LMFDB - Embedded newform 2640.2.a.be.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2640,2,Mod(1,2640)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2640, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2640.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2640 = 2^{4} \cdot 3 \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2640.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:	$21.0805061336$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.311108$ of defining polynomial
Character	$\chi$	$=$	2640.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} +4.42864 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.00000 q^{5} +4.42864 q^{7} +1.00000 q^{9} -1.00000 q^{11} -0.622216 q^{13} -1.00000 q^{15} -5.18421 q^{17} -7.05086 q^{19} -4.42864 q^{21} -8.85728 q^{23} +1.00000 q^{25} -1.00000 q^{27} -7.80642 q^{29} -2.75557 q^{31} +1.00000 q^{33} +4.42864 q^{35} -2.00000 q^{37} +0.622216 q^{39} -0.193576 q^{41} -5.67307 q^{43} +1.00000 q^{45} +2.75557 q^{47} +12.6128 q^{49} +5.18421 q^{51} -10.8573 q^{53} -1.00000 q^{55} +7.05086 q^{57} +4.85728 q^{59} +6.85728 q^{61} +4.42864 q^{63} -0.622216 q^{65} +1.24443 q^{67} +8.85728 q^{69} -2.75557 q^{71} +4.23506 q^{73} -1.00000 q^{75} -4.42864 q^{77} -8.56199 q^{79} +1.00000 q^{81} -0.133353 q^{83} -5.18421 q^{85} +7.80642 q^{87} +5.61285 q^{89} -2.75557 q^{91} +2.75557 q^{93} -7.05086 q^{95} +7.24443 q^{97} -1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + 3 q^{5} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 + 3 * q^5 + 3 * q^9	$ 3 q - 3 q^{3} + 3 q^{5} + 3 q^{9} - 3 q^{11} - 2 q^{13} - 3 q^{15} - 2 q^{17} - 8 q^{19} + 3 q^{25} - 3 q^{27} - 10 q^{29} - 8 q^{31} + 3 q^{33} - 6 q^{37} + 2 q^{39} - 14 q^{41} - 4 q^{43} + 3 q^{45} + 8 q^{47} + 11 q^{49} + 2 q^{51} - 6 q^{53} - 3 q^{55} + 8 q^{57} - 12 q^{59} - 6 q^{61} - 2 q^{65} + 4 q^{67} - 8 q^{71} - 14 q^{73} - 3 q^{75} - 12 q^{79} + 3 q^{81} - 2 q^{85} + 10 q^{87} - 10 q^{89} - 8 q^{91} + 8 q^{93} - 8 q^{95} + 22 q^{97} - 3 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 + 3 * q^5 + 3 * q^9 - 3 * q^11 - 2 * q^13 - 3 * q^15 - 2 * q^17 - 8 * q^19 + 3 * q^25 - 3 * q^27 - 10 * q^29 - 8 * q^31 + 3 * q^33 - 6 * q^37 + 2 * q^39 - 14 * q^41 - 4 * q^43 + 3 * q^45 + 8 * q^47 + 11 * q^49 + 2 * q^51 - 6 * q^53 - 3 * q^55 + 8 * q^57 - 12 * q^59 - 6 * q^61 - 2 * q^65 + 4 * q^67 - 8 * q^71 - 14 * q^73 - 3 * q^75 - 12 * q^79 + 3 * q^81 - 2 * q^85 + 10 * q^87 - 10 * q^89 - 8 * q^91 + 8 * q^93 - 8 * q^95 + 22 * q^97 - 3 * q^99

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
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	4.42864	1.67387	0.836934	−	0.547304i	$-0.184346\pi$
$7$	4.42864	1.67387	0.836934	+	0.547304i	$0.184346\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−0.622216	−0.172572	−0.0862858	−	0.996270i	$-0.527500\pi$
$13$	−0.622216	−0.172572	−0.0862858	+	0.996270i	$0.527500\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−5.18421	−1.25736	−0.628678	−	0.777666i	$-0.716403\pi$
$17$	−5.18421	−1.25736	−0.628678	+	0.777666i	$0.716403\pi$
$18$	0	0
$19$	−7.05086	−1.61758	−0.808789	−	0.588100i	$-0.799876\pi$
$19$	−7.05086	−1.61758	−0.808789	+	0.588100i	$0.799876\pi$
$20$	0	0
$21$	−4.42864	−0.966408
$22$	0	0
$23$	−8.85728	−1.84687	−0.923435	−	0.383754i	$-0.874631\pi$
$23$	−8.85728	−1.84687	−0.923435	+	0.383754i	$0.874631\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−7.80642	−1.44962	−0.724808	−	0.688951i	$-0.758072\pi$
$29$	−7.80642	−1.44962	−0.724808	+	0.688951i	$0.758072\pi$
$30$	0	0
$31$	−2.75557	−0.494915	−0.247457	−	0.968899i	$-0.579595\pi$
$31$	−2.75557	−0.494915	−0.247457	+	0.968899i	$0.579595\pi$
$32$	0	0
$33$	1.00000	0.174078
$34$	0	0
$35$	4.42864	0.748577
$36$	0	0
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	0	0
$39$	0.622216	0.0996342
$40$	0	0
$41$	−0.193576	−0.0302315	−0.0151158	−	0.999886i	$-0.504812\pi$
$41$	−0.193576	−0.0302315	−0.0151158	+	0.999886i	$0.504812\pi$
$42$	0	0
$43$	−5.67307	−0.865135	−0.432568	−	0.901602i	$-0.642392\pi$
$43$	−5.67307	−0.865135	−0.432568	+	0.901602i	$0.642392\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	2.75557	0.401941	0.200971	−	0.979597i	$-0.435590\pi$
$47$	2.75557	0.401941	0.200971	+	0.979597i	$0.435590\pi$
$48$	0	0
$49$	12.6128	1.80184
$50$	0	0
$51$	5.18421	0.725934
$52$	0	0
$53$	−10.8573	−1.49136	−0.745681	−	0.666303i	$-0.767876\pi$
$53$	−10.8573	−1.49136	−0.745681	+	0.666303i	$0.767876\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	0	0
$57$	7.05086	0.933909
$58$	0	0
$59$	4.85728	0.632364	0.316182	−	0.948699i	$-0.397599\pi$
$59$	4.85728	0.632364	0.316182	+	0.948699i	$0.397599\pi$
$60$	0	0
$61$	6.85728	0.877985	0.438992	−	0.898491i	$-0.355336\pi$
$61$	6.85728	0.877985	0.438992	+	0.898491i	$0.355336\pi$
$62$	0	0
$63$	4.42864	0.557956
