LMFDB - Embedded newform 4056.2.c.k.337.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4056,2,Mod(337,4056)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4056.337"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4056, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4056 = 2^{3} \cdot 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4056.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,-4,0,0,0,0,0,4,0,0,0,0,0,0,0,8,0,0,0,0,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)

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

Self dual:	no
Analytic conductor:	$32.3873230598$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 312)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.4
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	4056.337
Dual form			4056.2.c.k.337.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.73205i q^{5} -0.732051i q^{7} +1.00000 q^{9} -0.732051i q^{11} -1.73205i q^{15} +3.73205 q^{17} +1.26795i q^{19} +0.732051i q^{21} +2.19615 q^{23} +2.00000 q^{25} -1.00000 q^{27} -5.92820 q^{29} -5.46410i q^{31} +0.732051i q^{33} +1.26795 q^{35} +8.46410i q^{37} -5.00000i q^{41} -1.80385 q^{43} +1.73205i q^{45} -6.73205i q^{47} +6.46410 q^{49} -3.73205 q^{51} +5.92820 q^{53} +1.26795 q^{55} -1.26795i q^{57} -9.73205 q^{61} -0.732051i q^{63} -5.26795i q^{67} -2.19615 q^{69} +8.19615i q^{71} -1.19615i q^{73} -2.00000 q^{75} -0.535898 q^{77} +12.3923 q^{79} +1.00000 q^{81} -15.6603i q^{83} +6.46410i q^{85} +5.92820 q^{87} -2.53590i q^{89} +5.46410i q^{93} -2.19615 q^{95} +10.0000i q^{97} -0.732051i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} + 4 q^{9} + 8 q^{17} - 12 q^{23} + 8 q^{25} - 4 q^{27} + 4 q^{29} + 12 q^{35} - 28 q^{43} + 12 q^{49} - 8 q^{51} - 4 q^{53} + 12 q^{55} - 32 q^{61} + 12 q^{69} - 8 q^{75} - 16 q^{77} + 8 q^{79}+ \cdots + 12 q^{95}+O(q^{100}) $ 4 * q - 4 * q^3 + 4 * q^9 + 8 * q^17 - 12 * q^23 + 8 * q^25 - 4 * q^27 + 4 * q^29 + 12 * q^35 - 28 * q^43 + 12 * q^49 - 8 * q^51 - 4 * q^53 + 12 * q^55 - 32 * q^61 + 12 * q^69 - 8 * q^75 - 16 * q^77 + 8 * q^79 + 4 * q^81 - 4 * q^87 + 12 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4056\mathbb{Z}\right)^\times$.

$n$	$1015$	$2029$	$2705$	$3889$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	1.73205i	0.774597i	0.921954	+	0.387298i	$0.126592\pi$
$5$	1.73205i	0.774597i	−0.921954	+	0.387298i	$0.873408\pi$
$6$	0	0
$7$	− 0.732051i	− 0.276689i	−0.990384	−	0.138345i	$-0.955822\pi$
$7$	− 0.732051i	− 0.276689i	0.990384	−	0.138345i	$-0.0441781\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	− 0.732051i	− 0.220722i	−0.993892	−	0.110361i	$-0.964799\pi$
$11$	− 0.732051i	− 0.220722i	0.993892	−	0.110361i	$-0.0352006\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	− 1.73205i	− 0.447214i
$16$	0	0
$17$	3.73205	0.905155	0.452578	−	0.891725i	$-0.350505\pi$
$17$	3.73205	0.905155	0.452578	+	0.891725i	$0.350505\pi$
$18$	0	0
$19$	1.26795i	0.290887i	0.989367	+	0.145444i	$0.0464610\pi$
$19$	1.26795i	0.290887i	−0.989367	+	0.145444i	$0.953539\pi$
$20$	0	0
$21$	0.732051i	0.159747i
$22$	0	0
$23$	2.19615	0.457929	0.228965	−	0.973435i	$-0.426466\pi$
$23$	2.19615	0.457929	0.228965	+	0.973435i	$0.426466\pi$
$24$	0	0
$25$	2.00000	0.400000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−5.92820	−1.10084	−0.550420	−	0.834888i	$-0.685532\pi$
$29$	−5.92820	−1.10084	−0.550420	+	0.834888i	$0.685532\pi$
$30$	0	0
$31$	− 5.46410i	− 0.981382i	−0.871334	−	0.490691i	$-0.836744\pi$
$31$	− 5.46410i	− 0.981382i	0.871334	−	0.490691i	$-0.163256\pi$
$32$	0	0
$33$	0.732051i	0.127434i
$34$	0	0
$35$	1.26795	0.214323
$36$	0	0
$37$	8.46410i	1.39149i	0.718289	+	0.695745i	$0.244926\pi$
$37$	8.46410i	1.39149i	−0.718289	+	0.695745i	$0.755074\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 5.00000i	− 0.780869i	−0.920631	−	0.390434i	$-0.872325\pi$
$41$	− 5.00000i	− 0.780869i	0.920631	−	0.390434i	$-0.127675\pi$
$42$	0	0
$43$	−1.80385	−0.275084	−0.137542	−	0.990496i	$-0.543920\pi$
$43$	−1.80385	−0.275084	−0.137542	+	0.990496i	$0.543920\pi$
$44$	0	0
$45$	1.73205i	0.258199i
$46$	0	0
$47$	− 6.73205i	− 0.981971i	−0.871168	−	0.490985i	$-0.836637\pi$
$47$	− 6.73205i	− 0.981971i	0.871168	−	0.490985i	$-0.163363\pi$
$48$	0	0
$49$	6.46410	0.923443
$50$	0	0
$51$	−3.73205	−0.522592
$52$	0	0
$53$	5.92820	0.814301	0.407151	−	0.913361i	$-0.366522\pi$
$53$	5.92820	0.814301	0.407151	+	0.913361i	$0.366522\pi$
$54$	0	0
$55$	1.26795	0.170970
$56$	0	0
$57$	− 1.26795i	− 0.167944i
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	−9.73205	−1.24606	−0.623031	−	0.782197i	$-0.714099\pi$
$61$	−9.73205	−1.24606	−0.623031	+	0.782197i	$0.714099\pi$
$62$	0	0
$63$	− 0.732051i	− 0.0922297i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 5.26795i	− 0.643582i	−0.946811	−	0.321791i	$-0.895715\pi$
$67$	− 5.26795i	− 0.643582i	0.946811	−	0.321791i	$-0.104285\pi$
