LMFDB - Embedded newform 128.5.d.d.63.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [128,5,Mod(63,128)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(128, 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("128.63");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 128 = 2^{7} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	128.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:	$13.2313552747$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.1871773696.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 31x^{4} + 81 $ x^8 + 31*x^4 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{38} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			63.8
Root			$-1.62831 + 1.62831i$ of defining polynomial
Character	$\chi$	$=$	128.63
Dual form			128.5.d.d.63.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+15.8549 q^{3} +40.8444i q^{5} -40.7922i q^{7} +170.378 q^{9} +O(q^{10})$	$q+15.8549 q^{3} +40.8444i q^{5} -40.7922i q^{7} +170.378 q^{9} +42.9450 q^{11} +186.533i q^{13} +647.584i q^{15} -157.378 q^{17} +278.145 q^{19} -646.755i q^{21} -249.844i q^{23} -1043.27 q^{25} +1417.07 q^{27} -416.488i q^{29} +1086.12i q^{31} +680.888 q^{33} +1666.13 q^{35} -742.267i q^{37} +2957.46i q^{39} +1190.27 q^{41} -2250.03 q^{43} +6958.97i q^{45} -2799.39i q^{47} +737.000 q^{49} -2495.21 q^{51} -3876.67i q^{53} +1754.06i q^{55} +4409.95 q^{57} +5255.73 q^{59} -1364.93i q^{61} -6950.07i q^{63} -7618.84 q^{65} -316.581 q^{67} -3961.24i q^{69} -8969.32i q^{71} -9726.79 q^{73} -16540.9 q^{75} -1751.82i q^{77} +5466.15i q^{79} +8666.95 q^{81} -10615.1 q^{83} -6428.00i q^{85} -6603.37i q^{87} -6007.69 q^{89} +7609.09 q^{91} +17220.3i q^{93} +11360.6i q^{95} +7307.85 q^{97} +7316.87 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 440 q^{9}+O(q^{10}) $ 8 * q + 440 * q^9	$ 8 q + 440 q^{9} - 336 q^{17} - 2808 q^{25} + 832 q^{33} + 3984 q^{41} + 5896 q^{49} + 8512 q^{57} - 29568 q^{65} - 19664 q^{73} + 42568 q^{81} - 26832 q^{89} - 21840 q^{97}+O(q^{100}) $ 8 * q + 440 * q^9 - 336 * q^17 - 2808 * q^25 + 832 * q^33 + 3984 * q^41 + 5896 * q^49 + 8512 * q^57 - 29568 * q^65 - 19664 * q^73 + 42568 * q^81 - 26832 * q^89 - 21840 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/128\mathbb{Z}\right)^\times$.

$n$	$5$	$127$
$\chi(n)$	$-1$	$-1$

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
$3$	15.8549	1.76165	0.880827	−	0.473438i	$-0.156987\pi$
$3$	15.8549	1.76165	0.880827	+	0.473438i	$0.156987\pi$
$4$	0	0
$5$	40.8444i	1.63378i	0.576796	+	0.816888i	$0.304303\pi$
$5$	40.8444i	1.63378i	−0.576796	+	0.816888i	$0.695697\pi$
$6$	0	0
$7$	− 40.7922i	− 0.832493i	−0.909252	−	0.416246i	$-0.863345\pi$
$7$	− 40.7922i	− 0.832493i	0.909252	−	0.416246i	$-0.136655\pi$
$8$	0	0
$9$	170.378	2.10343
$10$	0	0
$11$	42.9450	0.354917	0.177459	−	0.984128i	$-0.443212\pi$
$11$	42.9450	0.354917	0.177459	+	0.984128i	$0.443212\pi$
$12$	0	0
$13$	186.533i	1.10375i	0.833928	+	0.551873i	$0.186087\pi$
$13$	186.533i	1.10375i	−0.833928	+	0.551873i	$0.813913\pi$
$14$	0	0
$15$	647.584i	2.87815i
$16$	0	0
$17$	−157.378	−0.544559	−0.272280	−	0.962218i	$-0.587778\pi$
$17$	−157.378	−0.544559	−0.272280	+	0.962218i	$0.587778\pi$
$18$	0	0
$19$	278.145	0.770483	0.385242	−	0.922816i	$-0.374118\pi$
$19$	278.145	0.770483	0.385242	+	0.922816i	$0.374118\pi$
$20$	0	0
$21$	− 646.755i	− 1.46657i
$22$	0	0
$23$	− 249.844i	− 0.472294i	−0.971717	−	0.236147i	$-0.924115\pi$
$23$	− 249.844i	− 0.472294i	0.971717	−	0.236147i	$-0.0758847\pi$
$24$	0	0
$25$	−1043.27	−1.66923
$26$	0	0
$27$	1417.07	1.94386
$28$	0	0
$29$	− 416.488i	− 0.495229i	−0.968859	−	0.247615i	$-0.920353\pi$
$29$	− 416.488i	− 0.495229i	0.968859	−	0.247615i	$-0.0796467\pi$
$30$	0	0
$31$	1086.12i	1.13019i	0.825025	+	0.565097i	$0.191161\pi$
$31$	1086.12i	1.13019i	−0.825025	+	0.565097i	$0.808839\pi$
$32$	0	0
$33$	680.888	0.625242
$34$	0	0
$35$	1666.13	1.36011
$36$	0	0
$37$	− 742.267i	− 0.542197i	−0.962552	−	0.271098i	$-0.912613\pi$
$37$	− 742.267i	− 0.542197i	0.962552	−	0.271098i	$-0.0873869\pi$
$38$	0	0
$39$	2957.46i	1.94442i
$40$	0	0
$41$	1190.27	0.708070	0.354035	−	0.935232i	$-0.384809\pi$
$41$	1190.27	0.708070	0.354035	+	0.935232i	$0.384809\pi$
$42$	0	0
$43$	−2250.03	−1.21689	−0.608444	−	0.793597i	$-0.708206\pi$
$43$	−2250.03	−1.21689	−0.608444	+	0.793597i	$0.708206\pi$
$44$	0	0
$45$	6958.97i	3.43653i
$46$	0	0
$47$	− 2799.39i	− 1.26726i	−0.773635	−	0.633632i	$-0.781563\pi$
$47$	− 2799.39i	− 1.26726i	0.773635	−	0.633632i	$-0.218437\pi$
$48$	0	0
$49$	737.000	0.306955
$50$	0	0
$51$	−2495.21	−0.959326
$52$	0	0
$53$	− 3876.67i	− 1.38009i	−0.723768	−	0.690044i	$-0.757591\pi$
$53$	− 3876.67i	− 1.38009i	0.723768	−	0.690044i	$-0.242409\pi$
$54$	0	0
$55$	1754.06i	0.579855i
$56$	0	0
$57$	4409.95	1.35733
$58$	0	0
$59$	5255.73	1.50983	0.754917	−	0.655821i	$-0.227677\pi$
$59$	5255.73	1.50983	0.754917	+	0.655821i	$0.227677\pi$
$60$	0	0