$64$	0	0
$65$	−0.622216	−0.0771764
$66$	0	0
$67$	1.24443	0.152031	0.0760157	−	0.997107i	$-0.475780\pi$
$67$	1.24443	0.152031	0.0760157	+	0.997107i	$0.475780\pi$
$68$	0	0
$69$	8.85728	1.06629
$70$	0	0
$71$	−2.75557	−0.327026	−0.163513	−	0.986541i	$-0.552283\pi$
$71$	−2.75557	−0.327026	−0.163513	+	0.986541i	$0.552283\pi$
$72$	0	0
$73$	4.23506	0.495677	0.247838	−	0.968801i	$-0.420280\pi$
$73$	4.23506	0.495677	0.247838	+	0.968801i	$0.420280\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	−4.42864	−0.504690
$78$	0	0
$79$	−8.56199	−0.963299	−0.481650	−	0.876364i	$-0.659962\pi$
$79$	−8.56199	−0.963299	−0.481650	+	0.876364i	$0.659962\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−0.133353	−0.0146374	−0.00731870	−	0.999973i	$-0.502330\pi$
$83$	−0.133353	−0.0146374	−0.00731870	+	0.999973i	$0.502330\pi$
$84$	0	0
$85$	−5.18421	−0.562306
$86$	0	0
$87$	7.80642	0.836936
$88$	0	0
$89$	5.61285	0.594961	0.297480	−	0.954728i	$-0.403854\pi$
$89$	5.61285	0.594961	0.297480	+	0.954728i	$0.403854\pi$
$90$	0	0
$91$	−2.75557	−0.288862
$92$	0	0
$93$	2.75557	0.285739
$94$	0	0
$95$	−7.05086	−0.723402
$96$	0	0
$97$	7.24443	0.735561	0.367780	−	0.929913i	$-0.380118\pi$
$97$	7.24443	0.735561	0.367780	+	0.929913i	$0.380118\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	0	0
$101$	4.66370	0.464056	0.232028	−	0.972709i	$-0.425464\pi$
$101$	4.66370	0.464056	0.232028	+	0.972709i	$0.425464\pi$
$102$	0	0
$103$	11.6128	1.14425	0.572124	−	0.820167i	$-0.306120\pi$
$103$	11.6128	1.14425	0.572124	+	0.820167i	$0.306120\pi$
$104$	0	0
$105$	−4.42864	−0.432191
$106$	0	0
$107$	−2.62222	−0.253499	−0.126750	−	0.991935i	$-0.540454\pi$
$107$	−2.62222	−0.253499	−0.126750	+	0.991935i	$0.540454\pi$
$108$	0	0
$109$	−19.7146	−1.88831	−0.944156	−	0.329499i	$-0.893120\pi$
$109$	−19.7146	−1.88831	−0.944156	+	0.329499i	$0.893120\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	−8.85728	−0.825946
$116$	0	0
$117$	−0.622216	−0.0575239
$118$	0	0
$119$	−22.9590	−2.10465
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0.193576	0.0174542
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	15.1842	1.34738	0.673690	−	0.739014i	$-0.264708\pi$
$127$	15.1842	1.34738	0.673690	+	0.739014i	$0.264708\pi$
$128$	0	0
$129$	5.67307	0.499486
$130$	0	0
$131$	−1.24443	−0.108726	−0.0543632	−	0.998521i	$-0.517313\pi$
$131$	−1.24443	−0.108726	−0.0543632	+	0.998521i	$0.517313\pi$
$132$	0	0
$133$	−31.2257	−2.70761
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−0.488863	−0.0417663	−0.0208832	−	0.999782i	$-0.506648\pi$
$137$	−0.488863	−0.0417663	−0.0208832	+	0.999782i	$0.506648\pi$
$138$	0	0
$139$	−17.8064	−1.51032	−0.755161	−	0.655540i	$-0.772441\pi$
$139$	−17.8064	−1.51032	−0.755161	+	0.655540i	$0.772441\pi$
$140$	0	0
$141$	−2.75557	−0.232061
$142$	0	0
$143$	0.622216	0.0520323
$144$	0	0
$145$	−7.80642	−0.648288
$146$	0	0
$147$	−12.6128	−1.04029
$148$	0	0
$149$	−1.43801	−0.117806	−0.0589031	−	0.998264i	$-0.518760\pi$
$149$	−1.43801	−0.117806	−0.0589031	+	0.998264i	$0.518760\pi$
$150$	0	0
$151$	12.1748	0.990774	0.495387	−	0.868672i	$-0.335026\pi$
$151$	12.1748	0.990774	0.495387	+	0.868672i	$0.335026\pi$
$152$	0	0
$153$	−5.18421	−0.419118
$154$	0	0
$155$	−2.75557	−0.221333
$156$	0	0
$157$	18.4701	1.47408	0.737038	−	0.675851i	$-0.236224\pi$
$157$	18.4701	1.47408	0.737038	+	0.675851i	$0.236224\pi$
$158$	0	0
$159$	10.8573	0.861038
$160$	0	0
$161$	−39.2257	−3.09142
$162$	0	0
$163$	10.1017	0.791227	0.395614	−	0.918417i	$-0.370532\pi$
$163$	10.1017	0.791227	0.395614	+	0.918417i	$0.370532\pi$
$164$	0	0
$165$	1.00000	0.0778499
$166$	0	0
$167$	16.3368	1.26418	0.632089	−	0.774896i	$-0.282198\pi$
$167$	16.3368	1.26418	0.632089	+	0.774896i	$0.282198\pi$
$168$	0	0
$169$	−12.6128	−0.970219
$170$	0	0
$171$	−7.05086	−0.539192
$172$	0	0
$173$	−9.18421	−0.698262	−0.349131	−	0.937074i	$-0.613523\pi$
$173$	−9.18421	−0.698262	−0.349131	+	0.937074i	$0.613523\pi$
$174$	0	0
$175$	4.42864	0.334774
$176$	0	0
$177$	−4.85728	−0.365095
$178$	0	0
$179$	25.3274	1.89306	0.946530	−	0.322617i	$-0.104563\pi$
$179$	25.3274	1.89306	0.946530	+	0.322617i	$0.104563\pi$
$180$	0	0
$181$	−13.6128	−1.01184	−0.505918	−	0.862582i	$-0.668846\pi$
$181$	−13.6128	−1.01184	−0.505918	+	0.862582i	$0.668846\pi$
$182$	0	0
$183$	−6.85728	−0.506905
$184$	0	0
$185$	−2.00000	−0.147043
$186$	0	0
$187$	5.18421	0.379107
$188$	0	0
$189$	−4.42864	−0.322136
$190$	0	0
$191$	−6.10171	−0.441504	−0.220752	−	0.975330i	$-0.570851\pi$
$191$	−6.10171	−0.441504	−0.220752	+	0.975330i	$0.570851\pi$
$192$	0	0
$193$	18.3368	1.31991	0.659955	−	0.751305i	$-0.270575\pi$
$193$	18.3368	1.31991	0.659955	+	0.751305i	$0.270575\pi$
$194$	0	0
$195$	0.622216	0.0445578
$196$	0	0
$197$	−6.69535	−0.477024	−0.238512	−	0.971140i	$-0.576660\pi$
$197$	−6.69535	−0.477024	−0.238512	+	0.971140i	$0.576660\pi$
$198$	0	0
