LMFDB - Embedded newform 6448.2.a.y.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6448,2,Mod(1,6448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6448, 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("6448.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6448 = 2^{4} \cdot 13 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6448.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:	$51.4875392233$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.5748973.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 6x^{4} + 4x^{3} + 8x^{2} - 4x - 1 $ x^6 - x^5 - 6x^4 + 4x^3 + 8x^2 - 4x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 403)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.699790$ of defining polynomial
Character	$\chi$	$=$	6448.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.300210 q^{3} +0.848172 q^{5} +1.74674 q^{7} -2.90987 q^{9} +O(q^{10})$	$q+0.300210 q^{3} +0.848172 q^{5} +1.74674 q^{7} -2.90987 q^{9} -0.172417 q^{11} +1.00000 q^{13} +0.254630 q^{15} -7.81383 q^{17} +7.12497 q^{19} +0.524389 q^{21} -2.39809 q^{23} -4.28060 q^{25} -1.77420 q^{27} +4.75312 q^{29} -1.00000 q^{31} -0.0517613 q^{33} +1.48153 q^{35} -0.630212 q^{37} +0.300210 q^{39} -5.65635 q^{41} -1.81589 q^{43} -2.46807 q^{45} -1.80389 q^{47} -3.94891 q^{49} -2.34579 q^{51} -1.19545 q^{53} -0.146239 q^{55} +2.13899 q^{57} -9.16997 q^{59} +12.0122 q^{61} -5.08279 q^{63} +0.848172 q^{65} +5.84643 q^{67} -0.719930 q^{69} -2.22849 q^{71} -6.64115 q^{73} -1.28508 q^{75} -0.301167 q^{77} +0.527743 q^{79} +8.19699 q^{81} -7.27854 q^{83} -6.62747 q^{85} +1.42694 q^{87} -12.9621 q^{89} +1.74674 q^{91} -0.300210 q^{93} +6.04320 q^{95} +0.368869 q^{97} +0.501711 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 5 q^{3} - 9 q^{5} - q^{9}+O(q^{10}) $ 6 * q + 5 * q^3 - 9 * q^5 - q^9	$ 6 q + 5 q^{3} - 9 q^{5} - q^{9} + 5 q^{11} + 6 q^{13} - 4 q^{15} - 23 q^{17} - 7 q^{19} + 2 q^{21} + 18 q^{23} + 11 q^{25} + 5 q^{27} - 18 q^{29} - 6 q^{31} - 2 q^{33} + q^{35} - 13 q^{37} + 5 q^{39} - 5 q^{41} + 7 q^{43} - 5 q^{45} + 9 q^{47} + 16 q^{49} - 26 q^{51} - 31 q^{53} - 7 q^{55} - 5 q^{57} + q^{59} - 15 q^{61} - 11 q^{63} - 9 q^{65} + 28 q^{67} + 5 q^{69} - q^{71} - 20 q^{73} - q^{75} - 29 q^{77} + 15 q^{79} + 2 q^{81} - q^{83} + 29 q^{85} - 10 q^{87} - q^{89} - 5 q^{93} + 13 q^{95} - 5 q^{97} + 15 q^{99}+O(q^{100}) $ 6 * q + 5 * q^3 - 9 * q^5 - q^9 + 5 * q^11 + 6 * q^13 - 4 * q^15 - 23 * q^17 - 7 * q^19 + 2 * q^21 + 18 * q^23 + 11 * q^25 + 5 * q^27 - 18 * q^29 - 6 * q^31 - 2 * q^33 + q^35 - 13 * q^37 + 5 * q^39 - 5 * q^41 + 7 * q^43 - 5 * q^45 + 9 * q^47 + 16 * q^49 - 26 * q^51 - 31 * q^53 - 7 * q^55 - 5 * q^57 + q^59 - 15 * q^61 - 11 * q^63 - 9 * q^65 + 28 * q^67 + 5 * q^69 - q^71 - 20 * q^73 - q^75 - 29 * q^77 + 15 * q^79 + 2 * q^81 - q^83 + 29 * q^85 - 10 * q^87 - q^89 - 5 * q^93 + 13 * q^95 - 5 * q^97 + 15 * 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
$3$	0.300210	0.173326	0.0866632	−	0.996238i	$-0.472380\pi$
$3$	0.300210	0.173326	0.0866632	+	0.996238i	$0.472380\pi$
$4$	0	0
$5$	0.848172	0.379314	0.189657	−	0.981850i	$-0.439262\pi$
$5$	0.848172	0.379314	0.189657	+	0.981850i	$0.439262\pi$
$6$	0	0
$7$	1.74674	0.660205	0.330102	−	0.943945i	$-0.392917\pi$
$7$	1.74674	0.660205	0.330102	+	0.943945i	$0.392917\pi$
$8$	0	0
$9$	−2.90987	−0.969958
$10$	0	0
$11$	−0.172417	−0.0519857	−0.0259928	−	0.999662i	$-0.508275\pi$
$11$	−0.172417	−0.0519857	−0.0259928	+	0.999662i	$0.508275\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0.254630	0.0657451
$16$	0	0
$17$	−7.81383	−1.89513	−0.947566	−	0.319561i	$-0.896464\pi$
$17$	−7.81383	−1.89513	−0.947566	+	0.319561i	$0.896464\pi$
$18$	0	0
$19$	7.12497	1.63458	0.817291	−	0.576226i	$-0.195475\pi$
$19$	7.12497	1.63458	0.817291	+	0.576226i	$0.195475\pi$
$20$	0	0
$21$	0.524389	0.114431
$22$	0	0
$23$	−2.39809	−0.500036	−0.250018	−	0.968241i	$-0.580436\pi$
$23$	−2.39809	−0.500036	−0.250018	+	0.968241i	$0.580436\pi$
$24$	0	0
$25$	−4.28060	−0.856121
$26$	0	0
$27$	−1.77420	−0.341446
$28$	0	0
$29$	4.75312	0.882632	0.441316	−	0.897352i	$-0.354512\pi$
$29$	4.75312	0.882632	0.441316	+	0.897352i	$0.354512\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	0	0
$33$	−0.0517613	−0.00901049
$34$	0	0
$35$	1.48153	0.250425
$36$	0	0
$37$	−0.630212	−0.103606	−0.0518031	−	0.998657i	$-0.516497\pi$
$37$	−0.630212	−0.103606	−0.0518031	+	0.998657i	$0.516497\pi$
$38$	0	0
$39$	0.300210	0.0480721
$40$	0	0
$41$	−5.65635	−0.883374	−0.441687	−	0.897169i	$-0.645620\pi$
$41$	−5.65635	−0.883374	−0.441687	+	0.897169i	$0.645620\pi$
$42$	0	0
$43$	−1.81589	−0.276921	−0.138460	−	0.990368i	$-0.544215\pi$
$43$	−1.81589	−0.276921	−0.138460	+	0.990368i	$0.544215\pi$
$44$	0	0
$45$	−2.46807	−0.367919
$46$	0	0
$47$	−1.80389	−0.263124	−0.131562	−	0.991308i	$-0.541999\pi$
$47$	−1.80389	−0.263124	−0.131562	+	0.991308i	$0.541999\pi$
