LMFDB - Embedded newform 32.6.b.a.17.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [32,6,Mod(17,32)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(32, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("32.17");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 32 = 2^{5} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	32.b (of order $2$, degree $1$, not 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:	$5.13228223402$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.218489.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 2x^{2} - 8x + 64 $ x^4 - x^3 - 2x^2 - 8x + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	no (minimal twist has level 8)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			17.2
Root			$-1.88600 + 2.10784i$ of defining polynomial
Character	$\chi$	$=$	32.17
Dual form			32.6.b.a.17.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.25452i q^{3} +73.9600i q^{5} +112.704 q^{7} +232.408 q^{9} +575.407i q^{11} +117.735i q^{13} +240.704 q^{15} -223.408 q^{17} -1752.26i q^{19} -366.797i q^{21} -2361.15 q^{23} -2345.08 q^{25} -1547.22i q^{27} -3865.52i q^{29} +1591.55 q^{31} +1872.67 q^{33} +8335.59i q^{35} +4736.44i q^{37} +383.172 q^{39} +8153.88 q^{41} -4920.23i q^{43} +17188.9i q^{45} +21062.0 q^{47} -4104.79 q^{49} +727.085i q^{51} -12709.0i q^{53} -42557.1 q^{55} -5702.75 q^{57} +14111.1i q^{59} -42030.6i q^{61} +26193.3 q^{63} -8707.71 q^{65} -54153.4i q^{67} +7684.41i q^{69} -43879.9 q^{71} -31290.6 q^{73} +7632.11i q^{75} +64850.7i q^{77} +50211.5 q^{79} +51439.7 q^{81} +43707.0i q^{83} -16523.3i q^{85} -12580.4 q^{87} +64418.7 q^{89} +13269.3i q^{91} -5179.73i q^{93} +129597. q^{95} -62350.9 q^{97} +133729. i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 96 q^{7} - 164 q^{9} + 416 q^{15} + 200 q^{17} - 2336 q^{23} + 1556 q^{25} + 12928 q^{31} - 2352 q^{33} - 35104 q^{39} - 4568 q^{41} + 54720 q^{47} + 9828 q^{49} - 85472 q^{55} - 2032 q^{57} + 153440 q^{63}+ \cdots - 99576 q^{97}+O(q^{100}) $ 4 * q - 96 * q^7 - 164 * q^9 + 416 * q^15 + 200 * q^17 - 2336 * q^23 + 1556 * q^25 + 12928 * q^31 - 2352 * q^33 - 35104 * q^39 - 4568 * q^41 + 54720 * q^47 + 9828 * q^49 - 85472 * q^55 - 2032 * q^57 + 153440 * q^63 - 19520 * q^65 - 206688 * q^71 + 39976 * q^73 + 247872 * q^79 + 29684 * q^81 - 307872 * q^87 - 84632 * q^89 + 259744 * q^95 - 99576 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/32\mathbb{Z}\right)^\times$.

$n$	$5$	$31$
$\chi(n)$	$-1$	$1$

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
$3$	− 3.25452i	− 0.208777i	−0.994537	−	0.104389i	$-0.966711\pi$
$3$	− 3.25452i	− 0.208777i	0.994537	−	0.104389i	$-0.0332886\pi$
$4$	0	0
$5$	73.9600i	1.32304i	0.749929	+	0.661518i	$0.230088\pi$
$5$	73.9600i	1.32304i	−0.749929	+	0.661518i	$0.769912\pi$
$6$	0	0
$7$	112.704	0.869350	0.434675	−	0.900587i	$-0.356863\pi$
$7$	112.704	0.869350	0.434675	+	0.900587i	$0.356863\pi$
$8$	0	0
$9$	232.408	0.956412
$10$	0	0
$11$	575.407i	1.43382i	0.697167	+	0.716909i	$0.254443\pi$
$11$	575.407i	1.43382i	−0.697167	+	0.716909i	$0.745557\pi$
$12$	0	0
$13$	117.735i	0.193219i	0.995322	+	0.0966093i	$0.0307997\pi$
$13$	117.735i	0.193219i	−0.995322	+	0.0966093i	$0.969200\pi$
$14$	0	0
$15$	240.704	0.276220
$16$	0	0
$17$	−223.408	−0.187489	−0.0937447	−	0.995596i	$-0.529884\pi$
$17$	−223.408	−0.187489	−0.0937447	+	0.995596i	$0.529884\pi$
$18$	0	0
$19$	− 1752.26i	− 1.11356i	−0.830660	−	0.556781i	$-0.812036\pi$
$19$	− 1752.26i	− 1.11356i	0.830660	−	0.556781i	$-0.187964\pi$
$20$	0	0
$21$	− 366.797i	− 0.181501i
$22$	0	0
$23$	−2361.15	−0.930689	−0.465344	−	0.885130i	$-0.654070\pi$
$23$	−2361.15	−0.930689	−0.465344	+	0.885130i	$0.654070\pi$
$24$	0	0
$25$	−2345.08	−0.750426
$26$	0	0
$27$	− 1547.22i	− 0.408455i
$28$	0	0
$29$	− 3865.52i	− 0.853518i	−0.904365	−	0.426759i	$-0.859655\pi$
$29$	− 3865.52i	− 0.853518i	0.904365	−	0.426759i	$-0.140345\pi$
$30$	0	0
$31$	1591.55	0.297452	0.148726	−	0.988878i	$-0.452483\pi$
$31$	1591.55	0.297452	0.148726	+	0.988878i	$0.452483\pi$
$32$	0	0
$33$	1872.67	0.299349
$34$	0	0
$35$	8335.59i	1.15018i
$36$	0	0
$37$	4736.44i	0.568785i	0.958708	+	0.284392i	$0.0917918\pi$
$37$	4736.44i	0.568785i	−0.958708	+	0.284392i	$0.908208\pi$
$38$	0	0
$39$	383.172	0.0403397
$40$	0	0
$41$	8153.88	0.757538	0.378769	−	0.925491i	$-0.376347\pi$
$41$	8153.88	0.757538	0.378769	+	0.925491i	$0.376347\pi$
$42$	0	0
$43$	− 4920.23i	− 0.405802i	−0.979199	−	0.202901i	$-0.934963\pi$
$43$	− 4920.23i	− 0.405802i	0.979199	−	0.202901i	$-0.0650370\pi$
$44$	0	0
$45$	17188.9i	1.26537i
$46$	0	0
$47$	21062.0	1.39077	0.695385	−	0.718637i	$-0.255234\pi$
$47$	21062.0	1.39077	0.695385	+	0.718637i	$0.255234\pi$
$48$	0	0
$49$	−4104.79	−0.244231
$50$	0	0
$51$	727.085i	0.0391435i
$52$	0	0
$53$	− 12709.0i	− 0.621470i	−0.950497	−	0.310735i	$-0.899425\pi$
$53$	− 12709.0i	− 0.621470i	0.950497	−	0.310735i	$-0.100575\pi$
$54$	0	0
$55$	−42557.1	−1.89699
$56$	0	0
$57$	−5702.75	−0.232486
$58$	0	0
$59$	14111.1i	0.527752i	0.964557	+	0.263876i	$0.0850010\pi$
$59$	14111.1i	0.527752i	−0.964557	+	0.263876i	$0.914999\pi$
$60$	0	0