$199$	−14.1017	−0.999644	−0.499822	−	0.866128i	$-0.666601\pi$
$199$	−14.1017	−0.999644	−0.499822	+	0.866128i	$0.666601\pi$
$200$	0	0
$201$	−1.24443	−0.0877754
$202$	0	0
$203$	−34.5718	−2.42647
$204$	0	0
$205$	−0.193576	−0.0135199
$206$	0	0
$207$	−8.85728	−0.615623
$208$	0	0
$209$	7.05086	0.487718
$210$	0	0
$211$	10.6637	0.734120	0.367060	−	0.930197i	$-0.380364\pi$
$211$	10.6637	0.734120	0.367060	+	0.930197i	$0.380364\pi$
$212$	0	0
$213$	2.75557	0.188808
$214$	0	0
$215$	−5.67307	−0.386900
$216$	0	0
$217$	−12.2034	−0.828422
$218$	0	0
$219$	−4.23506	−0.286179
$220$	0	0
$221$	3.22570	0.216984
$222$	0	0
$223$	−8.85728	−0.593127	−0.296564	−	0.955013i	$-0.595841\pi$
$223$	−8.85728	−0.593127	−0.296564	+	0.955013i	$0.595841\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−13.3778	−0.887915	−0.443957	−	0.896048i	$-0.646426\pi$
$227$	−13.3778	−0.887915	−0.443957	+	0.896048i	$0.646426\pi$
$228$	0	0
$229$	11.5111	0.760677	0.380339	−	0.924847i	$-0.375807\pi$
$229$	11.5111	0.760677	0.380339	+	0.924847i	$0.375807\pi$
$230$	0	0
$231$	4.42864	0.291383
$232$	0	0
$233$	−4.32693	−0.283467	−0.141733	−	0.989905i	$-0.545268\pi$
$233$	−4.32693	−0.283467	−0.141733	+	0.989905i	$0.545268\pi$
$234$	0	0
$235$	2.75557	0.179753
$236$	0	0
$237$	8.56199	0.556161
$238$	0	0
$239$	3.34614	0.216444	0.108222	−	0.994127i	$-0.465484\pi$
$239$	3.34614	0.216444	0.108222	+	0.994127i	$0.465484\pi$
$240$	0	0
$241$	−1.34614	−0.0867126	−0.0433563	−	0.999060i	$-0.513805\pi$
$241$	−1.34614	−0.0867126	−0.0433563	+	0.999060i	$0.513805\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	12.6128	0.805805
$246$	0	0
$247$	4.38715	0.279148
$248$	0	0
$249$	0.133353	0.00845091
$250$	0	0
$251$	−22.7556	−1.43632	−0.718159	−	0.695879i	$-0.755015\pi$
$251$	−22.7556	−1.43632	−0.718159	+	0.695879i	$0.755015\pi$
$252$	0	0
$253$	8.85728	0.556852
$254$	0	0
$255$	5.18421	0.324648
$256$	0	0
$257$	−6.85728	−0.427745	−0.213873	−	0.976862i	$-0.568608\pi$
$257$	−6.85728	−0.427745	−0.213873	+	0.976862i	$0.568608\pi$
$258$	0	0
$259$	−8.85728	−0.550365
$260$	0	0
$261$	−7.80642	−0.483206
$262$	0	0
$263$	29.5812	1.82406	0.912028	−	0.410129i	$-0.134516\pi$
$263$	29.5812	1.82406	0.912028	+	0.410129i	$0.134516\pi$
$264$	0	0
$265$	−10.8573	−0.666957
$266$	0	0
$267$	−5.61285	−0.343501
$268$	0	0
$269$	8.48886	0.517575	0.258788	−	0.965934i	$-0.416677\pi$
$269$	8.48886	0.517575	0.258788	+	0.965934i	$0.416677\pi$
$270$	0	0
$271$	−14.6637	−0.890757	−0.445378	−	0.895343i	$-0.646931\pi$
$271$	−14.6637	−0.890757	−0.445378	+	0.895343i	$0.646931\pi$
$272$	0	0
$273$	2.75557	0.166775
$274$	0	0
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	14.6035	0.877438	0.438719	−	0.898624i	$-0.355432\pi$
$277$	14.6035	0.877438	0.438719	+	0.898624i	$0.355432\pi$
$278$	0	0
$279$	−2.75557	−0.164972
$280$	0	0
$281$	−0.193576	−0.0115478	−0.00577389	−	0.999983i	$-0.501838\pi$
$281$	−0.193576	−0.0115478	−0.00577389	+	0.999983i	$0.501838\pi$
$282$	0	0
$283$	−27.1842	−1.61593	−0.807967	−	0.589228i	$-0.799432\pi$
$283$	−27.1842	−1.61593	−0.807967	+	0.589228i	$0.799432\pi$
$284$	0	0
$285$	7.05086	0.417657
$286$	0	0
$287$	−0.857279	−0.0506036
$288$	0	0
$289$	9.87601	0.580942
$290$	0	0
$291$	−7.24443	−0.424676
$292$	0	0
$293$	−2.81579	−0.164500	−0.0822502	−	0.996612i	$-0.526211\pi$
$293$	−2.81579	−0.164500	−0.0822502	+	0.996612i	$0.526211\pi$
$294$	0	0
$295$	4.85728	0.282802
$296$	0	0
$297$	1.00000	0.0580259
$298$	0	0
$299$	5.51114	0.318717
$300$	0	0
$301$	−25.1240	−1.44812
$302$	0	0
$303$	−4.66370	−0.267923
$304$	0	0
$305$	6.85728	0.392647
$306$	0	0
$307$	24.4286	1.39422	0.697108	−	0.716966i	$-0.254470\pi$
$307$	24.4286	1.39422	0.697108	+	0.716966i	$0.254470\pi$
$308$	0	0
$309$	−11.6128	−0.660632
$310$	0	0
$311$	−19.8796	−1.12727	−0.563633	−	0.826025i	$-0.690597\pi$
$311$	−19.8796	−1.12727	−0.563633	+	0.826025i	$0.690597\pi$
$312$	0	0
$313$	−15.7146	−0.888239	−0.444120	−	0.895967i	$-0.646483\pi$
$313$	−15.7146	−0.888239	−0.444120	+	0.895967i	$0.646483\pi$
$314$	0	0
$315$	4.42864	0.249526
$316$	0	0
$317$	16.4889	0.926107	0.463053	−	0.886330i	$-0.346754\pi$
$317$	16.4889	0.926107	0.463053	+	0.886330i	$0.346754\pi$
$318$	0	0
$319$	7.80642	0.437076
$320$	0	0
$321$	2.62222	0.146358
$322$	0	0
$323$	36.5531	2.03387
$324$	0	0
$325$	−0.622216	−0.0345143
$326$	0	0
$327$	19.7146	1.09022
$328$	0	0
$329$	12.2034	0.672796
$330$	0	0
$331$	−15.3461	−0.843500	−0.421750	−	0.906712i	$-0.638584\pi$
$331$	−15.3461	−0.843500	−0.421750	+	0.906712i	$0.638584\pi$
$332$	0	0
$333$	−2.00000	−0.109599
$334$	0	0
$335$	1.24443	0.0679905
$336$	0	0
$337$	28.2351	1.53806	0.769031	−	0.639212i	$-0.220739\pi$
$337$	28.2351	1.53806	0.769031	+	0.639212i	$0.220739\pi$
$338$	0	0
$339$	6.00000	0.325875
$340$	0	0
$341$	2.75557	0.149222
$342$	0	0
$343$	24.8573	1.34217
$344$	0	0
$345$	8.85728	0.476860
$346$	0	0