$61$	− 1364.93i	− 0.366818i	−0.983037	−	0.183409i	$-0.941287\pi$
$61$	− 1364.93i	− 0.366818i	0.983037	−	0.183409i	$-0.0587132\pi$
$62$	0	0
$63$	− 6950.07i	− 1.75109i
$64$	0	0
$65$	−7618.84	−1.80328
$66$	0	0
$67$	−316.581	−0.0705238	−0.0352619	−	0.999378i	$-0.511227\pi$
$67$	−316.581	−0.0705238	−0.0352619	+	0.999378i	$0.511227\pi$
$68$	0	0
$69$	− 3961.24i	− 0.832019i
$70$	0	0
$71$	− 8969.32i	− 1.77927i	−0.456669	−	0.889637i	$-0.650958\pi$
$71$	− 8969.32i	− 1.77927i	0.456669	−	0.889637i	$-0.349042\pi$
$72$	0	0
$73$	−9726.79	−1.82526	−0.912628	−	0.408791i	$-0.865951\pi$
$73$	−9726.79	−1.82526	−0.912628	+	0.408791i	$0.865951\pi$
$74$	0	0
$75$	−16540.9	−2.94060
$76$	0	0
$77$	− 1751.82i	− 0.295466i
$78$	0	0
$79$	5466.15i	0.875845i	0.899013	+	0.437923i	$0.144286\pi$
$79$	5466.15i	0.875845i	−0.899013	+	0.437923i	$0.855714\pi$
$80$	0	0
$81$	8666.95	1.32098
$82$	0	0
$83$	−10615.1	−1.54088	−0.770441	−	0.637512i	$-0.779964\pi$
$83$	−10615.1	−1.54088	−0.770441	+	0.637512i	$0.779964\pi$
$84$	0	0
$85$	− 6428.00i	− 0.889688i
$86$	0	0
$87$	− 6603.37i	− 0.872423i
$88$	0	0
$89$	−6007.69	−0.758450	−0.379225	−	0.925304i	$-0.623809\pi$
$89$	−6007.69	−0.758450	−0.379225	+	0.925304i	$0.623809\pi$
$90$	0	0
$91$	7609.09	0.918862
$92$	0	0
$93$	17220.3i	1.99101i
$94$	0	0
$95$	11360.6i	1.25880i
$96$	0	0
$97$	7307.85	0.776688	0.388344	−	0.921515i	$-0.373047\pi$
$97$	7307.85	0.776688	0.388344	+	0.921515i	$0.373047\pi$
$98$	0	0
$99$	7316.87	0.746543
$100$	0	0
$101$	4946.98i	0.484950i	0.970158	+	0.242475i	$0.0779593\pi$
$101$	4946.98i	0.484950i	−0.970158	+	0.242475i	$0.922041\pi$
$102$	0	0
$103$	− 11029.2i	− 1.03960i	−0.854287	−	0.519802i	$-0.826006\pi$
$103$	− 11029.2i	− 1.03960i	0.854287	−	0.519802i	$-0.173994\pi$
$104$	0	0
$105$	26416.3	2.39604
$106$	0	0
$107$	−4286.81	−0.374426	−0.187213	−	0.982319i	$-0.559946\pi$
$107$	−4286.81	−0.374426	−0.187213	+	0.982319i	$0.559946\pi$
$108$	0	0
$109$	− 10601.2i	− 0.892281i	−0.894963	−	0.446141i	$-0.852798\pi$
$109$	− 10601.2i	− 0.892281i	0.894963	−	0.446141i	$-0.147202\pi$
$110$	0	0
$111$	− 11768.6i	− 0.955164i
$112$	0	0
$113$	−542.701	−0.0425014	−0.0212507	−	0.999774i	$-0.506765\pi$
$113$	−542.701	−0.0425014	−0.0212507	+	0.999774i	$0.506765\pi$
$114$	0	0
$115$	10204.7	0.771623
$116$	0	0
$117$	31781.1i	2.32165i
$118$	0	0
$119$	6419.77i	0.453342i
$120$	0	0
$121$	−12796.7	−0.874034
$122$	0	0
$123$	18871.5	1.24738
$124$	0	0
$125$	− 17083.8i	− 1.09336i
$126$	0	0
$127$	14711.1i	0.912090i	0.889957	+	0.456045i	$0.150734\pi$
$127$	14711.1i	0.912090i	−0.889957	+	0.456045i	$0.849266\pi$
$128$	0	0
$129$	−35673.9	−2.14374
$130$	0	0
$131$	−6762.24	−0.394047	−0.197023	−	0.980399i	$-0.563128\pi$
$131$	−6762.24	−0.394047	−0.197023	+	0.980399i	$0.563128\pi$
$132$	0	0
$133$	− 11346.1i	− 0.641422i
$134$	0	0
$135$	57879.5i	3.17583i
$136$	0	0
$137$	7123.78	0.379550	0.189775	−	0.981828i	$-0.439224\pi$
$137$	7123.78	0.379550	0.189775	+	0.981828i	$0.439224\pi$
$138$	0	0
$139$	22622.3	1.17086	0.585432	−	0.810721i	$-0.300925\pi$
$139$	22622.3	1.17086	0.585432	+	0.810721i	$0.300925\pi$
$140$	0	0
$141$	− 44384.0i	− 2.23248i
$142$	0	0
$143$	8010.67i	0.391739i
$144$	0	0
$145$	17011.2	0.809094
$146$	0	0
$147$	11685.1	0.540750
$148$	0	0
$149$	− 5453.00i	− 0.245619i	−0.992430	−	0.122810i	$-0.960810\pi$
$149$	− 5453.00i	− 0.245619i	0.992430	−	0.122810i	$-0.0391905\pi$
$150$	0	0
$151$	22839.2i	1.00167i	0.865541	+	0.500837i	$0.166974\pi$
$151$	22839.2i	1.00167i	−0.865541	+	0.500837i	$0.833026\pi$
$152$	0	0
$153$	−26813.6	−1.14544
$154$	0	0
$155$	−44361.8	−1.84648
$156$	0	0
$157$	15364.6i	0.623337i	0.950191	+	0.311669i	$0.100888\pi$
$157$	15364.6i	0.623337i	−0.950191	+	0.311669i	$0.899112\pi$
$158$	0	0
$159$	− 61464.1i	− 2.43124i
$160$	0	0
$161$	−10191.7	−0.393182
$162$	0	0
$163$	40347.8	1.51861	0.759303	−	0.650738i	$-0.225540\pi$
$163$	40347.8	1.51861	0.759303	+	0.650738i	$0.225540\pi$
$164$	0	0
$165$	27810.5i	1.02151i
$166$	0	0
$167$	− 27514.7i	− 0.986580i	−0.869865	−	0.493290i	$-0.835794\pi$
$167$	− 27514.7i	− 0.986580i	0.869865	−	0.493290i	$-0.164206\pi$
$168$	0	0
$169$	−6233.65	−0.218257
$170$	0	0
$171$	47389.6	1.62066
$172$	0	0
$173$	8953.08i	0.299144i	0.988751	+	0.149572i	$0.0477896\pi$
$173$	8953.08i	0.299144i	−0.988751	+	0.149572i	$0.952210\pi$
$174$	0	0
$175$	42557.1i	1.38962i
$176$	0	0
$177$	83329.0	2.65981
$178$	0	0
$179$	2832.69	0.0884084	0.0442042	−	0.999023i	$-0.485925\pi$
$179$	2832.69	0.0884084	0.0442042	+	0.999023i	$0.485925\pi$
$180$	0	0
$181$	57763.3i	1.76317i	0.472023	+	0.881586i	$0.343524\pi$
$181$	57763.3i	1.76317i	−0.472023	+	0.881586i	$0.656476\pi$
$182$	0	0
$183$	− 21640.8i	− 0.646206i
$184$	0	0
$185$	30317.5	0.885828
$186$	0	0
$187$	−6758.58	−0.193273
$188$	0	0
$189$	− 57805.5i	− 1.61825i
$190$	0	0
$191$	33495.3i	0.918158i	0.888395	+	0.459079i	$0.151820\pi$
$191$	33495.3i	0.918158i	−0.888395	+	0.459079i	$0.848180\pi$
$192$	0	0