$61$	− 42030.6i	− 1.44624i	−0.690721	−	0.723121i	$-0.742707\pi$
$61$	− 42030.6i	− 1.44624i	0.690721	−	0.723121i	$-0.257293\pi$
$62$	0	0
$63$	26193.3	0.831456
$64$	0	0
$65$	−8707.71	−0.255635
$66$	0	0
$67$	− 54153.4i	− 1.47380i	−0.676001	−	0.736901i	$-0.736289\pi$
$67$	− 54153.4i	− 1.47380i	0.676001	−	0.736901i	$-0.263711\pi$
$68$	0	0
$69$	7684.41i	0.194307i
$70$	0	0
$71$	−43879.9	−1.03305	−0.516523	−	0.856273i	$-0.672774\pi$
$71$	−43879.9	−1.03305	−0.516523	+	0.856273i	$0.672774\pi$
$72$	0	0
$73$	−31290.6	−0.687238	−0.343619	−	0.939109i	$-0.611653\pi$
$73$	−31290.6	−0.687238	−0.343619	+	0.939109i	$0.611653\pi$
$74$	0	0
$75$	7632.11i	0.156672i
$76$	0	0
$77$	64850.7i	1.24649i
$78$	0	0
$79$	50211.5	0.905180	0.452590	−	0.891719i	$-0.350500\pi$
$79$	50211.5	0.905180	0.452590	+	0.891719i	$0.350500\pi$
$80$	0	0
$81$	51439.7	0.871136
$82$	0	0
$83$	43707.0i	0.696395i	0.937421	+	0.348197i	$0.113206\pi$
$83$	43707.0i	0.696395i	−0.937421	+	0.348197i	$0.886794\pi$
$84$	0	0
$85$	− 16523.3i	− 0.248055i
$86$	0	0
$87$	−12580.4	−0.178195
$88$	0	0
$89$	64418.7	0.862059	0.431030	−	0.902338i	$-0.358150\pi$
$89$	64418.7	0.862059	0.431030	+	0.902338i	$0.358150\pi$
$90$	0	0
$91$	13269.3i	0.167974i
$92$	0	0
$93$	− 5179.73i	− 0.0621012i
$94$	0	0
$95$	129597.	1.47328
$96$	0	0
$97$	−62350.9	−0.672843	−0.336421	−	0.941712i	$-0.609217\pi$
$97$	−62350.9	−0.672843	−0.336421	+	0.941712i	$0.609217\pi$
$98$	0	0
$99$	133729.i	1.37132i
$100$	0	0
$101$	49960.5i	0.487330i	0.969859	+	0.243665i	$0.0783497\pi$
$101$	49960.5i	0.487330i	−0.969859	+	0.243665i	$0.921650\pi$
$102$	0	0
$103$	−159260.	−1.47915	−0.739577	−	0.673072i	$-0.764974\pi$
$103$	−159260.	−1.47915	−0.739577	+	0.673072i	$0.764974\pi$
$104$	0	0
$105$	27128.3	0.240132
$106$	0	0
$107$	− 135565.i	− 1.14469i	−0.820012	−	0.572346i	$-0.806033\pi$
$107$	− 135565.i	− 1.14469i	0.820012	−	0.572346i	$-0.193967\pi$
$108$	0	0
$109$	− 115137.i	− 0.928218i	−0.885778	−	0.464109i	$-0.846375\pi$
$109$	− 115137.i	− 0.928218i	0.885778	−	0.464109i	$-0.153625\pi$
$110$	0	0
$111$	15414.8	0.118749
$112$	0	0
$113$	−19192.0	−0.141392	−0.0706960	−	0.997498i	$-0.522522\pi$
$113$	−19192.0	−0.141392	−0.0706960	+	0.997498i	$0.522522\pi$
$114$	0	0
$115$	− 174631.i	− 1.23134i
$116$	0	0
$117$	27362.7i	0.184797i
$118$	0	0
$119$	−25179.0	−0.162994
$120$	0	0
$121$	−170043.	−1.05583
$122$	0	0
$123$	− 26536.9i	− 0.158157i
$124$	0	0
$125$	57682.8i	0.330196i
$126$	0	0
$127$	−102000.	−0.561163	−0.280582	−	0.959830i	$-0.590527\pi$
$127$	−102000.	−0.561163	−0.280582	+	0.959830i	$0.590527\pi$
$128$	0	0
$129$	−16013.0	−0.0847223
$130$	0	0
$131$	− 11296.1i	− 0.0575111i	−0.999586	−	0.0287556i	$-0.990846\pi$
$131$	− 11296.1i	− 0.0575111i	0.999586	−	0.0287556i	$-0.00915444\pi$
$132$	0	0
$133$	− 197487.i	− 0.968074i
$134$	0	0
$135$	114433.	0.540400
$136$	0	0
$137$	102753.	0.467727	0.233863	−	0.972269i	$-0.424863\pi$
$137$	102753.	0.467727	0.233863	+	0.972269i	$0.424863\pi$
$138$	0	0
$139$	343928.i	1.50984i	0.655818	+	0.754919i	$0.272324\pi$
$139$	343928.i	1.50984i	−0.655818	+	0.754919i	$0.727676\pi$
$140$	0	0
$141$	− 68546.7i	− 0.290361i
$142$	0	0
$143$	−67745.8	−0.277040
$144$	0	0
$145$	285894.	1.12924
$146$	0	0
$147$	13359.1i	0.0509900i
$148$	0	0
$149$	186446.i	0.687998i	0.938970	+	0.343999i	$0.111782\pi$
$149$	186446.i	0.687998i	−0.938970	+	0.343999i	$0.888218\pi$
$150$	0	0
$151$	285769.	1.01994	0.509968	−	0.860194i	$-0.329657\pi$
$151$	285769.	1.01994	0.509968	+	0.860194i	$0.329657\pi$
$152$	0	0
$153$	−51921.9	−0.179317
$154$	0	0
$155$	117711.i	0.393539i
$156$	0	0
$157$	480654.i	1.55626i	0.628101	+	0.778132i	$0.283833\pi$
$157$	480654.i	1.55626i	−0.628101	+	0.778132i	$0.716167\pi$
$158$	0	0
$159$	−41361.5	−0.129749
$160$	0	0
$161$	−266112.	−0.809094
$162$	0	0
$163$	− 176613.i	− 0.520659i	−0.965520	−	0.260329i	$-0.916169\pi$
$163$	− 176613.i	− 0.520659i	0.965520	−	0.260329i	$-0.0838312\pi$
$164$	0	0
$165$	138503.i	0.396049i
$166$	0	0
$167$	218853.	0.607240	0.303620	−	0.952793i	$-0.401805\pi$
$167$	218853.	0.607240	0.303620	+	0.952793i	$0.401805\pi$
$168$	0	0
$169$	357431.	0.962667
$170$	0	0
$171$	− 407239.i	− 1.06502i
$172$	0	0
$173$	− 522746.i	− 1.32793i	−0.747764	−	0.663965i	$-0.768872\pi$
$173$	− 522746.i	− 1.32793i	0.747764	−	0.663965i	$-0.231128\pi$
$174$	0	0
$175$	−264300.	−0.652383
$176$	0	0
$177$	45924.7	0.110183
$178$	0	0
$179$	411059.i	0.958897i	0.877570	+	0.479449i	$0.159163\pi$
$179$	411059.i	0.958897i	−0.877570	+	0.479449i	$0.840837\pi$
$180$	0	0
$181$	− 133094.i	− 0.301970i	−0.988536	−	0.150985i	$-0.951755\pi$
$181$	− 133094.i	− 0.301970i	0.988536	−	0.150985i	$-0.0482445\pi$
$182$	0	0
$183$	−136789.	−0.301943
$184$	0	0
$185$	−350307.	−0.752523
$186$	0	0
$187$	− 128551.i	− 0.268825i
$188$	0	0
$189$	− 174378.i	− 0.355090i
$190$	0	0
$191$	−833447.	−1.65308	−0.826541	−	0.562877i	$-0.809695\pi$
$191$	−833447.	−1.65308	−0.826541	+	0.562877i	$0.809695\pi$
$192$	0	0
$193$	−597550.	−1.15473	−0.577366	−	0.816486i	$-0.695919\pi$
$193$	−597550.	−1.15473	−0.577366	+	0.816486i	$0.695919\pi$
$194$	0	0
$195$	28339.4i	0.0533709i
$196$	0	0
$197$	− 688179.i	− 1.26339i	−0.775219	−	0.631693i	$-0.782360\pi$