$347$	−2.62222	−0.140768	−0.0703840	−	0.997520i	$-0.522422\pi$
$347$	−2.62222	−0.140768	−0.0703840	+	0.997520i	$0.522422\pi$
$348$	0	0
$349$	5.14272	0.275284	0.137642	−	0.990482i	$-0.456048\pi$
$349$	5.14272	0.275284	0.137642	+	0.990482i	$0.456048\pi$
$350$	0	0
$351$	0.622216	0.0332114
$352$	0	0
$353$	−9.34614	−0.497445	−0.248722	−	0.968575i	$-0.580011\pi$
$353$	−9.34614	−0.497445	−0.248722	+	0.968575i	$0.580011\pi$
$354$	0	0
$355$	−2.75557	−0.146250
$356$	0	0
$357$	22.9590	1.21512
$358$	0	0
$359$	−10.7556	−0.567657	−0.283829	−	0.958875i	$-0.591605\pi$
$359$	−10.7556	−0.567657	−0.283829	+	0.958875i	$0.591605\pi$
$360$	0	0
$361$	30.7146	1.61656
$362$	0	0
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	4.23506	0.221673
$366$	0	0
$367$	−33.7975	−1.76422	−0.882108	−	0.471046i	$-0.843876\pi$
$367$	−33.7975	−1.76422	−0.882108	+	0.471046i	$0.843876\pi$
$368$	0	0
$369$	−0.193576	−0.0100772
$370$	0	0
$371$	−48.0830	−2.49634
$372$	0	0
$373$	33.9496	1.75784	0.878922	−	0.476965i	$-0.158263\pi$
$373$	33.9496	1.75784	0.878922	+	0.476965i	$0.158263\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	4.85728	0.250163
$378$	0	0
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	0	0
$381$	−15.1842	−0.777911
$382$	0	0
$383$	14.6351	0.747820	0.373910	−	0.927465i	$-0.378017\pi$
$383$	14.6351	0.747820	0.373910	+	0.927465i	$0.378017\pi$
$384$	0	0
$385$	−4.42864	−0.225704
$386$	0	0
$387$	−5.67307	−0.288378
$388$	0	0
$389$	−5.61285	−0.284583	−0.142291	−	0.989825i	$-0.545447\pi$
$389$	−5.61285	−0.284583	−0.142291	+	0.989825i	$0.545447\pi$
$390$	0	0
$391$	45.9180	2.32217
$392$	0	0
$393$	1.24443	0.0627733
$394$	0	0
$395$	−8.56199	−0.430801
$396$	0	0
$397$	−12.7556	−0.640184	−0.320092	−	0.947387i	$-0.603714\pi$
$397$	−12.7556	−0.640184	−0.320092	+	0.947387i	$0.603714\pi$
$398$	0	0
$399$	31.2257	1.56324
$400$	0	0
$401$	2.00000	0.0998752	0.0499376	−	0.998752i	$-0.484098\pi$
$401$	2.00000	0.0998752	0.0499376	+	0.998752i	$0.484098\pi$
$402$	0	0
$403$	1.71456	0.0854082
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	2.00000	0.0991363
$408$	0	0
$409$	−7.12399	−0.352258	−0.176129	−	0.984367i	$-0.556358\pi$
$409$	−7.12399	−0.352258	−0.176129	+	0.984367i	$0.556358\pi$
$410$	0	0
$411$	0.488863	0.0241138
$412$	0	0
$413$	21.5111	1.05849
$414$	0	0
$415$	−0.133353	−0.00654605
$416$	0	0
$417$	17.8064	0.871984
$418$	0	0
$419$	−15.6128	−0.762738	−0.381369	−	0.924423i	$-0.624547\pi$
$419$	−15.6128	−0.762738	−0.381369	+	0.924423i	$0.624547\pi$
$420$	0	0
$421$	7.89829	0.384939	0.192470	−	0.981303i	$-0.438350\pi$
$421$	7.89829	0.384939	0.192470	+	0.981303i	$0.438350\pi$
$422$	0	0
$423$	2.75557	0.133980
$424$	0	0
$425$	−5.18421	−0.251471
$426$	0	0
$427$	30.3684	1.46963
$428$	0	0
$429$	−0.622216	−0.0300409
$430$	0	0
$431$	−34.3051	−1.65242	−0.826210	−	0.563362i	$-0.809508\pi$
$431$	−34.3051	−1.65242	−0.826210	+	0.563362i	$0.809508\pi$
$432$	0	0
$433$	14.4701	0.695390	0.347695	−	0.937608i	$-0.386964\pi$
$433$	14.4701	0.695390	0.347695	+	0.937608i	$0.386964\pi$
$434$	0	0
$435$	7.80642	0.374289
$436$	0	0
$437$	62.4514	2.98746
$438$	0	0
$439$	19.3176	0.921977	0.460988	−	0.887406i	$-0.347495\pi$
$439$	19.3176	0.921977	0.460988	+	0.887406i	$0.347495\pi$
$440$	0	0
$441$	12.6128	0.600612
$442$	0	0
$443$	13.1240	0.623539	0.311770	−	0.950158i	$-0.399078\pi$
$443$	13.1240	0.623539	0.311770	+	0.950158i	$0.399078\pi$
$444$	0	0
$445$	5.61285	0.266074
$446$	0	0
$447$	1.43801	0.0680154
$448$	0	0
$449$	−32.3051	−1.52457	−0.762287	−	0.647240i	$-0.775923\pi$
$449$	−32.3051	−1.52457	−0.762287	+	0.647240i	$0.775923\pi$
$450$	0	0
$451$	0.193576	0.00911514
$452$	0	0
$453$	−12.1748	−0.572024
$454$	0	0
$455$	−2.75557	−0.129183
$456$	0	0
$457$	−23.4608	−1.09745	−0.548724	−	0.836004i	$-0.684886\pi$
$457$	−23.4608	−1.09745	−0.548724	+	0.836004i	$0.684886\pi$
$458$	0	0
$459$	5.18421	0.241978
$460$	0	0
$461$	−28.8671	−1.34448	−0.672238	−	0.740335i	$-0.734667\pi$
$461$	−28.8671	−1.34448	−0.672238	+	0.740335i	$0.734667\pi$
$462$	0	0
$463$	19.3461	0.899091	0.449546	−	0.893257i	$-0.351586\pi$
$463$	19.3461	0.899091	0.449546	+	0.893257i	$0.351586\pi$
$464$	0	0
$465$	2.75557	0.127786
$466$	0	0
$467$	−3.14272	−0.145428	−0.0727139	−	0.997353i	$-0.523166\pi$
$467$	−3.14272	−0.145428	−0.0727139	+	0.997353i	$0.523166\pi$
$468$	0	0
$469$	5.51114	0.254481
$470$	0	0
$471$	−18.4701	−0.851059
$472$	0	0
$473$	5.67307	0.260848
$474$	0	0
$475$	−7.05086	−0.323515
$476$	0	0
$477$	−10.8573	−0.497121
$478$	0	0
$479$	24.8573	1.13576	0.567879	−	0.823112i	$-0.307764\pi$
$479$	24.8573	1.13576	0.567879	+	0.823112i	$0.307764\pi$
$480$	0	0
$481$	1.24443	0.0567412
$482$	0	0
$483$	39.2257	1.78483
$484$	0	0
$485$	7.24443	0.328953
$486$	0	0
$487$	11.5299	0.522468	0.261234	−	0.965275i	$-0.415871\pi$
$487$	11.5299	0.522468	0.261234	+	0.965275i	$0.415871\pi$
$488$	0	0
$489$	−10.1017	−0.456815