$193$	−31766.3	−0.852809	−0.426404	−	0.904533i	$-0.640220\pi$
$193$	−31766.3	−0.852809	−0.426404	+	0.904533i	$0.640220\pi$
$194$	0	0
$195$	−120796.	−3.17675
$196$	0	0
$197$	65845.3i	1.69665i	0.529477	+	0.848325i	$0.322388\pi$
$197$	65845.3i	1.69665i	−0.529477	+	0.848325i	$0.677612\pi$
$198$	0	0
$199$	− 33745.6i	− 0.852139i	−0.904690	−	0.426070i	$-0.859898\pi$
$199$	− 33745.6i	− 0.852139i	0.904690	−	0.426070i	$-0.140102\pi$
$200$	0	0
$201$	−5019.37	−0.124239
$202$	0	0
$203$	−16989.4	−0.412275
$204$	0	0
$205$	48615.7i	1.15683i
$206$	0	0
$207$	− 42567.8i	− 0.993437i
$208$	0	0
$209$	11944.9	0.273458
$210$	0	0
$211$	10823.7	0.243115	0.121558	−	0.992584i	$-0.461211\pi$
$211$	10823.7	0.243115	0.121558	+	0.992584i	$0.461211\pi$
$212$	0	0
$213$	− 142208.i	− 3.13447i
$214$	0	0
$215$	− 91901.0i	− 1.98812i
$216$	0	0
$217$	44305.0	0.940878
$218$	0	0
$219$	−154217.	−3.21547
$220$	0	0
$221$	− 29356.2i	− 0.601056i
$222$	0	0
$223$	84353.9i	1.69627i	0.529778	+	0.848136i	$0.322275\pi$
$223$	84353.9i	1.69627i	−0.529778	+	0.848136i	$0.677725\pi$
$224$	0	0
$225$	−177749.	−3.51109
$226$	0	0
$227$	20200.7	0.392026	0.196013	−	0.980601i	$-0.437201\pi$
$227$	20200.7	0.392026	0.196013	+	0.980601i	$0.437201\pi$
$228$	0	0
$229$	− 75891.6i	− 1.44718i	−0.690230	−	0.723590i	$-0.742491\pi$
$229$	− 75891.6i	− 1.44718i	0.690230	−	0.723590i	$-0.257509\pi$
$230$	0	0
$231$	− 27774.9i	− 0.520509i
$232$	0	0
$233$	1141.85	0.0210328	0.0105164	−	0.999945i	$-0.496652\pi$
$233$	1141.85	0.0210328	0.0105164	+	0.999945i	$0.496652\pi$
$234$	0	0
$235$	114339.	2.07043
$236$	0	0
$237$	86665.2i	1.54294i
$238$	0	0
$239$	− 94904.2i	− 1.66146i	−0.556676	−	0.830730i	$-0.687923\pi$
$239$	− 94904.2i	− 1.66146i	0.556676	−	0.830730i	$-0.312077\pi$
$240$	0	0
$241$	29917.0	0.515090	0.257545	−	0.966266i	$-0.417086\pi$
$241$	29917.0	0.515090	0.257545	+	0.966266i	$0.417086\pi$
$242$	0	0
$243$	22630.7	0.383253
$244$	0	0
$245$	30102.3i	0.501497i
$246$	0	0
$247$	51883.2i	0.850419i
$248$	0	0
$249$	−168302.	−2.71450
$250$	0	0
$251$	38342.3	0.608598	0.304299	−	0.952577i	$-0.401578\pi$
$251$	38342.3	0.608598	0.304299	+	0.952577i	$0.401578\pi$
$252$	0	0
$253$	− 10729.5i	− 0.167625i
$254$	0	0
$255$	− 101915.i	− 1.56732i
$256$	0	0
$257$	110921.	1.67937	0.839687	−	0.543071i	$-0.182738\pi$
$257$	110921.	1.67937	0.839687	+	0.543071i	$0.182738\pi$
$258$	0	0
$259$	−30278.7	−0.451375
$260$	0	0
$261$	− 70960.2i	− 1.04168i
$262$	0	0
$263$	− 30640.3i	− 0.442978i	−0.975163	−	0.221489i	$-0.928908\pi$
$263$	− 30640.3i	− 0.442978i	0.975163	−	0.221489i	$-0.0710917\pi$
$264$	0	0
$265$	158340.	2.25475
$266$	0	0
$267$	−95251.2	−1.33613
$268$	0	0
$269$	− 6337.36i	− 0.0875798i	−0.999041	−	0.0437899i	$-0.986057\pi$
$269$	− 6337.36i	− 0.0875798i	0.999041	−	0.0437899i	$-0.0139432\pi$
$270$	0	0
$271$	77706.2i	1.05808i	0.848598	+	0.529038i	$0.177447\pi$
$271$	77706.2i	1.05808i	−0.848598	+	0.529038i	$0.822553\pi$
$272$	0	0
$273$	120641.	1.61872
$274$	0	0
$275$	−44803.0	−0.592437
$276$	0	0
$277$	51172.8i	0.666930i	0.942763	+	0.333465i	$0.108218\pi$
$277$	51172.8i	0.666930i	−0.942763	+	0.333465i	$0.891782\pi$
$278$	0	0
$279$	185050.i	2.37728i
$280$	0	0
$281$	−67120.5	−0.850047	−0.425023	−	0.905182i	$-0.639734\pi$
$281$	−67120.5	−0.850047	−0.425023	+	0.905182i	$0.639734\pi$
$282$	0	0
$283$	19713.8	0.246148	0.123074	−	0.992397i	$-0.460725\pi$
$283$	19713.8	0.246148	0.123074	+	0.992397i	$0.460725\pi$
$284$	0	0
$285$	180122.i	2.21757i
$286$	0	0
$287$	− 48553.5i	− 0.589463i
$288$	0	0
$289$	−58753.3	−0.703455
$290$	0	0
$291$	115865.	1.36826
$292$	0	0
$293$	58916.5i	0.686281i	0.939284	+	0.343141i	$0.111491\pi$
$293$	58916.5i	0.686281i	−0.939284	+	0.343141i	$0.888509\pi$
$294$	0	0
$295$	214667.i	2.46673i
$296$	0	0
$297$	60856.2	0.689909
$298$	0	0
$299$	46604.1	0.521293
$300$	0	0
$301$	91783.5i	1.01305i
$302$	0	0
$303$	78433.8i	0.854315i
$304$	0	0
$305$	55749.7	0.599298
$306$	0	0
$307$	64023.9	0.679305	0.339653	−	0.940551i	$-0.389690\pi$
$307$	64023.9	0.679305	0.339653	+	0.940551i	$0.389690\pi$
$308$	0	0
$309$	− 174866.i	− 1.83142i
$310$	0	0
$311$	− 6117.83i	− 0.0632523i	−0.999500	−	0.0316262i	$-0.989931\pi$
$311$	− 6117.83i	− 0.0632523i	0.999500	−	0.0316262i	$-0.0100686\pi$
$312$	0	0
$313$	−9395.35	−0.0959013	−0.0479506	−	0.998850i	$-0.515269\pi$
$313$	−9395.35	−0.0959013	−0.0479506	+	0.998850i	$0.515269\pi$
$314$	0	0
$315$	283872.	2.86089
$316$	0	0
$317$	44305.6i	0.440900i	0.975398	+	0.220450i	$0.0707525\pi$
$317$	44305.6i	0.440900i	−0.975398	+	0.220450i	$0.929247\pi$
$318$	0	0
$319$	− 17886.1i	− 0.175765i
$320$	0	0
$321$	−67966.9	−0.659610
$322$	0	0
$323$	−43773.7	−0.419574
$324$	0	0
$325$	− 194604.i	− 1.84240i
$326$	0	0
$327$	− 168081.i	− 1.57189i
$328$	0	0
$329$	−114193.	−1.05499
$330$	0	0
$331$	−68362.0	−0.623963	−0.311981	−	0.950088i	$-0.600993\pi$
$331$	−68362.0	−0.623963	−0.311981	+	0.950088i	$0.600993\pi$
$332$	0	0
$333$	− 126466.i	− 1.14047i
$334$	0	0