$197$	− 688179.i	− 1.26339i	0.775219	−	0.631693i	$-0.217640\pi$
$198$	0	0
$199$	977514.	1.74981	0.874904	−	0.484296i	$-0.160924\pi$
$199$	977514.	1.74981	0.874904	+	0.484296i	$0.160924\pi$
$200$	0	0
$201$	−176243.	−0.307696
$202$	0	0
$203$	− 435659.i	− 0.742005i
$204$	0	0
$205$	603061.i	1.00225i
$206$	0	0
$207$	−548751.	−0.890122
$208$	0	0
$209$	1.00826e6	1.59664
$210$	0	0
$211$	44234.7i	0.0684001i	0.999415	+	0.0342000i	$0.0108883\pi$
$211$	44234.7i	0.0684001i	−0.999415	+	0.0342000i	$0.989112\pi$
$212$	0	0
$213$	142808.i	0.215677i
$214$	0	0
$215$	363900.	0.536891
$216$	0	0
$217$	179374.	0.258589
$218$	0	0
$219$	101836.i	0.143480i
$220$	0	0
$221$	− 26303.1i	− 0.0362264i
$222$	0	0
$223$	73883.6	0.0994915	0.0497458	−	0.998762i	$-0.484159\pi$
$223$	73883.6	0.0994915	0.0497458	+	0.998762i	$0.484159\pi$
$224$	0	0
$225$	−545016.	−0.717716
$226$	0	0
$227$	− 195146.i	− 0.251359i	−0.992071	−	0.125680i	$-0.959889\pi$
$227$	− 195146.i	− 0.251359i	0.992071	−	0.125680i	$-0.0401111\pi$
$228$	0	0
$229$	1.55451e6i	1.95886i	0.201779	+	0.979431i	$0.435328\pi$
$229$	1.55451e6i	1.95886i	−0.201779	+	0.979431i	$0.564672\pi$
$230$	0	0
$231$	211058.	0.260239
$232$	0	0
$233$	−56457.4	−0.0681289	−0.0340644	−	0.999420i	$-0.510845\pi$
$233$	−56457.4	−0.0681289	−0.0340644	+	0.999420i	$0.510845\pi$
$234$	0	0
$235$	1.55775e6i	1.84004i
$236$	0	0
$237$	− 163414.i	− 0.188981i
$238$	0	0
$239$	551027.	0.623990	0.311995	−	0.950084i	$-0.399003\pi$
$239$	551027.	0.623990	0.311995	+	0.950084i	$0.399003\pi$
$240$	0	0
$241$	1.31586e6	1.45938	0.729689	−	0.683779i	$-0.239665\pi$
$241$	1.31586e6	1.45938	0.729689	+	0.683779i	$0.239665\pi$
$242$	0	0
$243$	− 543387.i	− 0.590328i
$244$	0	0
$245$	− 303591.i	− 0.323127i
$246$	0	0
$247$	206303.	0.215161
$248$	0	0
$249$	142245.	0.145391
$250$	0	0
$251$	95208.0i	0.0953870i	0.998862	+	0.0476935i	$0.0151871\pi$
$251$	95208.0i	0.0953870i	−0.998862	+	0.0476935i	$0.984813\pi$
$252$	0	0
$253$	− 1.35862e6i	− 1.33444i
$254$	0	0
$255$	−53775.2	−0.0517883
$256$	0	0
$257$	−1.73166e6	−1.63542	−0.817711	−	0.575630i	$-0.804757\pi$
$257$	−1.73166e6	−1.63542	−0.817711	+	0.575630i	$0.804757\pi$
$258$	0	0
$259$	533816.i	0.494473i
$260$	0	0
$261$	− 898377.i	− 0.816315i
$262$	0	0
$263$	−1.51692e6	−1.35230	−0.676150	−	0.736764i	$-0.736353\pi$
$263$	−1.51692e6	−1.35230	−0.676150	+	0.736764i	$0.736353\pi$
$264$	0	0
$265$	939954.	0.822227
$266$	0	0
$267$	− 209652.i	− 0.179978i
$268$	0	0
$269$	1.89610e6i	1.59765i	0.601565	+	0.798824i	$0.294544\pi$
$269$	1.89610e6i	1.59765i	−0.601565	+	0.798824i	$0.705456\pi$
$270$	0	0
$271$	−476326.	−0.393986	−0.196993	−	0.980405i	$-0.563118\pi$
$271$	−476326.	−0.393986	−0.196993	+	0.980405i	$0.563118\pi$
$272$	0	0
$273$	43185.0	0.0350693
$274$	0	0
$275$	− 1.34938e6i	− 1.07597i
$276$	0	0
$277$	− 842760.i	− 0.659940i	−0.943991	−	0.329970i	$-0.892961\pi$
$277$	− 842760.i	− 0.659940i	0.943991	−	0.329970i	$-0.107039\pi$
$278$	0	0
$279$	369889.	0.284486
$280$	0	0
$281$	−1.64465e6	−1.24254	−0.621268	−	0.783598i	$-0.713382\pi$
$281$	−1.64465e6	−1.24254	−0.621268	+	0.783598i	$0.713382\pi$
$282$	0	0
$283$	− 2.22864e6i	− 1.65415i	−0.562092	−	0.827074i	$-0.690003\pi$
$283$	− 2.22864e6i	− 1.65415i	0.562092	−	0.827074i	$-0.309997\pi$
$284$	0	0
$285$	− 421776.i	− 0.307588i
$286$	0	0
$287$	918975.	0.658565
$288$	0	0
$289$	−1.36995e6	−0.964848
$290$	0	0
$291$	202922.i	0.140474i
$292$	0	0
$293$	692080.i	0.470964i	0.971879	+	0.235482i	$0.0756668\pi$
$293$	692080.i	0.470964i	−0.971879	+	0.235482i	$0.924333\pi$
$294$	0	0
$295$	−1.04366e6	−0.698236
$296$	0	0
$297$	890284.	0.585649
$298$	0	0
$299$	− 277991.i	− 0.179826i
$300$	0	0
$301$	− 554530.i	− 0.352784i
$302$	0	0
$303$	162597.	0.101743
$304$	0	0
$305$	3.10858e6	1.91343
$306$	0	0
$307$	2.91707e6i	1.76645i	0.468952	+	0.883223i	$0.344632\pi$
$307$	2.91707e6i	1.76645i	−0.468952	+	0.883223i	$0.655368\pi$
$308$	0	0
$309$	518314.i	0.308814i
$310$	0	0
$311$	−1.10725e6	−0.649151	−0.324575	−	0.945860i	$-0.605221\pi$
$311$	−1.10725e6	−0.649151	−0.324575	+	0.945860i	$0.605221\pi$
$312$	0	0
$313$	−1.65389e6	−0.954213	−0.477107	−	0.878845i	$-0.658314\pi$
$313$	−1.65389e6	−0.954213	−0.477107	+	0.878845i	$0.658314\pi$
$314$	0	0
$315$	1.93726e6i	1.10005i
$316$	0	0
$317$	− 459259.i	− 0.256691i	−0.991730	−	0.128345i	$-0.959033\pi$
$317$	− 459259.i	− 0.256691i	0.991730	−	0.128345i	$-0.0409666\pi$
$318$	0	0
$319$	2.22425e6	1.22379
$320$	0	0
$321$	−441199.	−0.238986
$322$	0	0
$323$	391469.i	0.208781i
$324$	0	0
$325$	− 276099.i	− 0.144996i
$326$	0	0
$327$	−374717.	−0.193791
$328$	0	0
$329$	2.37378e6	1.20907
$330$	0	0
$331$	− 1.55618e6i	− 0.780711i	−0.920664	−	0.390356i	$-0.872352\pi$
$331$	− 1.55618e6i	− 0.780711i	0.920664	−	0.390356i	$-0.127648\pi$
$332$	0	0
$333$	1.10079e6i	0.543993i
$334$	0	0
$335$	4.00519e6	1.94989
$336$	0	0
$337$	919178.	0.440885	0.220442	−	0.975400i	$-0.429250\pi$
$337$	919178.	0.440885	0.220442	+	0.975400i	$0.429250\pi$
$338$	0	0
$339$	62460.8i	0.0295194i
$340$	0	0
$341$	915790.i	0.426491i
$342$	0	0
$343$	−2.35684e6	−1.08167
$344$	0	0
$345$	−568339.	−0.257075
$346$	0	0
$347$	1.87289e6i	0.835003i	0.908676	+	0.417502i	$0.137094\pi$