$490$	0	0
$491$	−16.3872	−0.739542	−0.369771	−	0.929123i	$-0.620564\pi$
$491$	−16.3872	−0.739542	−0.369771	+	0.929123i	$0.620564\pi$
$492$	0	0
$493$	40.4701	1.82268
$494$	0	0
$495$	−1.00000	−0.0449467
$496$	0	0
$497$	−12.2034	−0.547398
$498$	0	0
$499$	25.3274	1.13381	0.566905	−	0.823783i	$-0.308141\pi$
$499$	25.3274	1.13381	0.566905	+	0.823783i	$0.308141\pi$
$500$	0	0
$501$	−16.3368	−0.729873
$502$	0	0
$503$	−19.0923	−0.851285	−0.425643	−	0.904891i	$-0.639952\pi$
$503$	−19.0923	−0.851285	−0.425643	+	0.904891i	$0.639952\pi$
$504$	0	0
$505$	4.66370	0.207532
$506$	0	0
$507$	12.6128	0.560156
$508$	0	0
$509$	−32.4514	−1.43838	−0.719191	−	0.694812i	$-0.755488\pi$
$509$	−32.4514	−1.43838	−0.719191	+	0.694812i	$0.755488\pi$
$510$	0	0
$511$	18.7556	0.829698
$512$	0	0
$513$	7.05086	0.311303
$514$	0	0
$515$	11.6128	0.511723
$516$	0	0
$517$	−2.75557	−0.121190
$518$	0	0
$519$	9.18421	0.403142
$520$	0	0
$521$	−29.2257	−1.28040	−0.640200	−	0.768208i	$-0.721149\pi$
$521$	−29.2257	−1.28040	−0.640200	+	0.768208i	$0.721149\pi$
$522$	0	0
$523$	−6.71408	−0.293586	−0.146793	−	0.989167i	$-0.546895\pi$
$523$	−6.71408	−0.293586	−0.146793	+	0.989167i	$0.546895\pi$
$524$	0	0
$525$	−4.42864	−0.193282
$526$	0	0
$527$	14.2854	0.622284
$528$	0	0
$529$	55.4514	2.41093
$530$	0	0
$531$	4.85728	0.210788
$532$	0	0
$533$	0.120446	0.00521710
$534$	0	0
$535$	−2.62222	−0.113368
$536$	0	0
$537$	−25.3274	−1.09296
$538$	0	0
$539$	−12.6128	−0.543274
$540$	0	0
$541$	−18.0000	−0.773880	−0.386940	−	0.922105i	$-0.626468\pi$
$541$	−18.0000	−0.773880	−0.386940	+	0.922105i	$0.626468\pi$
$542$	0	0
$543$	13.6128	0.584183
$544$	0	0
$545$	−19.7146	−0.844479
$546$	0	0
$547$	−41.3689	−1.76881	−0.884403	−	0.466724i	$-0.845434\pi$
$547$	−41.3689	−1.76881	−0.884403	+	0.466724i	$0.845434\pi$
$548$	0	0
$549$	6.85728	0.292662
$550$	0	0
$551$	55.0420	2.34487
$552$	0	0
$553$	−37.9180	−1.61244
$554$	0	0
$555$	2.00000	0.0848953
$556$	0	0
$557$	−20.7971	−0.881200	−0.440600	−	0.897704i	$-0.645234\pi$
$557$	−20.7971	−0.881200	−0.440600	+	0.897704i	$0.645234\pi$
$558$	0	0
$559$	3.52987	0.149298
$560$	0	0
$561$	−5.18421	−0.218877
$562$	0	0
$563$	−37.7275	−1.59002	−0.795012	−	0.606594i	$-0.792535\pi$
$563$	−37.7275	−1.59002	−0.795012	+	0.606594i	$0.792535\pi$
$564$	0	0
$565$	−6.00000	−0.252422
$566$	0	0
$567$	4.42864	0.185985
$568$	0	0
$569$	−7.33630	−0.307554	−0.153777	−	0.988106i	$-0.549144\pi$
$569$	−7.33630	−0.307554	−0.153777	+	0.988106i	$0.549144\pi$
$570$	0	0
$571$	−36.6450	−1.53354	−0.766772	−	0.641919i	$-0.778138\pi$
$571$	−36.6450	−1.53354	−0.766772	+	0.641919i	$0.778138\pi$
$572$	0	0
$573$	6.10171	0.254903
$574$	0	0
$575$	−8.85728	−0.369374
$576$	0	0
$577$	4.22216	0.175771	0.0878853	−	0.996131i	$-0.471989\pi$
$577$	4.22216	0.175771	0.0878853	+	0.996131i	$0.471989\pi$
$578$	0	0
$579$	−18.3368	−0.762050
$580$	0	0
$581$	−0.590573	−0.0245011
$582$	0	0
$583$	10.8573	0.449663
$584$	0	0
$585$	−0.622216	−0.0257255
$586$	0	0
$587$	−34.3684	−1.41854	−0.709268	−	0.704939i	$-0.750974\pi$
$587$	−34.3684	−1.41854	−0.709268	+	0.704939i	$0.750974\pi$
$588$	0	0
$589$	19.4291	0.800563
$590$	0	0
$591$	6.69535	0.275410
$592$	0	0
$593$	27.9398	1.14735	0.573675	−	0.819083i	$-0.305517\pi$
$593$	27.9398	1.14735	0.573675	+	0.819083i	$0.305517\pi$
$594$	0	0
$595$	−22.9590	−0.941227
$596$	0	0
$597$	14.1017	0.577145
$598$	0	0
$599$	31.2257	1.27585	0.637924	−	0.770100i	$-0.279794\pi$
$599$	31.2257	1.27585	0.637924	+	0.770100i	$0.279794\pi$
$600$	0	0
$601$	−8.75557	−0.357147	−0.178574	−	0.983927i	$-0.557148\pi$
$601$	−8.75557	−0.357147	−0.178574	+	0.983927i	$0.557148\pi$
$602$	0	0
$603$	1.24443	0.0506772
$604$	0	0
$605$	1.00000	0.0406558
$606$	0	0
$607$	15.1842	0.616308	0.308154	−	0.951336i	$-0.400289\pi$
$607$	15.1842	0.616308	0.308154	+	0.951336i	$0.400289\pi$
$608$	0	0
$609$	34.5718	1.40092
$610$	0	0
$611$	−1.71456	−0.0693636
$612$	0	0
$613$	42.7239	1.72560	0.862802	−	0.505543i	$-0.168708\pi$
$613$	42.7239	1.72560	0.862802	+	0.505543i	$0.168708\pi$
$614$	0	0
$615$	0.193576	0.00780574
$616$	0	0
$617$	−3.51114	−0.141353	−0.0706765	−	0.997499i	$-0.522516\pi$
$617$	−3.51114	−0.141353	−0.0706765	+	0.997499i	$0.522516\pi$
$618$	0	0
$619$	−17.5941	−0.707167	−0.353584	−	0.935403i	$-0.615037\pi$
$619$	−17.5941	−0.707167	−0.353584	+	0.935403i	$0.615037\pi$
$620$	0	0
$621$	8.85728	0.355430
$622$	0	0
$623$	24.8573	0.995886
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−7.05086	−0.281584
$628$	0	0
$629$	10.3684	0.413416
$630$	0	0
$631$	−15.8163	−0.629636	−0.314818	−	0.949152i	$-0.601943\pi$
$631$	−15.8163	−0.629636	−0.314818	+	0.949152i	$0.601943\pi$
$632$	0	0
$633$	−10.6637	−0.423844
$634$	0	0
$635$	15.1842	0.602567
$636$	0	0
$637$	−7.84791	−0.310946
$638$	0	0
$639$	−2.75557	−0.109009
$640$	0	0
$641$	25.8163	1.01968	0.509841	−	0.860269i	$-0.329704\pi$