$68$	0	0
$69$	−2.19615	−0.264386
$70$	0	0
$71$	8.19615i	0.972704i	0.873763	+	0.486352i	$0.161673\pi$
$71$	8.19615i	0.972704i	−0.873763	+	0.486352i	$0.838327\pi$
$72$	0	0
$73$	− 1.19615i	− 0.139999i	−0.997547	−	0.0699995i	$-0.977700\pi$
$73$	− 1.19615i	− 0.139999i	0.997547	−	0.0699995i	$-0.0222998\pi$
$74$	0	0
$75$	−2.00000	−0.230940
$76$	0	0
$77$	−0.535898	−0.0610713
$78$	0	0
$79$	12.3923	1.39424	0.697122	−	0.716953i	$-0.254464\pi$
$79$	12.3923	1.39424	0.697122	+	0.716953i	$0.254464\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 15.6603i	− 1.71894i	−0.511189	−	0.859468i	$-0.670795\pi$
$83$	− 15.6603i	− 1.71894i	0.511189	−	0.859468i	$-0.329205\pi$
$84$	0	0
$85$	6.46410i	0.701130i
$86$	0	0
$87$	5.92820	0.635570
$88$	0	0
$89$	− 2.53590i	− 0.268805i	−0.990927	−	0.134402i	$-0.957089\pi$
$89$	− 2.53590i	− 0.268805i	0.990927	−	0.134402i	$-0.0429115\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	5.46410i	0.566601i
$94$	0	0
$95$	−2.19615	−0.225320
$96$	0	0
$97$	10.0000i	1.01535i	0.861550	+	0.507673i	$0.169494\pi$
$97$	10.0000i	1.01535i	−0.861550	+	0.507673i	$0.830506\pi$
$98$	0	0
$99$	− 0.732051i	− 0.0735739i
$100$	0	0
$101$	14.4641	1.43923	0.719616	−	0.694372i	$-0.244318\pi$
$101$	14.4641	1.43923	0.719616	+	0.694372i	$0.244318\pi$
$102$	0	0
$103$	2.73205	0.269197	0.134598	−	0.990900i	$-0.457026\pi$
$103$	2.73205	0.269197	0.134598	+	0.990900i	$0.457026\pi$
$104$	0	0
$105$	−1.26795	−0.123739
$106$	0	0
$107$	−15.1244	−1.46213	−0.731063	−	0.682310i	$-0.760976\pi$
$107$	−15.1244	−1.46213	−0.731063	+	0.682310i	$0.760976\pi$
$108$	0	0
$109$	0.392305i	0.0375760i	0.999823	+	0.0187880i	$0.00598076\pi$
$109$	0.392305i	0.0375760i	−0.999823	+	0.0187880i	$0.994019\pi$
$110$	0	0
$111$	− 8.46410i	− 0.803377i
$112$	0	0
$113$	4.66025	0.438400	0.219200	−	0.975680i	$-0.429655\pi$
$113$	4.66025	0.438400	0.219200	+	0.975680i	$0.429655\pi$
$114$	0	0
$115$	3.80385i	0.354711i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 2.73205i	− 0.250447i
$120$	0	0
$121$	10.4641	0.951282
$122$	0	0
$123$	5.00000i	0.450835i
$124$	0	0
$125$	12.1244i	1.08444i
$126$	0	0
$127$	−4.00000	−0.354943	−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000	−0.354943	−0.177471	+	0.984126i	$0.556792\pi$
$128$	0	0
$129$	1.80385	0.158820
$130$	0	0
$131$	19.3205	1.68804	0.844020	−	0.536311i	$-0.180183\pi$
$131$	19.3205	1.68804	0.844020	+	0.536311i	$0.180183\pi$
$132$	0	0
$133$	0.928203	0.0804854
$134$	0	0
$135$	− 1.73205i	− 0.149071i
$136$	0	0
$137$	12.4641i	1.06488i	0.846468	+	0.532440i	$0.178725\pi$
$137$	12.4641i	1.06488i	−0.846468	+	0.532440i	$0.821275\pi$
$138$	0	0
$139$	20.0000	1.69638	0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000	1.69638	0.848189	+	0.529694i	$0.177693\pi$
$140$	0	0
$141$	6.73205i	0.566941i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	− 10.2679i	− 0.852707i
$146$	0	0
$147$	−6.46410	−0.533150
$148$	0	0
$149$	− 9.58846i	− 0.785517i	−0.919642	−	0.392759i	$-0.871521\pi$
$149$	− 9.58846i	− 0.785517i	0.919642	−	0.392759i	$-0.128479\pi$
$150$	0	0
$151$	20.7321i	1.68715i	0.537011	+	0.843575i	$0.319553\pi$
$151$	20.7321i	1.68715i	−0.537011	+	0.843575i	$0.680447\pi$
$152$	0	0
$153$	3.73205	0.301718
$154$	0	0
$155$	9.46410	0.760175
$156$	0	0
$157$	15.0526	1.20132	0.600662	−	0.799503i	$-0.294904\pi$
$157$	15.0526	1.20132	0.600662	+	0.799503i	$0.294904\pi$
$158$	0	0
$159$	−5.92820	−0.470137
$160$	0	0
$161$	− 1.60770i	− 0.126704i
$162$	0	0
$163$	6.53590i	0.511931i	0.966686	+	0.255966i	$0.0823934\pi$
$163$	6.53590i	0.511931i	−0.966686	+	0.255966i	$0.917607\pi$
$164$	0	0
$165$	−1.26795	−0.0987097
$166$	0	0
$167$	− 5.46410i	− 0.422825i	−0.977397	−	0.211412i	$-0.932194\pi$
$167$	− 5.46410i	− 0.422825i	0.977397	−	0.211412i	$-0.0678063\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	1.26795i	0.0969625i
$172$	0	0
$173$	−10.5359	−0.801030	−0.400515	−	0.916290i	$-0.631169\pi$
$173$	−10.5359	−0.801030	−0.400515	+	0.916290i	$0.631169\pi$
$174$	0	0
$175$	− 1.46410i	− 0.110676i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−17.2679	−1.29067	−0.645334	−	0.763901i	$-0.723282\pi$
$179$	−17.2679	−1.29067	−0.645334	+	0.763901i	$0.723282\pi$
$180$	0	0
$181$	−8.80385	−0.654385	−0.327192	−	0.944958i	$-0.606103\pi$
$181$	−8.80385	−0.654385	−0.327192	+	0.944958i	$0.606103\pi$
$182$	0	0
$183$	9.73205	0.719414
$184$	0	0
$185$	−14.6603	−1.07784
$186$	0	0
$187$	− 2.73205i	− 0.199787i
$188$	0	0
$189$	0.732051i	0.0532489i
$190$	0	0
$191$	−1.07180	−0.0775525	−0.0387762	−	0.999248i	$-0.512346\pi$
$191$	−1.07180	−0.0775525	−0.0387762	+	0.999248i	$0.512346\pi$
$192$	0	0
$193$	8.26795i	0.595140i	0.954700	+	0.297570i	$0.0961762\pi$
$193$	8.26795i	0.595140i	−0.954700	+	0.297570i	$0.903824\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	14.9282i	1.06359i	0.846873	+	0.531795i	$0.178482\pi$