$48$	0	0
$49$	−3.94891	−0.564130
$50$	0	0
$51$	−2.34579	−0.328476
$52$	0	0
$53$	−1.19545	−0.164208	−0.0821042	−	0.996624i	$-0.526164\pi$
$53$	−1.19545	−0.164208	−0.0821042	+	0.996624i	$0.526164\pi$
$54$	0	0
$55$	−0.146239	−0.0197189
$56$	0	0
$57$	2.13899	0.283316
$58$	0	0
$59$	−9.16997	−1.19383	−0.596914	−	0.802305i	$-0.703607\pi$
$59$	−9.16997	−1.19383	−0.596914	+	0.802305i	$0.703607\pi$
$60$	0	0
$61$	12.0122	1.53800	0.769001	−	0.639248i	$-0.220754\pi$
$61$	12.0122	1.53800	0.769001	+	0.639248i	$0.220754\pi$
$62$	0	0
$63$	−5.08279	−0.640371
$64$	0	0
$65$	0.848172	0.105203
$66$	0	0
$67$	5.84643	0.714256	0.357128	−	0.934056i	$-0.383756\pi$
$67$	5.84643	0.714256	0.357128	+	0.934056i	$0.383756\pi$
$68$	0	0
$69$	−0.719930	−0.0866694
$70$	0	0
$71$	−2.22849	−0.264473	−0.132236	−	0.991218i	$-0.542216\pi$
$71$	−2.22849	−0.264473	−0.132236	+	0.991218i	$0.542216\pi$
$72$	0	0
$73$	−6.64115	−0.777288	−0.388644	−	0.921388i	$-0.627056\pi$
$73$	−6.64115	−0.777288	−0.388644	+	0.921388i	$0.627056\pi$
$74$	0	0
$75$	−1.28508	−0.148388
$76$	0	0
$77$	−0.301167	−0.0343212
$78$	0	0
$79$	0.527743	0.0593757	0.0296879	−	0.999559i	$-0.490549\pi$
$79$	0.527743	0.0593757	0.0296879	+	0.999559i	$0.490549\pi$
$80$	0	0
$81$	8.19699	0.910776
$82$	0	0
$83$	−7.27854	−0.798924	−0.399462	−	0.916750i	$-0.630803\pi$
$83$	−7.27854	−0.798924	−0.399462	+	0.916750i	$0.630803\pi$
$84$	0	0
$85$	−6.62747	−0.718850
$86$	0	0
$87$	1.42694	0.152984
$88$	0	0
$89$	−12.9621	−1.37398	−0.686989	−	0.726668i	$-0.741068\pi$
$89$	−12.9621	−1.37398	−0.686989	+	0.726668i	$0.741068\pi$
$90$	0	0
$91$	1.74674	0.183108
$92$	0	0
$93$	−0.300210	−0.0311304
$94$	0	0
$95$	6.04320	0.620019
$96$	0	0
$97$	0.368869	0.0374529	0.0187265	−	0.999825i	$-0.494039\pi$
$97$	0.368869	0.0374529	0.0187265	+	0.999825i	$0.494039\pi$
$98$	0	0
$99$	0.501711	0.0504239
$100$	0	0
$101$	−8.82133	−0.877755	−0.438878	−	0.898547i	$-0.644624\pi$
$101$	−8.82133	−0.877755	−0.438878	+	0.898547i	$0.644624\pi$
$102$	0	0
$103$	−10.8093	−1.06507	−0.532535	−	0.846408i	$-0.678760\pi$
$103$	−10.8093	−1.06507	−0.532535	+	0.846408i	$0.678760\pi$
$104$	0	0
$105$	0.444772	0.0434053
$106$	0	0
$107$	4.55472	0.440321	0.220160	−	0.975464i	$-0.429342\pi$
$107$	4.55472	0.440321	0.220160	+	0.975464i	$0.429342\pi$
$108$	0	0
$109$	16.4271	1.57343	0.786717	−	0.617314i	$-0.211779\pi$
$109$	16.4271	1.57343	0.786717	+	0.617314i	$0.211779\pi$
$110$	0	0
$111$	−0.189196	−0.0179577
$112$	0	0
$113$	−4.47812	−0.421267	−0.210633	−	0.977565i	$-0.567553\pi$
$113$	−4.47812	−0.421267	−0.210633	+	0.977565i	$0.567553\pi$
$114$	0	0
$115$	−2.03399	−0.189671
$116$	0	0
$117$	−2.90987	−0.269018
$118$	0	0
$119$	−13.6487	−1.25117
$120$	0	0
$121$	−10.9703	−0.997297
$122$	0	0
$123$	−1.69809	−0.153112
$124$	0	0
$125$	−7.87155	−0.704053
$126$	0	0
$127$	−6.43512	−0.571025	−0.285512	−	0.958375i	$-0.592164\pi$
$127$	−6.43512	−0.571025	−0.285512	+	0.958375i	$0.592164\pi$
$128$	0	0
$129$	−0.545149	−0.0479977
$130$	0	0
$131$	12.3895	1.08247	0.541236	−	0.840871i	$-0.317957\pi$
$131$	12.3895	1.08247	0.541236	+	0.840871i	$0.317957\pi$
$132$	0	0
$133$	12.4455	1.07916
$134$	0	0
$135$	−1.50483	−0.129515
$136$	0	0
$137$	−18.7925	−1.60555	−0.802776	−	0.596281i	$-0.796645\pi$
$137$	−18.7925	−1.60555	−0.802776	+	0.596281i	$0.796645\pi$
$138$	0	0
$139$	−8.03922	−0.681878	−0.340939	−	0.940085i	$-0.610745\pi$
$139$	−8.03922	−0.681878	−0.340939	+	0.940085i	$0.610745\pi$
$140$	0	0
$141$	−0.541545	−0.0456063
$142$	0	0
$143$	−0.172417	−0.0144182
$144$	0	0
$145$	4.03146	0.334795
$146$	0	0
$147$	−1.18550	−0.0977786
$148$	0	0
$149$	−2.89530	−0.237192	−0.118596	−	0.992943i	$-0.537839\pi$
$149$	−2.89530	−0.237192	−0.118596	+	0.992943i	$0.537839\pi$
$150$	0	0
$151$	9.56165	0.778116	0.389058	−	0.921213i	$-0.372801\pi$
$151$	9.56165	0.778116	0.389058	+	0.921213i	$0.372801\pi$
$152$	0	0
$153$	22.7372	1.83820
$154$	0	0
$155$	−0.848172	−0.0681268
$156$	0	0
$157$	−20.4859	−1.63496	−0.817478	−	0.575960i	$-0.804629\pi$
$157$	−20.4859	−1.63496	−0.817478	+	0.575960i	$0.804629\pi$
$158$	0	0
$159$	−0.358888	−0.0284617
$160$	0	0
$161$	−4.18883	−0.330126
$162$	0	0
$163$	−10.3574	−0.811258	−0.405629	−	0.914038i	$-0.632948\pi$
$163$	−10.3574	−0.811258	−0.405629	+	0.914038i	$0.632948\pi$
$164$	0	0
$165$	−0.0439025	−0.00341780
$166$	0	0
$167$	−0.842674	−0.0652081	−0.0326040	−	0.999468i	$-0.510380\pi$
$167$	−0.842674	−0.0652081	−0.0326040	+	0.999468i	$0.510380\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−20.7328	−1.58547
$172$	0	0
$173$	−5.84433	−0.444336	−0.222168	−	0.975008i	$-0.571313\pi$
$173$	−5.84433	−0.444336	−0.222168	+	0.975008i	$0.571313\pi$
$174$	0	0
$175$	−7.47709	−0.565215
$176$	0	0
$177$	−2.75292	−0.206922
$178$	0	0
$179$	22.3606	1.67131	0.835653	−	0.549258i	$-0.185089\pi$
$179$	22.3606	1.67131	0.835653	+	0.549258i	$0.185089\pi$
$180$	0	0
$181$	−1.17586	−0.0874012	−0.0437006	−	0.999045i	$-0.513915\pi$
