LMFDB - Embedded newform 64.5.d.a.31.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [64,5,Mod(31,64)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(64, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("64.31");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 64 = 2^{6} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	64.d (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$6.61567763737$
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^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			31.2
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	64.31
Dual form			64.5.d.a.31.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.46410 q^{3} +20.7846i q^{5} -40.0000i q^{7} -69.0000 q^{9} +O(q^{10})$	$q-3.46410 q^{3} +20.7846i q^{5} -40.0000i q^{7} -69.0000 q^{9} -197.454 q^{11} +256.344i q^{13} -72.0000i q^{15} -198.000 q^{17} -252.879 q^{19} +138.564i q^{21} -504.000i q^{23} +193.000 q^{25} +519.615 q^{27} -436.477i q^{29} +1696.00i q^{31} +684.000 q^{33} +831.384 q^{35} -1752.84i q^{37} -888.000i q^{39} +18.0000 q^{41} +1548.45 q^{43} -1434.14i q^{45} +2448.00i q^{47} +801.000 q^{49} +685.892 q^{51} +3263.18i q^{53} -4104.00i q^{55} +876.000 q^{57} -5268.90 q^{59} +1004.59i q^{61} +2760.00i q^{63} -5328.00 q^{65} -1804.80 q^{67} +1745.91i q^{69} -7848.00i q^{71} -4310.00 q^{73} -668.572 q^{75} +7898.15i q^{77} +4144.00i q^{79} +3789.00 q^{81} -9093.27 q^{83} -4115.35i q^{85} +1512.00i q^{87} +3114.00 q^{89} +10253.7 q^{91} -5875.12i q^{93} -5256.00i q^{95} +4730.00 q^{97} +13624.3 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 276 q^{9}+O(q^{10}) $ 4 * q - 276 * q^9	$ 4 q - 276 q^{9} - 792 q^{17} + 772 q^{25} + 2736 q^{33} + 72 q^{41} + 3204 q^{49} + 3504 q^{57} - 21312 q^{65} - 17240 q^{73} + 15156 q^{81} + 12456 q^{89} + 18920 q^{97}+O(q^{100}) $ 4 * q - 276 * q^9 - 792 * q^17 + 772 * q^25 + 2736 * q^33 + 72 * q^41 + 3204 * q^49 + 3504 * q^57 - 21312 * q^65 - 17240 * q^73 + 15156 * q^81 + 12456 * q^89 + 18920 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/64\mathbb{Z}\right)^\times$.

$n$	$5$	$63$
$\chi(n)$	$-1$	$-1$

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
$3$	−3.46410	−0.384900	−0.192450	−	0.981307i	$-0.561643\pi$
$3$	−3.46410	−0.384900	−0.192450	+	0.981307i	$0.561643\pi$
$4$	0	0
$5$	20.7846i	0.831384i	0.909505	+	0.415692i	$0.136461\pi$
$5$	20.7846i	0.831384i	−0.909505	+	0.415692i	$0.863539\pi$
$6$	0	0
$7$	− 40.0000i	− 0.816327i	−0.912909	−	0.408163i	$-0.866169\pi$
$7$	− 40.0000i	− 0.816327i	0.912909	−	0.408163i	$-0.133831\pi$
$8$	0	0
$9$	−69.0000	−0.851852
$10$	0	0
$11$	−197.454	−1.63185	−0.815925	−	0.578158i	$-0.803772\pi$
$11$	−197.454	−1.63185	−0.815925	+	0.578158i	$0.803772\pi$
$12$	0	0
$13$	256.344i	1.51683i	0.651775	+	0.758413i	$0.274025\pi$
$13$	256.344i	1.51683i	−0.651775	+	0.758413i	$0.725975\pi$
$14$	0	0
$15$	− 72.0000i	− 0.320000i
$16$	0	0
$17$	−198.000	−0.685121	−0.342561	−	0.939496i	$-0.611294\pi$
$17$	−198.000	−0.685121	−0.342561	+	0.939496i	$0.611294\pi$
$18$	0	0
$19$	−252.879	−0.700497	−0.350249	−	0.936657i	$-0.613903\pi$
$19$	−252.879	−0.700497	−0.350249	+	0.936657i	$0.613903\pi$
$20$	0	0
$21$	138.564i	0.314204i
$22$	0	0
$23$	− 504.000i	− 0.952741i	−0.879245	−	0.476371i	$-0.841952\pi$
$23$	− 504.000i	− 0.952741i	0.879245	−	0.476371i	$-0.158048\pi$
$24$	0	0
$25$	193.000	0.308800
$26$	0	0
$27$	519.615	0.712778
$28$	0	0
$29$	− 436.477i	− 0.518997i	−0.965743	−	0.259499i	$-0.916443\pi$
$29$	− 436.477i	− 0.518997i	0.965743	−	0.259499i	$-0.0835573\pi$
$30$	0	0
$31$	1696.00i	1.76483i	0.470473	+	0.882414i	$0.344083\pi$
$31$	1696.00i	1.76483i	−0.470473	+	0.882414i	$0.655917\pi$
$32$	0	0
$33$	684.000	0.628099
$34$	0	0
$35$	831.384	0.678681
$36$	0	0
$37$	− 1752.84i	− 1.28038i	−0.768218	−	0.640188i	$-0.778856\pi$
$37$	− 1752.84i	− 1.28038i	0.768218	−	0.640188i	$-0.221144\pi$
$38$	0	0
$39$	− 888.000i	− 0.583826i
$40$	0	0
$41$	18.0000	0.0107079	0.00535396	−	0.999986i	$-0.498296\pi$
$41$	18.0000	0.0107079	0.00535396	+	0.999986i	$0.498296\pi$
$42$	0	0
$43$	1548.45	0.837455	0.418727	−	0.908112i	$-0.362476\pi$
$43$	1548.45	0.837455	0.418727	+	0.908112i	$0.362476\pi$
$44$	0	0
$45$	− 1434.14i	− 0.708216i
$46$	0	0
$47$	2448.00i	1.10819i	0.832452	+	0.554097i	$0.186936\pi$
$47$	2448.00i	1.10819i	−0.832452	+	0.554097i	$0.813064\pi$
$48$	0	0
$49$	801.000	0.333611
$50$	0	0
$51$	685.892	0.263703
$52$	0	0
$53$	3263.18i	1.16169i	0.814015	+	0.580844i	$0.197278\pi$
$53$	3263.18i	1.16169i	−0.814015	+	0.580844i	$0.802722\pi$
$54$	0	0
$55$	− 4104.00i	− 1.35669i
$56$	0	0
$57$	876.000	0.269621
$58$	0	0
$59$	−5268.90	−1.51362	−0.756808	−	0.653637i	$-0.773242\pi$
$59$	−5268.90	−1.51362	−0.756808	+	0.653637i	$0.773242\pi$
$60$	0	0
$61$	1004.59i	0.269978i	0.990847	+	0.134989i	$0.0431000\pi$
$61$	1004.59i	0.269978i	−0.990847	+	0.134989i	$0.956900\pi$
$62$	0	0
$63$	2760.00i	0.695389i