$347$	1.87289e6i	0.835003i	−0.908676	+	0.417502i	$0.862906\pi$
$348$	0	0
$349$	2.41450e6i	1.06112i	0.847649	+	0.530558i	$0.178018\pi$
$349$	2.41450e6i	1.06112i	−0.847649	+	0.530558i	$0.821982\pi$
$350$	0	0
$351$	182163.	0.0789210
$352$	0	0
$353$	−2.43268e6	−1.03908	−0.519539	−	0.854447i	$-0.673896\pi$
$353$	−2.43268e6	−1.03908	−0.519539	+	0.854447i	$0.673896\pi$
$354$	0	0
$355$	− 3.24535e6i	− 1.36676i
$356$	0	0
$357$	81945.5i	0.0340294i
$358$	0	0
$359$	−2.14148e6	−0.876958	−0.438479	−	0.898741i	$-0.644483\pi$
$359$	−2.14148e6	−0.876958	−0.438479	+	0.898741i	$0.644483\pi$
$360$	0	0
$361$	−594310.	−0.240019
$362$	0	0
$363$	553407.i	0.220434i
$364$	0	0
$365$	− 2.31425e6i	− 0.909241i
$366$	0	0
$367$	−1.43273e6	−0.555262	−0.277631	−	0.960688i	$-0.589549\pi$
$367$	−1.43273e6	−0.555262	−0.277631	+	0.960688i	$0.589549\pi$
$368$	0	0
$369$	1.89503e6	0.724519
$370$	0	0
$371$	− 1.43235e6i	− 0.540275i
$372$	0	0
$373$	− 2.57608e6i	− 0.958711i	−0.877621	−	0.479356i	$-0.840870\pi$
$373$	− 2.57608e6i	− 0.958711i	0.877621	−	0.479356i	$-0.159130\pi$
$374$	0	0
$375$	187730.	0.0689374
$376$	0	0
$377$	455108.	0.164915
$378$	0	0
$379$	− 1.88270e6i	− 0.673260i	−0.941637	−	0.336630i	$-0.890713\pi$
$379$	− 1.88270e6i	− 0.673260i	0.941637	−	0.336630i	$-0.109287\pi$
$380$	0	0
$381$	331960.i	0.117158i
$382$	0	0
$383$	929245.	0.323693	0.161847	−	0.986816i	$-0.448255\pi$
$383$	929245.	0.323693	0.161847	+	0.986816i	$0.448255\pi$
$384$	0	0
$385$	−4.79636e6	−1.64915
$386$	0	0
$387$	− 1.14350e6i	− 0.388114i
$388$	0	0
$389$	− 1.88218e6i	− 0.630647i	−0.948984	−	0.315324i	$-0.897887\pi$
$389$	− 1.88218e6i	− 0.630647i	0.948984	−	0.315324i	$-0.102113\pi$
$390$	0	0
$391$	527501.	0.174494
$392$	0	0
$393$	−36763.5	−0.0120070
$394$	0	0
$395$	3.71364e6i	1.19759i
$396$	0	0
$397$	− 4.38186e6i	− 1.39535i	−0.716417	−	0.697673i	$-0.754219\pi$
$397$	− 4.38186e6i	− 1.39535i	0.716417	−	0.697673i	$-0.245781\pi$
$398$	0	0
$399$	−642724.	−0.202112
$400$	0	0
$401$	1.43544e6	0.445784	0.222892	−	0.974843i	$-0.428450\pi$
$401$	1.43544e6	0.445784	0.222892	+	0.974843i	$0.428450\pi$
$402$	0	0
$403$	187382.i	0.0574732i
$404$	0	0
$405$	3.80448e6i	1.15254i
$406$	0	0
$407$	−2.72538e6	−0.815533
$408$	0	0
$409$	−479384.	−0.141702	−0.0708509	−	0.997487i	$-0.522571\pi$
$409$	−479384.	−0.141702	−0.0708509	+	0.997487i	$0.522571\pi$
$410$	0	0
$411$	− 334411.i	− 0.0976508i
$412$	0	0
$413$	1.59038e6i	0.458801i
$414$	0	0
$415$	−3.23257e6	−0.921356
$416$	0	0
$417$	1.11932e6	0.315220
$418$	0	0
$419$	4.82901e6i	1.34376i	0.740658	+	0.671882i	$0.234514\pi$
$419$	4.82901e6i	1.34376i	−0.740658	+	0.671882i	$0.765486\pi$
$420$	0	0
$421$	918869.i	0.252667i	0.991988	+	0.126333i	$0.0403209\pi$
$421$	918869.i	0.252667i	−0.991988	+	0.126333i	$0.959679\pi$
$422$	0	0
$423$	4.89498e6	1.33015
$424$	0	0
$425$	523910.	0.140697
$426$	0	0
$427$	− 4.73702e6i	− 1.25729i
$428$	0	0
$429$	220480.i	0.0578397i
$430$	0	0
$431$	4.81272e6	1.24795	0.623975	−	0.781444i	$-0.285517\pi$
$431$	4.81272e6	1.24795	0.623975	+	0.781444i	$0.285517\pi$
$432$	0	0
$433$	5.84598e6	1.49843	0.749217	−	0.662324i	$-0.230430\pi$
$433$	5.84598e6	1.49843	0.749217	+	0.662324i	$0.230430\pi$
$434$	0	0
$435$	− 930445.i	− 0.235759i
$436$	0	0
$437$	4.13735e6i	1.03638i
$438$	0	0
$439$	2.28777e6	0.566568	0.283284	−	0.959036i	$-0.408576\pi$
$439$	2.28777e6	0.566568	0.283284	+	0.959036i	$0.408576\pi$
$440$	0	0
$441$	−953988.	−0.233586
$442$	0	0
$443$	− 982970.i	− 0.237975i	−0.992896	−	0.118987i	$-0.962035\pi$
$443$	− 982970.i	− 0.237975i	0.992896	−	0.118987i	$-0.0379648\pi$
$444$	0	0
$445$	4.76441e6i	1.14054i
$446$	0	0
$447$	606791.	0.143638
$448$	0	0
$449$	−2.86769e6	−0.671299	−0.335650	−	0.941987i	$-0.608956\pi$
$449$	−2.86769e6	−0.671299	−0.335650	+	0.941987i	$0.608956\pi$
$450$	0	0
$451$	4.69180e6i	1.08617i
$452$	0	0
$453$	− 930040.i	− 0.212939i
$454$	0	0
$455$	−981395.	−0.222236
$456$	0	0
$457$	−451218.	−0.101064	−0.0505319	−	0.998722i	$-0.516092\pi$
$457$	−451218.	−0.101064	−0.0505319	+	0.998722i	$0.516092\pi$
$458$	0	0
$459$	345662.i	0.0765809i
$460$	0	0
$461$	242143.i	0.0530665i	0.999648	+	0.0265332i	$0.00844678\pi$
$461$	242143.i	0.0530665i	−0.999648	+	0.0265332i	$0.991553\pi$
$462$	0	0
$463$	−4.80106e6	−1.04084	−0.520421	−	0.853910i	$-0.674225\pi$
$463$	−4.80106e6	−1.04084	−0.520421	+	0.853910i	$0.674225\pi$
$464$	0	0
$465$	383093.	0.0821621
$466$	0	0
$467$	− 306183.i	− 0.0649664i	−0.999472	−	0.0324832i	$-0.989658\pi$
$467$	− 306183.i	− 0.0649664i	0.999472	−	0.0324832i	$-0.0103415\pi$
$468$	0	0
$469$	− 6.10331e6i	− 1.28125i
$470$	0	0
$471$	1.56430e6	0.324913
$472$	0	0
$473$	2.83114e6	0.581846
$474$	0	0
$475$	4.10919e6i	0.835645i
$476$	0	0
$477$	− 2.95366e6i	− 0.594381i
$478$	0	0
$479$	−3.99196e6	−0.794964	−0.397482	−	0.917610i	$-0.630116\pi$
$479$	−3.99196e6	−0.794964	−0.397482	+	0.917610i	$0.630116\pi$
$480$	0	0
$481$	−557647.	−0.109900
$482$	0	0
$483$	866064.i	0.168920i
$484$	0	0
$485$	− 4.61147e6i	− 0.890195i
$486$	0	0
$487$	6.45190e6	1.23272	0.616361	−	0.787464i	$-0.288606\pi$
$487$	6.45190e6	1.23272	0.616361	+	0.787464i	$0.288606\pi$
$488$	0	0
$489$	−574790.	−0.108702
$490$	0	0