$181$	−1.17586	−0.0874012	−0.0437006	+	0.999045i	$0.513915\pi$
$182$	0	0
$183$	3.60618	0.266576
$184$	0	0
$185$	−0.534528	−0.0392993
$186$	0	0
$187$	1.34724	0.0985196
$188$	0	0
$189$	−3.09907	−0.225424
$190$	0	0
$191$	25.1648	1.82086	0.910431	−	0.413662i	$-0.135750\pi$
$191$	25.1648	1.82086	0.910431	+	0.413662i	$0.135750\pi$
$192$	0	0
$193$	−4.55946	−0.328197	−0.164099	−	0.986444i	$-0.552471\pi$
$193$	−4.55946	−0.328197	−0.164099	+	0.986444i	$0.552471\pi$
$194$	0	0
$195$	0.254630	0.0182344
$196$	0	0
$197$	−13.9405	−0.993220	−0.496610	−	0.867974i	$-0.665422\pi$
$197$	−13.9405	−0.993220	−0.496610	+	0.867974i	$0.665422\pi$
$198$	0	0
$199$	19.5841	1.38828	0.694141	−	0.719839i	$-0.255784\pi$
$199$	19.5841	1.38828	0.694141	+	0.719839i	$0.255784\pi$
$200$	0	0
$201$	1.75516	0.123799
$202$	0	0
$203$	8.30245	0.582718
$204$	0	0
$205$	−4.79756	−0.335076
$206$	0	0
$207$	6.97813	0.485014
$208$	0	0
$209$	−1.22847	−0.0849748
$210$	0	0
$211$	−26.0696	−1.79470	−0.897352	−	0.441315i	$-0.854512\pi$
$211$	−26.0696	−1.79470	−0.897352	+	0.441315i	$0.854512\pi$
$212$	0	0
$213$	−0.669015	−0.0458401
$214$	0	0
$215$	−1.54019	−0.105040
$216$	0	0
$217$	−1.74674	−0.118576
$218$	0	0
$219$	−1.99374	−0.134725
$220$	0	0
$221$	−7.81383	−0.525615
$222$	0	0
$223$	25.0024	1.67428	0.837142	−	0.546985i	$-0.184225\pi$
$223$	25.0024	1.67428	0.837142	+	0.546985i	$0.184225\pi$
$224$	0	0
$225$	12.4560	0.830401
$226$	0	0
$227$	−20.9389	−1.38976	−0.694882	−	0.719124i	$-0.744543\pi$
$227$	−20.9389	−1.38976	−0.694882	+	0.719124i	$0.744543\pi$
$228$	0	0
$229$	2.42980	0.160566	0.0802828	−	0.996772i	$-0.474418\pi$
$229$	2.42980	0.160566	0.0802828	+	0.996772i	$0.474418\pi$
$230$	0	0
$231$	−0.0904135	−0.00594877
$232$	0	0
$233$	−1.36125	−0.0891786	−0.0445893	−	0.999005i	$-0.514198\pi$
$233$	−1.36125	−0.0891786	−0.0445893	+	0.999005i	$0.514198\pi$
$234$	0	0
$235$	−1.53000	−0.0998065
$236$	0	0
$237$	0.158434	0.0102914
$238$	0	0
$239$	−20.7779	−1.34401	−0.672006	−	0.740545i	$-0.734567\pi$
$239$	−20.7779	−1.34401	−0.672006	+	0.740545i	$0.734567\pi$
$240$	0	0
$241$	19.0861	1.22945	0.614723	−	0.788743i	$-0.289268\pi$
$241$	19.0861	1.22945	0.614723	+	0.788743i	$0.289268\pi$
$242$	0	0
$243$	7.78343	0.499308
$244$	0	0
$245$	−3.34935	−0.213982
$246$	0	0
$247$	7.12497	0.453351
$248$	0	0
$249$	−2.18509	−0.138475
$250$	0	0
$251$	9.92063	0.626185	0.313092	−	0.949723i	$-0.398635\pi$
$251$	9.92063	0.626185	0.313092	+	0.949723i	$0.398635\pi$
$252$	0	0
$253$	0.413471	0.0259947
$254$	0	0
$255$	−1.98963	−0.124596
$256$	0	0
$257$	−14.1407	−0.882074	−0.441037	−	0.897489i	$-0.645389\pi$
$257$	−14.1407	−0.882074	−0.441037	+	0.897489i	$0.645389\pi$
$258$	0	0
$259$	−1.10081	−0.0684013
$260$	0	0
$261$	−13.8310	−0.856116
$262$	0	0
$263$	−0.451850	−0.0278622	−0.0139311	−	0.999903i	$-0.504435\pi$
$263$	−0.451850	−0.0278622	−0.0139311	+	0.999903i	$0.504435\pi$
$264$	0	0
$265$	−1.01395	−0.0622865
$266$	0	0
$267$	−3.89135	−0.238147
$268$	0	0
$269$	−0.0631497	−0.00385031	−0.00192515	−	0.999998i	$-0.500613\pi$
$269$	−0.0631497	−0.00385031	−0.00192515	+	0.999998i	$0.500613\pi$
$270$	0	0
$271$	−29.8153	−1.81115	−0.905575	−	0.424187i	$-0.860560\pi$
$271$	−29.8153	−1.81115	−0.905575	+	0.424187i	$0.860560\pi$
$272$	0	0
$273$	0.524389	0.0317374
$274$	0	0
$275$	0.738049	0.0445060
$276$	0	0
$277$	13.8071	0.829588	0.414794	−	0.909915i	$-0.363854\pi$
$277$	13.8071	0.829588	0.414794	+	0.909915i	$0.363854\pi$
$278$	0	0
$279$	2.90987	0.174210
$280$	0	0
$281$	2.53623	0.151299	0.0756493	−	0.997134i	$-0.475897\pi$
$281$	2.53623	0.151299	0.0756493	+	0.997134i	$0.475897\pi$
$282$	0	0
$283$	12.5971	0.748821	0.374410	−	0.927263i	$-0.377845\pi$
$283$	12.5971	0.748821	0.374410	+	0.927263i	$0.377845\pi$
$284$	0	0
$285$	1.81423	0.107466
$286$	0	0
$287$	−9.88016	−0.583208
$288$	0	0
$289$	44.0559	2.59152
$290$	0	0
$291$	0.110738	0.00649159
$292$	0	0
$293$	−17.6688	−1.03222	−0.516110	−	0.856522i	$-0.672620\pi$
$293$	−17.6688	−1.03222	−0.516110	+	0.856522i	$0.672620\pi$
$294$	0	0
$295$	−7.77771	−0.452836
$296$	0	0
$297$	0.305903	0.0177503
$298$	0	0
$299$	−2.39809	−0.138685
$300$	0	0
$301$	−3.17188	−0.182824
$302$	0	0
$303$	−2.64825	−0.152138
$304$	0	0
$305$	10.1884	0.583386
$306$	0	0
$307$	−11.7570	−0.671010	−0.335505	−	0.942038i	$-0.608907\pi$
$307$	−11.7570	−0.671010	−0.335505	+	0.942038i	$0.608907\pi$
$308$	0	0
$309$	−3.24505	−0.184605
$310$	0	0
$311$	−16.4836	−0.934698	−0.467349	−	0.884073i	$-0.654791\pi$
$311$	−16.4836	−0.934698	−0.467349	+	0.884073i	$0.654791\pi$
$312$	0	0
$313$	−27.1494	−1.53458	−0.767288	−	0.641302i	$-0.778394\pi$
$313$	−27.1494	−1.53458	−0.767288	+	0.641302i	$0.778394\pi$
$314$	0	0
$315$	−4.31107	−0.242902
$316$	0	0
$317$	23.1731	1.30153	0.650766	−	0.759278i	$-0.274448\pi$
$317$	23.1731	1.30153	0.650766	+	0.759278i	$0.274448\pi$
$318$	0	0
$319$	−0.819518	−0.0458842
$320$	0	0
$321$	1.36737	0.0763193