$335$	− 12930.6i	− 0.115220i
$336$	0	0
$337$	−172000.	−1.51450	−0.757249	−	0.653126i	$-0.773457\pi$
$337$	−172000.	−1.51450	−0.757249	+	0.653126i	$0.773457\pi$
$338$	0	0
$339$	−8604.46	−0.0748728
$340$	0	0
$341$	46643.2i	0.401125i
$342$	0	0
$343$	− 128006.i	− 1.08803i
$344$	0	0
$345$	161795.	1.35933
$346$	0	0
$347$	−80821.6	−0.671225	−0.335613	−	0.942000i	$-0.608943\pi$
$347$	−80821.6	−0.671225	−0.335613	+	0.942000i	$0.608943\pi$
$348$	0	0
$349$	− 29674.7i	− 0.243633i	−0.992553	−	0.121817i	$-0.961128\pi$
$349$	− 29674.7i	− 0.243633i	0.992553	−	0.121817i	$-0.0388719\pi$
$350$	0	0
$351$	264331.i	2.14553i
$352$	0	0
$353$	−214207.	−1.71904	−0.859518	−	0.511106i	$-0.829236\pi$
$353$	−214207.	−1.71904	−0.859518	+	0.511106i	$0.829236\pi$
$354$	0	0
$355$	366346.	2.90693
$356$	0	0
$357$	101785.i	0.798632i
$358$	0	0
$359$	144490.i	1.12111i	0.828116	+	0.560557i	$0.189413\pi$
$359$	144490.i	1.12111i	−0.828116	+	0.560557i	$0.810587\pi$
$360$	0	0
$361$	−52956.6	−0.406355
$362$	0	0
$363$	−202891.	−1.53975
$364$	0	0
$365$	− 397285.i	− 2.98206i
$366$	0	0
$367$	87544.2i	0.649973i	0.945719	+	0.324986i	$0.105360\pi$
$367$	87544.2i	0.649973i	−0.945719	+	0.324986i	$0.894640\pi$
$368$	0	0
$369$	202795.	1.48937
$370$	0	0
$371$	−158138.	−1.14891
$372$	0	0
$373$	89451.2i	0.642937i	0.946920	+	0.321469i	$0.104177\pi$
$373$	89451.2i	0.642937i	−0.946920	+	0.321469i	$0.895823\pi$
$374$	0	0
$375$	− 270862.i	− 1.92613i
$376$	0	0
$377$	77688.8	0.546608
$378$	0	0
$379$	202128.	1.40718	0.703588	−	0.710609i	$-0.251580\pi$
$379$	202128.	1.40718	0.703588	+	0.710609i	$0.251580\pi$
$380$	0	0
$381$	233243.i	1.60679i
$382$	0	0
$383$	131942.i	0.899466i	0.893163	+	0.449733i	$0.148481\pi$
$383$	131942.i	0.899466i	−0.893163	+	0.449733i	$0.851519\pi$
$384$	0	0
$385$	71552.0	0.482726
$386$	0	0
$387$	−383354.	−2.55964
$388$	0	0
$389$	180125.i	1.19035i	0.803596	+	0.595175i	$0.202917\pi$
$389$	180125.i	1.19035i	−0.803596	+	0.595175i	$0.797083\pi$
$390$	0	0
$391$	39319.8i	0.257192i
$392$	0	0
$393$	−107215.	−0.694174
$394$	0	0
$395$	−223262.	−1.43093
$396$	0	0
$397$	157223.i	0.997548i	0.866732	+	0.498774i	$0.166216\pi$
$397$	157223.i	0.997548i	−0.866732	+	0.498774i	$0.833784\pi$
$398$	0	0
$399$	− 179891.i	− 1.12996i
$400$	0	0
$401$	106822.	0.664312	0.332156	−	0.943224i	$-0.392224\pi$
$401$	106822.	0.664312	0.332156	+	0.943224i	$0.392224\pi$
$402$	0	0
$403$	−202597.	−1.24745
$404$	0	0
$405$	353997.i	2.15819i
$406$	0	0
$407$	− 31876.7i	− 0.192435i
$408$	0	0
$409$	−120610.	−0.721003	−0.360502	−	0.932759i	$-0.617394\pi$
$409$	−120610.	−0.721003	−0.360502	+	0.932759i	$0.617394\pi$
$410$	0	0
$411$	112947.	0.668636
$412$	0	0
$413$	− 214393.i	− 1.25693i
$414$	0	0
$415$	− 433569.i	− 2.51746i
$416$	0	0
$417$	358674.	2.06266
$418$	0	0
$419$	61097.6	0.348014	0.174007	−	0.984744i	$-0.444329\pi$
$419$	61097.6	0.348014	0.174007	+	0.984744i	$0.444329\pi$
$420$	0	0
$421$	− 254570.i	− 1.43629i	−0.695893	−	0.718146i	$-0.744991\pi$
$421$	− 254570.i	− 1.43629i	0.695893	−	0.718146i	$-0.255009\pi$
$422$	0	0
$423$	− 476953.i	− 2.66560i
$424$	0	0
$425$	164187.	0.908992
$426$	0	0
$427$	−55678.4	−0.305373
$428$	0	0
$429$	127008.i	0.690109i
$430$	0	0
$431$	− 57179.8i	− 0.307814i	−0.988085	−	0.153907i	$-0.950814\pi$
$431$	− 57179.8i	− 0.307814i	0.988085	−	0.153907i	$-0.0491856\pi$
$432$	0	0
$433$	76952.6	0.410438	0.205219	−	0.978716i	$-0.434209\pi$
$433$	76952.6	0.410438	0.205219	+	0.978716i	$0.434209\pi$
$434$	0	0
$435$	269711.	1.42534
$436$	0	0
$437$	− 69492.6i	− 0.363895i
$438$	0	0
$439$	310714.i	1.61225i	0.591747	+	0.806123i	$0.298438\pi$
$439$	310714.i	1.61225i	−0.591747	+	0.806123i	$0.701562\pi$
$440$	0	0
$441$	125568.	0.645659
$442$	0	0
$443$	−36718.0	−0.187099	−0.0935494	−	0.995615i	$-0.529821\pi$
$443$	−36718.0	−0.187099	−0.0935494	+	0.995615i	$0.529821\pi$
$444$	0	0
$445$	− 245380.i	− 1.23914i
$446$	0	0
$447$	− 86456.7i	− 0.432697i
$448$	0	0
$449$	−212775.	−1.05543	−0.527713	−	0.849423i	$-0.676950\pi$
$449$	−212775.	−1.05543	−0.527713	+	0.849423i	$0.676950\pi$
$450$	0	0
$451$	51116.0	0.251306
$452$	0	0
$453$	362113.i	1.76460i
$454$	0	0
$455$	310789.i	1.50121i
$456$	0	0
$457$	−386897.	−1.85252	−0.926260	−	0.376885i	$-0.876995\pi$
$457$	−386897.	−1.85252	−0.926260	+	0.376885i	$0.876995\pi$
$458$	0	0
$459$	−223016.	−1.05855
$460$	0	0
$461$	− 213221.i	− 1.00329i	−0.865072	−	0.501647i	$-0.832728\pi$
$461$	− 213221.i	− 1.00329i	0.865072	−	0.501647i	$-0.167272\pi$
$462$	0	0
$463$	− 97351.8i	− 0.454132i	−0.973879	−	0.227066i	$-0.927087\pi$
$463$	− 97351.8i	− 0.454132i	0.973879	−	0.227066i	$-0.0729133\pi$
$464$	0	0
$465$	−703351.	−3.25287
$466$	0	0
$467$	−232169.	−1.06456	−0.532281	−	0.846568i	$-0.678665\pi$
$467$	−232169.	−1.06456	−0.532281	+	0.846568i	$0.678665\pi$
$468$	0	0
$469$	12914.0i	0.0587106i
$470$	0	0
$471$	243605.i	1.09811i
$472$	0	0
$473$	−96627.4	−0.431895
$474$	0	0
$475$	−290179.	−1.28611
$476$	0	0
