
$464$	0	0
$465$	−18.2788	−0.847660
$466$	0	0
$467$	−1.00470	−0.0464917	−0.0232459	−	0.999730i	$-0.507400\pi$
$467$	−1.00470	−0.0464917	−0.0232459	+	0.999730i	$0.507400\pi$
$468$	0	0
$469$	67.5288	3.11819
$470$	0	0
$471$	−4.27412	−0.196941
$472$	0	0
$473$	0.958007	0.0440492
$474$	0	0
$475$	56.3145	2.58389
$476$	0	0
$477$	9.41353	0.431016
$478$	0	0
$479$	−23.8878	−1.09146	−0.545732	−	0.837960i	$-0.683748\pi$
$479$	−23.8878	−1.09146	−0.545732	+	0.837960i	$0.683748\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	1.72118	0.0783164
$484$	0	0
$485$	41.1059	1.86652
$486$	0	0
$487$	−29.9439	−1.35689	−0.678444	−	0.734652i	$-0.737346\pi$
$487$	−29.9439	−1.35689	−0.678444	+	0.734652i	$0.737346\pi$
$488$	0	0
$489$	3.19549	0.144505
$490$	0	0
$491$	−22.2134	−1.00247	−0.501237	−	0.865310i	$-0.667122\pi$
$491$	−22.2134	−1.00247	−0.501237	+	0.865310i	$0.667122\pi$
$492$	0	0
$493$	−0.809207	−0.0364448
$494$	0	0
$495$	1.27412	0.0572676
$496$	0	0
$497$	22.2788	0.999342
$498$	0	0
$499$	−4.66980	−0.209049	−0.104524	−	0.994522i	$-0.533332\pi$
$499$	−4.66980	−0.209049	−0.104524	+	0.994522i	$0.533332\pi$
$500$	0	0
$501$	−25.2274	−1.12708
$502$	0	0
$503$	−34.2134	−1.52550	−0.762749	−	0.646695i	$-0.776151\pi$
$503$	−34.2134	−1.52550	−0.762749	+	0.646695i	$0.776151\pi$
$504$	0	0
$505$	61.1892	2.72288
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−1.81390	−0.0803998	−0.0401999	−	0.999192i	$-0.512799\pi$
$509$	−1.81390	−0.0803998	−0.0401999	+	0.999192i	$0.512799\pi$
$510$	0	0
$511$	−54.5241	−2.41201
$512$	0	0
$513$	−5.94392	−0.262431
$514$	0	0
$515$	−17.0047	−0.749316
$516$	0	0
$517$	−1.22744	−0.0539826
$518$	0	0
$519$	−21.8878	−0.960770
$520$	0	0
$521$	−24.9759	−1.09421	−0.547106	−	0.837063i	$-0.684271\pi$
$521$	−24.9759	−1.09421	−0.547106	+	0.837063i	$0.684271\pi$
$522$	0	0
$523$	−3.39567	−0.148482	−0.0742412	−	0.997240i	$-0.523653\pi$
$523$	−3.39567	−0.148482	−0.0742412	+	0.997240i	$0.523653\pi$
$524$	0	0
$525$	48.6924	2.12511
$526$	0	0
$527$	−19.8878	−0.866328
$528$	0	0
$529$	−22.8878	−0.995124
$530$	0	0
$531$	6.27882	0.272478
$532$	0	0
$533$	0	0
$534$	0	0
$535$	13.9439	0.602848
$536$	0	0
$537$	13.9439	0.601725
$538$	0	0
$539$	−6.50156	−0.280042
$540$	0	0
$541$	29.7804	1.28036	0.640179	−	0.768226i	$-0.278860\pi$
$541$	29.7804	1.28036	0.640179	+	0.768226i	$0.278860\pi$
$542$	0	0
$543$	−9.86059	−0.423158
$544$	0	0
$545$	−34.5576	−1.48029
$546$	0	0
$547$	22.6363	0.967857	0.483929	−	0.875107i	$-0.339209\pi$