$491$	− 4.37221e6i	− 0.818459i	−0.912432	−	0.409229i	$-0.865798\pi$
$491$	− 4.37221e6i	− 0.818459i	0.912432	−	0.409229i	$-0.134202\pi$
$492$	0	0
$493$	863588.i	0.160025i
$494$	0	0
$495$	−9.89062e6	−1.81431
$496$	0	0
$497$	−4.94544e6	−0.898078
$498$	0	0
$499$	− 5.17114e6i	− 0.929683i	−0.885394	−	0.464842i	$-0.846111\pi$
$499$	− 5.17114e6i	− 0.929683i	0.885394	−	0.464842i	$-0.153889\pi$
$500$	0	0
$501$	− 712260.i	− 0.126778i
$502$	0	0
$503$	4.16421e6	0.733859	0.366929	−	0.930249i	$-0.380409\pi$
$503$	4.16421e6	0.733859	0.366929	+	0.930249i	$0.380409\pi$
$504$	0	0
$505$	−3.69508e6	−0.644755
$506$	0	0
$507$	− 1.16327e6i	− 0.200983i
$508$	0	0
$509$	− 4.71340e6i	− 0.806381i	−0.915116	−	0.403190i	$-0.867901\pi$
$509$	− 4.71340e6i	− 0.806381i	0.915116	−	0.403190i	$-0.132099\pi$
$510$	0	0
$511$	−3.52658e6	−0.597450
$512$	0	0
$513$	−2.71114e6	−0.454839
$514$	0	0
$515$	− 1.17789e7i	− 1.95697i
$516$	0	0
$517$	1.21192e7i	1.99411i
$518$	0	0
$519$	−1.70128e6	−0.277242
$520$	0	0
$521$	2.34479e6	0.378451	0.189225	−	0.981934i	$-0.439402\pi$
$521$	2.34479e6	0.378451	0.189225	+	0.981934i	$0.439402\pi$
$522$	0	0
$523$	8.17020e6i	1.30611i	0.757312	+	0.653053i	$0.226512\pi$
$523$	8.17020e6i	1.30611i	−0.757312	+	0.653053i	$0.773488\pi$
$524$	0	0
$525$	860169.i	0.136203i
$526$	0	0
$527$	−355565.	−0.0557690
$528$	0	0
$529$	−861301.	−0.133818
$530$	0	0
$531$	3.27953e6i	0.504749i
$532$	0	0
$533$	960000.i	0.146370i
$534$	0	0
$535$	1.00264e7	1.51447
$536$	0	0
$537$	1.33780e6	0.200196
$538$	0	0
$539$	− 2.36193e6i	− 0.350183i
$540$	0	0
$541$	− 3.53355e6i	− 0.519060i	−0.965735	−	0.259530i	$-0.916432\pi$
$541$	− 3.53355e6i	− 0.519060i	0.965735	−	0.259530i	$-0.0835677\pi$
$542$	0	0
$543$	−433158.	−0.0630445
$544$	0	0
$545$	8.51556e6	1.22807
$546$	0	0
$547$	− 657235.i	− 0.0939187i	−0.998897	−	0.0469594i	$-0.985047\pi$
$547$	− 657235.i	− 0.0939187i	0.998897	−	0.0469594i	$-0.0149531\pi$
$548$	0	0
$549$	− 9.76826e6i	− 1.38320i
$550$	0	0
$551$	−6.77338e6	−0.950444
$552$	0	0
$553$	5.65903e6	0.786918
$554$	0	0
$555$	1.14008e6i	0.157110i
$556$	0	0
$557$	1.35929e7i	1.85641i	0.372065	+	0.928207i	$0.378650\pi$
$557$	1.35929e7i	1.85641i	−0.372065	+	0.928207i	$0.621350\pi$
$558$	0	0
$559$	579286.	0.0784085
$560$	0	0
$561$	−418370.	−0.0561247
$562$	0	0
$563$	1.18702e7i	1.57829i	0.614206	+	0.789146i	$0.289477\pi$
$563$	1.18702e7i	1.57829i	−0.614206	+	0.789146i	$0.710523\pi$
$564$	0	0
$565$	− 1.41944e6i	− 0.187067i
$566$	0	0
$567$	5.79746e6	0.757322
$568$	0	0
$569$	−1.16590e7	−1.50966	−0.754831	−	0.655920i	$-0.772281\pi$
$569$	−1.16590e7	−1.50966	−0.754831	+	0.655920i	$0.772281\pi$
$570$	0	0
$571$	1.14849e7i	1.47414i	0.675819	+	0.737068i	$0.263790\pi$
$571$	1.14849e7i	1.47414i	−0.675819	+	0.737068i	$0.736210\pi$
$572$	0	0
$573$	2.71247e6i	0.345126i
$574$	0	0
$575$	5.53709e6	0.698413
$576$	0	0
$577$	7.93609e6	0.992355	0.496178	−	0.868221i	$-0.334736\pi$
$577$	7.93609e6	0.992355	0.496178	+	0.868221i	$0.334736\pi$
$578$	0	0
$579$	1.94474e6i	0.241082i
$580$	0	0
$581$	4.92595e6i	0.605411i
$582$	0	0
$583$	7.31282e6	0.891074
$584$	0	0
$585$	−2.02374e6	−0.244493
$586$	0	0
$587$	− 7.16597e6i	− 0.858381i	−0.903214	−	0.429190i	$-0.858799\pi$
$587$	− 7.16597e6i	− 0.858381i	0.903214	−	0.429190i	$-0.141201\pi$
$588$	0	0
$589$	− 2.78881e6i	− 0.331231i
$590$	0	0
$591$	−2.23969e6	−0.263766
$592$	0	0
$593$	−1.99624e6	−0.233118	−0.116559	−	0.993184i	$-0.537186\pi$
$593$	−1.99624e6	−0.233118	−0.116559	+	0.993184i	$0.537186\pi$
$594$	0	0
$595$	− 1.86224e6i	− 0.215647i
$596$	0	0
$597$	− 3.18134e6i	− 0.365320i
$598$	0	0
$599$	−1.20579e7	−1.37311	−0.686555	−	0.727078i	$-0.740878\pi$
$599$	−1.20579e7	−1.37311	−0.686555	+	0.727078i	$0.740878\pi$
$600$	0	0
$601$	1.50698e7	1.70185	0.850924	−	0.525289i	$-0.176043\pi$
$601$	1.50698e7	1.70185	0.850924	+	0.525289i	$0.176043\pi$
$602$	0	0
$603$	− 1.25857e7i	− 1.40956i
$604$	0	0
$605$	− 1.25764e7i	− 1.39690i
$606$	0	0
$607$	−4.03809e6	−0.444841	−0.222420	−	0.974951i	$-0.571396\pi$
$607$	−4.03809e6	−0.444841	−0.222420	+	0.974951i	$0.571396\pi$
$608$	0	0
$609$	−1.41786e6	−0.154914
$610$	0	0
$611$	2.47975e6i	0.268723i
$612$	0	0
$613$	1.07407e7i	1.15447i	0.816578	+	0.577236i	$0.195869\pi$
$613$	1.07407e7i	1.15447i	−0.816578	+	0.577236i	$0.804131\pi$
$614$	0	0
$615$	1.96267e6	0.209247
$616$	0	0
$617$	−9.37637e6	−0.991567	−0.495783	−	0.868446i	$-0.665119\pi$
$617$	−9.37637e6	−0.991567	−0.495783	+	0.868446i	$0.665119\pi$
$618$	0	0
$619$	− 4.03378e6i	− 0.423141i	−0.977363	−	0.211571i	$-0.932142\pi$
$619$	− 4.03378e6i	− 0.423141i	0.977363	−	0.211571i	$-0.0678579\pi$
$620$	0	0
$621$	3.65323e6i	0.380144i
$622$	0	0
$623$	7.26025e6	0.749431
$624$	0	0
$625$	−1.15946e7	−1.18729
$626$	0	0
$627$	− 3.28141e6i	− 0.333343i
$628$	0	0
$629$	− 1.05816e6i	− 0.106641i
$630$	0	0
$631$	−3.01348e6	−0.301297	−0.150648	−	0.988587i	$-0.548136\pi$
$631$	−3.01348e6	−0.301297	−0.150648	+	0.988587i	$0.548136\pi$
$632$	0	0
$633$	143962.	0.0142804
$634$	0	0
$635$	− 7.54389e6i	− 0.742440i
$636$	0	0
$637$	− 483280.i	− 0.0471900i
$638$	0	0
$639$	−1.01980e7	−0.988017
$640$	0	0
$641$	−2.09755e6	−0.201636	−0.100818	−	0.994905i	$-0.532146\pi$