$64$	0	0
$65$	−5328.00	−1.26107
$66$	0	0
$67$	−1804.80	−0.402049	−0.201024	−	0.979586i	$-0.564427\pi$
$67$	−1804.80	−0.402049	−0.201024	+	0.979586i	$0.564427\pi$
$68$	0	0
$69$	1745.91i	0.366710i
$70$	0	0
$71$	− 7848.00i	− 1.55683i	−0.627748	−	0.778417i	$-0.716023\pi$
$71$	− 7848.00i	− 1.55683i	0.627748	−	0.778417i	$-0.283977\pi$
$72$	0	0
$73$	−4310.00	−0.808782	−0.404391	−	0.914586i	$-0.632517\pi$
$73$	−4310.00	−0.808782	−0.404391	+	0.914586i	$0.632517\pi$
$74$	0	0
$75$	−668.572	−0.118857
$76$	0	0
$77$	7898.15i	1.33212i
$78$	0	0
$79$	4144.00i	0.663996i	0.943280	+	0.331998i	$0.107723\pi$
$79$	4144.00i	0.663996i	−0.943280	+	0.331998i	$0.892277\pi$
$80$	0	0
$81$	3789.00	0.577503
$82$	0	0
$83$	−9093.27	−1.31997	−0.659985	−	0.751279i	$-0.729437\pi$
$83$	−9093.27	−1.31997	−0.659985	+	0.751279i	$0.729437\pi$
$84$	0	0
$85$	− 4115.35i	− 0.569599i
$86$	0	0
$87$	1512.00i	0.199762i
$88$	0	0
$89$	3114.00	0.393132	0.196566	−	0.980491i	$-0.437021\pi$
$89$	3114.00	0.393132	0.196566	+	0.980491i	$0.437021\pi$
$90$	0	0
$91$	10253.7	1.23822
$92$	0	0
$93$	− 5875.12i	− 0.679283i
$94$	0	0
$95$	− 5256.00i	− 0.582382i
$96$	0	0
$97$	4730.00	0.502710	0.251355	−	0.967895i	$-0.419124\pi$
$97$	4730.00	0.502710	0.251355	+	0.967895i	$0.419124\pi$
$98$	0	0
$99$	13624.3	1.39009
$100$	0	0
$101$	20.7846i	0.00203751i	0.999999	+	0.00101875i	$0.000324279\pi$
$101$	20.7846i	0.00203751i	−0.999999	+	0.00101875i	$0.999676\pi$
$102$	0	0
$103$	9016.00i	0.849844i	0.905230	+	0.424922i	$0.139699\pi$
$103$	9016.00i	0.849844i	−0.905230	+	0.424922i	$0.860301\pi$
$104$	0	0
$105$	−2880.00	−0.261224
$106$	0	0
$107$	−2359.05	−0.206049	−0.103024	−	0.994679i	$-0.532852\pi$
$107$	−2359.05	−0.206049	−0.103024	+	0.994679i	$0.532852\pi$
$108$	0	0
$109$	− 14486.9i	− 1.21933i	−0.792659	−	0.609666i	$-0.791304\pi$
$109$	− 14486.9i	− 1.21933i	0.792659	−	0.609666i	$-0.208696\pi$
$110$	0	0
$111$	6072.00i	0.492817i
$112$	0	0
$113$	−2142.00	−0.167750	−0.0838750	−	0.996476i	$-0.526730\pi$
$113$	−2142.00	−0.167750	−0.0838750	+	0.996476i	$0.526730\pi$
$114$	0	0
$115$	10475.4	0.792094
$116$	0	0
$117$	− 17687.7i	− 1.29211i
$118$	0	0
$119$	7920.00i	0.559283i
$120$	0	0
$121$	24347.0	1.66293
$122$	0	0
$123$	−62.3538	−0.00412148
$124$	0	0
$125$	17001.8i	1.08812i
$126$	0	0
$127$	− 5056.00i	− 0.313473i	−0.987640	−	0.156736i	$-0.949903\pi$
$127$	− 5056.00i	− 0.313473i	0.987640	−	0.156736i	$-0.0500973\pi$
$128$	0	0
$129$	−5364.00	−0.322336
$130$	0	0
$131$	−9176.41	−0.534724	−0.267362	−	0.963596i	$-0.586152\pi$
$131$	−9176.41	−0.534724	−0.267362	+	0.963596i	$0.586152\pi$
$132$	0	0
$133$	10115.2i	0.571834i
$134$	0	0
$135$	10800.0i	0.592593i
$136$	0	0
$137$	−32526.0	−1.73296	−0.866482	−	0.499208i	$-0.833624\pi$
$137$	−32526.0	−1.73296	−0.866482	+	0.499208i	$0.833624\pi$
$138$	0	0
$139$	1105.05	0.0571942	0.0285971	−	0.999591i	$-0.490896\pi$
$139$	1105.05	0.0571942	0.0285971	+	0.999591i	$0.490896\pi$
$140$	0	0
$141$	− 8480.12i	− 0.426544i
$142$	0	0
$143$	− 50616.0i	− 2.47523i
$144$	0	0
$145$	9072.00	0.431486
$146$	0	0
$147$	−2774.75	−0.128407
$148$	0	0
$149$	34523.2i	1.55503i	0.628864	+	0.777515i	$0.283520\pi$
$149$	34523.2i	1.55503i	−0.628864	+	0.777515i	$0.716480\pi$
$150$	0	0
$151$	14216.0i	0.623481i	0.950167	+	0.311741i	$0.100912\pi$
$151$	14216.0i	0.623481i	−0.950167	+	0.311741i	$0.899088\pi$
$152$	0	0
$153$	13662.0	0.583622
$154$	0	0
$155$	−35250.7	−1.46725
$156$	0	0
$157$	− 270.200i	− 0.0109619i	−0.999985	−	0.00548095i	$-0.998255\pi$
$157$	− 270.200i	− 0.0109619i	0.999985	−	0.00548095i	$-0.00174465\pi$
$158$	0	0
$159$	− 11304.0i	− 0.447134i
$160$	0	0
$161$	−20160.0	−0.777748
$162$	0	0
$163$	−28658.5	−1.07864	−0.539322	−	0.842099i	$-0.681320\pi$
$163$	−28658.5	−1.07864	−0.539322	+	0.842099i	$0.681320\pi$
$164$	0	0
$165$	14216.7i	0.522192i
$166$	0	0
$167$	21816.0i	0.782244i	0.920339	+	0.391122i	$0.127913\pi$
$167$	21816.0i	0.782244i	−0.920339	+	0.391122i	$0.872087\pi$
$168$	0	0
$169$	−37151.0	−1.30076
$170$	0	0
$171$	17448.7	0.596720
$172$	0	0
$173$	31904.4i	1.06600i	0.846115	+	0.533001i	$0.178936\pi$
$173$	31904.4i	1.06600i	−0.846115	+	0.533001i	$0.821064\pi$
$174$	0	0
$175$	− 7720.00i	− 0.252082i
$176$	0	0
$177$	18252.0	0.582591
$178$	0	0
$179$	−10589.8	−0.330506	−0.165253	−	0.986251i	$-0.552844\pi$
$179$	−10589.8	−0.330506	−0.165253	+	0.986251i	$0.552844\pi$
$180$	0	0
$181$	− 36920.4i	− 1.12696i	−0.826129	−	0.563481i	$-0.809462\pi$
$181$	− 36920.4i	− 1.12696i	0.826129	−	0.563481i	$-0.190538\pi$
$182$	0	0
$183$	− 3480.00i	− 0.103915i
$184$	0	0
$185$	36432.0	1.06449
$186$	0	0
$187$	39095.9	1.11801
$188$	0	0
$189$	− 20784.6i	− 0.581860i
$190$	0	0
$191$	− 6912.00i	− 0.189468i	−0.995503	−	0.0947342i	$-0.969800\pi$
$191$	− 6912.00i	− 0.189468i	0.995503	−	0.0947342i	$-0.0302001\pi$
$192$	0	0
$193$	−29446.0	−0.790518	−0.395259	−	0.918570i	$-0.629345\pi$
$193$	−29446.0	−0.790518	−0.395259	+	0.918570i	$0.629345\pi$