$197$	14.9282i	1.06359i	−0.846873	+	0.531795i	$0.821518\pi$
$198$	0	0
$199$	17.6603	1.25190	0.625951	−	0.779862i	$-0.284711\pi$
$199$	17.6603	1.25190	0.625951	+	0.779862i	$0.284711\pi$
$200$	0	0
$201$	5.26795i	0.371572i
$202$	0	0
$203$	4.33975i	0.304590i
$204$	0	0
$205$	8.66025	0.604858
$206$	0	0
$207$	2.19615	0.152643
$208$	0	0
$209$	0.928203	0.0642052
$210$	0	0
$211$	23.3205	1.60545	0.802725	−	0.596349i	$-0.203383\pi$
$211$	23.3205	1.60545	0.802725	+	0.596349i	$0.203383\pi$
$212$	0	0
$213$	− 8.19615i	− 0.561591i
$214$	0	0
$215$	− 3.12436i	− 0.213079i
$216$	0	0
$217$	−4.00000	−0.271538
$218$	0	0
$219$	1.19615i	0.0808285i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 24.7846i	− 1.65970i	−0.557986	−	0.829850i	$-0.688426\pi$
$223$	− 24.7846i	− 1.65970i	0.557986	−	0.829850i	$-0.311574\pi$
$224$	0	0
$225$	2.00000	0.133333
$226$	0	0
$227$	16.0526i	1.06545i	0.846290	+	0.532723i	$0.178831\pi$
$227$	16.0526i	1.06545i	−0.846290	+	0.532723i	$0.821169\pi$
$228$	0	0
$229$	27.8564i	1.84080i	0.390974	+	0.920402i	$0.372138\pi$
$229$	27.8564i	1.84080i	−0.390974	+	0.920402i	$0.627862\pi$
$230$	0	0
$231$	0.535898	0.0352595
$232$	0	0
$233$	2.00000	0.131024	0.0655122	−	0.997852i	$-0.479132\pi$
$233$	2.00000	0.131024	0.0655122	+	0.997852i	$0.479132\pi$
$234$	0	0
$235$	11.6603	0.760631
$236$	0	0
$237$	−12.3923	−0.804967
$238$	0	0
$239$	− 26.7321i	− 1.72915i	−0.502502	−	0.864576i	$-0.667587\pi$
$239$	− 26.7321i	− 1.72915i	0.502502	−	0.864576i	$-0.332413\pi$
$240$	0	0
$241$	22.6603i	1.45968i	0.683621	+	0.729838i	$0.260404\pi$
$241$	22.6603i	1.45968i	−0.683621	+	0.729838i	$0.739596\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	11.1962i	0.715296i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	15.6603i	0.992428i
$250$	0	0
$251$	9.46410	0.597369	0.298684	−	0.954352i	$-0.403452\pi$
$251$	9.46410	0.597369	0.298684	+	0.954352i	$0.403452\pi$
$252$	0	0
$253$	− 1.60770i	− 0.101075i
$254$	0	0
$255$	− 6.46410i	− 0.404798i
$256$	0	0
$257$	17.7321	1.10609	0.553047	−	0.833150i	$-0.313465\pi$
$257$	17.7321	1.10609	0.553047	+	0.833150i	$0.313465\pi$
$258$	0	0
$259$	6.19615	0.385010
$260$	0	0
$261$	−5.92820	−0.366947
$262$	0	0
$263$	12.3397	0.760901	0.380451	−	0.924801i	$-0.375769\pi$
$263$	12.3397	0.760901	0.380451	+	0.924801i	$0.375769\pi$
$264$	0	0
$265$	10.2679i	0.630755i
$266$	0	0
$267$	2.53590i	0.155194i
$268$	0	0
$269$	9.46410	0.577036	0.288518	−	0.957474i	$-0.406837\pi$
$269$	9.46410	0.577036	0.288518	+	0.957474i	$0.406837\pi$
$270$	0	0
$271$	16.0000i	0.971931i	0.873978	+	0.485965i	$0.161532\pi$
$271$	16.0000i	0.971931i	−0.873978	+	0.485965i	$0.838468\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 1.46410i	− 0.0882886i
$276$	0	0
$277$	8.12436	0.488145	0.244073	−	0.969757i	$-0.421516\pi$
$277$	8.12436	0.488145	0.244073	+	0.969757i	$0.421516\pi$
$278$	0	0
$279$	− 5.46410i	− 0.327127i
$280$	0	0
$281$	− 24.7128i	− 1.47424i	−0.675760	−	0.737121i	$-0.736185\pi$
$281$	− 24.7128i	− 1.47424i	0.675760	−	0.737121i	$-0.263815\pi$
$282$	0	0
$283$	21.5167	1.27903	0.639516	−	0.768777i	$-0.279135\pi$
$283$	21.5167	1.27903	0.639516	+	0.768777i	$0.279135\pi$
$284$	0	0
$285$	2.19615	0.130089
$286$	0	0
$287$	−3.66025	−0.216058
$288$	0	0
$289$	−3.07180	−0.180694
$290$	0	0
$291$	− 10.0000i	− 0.586210i
$292$	0	0
$293$	2.12436i	0.124106i	0.998073	+	0.0620531i	$0.0197648\pi$
$293$	2.12436i	0.124106i	−0.998073	+	0.0620531i	$0.980235\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0.732051i	0.0424779i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	1.32051i	0.0761128i
$302$	0	0
$303$	−14.4641	−0.830941
$304$	0	0
$305$	− 16.8564i	− 0.965195i
$306$	0	0
$307$	− 10.7321i	− 0.612510i	−0.951949	−	0.306255i	$-0.900924\pi$
$307$	− 10.7321i	− 0.612510i	0.951949	−	0.306255i	$-0.0990761\pi$
$308$	0	0
$309$	−2.73205	−0.155421
$310$	0	0
$311$	4.33975	0.246084	0.123042	−	0.992401i	$-0.460735\pi$
$311$	4.33975	0.246084	0.123042	+	0.992401i	$0.460735\pi$
$312$	0	0
$313$	3.32051	0.187686	0.0938431	−	0.995587i	$-0.470085\pi$
$313$	3.32051	0.187686	0.0938431	+	0.995587i	$0.470085\pi$
$314$	0	0
$315$	1.26795	0.0714408
$316$	0	0
$317$	− 11.1962i	− 0.628839i	−0.949284	−	0.314419i	$-0.898190\pi$
$317$	− 11.1962i	− 0.628839i	0.949284	−	0.314419i	$-0.101810\pi$
$318$	0	0
$319$	4.33975i	0.242979i
$320$	0	0
$321$	15.1244	0.844159
$322$	0	0
$323$	4.73205i	0.263298i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 0.392305i	− 0.0216945i
$328$	0	0
$329$	−4.92820	−0.271701
$330$	0	0
$331$	− 12.0000i	− 0.659580i	−0.944054	−	0.329790i	$-0.893022\pi$
$331$	− 12.0000i	− 0.659580i	0.944054	−	0.329790i	$-0.106978\pi$
$332$	0	0
$333$	8.46410i	0.463830i
$334$	0	0
$335$	9.12436	0.498517
$336$	0	0
$337$	17.9282	0.976611	0.488306	−	0.872673i	$-0.337615\pi$
$337$	17.9282	0.976611	0.488306	+	0.872673i	$0.337615\pi$
$338$	0	0
$339$	−4.66025	−0.253110
$340$	0	0
$341$	−4.00000	−0.216612
$342$	0	0