$641$	−2.09755e6	−0.201636	−0.100818	+	0.994905i	$0.532146\pi$
$642$	0	0
$643$	3.48456e6i	0.332369i	0.986095	+	0.166185i	$0.0531448\pi$
$643$	3.48456e6i	0.332369i	−0.986095	+	0.166185i	$0.946855\pi$
$644$	0	0
$645$	− 1.18432e6i	− 0.112091i
$646$	0	0
$647$	−9.25999e6	−0.869660	−0.434830	−	0.900513i	$-0.643192\pi$
$647$	−9.25999e6	−0.869660	−0.434830	+	0.900513i	$0.643192\pi$
$648$	0	0
$649$	−8.11962e6	−0.756700
$650$	0	0
$651$	− 583777.i	− 0.0539876i
$652$	0	0
$653$	− 1.12475e7i	− 1.03222i	−0.856523	−	0.516109i	$-0.827380\pi$
$653$	− 1.12475e7i	− 1.03222i	0.856523	−	0.516109i	$-0.172620\pi$
$654$	0	0
$655$	835462.	0.0760893
$656$	0	0
$657$	−7.27220e6	−0.657283
$658$	0	0
$659$	− 6.37278e6i	− 0.571630i	−0.958285	−	0.285815i	$-0.907736\pi$
$659$	− 6.37278e6i	− 0.571630i	0.958285	−	0.285815i	$-0.0922643\pi$
$660$	0	0
$661$	− 4.13736e6i	− 0.368315i	−0.982897	−	0.184158i	$-0.941044\pi$
$661$	− 4.13736e6i	− 0.368315i	0.982897	−	0.184158i	$-0.0589556\pi$
$662$	0	0
$663$	−85603.7	−0.00756326
$664$	0	0
$665$	1.46061e7	1.28080
$666$	0	0
$667$	9.12707e6i	0.794359i
$668$	0	0
$669$	− 240456.i	− 0.0207716i
$670$	0	0
$671$	2.41847e7	2.07365
$672$	0	0
$673$	1.50812e7	1.28350	0.641752	−	0.766913i	$-0.278208\pi$
$673$	1.50812e7	1.28350	0.641752	+	0.766913i	$0.278208\pi$
$674$	0	0
$675$	3.62837e6i	0.306515i
$676$	0	0
$677$	− 1.85553e7i	− 1.55595i	−0.628295	−	0.777975i	$-0.716247\pi$
$677$	− 1.85553e7i	− 1.55595i	0.628295	−	0.777975i	$-0.283753\pi$
$678$	0	0
$679$	−7.02720e6	−0.584935
$680$	0	0
$681$	−635105.	−0.0524781
$682$	0	0
$683$	2.31850e7i	1.90176i	0.309558	+	0.950881i	$0.399819\pi$
$683$	2.31850e7i	1.90176i	−0.309558	+	0.950881i	$0.600181\pi$
$684$	0	0
$685$	7.59960e6i	0.618820i
$686$	0	0
$687$	5.05917e6	0.408966
$688$	0	0
$689$	1.49629e6	0.120080
$690$	0	0
$691$	− 1.04315e7i	− 0.831099i	−0.909571	−	0.415549i	$-0.863589\pi$
$691$	− 1.04315e7i	− 0.831099i	0.909571	−	0.415549i	$-0.136411\pi$
$692$	0	0
$693$	1.50718e7i	1.19216i
$694$	0	0
$695$	−2.54369e7	−1.99757
$696$	0	0
$697$	−1.82164e6	−0.142030
$698$	0	0
$699$	183742.i	0.0142238i
$700$	0	0
$701$	1.93026e7i	1.48362i	0.670613	+	0.741808i	$0.266031\pi$
$701$	1.93026e7i	1.48362i	−0.670613	+	0.741808i	$0.733969\pi$
$702$	0	0
$703$	8.29947e6	0.633377
$704$	0	0
$705$	5.06971e6	0.384159
$706$	0	0
$707$	5.63075e6i	0.423660i
$708$	0	0
$709$	5.90966e6i	0.441517i	0.975329	+	0.220758i	$0.0708532\pi$
$709$	5.90966e6i	0.441517i	−0.975329	+	0.220758i	$0.929147\pi$
$710$	0	0
$711$	1.16695e7	0.865725
$712$	0	0
$713$	−3.75790e6	−0.276835
$714$	0	0
$715$	− 5.01048e6i	− 0.366534i
$716$	0	0
$717$	− 1.79333e6i	− 0.130275i
$718$	0	0
$719$	2.58536e7	1.86509	0.932543	−	0.361058i	$-0.117584\pi$
$719$	2.58536e7	1.86509	0.932543	+	0.361058i	$0.117584\pi$
$720$	0	0
$721$	−1.79492e7	−1.28590
$722$	0	0
$723$	− 4.28250e6i	− 0.304685i
$724$	0	0
$725$	9.06495e6i	0.640502i
$726$	0	0
$727$	497513.	0.0349115	0.0174558	−	0.999848i	$-0.494443\pi$
$727$	497513.	0.0349115	0.0174558	+	0.999848i	$0.494443\pi$
$728$	0	0
$729$	1.07314e7	0.747889
$730$	0	0
$731$	1.09922e6i	0.0760836i
$732$	0	0
$733$	9.66956e6i	0.664732i	0.943150	+	0.332366i	$0.107847\pi$
$733$	9.66956e6i	0.664732i	−0.943150	+	0.332366i	$0.892153\pi$
$734$	0	0
$735$	−988041.	−0.0674616
$736$	0	0
$737$	3.11603e7	2.11316
$738$	0	0
$739$	− 1.83759e7i	− 1.23776i	−0.785485	−	0.618881i	$-0.787586\pi$
$739$	− 1.83759e7i	− 1.23776i	0.785485	−	0.618881i	$-0.212414\pi$
$740$	0	0
$741$	− 671416.i	− 0.0449207i
$742$	0	0
$743$	−1.51555e7	−1.00716	−0.503578	−	0.863950i	$-0.667983\pi$
$743$	−1.51555e7	−1.00716	−0.503578	+	0.863950i	$0.667983\pi$
$744$	0	0
$745$	−1.37895e7	−0.910246
$746$	0	0
$747$	1.01579e7i	0.666040i
$748$	0	0
$749$	− 1.52788e7i	− 0.995138i
$750$	0	0
$751$	−7.70448e6	−0.498475	−0.249238	−	0.968442i	$-0.580180\pi$
$751$	−7.70448e6	−0.498475	−0.249238	+	0.968442i	$0.580180\pi$
$752$	0	0
$753$	309856.	0.0199147
$754$	0	0
$755$	2.11355e7i	1.34941i
$756$	0	0
$757$	− 1.96934e7i	− 1.24905i	−0.781005	−	0.624525i	$-0.785293\pi$
$757$	− 1.96934e7i	− 1.24905i	0.781005	−	0.624525i	$-0.214707\pi$
$758$	0	0
$759$	−4.42167e6	−0.278600
$760$	0	0
$761$	−6.73562e6	−0.421615	−0.210807	−	0.977528i	$-0.567609\pi$
$761$	−6.73562e6	−0.421615	−0.210807	+	0.977528i	$0.567609\pi$
$762$	0	0
$763$	− 1.29765e7i	− 0.806946i
$764$	0	0
$765$	− 3.84014e6i	− 0.237243i
$766$	0	0
$767$	−1.66137e6	−0.101972
$768$	0	0
$769$	−6.67796e6	−0.407219	−0.203609	−	0.979052i	$-0.565267\pi$
$769$	−6.67796e6	−0.407219	−0.203609	+	0.979052i	$0.565267\pi$
$770$	0	0
$771$	5.63571e6i	0.341439i
$772$	0	0
$773$	2.85469e6i	0.171835i	0.996302	+	0.0859173i	$0.0273821\pi$
$773$	2.85469e6i	0.171835i	−0.996302	+	0.0859173i	$0.972618\pi$
$774$	0	0
$775$	−3.73232e6	−0.223215
$776$	0	0
$777$	1.73731e6	0.103235
$778$	0	0
$779$	− 1.42877e7i	− 0.843565i
$780$	0	0
$781$	− 2.52488e7i	− 1.48120i
$782$	0	0
$783$	−5.98082e6	−0.348623
$784$	0	0
$785$	−3.55492e7	−2.05899
$786$	0	0
$787$	1.38785e7i	0.798738i	0.916790	+	0.399369i	$0.130771\pi$
$787$	1.38785e7i	0.798738i	−0.916790	+	0.399369i	$0.869229\pi$
$788$	0	0
$789$	4.93684e6i	0.282330i
$790$	0	0
$791$	−2.16302e6	−0.122919