$194$	0	0
$195$	18456.7	0.485384
$196$	0	0
$197$	− 40550.8i	− 1.04488i	−0.852676	−	0.522440i	$-0.825022\pi$
$197$	− 40550.8i	− 1.04488i	0.852676	−	0.522440i	$-0.174978\pi$
$198$	0	0
$199$	1432.00i	0.0361607i	0.999837	+	0.0180804i	$0.00575547\pi$
$199$	1432.00i	0.0361607i	−0.999837	+	0.0180804i	$0.994245\pi$
$200$	0	0
$201$	6252.00	0.154749
$202$	0	0
$203$	−17459.1	−0.423671
$204$	0	0
$205$	374.123i	0.00890239i
$206$	0	0
$207$	34776.0i	0.811594i
$208$	0	0
$209$	49932.0	1.14311
$210$	0	0
$211$	75985.1	1.70672	0.853362	−	0.521319i	$-0.174560\pi$
$211$	75985.1	1.70672	0.853362	+	0.521319i	$0.174560\pi$
$212$	0	0
$213$	27186.3i	0.599226i
$214$	0	0
$215$	32184.0i	0.696247i
$216$	0	0
$217$	67840.0	1.44068
$218$	0	0
$219$	14930.3	0.311300
$220$	0	0
$221$	− 50756.0i	− 1.03921i
$222$	0	0
$223$	19808.0i	0.398319i	0.979967	+	0.199159i	$0.0638212\pi$
$223$	19808.0i	0.398319i	−0.979967	+	0.199159i	$0.936179\pi$
$224$	0	0
$225$	−13317.0	−0.263052
$226$	0	0
$227$	28152.8	0.546348	0.273174	−	0.961965i	$-0.411927\pi$
$227$	28152.8	0.546348	0.273174	+	0.961965i	$0.411927\pi$
$228$	0	0
$229$	− 45372.8i	− 0.865216i	−0.901582	−	0.432608i	$-0.857593\pi$
$229$	− 45372.8i	− 0.865216i	0.901582	−	0.432608i	$-0.142407\pi$
$230$	0	0
$231$	− 27360.0i	− 0.512734i
$232$	0	0
$233$	−105750.	−1.94791	−0.973954	−	0.226745i	$-0.927192\pi$
$233$	−105750.	−1.94791	−0.973954	+	0.226745i	$0.927192\pi$
$234$	0	0
$235$	−50880.7	−0.921335
$236$	0	0
$237$	− 14355.2i	− 0.255572i
$238$	0	0
$239$	23184.0i	0.405875i	0.979192	+	0.202938i	$0.0650489\pi$
$239$	23184.0i	0.405875i	−0.979192	+	0.202938i	$0.934951\pi$
$240$	0	0
$241$	13306.0	0.229094	0.114547	−	0.993418i	$-0.463458\pi$
$241$	13306.0	0.229094	0.114547	+	0.993418i	$0.463458\pi$
$242$	0	0
$243$	−55214.3	−0.935059
$244$	0	0
$245$	16648.5i	0.277359i
$246$	0	0
$247$	− 64824.0i	− 1.06253i
$248$	0	0
$249$	31500.0	0.508056
$250$	0	0
$251$	63569.7	1.00903	0.504514	−	0.863404i	$-0.331672\pi$
$251$	63569.7	1.00903	0.504514	+	0.863404i	$0.331672\pi$
$252$	0	0
$253$	99516.7i	1.55473i
$254$	0	0
$255$	14256.0i	0.219239i
$256$	0	0
$257$	−2142.00	−0.0324305	−0.0162152	−	0.999869i	$-0.505162\pi$
$257$	−2142.00	−0.0324305	−0.0162152	+	0.999869i	$0.505162\pi$
$258$	0	0
$259$	−70113.4	−1.04521
$260$	0	0
$261$	30116.9i	0.442109i
$262$	0	0
$263$	− 59112.0i	− 0.854602i	−0.904109	−	0.427301i	$-0.859464\pi$
$263$	− 59112.0i	− 0.854602i	0.904109	−	0.427301i	$-0.140536\pi$
$264$	0	0
$265$	−67824.0	−0.965810
$266$	0	0
$267$	−10787.2	−0.151317
$268$	0	0
$269$	8293.06i	0.114607i	0.998357	+	0.0573034i	$0.0182502\pi$
$269$	8293.06i	0.114607i	−0.998357	+	0.0573034i	$0.981750\pi$
$270$	0	0
$271$	97712.0i	1.33048i	0.746628	+	0.665241i	$0.231671\pi$
$271$	97712.0i	1.33048i	−0.746628	+	0.665241i	$0.768329\pi$
$272$	0	0
$273$	−35520.0	−0.476593
$274$	0	0
$275$	−38108.6	−0.503915
$276$	0	0
$277$	82577.3i	1.07622i	0.842875	+	0.538110i	$0.180861\pi$
$277$	82577.3i	1.07622i	−0.842875	+	0.538110i	$0.819139\pi$
$278$	0	0
$279$	− 117024.i	− 1.50337i
$280$	0	0
$281$	116586.	1.47650	0.738251	−	0.674527i	$-0.235652\pi$
$281$	116586.	1.47650	0.738251	+	0.674527i	$0.235652\pi$
$282$	0	0
$283$	−142032.	−1.77342	−0.886711	−	0.462324i	$-0.847016\pi$
$283$	−142032.	−1.77342	−0.886711	+	0.462324i	$0.847016\pi$
$284$	0	0
$285$	18207.3i	0.224159i
$286$	0	0
$287$	− 720.000i	− 0.00874115i
$288$	0	0
$289$	−44317.0	−0.530609
$290$	0	0
$291$	−16385.2	−0.193493
$292$	0	0
$293$	42088.8i	0.490266i	0.969489	+	0.245133i	$0.0788316\pi$
$293$	42088.8i	0.490266i	−0.969489	+	0.245133i	$0.921168\pi$
$294$	0	0
$295$	− 109512.i	− 1.25840i
$296$	0	0
$297$	−102600.	−1.16315
$298$	0	0
$299$	129197.	1.44514
$300$	0	0
$301$	− 61938.1i	− 0.683636i
$302$	0	0
$303$	− 72.0000i	0 0.000784237i
$304$	0	0
$305$	−20880.0	−0.224456
$306$	0	0
$307$	39875.3	0.423084	0.211542	−	0.977369i	$-0.432151\pi$
$307$	39875.3	0.423084	0.211542	+	0.977369i	$0.432151\pi$
$308$	0	0
$309$	− 31232.3i	− 0.327105i
$310$	0	0
$311$	143784.i	1.48659i	0.668966	+	0.743293i	$0.266737\pi$
$311$	143784.i	1.48659i	−0.668966	+	0.743293i	$0.733263\pi$
$312$	0	0
$313$	15826.0	0.161541	0.0807704	−	0.996733i	$-0.474262\pi$
$313$	15826.0	0.161541	0.0807704	+	0.996733i	$0.474262\pi$
$314$	0	0
$315$	−57365.5	−0.578136
$316$	0	0
$317$	− 113006.i	− 1.12456i	−0.826947	−	0.562280i	$-0.809924\pi$
$317$	− 113006.i	− 1.12456i	0.826947	−	0.562280i	$-0.190076\pi$
$318$	0	0
$319$	86184.0i	0.846926i
$320$	0	0
$321$	8172.00	0.0793082
$322$	0	0
$323$	50070.1	0.479925
$324$	0	0
$325$	49474.3i	0.468396i
$326$	0	0
$327$	50184.0i	0.469321i
$328$	0	0
$329$	97920.0	0.904648
$330$	0	0
$331$	34776.1	0.317413	0.158707	−	0.987326i	$-0.449268\pi$
$331$	34776.1	0.317413	0.158707	+	0.987326i	$0.449268\pi$
$332$	0	0
$333$	120946.i	1.09069i
$334$	0	0
$335$	− 37512.0i	− 0.334257i
$336$	0	0
$337$	158114.	1.39223	0.696114	−	0.717931i	$-0.254911\pi$