$641$	25.8163	1.01968	0.509841	+	0.860269i	$0.329704\pi$
$642$	0	0
$643$	−18.1017	−0.713862	−0.356931	−	0.934131i	$-0.616177\pi$
$643$	−18.1017	−0.713862	−0.356931	+	0.934131i	$0.616177\pi$
$644$	0	0
$645$	5.67307	0.223377
$646$	0	0
$647$	47.0420	1.84941	0.924705	−	0.380684i	$-0.124311\pi$
$647$	47.0420	1.84941	0.924705	+	0.380684i	$0.124311\pi$
$648$	0	0
$649$	−4.85728	−0.190665
$650$	0	0
$651$	12.2034	0.478290
$652$	0	0
$653$	30.0830	1.17724	0.588619	−	0.808411i	$-0.299672\pi$
$653$	30.0830	1.17724	0.588619	+	0.808411i	$0.299672\pi$
$654$	0	0
$655$	−1.24443	−0.0486240
$656$	0	0
$657$	4.23506	0.165226
$658$	0	0
$659$	10.2854	0.400664	0.200332	−	0.979728i	$-0.435798\pi$
$659$	10.2854	0.400664	0.200332	+	0.979728i	$0.435798\pi$
$660$	0	0
$661$	−27.7146	−1.07797	−0.538986	−	0.842315i	$-0.681192\pi$
$661$	−27.7146	−1.07797	−0.538986	+	0.842315i	$0.681192\pi$
$662$	0	0
$663$	−3.22570	−0.125276
$664$	0	0
$665$	−31.2257	−1.21088
$666$	0	0
$667$	69.1437	2.67725
$668$	0	0
$669$	8.85728	0.342442
$670$	0	0
$671$	−6.85728	−0.264722
$672$	0	0
$673$	−9.86665	−0.380331	−0.190166	−	0.981752i	$-0.560903\pi$
$673$	−9.86665	−0.380331	−0.190166	+	0.981752i	$0.560903\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−5.65433	−0.217314	−0.108657	−	0.994079i	$-0.534655\pi$
$677$	−5.65433	−0.217314	−0.108657	+	0.994079i	$0.534655\pi$
$678$	0	0
$679$	32.0830	1.23123
$680$	0	0
$681$	13.3778	0.512638
$682$	0	0
$683$	−34.1847	−1.30804	−0.654020	−	0.756477i	$-0.726919\pi$
$683$	−34.1847	−1.30804	−0.654020	+	0.756477i	$0.726919\pi$
$684$	0	0
$685$	−0.488863	−0.0186785
$686$	0	0
$687$	−11.5111	−0.439177
$688$	0	0
$689$	6.75557	0.257367
$690$	0	0
$691$	19.2257	0.731380	0.365690	−	0.930737i	$-0.380833\pi$
$691$	19.2257	0.731380	0.365690	+	0.930737i	$0.380833\pi$
$692$	0	0
$693$	−4.42864	−0.168230
$694$	0	0
$695$	−17.8064	−0.675436
$696$	0	0
$697$	1.00354	0.0380118
$698$	0	0
$699$	4.32693	0.163659
$700$	0	0
$701$	−29.9081	−1.12961	−0.564807	−	0.825223i	$-0.691050\pi$
$701$	−29.9081	−1.12961	−0.564807	+	0.825223i	$0.691050\pi$
$702$	0	0
$703$	14.1017	0.531856
$704$	0	0
$705$	−2.75557	−0.103781
$706$	0	0
$707$	20.6539	0.776768
$708$	0	0
$709$	−15.3274	−0.575633	−0.287816	−	0.957686i	$-0.592929\pi$
$709$	−15.3274	−0.575633	−0.287816	+	0.957686i	$0.592929\pi$
$710$	0	0
$711$	−8.56199	−0.321100
$712$	0	0
$713$	24.4068	0.914043
$714$	0	0
$715$	0.622216	0.0232695
$716$	0	0
$717$	−3.34614	−0.124964
$718$	0	0
$719$	−23.8163	−0.888197	−0.444098	−	0.895978i	$-0.646476\pi$
$719$	−23.8163	−0.888197	−0.444098	+	0.895978i	$0.646476\pi$
$720$	0	0
$721$	51.4291	1.91532
$722$	0	0
$723$	1.34614	0.0500635
$724$	0	0
$725$	−7.80642	−0.289923
$726$	0	0
$727$	32.9403	1.22169	0.610843	−	0.791752i	$-0.290831\pi$
$727$	32.9403	1.22169	0.610843	+	0.791752i	$0.290831\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	29.4104	1.08778
$732$	0	0
$733$	−29.8666	−1.10315	−0.551575	−	0.834125i	$-0.685973\pi$
$733$	−29.8666	−1.10315	−0.551575	+	0.834125i	$0.685973\pi$
$734$	0	0
$735$	−12.6128	−0.465232
$736$	0	0
$737$	−1.24443	−0.0458392
$738$	0	0
$739$	5.06959	0.186488	0.0932440	−	0.995643i	$-0.470276\pi$
$739$	5.06959	0.186488	0.0932440	+	0.995643i	$0.470276\pi$
$740$	0	0
$741$	−4.38715	−0.161166
$742$	0	0
$743$	−22.4385	−0.823188	−0.411594	−	0.911367i	$-0.635028\pi$
$743$	−22.4385	−0.823188	−0.411594	+	0.911367i	$0.635028\pi$
$744$	0	0
$745$	−1.43801	−0.0526845
$746$	0	0
$747$	−0.133353	−0.00487913
$748$	0	0
$749$	−11.6128	−0.424324
$750$	0	0
$751$	6.63512	0.242119	0.121060	−	0.992645i	$-0.461371\pi$
$751$	6.63512	0.242119	0.121060	+	0.992645i	$0.461371\pi$
$752$	0	0
$753$	22.7556	0.829259
$754$	0	0
$755$	12.1748	0.443088
$756$	0	0
$757$	8.75557	0.318227	0.159113	−	0.987260i	$-0.449136\pi$
$757$	8.75557	0.318227	0.159113	+	0.987260i	$0.449136\pi$
$758$	0	0
$759$	−8.85728	−0.321499
$760$	0	0
$761$	3.15257	0.114280	0.0571402	−	0.998366i	$-0.481802\pi$
$761$	3.15257	0.114280	0.0571402	+	0.998366i	$0.481802\pi$
$762$	0	0
$763$	−87.3087	−3.16079
$764$	0	0
$765$	−5.18421	−0.187435
$766$	0	0
$767$	−3.02227	−0.109128
$768$	0	0
$769$	−28.9590	−1.04429	−0.522144	−	0.852857i	$-0.674868\pi$
$769$	−28.9590	−1.04429	−0.522144	+	0.852857i	$0.674868\pi$
$770$	0	0
$771$	6.85728	0.246959
$772$	0	0
$773$	29.1427	1.04819	0.524095	−	0.851660i	$-0.324404\pi$
$773$	29.1427	1.04819	0.524095	+	0.851660i	$0.324404\pi$
$774$	0	0
$775$	−2.75557	−0.0989830
$776$	0	0
$777$	8.85728	0.317753
$778$	0	0
$779$	1.36488	0.0489018
$780$	0	0
$781$	2.75557	0.0986020
$782$	0	0
$783$	7.80642	0.278979
$784$	0	0
$785$	18.4701	0.659227
$786$	0	0
$787$	11.2672	0.401632	0.200816	−	0.979629i	$-0.435641\pi$
$787$	11.2672	0.401632	0.200816	+	0.979629i	$0.435641\pi$
$788$	0	0
$789$	−29.5812	−1.05312
$790$	0	0
$791$	−26.5718	−0.944786
$792$	0	0
$793$	−4.26671	−0.151515
$794$	0	0