$343$	− 9.85641i	− 0.532196i
$344$	0	0
$345$	− 3.80385i	− 0.204792i
$346$	0	0
$347$	27.5167	1.47717	0.738586	−	0.674159i	$-0.235494\pi$
$347$	27.5167	1.47717	0.738586	+	0.674159i	$0.235494\pi$
$348$	0	0
$349$	− 8.39230i	− 0.449230i	−0.974448	−	0.224615i	$-0.927888\pi$
$349$	− 8.39230i	− 0.449230i	0.974448	−	0.224615i	$-0.0721124\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	24.4641i	1.30209i	0.759038	+	0.651046i	$0.225670\pi$
$353$	24.4641i	1.30209i	−0.759038	+	0.651046i	$0.774330\pi$
$354$	0	0
$355$	−14.1962	−0.753454
$356$	0	0
$357$	2.73205i	0.144595i
$358$	0	0
$359$	19.5167i	1.03005i	0.857175	+	0.515025i	$0.172217\pi$
$359$	19.5167i	1.03005i	−0.857175	+	0.515025i	$0.827783\pi$
$360$	0	0
$361$	17.3923	0.915384
$362$	0	0
$363$	−10.4641	−0.549223
$364$	0	0
$365$	2.07180	0.108443
$366$	0	0
$367$	23.1244	1.20708	0.603541	−	0.797332i	$-0.293756\pi$
$367$	23.1244	1.20708	0.603541	+	0.797332i	$0.293756\pi$
$368$	0	0
$369$	− 5.00000i	− 0.260290i
$370$	0	0
$371$	− 4.33975i	− 0.225308i
$372$	0	0
$373$	−15.3397	−0.794262	−0.397131	−	0.917762i	$-0.629994\pi$
$373$	−15.3397	−0.794262	−0.397131	+	0.917762i	$0.629994\pi$
$374$	0	0
$375$	− 12.1244i	− 0.626099i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	− 36.3923i	− 1.86935i	−0.355507	−	0.934674i	$-0.615692\pi$
$379$	− 36.3923i	− 1.86935i	0.355507	−	0.934674i	$-0.384308\pi$
$380$	0	0
$381$	4.00000	0.204926
$382$	0	0
$383$	35.3205i	1.80479i	0.430906	+	0.902397i	$0.358194\pi$
$383$	35.3205i	1.80479i	−0.430906	+	0.902397i	$0.641806\pi$
$384$	0	0
$385$	− 0.928203i	− 0.0473056i
$386$	0	0
$387$	−1.80385	−0.0916947
$388$	0	0
$389$	−24.4641	−1.24038	−0.620190	−	0.784452i	$-0.712944\pi$
$389$	−24.4641	−1.24038	−0.620190	+	0.784452i	$0.712944\pi$
$390$	0	0
$391$	8.19615	0.414497
$392$	0	0
$393$	−19.3205	−0.974591
$394$	0	0
$395$	21.4641i	1.07998i
$396$	0	0
$397$	10.5359i	0.528782i	0.964416	+	0.264391i	$0.0851709\pi$
$397$	10.5359i	0.528782i	−0.964416	+	0.264391i	$0.914829\pi$
$398$	0	0
$399$	−0.928203	−0.0464683
$400$	0	0
$401$	15.5359i	0.775826i	0.921696	+	0.387913i	$0.126804\pi$
$401$	15.5359i	0.775826i	−0.921696	+	0.387913i	$0.873196\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	1.73205i	0.0860663i
$406$	0	0
$407$	6.19615	0.307132
$408$	0	0
$409$	− 22.8038i	− 1.12758i	−0.825919	−	0.563789i	$-0.809343\pi$
$409$	− 22.8038i	− 1.12758i	0.825919	−	0.563789i	$-0.190657\pi$
$410$	0	0
$411$	− 12.4641i	− 0.614809i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	27.1244	1.33148
$416$	0	0
$417$	−20.0000	−0.979404
$418$	0	0
$419$	−32.3923	−1.58247	−0.791234	−	0.611514i	$-0.790561\pi$
$419$	−32.3923	−1.58247	−0.791234	+	0.611514i	$0.790561\pi$
$420$	0	0
$421$	5.00000i	0.243685i	0.992549	+	0.121843i	$0.0388803\pi$
$421$	5.00000i	0.243685i	−0.992549	+	0.121843i	$0.961120\pi$
$422$	0	0
$423$	− 6.73205i	− 0.327324i
$424$	0	0
$425$	7.46410	0.362062
$426$	0	0
$427$	7.12436i	0.344772i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 10.0526i	− 0.484215i	−0.970249	−	0.242107i	$-0.922161\pi$
$431$	− 10.0526i	− 0.484215i	0.970249	−	0.242107i	$-0.0778386\pi$
$432$	0	0
$433$	30.8564	1.48286	0.741432	−	0.671028i	$-0.234147\pi$
$433$	30.8564	1.48286	0.741432	+	0.671028i	$0.234147\pi$
$434$	0	0
$435$	10.2679i	0.492310i
$436$	0	0
$437$	2.78461i	0.133206i
$438$	0	0
$439$	−20.1962	−0.963910	−0.481955	−	0.876196i	$-0.660073\pi$
$439$	−20.1962	−0.963910	−0.481955	+	0.876196i	$0.660073\pi$
$440$	0	0
$441$	6.46410	0.307814
$442$	0	0
$443$	−15.3205	−0.727899	−0.363950	−	0.931419i	$-0.618572\pi$
$443$	−15.3205	−0.727899	−0.363950	+	0.931419i	$0.618572\pi$
$444$	0	0
$445$	4.39230	0.208215
$446$	0	0
$447$	9.58846i	0.453518i
$448$	0	0
$449$	− 31.3205i	− 1.47811i	−0.673647	−	0.739053i	$-0.735273\pi$
$449$	− 31.3205i	− 1.47811i	0.673647	−	0.739053i	$-0.264727\pi$
$450$	0	0
$451$	−3.66025	−0.172355
$452$	0	0
$453$	− 20.7321i	− 0.974077i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	19.4449i	0.909592i	0.890596	+	0.454796i	$0.150288\pi$
$457$	19.4449i	0.909592i	−0.890596	+	0.454796i	$0.849712\pi$
$458$	0	0
$459$	−3.73205	−0.174197
$460$	0	0
$461$	2.41154i	0.112317i	0.998422	+	0.0561584i	$0.0178852\pi$
$461$	2.41154i	0.112317i	−0.998422	+	0.0561584i	$0.982115\pi$
$462$	0	0
$463$	− 23.6603i	− 1.09959i	−0.835301	−	0.549793i	$-0.814707\pi$
$463$	− 23.6603i	− 1.09959i	0.835301	−	0.549793i	$-0.185293\pi$
$464$	0	0
$465$	−9.46410	−0.438887
$466$	0	0
$467$	5.66025	0.261925	0.130963	−	0.991387i	$-0.458193\pi$
$467$	5.66025	0.261925	0.130963	+	0.991387i	$0.458193\pi$
$468$	0	0
$469$	−3.85641	−0.178072
$470$	0	0
$471$	−15.0526	−0.693585
$472$	0	0
$473$	1.32051i	0.0607170i
$474$	0	0
$475$	2.53590i	0.116355i
$476$	0	0
$477$	5.92820	0.271434
$478$	0	0
$479$	5.46410i	0.249661i	0.992178	+	0.124831i	$0.0398387\pi$
$479$	5.46410i	0.249661i	−0.992178	+	0.124831i	$0.960161\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	1.60770i	0.0731527i