$322$	0	0
$323$	−55.6733	−3.09775
$324$	0	0
$325$	−4.28060	−0.237445
$326$	0	0
$327$	4.93159	0.272718
$328$	0	0
$329$	−3.15091	−0.173715
$330$	0	0
$331$	3.68237	0.202402	0.101201	−	0.994866i	$-0.467732\pi$
$331$	3.68237	0.202402	0.101201	+	0.994866i	$0.467732\pi$
$332$	0	0
$333$	1.83384	0.100494
$334$	0	0
$335$	4.95878	0.270927
$336$	0	0
$337$	0.610718	0.0332679	0.0166340	−	0.999862i	$-0.494705\pi$
$337$	0.610718	0.0332679	0.0166340	+	0.999862i	$0.494705\pi$
$338$	0	0
$339$	−1.34438	−0.0730166
$340$	0	0
$341$	0.172417	0.00933690
$342$	0	0
$343$	−19.1249	−1.03265
$344$	0	0
$345$	−0.610625	−0.0328749
$346$	0	0
$347$	34.3743	1.84531	0.922654	−	0.385630i	$-0.126016\pi$
$347$	34.3743	1.84531	0.922654	+	0.385630i	$0.126016\pi$
$348$	0	0
$349$	−0.794547	−0.0425311	−0.0212656	−	0.999774i	$-0.506770\pi$
$349$	−0.794547	−0.0425311	−0.0212656	+	0.999774i	$0.506770\pi$
$350$	0	0
$351$	−1.77420	−0.0947000
$352$	0	0
$353$	−13.4134	−0.713924	−0.356962	−	0.934119i	$-0.616188\pi$
$353$	−13.4134	−0.713924	−0.356962	+	0.934119i	$0.616188\pi$
$354$	0	0
$355$	−1.89014	−0.100318
$356$	0	0
$357$	−4.09748	−0.216862
$358$	0	0
$359$	12.5532	0.662534	0.331267	−	0.943537i	$-0.392524\pi$
$359$	12.5532	0.662534	0.331267	+	0.943537i	$0.392524\pi$
$360$	0	0
$361$	31.7653	1.67186
$362$	0	0
$363$	−3.29339	−0.172858
$364$	0	0
$365$	−5.63284	−0.294836
$366$	0	0
$367$	29.2060	1.52454	0.762272	−	0.647257i	$-0.224084\pi$
$367$	29.2060	1.52454	0.762272	+	0.647257i	$0.224084\pi$
$368$	0	0
$369$	16.4593	0.856835
$370$	0	0
$371$	−2.08815	−0.108411
$372$	0	0
$373$	−0.683653	−0.0353982	−0.0176991	−	0.999843i	$-0.505634\pi$
$373$	−0.683653	−0.0353982	−0.0176991	+	0.999843i	$0.505634\pi$
$374$	0	0
$375$	−2.36312	−0.122031
$376$	0	0
$377$	4.75312	0.244798
$378$	0	0
$379$	−33.9242	−1.74257	−0.871284	−	0.490779i	$-0.836712\pi$
$379$	−33.9242	−1.74257	−0.871284	+	0.490779i	$0.836712\pi$
$380$	0	0
$381$	−1.93189	−0.0989737
$382$	0	0
$383$	−14.9792	−0.765403	−0.382702	−	0.923872i	$-0.625006\pi$
$383$	−14.9792	−0.765403	−0.382702	+	0.923872i	$0.625006\pi$
$384$	0	0
$385$	−0.255441	−0.0130185
$386$	0	0
$387$	5.28401	0.268601
$388$	0	0
$389$	−32.8943	−1.66781	−0.833903	−	0.551911i	$-0.813899\pi$
$389$	−32.8943	−1.66781	−0.833903	+	0.551911i	$0.813899\pi$
$390$	0	0
$391$	18.7382	0.947633
$392$	0	0
$393$	3.71944	0.187621
$394$	0	0
$395$	0.447617	0.0225220
$396$	0	0
$397$	35.6692	1.79019	0.895093	−	0.445879i	$-0.147109\pi$
$397$	35.6692	1.79019	0.895093	+	0.445879i	$0.147109\pi$
$398$	0	0
$399$	3.73625	0.187047
$400$	0	0
$401$	18.7092	0.934295	0.467147	−	0.884180i	$-0.345282\pi$
$401$	18.7092	0.934295	0.467147	+	0.884180i	$0.345282\pi$
$402$	0	0
$403$	−1.00000	−0.0498135
$404$	0	0
$405$	6.95245	0.345470
$406$	0	0
$407$	0.108659	0.00538604
$408$	0	0
$409$	−6.75543	−0.334035	−0.167017	−	0.985954i	$-0.553414\pi$
$409$	−6.75543	−0.334035	−0.167017	+	0.985954i	$0.553414\pi$
$410$	0	0
$411$	−5.64170	−0.278285
$412$	0	0
$413$	−16.0175	−0.788171
$414$	0	0
$415$	−6.17345	−0.303043
$416$	0	0
$417$	−2.41346	−0.118188
$418$	0	0
$419$	1.17977	0.0576358	0.0288179	−	0.999585i	$-0.490826\pi$
$419$	1.17977	0.0576358	0.0288179	+	0.999585i	$0.490826\pi$
$420$	0	0
$421$	18.9358	0.922875	0.461437	−	0.887173i	$-0.347334\pi$
$421$	18.9358	0.922875	0.461437	+	0.887173i	$0.347334\pi$
$422$	0	0
$423$	5.24908	0.255219
$424$	0	0
$425$	33.4479	1.62246
$426$	0	0
$427$	20.9821	1.01540
$428$	0	0
$429$	−0.0517613	−0.00249906
$430$	0	0
$431$	−29.1549	−1.40434	−0.702171	−	0.712008i	$-0.747786\pi$
$431$	−29.1549	−1.40434	−0.702171	+	0.712008i	$0.747786\pi$
$432$	0	0
$433$	−11.9053	−0.572132	−0.286066	−	0.958210i	$-0.592348\pi$
$433$	−11.9053	−0.572132	−0.286066	+	0.958210i	$0.592348\pi$
$434$	0	0
$435$	1.21029	0.0580288
$436$	0	0
$437$	−17.0863	−0.817349
$438$	0	0
$439$	−18.5713	−0.886361	−0.443180	−	0.896432i	$-0.646150\pi$
$439$	−18.5713	−0.886361	−0.443180	+	0.896432i	$0.646150\pi$
$440$	0	0
$441$	11.4908	0.547182
$442$	0	0
$443$	−20.7364	−0.985216	−0.492608	−	0.870251i	$-0.663956\pi$
$443$	−20.7364	−0.985216	−0.492608	+	0.870251i	$0.663956\pi$
$444$	0	0
$445$	−10.9941	−0.521169
$446$	0	0
$447$	−0.869198	−0.0411117
$448$	0	0
$449$	21.5943	1.01910	0.509549	−	0.860441i	$-0.329812\pi$
$449$	21.5943	1.01910	0.509549	+	0.860441i	$0.329812\pi$
$450$	0	0
$451$	0.975251	0.0459228
$452$	0	0
$453$	2.87050	0.134868
$454$	0	0
$455$	1.48153	0.0694554
$456$	0	0
$457$	3.58158	0.167539	0.0837695	−	0.996485i	$-0.473304\pi$
$457$	3.58158	0.167539	0.0837695	+	0.996485i	$0.473304\pi$
$458$	0	0
$459$	13.8633	0.647085
$460$	0	0
$461$	−35.3277	−1.64537	−0.822687	−	0.568495i	$-0.807526\pi$
$461$	−35.3277	−1.64537	−0.822687	+	0.568495i	$0.807526\pi$
$462$	0	0
$463$	−15.4893	−0.719849	−0.359924	−	0.932981i	$-0.617198\pi$
$463$	−15.4893	−0.719849	−0.359924	+	0.932981i	$0.617198\pi$
$464$	0	0
$465$	−0.254630	−0.0118082
$466$	0	0