$337$	158114.	1.39223	0.696114	+	0.717931i	$0.254911\pi$
$338$	0	0
$339$	7420.11	0.0645670
$340$	0	0
$341$	− 334882.i	− 2.87993i
$342$	0	0
$343$	− 128080.i	− 1.08866i
$344$	0	0
$345$	−36288.0	−0.304877
$346$	0	0
$347$	122598.	1.01818	0.509090	−	0.860713i	$-0.329982\pi$
$347$	122598.	1.01818	0.509090	+	0.860713i	$0.329982\pi$
$348$	0	0
$349$	1780.55i	0.0146185i	0.999973	+	0.00730925i	$0.00232663\pi$
$349$	1780.55i	0.0146185i	−0.999973	+	0.00730925i	$0.997673\pi$
$350$	0	0
$351$	133200.i	1.08116i
$352$	0	0
$353$	41346.0	0.331806	0.165903	−	0.986142i	$-0.446946\pi$
$353$	41346.0	0.331806	0.165903	+	0.986142i	$0.446946\pi$
$354$	0	0
$355$	163118.	1.29433
$356$	0	0
$357$	− 27435.7i	− 0.215268i
$358$	0	0
$359$	150840.i	1.17038i	0.810895	+	0.585191i	$0.198980\pi$
$359$	150840.i	1.17038i	−0.810895	+	0.585191i	$0.801020\pi$
$360$	0	0
$361$	−66373.0	−0.509304
$362$	0	0
$363$	−84340.5	−0.640063
$364$	0	0
$365$	− 89581.7i	− 0.672409i
$366$	0	0
$367$	− 79664.0i	− 0.591466i	−0.955271	−	0.295733i	$-0.904436\pi$
$367$	− 79664.0i	− 0.591466i	0.955271	−	0.295733i	$-0.0955639\pi$
$368$	0	0
$369$	−1242.00	−0.00912155
$370$	0	0
$371$	130527.	0.948317
$372$	0	0
$373$	135176.i	0.971589i	0.874073	+	0.485794i	$0.161470\pi$
$373$	135176.i	0.971589i	−0.874073	+	0.485794i	$0.838530\pi$
$374$	0	0
$375$	− 58896.0i	− 0.418816i
$376$	0	0
$377$	111888.	0.787229
$378$	0	0
$379$	16735.1	0.116506	0.0582531	−	0.998302i	$-0.481447\pi$
$379$	16735.1	0.116506	0.0582531	+	0.998302i	$0.481447\pi$
$380$	0	0
$381$	17514.5i	0.120656i
$382$	0	0
$383$	− 157824.i	− 1.07591i	−0.842974	−	0.537954i	$-0.819197\pi$
$383$	− 157824.i	− 1.07591i	0.842974	−	0.537954i	$-0.180803\pi$
$384$	0	0
$385$	−164160.	−1.10751
$386$	0	0
$387$	−106843.	−0.713387
$388$	0	0
$389$	50735.2i	0.335282i	0.985848	+	0.167641i	$0.0536150\pi$
$389$	50735.2i	0.335282i	−0.985848	+	0.167641i	$0.946385\pi$
$390$	0	0
$391$	99792.0i	0.652743i
$392$	0	0
$393$	31788.0	0.205816
$394$	0	0
$395$	−86131.4	−0.552036
$396$	0	0
$397$	− 81829.0i	− 0.519190i	−0.965718	−	0.259595i	$-0.916411\pi$
$397$	− 81829.0i	− 0.519190i	0.965718	−	0.259595i	$-0.0835891\pi$
$398$	0	0
$399$	− 35040.0i	− 0.220099i
$400$	0	0
$401$	33210.0	0.206529	0.103264	−	0.994654i	$-0.467071\pi$
$401$	33210.0	0.206529	0.103264	+	0.994654i	$0.467071\pi$
$402$	0	0
$403$	−434759.	−2.67694
$404$	0	0
$405$	78752.9i	0.480127i
$406$	0	0
$407$	346104.i	2.08938i
$408$	0	0
$409$	−123950.	−0.740969	−0.370484	−	0.928839i	$-0.620808\pi$
$409$	−123950.	−0.740969	−0.370484	+	0.928839i	$0.620808\pi$
$410$	0	0
$411$	112673.	0.667018
$412$	0	0
$413$	210756.i	1.23561i
$414$	0	0
$415$	− 189000.i	− 1.09740i
$416$	0	0
$417$	−3828.00	−0.0220140
$418$	0	0
$419$	24910.4	0.141890	0.0709450	−	0.997480i	$-0.477399\pi$
$419$	24910.4	0.141890	0.0709450	+	0.997480i	$0.477399\pi$
$420$	0	0
$421$	− 18990.2i	− 0.107143i	−0.998564	−	0.0535717i	$-0.982939\pi$
$421$	− 18990.2i	− 0.107143i	0.998564	−	0.0535717i	$-0.0170606\pi$
$422$	0	0
$423$	− 168912.i	− 0.944017i
$424$	0	0
$425$	−38214.0	−0.211565
$426$	0	0
$427$	40183.6	0.220390
$428$	0	0
$429$	175339.i	0.952717i
$430$	0	0
$431$	− 304560.i	− 1.63953i	−0.572703	−	0.819763i	$-0.694105\pi$
$431$	− 304560.i	− 1.63953i	0.572703	−	0.819763i	$-0.305895\pi$
$432$	0	0
$433$	−237062.	−1.26440	−0.632202	−	0.774803i	$-0.717849\pi$
$433$	−237062.	−1.26440	−0.632202	+	0.774803i	$0.717849\pi$
$434$	0	0
$435$	−31426.3	−0.166079
$436$	0	0
$437$	127451.i	0.667392i
$438$	0	0
$439$	8488.00i	0.0440429i	0.999757	+	0.0220215i	$0.00701022\pi$
$439$	8488.00i	0.0440429i	−0.999757	+	0.0220215i	$0.992990\pi$
$440$	0	0
$441$	−55269.0	−0.284187
$442$	0	0
$443$	−227997.	−1.16177	−0.580886	−	0.813985i	$-0.697294\pi$
$443$	−227997.	−1.16177	−0.580886	+	0.813985i	$0.697294\pi$
$444$	0	0
$445$	64723.3i	0.326844i
$446$	0	0
$447$	− 119592.i	− 0.598532i
$448$	0	0
$449$	49338.0	0.244731	0.122365	−	0.992485i	$-0.460952\pi$
$449$	49338.0	0.244731	0.122365	+	0.992485i	$0.460952\pi$
$450$	0	0
$451$	−3554.17	−0.0174737
$452$	0	0
$453$	− 49245.7i	− 0.239978i
$454$	0	0
$455$	213120.i	1.02944i
$456$	0	0
$457$	206866.	0.990505	0.495253	−	0.868749i	$-0.335076\pi$
$457$	206866.	0.990505	0.495253	+	0.868749i	$0.335076\pi$
$458$	0	0
$459$	−102884.	−0.488339
$460$	0	0
$461$	− 194897.i	− 0.917073i	−0.888676	−	0.458537i	$-0.848374\pi$
$461$	− 194897.i	− 0.917073i	0.888676	−	0.458537i	$-0.151626\pi$
$462$	0	0
$463$	137648.i	0.642108i	0.947061	+	0.321054i	$0.104037\pi$
$463$	137648.i	0.642108i	−0.947061	+	0.321054i	$0.895963\pi$
$464$	0	0
$465$	122112.	0.564745
$466$	0	0
$467$	270501.	1.24033	0.620163	−	0.784473i	$-0.287067\pi$
$467$	270501.	1.24033	0.620163	+	0.784473i	$0.287067\pi$
$468$	0	0
$469$	72191.9i	0.328203i
$470$	0	0
$471$	936.000i	0.00421924i
$472$	0	0
$473$	−305748.	−1.36660
$474$	0	0
$475$	−48805.7	−0.216313
$476$	0	0
$477$	− 225160.i	− 0.989587i
$478$	0	0