$484$	0	0
$485$	−17.3205	−0.786484
$486$	0	0
$487$	− 30.5885i	− 1.38610i	−0.720892	−	0.693048i	$-0.756268\pi$
$487$	− 30.5885i	− 1.38610i	0.720892	−	0.693048i	$-0.243732\pi$
$488$	0	0
$489$	− 6.53590i	− 0.295564i
$490$	0	0
$491$	−39.9090	−1.80107	−0.900533	−	0.434787i	$-0.856824\pi$
$491$	−39.9090	−1.80107	−0.900533	+	0.434787i	$0.856824\pi$
$492$	0	0
$493$	−22.1244	−0.996431
$494$	0	0
$495$	1.26795	0.0569901
$496$	0	0
$497$	6.00000	0.269137
$498$	0	0
$499$	13.8564i	0.620298i	0.950688	+	0.310149i	$0.100379\pi$
$499$	13.8564i	0.620298i	−0.950688	+	0.310149i	$0.899621\pi$
$500$	0	0
$501$	5.46410i	0.244118i
$502$	0	0
$503$	12.7321	0.567694	0.283847	−	0.958870i	$-0.408389\pi$
$503$	12.7321	0.567694	0.283847	+	0.958870i	$0.408389\pi$
$504$	0	0
$505$	25.0526i	1.11482i
$506$	0	0
$507$	0	0
$508$	0	0
$509$	1.58846i	0.0704071i	0.999380	+	0.0352036i	$0.0112080\pi$
$509$	1.58846i	0.0704071i	−0.999380	+	0.0352036i	$0.988792\pi$
$510$	0	0
$511$	−0.875644	−0.0387362
$512$	0	0
$513$	− 1.26795i	− 0.0559813i
$514$	0	0
$515$	4.73205i	0.208519i
$516$	0	0
$517$	−4.92820	−0.216742
$518$	0	0
$519$	10.5359	0.462475
$520$	0	0
$521$	−31.9808	−1.40110	−0.700551	−	0.713602i	$-0.747063\pi$
$521$	−31.9808	−1.40110	−0.700551	+	0.713602i	$0.747063\pi$
$522$	0	0
$523$	22.5885	0.987724	0.493862	−	0.869540i	$-0.335585\pi$
$523$	22.5885	0.987724	0.493862	+	0.869540i	$0.335585\pi$
$524$	0	0
$525$	1.46410i	0.0638986i
$526$	0	0
$527$	− 20.3923i	− 0.888303i
$528$	0	0
$529$	−18.1769	−0.790301
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	− 26.1962i	− 1.13256i
$536$	0	0
$537$	17.2679	0.745167
$538$	0	0
$539$	− 4.73205i	− 0.203824i
$540$	0	0
$541$	3.39230i	0.145847i	0.997338	+	0.0729233i	$0.0232328\pi$
$541$	3.39230i	0.145847i	−0.997338	+	0.0729233i	$0.976767\pi$
$542$	0	0
$543$	8.80385	0.377809
$544$	0	0
$545$	−0.679492	−0.0291062
$546$	0	0
$547$	36.0526	1.54150	0.770748	−	0.637140i	$-0.219883\pi$
$547$	36.0526	1.54150	0.770748	+	0.637140i	$0.219883\pi$
$548$	0	0
$549$	−9.73205	−0.415354
$550$	0	0
$551$	− 7.51666i	− 0.320221i
$552$	0	0
$553$	− 9.07180i	− 0.385772i
$554$	0	0
$555$	14.6603	0.622293
$556$	0	0
$557$	− 31.5885i	− 1.33845i	−0.743062	−	0.669223i	$-0.766627\pi$
$557$	− 31.5885i	− 1.33845i	0.743062	−	0.669223i	$-0.233373\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	2.73205i	0.115347i
$562$	0	0
$563$	5.07180	0.213751	0.106875	−	0.994272i	$-0.465915\pi$
$563$	5.07180	0.213751	0.106875	+	0.994272i	$0.465915\pi$
$564$	0	0
$565$	8.07180i	0.339583i
$566$	0	0
$567$	− 0.732051i	− 0.0307432i
$568$	0	0
$569$	16.0000	0.670755	0.335377	−	0.942084i	$-0.391136\pi$
$569$	16.0000	0.670755	0.335377	+	0.942084i	$0.391136\pi$
$570$	0	0
$571$	−3.66025	−0.153177	−0.0765884	−	0.997063i	$-0.524403\pi$
$571$	−3.66025	−0.153177	−0.0765884	+	0.997063i	$0.524403\pi$
$572$	0	0
$573$	1.07180	0.0447750
$574$	0	0
$575$	4.39230	0.183172
$576$	0	0
$577$	− 5.58846i	− 0.232651i	−0.993211	−	0.116325i	$-0.962889\pi$
$577$	− 5.58846i	− 0.232651i	0.993211	−	0.116325i	$-0.0371115\pi$
$578$	0	0
$579$	− 8.26795i	− 0.343604i
$580$	0	0
$581$	−11.4641	−0.475611
$582$	0	0
$583$	− 4.33975i	− 0.179734i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	2.14359i	0.0884756i	0.999021	+	0.0442378i	$0.0140859\pi$
$587$	2.14359i	0.0884756i	−0.999021	+	0.0442378i	$0.985914\pi$
$588$	0	0
$589$	6.92820	0.285472
$590$	0	0
$591$	− 14.9282i	− 0.614064i
$592$	0	0
$593$	− 34.1769i	− 1.40348i	−0.712434	−	0.701739i	$-0.752407\pi$
$593$	− 34.1769i	− 1.40348i	0.712434	−	0.701739i	$-0.247593\pi$
$594$	0	0
$595$	4.73205	0.193995
$596$	0	0
$597$	−17.6603	−0.722786
$598$	0	0
$599$	−33.4641	−1.36731	−0.683653	−	0.729807i	$-0.739610\pi$
$599$	−33.4641	−1.36731	−0.683653	+	0.729807i	$0.739610\pi$
$600$	0	0
$601$	27.6410	1.12750	0.563750	−	0.825945i	$-0.309358\pi$
$601$	27.6410	1.12750	0.563750	+	0.825945i	$0.309358\pi$
$602$	0	0
$603$	− 5.26795i	− 0.214527i
$604$	0	0
$605$	18.1244i	0.736860i
$606$	0	0
$607$	−40.7846	−1.65540	−0.827698	−	0.561174i	$-0.810350\pi$
$607$	−40.7846	−1.65540	−0.827698	+	0.561174i	$0.810350\pi$
$608$	0	0
$609$	− 4.33975i	− 0.175855i
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 1.67949i	− 0.0678340i	−0.999425	−	0.0339170i	$-0.989202\pi$
$613$	− 1.67949i	− 0.0678340i	0.999425	−	0.0339170i	$-0.0107982\pi$
$614$	0	0
$615$	−8.66025	−0.349215
$616$	0	0
$617$	− 3.78461i	− 0.152363i	−0.997094	−	0.0761813i	$-0.975727\pi$
$617$	− 3.78461i	− 0.152363i	0.997094	−	0.0761813i	$-0.0242728\pi$
$618$	0	0
$619$	23.6077i	0.948873i	0.880290	+	0.474437i	$0.157348\pi$
$619$	23.6077i	0.948873i	−0.880290	+	0.474437i	$0.842652\pi$
$620$	0	0
$621$	−2.19615	−0.0881286
$622$	0	0
$623$	−1.85641	−0.0743754
$624$	0	0
$625$	−11.0000	−0.440000
$626$	0	0
$627$	−0.928203	−0.0370689
$628$	0	0
$629$	31.5885i	1.25951i
$630$	0	0