$547$	22.6363	0.967857	0.483929	+	0.875107i	$0.339209\pi$
$548$	0	0
$549$	−6.94392	−0.296359
$550$	0	0
$551$	−1.16197	−0.0495015
$552$	0	0
$553$	0.692355	0.0294419
$554$	0	0
$555$	8.13941	0.345499
$556$	0	0
$557$	19.7017	0.834790	0.417395	−	0.908725i	$-0.362943\pi$
$557$	19.7017	0.834790	0.417395	+	0.908725i	$0.362943\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	1.38628	0.0585289
$562$	0	0
$563$	−38.0545	−1.60381	−0.801903	−	0.597454i	$-0.796179\pi$
$563$	−38.0545	−1.60381	−0.801903	+	0.597454i	$0.796179\pi$
$564$	0	0
$565$	−37.5147	−1.57826
$566$	0	0
$567$	−5.13941	−0.215835
$568$	0	0
$569$	11.2274	0.470679	0.235339	−	0.971913i	$-0.424380\pi$
$569$	11.2274	0.470679	0.235339	+	0.971913i	$0.424380\pi$
$570$	0	0
$571$	−37.9439	−1.58790	−0.793952	−	0.607981i	$-0.791980\pi$
$571$	−37.9439	−1.58790	−0.793952	+	0.607981i	$0.791980\pi$
$572$	0	0
$573$	10.2788	0.429404
$574$	0	0
$575$	3.17293	0.132320
$576$	0	0
$577$	−33.6830	−1.40224	−0.701120	−	0.713043i	$-0.747316\pi$
$577$	−33.6830	−1.40224	−0.701120	+	0.713043i	$0.747316\pi$
$578$	0	0
$579$	3.66980	0.152512
$580$	0	0
$581$	−68.2212	−2.83029
$582$	0	0
$583$	−3.15258	−0.130566
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−9.60902	−0.396607	−0.198303	−	0.980141i	$-0.563543\pi$
$587$	−9.60902	−0.396607	−0.198303	+	0.980141i	$0.563543\pi$
$588$	0	0
$589$	−28.5576	−1.17670
$590$	0	0
$591$	20.1667	0.829546
$592$	0	0
$593$	38.0273	1.56159	0.780796	−	0.624786i	$-0.214814\pi$
$593$	38.0273	1.56159	0.780796	+	0.624786i	$0.214814\pi$
$594$	0	0
$595$	80.9377	3.31812
$596$	0	0
$597$	−2.74843	−0.112486
$598$	0	0
$599$	−4.82707	−0.197229	−0.0986144	−	0.995126i	$-0.531441\pi$
$599$	−4.82707	−0.197229	−0.0986144	+	0.995126i	$0.531441\pi$
$600$	0	0
$601$	−2.74374	−0.111919	−0.0559597	−	0.998433i	$-0.517822\pi$
$601$	−2.74374	−0.111919	−0.0559597	+	0.998433i	$0.517822\pi$
$602$	0	0
$603$	−13.1394	−0.535078
$604$	0	0
$605$	41.4229	1.68408
$606$	0	0
$607$	35.1059	1.42490	0.712452	−	0.701721i	$-0.247585\pi$
$607$	35.1059	1.42490	0.712452	+	0.701721i	$0.247585\pi$
$608$	0	0
$609$	−1.00470	−0.0407123
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−24.1620	−0.975893	−0.487946	−	0.872874i	$-0.662254\pi$
$613$	−24.1620	−0.975893	−0.487946	+	0.872874i	$0.662254\pi$
$614$	0	0
$615$	−13.2002	−0.532283
$616$	0	0
$617$	−17.6363	−0.710010	−0.355005	−	0.934864i	$-0.615521\pi$
$617$	−17.6363	−0.710010	−0.355005	+	0.934864i	$0.615521\pi$
$618$	0	0