$795$	10.8573	0.385068
$796$	0	0
$797$	41.9625	1.48639	0.743195	−	0.669075i	$-0.233310\pi$
$797$	41.9625	1.48639	0.743195	+	0.669075i	$0.233310\pi$
$798$	0	0
$799$	−14.2854	−0.505383
$800$	0	0
$801$	5.61285	0.198320
$802$	0	0
$803$	−4.23506	−0.149452
$804$	0	0
$805$	−39.2257	−1.38252
$806$	0	0
$807$	−8.48886	−0.298822
$808$	0	0
$809$	−27.8064	−0.977622	−0.488811	−	0.872390i	$-0.662569\pi$
$809$	−27.8064	−0.977622	−0.488811	+	0.872390i	$0.662569\pi$
$810$	0	0
$811$	−6.78415	−0.238224	−0.119112	−	0.992881i	$-0.538005\pi$
$811$	−6.78415	−0.238224	−0.119112	+	0.992881i	$0.538005\pi$
$812$	0	0
$813$	14.6637	0.514279
$814$	0	0
$815$	10.1017	0.353847
$816$	0	0
$817$	40.0000	1.39942
$818$	0	0
$819$	−2.75557	−0.0962874
$820$	0	0
$821$	3.62269	0.126433	0.0632164	−	0.998000i	$-0.479864\pi$
$821$	3.62269	0.126433	0.0632164	+	0.998000i	$0.479864\pi$
$822$	0	0
$823$	−42.0642	−1.46627	−0.733134	−	0.680085i	$-0.761943\pi$
$823$	−42.0642	−1.46627	−0.733134	+	0.680085i	$0.761943\pi$
$824$	0	0
$825$	1.00000	0.0348155
$826$	0	0
$827$	−30.8256	−1.07191	−0.535956	−	0.844246i	$-0.680049\pi$
$827$	−30.8256	−1.07191	−0.535956	+	0.844246i	$0.680049\pi$
$828$	0	0
$829$	7.12399	0.247426	0.123713	−	0.992318i	$-0.460520\pi$
$829$	7.12399	0.247426	0.123713	+	0.992318i	$0.460520\pi$
$830$	0	0
$831$	−14.6035	−0.506589
$832$	0	0
$833$	−65.3876	−2.26555
$834$	0	0
$835$	16.3368	0.565357
$836$	0	0
$837$	2.75557	0.0952464
$838$	0	0
$839$	−3.34614	−0.115522	−0.0577608	−	0.998330i	$-0.518396\pi$
$839$	−3.34614	−0.115522	−0.0577608	+	0.998330i	$0.518396\pi$
$840$	0	0
$841$	31.9403	1.10139
$842$	0	0
$843$	0.193576	0.00666712
$844$	0	0
$845$	−12.6128	−0.433895
$846$	0	0
$847$	4.42864	0.152170
$848$	0	0
$849$	27.1842	0.932960
$850$	0	0
$851$	17.7146	0.607247
$852$	0	0
$853$	−26.4197	−0.904595	−0.452297	−	0.891867i	$-0.649395\pi$
$853$	−26.4197	−0.904595	−0.452297	+	0.891867i	$0.649395\pi$
$854$	0	0
$855$	−7.05086	−0.241134
$856$	0	0
$857$	38.7783	1.32464	0.662321	−	0.749220i	$-0.269571\pi$
$857$	38.7783	1.32464	0.662321	+	0.749220i	$0.269571\pi$
$858$	0	0
$859$	27.3087	0.931760	0.465880	−	0.884848i	$-0.345738\pi$
$859$	27.3087	0.931760	0.465880	+	0.884848i	$0.345738\pi$
$860$	0	0
$861$	0.857279	0.0292160
$862$	0	0
$863$	49.5308	1.68605	0.843024	−	0.537875i	$-0.180773\pi$
$863$	49.5308	1.68605	0.843024	+	0.537875i	$0.180773\pi$
$864$	0	0
$865$	−9.18421	−0.312272
$866$	0	0
$867$	−9.87601	−0.335407
$868$	0	0
$869$	8.56199	0.290446
$870$	0	0
$871$	−0.774305	−0.0262363
$872$	0	0
$873$	7.24443	0.245187
$874$	0	0
$875$	4.42864	0.149715
$876$	0	0
$877$	−4.50177	−0.152014	−0.0760070	−	0.997107i	$-0.524217\pi$
$877$	−4.50177	−0.152014	−0.0760070	+	0.997107i	$0.524217\pi$
$878$	0	0
$879$	2.81579	0.0949743
$880$	0	0
$881$	−15.1240	−0.509540	−0.254770	−	0.967002i	$-0.582000\pi$
$881$	−15.1240	−0.509540	−0.254770	+	0.967002i	$0.582000\pi$
$882$	0	0
$883$	30.2480	1.01793	0.508963	−	0.860789i	$-0.330029\pi$
$883$	30.2480	1.01793	0.508963	+	0.860789i	$0.330029\pi$
$884$	0	0
$885$	−4.85728	−0.163276
$886$	0	0
$887$	−57.1941	−1.92039	−0.960194	−	0.279333i	$-0.909887\pi$
$887$	−57.1941	−1.92039	−0.960194	+	0.279333i	$0.909887\pi$
$888$	0	0
$889$	67.2454	2.25534
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	−19.4291	−0.650171
$894$	0	0
$895$	25.3274	0.846602
$896$	0	0
$897$	−5.51114	−0.184012
$898$	0	0
$899$	21.5111	0.717437
$900$	0	0
$901$	56.2864	1.87517
$902$	0	0
$903$	25.1240	0.836074
$904$	0	0
$905$	−13.6128	−0.452506
$906$	0	0
$907$	53.2641	1.76861	0.884303	−	0.466913i	$-0.154634\pi$
$907$	53.2641	1.76861	0.884303	+	0.466913i	$0.154634\pi$
$908$	0	0
$909$	4.66370	0.154685
$910$	0	0
$911$	0.590573	0.0195665	0.00978327	−	0.999952i	$-0.496886\pi$
$911$	0.590573	0.0195665	0.00978327	+	0.999952i	$0.496886\pi$
$912$	0	0
$913$	0.133353	0.00441334
$914$	0	0
$915$	−6.85728	−0.226695
$916$	0	0
$917$	−5.51114	−0.181994
$918$	0	0
$919$	−55.8707	−1.84300	−0.921502	−	0.388375i	$-0.873037\pi$
$919$	−55.8707	−1.84300	−0.921502	+	0.388375i	$0.873037\pi$
$920$	0	0
$921$	−24.4286	−0.804951
$922$	0	0
$923$	1.71456	0.0564354
$924$	0	0
$925$	−2.00000	−0.0657596
$926$	0	0
$927$	11.6128	0.381416
$928$	0	0
$929$	15.3274	0.502876	0.251438	−	0.967873i	$-0.419097\pi$
$929$	15.3274	0.502876	0.251438	+	0.967873i	$0.419097\pi$
$930$	0	0
$931$	−88.9314	−2.91461
$932$	0	0
$933$	19.8796	0.650827
$934$	0	0
$935$	5.18421	0.169542
$936$	0	0
$937$	−27.8479	−0.909752	−0.454876	−	0.890555i	$-0.650316\pi$
$937$	−27.8479	−0.909752	−0.454876	+	0.890555i	$0.650316\pi$
$938$	0	0
$939$	15.7146	0.512825
$940$	0	0
$941$	10.4157	0.339543	0.169772	−	0.985483i	$-0.445697\pi$
$941$	10.4157	0.339543	0.169772	+	0.985483i	$0.445697\pi$
$942$	0	0
$943$	1.71456	0.0558337
$944$	0	0
$945$	−4.42864	−0.144064
$946$	0	0
$947$	−8.47013	−0.275242	−0.137621	−	0.990485i	$-0.543946\pi$