$631$	− 6.14359i	− 0.244573i	−0.992495	−	0.122286i	$-0.960977\pi$
$631$	− 6.14359i	− 0.244573i	0.992495	−	0.122286i	$-0.0390226\pi$
$632$	0	0
$633$	−23.3205	−0.926907
$634$	0	0
$635$	− 6.92820i	− 0.274937i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	8.19615i	0.324235i
$640$	0	0
$641$	−32.5167	−1.28433	−0.642165	−	0.766566i	$-0.721964\pi$
$641$	−32.5167	−1.28433	−0.642165	+	0.766566i	$0.721964\pi$
$642$	0	0
$643$	27.7128i	1.09289i	0.837496	+	0.546443i	$0.184019\pi$
$643$	27.7128i	1.09289i	−0.837496	+	0.546443i	$0.815981\pi$
$644$	0	0
$645$	3.12436i	0.123021i
$646$	0	0
$647$	−34.2487	−1.34646	−0.673228	−	0.739435i	$-0.735093\pi$
$647$	−34.2487	−1.34646	−0.673228	+	0.739435i	$0.735093\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	4.00000	0.156772
$652$	0	0
$653$	26.5359	1.03843	0.519215	−	0.854644i	$-0.326224\pi$
$653$	26.5359	1.03843	0.519215	+	0.854644i	$0.326224\pi$
$654$	0	0
$655$	33.4641i	1.30755i
$656$	0	0
$657$	− 1.19615i	− 0.0466664i
$658$	0	0
$659$	−43.3205	−1.68753	−0.843764	−	0.536715i	$-0.819665\pi$
$659$	−43.3205	−1.68753	−0.843764	+	0.536715i	$0.819665\pi$
$660$	0	0
$661$	− 15.3923i	− 0.598691i	−0.954145	−	0.299346i	$-0.903232\pi$
$661$	− 15.3923i	− 0.598691i	0.954145	−	0.299346i	$-0.0967684\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.60770i	0.0623437i
$666$	0	0
$667$	−13.0192	−0.504107
$668$	0	0
$669$	24.7846i	0.958228i
$670$	0	0
$671$	7.12436i	0.275033i
$672$	0	0
$673$	6.21539	0.239586	0.119793	−	0.992799i	$-0.461777\pi$
$673$	6.21539	0.239586	0.119793	+	0.992799i	$0.461777\pi$
$674$	0	0
$675$	−2.00000	−0.0769800
$676$	0	0
$677$	14.5359	0.558660	0.279330	−	0.960195i	$-0.409888\pi$
$677$	14.5359	0.558660	0.279330	+	0.960195i	$0.409888\pi$
$678$	0	0
$679$	7.32051	0.280935
$680$	0	0
$681$	− 16.0526i	− 0.615135i
$682$	0	0
$683$	− 10.1436i	− 0.388134i	−0.980988	−	0.194067i	$-0.937832\pi$
$683$	− 10.1436i	− 0.388134i	0.980988	−	0.194067i	$-0.0621679\pi$
$684$	0	0
$685$	−21.5885	−0.824853
$686$	0	0
$687$	− 27.8564i	− 1.06279i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	10.3397i	0.393342i	0.980469	+	0.196671i	$0.0630132\pi$
$691$	10.3397i	0.393342i	−0.980469	+	0.196671i	$0.936987\pi$
$692$	0	0
$693$	−0.535898	−0.0203571
$694$	0	0
$695$	34.6410i	1.31401i
$696$	0	0
$697$	− 18.6603i	− 0.706808i
$698$	0	0
$699$	−2.00000	−0.0756469
$700$	0	0
$701$	−32.1051	−1.21259	−0.606297	−	0.795238i	$-0.707346\pi$
$701$	−32.1051	−1.21259	−0.606297	+	0.795238i	$0.707346\pi$
$702$	0	0
$703$	−10.7321	−0.404767
$704$	0	0
$705$	−11.6603	−0.439151
$706$	0	0
$707$	− 10.5885i	− 0.398220i
$708$	0	0
$709$	34.8564i	1.30906i	0.756036	+	0.654530i	$0.227133\pi$
$709$	34.8564i	1.30906i	−0.756036	+	0.654530i	$0.772867\pi$
$710$	0	0
$711$	12.3923	0.464748
$712$	0	0
$713$	− 12.0000i	− 0.449404i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	26.7321i	0.998327i
$718$	0	0
$719$	−37.1769	−1.38646	−0.693232	−	0.720714i	$-0.743814\pi$
$719$	−37.1769	−1.38646	−0.693232	+	0.720714i	$0.743814\pi$
$720$	0	0
$721$	− 2.00000i	− 0.0744839i
$722$	0	0
$723$	− 22.6603i	− 0.842744i
$724$	0	0
$725$	−11.8564	−0.440336
$726$	0	0
$727$	−1.26795	−0.0470256	−0.0235128	−	0.999724i	$-0.507485\pi$
$727$	−1.26795	−0.0470256	−0.0235128	+	0.999724i	$0.507485\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−6.73205	−0.248994
$732$	0	0
$733$	35.0000i	1.29275i	0.763018	+	0.646377i	$0.223717\pi$
$733$	35.0000i	1.29275i	−0.763018	+	0.646377i	$0.776283\pi$
$734$	0	0
$735$	− 11.1962i	− 0.412976i
$736$	0	0
$737$	−3.85641	−0.142053
$738$	0	0
$739$	− 24.7846i	− 0.911717i	−0.890052	−	0.455858i	$-0.849332\pi$
$739$	− 24.7846i	− 0.911717i	0.890052	−	0.455858i	$-0.150668\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	32.1051i	1.17782i	0.808198	+	0.588911i	$0.200443\pi$
$743$	32.1051i	1.17782i	−0.808198	+	0.588911i	$0.799557\pi$
$744$	0	0
$745$	16.6077	0.608459
$746$	0	0
$747$	− 15.6603i	− 0.572979i
$748$	0	0
$749$	11.0718i	0.404555i
$750$	0	0
$751$	1.66025	0.0605835	0.0302918	−	0.999541i	$-0.490356\pi$
$751$	1.66025	0.0605835	0.0302918	+	0.999541i	$0.490356\pi$
$752$	0	0
$753$	−9.46410	−0.344891
$754$	0	0
$755$	−35.9090	−1.30686
$756$	0	0
$757$	34.7846	1.26427	0.632134	−	0.774859i	$-0.282179\pi$
$757$	34.7846	1.26427	0.632134	+	0.774859i	$0.282179\pi$
$758$	0	0
$759$	1.60770i	0.0583556i
$760$	0	0
$761$	− 9.46410i	− 0.343073i	−0.985178	−	0.171537i	$-0.945127\pi$
$761$	− 9.46410i	− 0.343073i	0.985178	−	0.171537i	$-0.0548732\pi$
$762$	0	0
$763$	0.287187	0.0103969
$764$	0	0
$765$	6.46410i	0.233710i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	− 39.5692i	− 1.42690i	−0.700705	−	0.713451i	$-0.747131\pi$
$769$	− 39.5692i	− 1.42690i	0.700705	−	0.713451i	$-0.252869\pi$
$770$	0	0
$771$	−17.7321	−0.638604
$772$	0	0
$773$	− 46.7846i	− 1.68273i	−0.540471	−	0.841363i	$-0.681754\pi$
$773$	− 46.7846i	− 1.68273i	0.540471	−	0.841363i	$-0.318246\pi$
$774$	0	0