$479$	− 183456.i	− 0.799578i	−0.916607	−	0.399789i	$-0.869083\pi$
$479$	− 183456.i	− 0.799578i	0.916607	−	0.399789i	$-0.130917\pi$
$480$	0	0
$481$	449328.	1.94211
$482$	0	0
$483$	69836.3	0.299355
$484$	0	0
$485$	98311.2i	0.417945i
$486$	0	0
$487$	108152.i	0.456012i	0.973660	+	0.228006i	$0.0732206\pi$
$487$	108152.i	0.456012i	−0.973660	+	0.228006i	$0.926779\pi$
$488$	0	0
$489$	99276.0	0.415171
$490$	0	0
$491$	170652.	0.707862	0.353931	−	0.935272i	$-0.384845\pi$
$491$	170652.	0.707862	0.353931	+	0.935272i	$0.384845\pi$
$492$	0	0
$493$	86422.4i	0.355576i
$494$	0	0
$495$	283176.i	1.15570i
$496$	0	0
$497$	−313920.	−1.27088
$498$	0	0
$499$	146874.	0.589855	0.294927	−	0.955520i	$-0.404705\pi$
$499$	146874.	0.589855	0.294927	+	0.955520i	$0.404705\pi$
$500$	0	0
$501$	− 75572.8i	− 0.301086i
$502$	0	0
$503$	184104.i	0.727658i	0.931466	+	0.363829i	$0.118531\pi$
$503$	184104.i	0.727658i	−0.931466	+	0.363829i	$0.881469\pi$
$504$	0	0
$505$	−432.000	−0.00169395
$506$	0	0
$507$	128695.	0.500663
$508$	0	0
$509$	− 441070.i	− 1.70244i	−0.524808	−	0.851221i	$-0.675863\pi$
$509$	− 441070.i	− 1.70244i	0.524808	−	0.851221i	$-0.324137\pi$
$510$	0	0
$511$	172400.i	0.660230i
$512$	0	0
$513$	−131400.	−0.499299
$514$	0	0
$515$	−187394.	−0.706547
$516$	0	0
$517$	− 483367.i	− 1.80841i
$518$	0	0
$519$	− 110520.i	− 0.410304i
$520$	0	0
$521$	332658.	1.22553	0.612763	−	0.790267i	$-0.290058\pi$
$521$	332658.	1.22553	0.612763	+	0.790267i	$0.290058\pi$
$522$	0	0
$523$	350037.	1.27971	0.639854	−	0.768497i	$-0.278995\pi$
$523$	350037.	1.27971	0.639854	+	0.768497i	$0.278995\pi$
$524$	0	0
$525$	26742.9i	0.0970263i
$526$	0	0
$527$	− 335808.i	− 1.20912i
$528$	0	0
$529$	25825.0	0.0922845
$530$	0	0
$531$	363554.	1.28938
$532$	0	0
$533$	4614.18i	0.0162420i
$534$	0	0
$535$	− 49032.0i	− 0.171306i
$536$	0	0
$537$	36684.0	0.127212
$538$	0	0
$539$	−158160.	−0.544403
$540$	0	0
$541$	158136.i	0.540302i	0.962818	+	0.270151i	$0.0870737\pi$
$541$	158136.i	0.540302i	−0.962818	+	0.270151i	$0.912926\pi$
$542$	0	0
$543$	127896.i	0.433768i
$544$	0	0
$545$	301104.	1.01373
$546$	0	0
$547$	−38385.7	−0.128291	−0.0641453	−	0.997941i	$-0.520432\pi$
$547$	−38385.7	−0.128291	−0.0641453	+	0.997941i	$0.520432\pi$
$548$	0	0
$549$	− 69316.7i	− 0.229982i
$550$	0	0
$551$	110376.i	0.363556i
$552$	0	0
$553$	165760.	0.542038
$554$	0	0
$555$	−126204.	−0.409720
$556$	0	0
$557$	540379.i	1.74176i	0.491496	+	0.870880i	$0.336450\pi$
$557$	540379.i	1.74176i	−0.491496	+	0.870880i	$0.663550\pi$
$558$	0	0
$559$	396936.i	1.27027i
$560$	0	0
$561$	−135432.	−0.430324
$562$	0	0
$563$	−390782.	−1.23287	−0.616435	−	0.787405i	$-0.711424\pi$
$563$	−390782.	−1.23287	−0.616435	+	0.787405i	$0.711424\pi$
$564$	0	0
$565$	− 44520.6i	− 0.139465i
$566$	0	0
$567$	− 151560.i	− 0.471431i
$568$	0	0
$569$	−461646.	−1.42589	−0.712943	−	0.701222i	$-0.752638\pi$
$569$	−461646.	−1.42589	−0.712943	+	0.701222i	$0.752638\pi$
$570$	0	0
$571$	215076.	0.659658	0.329829	−	0.944041i	$-0.393009\pi$
$571$	215076.	0.659658	0.329829	+	0.944041i	$0.393009\pi$
$572$	0	0
$573$	23943.9i	0.0729265i
$574$	0	0
$575$	− 97272.0i	− 0.294206i
$576$	0	0
$577$	−282590.	−0.848800	−0.424400	−	0.905475i	$-0.639515\pi$
$577$	−282590.	−0.848800	−0.424400	+	0.905475i	$0.639515\pi$
$578$	0	0
$579$	102004.	0.304270
$580$	0	0
$581$	363731.i	1.07753i
$582$	0	0
$583$	− 644328.i	− 1.89570i
$584$	0	0
$585$	367632.	1.07424
$586$	0	0
$587$	−355032.	−1.03037	−0.515183	−	0.857080i	$-0.672276\pi$
$587$	−355032.	−1.03037	−0.515183	+	0.857080i	$0.672276\pi$
$588$	0	0
$589$	− 428883.i	− 1.23626i
$590$	0	0
$591$	140472.i	0.402175i
$592$	0	0
$593$	119682.	0.340345	0.170173	−	0.985414i	$-0.445567\pi$
$593$	119682.	0.340345	0.170173	+	0.985414i	$0.445567\pi$
$594$	0	0
$595$	−164614.	−0.464979
$596$	0	0
$597$	− 4960.59i	− 0.0139183i
$598$	0	0
$599$	− 505080.i	− 1.40769i	−0.710354	−	0.703844i	$-0.751465\pi$
$599$	− 505080.i	− 1.40769i	0.710354	−	0.703844i	$-0.248535\pi$
$600$	0	0
$601$	−576406.	−1.59580	−0.797902	−	0.602787i	$-0.794057\pi$
$601$	−576406.	−1.59580	−0.797902	+	0.602787i	$0.794057\pi$
$602$	0	0
$603$	124531.	0.342486
$604$	0	0
$605$	506043.i	1.38254i
$606$	0	0
$607$	464672.i	1.26116i	0.776126	+	0.630578i	$0.217182\pi$
$607$	464672.i	1.26116i	−0.776126	+	0.630578i	$0.782818\pi$
$608$	0	0
$609$	60480.0	0.163071
$610$	0	0
$611$	−627529.	−1.68094
$612$	0	0
$613$	292613.i	0.778704i	0.921089	+	0.389352i	$0.127301\pi$
$613$	292613.i	0.778704i	−0.921089	+	0.389352i	$0.872699\pi$
$614$	0	0
$615$	− 1296.00i	− 0.00342653i
$616$	0	0
$617$	−294678.	−0.774065	−0.387032	−	0.922066i	$-0.626500\pi$
$617$	−294678.	−0.774065	−0.387032	+	0.922066i	$0.626500\pi$
$618$	0	0
$619$	−347273.	−0.906336	−0.453168	−	0.891425i	$-0.649706\pi$
$619$	−347273.	−0.906336	−0.453168	+	0.891425i	$0.649706\pi$
$620$	0	0
$621$	− 261886.i	− 0.679093i
$622$	0	0
$623$	− 124560.i	− 0.320924i
$624$	0	0
$625$	−232751.	−0.595843
$626$	0	0