$467$	12.7713	0.590984	0.295492	−	0.955345i	$-0.404516\pi$
$467$	12.7713	0.590984	0.295492	+	0.955345i	$0.404516\pi$
$468$	0	0
$469$	10.2122	0.471555
$470$	0	0
$471$	−6.15009	−0.283381
$472$	0	0
$473$	0.313090	0.0143959
$474$	0	0
$475$	−30.4992	−1.39940
$476$	0	0
$477$	3.47862	0.159275
$478$	0	0
$479$	−12.1752	−0.556299	−0.278150	−	0.960538i	$-0.589721\pi$
$479$	−12.1752	−0.556299	−0.278150	+	0.960538i	$0.589721\pi$
$480$	0	0
$481$	−0.630212	−0.0287352
$482$	0	0
$483$	−1.25753	−0.0572196
$484$	0	0
$485$	0.312864	0.0142064
$486$	0	0
$487$	−18.5771	−0.841808	−0.420904	−	0.907105i	$-0.638287\pi$
$487$	−18.5771	−0.841808	−0.420904	+	0.907105i	$0.638287\pi$
$488$	0	0
$489$	−3.10941	−0.140612
$490$	0	0
$491$	−22.6700	−1.02308	−0.511540	−	0.859259i	$-0.670925\pi$
$491$	−22.6700	−1.02308	−0.511540	+	0.859259i	$0.670925\pi$
$492$	0	0
$493$	−37.1401	−1.67270
$494$	0	0
$495$	0.425537	0.0191265
$496$	0	0
$497$	−3.89258	−0.174606
$498$	0	0
$499$	27.0742	1.21201	0.606003	−	0.795462i	$-0.292772\pi$
$499$	27.0742	1.21201	0.606003	+	0.795462i	$0.292772\pi$
$500$	0	0
$501$	−0.252979	−0.0113023
$502$	0	0
$503$	10.1571	0.452882	0.226441	−	0.974025i	$-0.427291\pi$
$503$	10.1571	0.452882	0.226441	+	0.974025i	$0.427291\pi$
$504$	0	0
$505$	−7.48200	−0.332945
$506$	0	0
$507$	0.300210	0.0133328
$508$	0	0
$509$	11.3725	0.504076	0.252038	−	0.967717i	$-0.418899\pi$
$509$	11.3725	0.504076	0.252038	+	0.967717i	$0.418899\pi$
$510$	0	0
$511$	−11.6003	−0.513169
$512$	0	0
$513$	−12.6412	−0.558121
$514$	0	0
$515$	−9.16812	−0.403996
$516$	0	0
$517$	0.311020	0.0136787
$518$	0	0
$519$	−1.75453	−0.0770152
$520$	0	0
$521$	41.1410	1.80242	0.901210	−	0.433382i	$-0.142680\pi$
$521$	41.1410	1.80242	0.901210	+	0.433382i	$0.142680\pi$
$522$	0	0
$523$	−24.7940	−1.08417	−0.542083	−	0.840325i	$-0.682364\pi$
$523$	−24.7940	−1.08417	−0.542083	+	0.840325i	$0.682364\pi$
$524$	0	0
$525$	−2.24470	−0.0979667
$526$	0	0
$527$	7.81383	0.340376
$528$	0	0
$529$	−17.2492	−0.749964
$530$	0	0
$531$	26.6834	1.15796
$532$	0	0
$533$	−5.65635	−0.245004
$534$	0	0
$535$	3.86318	0.167020
$536$	0	0
$537$	6.71287	0.289682
$538$	0	0
$539$	0.680859	0.0293267
$540$	0	0
$541$	1.40582	0.0604409	0.0302204	−	0.999543i	$-0.490379\pi$
$541$	1.40582	0.0604409	0.0302204	+	0.999543i	$0.490379\pi$
$542$	0	0
$543$	−0.353006	−0.0151489
$544$	0	0
$545$	13.9330	0.596825
$546$	0	0
$547$	13.2873	0.568126	0.284063	−	0.958806i	$-0.408318\pi$
$547$	13.2873	0.568126	0.284063	+	0.958806i	$0.408318\pi$
$548$	0	0
$549$	−34.9539	−1.49180
$550$	0	0
$551$	33.8659	1.44273
$552$	0	0
$553$	0.921829	0.0392001
$554$	0	0
$555$	−0.160471	−0.00681161
$556$	0	0
$557$	−20.1892	−0.855444	−0.427722	−	0.903910i	$-0.640684\pi$
$557$	−20.1892	−0.855444	−0.427722	+	0.903910i	$0.640684\pi$
$558$	0	0
$559$	−1.81589	−0.0768040
$560$	0	0
$561$	0.404454	0.0170761
$562$	0	0
$563$	41.0121	1.72846	0.864228	−	0.503100i	$-0.167807\pi$
$563$	41.0121	1.72846	0.864228	+	0.503100i	$0.167807\pi$
$564$	0	0
$565$	−3.79822	−0.159792
$566$	0	0
$567$	14.3180	0.601299
$568$	0	0
$569$	−19.6986	−0.825809	−0.412904	−	0.910774i	$-0.635486\pi$
$569$	−19.6986	−0.825809	−0.412904	+	0.910774i	$0.635486\pi$
$570$	0	0
$571$	11.9584	0.500445	0.250223	−	0.968188i	$-0.419496\pi$
$571$	11.9584	0.500445	0.250223	+	0.968188i	$0.419496\pi$
$572$	0	0
$573$	7.55473	0.315604
$574$	0	0
$575$	10.2653	0.428091
$576$	0	0
$577$	−35.3881	−1.47322	−0.736612	−	0.676315i	$-0.763576\pi$
$577$	−35.3881	−1.47322	−0.736612	+	0.676315i	$0.763576\pi$
$578$	0	0
$579$	−1.36880	−0.0568852
$580$	0	0
$581$	−12.7137	−0.527453
$582$	0	0
$583$	0.206117	0.00853648
$584$	0	0
$585$	−2.46807	−0.102042
$586$	0	0
$587$	12.2163	0.504222	0.252111	−	0.967698i	$-0.418875\pi$
$587$	12.2163	0.504222	0.252111	+	0.967698i	$0.418875\pi$
$588$	0	0
$589$	−7.12497	−0.293579
$590$	0	0
$591$	−4.18509	−0.172151
$592$	0	0
$593$	−40.2153	−1.65144	−0.825722	−	0.564077i	$-0.809232\pi$
$593$	−40.2153	−1.65144	−0.825722	+	0.564077i	$0.809232\pi$
$594$	0	0
$595$	−11.5764	−0.474588
$596$	0	0
$597$	5.87936	0.240626
$598$	0	0
$599$	−25.1290	−1.02674	−0.513371	−	0.858167i	$-0.671604\pi$
$599$	−25.1290	−1.02674	−0.513371	+	0.858167i	$0.671604\pi$
$600$	0	0
$601$	−37.4928	−1.52936	−0.764681	−	0.644409i	$-0.777103\pi$
$601$	−37.4928	−1.52936	−0.764681	+	0.644409i	$0.777103\pi$
$602$	0	0
$603$	−17.0124	−0.692798
$604$	0	0
$605$	−9.30467	−0.378289
$606$	0	0
$607$	−27.6061	−1.12050	−0.560249	−	0.828324i	$-0.689295\pi$
$607$	−27.6061	−1.12050	−0.560249	+	0.828324i	$0.689295\pi$
$608$	0	0
$609$	2.49248	0.101000
$610$	0	0
$611$	−1.80389	−0.0729774
$612$	0	0
$613$	17.2168	0.695381	0.347690	−	0.937609i	$-0.386966\pi$
$613$	17.2168	0.695381	0.347690	+	0.937609i	$0.386966\pi$
$614$	0	0
$615$	−1.44028	−0.0580775
$616$	0	0
$617$	−30.9354	−1.24541	−0.622706	−	0.782456i	$-0.713967\pi$
$617$	−30.9354	−1.24541	−0.622706	+	0.782456i	$0.713967\pi$
$618$	0	0