$775$	− 10.9282i	− 0.392553i
$776$	0	0
$777$	−6.19615	−0.222286
$778$	0	0
$779$	6.33975	0.227145
$780$	0	0
$781$	6.00000	0.214697
$782$	0	0
$783$	5.92820	0.211857
$784$	0	0
$785$	26.0718i	0.930542i
$786$	0	0
$787$	8.67949i	0.309390i	0.987962	+	0.154695i	$0.0494396\pi$
$787$	8.67949i	0.309390i	−0.987962	+	0.154695i	$0.950560\pi$
$788$	0	0
$789$	−12.3397	−0.439307
$790$	0	0
$791$	− 3.41154i	− 0.121300i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	− 10.2679i	− 0.364167i
$796$	0	0
$797$	−6.00000	−0.212531	−0.106265	−	0.994338i	$-0.533889\pi$
$797$	−6.00000	−0.212531	−0.106265	+	0.994338i	$0.533889\pi$
$798$	0	0
$799$	− 25.1244i	− 0.888836i
$800$	0	0
$801$	− 2.53590i	− 0.0896016i
$802$	0	0
$803$	−0.875644	−0.0309008
$804$	0	0
$805$	2.78461	0.0981446
$806$	0	0
$807$	−9.46410	−0.333152
$808$	0	0
$809$	2.26795	0.0797368	0.0398684	−	0.999205i	$-0.487306\pi$
$809$	2.26795	0.0797368	0.0398684	+	0.999205i	$0.487306\pi$
$810$	0	0
$811$	50.2487i	1.76447i	0.470808	+	0.882235i	$0.343962\pi$
$811$	50.2487i	1.76447i	−0.470808	+	0.882235i	$0.656038\pi$
$812$	0	0
$813$	− 16.0000i	− 0.561144i
$814$	0	0
$815$	−11.3205	−0.396540
$816$	0	0
$817$	− 2.28719i	− 0.0800185i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	28.9282i	1.00960i	0.863236	+	0.504801i	$0.168434\pi$
$821$	28.9282i	1.00960i	−0.863236	+	0.504801i	$0.831566\pi$
$822$	0	0
$823$	24.0000	0.836587	0.418294	−	0.908312i	$-0.362628\pi$
$823$	24.0000	0.836587	0.418294	+	0.908312i	$0.362628\pi$
$824$	0	0
$825$	1.46410i	0.0509735i
$826$	0	0
$827$	− 35.6077i	− 1.23820i	−0.785312	−	0.619100i	$-0.787497\pi$
$827$	− 35.6077i	− 1.23820i	0.785312	−	0.619100i	$-0.212503\pi$
$828$	0	0
$829$	−22.2679	−0.773398	−0.386699	−	0.922206i	$-0.626385\pi$
$829$	−22.2679	−0.773398	−0.386699	+	0.922206i	$0.626385\pi$
$830$	0	0
$831$	−8.12436	−0.281831
$832$	0	0
$833$	24.1244	0.835859
$834$	0	0
$835$	9.46410	0.327519
$836$	0	0
$837$	5.46410i	0.188867i
$838$	0	0
$839$	− 9.85641i	− 0.340281i	−0.985420	−	0.170140i	$-0.945578\pi$
$839$	− 9.85641i	− 0.340281i	0.985420	−	0.170140i	$-0.0544221\pi$
$840$	0	0
$841$	6.14359	0.211848
$842$	0	0
$843$	24.7128i	0.851154i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 7.66025i	− 0.263209i
$848$	0	0
$849$	−21.5167	−0.738450
$850$	0	0
$851$	18.5885i	0.637204i
$852$	0	0
$853$	56.0333i	1.91854i	0.282483	+	0.959272i	$0.408842\pi$
$853$	56.0333i	1.91854i	−0.282483	+	0.959272i	$0.591158\pi$
$854$	0	0
$855$	−2.19615	−0.0751068
$856$	0	0
$857$	−10.5167	−0.359242	−0.179621	−	0.983736i	$-0.557487\pi$
$857$	−10.5167	−0.359242	−0.179621	+	0.983736i	$0.557487\pi$
$858$	0	0
$859$	−24.4449	−0.834048	−0.417024	−	0.908895i	$-0.636927\pi$
$859$	−24.4449	−0.834048	−0.417024	+	0.908895i	$0.636927\pi$
$860$	0	0
$861$	3.66025	0.124741
$862$	0	0
$863$	10.3397i	0.351969i	0.984393	+	0.175985i	$0.0563109\pi$
$863$	10.3397i	0.351969i	−0.984393	+	0.175985i	$0.943689\pi$
$864$	0	0
$865$	− 18.2487i	− 0.620475i
$866$	0	0
$867$	3.07180	0.104324
$868$	0	0
$869$	− 9.07180i	− 0.307740i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	10.0000i	0.338449i
$874$	0	0
$875$	8.87564	0.300052
$876$	0	0
$877$	− 19.0000i	− 0.641584i	−0.947150	−	0.320792i	$-0.896051\pi$
$877$	− 19.0000i	− 0.641584i	0.947150	−	0.320792i	$-0.103949\pi$
$878$	0	0
$879$	− 2.12436i	− 0.0716527i
$880$	0	0
$881$	−20.5167	−0.691224	−0.345612	−	0.938378i	$-0.612329\pi$
$881$	−20.5167	−0.691224	−0.345612	+	0.938378i	$0.612329\pi$
$882$	0	0
$883$	−32.7846	−1.10329	−0.551645	−	0.834079i	$-0.686000\pi$
$883$	−32.7846	−1.10329	−0.551645	+	0.834079i	$0.686000\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−2.14359	−0.0719748	−0.0359874	−	0.999352i	$-0.511458\pi$
$887$	−2.14359	−0.0719748	−0.0359874	+	0.999352i	$0.511458\pi$
$888$	0	0
$889$	2.92820i	0.0982088i
$890$	0	0
$891$	− 0.732051i	− 0.0245246i
$892$	0	0
$893$	8.53590	0.285643
$894$	0	0
$895$	− 29.9090i	− 0.999746i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	32.3923i	1.08034i
$900$	0	0
$901$	22.1244	0.737069
$902$	0	0
$903$	− 1.32051i	− 0.0439438i
$904$	0	0
$905$	− 15.2487i	− 0.506884i
$906$	0	0
$907$	−39.0333	−1.29608	−0.648040	−	0.761606i	$-0.724411\pi$
$907$	−39.0333	−1.29608	−0.648040	+	0.761606i	$0.724411\pi$
$908$	0	0
$909$	14.4641	0.479744
$910$	0	0
$911$	−9.46410	−0.313560	−0.156780	−	0.987634i	$-0.550111\pi$
$911$	−9.46410	−0.313560	−0.156780	+	0.987634i	$0.550111\pi$
$912$	0	0
$913$	−11.4641	−0.379406
$914$	0	0
$915$	16.8564i	0.557256i
$916$	0	0
$917$	− 14.1436i	− 0.467063i
$918$	0	0
$919$	−8.39230	−0.276837	−0.138418	−	0.990374i	$-0.544202\pi$
$919$	−8.39230	−0.276837	−0.138418	+	0.990374i	$0.544202\pi$
$920$	0	0
$921$	10.7321i	0.353633i
$922$	0	0
$923$	0	0
$924$	0	0
$925$	16.9282i	0.556596i
$926$	0	0
$927$	2.73205	0.0897323
$928$	0	0
$929$	− 26.0718i	− 0.855388i	−0.903924	−	0.427694i	$-0.859326\pi$
$929$	− 26.0718i	− 0.855388i	0.903924	−	0.427694i	$-0.140674\pi$