$947$	−8.47013	−0.275242	−0.137621	+	0.990485i	$0.543946\pi$
$948$	0	0
$949$	−2.63512	−0.0855397
$950$	0	0
$951$	−16.4889	−0.534688
$952$	0	0
$953$	−8.71408	−0.282277	−0.141138	−	0.989990i	$-0.545076\pi$
$953$	−8.71408	−0.282277	−0.141138	+	0.989990i	$0.545076\pi$
$954$	0	0
$955$	−6.10171	−0.197447
$956$	0	0
$957$	−7.80642	−0.252346
$958$	0	0
$959$	−2.16500	−0.0699114
$960$	0	0
$961$	−23.4068	−0.755059
$962$	0	0
$963$	−2.62222	−0.0844997
$964$	0	0
$965$	18.3368	0.590282
$966$	0	0
$967$	−44.2449	−1.42282	−0.711410	−	0.702777i	$-0.751943\pi$
$967$	−44.2449	−1.42282	−0.711410	+	0.702777i	$0.751943\pi$
$968$	0	0
$969$	−36.5531	−1.17425
$970$	0	0
$971$	57.1437	1.83383	0.916914	−	0.399085i	$-0.130672\pi$
$971$	57.1437	1.83383	0.916914	+	0.399085i	$0.130672\pi$
$972$	0	0
$973$	−78.8582	−2.52808
$974$	0	0
$975$	0.622216	0.0199268
$976$	0	0
$977$	−16.2480	−0.519819	−0.259909	−	0.965633i	$-0.583693\pi$
$977$	−16.2480	−0.519819	−0.259909	+	0.965633i	$0.583693\pi$
$978$	0	0
$979$	−5.61285	−0.179387
$980$	0	0
$981$	−19.7146	−0.629437
$982$	0	0
$983$	−1.12399	−0.0358496	−0.0179248	−	0.999839i	$-0.505706\pi$
$983$	−1.12399	−0.0358496	−0.0179248	+	0.999839i	$0.505706\pi$
$984$	0	0
$985$	−6.69535	−0.213331
$986$	0	0
$987$	−12.2034	−0.388439
$988$	0	0
$989$	50.2480	1.59779
$990$	0	0
$991$	53.6513	1.70429	0.852144	−	0.523307i	$-0.175302\pi$
$991$	53.6513	1.70429	0.852144	+	0.523307i	$0.175302\pi$
$992$	0	0
$993$	15.3461	0.486995
$994$	0	0
$995$	−14.1017	−0.447054
$996$	0	0
$997$	−35.7275	−1.13150	−0.565750	−	0.824577i	$-0.691413\pi$
$997$	−35.7275	−1.13150	−0.565750	+	0.824577i	$0.691413\pi$
$998$	0	0
$999$	2.00000	0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2640.2.a.be.1.3		3
3.2	odd	2		7920.2.a.cj.1.3		3
4.3	odd	2		165.2.a.c.1.3	✓	3
12.11	even	2		495.2.a.e.1.1		3
20.3	even	4		825.2.c.g.199.2		6
20.7	even	4		825.2.c.g.199.5		6
20.19	odd	2		825.2.a.k.1.1		3
28.27	even	2		8085.2.a.bk.1.3		3
44.43	even	2		1815.2.a.m.1.1		3
60.23	odd	4		2475.2.c.r.199.5		6
60.47	odd	4		2475.2.c.r.199.2		6
60.59	even	2		2475.2.a.bb.1.3		3
132.131	odd	2		5445.2.a.z.1.3		3
220.219	even	2		9075.2.a.cf.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.2.a.c.1.3	✓	3	4.3	odd	2
495.2.a.e.1.1		3	12.11	even	2
825.2.a.k.1.1		3	20.19	odd	2
825.2.c.g.199.2		6	20.3	even	4
825.2.c.g.199.5		6	20.7	even	4
1815.2.a.m.1.1		3	44.43	even	2
2475.2.a.bb.1.3		3	60.59	even	2
2475.2.c.r.199.2		6	60.47	odd	4
2475.2.c.r.199.5		6	60.23	odd	4
2640.2.a.be.1.3		3	1.1	even	1	trivial
5445.2.a.z.1.3		3	132.131	odd	2
7920.2.a.cj.1.3		3	3.2	odd	2
8085.2.a.bk.1.3		3	28.27	even	2
9075.2.a.cf.1.3		3	220.219	even	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:	\( N \)	\(=\)	\( 2640 = 2^{4} \cdot 3 \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2640.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} +4.42864 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.00000 q^{5} +4.42864 q^{7} +1.00000 q^{9} -1.00000 q^{11} -0.622216 q^{13} -1.00000 q^{15} -5.18421 q^{17} -7.05086 q^{19} -4.42864 q^{21} -8.85728 q^{23} +1.00000 q^{25} -1.00000 q^{27} -7.80642 q^{29} -2.75557 q^{31} +1.00000 q^{33} +4.42864 q^{35} -2.00000 q^{37} +0.622216 q^{39} -0.193576 q^{41} -5.67307 q^{43} +1.00000 q^{45} +2.75557 q^{47} +12.6128 q^{49} +5.18421 q^{51} -10.8573 q^{53} -1.00000 q^{55} +7.05086 q^{57} +4.85728 q^{59} +6.85728 q^{61} +4.42864 q^{63} -0.622216 q^{65} +1.24443 q^{67} +8.85728 q^{69} -2.75557 q^{71} +4.23506 q^{73} -1.00000 q^{75} -4.42864 q^{77} -8.56199 q^{79} +1.00000 q^{81} -0.133353 q^{83} -5.18421 q^{85} +7.80642 q^{87} +5.61285 q^{89} -2.75557 q^{91} +2.75557 q^{93} -7.05086 q^{95} +7.24443 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 3 q^{5} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 + 3 * q^5 + 3 * q^9	\( 3 q - 3 q^{3} + 3 q^{5} + 3 q^{9} - 3 q^{11} - 2 q^{13} - 3 q^{15} - 2 q^{17} - 8 q^{19} + 3 q^{25} - 3 q^{27} - 10 q^{29} - 8 q^{31} + 3 q^{33} - 6 q^{37} + 2 q^{39} - 14 q^{41} - 4 q^{43} + 3 q^{45} + 8 q^{47} + 11 q^{49} + 2 q^{51} - 6 q^{53} - 3 q^{55} + 8 q^{57} - 12 q^{59} - 6 q^{61} - 2 q^{65} + 4 q^{67} - 8 q^{71} - 14 q^{73} - 3 q^{75} - 12 q^{79} + 3 q^{81} - 2 q^{85} + 10 q^{87} - 10 q^{89} - 8 q^{91} + 8 q^{93} - 8 q^{95} + 22 q^{97} - 3 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 + 3 * q^5 + 3 * q^9 - 3 * q^11 - 2 * q^13 - 3 * q^15 - 2 * q^17 - 8 * q^19 + 3 * q^25 - 3 * q^27 - 10 * q^29 - 8 * q^31 + 3 * q^33 - 6 * q^37 + 2 * q^39 - 14 * q^41 - 4 * q^43 + 3 * q^45 + 8 * q^47 + 11 * q^49 + 2 * q^51 - 6 * q^53 - 3 * q^55 + 8 * q^57 - 12 * q^59 - 6 * q^61 - 2 * q^65 + 4 * q^67 - 8 * q^71 - 14 * q^73 - 3 * q^75 - 12 * q^79 + 3 * q^81 - 2 * q^85 + 10 * q^87 - 10 * q^89 - 8 * q^91 + 8 * q^93 - 8 * q^95 + 22 * q^97 - 3 * q^99

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