$792$	0	0
$793$	4.94849e6	0.279441
$794$	0	0
$795$	− 3.05910e6i	− 0.171662i
$796$	0	0
$797$	8.67764e6i	0.483900i	0.970289	+	0.241950i	$0.0777871\pi$
$797$	8.67764e6i	0.483900i	−0.970289	+	0.241950i	$0.922213\pi$
$798$	0	0
$799$	−4.70543e6	−0.260755
$800$	0	0
$801$	1.49714e7	0.824484
$802$	0	0
$803$	− 1.80049e7i	− 0.985373i
$804$	0	0
$805$	− 1.96816e7i	− 1.07046i
$806$	0	0
$807$	6.17089e6	0.333553
$808$	0	0
$809$	7.54612e6	0.405371	0.202685	−	0.979244i	$-0.435033\pi$
$809$	7.54612e6	0.405371	0.202685	+	0.979244i	$0.435033\pi$
$810$	0	0
$811$	− 2.53731e7i	− 1.35463i	−0.735692	−	0.677316i	$-0.763143\pi$
$811$	− 2.53731e7i	− 1.35463i	0.735692	−	0.677316i	$-0.236857\pi$
$812$	0	0
$813$	1.55021e6i	0.0822554i
$814$	0	0
$815$	1.30623e7	0.688851
$816$	0	0
$817$	−8.62152e6	−0.451886
$818$	0	0
$819$	3.08388e6i	0.160653i
$820$	0	0
$821$	1.52925e7i	0.791812i	0.918291	+	0.395906i	$0.129569\pi$
$821$	1.52925e7i	0.791812i	−0.918291	+	0.395906i	$0.870431\pi$
$822$	0	0
$823$	−4.47750e6	−0.230428	−0.115214	−	0.993341i	$-0.536755\pi$
$823$	−4.47750e6	−0.230428	−0.115214	+	0.993341i	$0.536755\pi$
$824$	0	0
$825$	−4.39157e6	−0.224639
$826$	0	0
$827$	8.21663e6i	0.417763i	0.977941	+	0.208882i	$0.0669823\pi$
$827$	8.21663e6i	0.417763i	−0.977941	+	0.208882i	$0.933018\pi$
$828$	0	0
$829$	1.58318e7i	0.800098i	0.916494	+	0.400049i	$0.131007\pi$
$829$	1.58318e7i	0.800098i	−0.916494	+	0.400049i	$0.868993\pi$
$830$	0	0
$831$	−2.74278e6	−0.137781
$832$	0	0
$833$	917045.	0.0457908
$834$	0	0
$835$	1.61863e7i	0.803401i
$836$	0	0
$837$	− 2.46249e6i	− 0.121495i
$838$	0	0
$839$	−1.29263e7	−0.633970	−0.316985	−	0.948430i	$-0.602671\pi$
$839$	−1.29263e7	−0.633970	−0.316985	+	0.948430i	$0.602671\pi$
$840$	0	0
$841$	5.56893e6	0.271508
$842$	0	0
$843$	5.35256e6i	0.259413i
$844$	0	0
$845$	2.64356e7i	1.27364i
$846$	0	0
$847$	−1.91645e7	−0.917886
$848$	0	0
$849$	−7.25316e6	−0.345349
$850$	0	0
$851$	− 1.11835e7i	− 0.529362i
$852$	0	0
$853$	− 8.07995e6i	− 0.380221i	−0.981763	−	0.190110i	$-0.939115\pi$
$853$	− 8.07995e6i	− 0.380221i	0.981763	−	0.190110i	$-0.0608846\pi$
$854$	0	0
$855$	3.01194e7	1.40906
$856$	0	0
$857$	1.51474e7	0.704510	0.352255	−	0.935904i	$-0.385415\pi$
$857$	1.51474e7	0.704510	0.352255	+	0.935904i	$0.385415\pi$
$858$	0	0
$859$	1.27089e7i	0.587658i	0.955858	+	0.293829i	$0.0949297\pi$
$859$	1.27089e7i	0.587658i	−0.955858	+	0.293829i	$0.905070\pi$
$860$	0	0
$861$	− 2.99082e6i	− 0.137494i
$862$	0	0
$863$	3.10500e7	1.41917	0.709585	−	0.704620i	$-0.248882\pi$
$863$	3.10500e7	1.41917	0.709585	+	0.704620i	$0.248882\pi$
$864$	0	0
$865$	3.86623e7	1.75690
$866$	0	0
$867$	4.45851e6i	0.201438i
$868$	0	0
$869$	2.88920e7i	1.29786i
$870$	0	0
$871$	6.37578e6	0.284766
$872$	0	0
$873$	−1.44909e7	−0.643515
$874$	0	0
$875$	6.50108e6i	0.287055i
$876$	0	0
$877$	− 1.45850e7i	− 0.640336i	−0.947361	−	0.320168i	$-0.896261\pi$
$877$	− 1.45850e7i	− 0.640336i	0.947361	−	0.320168i	$-0.103739\pi$
$878$	0	0
$879$	2.25239e6	0.0983265
$880$	0	0
$881$	5.73224e6	0.248820	0.124410	−	0.992231i	$-0.460296\pi$
$881$	5.73224e6	0.248820	0.124410	+	0.992231i	$0.460296\pi$
$882$	0	0
$883$	522077.i	0.0225337i	0.999937	+	0.0112669i	$0.00358643\pi$
$883$	522077.i	0.0225337i	−0.999937	+	0.0112669i	$0.996414\pi$
$884$	0	0
$885$	3.39659e6i	0.145776i
$886$	0	0
$887$	2.52815e7	1.07893	0.539466	−	0.842007i	$-0.318626\pi$
$887$	2.52815e7	1.07893	0.539466	+	0.842007i	$0.318626\pi$
$888$	0	0
$889$	−1.14958e7	−0.487847
$890$	0	0
$891$	2.95988e7i	1.24905i
$892$	0	0
$893$	− 3.69061e7i	− 1.54871i
$894$	0	0
$895$	−3.04020e7	−1.26866
$896$	0	0
$897$	−904728.	−0.0375437
$898$	0	0
$899$	− 6.15217e6i	− 0.253880i
$900$	0	0
$901$	2.83928e6i	0.116519i
$902$	0	0
$903$	−1.80473e6	−0.0736533
$904$	0	0
$905$	9.84367e6	0.399517
$906$	0	0
$907$	1.27692e7i	0.515402i	0.966225	+	0.257701i	$0.0829649\pi$
$907$	1.27692e7i	0.515402i	−0.966225	+	0.257701i	$0.917035\pi$
$908$	0	0
$909$	1.16112e7i	0.466088i
$910$	0	0
$911$	−3.33197e7	−1.33017	−0.665083	−	0.746770i	$-0.731604\pi$
$911$	−3.33197e7	−1.33017	−0.665083	+	0.746770i	$0.731604\pi$
$912$	0	0
$913$	−2.51493e7	−0.998503
$914$	0	0
$915$	− 1.01169e7i	− 0.399481i
$916$	0	0
$917$	− 1.27312e6i	− 0.0499973i
$918$	0	0
$919$	2.91253e7	1.13758	0.568789	−	0.822484i	$-0.307412\pi$
$919$	2.91253e7	1.13758	0.568789	+	0.822484i	$0.307412\pi$
$920$	0	0
$921$	9.49365e6	0.368794
$922$	0	0
$923$	− 5.16622e6i	− 0.199604i
$924$	0	0
$925$	− 1.11073e7i	− 0.426831i
$926$	0	0
$927$	−3.70133e7	−1.41468
$928$	0	0
$929$	1.84988e7	0.703242	0.351621	−	0.936143i	$-0.385631\pi$
$929$	1.84988e7	0.703242	0.351621	+	0.936143i	$0.385631\pi$
$930$	0	0
$931$	7.19266e6i	0.271966i
$932$	0	0
$933$	3.60357e6i	0.135528i
$934$	0	0
$935$	9.50761e6	0.355666
$936$	0	0
$937$	−3.42891e7	−1.27587	−0.637936	−	0.770089i	$-0.720212\pi$
$937$	−3.42891e7	−1.27587	−0.637936	+	0.770089i	$0.720212\pi$
$938$	0	0
$939$	5.38261e6i	0.199218i
$940$	0	0
$941$	1.23906e7i	0.456161i	0.973642	+	0.228080i	$0.0732449\pi$
$941$	1.23906e7i	0.456161i	−0.973642	+	0.228080i	$0.926755\pi$
$942$	0	0
$943$	−1.92525e7	−0.705032
$944$	0	0
$945$	1.28970e7	0.469797
$946$	0	0