$930$	0	0
$931$	8.19615i	0.268618i
$932$	0	0
$933$	−4.33975	−0.142077
$934$	0	0
$935$	4.73205	0.154755
$936$	0	0
$937$	−54.3205	−1.77457	−0.887287	−	0.461218i	$-0.847413\pi$
$937$	−54.3205	−1.77457	−0.887287	+	0.461218i	$0.847413\pi$
$938$	0	0
$939$	−3.32051	−0.108361
$940$	0	0
$941$	14.7846i	0.481965i	0.970530	+	0.240982i	$0.0774696\pi$
$941$	14.7846i	0.481965i	−0.970530	+	0.240982i	$0.922530\pi$
$942$	0	0
$943$	− 10.9808i	− 0.357583i
$944$	0	0
$945$	−1.26795	−0.0412464
$946$	0	0
$947$	− 13.0718i	− 0.424776i	−0.977185	−	0.212388i	$-0.931876\pi$
$947$	− 13.0718i	− 0.424776i	0.977185	−	0.212388i	$-0.0681241\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	11.1962i	0.363060i
$952$	0	0
$953$	8.78461	0.284561	0.142281	−	0.989826i	$-0.454556\pi$
$953$	8.78461	0.284561	0.142281	+	0.989826i	$0.454556\pi$
$954$	0	0
$955$	− 1.85641i	− 0.0600719i
$956$	0	0
$957$	− 4.33975i	− 0.140284i
$958$	0	0
$959$	9.12436	0.294641
$960$	0	0
$961$	1.14359	0.0368901
$962$	0	0
$963$	−15.1244	−0.487376
$964$	0	0
$965$	−14.3205	−0.460993
$966$	0	0
$967$	− 0.732051i	− 0.0235412i	−0.999931	−	0.0117706i	$-0.996253\pi$
$967$	− 0.732051i	− 0.0235412i	0.999931	−	0.0117706i	$-0.00374678\pi$
$968$	0	0
$969$	− 4.73205i	− 0.152015i
$970$	0	0
$971$	−20.3923	−0.654420	−0.327210	−	0.944952i	$-0.606108\pi$
$971$	−20.3923	−0.654420	−0.327210	+	0.944952i	$0.606108\pi$
$972$	0	0
$973$	− 14.6410i	− 0.469369i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 54.9615i	− 1.75837i	−0.476476	−	0.879187i	$-0.658086\pi$
$977$	− 54.9615i	− 1.75837i	0.476476	−	0.879187i	$-0.341914\pi$
$978$	0	0
$979$	−1.85641	−0.0593310
$980$	0	0
$981$	0.392305i	0.0125253i
$982$	0	0
$983$	26.6410i	0.849716i	0.905260	+	0.424858i	$0.139676\pi$
$983$	26.6410i	0.849716i	−0.905260	+	0.424858i	$0.860324\pi$
$984$	0	0
$985$	−25.8564	−0.823854
$986$	0	0
$987$	4.92820	0.156866
$988$	0	0
$989$	−3.96152	−0.125969
$990$	0	0
$991$	−12.8756	−0.409008	−0.204504	−	0.978866i	$-0.565558\pi$
$991$	−12.8756	−0.409008	−0.204504	+	0.978866i	$0.565558\pi$
$992$	0	0
$993$	12.0000i	0.380808i
$994$	0	0
$995$	30.5885i	0.969719i
$996$	0	0
$997$	−31.9808	−1.01284	−0.506420	−	0.862287i	$-0.669032\pi$
$997$	−31.9808	−1.01284	−0.506420	+	0.862287i	$0.669032\pi$
$998$	0	0
$999$	− 8.46410i	− 0.267792i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4056.2.c.k.337.4		4
13.5	odd	4		4056.2.a.t.1.2		2
13.8	odd	4		4056.2.a.u.1.1		2
13.9	even	3		312.2.bf.a.49.1	✓	4
13.10	even	6		312.2.bf.a.121.1	yes	4
13.12	even	2	inner	4056.2.c.k.337.1		4
39.23	odd	6		936.2.bi.a.433.1		4
39.35	odd	6		936.2.bi.a.361.1		4
52.23	odd	6		624.2.bv.c.433.2		4
52.31	even	4		8112.2.a.bw.1.2		2
52.35	odd	6		624.2.bv.c.49.2		4
52.47	even	4		8112.2.a.br.1.1		2
156.23	even	6		1872.2.by.g.433.2		4
156.35	even	6		1872.2.by.g.1297.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.2.bf.a.49.1	✓	4	13.9	even	3
312.2.bf.a.121.1	yes	4	13.10	even	6
624.2.bv.c.49.2		4	52.35	odd	6
624.2.bv.c.433.2		4	52.23	odd	6
936.2.bi.a.361.1		4	39.35	odd	6
936.2.bi.a.433.1		4	39.23	odd	6
1872.2.by.g.433.2		4	156.23	even	6
1872.2.by.g.1297.2		4	156.35	even	6
4056.2.a.t.1.2		2	13.5	odd	4
4056.2.a.u.1.1		2	13.8	odd	4
4056.2.c.k.337.1		4	13.12	even	2	inner
4056.2.c.k.337.4		4	1.1	even	1	trivial
8112.2.a.br.1.1		2	52.47	even	4
8112.2.a.bw.1.2		2	52.31	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4056 = 2^{3} \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4056.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.73205i q^{5} -0.732051i q^{7} +1.00000 q^{9} -0.732051i q^{11} -1.73205i q^{15} +3.73205 q^{17} +1.26795i q^{19} +0.732051i q^{21} +2.19615 q^{23} +2.00000 q^{25} -1.00000 q^{27} -5.92820 q^{29} -5.46410i q^{31} +0.732051i q^{33} +1.26795 q^{35} +8.46410i q^{37} -5.00000i q^{41} -1.80385 q^{43} +1.73205i q^{45} -6.73205i q^{47} +6.46410 q^{49} -3.73205 q^{51} +5.92820 q^{53} +1.26795 q^{55} -1.26795i q^{57} -9.73205 q^{61} -0.732051i q^{63} -5.26795i q^{67} -2.19615 q^{69} +8.19615i q^{71} -1.19615i q^{73} -2.00000 q^{75} -0.535898 q^{77} +12.3923 q^{79} +1.00000 q^{81} -15.6603i q^{83} +6.46410i q^{85} +5.92820 q^{87} -2.53590i q^{89} +5.46410i q^{93} -2.19615 q^{95} +10.0000i q^{97} -0.732051i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + 4 q^{9} + 8 q^{17} - 12 q^{23} + 8 q^{25} - 4 q^{27} + 4 q^{29} + 12 q^{35} - 28 q^{43} + 12 q^{49} - 8 q^{51} - 4 q^{53} + 12 q^{55} - 32 q^{61} + 12 q^{69} - 8 q^{75} - 16 q^{77} + 8 q^{79}+ \cdots + 12 q^{95}+O(q^{100}) \) 4 * q - 4 * q^3 + 4 * q^9 + 8 * q^17 - 12 * q^23 + 8 * q^25 - 4 * q^27 + 4 * q^29 + 12 * q^35 - 28 * q^43 + 12 * q^49 - 8 * q^51 - 4 * q^53 + 12 * q^55 - 32 * q^61 + 12 * q^69 - 8 * q^75 - 16 * q^77 + 8 * q^79 + 4 * q^81 - 4 * q^87 + 12 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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