$619$	−34.1441	−1.37237	−0.686184	−	0.727428i	$-0.740715\pi$
$619$	−34.1441	−1.37237	−0.686184	+	0.727428i	$0.740715\pi$
$620$	0	0
$621$	−0.334898	−0.0134390
$622$	0	0
$623$	2.00939	0.0805045
$624$	0	0
$625$	17.3910	0.695639
$626$	0	0
$627$	1.99061	0.0794973
$628$	0	0
$629$	8.85589	0.353108
$630$	0	0
$631$	−26.9712	−1.07371	−0.536853	−	0.843676i	$-0.680387\pi$
$631$	−26.9712	−1.07371	−0.536853	+	0.843676i	$0.680387\pi$
$632$	0	0
$633$	12.1347	0.482312
$634$	0	0
$635$	0.512524	0.0203389
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−4.33490	−0.171486
$640$	0	0
$641$	−28.6877	−1.13309	−0.566547	−	0.824029i	$-0.691721\pi$
$641$	−28.6877	−1.13309	−0.566547	+	0.824029i	$0.691721\pi$
$642$	0	0
$643$	20.0226	0.789613	0.394806	−	0.918764i	$-0.370812\pi$
$643$	20.0226	0.789613	0.394806	+	0.918764i	$0.370812\pi$
$644$	0	0
$645$	−10.8831	−0.428524
$646$	0	0
$647$	−6.16666	−0.242437	−0.121218	−	0.992626i	$-0.538680\pi$
$647$	−6.16666	−0.242437	−0.121218	+	0.992626i	$0.538680\pi$
$648$	0	0
$649$	−2.10277	−0.0825408
$650$	0	0
$651$	−24.6924	−0.967770
$652$	0	0
$653$	−21.5996	−0.845259	−0.422629	−	0.906303i	$-0.638893\pi$
$653$	−21.5996	−0.845259	−0.422629	+	0.906303i	$0.638893\pi$
$654$	0	0
$655$	72.0902	2.81680
$656$	0	0
$657$	10.6090	0.413897
$658$	0	0
$659$	−11.2180	−0.436993	−0.218497	−	0.975838i	$-0.570115\pi$
$659$	−11.2180	−0.436993	−0.218497	+	0.975838i	$0.570115\pi$
$660$	0	0
$661$	−44.7196	−1.73939	−0.869696	−	0.493589i	$-0.835685\pi$
$661$	−44.7196	−1.73939	−0.869696	+	0.493589i	$0.835685\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	116.221	4.50686
$666$	0	0
$667$	−0.0654688	−0.00253496
$668$	0	0
$669$	16.0000	0.618596
$670$	0	0
$671$	2.32551	0.0897752
$672$	0	0
$673$	−15.4361	−0.595018	−0.297509	−	0.954719i	$-0.596156\pi$
$673$	−15.4361	−0.595018	−0.297509	+	0.954719i	$0.596156\pi$
$674$	0	0
$675$	−9.47431	−0.364666
$676$	0	0
$677$	−35.4969	−1.36426	−0.682128	−	0.731233i	$-0.738945\pi$
$677$	−35.4969	−1.36426	−0.682128	+	0.731233i	$0.738945\pi$
$678$	0	0
$679$	55.5288	2.13100
$680$	0	0
$681$	−11.2835	−0.432385
$682$	0	0
$683$	−9.60902	−0.367679	−0.183840	−	0.982956i	$-0.558853\pi$
$683$	−9.60902	−0.367679	−0.183840	+	0.982956i	$0.558853\pi$
$684$	0	0
$685$	−57.3668	−2.19187
$686$	0	0
$687$	2.66980	0.101859
$688$	0	0
$689$	0	0
$690$	0	0
$691$	37.0273	1.40858	0.704292	−	0.709911i	$-0.251265\pi$
$691$	37.0273	1.40858	0.704292	+	0.709911i	$0.251265\pi$
$692$	0	0
$693$	1.72118	0.0653822
$694$	0	0
$695$	54.8365	2.08007