$619$	26.4991	1.06509	0.532544	−	0.846402i	$-0.321236\pi$
$619$	26.4991	1.06509	0.532544	+	0.846402i	$0.321236\pi$
$620$	0	0
$621$	4.25470	0.170735
$622$	0	0
$623$	−22.6414	−0.907107
$624$	0	0
$625$	14.7266	0.589064
$626$	0	0
$627$	−0.368798	−0.0147284
$628$	0	0
$629$	4.92437	0.196347
$630$	0	0
$631$	−16.3209	−0.649726	−0.324863	−	0.945761i	$-0.605318\pi$
$631$	−16.3209	−0.649726	−0.324863	+	0.945761i	$0.605318\pi$
$632$	0	0
$633$	−7.82636	−0.311070
$634$	0	0
$635$	−5.45809	−0.216598
$636$	0	0
$637$	−3.94891	−0.156461
$638$	0	0
$639$	6.48462	0.256527
$640$	0	0
$641$	13.8537	0.547187	0.273594	−	0.961845i	$-0.411788\pi$
$641$	13.8537	0.547187	0.273594	+	0.961845i	$0.411788\pi$
$642$	0	0
$643$	45.4581	1.79269	0.896346	−	0.443356i	$-0.146212\pi$
$643$	45.4581	1.79269	0.896346	+	0.443356i	$0.146212\pi$
$644$	0	0
$645$	−0.462380	−0.0182062
$646$	0	0
$647$	25.7784	1.01345	0.506727	−	0.862107i	$-0.330855\pi$
$647$	25.7784	1.01345	0.506727	+	0.862107i	$0.330855\pi$
$648$	0	0
$649$	1.58106	0.0620619
$650$	0	0
$651$	−0.524389	−0.0205524
$652$	0	0
$653$	−28.8461	−1.12883	−0.564417	−	0.825490i	$-0.690899\pi$
$653$	−28.8461	−1.12883	−0.564417	+	0.825490i	$0.690899\pi$
$654$	0	0
$655$	10.5084	0.410597
$656$	0	0
$657$	19.3249	0.753936
$658$	0	0
$659$	−6.13123	−0.238839	−0.119419	−	0.992844i	$-0.538103\pi$
$659$	−6.13123	−0.238839	−0.119419	+	0.992844i	$0.538103\pi$
$660$	0	0
$661$	9.46628	0.368196	0.184098	−	0.982908i	$-0.441064\pi$
$661$	9.46628	0.368196	0.184098	+	0.982908i	$0.441064\pi$
$662$	0	0
$663$	−2.34579	−0.0911030
$664$	0	0
$665$	10.5559	0.409340
$666$	0	0
$667$	−11.3984	−0.441348
$668$	0	0
$669$	7.50598	0.290198
$670$	0	0
$671$	−2.07110	−0.0799540
$672$	0	0
$673$	25.2526	0.973416	0.486708	−	0.873565i	$-0.338198\pi$
$673$	25.2526	0.973416	0.486708	+	0.873565i	$0.338198\pi$
$674$	0	0
$675$	7.59467	0.292319
$676$	0	0
$677$	−7.53287	−0.289512	−0.144756	−	0.989467i	$-0.546240\pi$
$677$	−7.53287	−0.289512	−0.144756	+	0.989467i	$0.546240\pi$
$678$	0	0
$679$	0.644317	0.0247266
$680$	0	0
$681$	−6.28607	−0.240883
$682$	0	0
$683$	31.5199	1.20607	0.603037	−	0.797713i	$-0.293957\pi$
$683$	31.5199	1.20607	0.603037	+	0.797713i	$0.293957\pi$
$684$	0	0
$685$	−15.9393	−0.609008
$686$	0	0
$687$	0.729451	0.0278303
$688$	0	0
$689$	−1.19545	−0.0455432
$690$	0	0
$691$	12.6388	0.480801	0.240401	−	0.970674i	$-0.422721\pi$
$691$	12.6388	0.480801	0.240401	+	0.970674i	$0.422721\pi$
$692$	0	0
$693$	0.876358	0.0332901
$694$	0	0
$695$	−6.81864	−0.258646
$696$	0	0
$697$	44.1978	1.67411
$698$	0	0
$699$	−0.408662	−0.0154570
$700$	0	0
$701$	7.70250	0.290919	0.145460	−	0.989364i	$-0.453534\pi$
$701$	7.70250	0.290919	0.145460	+	0.989364i	$0.453534\pi$
$702$	0	0
$703$	−4.49024	−0.169353
$704$	0	0
$705$	−0.459323	−0.0172991
$706$	0	0
$707$	−15.4085	−0.579498
$708$	0	0
$709$	24.9575	0.937298	0.468649	−	0.883384i	$-0.344741\pi$
$709$	24.9575	0.937298	0.468649	+	0.883384i	$0.344741\pi$
$710$	0	0
$711$	−1.53567	−0.0575920
$712$	0	0
$713$	2.39809	0.0898091
$714$	0	0
$715$	−0.146239	−0.00546903
$716$	0	0
$717$	−6.23775	−0.232953
$718$	0	0
$719$	1.16871	0.0435854	0.0217927	−	0.999763i	$-0.493063\pi$
$719$	1.16871	0.0435854	0.0217927	+	0.999763i	$0.493063\pi$
$720$	0	0
$721$	−18.8810	−0.703164
$722$	0	0
$723$	5.72985	0.213096
$724$	0	0
$725$	−20.3462	−0.755640
$726$	0	0
$727$	−25.2048	−0.934796	−0.467398	−	0.884047i	$-0.654808\pi$
$727$	−25.2048	−0.934796	−0.467398	+	0.884047i	$0.654808\pi$
$728$	0	0
$729$	−22.2543	−0.824233
$730$	0	0
$731$	14.1891	0.524801
$732$	0	0
$733$	−8.03218	−0.296675	−0.148338	−	0.988937i	$-0.547392\pi$
$733$	−8.03218	−0.296675	−0.148338	+	0.988937i	$0.547392\pi$
$734$	0	0
$735$	−1.00551	−0.0370888
$736$	0	0
$737$	−1.00802	−0.0371310
$738$	0	0
$739$	−3.29736	−0.121295	−0.0606477	−	0.998159i	$-0.519317\pi$
$739$	−3.29736	−0.121295	−0.0606477	+	0.998159i	$0.519317\pi$
$740$	0	0
$741$	2.13899	0.0785778
$742$	0	0
$743$	7.63134	0.279967	0.139983	−	0.990154i	$-0.455295\pi$
$743$	7.63134	0.279967	0.139983	+	0.990154i	$0.455295\pi$
$744$	0	0
$745$	−2.45571	−0.0899703
$746$	0	0
$747$	21.1796	0.774922
$748$	0	0
$749$	7.95589	0.290702
$750$	0	0
$751$	−27.7614	−1.01303	−0.506513	−	0.862232i	$-0.669066\pi$
$751$	−27.7614	−1.01303	−0.506513	+	0.862232i	$0.669066\pi$
$752$	0	0
$753$	2.97828	0.108534
$754$	0	0
$755$	8.10992	0.295150
$756$	0	0
$757$	−33.7745	−1.22755	−0.613777	−	0.789479i	$-0.710351\pi$
$757$	−33.7745	−1.22755	−0.613777	+	0.789479i	$0.710351\pi$
$758$	0	0
$759$	0.124128	0.00450557
$760$	0	0
$761$	41.8501	1.51706	0.758532	−	0.651636i	$-0.225917\pi$
$761$	41.8501	1.51706	0.758532	+	0.651636i	$0.225917\pi$
$762$	0	0
$763$	28.6939	1.03879
$764$	0	0
$765$	19.2851	0.697254
$766$	0	0
$767$	−9.16997	−0.331108
$768$	0	0
$769$	1.68878	0.0608990	0.0304495	−	0.999536i	$-0.490306\pi$
$769$	1.68878	0.0608990	0.0304495	+	0.999536i	$0.490306\pi$
$770$	0	0