$477$	− 660497.i	− 2.90291i
$478$	0	0
$479$	− 396652.i	− 1.72878i	−0.502824	−	0.864389i	$-0.667706\pi$
$479$	− 396652.i	− 1.72878i	0.502824	−	0.864389i	$-0.332294\pi$
$480$	0	0
$481$	138458.	0.598448
$482$	0	0
$483$	−161588.	−0.692650
$484$	0	0
$485$	298485.i	1.26893i
$486$	0	0
$487$	− 50329.3i	− 0.212208i	−0.994355	−	0.106104i	$-0.966162\pi$
$487$	− 50329.3i	− 0.212208i	0.994355	−	0.106104i	$-0.0338377\pi$
$488$	0	0
$489$	639711.	2.67526
$490$	0	0
$491$	440519.	1.82727	0.913633	−	0.406539i	$-0.133264\pi$
$491$	440519.	1.82727	0.913633	+	0.406539i	$0.133264\pi$
$492$	0	0
$493$	65545.9i	0.269682i
$494$	0	0
$495$	298853.i	1.21968i
$496$	0	0
$497$	−365878.	−1.48123
$498$	0	0
$499$	−254053.	−1.02029	−0.510145	−	0.860088i	$-0.670408\pi$
$499$	−254053.	−1.02029	−0.510145	+	0.860088i	$0.670408\pi$
$500$	0	0
$501$	− 436243.i	− 1.73801i
$502$	0	0
$503$	50742.6i	0.200557i	0.994959	+	0.100278i	$0.0319733\pi$
$503$	50742.6i	0.200557i	−0.994959	+	0.100278i	$0.968027\pi$
$504$	0	0
$505$	−202056.	−0.792300
$506$	0	0
$507$	−98833.8	−0.384494
$508$	0	0
$509$	141674.i	0.546832i	0.961896	+	0.273416i	$0.0881536\pi$
$509$	141674.i	0.546832i	−0.961896	+	0.273416i	$0.911846\pi$
$510$	0	0
$511$	396777.i	1.51951i
$512$	0	0
$513$	394151.	1.49771
$514$	0	0
$515$	450479.	1.69848
$516$	0	0
$517$	− 120220.i	− 0.449774i
$518$	0	0
$519$	141950.i	0.526989i
$520$	0	0
$521$	−136265.	−0.502005	−0.251003	−	0.967986i	$-0.580760\pi$
$521$	−136265.	−0.502005	−0.251003	+	0.967986i	$0.580760\pi$
$522$	0	0
$523$	457422.	1.67230	0.836148	−	0.548503i	$-0.184802\pi$
$523$	457422.	1.67230	0.836148	+	0.548503i	$0.184802\pi$
$524$	0	0
$525$	674738.i	2.44803i
$526$	0	0
$527$	− 170930.i	− 0.615457i
$528$	0	0
$529$	217419.	0.776938
$530$	0	0
$531$	895459.	3.17583
$532$	0	0
$533$	222024.i	0.781530i
$534$	0	0
$535$	− 175092.i	− 0.611729i
$536$	0	0
$537$	44912.1	0.155745
$538$	0	0
$539$	31650.5	0.108944
$540$	0	0
$541$	315987.i	1.07963i	0.841784	+	0.539814i	$0.181505\pi$
$541$	315987.i	1.07963i	−0.841784	+	0.539814i	$0.818495\pi$
$542$	0	0
$543$	915831.i	3.10610i
$544$	0	0
$545$	433000.	1.45779
$546$	0	0
$547$	−377170.	−1.26056	−0.630279	−	0.776369i	$-0.717059\pi$
$547$	−377170.	−1.26056	−0.630279	+	0.776369i	$0.717059\pi$
$548$	0	0
$549$	− 232553.i	− 0.771575i
$550$	0	0
$551$	− 115844.i	− 0.381566i
$552$	0	0
$553$	222976.	0.729135
$554$	0	0
$555$	480680.	1.56052
$556$	0	0
$557$	− 279142.i	− 0.899735i	−0.893095	−	0.449867i	$-0.851471\pi$
$557$	− 279142.i	− 0.899735i	0.893095	−	0.449867i	$-0.148529\pi$
$558$	0	0
$559$	− 419705.i	− 1.34314i
$560$	0	0
$561$	−107157.	−0.340481
$562$	0	0
$563$	228763.	0.721719	0.360860	−	0.932620i	$-0.382483\pi$
$563$	228763.	0.721719	0.360860	+	0.932620i	$0.382483\pi$
$564$	0	0
$565$	− 22166.3i	− 0.0694378i
$566$	0	0
$567$	− 353544.i	− 1.09971i
$568$	0	0
$569$	463126.	1.43046	0.715229	−	0.698890i	$-0.246322\pi$
$569$	463126.	1.43046	0.715229	+	0.698890i	$0.246322\pi$
$570$	0	0
$571$	−377507.	−1.15785	−0.578925	−	0.815381i	$-0.696528\pi$
$571$	−377507.	−1.15785	−0.578925	+	0.815381i	$0.696528\pi$
$572$	0	0
$573$	531065.i	1.61748i
$574$	0	0
$575$	260653.i	0.788366i
$576$	0	0
$577$	255507.	0.767452	0.383726	−	0.923447i	$-0.374641\pi$
$577$	255507.	0.767452	0.383726	+	0.923447i	$0.374641\pi$
$578$	0	0
$579$	−503651.	−1.50235
$580$	0	0
$581$	433014.i	1.28277i
$582$	0	0
$583$	− 166483.i	− 0.489817i
$584$	0	0
$585$	−1.29808e6	−3.79306
$586$	0	0
$587$	319485.	0.927203	0.463601	−	0.886044i	$-0.346557\pi$
$587$	319485.	0.927203	0.463601	+	0.886044i	$0.346557\pi$
$588$	0	0
$589$	302097.i	0.870795i
$590$	0	0
$591$	1.04397e6i	2.98891i
$592$	0	0
$593$	−137042.	−0.389713	−0.194856	−	0.980832i	$-0.562424\pi$
$593$	−137042.	−0.389713	−0.194856	+	0.980832i	$0.562424\pi$
$594$	0	0
$595$	−262212.	−0.740659
$596$	0	0
$597$	− 535032.i	− 1.50117i
$598$	0	0
$599$	160042.i	0.446047i	0.974813	+	0.223023i	$0.0715926\pi$
$599$	160042.i	0.446047i	−0.974813	+	0.223023i	$0.928407\pi$
$600$	0	0
$601$	375148.	1.03861	0.519306	−	0.854588i	$-0.326191\pi$
$601$	375148.	1.03861	0.519306	+	0.854588i	$0.326191\pi$
$602$	0	0
$603$	−53938.4	−0.148342
$604$	0	0
$605$	− 522675.i	− 1.42798i
$606$	0	0
$607$	− 233576.i	− 0.633943i	−0.948435	−	0.316972i	$-0.897334\pi$
$607$	− 233576.i	− 0.633943i	0.948435	−	0.316972i	$-0.102666\pi$
$608$	0	0
$609$	−269366.	−0.726286
$610$	0	0
$611$	522179.	1.39874
$612$	0	0
$613$	− 582414.i	− 1.54993i	−0.632007	−	0.774963i	$-0.717768\pi$
$613$	− 582414.i	− 1.54993i	0.632007	−	0.774963i	$-0.282232\pi$
$614$	0	0
$615$	770797.i	2.03793i
$616$	0	0
$617$	−375835.	−0.987249	−0.493625	−	0.869675i	$-0.664328\pi$
$617$	−375835.	−0.987249	−0.493625	+	0.869675i	$0.664328\pi$
$618$	0	0
$619$	−589437.	−1.53835	−0.769177	−	0.639035i	$-0.779334\pi$
$619$	−589437.	−1.53835	−0.769177	+	0.639035i	$0.779334\pi$
$620$	0	0
$621$	− 354047.i	− 0.918073i
$622$	0	0
$623$	245066.i	0.631405i
$624$	0	0