$696$	0	0
$697$	−14.3622	−0.544005
$698$	0	0
$699$	−21.2180	−0.802540
$700$	0	0
$701$	−19.7212	−0.744859	−0.372429	−	0.928061i	$-0.621475\pi$
$701$	−19.7212	−0.744859	−0.372429	+	0.928061i	$0.621475\pi$
$702$	0	0
$703$	12.7165	0.479611
$704$	0	0
$705$	13.9439	0.525158
$706$	0	0
$707$	82.6588	3.10871
$708$	0	0
$709$	−21.3443	−0.801602	−0.400801	−	0.916165i	$-0.631268\pi$
$709$	−21.3443	−0.801602	−0.400801	+	0.916165i	$0.631268\pi$
$710$	0	0
$711$	−0.134715	−0.00505220
$712$	0	0
$713$	−1.60902	−0.0602584
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−13.2741	−0.495731
$718$	0	0
$719$	23.6184	0.880818	0.440409	−	0.897797i	$-0.354833\pi$
$719$	23.6184	0.880818	0.440409	+	0.897797i	$0.354833\pi$
$720$	0	0
$721$	−22.9712	−0.855491
$722$	0	0
$723$	0.586465	0.0218109
$724$	0	0
$725$	−1.85212	−0.0687859
$726$	0	0
$727$	40.8029	1.51330	0.756649	−	0.653822i	$-0.226835\pi$
$727$	40.8029	1.51330	0.756649	+	0.653822i	$0.226835\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−11.8412	−0.437961
$732$	0	0
$733$	33.3349	1.23125	0.615626	−	0.788038i	$-0.288903\pi$
$733$	33.3349	1.23125	0.615626	+	0.788038i	$0.288903\pi$
$734$	0	0
$735$	73.8590	2.72433
$736$	0	0
$737$	4.40037	0.162090
$738$	0	0
$739$	3.21805	0.118378	0.0591889	−	0.998247i	$-0.481149\pi$
$739$	3.21805	0.118378	0.0591889	+	0.998247i	$0.481149\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−13.4969	−0.495152	−0.247576	−	0.968868i	$-0.579634\pi$
$743$	−13.4969	−0.495152	−0.247576	+	0.968868i	$0.579634\pi$
$744$	0	0
$745$	68.7982	2.52057
$746$	0	0
$747$	13.2741	0.485675
$748$	0	0
$749$	18.8365	0.688269
$750$	0	0
$751$	−15.8224	−0.577367	−0.288683	−	0.957425i	$-0.593217\pi$
$751$	−15.8224	−0.577367	−0.288683	+	0.957425i	$0.593217\pi$
$752$	0	0
$753$	−0.669797	−0.0244088
$754$	0	0
$755$	20.0655	0.730257
$756$	0	0
$757$	−0.660406	−0.0240029	−0.0120014	−	0.999928i	$-0.503820\pi$
$757$	−0.660406	−0.0240029	−0.0120014	+	0.999928i	$0.503820\pi$
$758$	0	0
$759$	0.112157	0.00407104
$760$	0	0
$761$	15.6090	0.565827	0.282913	−	0.959145i	$-0.408699\pi$
$761$	15.6090	0.565827	0.282913	+	0.959145i	$0.408699\pi$
$762$	0	0
$763$	−46.6830	−1.69004
$764$	0	0
$765$	−15.7484	−0.569386
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−40.3239	−1.45412	−0.727059	−	0.686575i	$-0.759113\pi$
$769$	−40.3239	−1.45412	−0.727059	+	0.686575i	$0.759113\pi$
$770$	0	0
$771$	0.530387	0.0191014
$772$	0	0
$773$	13.3302	0.479454	0.239727	−	0.970840i	$-0.422942\pi$