$771$	−4.24519	−0.152887
$772$	0	0
$773$	38.2742	1.37663	0.688314	−	0.725413i	$-0.258351\pi$
$773$	38.2742	1.37663	0.688314	+	0.725413i	$0.258351\pi$
$774$	0	0
$775$	4.28060	0.153764
$776$	0	0
$777$	−0.330476	−0.0118558
$778$	0	0
$779$	−40.3014	−1.44395
$780$	0	0
$781$	0.384229	0.0137488
$782$	0	0
$783$	−8.43301	−0.301371
$784$	0	0
$785$	−17.3756	−0.620162
$786$	0	0
$787$	7.90010	0.281608	0.140804	−	0.990037i	$-0.455031\pi$
$787$	7.90010	0.281608	0.140804	+	0.990037i	$0.455031\pi$
$788$	0	0
$789$	−0.135650	−0.00482926
$790$	0	0
$791$	−7.82211	−0.278122
$792$	0	0
$793$	12.0122	0.426565
$794$	0	0
$795$	−0.304398	−0.0107959
$796$	0	0
$797$	−31.1460	−1.10325	−0.551625	−	0.834092i	$-0.685992\pi$
$797$	−31.1460	−1.10325	−0.551625	+	0.834092i	$0.685992\pi$
$798$	0	0
$799$	14.0952	0.498654
$800$	0	0
$801$	37.7180	1.33270
$802$	0	0
$803$	1.14505	0.0404078
$804$	0	0
$805$	−3.55285	−0.125221
$806$	0	0
$807$	−0.0189582	−0.000667360 0
$808$	0	0
$809$	19.2376	0.676358	0.338179	−	0.941082i	$-0.390189\pi$
$809$	19.2376	0.676358	0.338179	+	0.941082i	$0.390189\pi$
$810$	0	0
$811$	−8.38444	−0.294417	−0.147209	−	0.989105i	$-0.547029\pi$
$811$	−8.38444	−0.294417	−0.147209	+	0.989105i	$0.547029\pi$
$812$	0	0
$813$	−8.95085	−0.313920
$814$	0	0
$815$	−8.78490	−0.307721
$816$	0	0
$817$	−12.9382	−0.452649
$818$	0	0
$819$	−5.08279	−0.177607
$820$	0	0
$821$	19.9624	0.696691	0.348346	−	0.937366i	$-0.386744\pi$
$821$	19.9624	0.696691	0.348346	+	0.937366i	$0.386744\pi$
$822$	0	0
$823$	29.0804	1.01368	0.506840	−	0.862040i	$-0.330814\pi$
$823$	29.0804	1.01368	0.506840	+	0.862040i	$0.330814\pi$
$824$	0	0
$825$	0.221570	0.00771407
$826$	0	0
$827$	−16.7954	−0.584032	−0.292016	−	0.956413i	$-0.594326\pi$
$827$	−16.7954	−0.584032	−0.292016	+	0.956413i	$0.594326\pi$
$828$	0	0
$829$	−15.6959	−0.545142	−0.272571	−	0.962136i	$-0.587874\pi$
$829$	−15.6959	−0.545142	−0.272571	+	0.962136i	$0.587874\pi$
$830$	0	0
$831$	4.14503	0.143790
$832$	0	0
$833$	30.8561	1.06910
$834$	0	0
$835$	−0.714733	−0.0247343
$836$	0	0
$837$	1.77420	0.0613255
$838$	0	0
$839$	36.4473	1.25830	0.629150	−	0.777284i	$-0.283403\pi$
$839$	36.4473	1.25830	0.629150	+	0.777284i	$0.283403\pi$
$840$	0	0
$841$	−6.40784	−0.220960
$842$	0	0
$843$	0.761401	0.0262241
$844$	0	0
$845$	0.848172	0.0291780
$846$	0	0
$847$	−19.1622	−0.658420
$848$	0	0
$849$	3.78178	0.129790
$850$	0	0
$851$	1.51130	0.0518068
$852$	0	0
$853$	−56.3291	−1.92867	−0.964336	−	0.264679i	$-0.914734\pi$
$853$	−56.3291	−1.92867	−0.964336	+	0.264679i	$0.914734\pi$
$854$	0	0
$855$	−17.5850	−0.601393
$856$	0	0
$857$	20.1096	0.686929	0.343465	−	0.939166i	$-0.388399\pi$
$857$	20.1096	0.686929	0.343465	+	0.939166i	$0.388399\pi$
$858$	0	0
$859$	−22.4894	−0.767329	−0.383664	−	0.923473i	$-0.625338\pi$
$859$	−22.4894	−0.767329	−0.383664	+	0.923473i	$0.625338\pi$
$860$	0	0
$861$	−2.96613	−0.101085
$862$	0	0
$863$	51.3272	1.74720	0.873599	−	0.486647i	$-0.161780\pi$
$863$	51.3272	1.74720	0.873599	+	0.486647i	$0.161780\pi$
$864$	0	0
$865$	−4.95699	−0.168543
$866$	0	0
$867$	13.2260	0.449179
$868$	0	0
$869$	−0.0909918	−0.00308669
$870$	0	0
$871$	5.84643	0.198099
$872$	0	0
$873$	−1.07336	−0.0363278
$874$	0	0
$875$	−13.7495	−0.464819
$876$	0	0
$877$	−38.2648	−1.29211	−0.646055	−	0.763291i	$-0.723582\pi$
$877$	−38.2648	−1.29211	−0.646055	+	0.763291i	$0.723582\pi$
$878$	0	0
$879$	−5.30434	−0.178911
$880$	0	0
$881$	30.2398	1.01881	0.509403	−	0.860528i	$-0.329866\pi$
$881$	30.2398	1.01881	0.509403	+	0.860528i	$0.329866\pi$
$882$	0	0
$883$	−4.67446	−0.157308	−0.0786541	−	0.996902i	$-0.525062\pi$
$883$	−4.67446	−0.157308	−0.0786541	+	0.996902i	$0.525062\pi$
$884$	0	0
$885$	−2.33495	−0.0784884
$886$	0	0
$887$	22.4333	0.753235	0.376618	−	0.926369i	$-0.377087\pi$
$887$	22.4333	0.753235	0.376618	+	0.926369i	$0.377087\pi$
$888$	0	0
$889$	−11.2405	−0.376993
$890$	0	0
$891$	−1.41330	−0.0473473
$892$	0	0
$893$	−12.8526	−0.430097
$894$	0	0
$895$	18.9656	0.633950
$896$	0	0
$897$	−0.719930	−0.0240378
$898$	0	0
$899$	−4.75312	−0.158525
$900$	0	0
$901$	9.34108	0.311196
$902$	0	0
$903$	−0.952232	−0.0316883
$904$	0	0
$905$	−0.997334	−0.0331525
$906$	0	0
$907$	15.9983	0.531214	0.265607	−	0.964081i	$-0.414428\pi$
$907$	15.9983	0.531214	0.265607	+	0.964081i	$0.414428\pi$
$908$	0	0
$909$	25.6690	0.851386
$910$	0	0
$911$	−33.8374	−1.12108	−0.560541	−	0.828127i	$-0.689407\pi$
$911$	−33.8374	−1.12108	−0.560541	+	0.828127i	$0.689407\pi$
$912$	0	0
$913$	1.25494	0.0415326
$914$	0	0
$915$	3.05866	0.101116
$916$	0	0
$917$	21.6411	0.714653
$918$	0	0
$919$	−9.19972	−0.303471	−0.151735	−	0.988421i	$-0.548486\pi$
$919$	−9.19972	−0.303471	−0.151735	+	0.988421i	$0.548486\pi$
$920$	0	0
$921$	−3.52958	−0.116304
$922$	0	0
$923$	−2.22849	−0.0733515
$924$	0	0
$925$	2.69769	0.0886994
$926$	0	0
$927$	31.4536	1.03307
$928$	0	0
$929$	38.0189	1.24736	0.623679	−	0.781680i	$-0.285637\pi$
$929$	38.0189	1.24736	0.623679	+	0.781680i	$0.285637\pi$