$625$	45737.5	0.117088
$626$	0	0
$627$	189385.	0.481738
$628$	0	0
$629$	116816.i	0.295258i
$630$	0	0
$631$	31162.4i	0.0782658i	0.999234	+	0.0391329i	$0.0124596\pi$
$631$	31162.4i	0.0782658i	−0.999234	+	0.0391329i	$0.987540\pi$
$632$	0	0
$633$	171609.	0.428285
$634$	0	0
$635$	−600866.	−1.49015
$636$	0	0
$637$	137475.i	0.338801i
$638$	0	0
$639$	− 1.52817e6i	− 3.74257i
$640$	0	0
$641$	222106.	0.540561	0.270281	−	0.962782i	$-0.412884\pi$
$641$	222106.	0.540561	0.270281	+	0.962782i	$0.412884\pi$
$642$	0	0
$643$	322522.	0.780078	0.390039	−	0.920798i	$-0.372462\pi$
$643$	322522.	0.780078	0.390039	+	0.920798i	$0.372462\pi$
$644$	0	0
$645$	− 1.45708e6i	− 3.50239i
$646$	0	0
$647$	− 583042.i	− 1.39281i	−0.717650	−	0.696404i	$-0.754782\pi$
$647$	− 583042.i	− 1.39281i	0.717650	−	0.696404i	$-0.245218\pi$
$648$	0	0
$649$	225707.	0.535866
$650$	0	0
$651$	702451.	1.65750
$652$	0	0
$653$	227229.i	0.532890i	0.963850	+	0.266445i	$0.0858491\pi$
$653$	227229.i	0.532890i	−0.963850	+	0.266445i	$0.914151\pi$
$654$	0	0
$655$	− 276200.i	− 0.643784i
$656$	0	0
$657$	−1.65723e6	−3.83929
$658$	0	0
$659$	−115150.	−0.265151	−0.132575	−	0.991173i	$-0.542325\pi$
$659$	−115150.	−0.265151	−0.132575	+	0.991173i	$0.542325\pi$
$660$	0	0
$661$	284004.i	0.650012i	0.945712	+	0.325006i	$0.105366\pi$
$661$	284004.i	0.650012i	−0.945712	+	0.325006i	$0.894634\pi$
$662$	0	0
$663$	− 465439.i	− 1.05885i
$664$	0	0
$665$	463425.	1.04794
$666$	0	0
$667$	−104057.	−0.233894
$668$	0	0
$669$	1.33742e6i	2.98825i
$670$	0	0
$671$	− 58616.9i	− 0.130190i
$672$	0	0
$673$	−390379.	−0.861899	−0.430950	−	0.902376i	$-0.641821\pi$
$673$	−390379.	−0.861899	−0.430950	+	0.902376i	$0.641821\pi$
$674$	0	0
$675$	−1.47838e6	−3.24474
$676$	0	0
$677$	329692.i	0.719335i	0.933081	+	0.359667i	$0.117110\pi$
$677$	329692.i	0.719335i	−0.933081	+	0.359667i	$0.882890\pi$
$678$	0	0
$679$	− 298103.i	− 0.646587i
$680$	0	0
$681$	320280.	0.690614
$682$	0	0
$683$	7064.69	0.0151444	0.00757220	−	0.999971i	$-0.497590\pi$
$683$	7064.69	0.0151444	0.00757220	+	0.999971i	$0.497590\pi$
$684$	0	0
$685$	290966.i	0.620100i
$686$	0	0
$687$	− 1.20325e6i	− 2.54943i
$688$	0	0
$689$	723127.	1.52327
$690$	0	0
$691$	−444613.	−0.931165	−0.465582	−	0.885005i	$-0.654155\pi$
$691$	−444613.	−0.931165	−0.465582	+	0.885005i	$0.654155\pi$
$692$	0	0
$693$	− 298471.i	− 0.621492i
$694$	0	0
$695$	923994.i	1.91293i
$696$	0	0
$697$	−187321.	−0.385586
$698$	0	0
$699$	18103.9	0.0370525
$700$	0	0
$701$	245215.i	0.499011i	0.968373	+	0.249506i	$0.0802681\pi$
$701$	245215.i	0.499011i	−0.968373	+	0.249506i	$0.919732\pi$
$702$	0	0
$703$	− 206458.i	− 0.417754i
$704$	0	0
$705$	1.81284e6	3.64738
$706$	0	0
$707$	201798.	0.403718
$708$	0	0
$709$	530467.i	1.05527i	0.849470	+	0.527637i	$0.176922\pi$
$709$	530467.i	1.05527i	−0.849470	+	0.527637i	$0.823078\pi$
$710$	0	0
$711$	931310.i	1.84228i
$712$	0	0
$713$	271359.	0.533784
$714$	0	0
$715$	−327191.	−0.640014
$716$	0	0
$717$	− 1.50470e6i	− 2.92692i
$718$	0	0
$719$	266398.i	0.515316i	0.966236	+	0.257658i	$0.0829508\pi$
$719$	266398.i	0.515316i	−0.966236	+	0.257658i	$0.917049\pi$
$720$	0	0
$721$	−449903.	−0.865463
$722$	0	0
$723$	474330.	0.907412
$724$	0	0
$725$	434508.i	0.826649i
$726$	0	0
$727$	− 546127.i	− 1.03330i	−0.856198	−	0.516649i	$-0.827179\pi$
$727$	− 546127.i	− 1.03330i	0.856198	−	0.516649i	$-0.172821\pi$
$728$	0	0
$729$	−343216.	−0.645821
$730$	0	0
$731$	354104.	0.662668
$732$	0	0
$733$	611757.i	1.13860i	0.822130	+	0.569300i	$0.192786\pi$
$733$	611757.i	1.13860i	−0.822130	+	0.569300i	$0.807214\pi$
$734$	0	0
$735$	477269.i	0.883464i
$736$	0	0
$737$	−13595.6	−0.0250301
$738$	0	0
$739$	−60509.1	−0.110798	−0.0553990	−	0.998464i	$-0.517643\pi$
$739$	−60509.1	−0.110798	−0.0553990	+	0.998464i	$0.517643\pi$
$740$	0	0
$741$	822603.i	1.49814i
$742$	0	0
$743$	− 143880.i	− 0.260628i	−0.991473	−	0.130314i	$-0.958401\pi$
$743$	− 143880.i	− 0.260628i	0.991473	−	0.130314i	$-0.0415986\pi$
$744$	0	0
$745$	222724.	0.401287
$746$	0	0
$747$	−1.80858e6	−3.24113
$748$	0	0
$749$	174868.i	0.311707i
$750$	0	0
$751$	− 358084.i	− 0.634900i	−0.948275	−	0.317450i	$-0.897173\pi$
$751$	− 358084.i	− 0.634900i	0.948275	−	0.317450i	$-0.102827\pi$
$752$	0	0
$753$	607913.	1.07214
$754$	0	0
$755$	−932853.	−1.63651
$756$	0	0
$757$	309368.i	0.539862i	0.962880	+	0.269931i	$0.0870010\pi$
$757$	309368.i	0.539862i	−0.962880	+	0.269931i	$0.912999\pi$
$758$	0	0
$759$	− 170116.i	− 0.295298i
$760$	0	0
$761$	−922637.	−1.59317	−0.796584	−	0.604527i	$-0.793362\pi$
$761$	−922637.	−1.59317	−0.796584	+	0.604527i	$0.793362\pi$
$762$	0	0
$763$	−432446.	−0.742818
$764$	0	0
$765$	− 1.09519e6i	− 1.87139i
$766$	0	0
$767$	980368.i	1.66647i
$768$	0	0
$769$	301222.	0.509370	0.254685	−	0.967024i	$-0.418028\pi$
$769$	301222.	0.509370	0.254685	+	0.967024i	$0.418028\pi$
$770$	0	0
$771$	1.75864e6	2.95848
$772$	0	0
$773$	− 460864.i	− 0.771282i	−0.922649	−	0.385641i	$-0.873980\pi$