$773$	13.3302	0.479454	0.239727	+	0.970840i	$0.422942\pi$
$774$	0	0
$775$	−45.5194	−1.63511
$776$	0	0
$777$	10.9953	0.394454
$778$	0	0
$779$	−20.6231	−0.738900
$780$	0	0
$781$	1.45175	0.0519477
$782$	0	0
$783$	0.195488	0.00698619
$784$	0	0
$785$	−16.2610	−0.580378
$786$	0	0
$787$	−9.85589	−0.351325	−0.175662	−	0.984450i	$-0.556207\pi$
$787$	−9.85589	−0.351325	−0.175662	+	0.984450i	$0.556207\pi$
$788$	0	0
$789$	2.61372	0.0930508
$790$	0	0
$791$	−50.6776	−1.80189
$792$	0	0
$793$	0	0
$794$	0	0
$795$	35.8139	1.27019
$796$	0	0
$797$	−45.4875	−1.61125	−0.805625	−	0.592426i	$-0.798170\pi$
$797$	−45.4875	−1.61125	−0.805625	+	0.592426i	$0.798170\pi$
$798$	0	0
$799$	15.1714	0.536724
$800$	0	0
$801$	−0.390977	−0.0138145
$802$	0	0
$803$	−3.55294	−0.125381
$804$	0	0
$805$	6.54825	0.230795
$806$	0	0
$807$	−12.2788	−0.432235
$808$	0	0
$809$	−19.5241	−0.686431	−0.343216	−	0.939257i	$-0.611516\pi$
$809$	−19.5241	−0.686431	−0.343216	+	0.939257i	$0.611516\pi$
$810$	0	0
$811$	−33.3622	−1.17150	−0.585752	−	0.810490i	$-0.699201\pi$
$811$	−33.3622	−1.17150	−0.585752	+	0.810490i	$0.699201\pi$
$812$	0	0
$813$	2.14411	0.0751970
$814$	0	0
$815$	12.1573	0.425851
$816$	0	0
$817$	−17.0031	−0.594864
$818$	0	0
$819$	0	0
$820$	0	0
$821$	43.2274	1.50865	0.754324	−	0.656502i	$-0.227965\pi$
$821$	43.2274	1.50865	0.754324	+	0.656502i	$0.227965\pi$
$822$	0	0
$823$	22.0094	0.767199	0.383600	−	0.923500i	$-0.374684\pi$
$823$	22.0094	0.767199	0.383600	+	0.923500i	$0.374684\pi$
$824$	0	0
$825$	3.17293	0.110467
$826$	0	0
$827$	−38.5482	−1.34045	−0.670227	−	0.742156i	$-0.733803\pi$
$827$	−38.5482	−1.34045	−0.670227	+	0.742156i	$0.733803\pi$
$828$	0	0
$829$	30.4408	1.05725	0.528626	−	0.848855i	$-0.322707\pi$
$829$	30.4408	1.05725	0.528626	+	0.848855i	$0.322707\pi$
$830$	0	0
$831$	−6.53039	−0.226537
$832$	0	0
$833$	80.3606	2.78433
$834$	0	0
$835$	−95.9781	−3.32146
$836$	0	0
$837$	4.80451	0.166068
$838$	0	0
$839$	13.3847	0.462091	0.231046	−	0.972943i	$-0.425785\pi$
$839$	13.3847	0.462091	0.231046	+	0.972943i	$0.425785\pi$
$840$	0	0
$841$	−28.9618	−0.998682
$842$	0	0
$843$	−29.3575	−1.01112
$844$	0	0
$845$	0	0
$846$	0	0
$847$	55.9571	1.92271
$848$	0	0
$849$	12.0880	0.414860
$850$	0	0
$851$	0.716485	0.0245608
$852$	0	0
$853$	−44.7196	−1.53117	−0.765585	−	0.643335i	$-0.777550\pi$
$853$	−44.7196	−1.53117	−0.765585	+	0.643335i	$0.777550\pi$
$854$	0	0
$855$	−22.6137	−0.773373
$856$	0	0
$857$	17.6363	0.602444	0.301222	−	0.953554i	$-0.402606\pi$