$627$	−172970.	−0.439982
$628$	0	0
$629$	347061.i	0.877213i
$630$	0	0
$631$	− 434008.i	− 1.09003i	−0.838426	−	0.545016i	$-0.816524\pi$
$631$	− 434008.i	− 1.09003i	0.838426	−	0.545016i	$-0.183476\pi$
$632$	0	0
$633$	−263220.	−0.656918
$634$	0	0
$635$	105087.	0.260616
$636$	0	0
$637$	205331.i	0.506030i
$638$	0	0
$639$	541512.i	1.32619i
$640$	0	0
$641$	−68742.0	−0.167304	−0.0836520	−	0.996495i	$-0.526658\pi$
$641$	−68742.0	−0.167304	−0.0836520	+	0.996495i	$0.526658\pi$
$642$	0	0
$643$	610538.	1.47669	0.738347	−	0.674421i	$-0.235607\pi$
$643$	610538.	1.47669	0.738347	+	0.674421i	$0.235607\pi$
$644$	0	0
$645$	− 111489.i	− 0.267985i
$646$	0	0
$647$	114840.i	0.274337i	0.990548	+	0.137169i	$0.0438002\pi$
$647$	114840.i	0.274337i	−0.990548	+	0.137169i	$0.956200\pi$
$648$	0	0
$649$	1.04036e6	2.46999
$650$	0	0
$651$	−235005.	−0.554517
$652$	0	0
$653$	294289.i	0.690157i	0.938574	+	0.345079i	$0.112148\pi$
$653$	294289.i	0.690157i	−0.938574	+	0.345079i	$0.887852\pi$
$654$	0	0
$655$	− 190728.i	− 0.444562i
$656$	0	0
$657$	297390.	0.688963
$658$	0	0
$659$	333105.	0.767025	0.383513	−	0.923536i	$-0.374714\pi$
$659$	333105.	0.767025	0.383513	+	0.923536i	$0.374714\pi$
$660$	0	0
$661$	− 296589.i	− 0.678817i	−0.940639	−	0.339409i	$-0.889773\pi$
$661$	− 296589.i	− 0.678817i	0.940639	−	0.339409i	$-0.110227\pi$
$662$	0	0
$663$	175824.i	0.399992i
$664$	0	0
$665$	−210240.	−0.475414
$666$	0	0
$667$	−219984.	−0.494470
$668$	0	0
$669$	− 68616.9i	− 0.153313i
$670$	0	0
$671$	− 198360.i	− 0.440564i
$672$	0	0
$673$	379066.	0.836921	0.418461	−	0.908235i	$-0.362570\pi$
$673$	379066.	0.836921	0.418461	+	0.908235i	$0.362570\pi$
$674$	0	0
$675$	100286.	0.220106
$676$	0	0
$677$	− 204832.i	− 0.446911i	−0.974714	−	0.223456i	$-0.928266\pi$
$677$	− 204832.i	− 0.446911i	0.974714	−	0.223456i	$-0.0717338\pi$
$678$	0	0
$679$	− 189200.i	− 0.410376i
$680$	0	0
$681$	−97524.0	−0.210289
$682$	0	0
$683$	−13250.2	−0.0284041	−0.0142020	−	0.999899i	$-0.504521\pi$
$683$	−13250.2	−0.0284041	−0.0142020	+	0.999899i	$0.504521\pi$
$684$	0	0
$685$	− 676040.i	− 1.44076i
$686$	0	0
$687$	157176.i	0.333022i
$688$	0	0
$689$	−836496.	−1.76208
$690$	0	0
$691$	485913.	1.01766	0.508830	−	0.860867i	$-0.330078\pi$
$691$	485913.	1.01766	0.508830	+	0.860867i	$0.330078\pi$
$692$	0	0
$693$	− 544972.i	− 1.13477i
$694$	0	0
$695$	22968.0i	0.0475503i
$696$	0	0
$697$	−3564.00	−0.00733622
$698$	0	0
$699$	366329.	0.749750
$700$	0	0
$701$	− 119823.i	− 0.243840i	−0.992540	−	0.121920i	$-0.961095\pi$
$701$	− 119823.i	− 0.243840i	0.992540	−	0.121920i	$-0.0389052\pi$
$702$	0	0
$703$	443256.i	0.896900i
$704$	0	0
$705$	176256.	0.354622
$706$	0	0
$707$	831.384	0.00166327
$708$	0	0
$709$	564974.i	1.12392i	0.827164	+	0.561961i	$0.189953\pi$
$709$	564974.i	1.12392i	−0.827164	+	0.561961i	$0.810047\pi$
$710$	0	0
$711$	− 285936.i	− 0.565626i
$712$	0	0
$713$	854784.	1.68142
$714$	0	0
$715$	1.05203e6	2.05787
$716$	0	0
$717$	− 80311.7i	− 0.156221i
$718$	0	0
$719$	453168.i	0.876600i	0.898829	+	0.438300i	$0.144419\pi$
$719$	453168.i	0.876600i	−0.898829	+	0.438300i	$0.855581\pi$
$720$	0	0
$721$	360640.	0.693751
$722$	0	0
$723$	−46093.3	−0.0881783
$724$	0	0
$725$	− 84240.0i	− 0.160266i
$726$	0	0
$727$	604808.i	1.14432i	0.820141	+	0.572162i	$0.193895\pi$
$727$	604808.i	1.14432i	−0.820141	+	0.572162i	$0.806105\pi$
$728$	0	0
$729$	−115641.	−0.217599
$730$	0	0
$731$	−306594.	−0.573758
$732$	0	0
$733$	− 1.05230e6i	− 1.95854i	−0.202550	−	0.979272i	$-0.564923\pi$
$733$	− 1.05230e6i	− 1.95854i	0.202550	−	0.979272i	$-0.435077\pi$
$734$	0	0
$735$	− 57672.0i	− 0.106756i
$736$	0	0
$737$	356364.	0.656083
$738$	0	0
$739$	−502936.	−0.920923	−0.460462	−	0.887680i	$-0.652316\pi$
$739$	−502936.	−0.920923	−0.460462	+	0.887680i	$0.652316\pi$
$740$	0	0
$741$	224557.i	0.408969i
$742$	0	0
$743$	499320.i	0.904485i	0.891895	+	0.452242i	$0.149376\pi$
$743$	499320.i	0.904485i	−0.891895	+	0.452242i	$0.850624\pi$
$744$	0	0
$745$	−717552.	−1.29283
$746$	0	0
$747$	627435.	1.12442
$748$	0	0
$749$	94362.1i	0.168203i
$750$	0	0
$751$	207376.i	0.367687i	0.982955	+	0.183844i	$0.0588540\pi$
$751$	207376.i	0.367687i	−0.982955	+	0.183844i	$0.941146\pi$
$752$	0	0
$753$	−220212.	−0.388375
$754$	0	0
$755$	−295474.	−0.518353
$756$	0	0
$757$	− 175041.i	− 0.305456i	−0.988268	−	0.152728i	$-0.951194\pi$
$757$	− 175041.i	− 0.305456i	0.988268	−	0.152728i	$-0.0488058\pi$
$758$	0	0
$759$	− 344736.i	− 0.598416i
$760$	0	0
$761$	93906.0	0.162153	0.0810763	−	0.996708i	$-0.474164\pi$
$761$	93906.0	0.162153	0.0810763	+	0.996708i	$0.474164\pi$
$762$	0	0
$763$	−579475.	−0.995372
$764$	0	0
$765$	283959.i	0.485214i
$766$	0	0
$767$	− 1.35065e6i	− 2.29589i
$768$	0	0
$769$	162362.	0.274556	0.137278	−	0.990533i	$-0.456165\pi$
$769$	162362.	0.274556	0.137278	+	0.990533i	$0.456165\pi$
$770$	0	0
$771$	7420.11	0.0124825
$772$	0	0
$773$	− 504630.i	− 0.844527i	−0.906473	−	0.422264i	$-0.861236\pi$