$773$	− 460864.i	− 0.771282i	0.922649	−	0.385641i	$-0.126020\pi$
$774$	0	0
$775$	− 1.13311e6i	− 1.88655i
$776$	0	0
$777$	−480065.	−0.795167
$778$	0	0
$779$	331066.	0.545556
$780$	0	0
$781$	− 385187.i	− 0.631495i
$782$	0	0
$783$	− 590194.i	− 0.962656i
$784$	0	0
$785$	−627560.	−1.01839
$786$	0	0
$787$	−663401.	−1.07109	−0.535546	−	0.844506i	$-0.679894\pi$
$787$	−663401.	−1.07109	−0.535546	+	0.844506i	$0.679894\pi$
$788$	0	0
$789$	− 485799.i	− 0.780374i
$790$	0	0
$791$	22137.9i	0.0353821i
$792$	0	0
$793$	254605.	0.404874
$794$	0	0
$795$	2.51047e6	3.97210
$796$	0	0
$797$	− 946927.i	− 1.49073i	−0.666655	−	0.745367i	$-0.732274\pi$
$797$	− 946927.i	− 1.49073i	0.666655	−	0.745367i	$-0.267726\pi$
$798$	0	0
$799$	440561.i	0.690101i
$800$	0	0
$801$	−1.02358e6	−1.59535
$802$	0	0
$803$	−417717.	−0.647815
$804$	0	0
$805$	− 416272.i	− 0.642371i
$806$	0	0
$807$	− 100478.i	− 0.154285i
$808$	0	0
$809$	−144946.	−0.221466	−0.110733	−	0.993850i	$-0.535320\pi$
$809$	−144946.	−0.221466	−0.110733	+	0.993850i	$0.535320\pi$
$810$	0	0
$811$	569764.	0.866270	0.433135	−	0.901329i	$-0.357407\pi$
$811$	569764.	0.866270	0.433135	+	0.901329i	$0.357407\pi$
$812$	0	0
$813$	1.23202e6i	1.86397i
$814$	0	0
$815$	1.64798e6i	2.48106i
$816$	0	0
$817$	−625833.	−0.937593
$818$	0	0
$819$	1.29642e6	1.93276
$820$	0	0
$821$	82359.5i	0.122188i	0.998132	+	0.0610938i	$0.0194589\pi$
$821$	82359.5i	0.122188i	−0.998132	+	0.0610938i	$0.980541\pi$
$822$	0	0
$823$	− 540722.i	− 0.798315i	−0.916882	−	0.399157i	$-0.869303\pi$
$823$	− 540722.i	− 0.798315i	0.916882	−	0.399157i	$-0.130697\pi$
$824$	0	0
$825$	−710347.	−1.04367
$826$	0	0
$827$	266668.	0.389905	0.194953	−	0.980813i	$-0.437545\pi$
$827$	266668.	0.389905	0.194953	+	0.980813i	$0.437545\pi$
$828$	0	0
$829$	− 912028.i	− 1.32709i	−0.748138	−	0.663543i	$-0.769052\pi$
$829$	− 912028.i	− 1.32709i	0.748138	−	0.663543i	$-0.230948\pi$
$830$	0	0
$831$	811340.i	1.17490i
$832$	0	0
$833$	−115987.	−0.167155
$834$	0	0
$835$	1.12382e6	1.61185
$836$	0	0
$837$	1.53911e6i	2.19694i
$838$	0	0
$839$	82205.2i	0.116782i	0.998294	+	0.0583909i	$0.0185970\pi$
$839$	82205.2i	0.116782i	−0.998294	+	0.0583909i	$0.981403\pi$
$840$	0	0
$841$	533819.	0.754748
$842$	0	0
$843$	−1.06419e6	−1.49749
$844$	0	0
$845$	− 254610.i	− 0.356584i
$846$	0	0
$847$	522006.i	0.727627i
$848$	0	0
$849$	312560.	0.433628
$850$	0	0
$851$	−185451.	−0.256076
$852$	0	0
$853$	− 1.07479e6i	− 1.47715i	−0.674171	−	0.738576i	$-0.735499\pi$
$853$	− 1.07479e6i	− 1.47715i	0.674171	−	0.738576i	$-0.264501\pi$
$854$	0	0
$855$	1.93560e6i	2.64779i
$856$	0	0
$857$	1.33792e6	1.82167	0.910833	−	0.412776i	$-0.135441\pi$
$857$	1.33792e6	1.82167	0.910833	+	0.412776i	$0.135441\pi$
$858$	0	0
$859$	203953.	0.276403	0.138202	−	0.990404i	$-0.455868\pi$
$859$	203953.	0.276403	0.138202	+	0.990404i	$0.455868\pi$
$860$	0	0
$861$	− 769811.i	− 1.03843i
$862$	0	0
$863$	− 1.11902e6i	− 1.50250i	−0.660016	−	0.751251i	$-0.729451\pi$
$863$	− 1.11902e6i	− 1.50250i	0.660016	−	0.751251i	$-0.270549\pi$
$864$	0	0
$865$	−365683.	−0.488735
$866$	0	0
$867$	−931527.	−1.23925
$868$	0	0
$869$	234744.i	0.310853i
$870$	0	0
$871$	− 59053.0i	− 0.0778405i
$872$	0	0
$873$	1.24510e6	1.63371
$874$	0	0
$875$	−696886.	−0.910218
$876$	0	0
$877$	− 370325.i	− 0.481486i	−0.970589	−	0.240743i	$-0.922609\pi$
$877$	− 370325.i	− 0.481486i	0.970589	−	0.240743i	$-0.0773912\pi$
$878$	0	0
$879$	934115.i	1.20899i
$880$	0	0
$881$	1.03969e6	1.33953	0.669765	−	0.742573i	$-0.266395\pi$
$881$	1.03969e6	1.33953	0.669765	+	0.742573i	$0.266395\pi$
$882$	0	0
$883$	−884732.	−1.13472	−0.567362	−	0.823469i	$-0.692036\pi$
$883$	−884732.	−1.13472	−0.567362	+	0.823469i	$0.692036\pi$
$884$	0	0
$885$	3.40353e6i	4.34553i
$886$	0	0
$887$	− 530395.i	− 0.674143i	−0.941479	−	0.337071i	$-0.890564\pi$
$887$	− 530395.i	− 0.674143i	0.941479	−	0.337071i	$-0.109436\pi$
$888$	0	0
$889$	600097.	0.759308
$890$	0	0
$891$	372202.	0.468839
$892$	0	0
$893$	− 778634.i	− 0.976406i
$894$	0	0
$895$	115700.i	0.144440i
$896$	0	0
$897$	738904.	0.918339
$898$	0	0
$899$	452354.	0.559705
$900$	0	0
$901$	610101.i	0.751540i
$902$	0	0
$903$	1.45522e6i	1.78465i
$904$	0	0
$905$	−2.35931e6	−2.88063
$906$	0	0
$907$	402345.	0.489085	0.244542	−	0.969639i	$-0.421362\pi$
$907$	402345.	0.489085	0.244542	+	0.969639i	$0.421362\pi$
$908$	0	0
$909$	842854.i	1.02006i
$910$	0	0
$911$	− 180674.i	− 0.217700i	−0.994058	−	0.108850i	$-0.965283\pi$
$911$	− 180674.i	− 0.217700i	0.994058	−	0.108850i	$-0.0347168\pi$
$912$	0	0
$913$	−455867.	−0.546885
$914$	0	0
$915$	883906.	1.05576
$916$	0	0
$917$	275846.i	0.328041i
$918$	0	0
$919$	191824.i	0.227128i	0.993531	+	0.113564i	$0.0362267\pi$
$919$	191824.i	0.227128i	−0.993531	+	0.113564i	$0.963773\pi$
$920$	0	0
$921$	1.01509e6	1.19670
$922$	0	0
$923$	1.67308e6	1.96387
$924$	0	0
$925$	774382.i	0.905049i
$926$	0	0
$927$	− 1.87912e6i	− 2.18673i
$928$	0	0
$929$	−1.42402e6	−1.65000	−0.824999	−	0.565134i	$-0.808824\pi$