$857$	17.6363	0.602444	0.301222	+	0.953554i	$0.402606\pi$
$858$	0	0
$859$	12.4696	0.425458	0.212729	−	0.977111i	$-0.431765\pi$
$859$	12.4696	0.425458	0.212729	+	0.977111i	$0.431765\pi$
$860$	0	0
$861$	−17.8318	−0.607705
$862$	0	0
$863$	30.2134	1.02847	0.514237	−	0.857648i	$-0.328075\pi$
$863$	30.2134	1.02847	0.514237	+	0.857648i	$0.328075\pi$
$864$	0	0
$865$	−83.2726	−2.83135
$866$	0	0
$867$	−0.134715	−0.00457515
$868$	0	0
$869$	0.0451158	0.00153045
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−10.8045	−0.365677
$874$	0	0
$875$	87.4859	2.95756
$876$	0	0
$877$	15.3668	0.518902	0.259451	−	0.965756i	$-0.416458\pi$
$877$	15.3668	0.518902	0.259451	+	0.965756i	$0.416458\pi$
$878$	0	0
$879$	3.69235	0.124540
$880$	0	0
$881$	3.35746	0.113116	0.0565578	−	0.998399i	$-0.481987\pi$
$881$	3.35746	0.113116	0.0565578	+	0.998399i	$0.481987\pi$
$882$	0	0
$883$	−29.6316	−0.997182	−0.498591	−	0.866837i	$-0.666149\pi$
$883$	−29.6316	−0.997182	−0.498591	+	0.866837i	$0.666149\pi$
$884$	0	0
$885$	23.8878	0.802981
$886$	0	0
$887$	17.4969	0.587487	0.293744	−	0.955884i	$-0.405099\pi$
$887$	17.4969	0.587487	0.293744	+	0.955884i	$0.405099\pi$
$888$	0	0
$889$	0.692355	0.0232208
$890$	0	0
$891$	−0.334898	−0.0112195
$892$	0	0
$893$	21.7851	0.729010
$894$	0	0
$895$	53.0498	1.77326
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0.939226	0.0313250
$900$	0	0
$901$	38.9665	1.29816
$902$	0	0
$903$	−14.7017	−0.489243
$904$	0	0
$905$	−37.5147	−1.24703
$906$	0	0
$907$	3.34898	0.111201	0.0556006	−	0.998453i	$-0.482293\pi$
$907$	3.34898	0.111201	0.0556006	+	0.998453i	$0.482293\pi$
$908$	0	0
$909$	−16.0833	−0.533450
$910$	0	0
$911$	−22.0094	−0.729204	−0.364602	−	0.931164i	$-0.618795\pi$
$911$	−22.0094	−0.729204	−0.364602	+	0.931164i	$0.618795\pi$
$912$	0	0
$913$	−4.44548	−0.147124
$914$	0	0
$915$	−26.4182	−0.873360
$916$	0	0
$917$	97.3847	3.21593
$918$	0	0
$919$	43.1059	1.42193	0.710966	−	0.703226i	$-0.248258\pi$
$919$	43.1059	1.42193	0.710966	+	0.703226i	$0.248258\pi$
$920$	0	0
$921$	−6.07864	−0.200298
$922$	0	0
$923$	0	0
$924$	0	0
$925$	20.2694	0.666455
$926$	0	0
$927$	4.46961	0.146801
$928$	0	0
$929$	44.3061	1.45364	0.726818	−	0.686831i	$-0.240999\pi$
$929$	44.3061	1.45364	0.726818	+	0.686831i	$0.240999\pi$
$930$	0	0
$931$	115.393	3.78184
$932$	0	0
$933$	−10.0561	−0.329221
$934$	0	0
$935$	5.27412	0.172482
$936$	0	0
$937$	2.46492	0.0805254	0.0402627	−	0.999189i	$-0.487181\pi$
$937$	2.46492	0.0805254	0.0402627	+	0.999189i	$0.487181\pi$