$773$	− 504630.i	− 0.844527i	0.906473	−	0.422264i	$-0.138764\pi$
$774$	0	0
$775$	327328.i	0.544979i
$776$	0	0
$777$	242880.	0.402300
$778$	0	0
$779$	−4551.83	−0.00750086
$780$	0	0
$781$	1.54962e6i	2.54052i
$782$	0	0
$783$	− 226800.i	− 0.369930i
$784$	0	0
$785$	5616.00	0.00911355
$786$	0	0
$787$	−38413.4	−0.0620203	−0.0310101	−	0.999519i	$-0.509872\pi$
$787$	−38413.4	−0.0620203	−0.0310101	+	0.999519i	$0.509872\pi$
$788$	0	0
$789$	204770.i	0.328937i
$790$	0	0
$791$	85680.0i	0.136939i
$792$	0	0
$793$	−257520.	−0.409510
$794$	0	0
$795$	234949.	0.371740
$796$	0	0
$797$	269098.i	0.423637i	0.977309	+	0.211819i	$0.0679386\pi$
$797$	269098.i	0.423637i	−0.977309	+	0.211819i	$0.932061\pi$
$798$	0	0
$799$	− 484704.i	− 0.759247i
$800$	0	0
$801$	−214866.	−0.334890
$802$	0	0
$803$	851026.	1.31981
$804$	0	0
$805$	− 419018.i	− 0.646607i
$806$	0	0
$807$	− 28728.0i	− 0.0441122i
$808$	0	0
$809$	−813294.	−1.24265	−0.621327	−	0.783551i	$-0.713406\pi$
$809$	−813294.	−1.24265	−0.621327	+	0.783551i	$0.713406\pi$
$810$	0	0
$811$	−1.19639e6	−1.81900	−0.909499	−	0.415706i	$-0.863535\pi$
$811$	−1.19639e6	−1.81900	−0.909499	+	0.415706i	$0.863535\pi$
$812$	0	0
$813$	− 338484.i	− 0.512103i
$814$	0	0
$815$	− 595656.i	− 0.896768i
$816$	0	0
$817$	−391572.	−0.586634
$818$	0	0
$819$	−707508.	−1.05478
$820$	0	0
$821$	681340.i	1.01083i	0.862877	+	0.505415i	$0.168660\pi$
$821$	681340.i	1.01083i	−0.862877	+	0.505415i	$0.831340\pi$
$822$	0	0
$823$	688808.i	1.01695i	0.861078	+	0.508474i	$0.169790\pi$
$823$	688808.i	1.01695i	−0.861078	+	0.508474i	$0.830210\pi$
$824$	0	0
$825$	132012.	0.193957
$826$	0	0
$827$	−518815.	−0.758580	−0.379290	−	0.925278i	$-0.623832\pi$
$827$	−518815.	−0.758580	−0.379290	+	0.925278i	$0.623832\pi$
$828$	0	0
$829$	1.00144e6i	1.45718i	0.684948	+	0.728592i	$0.259825\pi$
$829$	1.00144e6i	1.45718i	−0.684948	+	0.728592i	$0.740175\pi$
$830$	0	0
$831$	− 286056.i	− 0.414237i
$832$	0	0
$833$	−158598.	−0.228564
$834$	0	0
$835$	−453437.	−0.650345
$836$	0	0
$837$	881267.i	1.25793i
$838$	0	0
$839$	520344.i	0.739208i	0.929189	+	0.369604i	$0.120507\pi$
$839$	520344.i	0.739208i	−0.929189	+	0.369604i	$0.879493\pi$
$840$	0	0
$841$	516769.	0.730642
$842$	0	0
$843$	−403866.	−0.568306
$844$	0	0
$845$	− 772169.i	− 1.08143i
$846$	0	0
$847$	− 973880.i	− 1.35750i
$848$	0	0
$849$	492012.	0.682591
$850$	0	0
$851$	−883429.	−1.21987
$852$	0	0
$853$	271357.i	0.372943i	0.982460	+	0.186472i	$0.0597053\pi$
$853$	271357.i	0.372943i	−0.982460	+	0.186472i	$0.940295\pi$
$854$	0	0
$855$	362664.i	0.496103i
$856$	0	0
$857$	324018.	0.441172	0.220586	−	0.975368i	$-0.429203\pi$
$857$	324018.	0.441172	0.220586	+	0.975368i	$0.429203\pi$
$858$	0	0
$859$	638832.	0.865766	0.432883	−	0.901450i	$-0.357496\pi$
$859$	638832.	0.865766	0.432883	+	0.901450i	$0.357496\pi$
$860$	0	0
$861$	2494.15i	0.00336447i
$862$	0	0
$863$	399456.i	0.536349i	0.963370	+	0.268174i	$0.0864204\pi$
$863$	399456.i	0.536349i	−0.963370	+	0.268174i	$0.913580\pi$
$864$	0	0
$865$	−663120.	−0.886257
$866$	0	0
$867$	153519.	0.204232
$868$	0	0
$869$	− 818249.i	− 1.08354i
$870$	0	0
$871$	− 462648.i	− 0.609838i
$872$	0	0
$873$	−326370.	−0.428235
$874$	0	0
$875$	680072.	0.888258
$876$	0	0
$877$	− 35216.1i	− 0.0457869i	−0.999738	−	0.0228935i	$-0.992712\pi$
$877$	− 35216.1i	− 0.0457869i	0.999738	−	0.0228935i	$-0.00728785\pi$
$878$	0	0
$879$	− 145800.i	− 0.188703i
$880$	0	0
$881$	−876510.	−1.12929	−0.564644	−	0.825334i	$-0.690987\pi$
$881$	−876510.	−1.12929	−0.564644	+	0.825334i	$0.690987\pi$
$882$	0	0
$883$	1.21983e6	1.56451	0.782255	−	0.622958i	$-0.214069\pi$
$883$	1.21983e6	1.56451	0.782255	+	0.622958i	$0.214069\pi$
$884$	0	0
$885$	379361.i	0.484357i
$886$	0	0
$887$	853992.i	1.08544i	0.839913	+	0.542721i	$0.182606\pi$
$887$	853992.i	1.08544i	−0.839913	+	0.542721i	$0.817394\pi$
$888$	0	0
$889$	−202240.	−0.255896
$890$	0	0
$891$	−748152.	−0.942399
$892$	0	0
$893$	− 619049.i	− 0.776286i
$894$	0	0
$895$	− 220104.i	− 0.274778i
$896$	0	0
$897$	−447552.	−0.556235
$898$	0	0
$899$	740265.	0.915941
$900$	0	0
$901$	− 646110.i	− 0.795897i
$902$	0	0
$903$	214560.i	0.263132i
$904$	0	0
$905$	767376.	0.936938
$906$	0	0
$907$	−804506.	−0.977946	−0.488973	−	0.872299i	$-0.662628\pi$
$907$	−804506.	−0.977946	−0.488973	+	0.872299i	$0.662628\pi$
$908$	0	0
$909$	− 1434.14i	− 0.00173565i
$910$	0	0
$911$	617904.i	0.744534i	0.928126	+	0.372267i	$0.121419\pi$
$911$	617904.i	0.744534i	−0.928126	+	0.372267i	$0.878581\pi$
$912$	0	0
$913$	1.79550e6	2.15399
$914$	0	0
$915$	72330.4	0.0863931
$916$	0	0
$917$	367056.i	0.436510i
$918$	0	0
$919$	− 930040.i	− 1.10121i	−0.834766	−	0.550606i	$-0.814397\pi$
$919$	− 930040.i	− 1.10121i	0.834766	−	0.550606i	$-0.185603\pi$
$920$	0	0
$921$	−138132.	−0.162845
$922$	0	0
$923$	2.01178e6	2.36145
$924$	0	0
$925$	− 338297.i	− 0.395380i
$926$	0	0
$927$	− 622104.i	− 0.723942i
$928$	0	0
$929$	1.63507e6	1.89454	0.947270	−	0.320436i	$-0.103829\pi$