$929$	−1.42402e6	−1.65000	−0.824999	+	0.565134i	$0.808824\pi$
$930$	0	0
$931$	204993.	0.236504
$932$	0	0
$933$	− 96997.5i	− 0.111429i
$934$	0	0
$935$	− 276050.i	− 0.315766i
$936$	0	0
$937$	292158.	0.332765	0.166383	−	0.986061i	$-0.446791\pi$
$937$	292158.	0.332765	0.166383	+	0.986061i	$0.446791\pi$
$938$	0	0
$939$	−148962.	−0.168945
$940$	0	0
$941$	726562.i	0.820528i	0.911967	+	0.410264i	$0.134563\pi$
$941$	726562.i	0.820528i	−0.911967	+	0.410264i	$0.865437\pi$
$942$	0	0
$943$	− 297380.i	− 0.334417i
$944$	0	0
$945$	2.36103e6	2.64386
$946$	0	0
$947$	568295.	0.633686	0.316843	−	0.948478i	$-0.397377\pi$
$947$	568295.	0.633686	0.316843	+	0.948478i	$0.397377\pi$
$948$	0	0
$949$	− 1.81437e6i	− 2.01462i
$950$	0	0
$951$	702460.i	0.776713i
$952$	0	0
$953$	−289189.	−0.318417	−0.159208	−	0.987245i	$-0.550894\pi$
$953$	−289189.	−0.318417	−0.159208	+	0.987245i	$0.550894\pi$
$954$	0	0
$955$	−1.36810e6	−1.50006
$956$	0	0
$957$	− 283582.i	− 0.309638i
$958$	0	0
$959$	− 290594.i	− 0.315973i
$960$	0	0
$961$	−256127.	−0.277337
$962$	0	0
$963$	−730376.	−0.787579
$964$	0	0
$965$	− 1.29747e6i	− 1.39330i
$966$	0	0
$967$	925276.i	0.989505i	0.869034	+	0.494753i	$0.164741\pi$
$967$	925276.i	0.989505i	−0.869034	+	0.494753i	$0.835259\pi$
$968$	0	0
$969$	−694028.	−0.739144
$970$	0	0
$971$	73831.0	0.0783069	0.0391535	−	0.999233i	$-0.487534\pi$
$971$	73831.0	0.0783069	0.0391535	+	0.999233i	$0.487534\pi$
$972$	0	0
$973$	− 922812.i	− 0.974737i
$974$	0	0
$975$	− 3.08542e6i	− 3.24568i
$976$	0	0
$977$	836035.	0.875861	0.437931	−	0.899009i	$-0.355712\pi$
$977$	836035.	0.875861	0.437931	+	0.899009i	$0.355712\pi$
$978$	0	0
$979$	−258000.	−0.269187
$980$	0	0
$981$	− 1.80621e6i	− 1.87685i
$982$	0	0
$983$	− 1.56021e6i	− 1.61464i	−0.590115	−	0.807319i	$-0.700918\pi$
$983$	− 1.56021e6i	− 1.61464i	0.590115	−	0.807319i	$-0.299082\pi$
$984$	0	0
$985$	−2.68941e6	−2.77195
$986$	0	0
$987$	−1.81052e6	−1.85853
$988$	0	0
$989$	562155.i	0.574730i
$990$	0	0
$991$	1.04005e6i	1.05903i	0.848300	+	0.529515i	$0.177626\pi$
$991$	1.04005e6i	1.05903i	−0.848300	+	0.529515i	$0.822374\pi$
$992$	0	0
$993$	−1.08387e6	−1.09921
$994$	0	0
$995$	1.37832e6	1.39220
$996$	0	0
$997$	1.64701e6i	1.65694i	0.560033	+	0.828470i	$0.310788\pi$
$997$	1.64701e6i	1.65694i	−0.560033	+	0.828470i	$0.689212\pi$
$998$	0	0
$999$	− 1.05185e6i	− 1.05395i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	128.5.d.d.63.8	yes	8
3.2	odd	2		1152.5.b.l.703.1		8
4.3	odd	2	inner	128.5.d.d.63.2	yes	8
8.3	odd	2	inner	128.5.d.d.63.7	yes	8
8.5	even	2	inner	128.5.d.d.63.1	✓	8
12.11	even	2		1152.5.b.l.703.2		8
16.3	odd	4		256.5.c.k.255.1		4
16.5	even	4		256.5.c.g.255.1		4
16.11	odd	4		256.5.c.g.255.4		4
16.13	even	4		256.5.c.k.255.4		4
24.5	odd	2		1152.5.b.l.703.7		8
24.11	even	2		1152.5.b.l.703.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.5.d.d.63.1	✓	8	8.5	even	2	inner
128.5.d.d.63.2	yes	8	4.3	odd	2	inner
128.5.d.d.63.7	yes	8	8.3	odd	2	inner
128.5.d.d.63.8	yes	8	1.1	even	1	trivial
256.5.c.g.255.1		4	16.5	even	4
256.5.c.g.255.4		4	16.11	odd	4
256.5.c.k.255.1		4	16.3	odd	4
256.5.c.k.255.4		4	16.13	even	4
1152.5.b.l.703.1		8	3.2	odd	2
1152.5.b.l.703.2		8	12.11	even	2
1152.5.b.l.703.7		8	24.5	odd	2
1152.5.b.l.703.8		8	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:	\( N \)	\(=\)	\( 128 = 2^{7} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	128.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+15.8549 q^{3} +40.8444i q^{5} -40.7922i q^{7} +170.378 q^{9} +O(q^{10})\)	\(q+15.8549 q^{3} +40.8444i q^{5} -40.7922i q^{7} +170.378 q^{9} +42.9450 q^{11} +186.533i q^{13} +647.584i q^{15} -157.378 q^{17} +278.145 q^{19} -646.755i q^{21} -249.844i q^{23} -1043.27 q^{25} +1417.07 q^{27} -416.488i q^{29} +1086.12i q^{31} +680.888 q^{33} +1666.13 q^{35} -742.267i q^{37} +2957.46i q^{39} +1190.27 q^{41} -2250.03 q^{43} +6958.97i q^{45} -2799.39i q^{47} +737.000 q^{49} -2495.21 q^{51} -3876.67i q^{53} +1754.06i q^{55} +4409.95 q^{57} +5255.73 q^{59} -1364.93i q^{61} -6950.07i q^{63} -7618.84 q^{65} -316.581 q^{67} -3961.24i q^{69} -8969.32i q^{71} -9726.79 q^{73} -16540.9 q^{75} -1751.82i q^{77} +5466.15i q^{79} +8666.95 q^{81} -10615.1 q^{83} -6428.00i q^{85} -6603.37i q^{87} -6007.69 q^{89} +7609.09 q^{91} +17220.3i q^{93} +11360.6i q^{95} +7307.85 q^{97} +7316.87 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 440 q^{9}+O(q^{10}) \) 8 * q + 440 * q^9	\( 8 q + 440 q^{9} - 336 q^{17} - 2808 q^{25} + 832 q^{33} + 3984 q^{41} + 5896 q^{49} + 8512 q^{57} - 29568 q^{65} - 19664 q^{73} + 42568 q^{81} - 26832 q^{89} - 21840 q^{97}+O(q^{100}) \) 8 * q + 440 * q^9 - 336 * q^17 - 2808 * q^25 + 832 * q^33 + 3984 * q^41 + 5896 * q^49 + 8512 * q^57 - 29568 * q^65 - 19664 * q^73 + 42568 * q^81 - 26832 * q^89 - 21840 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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