$938$	0	0
$939$	11.3622	0.370790
$940$	0	0
$941$	−44.3239	−1.44492	−0.722460	−	0.691413i	$-0.756988\pi$
$941$	−44.3239	−1.44492	−0.722460	+	0.691413i	$0.756988\pi$
$942$	0	0
$943$	−1.16197	−0.0378389
$944$	0	0
$945$	−19.5529	−0.636057
$946$	0	0
$947$	8.11216	0.263610	0.131805	−	0.991276i	$-0.457923\pi$
$947$	8.11216	0.263610	0.131805	+	0.991276i	$0.457923\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	19.4229	0.629831
$952$	0	0
$953$	−39.0031	−1.26344	−0.631718	−	0.775199i	$-0.717650\pi$
$953$	−39.0031	−1.26344	−0.631718	+	0.775199i	$0.717650\pi$
$954$	0	0
$955$	39.1059	1.26544
$956$	0	0
$957$	−0.0654688	−0.00211630
$958$	0	0
$959$	−77.4953	−2.50245
$960$	0	0
$961$	−7.91667	−0.255376
$962$	0	0
$963$	−3.66510	−0.118106
$964$	0	0
$965$	13.9618	0.449446
$966$	0	0
$967$	33.1807	1.06702	0.533510	−	0.845793i	$-0.320872\pi$
$967$	33.1807	1.06702	0.533510	+	0.845793i	$0.320872\pi$
$968$	0	0
$969$	−24.6043	−0.790405
$970$	0	0
$971$	−27.2368	−0.874071	−0.437036	−	0.899444i	$-0.643972\pi$
$971$	−27.2368	−0.874071	−0.437036	+	0.899444i	$0.643972\pi$
$972$	0	0
$973$	74.0771	2.37480
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−51.1331	−1.63589	−0.817947	−	0.575293i	$-0.804888\pi$
$977$	−51.1331	−1.63589	−0.817947	+	0.575293i	$0.804888\pi$
$978$	0	0
$979$	0.130938	0.00418478
$980$	0	0
$981$	9.08333	0.290008
$982$	0	0
$983$	54.8365	1.74901	0.874506	−	0.485015i	$-0.161186\pi$
$983$	54.8365	1.74901	0.874506	+	0.485015i	$0.161186\pi$
$984$	0	0
$985$	76.7243	2.44464
$986$	0	0
$987$	18.8365	0.599571
$988$	0	0
$989$	−0.958007	−0.0304628
$990$	0	0
$991$	40.3800	1.28271	0.641357	−	0.767243i	$-0.278372\pi$
$991$	40.3800	1.28271	0.641357	+	0.767243i	$0.278372\pi$
$992$	0	0
$993$	28.8045	0.914084
$994$	0	0
$995$	−10.4564	−0.331492
$996$	0	0
$997$	−14.2741	−0.452066	−0.226033	−	0.974120i	$-0.572576\pi$
$997$	−14.2741	−0.452066	−0.226033	+	0.974120i	$0.572576\pi$
$998$	0	0
$999$	−2.13941	−0.0676879

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4056.2.a.y.1.1		3
4.3	odd	2		8112.2.a.cl.1.1		3
13.4	even	6		312.2.q.e.289.3	yes	6
13.5	odd	4		4056.2.c.m.337.5		6
13.8	odd	4		4056.2.c.m.337.2		6
13.10	even	6		312.2.q.e.217.3	✓	6
13.12	even	2		4056.2.a.z.1.3		3
39.17	odd	6		936.2.t.h.289.1		6
39.23	odd	6		936.2.t.h.217.1		6
52.23	odd	6		624.2.q.j.529.3		6
52.43	odd	6		624.2.q.j.289.3		6
52.51	odd	2		8112.2.a.ck.1.3		3
156.23	even	6		1872.2.t.u.1153.1		6