$930$	0	0
$931$	−28.1359	−0.922116
$932$	0	0
$933$	−4.94854	−0.162008
$934$	0	0
$935$	1.14269	0.0373699
$936$	0	0
$937$	−12.8024	−0.418236	−0.209118	−	0.977890i	$-0.567059\pi$
$937$	−12.8024	−0.418236	−0.209118	+	0.977890i	$0.567059\pi$
$938$	0	0
$939$	−8.15054	−0.265983
$940$	0	0
$941$	19.0245	0.620181	0.310090	−	0.950707i	$-0.399641\pi$
$941$	19.0245	0.620181	0.310090	+	0.950707i	$0.399641\pi$
$942$	0	0
$943$	13.5644	0.441718
$944$	0	0
$945$	−2.62854	−0.0855065
$946$	0	0
$947$	2.15887	0.0701539	0.0350770	−	0.999385i	$-0.488832\pi$
$947$	2.15887	0.0701539	0.0350770	+	0.999385i	$0.488832\pi$
$948$	0	0
$949$	−6.64115	−0.215581
$950$	0	0
$951$	6.95681	0.225590
$952$	0	0
$953$	18.2117	0.589936	0.294968	−	0.955507i	$-0.404691\pi$
$953$	18.2117	0.589936	0.294968	+	0.955507i	$0.404691\pi$
$954$	0	0
$955$	21.3441	0.690678
$956$	0	0
$957$	−0.246028	−0.00795295
$958$	0	0
$959$	−32.8256	−1.05999
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	−13.2536	−0.427093
$964$	0	0
$965$	−3.86720	−0.124490
$966$	0	0
$967$	−17.9039	−0.575751	−0.287876	−	0.957668i	$-0.592949\pi$
$967$	−17.9039	−0.575751	−0.287876	+	0.957668i	$0.592949\pi$
$968$	0	0
$969$	−16.7137	−0.536921
$970$	0	0
$971$	2.17673	0.0698546	0.0349273	−	0.999390i	$-0.488880\pi$
$971$	2.17673	0.0698546	0.0349273	+	0.999390i	$0.488880\pi$
$972$	0	0
$973$	−14.0424	−0.450179
$974$	0	0
$975$	−1.28508	−0.0411555
$976$	0	0
$977$	24.3730	0.779762	0.389881	−	0.920865i	$-0.372516\pi$
$977$	24.3730	0.779762	0.389881	+	0.920865i	$0.372516\pi$
$978$	0	0
$979$	2.23488	0.0714271
$980$	0	0
$981$	−47.8009	−1.52616
$982$	0	0
$983$	−0.0380046	−0.00121216	−0.000606080	−	1.00000i	$-0.500193\pi$
$983$	−0.0380046	−0.00121216	−0.000606080		1.00000i	$0.500193\pi$
$984$	0	0
$985$	−11.8239	−0.376742
$986$	0	0
$987$	−0.945937	−0.0301095
$988$	0	0
$989$	4.35466	0.138470
$990$	0	0
$991$	6.63973	0.210918	0.105459	−	0.994424i	$-0.466369\pi$
$991$	6.63973	0.210918	0.105459	+	0.994424i	$0.466369\pi$
$992$	0	0
$993$	1.10549	0.0350816
$994$	0	0
$995$	16.6107	0.526595
$996$	0	0
$997$	30.6435	0.970490	0.485245	−	0.874378i	$-0.338730\pi$
$997$	30.6435	0.970490	0.485245	+	0.874378i	$0.338730\pi$
$998$	0	0
$999$	1.11813	0.0353759

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6448.2.a.y.1.3		6
4.3	odd	2		403.2.a.b.1.4	✓	6
12.11	even	2		3627.2.a.m.1.3		6
52.51	odd	2		5239.2.a.g.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.a.b.1.4	✓	6	4.3	odd	2
3627.2.a.m.1.3		6	12.11	even	2
5239.2.a.g.1.3		6	52.51	odd	2
6448.2.a.y.1.3		6	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6448 = 2^{4} \cdot 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6448.a (trivial)

\(f(q)\)	\(=\)	\(q+0.300210 q^{3} +0.848172 q^{5} +1.74674 q^{7} -2.90987 q^{9} +O(q^{10})\)	\(q+0.300210 q^{3} +0.848172 q^{5} +1.74674 q^{7} -2.90987 q^{9} -0.172417 q^{11} +1.00000 q^{13} +0.254630 q^{15} -7.81383 q^{17} +7.12497 q^{19} +0.524389 q^{21} -2.39809 q^{23} -4.28060 q^{25} -1.77420 q^{27} +4.75312 q^{29} -1.00000 q^{31} -0.0517613 q^{33} +1.48153 q^{35} -0.630212 q^{37} +0.300210 q^{39} -5.65635 q^{41} -1.81589 q^{43} -2.46807 q^{45} -1.80389 q^{47} -3.94891 q^{49} -2.34579 q^{51} -1.19545 q^{53} -0.146239 q^{55} +2.13899 q^{57} -9.16997 q^{59} +12.0122 q^{61} -5.08279 q^{63} +0.848172 q^{65} +5.84643 q^{67} -0.719930 q^{69} -2.22849 q^{71} -6.64115 q^{73} -1.28508 q^{75} -0.301167 q^{77} +0.527743 q^{79} +8.19699 q^{81} -7.27854 q^{83} -6.62747 q^{85} +1.42694 q^{87} -12.9621 q^{89} +1.74674 q^{91} -0.300210 q^{93} +6.04320 q^{95} +0.368869 q^{97} +0.501711 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 5 q^{3} - 9 q^{5} - q^{9}+O(q^{10}) \) 6 * q + 5 * q^3 - 9 * q^5 - q^9	\( 6 q + 5 q^{3} - 9 q^{5} - q^{9} + 5 q^{11} + 6 q^{13} - 4 q^{15} - 23 q^{17} - 7 q^{19} + 2 q^{21} + 18 q^{23} + 11 q^{25} + 5 q^{27} - 18 q^{29} - 6 q^{31} - 2 q^{33} + q^{35} - 13 q^{37} + 5 q^{39} - 5 q^{41} + 7 q^{43} - 5 q^{45} + 9 q^{47} + 16 q^{49} - 26 q^{51} - 31 q^{53} - 7 q^{55} - 5 q^{57} + q^{59} - 15 q^{61} - 11 q^{63} - 9 q^{65} + 28 q^{67} + 5 q^{69} - q^{71} - 20 q^{73} - q^{75} - 29 q^{77} + 15 q^{79} + 2 q^{81} - q^{83} + 29 q^{85} - 10 q^{87} - q^{89} - 5 q^{93} + 13 q^{95} - 5 q^{97} + 15 q^{99}+O(q^{100}) \) 6 * q + 5 * q^3 - 9 * q^5 - q^9 + 5 * q^11 + 6 * q^13 - 4 * q^15 - 23 * q^17 - 7 * q^19 + 2 * q^21 + 18 * q^23 + 11 * q^25 + 5 * q^27 - 18 * q^29 - 6 * q^31 - 2 * q^33 + q^35 - 13 * q^37 + 5 * q^39 - 5 * q^41 + 7 * q^43 - 5 * q^45 + 9 * q^47 + 16 * q^49 - 26 * q^51 - 31 * q^53 - 7 * q^55 - 5 * q^57 + q^59 - 15 * q^61 - 11 * q^63 - 9 * q^65 + 28 * q^67 + 5 * q^69 - q^71 - 20 * q^73 - q^75 - 29 * q^77 + 15 * q^79 + 2 * q^81 - q^83 + 29 * q^85 - 10 * q^87 - q^89 - 5 * q^93 + 13 * q^95 - 5 * q^97 + 15 * q^99

Introduction

L-functions

Modular forms

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