$947$	− 5.12809e7i	− 1.85815i	−0.369893	−	0.929074i	$-0.620606\pi$
$947$	− 5.12809e7i	− 1.85815i	0.369893	−	0.929074i	$-0.379394\pi$
$948$	0	0
$949$	− 3.68402e6i	− 0.132787i
$950$	0	0
$951$	−1.49467e6	−0.0535912
$952$	0	0
$953$	−1.10363e7	−0.393634	−0.196817	−	0.980440i	$-0.563060\pi$
$953$	−1.10363e7	−0.393634	−0.196817	+	0.980440i	$0.563060\pi$
$954$	0	0
$955$	− 6.16417e7i	− 2.18709i
$956$	0	0
$957$	− 7.23885e6i	− 0.255499i
$958$	0	0
$959$	1.15807e7	0.406618
$960$	0	0
$961$	−2.60961e7	−0.911523
$962$	0	0
$963$	− 3.15065e7i	− 1.09480i
$964$	0	0
$965$	− 4.41948e7i	− 1.52775i
$966$	0	0
$967$	4.54331e7	1.56245	0.781225	−	0.624249i	$-0.214595\pi$
$967$	4.54331e7	1.56245	0.781225	+	0.624249i	$0.214595\pi$
$968$	0	0
$969$	1.27404e6	0.0435887
$970$	0	0
$971$	6.77731e6i	0.230680i	0.993326	+	0.115340i	$0.0367957\pi$
$971$	6.77731e6i	0.230680i	−0.993326	+	0.115340i	$0.963204\pi$
$972$	0	0
$973$	3.87621e7i	1.31258i
$974$	0	0
$975$	−898569.	−0.0302719
$976$	0	0
$977$	−4.58953e7	−1.53827	−0.769134	−	0.639087i	$-0.779312\pi$
$977$	−4.58953e7	−1.53827	−0.769134	+	0.639087i	$0.779312\pi$
$978$	0	0
$979$	3.70670e7i	1.23604i
$980$	0	0
$981$	− 2.67589e7i	− 0.887759i
$982$	0	0
$983$	−4.96160e7	−1.63771	−0.818857	−	0.573998i	$-0.805392\pi$
$983$	−4.96160e7	−1.63771	−0.818857	+	0.573998i	$0.805392\pi$
$984$	0	0
$985$	5.08977e7	1.67151
$986$	0	0
$987$	− 7.72549e6i	− 0.252425i
$988$	0	0
$989$	1.16174e7i	0.377676i
$990$	0	0
$991$	−2.24227e7	−0.725276	−0.362638	−	0.931930i	$-0.618124\pi$
$991$	−2.24227e7	−0.725276	−0.362638	+	0.931930i	$0.618124\pi$
$992$	0	0
$993$	−5.06462e6	−0.162995
$994$	0	0
$995$	7.22970e7i	2.31506i
$996$	0	0
$997$	5.13276e7i	1.63536i	0.575675	+	0.817679i	$0.304740\pi$
$997$	5.13276e7i	1.63536i	−0.575675	+	0.817679i	$0.695260\pi$
$998$	0	0
$999$	7.32834e6	0.232323

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	32.6.b.a.17.2		4
3.2	odd	2		288.6.d.b.145.1		4
4.3	odd	2		8.6.b.a.5.3	✓	4
5.2	odd	4		800.6.f.a.49.4		8
5.3	odd	4		800.6.f.a.49.5		8
5.4	even	2		800.6.d.a.401.3		4
8.3	odd	2		8.6.b.a.5.4	yes	4
8.5	even	2	inner	32.6.b.a.17.3		4
12.11	even	2		72.6.d.b.37.2		4
16.3	odd	4		256.6.a.k.1.2		4
16.5	even	4		256.6.a.n.1.2		4
16.11	odd	4		256.6.a.k.1.3		4
16.13	even	4		256.6.a.n.1.3		4
20.3	even	4		200.6.f.a.149.1		8
20.7	even	4		200.6.f.a.149.8		8
20.19	odd	2		200.6.d.a.101.2		4
24.5	odd	2		288.6.d.b.145.4		4
24.11	even	2		72.6.d.b.37.1		4
40.3	even	4		200.6.f.a.149.7		8
40.13	odd	4		800.6.f.a.49.3		8
40.19	odd	2		200.6.d.a.101.1		4
40.27	even	4		200.6.f.a.149.2		8
40.29	even	2		800.6.d.a.401.2		4
40.37	odd	4		800.6.f.a.49.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.6.b.a.5.3	✓	4	4.3	odd	2
8.6.b.a.5.4	yes	4	8.3	odd	2
32.6.b.a.17.2		4	1.1	even	1	trivial
32.6.b.a.17.3		4	8.5	even	2	inner
72.6.d.b.37.1		4	24.11	even	2
72.6.d.b.37.2		4	12.11	even	2
200.6.d.a.101.1		4	40.19	odd	2
200.6.d.a.101.2		4	20.19	odd	2
200.6.f.a.149.1		8	20.3	even	4
200.6.f.a.149.2		8	40.27	even	4
200.6.f.a.149.7		8	40.3	even	4
200.6.f.a.149.8		8	20.7	even	4
256.6.a.k.1.2		4	16.3	odd	4
256.6.a.k.1.3		4	16.11	odd	4
256.6.a.n.1.2		4	16.5	even	4
256.6.a.n.1.3		4	16.13	even	4
288.6.d.b.145.1		4	3.2	odd	2
288.6.d.b.145.4		4	24.5	odd	2
800.6.d.a.401.2		4	40.29	even	2
800.6.d.a.401.3		4	5.4	even	2
800.6.f.a.49.3		8	40.13	odd	4
800.6.f.a.49.4		8	5.2	odd	4
800.6.f.a.49.5		8	5.3	odd	4
800.6.f.a.49.6		8	40.37	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 32 = 2^{5} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	32.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.25452i q^{3} +73.9600i q^{5} +112.704 q^{7} +232.408 q^{9} +575.407i q^{11} +117.735i q^{13} +240.704 q^{15} -223.408 q^{17} -1752.26i q^{19} -366.797i q^{21} -2361.15 q^{23} -2345.08 q^{25} -1547.22i q^{27} -3865.52i q^{29} +1591.55 q^{31} +1872.67 q^{33} +8335.59i q^{35} +4736.44i q^{37} +383.172 q^{39} +8153.88 q^{41} -4920.23i q^{43} +17188.9i q^{45} +21062.0 q^{47} -4104.79 q^{49} +727.085i q^{51} -12709.0i q^{53} -42557.1 q^{55} -5702.75 q^{57} +14111.1i q^{59} -42030.6i q^{61} +26193.3 q^{63} -8707.71 q^{65} -54153.4i q^{67} +7684.41i q^{69} -43879.9 q^{71} -31290.6 q^{73} +7632.11i q^{75} +64850.7i q^{77} +50211.5 q^{79} +51439.7 q^{81} +43707.0i q^{83} -16523.3i q^{85} -12580.4 q^{87} +64418.7 q^{89} +13269.3i q^{91} -5179.73i q^{93} +129597. q^{95} -62350.9 q^{97} +133729. i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 96 q^{7} - 164 q^{9} + 416 q^{15} + 200 q^{17} - 2336 q^{23} + 1556 q^{25} + 12928 q^{31} - 2352 q^{33} - 35104 q^{39} - 4568 q^{41} + 54720 q^{47} + 9828 q^{49} - 85472 q^{55} - 2032 q^{57} + 153440 q^{63}+ \cdots - 99576 q^{97}+O(q^{100}) \) 4 * q - 96 * q^7 - 164 * q^9 + 416 * q^15 + 200 * q^17 - 2336 * q^23 + 1556 * q^25 + 12928 * q^31 - 2352 * q^33 - 35104 * q^39 - 4568 * q^41 + 54720 * q^47 + 9828 * q^49 - 85472 * q^55 - 2032 * q^57 + 153440 * q^63 - 19520 * q^65 - 206688 * q^71 + 39976 * q^73 + 247872 * q^79 + 29684 * q^81 - 307872 * q^87 - 84632 * q^89 + 259744 * q^95 - 99576 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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