156.95	even	6		1872.2.t.u.289.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.2.q.e.217.3	✓	6	13.10	even	6
312.2.q.e.289.3	yes	6	13.4	even	6
624.2.q.j.289.3		6	52.43	odd	6
624.2.q.j.529.3		6	52.23	odd	6
936.2.t.h.217.1		6	39.23	odd	6
936.2.t.h.289.1		6	39.17	odd	6
1872.2.t.u.289.1		6	156.95	even	6
1872.2.t.u.1153.1		6	156.23	even	6
4056.2.a.y.1.1		3	1.1	even	1	trivial
4056.2.a.z.1.3		3	13.12	even	2
4056.2.c.m.337.2		6	13.8	odd	4
4056.2.c.m.337.5		6	13.5	odd	4
8112.2.a.ck.1.3		3	52.51	odd	2
8112.2.a.cl.1.1		3	4.3	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:	\( N \)	\(=\)	\( 4056 = 2^{3} \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4056.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -3.80451 q^{5} -5.13941 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -3.80451 q^{5} -5.13941 q^{7} +1.00000 q^{9} -0.334898 q^{11} +3.80451 q^{15} +4.13941 q^{17} +5.94392 q^{19} +5.13941 q^{21} +0.334898 q^{23} +9.47431 q^{25} -1.00000 q^{27} -0.195488 q^{29} -4.80451 q^{31} +0.334898 q^{33} +19.5529 q^{35} +2.13941 q^{37} -3.46961 q^{41} -2.86059 q^{43} -3.80451 q^{45} +3.66510 q^{47} +19.4135 q^{49} -4.13941 q^{51} +9.41353 q^{53} +1.27412 q^{55} -5.94392 q^{57} +6.27882 q^{59} -6.94392 q^{61} -5.13941 q^{63} -13.1394 q^{67} -0.334898 q^{69} -4.33490 q^{71} +10.6090 q^{73} -9.47431 q^{75} +1.72118 q^{77} -0.134715 q^{79} +1.00000 q^{81} +13.2741 q^{83} -15.7484 q^{85} +0.195488 q^{87} -0.390977 q^{89} +4.80451 q^{93} -22.6137 q^{95} -10.8045 q^{97} -0.334898 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} - 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 - 3 * q^7 + 3 * q^9	\( 3 q - 3 q^{3} - 3 q^{7} + 3 q^{9} - 6 q^{19} + 3 q^{21} + 15 q^{25} - 3 q^{27} - 12 q^{29} - 3 q^{31} + 12 q^{35} - 6 q^{37} - 21 q^{43} + 12 q^{47} + 24 q^{49} - 6 q^{53} - 18 q^{55} + 6 q^{57} - 6 q^{59} + 3 q^{61} - 3 q^{63} - 27 q^{67} - 12 q^{71} + 9 q^{73} - 15 q^{75} + 30 q^{77} + 9 q^{79} + 3 q^{81} + 18 q^{83} - 12 q^{85} + 12 q^{87} - 24 q^{89} + 3 q^{93} - 42 q^{95} - 21 q^{97}+O(q^{100}) \) 3 * q - 3 * q^3 - 3 * q^7 + 3 * q^9 - 6 * q^19 + 3 * q^21 + 15 * q^25 - 3 * q^27 - 12 * q^29 - 3 * q^31 + 12 * q^35 - 6 * q^37 - 21 * q^43 + 12 * q^47 + 24 * q^49 - 6 * q^53 - 18 * q^55 + 6 * q^57 - 6 * q^59 + 3 * q^61 - 3 * q^63 - 27 * q^67 - 12 * q^71 + 9 * q^73 - 15 * q^75 + 30 * q^77 + 9 * q^79 + 3 * q^81 + 18 * q^83 - 12 * q^85 + 12 * q^87 - 24 * q^89 + 3 * q^93 - 42 * q^95 - 21 * q^97

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