$929$	1.63507e6	1.89454	0.947270	+	0.320436i	$0.103829\pi$
$930$	0	0
$931$	−202556.	−0.233694
$932$	0	0
$933$	− 498082.i	− 0.572187i
$934$	0	0
$935$	812592.i	0.929500i
$936$	0	0
$937$	−1.16956e6	−1.33212	−0.666059	−	0.745899i	$-0.732020\pi$
$937$	−1.16956e6	−1.33212	−0.666059	+	0.745899i	$0.732020\pi$
$938$	0	0
$939$	−54822.9	−0.0621771
$940$	0	0
$941$	− 68693.1i	− 0.0775772i	−0.999247	−	0.0387886i	$-0.987650\pi$
$941$	− 68693.1i	− 0.0775772i	0.999247	−	0.0387886i	$-0.0123499\pi$
$942$	0	0
$943$	− 9072.00i	− 0.0102019i
$944$	0	0
$945$	432000.	0.483749
$946$	0	0
$947$	195178.	0.217636	0.108818	−	0.994062i	$-0.465293\pi$
$947$	195178.	0.217636	0.108818	+	0.994062i	$0.465293\pi$
$948$	0	0
$949$	− 1.10484e6i	− 1.22678i
$950$	0	0
$951$	391464.i	0.432843i
$952$	0	0
$953$	1.61311e6	1.77614	0.888070	−	0.459709i	$-0.152046\pi$
$953$	1.61311e6	1.77614	0.888070	+	0.459709i	$0.152046\pi$
$954$	0	0
$955$	143663.	0.157521
$956$	0	0
$957$	− 298550.i	− 0.325982i
$958$	0	0
$959$	1.30104e6i	1.41466i
$960$	0	0
$961$	−1.95290e6	−2.11462
$962$	0	0
$963$	162775.	0.175523
$964$	0	0
$965$	− 612024.i	− 0.657224i
$966$	0	0
$967$	1.15382e6i	1.23391i	0.786998	+	0.616955i	$0.211634\pi$
$967$	1.15382e6i	1.23391i	−0.786998	+	0.616955i	$0.788366\pi$
$968$	0	0
$969$	−173448.	−0.184723
$970$	0	0
$971$	1.34490e6	1.42643	0.713216	−	0.700944i	$-0.247238\pi$
$971$	1.34490e6	1.42643	0.713216	+	0.700944i	$0.247238\pi$
$972$	0	0
$973$	− 44201.9i	− 0.0466891i
$974$	0	0
$975$	− 171384.i	− 0.180286i
$976$	0	0
$977$	803322.	0.841590	0.420795	−	0.907156i	$-0.361751\pi$
$977$	803322.	0.841590	0.420795	+	0.907156i	$0.361751\pi$
$978$	0	0
$979$	−614871.	−0.641533
$980$	0	0
$981$	999594.i	1.03869i
$982$	0	0
$983$	− 590328.i	− 0.610923i	−0.952204	−	0.305461i	$-0.901189\pi$
$983$	− 590328.i	− 0.610923i	0.952204	−	0.305461i	$-0.0988107\pi$
$984$	0	0
$985$	842832.	0.868697
$986$	0	0
$987$	−339205.	−0.348199
$988$	0	0
$989$	− 780421.i	− 0.797877i
$990$	0	0
$991$	− 182240.i	− 0.185565i	−0.995686	−	0.0927826i	$-0.970424\pi$
$991$	− 182240.i	− 0.185565i	0.995686	−	0.0927826i	$-0.0295761\pi$
$992$	0	0
$993$	−120468.	−0.122172
$994$	0	0
$995$	−29763.6	−0.0300634
$996$	0	0
$997$	1.90123e6i	1.91269i	0.292241	+	0.956345i	$0.405599\pi$
$997$	1.90123e6i	1.91269i	−0.292241	+	0.956345i	$0.594401\pi$
$998$	0	0
$999$	− 910800.i	− 0.912624i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	64.5.d.a.31.2	yes	4
3.2	odd	2		576.5.b.e.415.1		4
4.3	odd	2	inner	64.5.d.a.31.4	yes	4
8.3	odd	2	inner	64.5.d.a.31.1	✓	4
8.5	even	2	inner	64.5.d.a.31.3	yes	4
12.11	even	2		576.5.b.e.415.2		4
16.3	odd	4		256.5.c.j.255.4		4
16.5	even	4		256.5.c.j.255.3		4
16.11	odd	4		256.5.c.j.255.1		4
16.13	even	4		256.5.c.j.255.2		4
24.5	odd	2		576.5.b.e.415.3		4
24.11	even	2		576.5.b.e.415.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
64.5.d.a.31.1	✓	4	8.3	odd	2	inner
64.5.d.a.31.2	yes	4	1.1	even	1	trivial
64.5.d.a.31.3	yes	4	8.5	even	2	inner
64.5.d.a.31.4	yes	4	4.3	odd	2	inner
256.5.c.j.255.1		4	16.11	odd	4
256.5.c.j.255.2		4	16.13	even	4
256.5.c.j.255.3		4	16.5	even	4
256.5.c.j.255.4		4	16.3	odd	4
576.5.b.e.415.1		4	3.2	odd	2
576.5.b.e.415.2		4	12.11	even	2
576.5.b.e.415.3		4	24.5	odd	2
576.5.b.e.415.4		4	24.11	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:	\( N \)	\(=\)	\( 64 = 2^{6} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	64.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.46410 q^{3} +20.7846i q^{5} -40.0000i q^{7} -69.0000 q^{9} +O(q^{10})\)	\(q-3.46410 q^{3} +20.7846i q^{5} -40.0000i q^{7} -69.0000 q^{9} -197.454 q^{11} +256.344i q^{13} -72.0000i q^{15} -198.000 q^{17} -252.879 q^{19} +138.564i q^{21} -504.000i q^{23} +193.000 q^{25} +519.615 q^{27} -436.477i q^{29} +1696.00i q^{31} +684.000 q^{33} +831.384 q^{35} -1752.84i q^{37} -888.000i q^{39} +18.0000 q^{41} +1548.45 q^{43} -1434.14i q^{45} +2448.00i q^{47} +801.000 q^{49} +685.892 q^{51} +3263.18i q^{53} -4104.00i q^{55} +876.000 q^{57} -5268.90 q^{59} +1004.59i q^{61} +2760.00i q^{63} -5328.00 q^{65} -1804.80 q^{67} +1745.91i q^{69} -7848.00i q^{71} -4310.00 q^{73} -668.572 q^{75} +7898.15i q^{77} +4144.00i q^{79} +3789.00 q^{81} -9093.27 q^{83} -4115.35i q^{85} +1512.00i q^{87} +3114.00 q^{89} +10253.7 q^{91} -5875.12i q^{93} -5256.00i q^{95} +4730.00 q^{97} +13624.3 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 276 q^{9}+O(q^{10}) \) 4 * q - 276 * q^9	\( 4 q - 276 q^{9} - 792 q^{17} + 772 q^{25} + 2736 q^{33} + 72 q^{41} + 3204 q^{49} + 3504 q^{57} - 21312 q^{65} - 17240 q^{73} + 15156 q^{81} + 12456 q^{89} + 18920 q^{97}+O(q^{100}) \) 4 * q - 276 * q^9 - 792 * q^17 + 772 * q^25 + 2736 * q^33 + 72 * q^41 + 3204 * q^49 + 3504 * q^57 - 21312 * q^65 - 17240 * q^73 + 15156 * q^81 + 12456 * q^89 + 18920 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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