LMFDB - Embedded newform 128.5.d.d.63.1

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			63.1
Root			$1.62831 + 1.62831i$ of defining polynomial
Character	$\chi$	$=$	128.63
Dual form			128.5.d.d.63.2

$q$-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	$-0.843013\pi$
$3$	−15.8549	−1.76165	−0.880827	+	0.473438i	$0.843013\pi$
$4$	0	0
$5$	− 40.8444i	− 1.63378i	−0.576796	−	0.816888i	$-0.695697\pi$
$5$	− 40.8444i	− 1.63378i	0.576796	−	0.816888i	$-0.304303\pi$
$6$	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	$-0.556788\pi$
$11$	−42.9450	−0.354917	−0.177459	+	0.984128i	$0.556788\pi$
$12$	0	0
$13$	− 186.533i	− 1.10375i	−0.833928	−	0.551873i	$-0.813913\pi$
$13$	− 186.533i	− 1.10375i	0.833928	−	0.551873i	$-0.186087\pi$
$14$	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	$-0.625882\pi$
$19$	−278.145	−0.770483	−0.385242	+	0.922816i	$0.625882\pi$
$20$	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	$0.0796467\pi$
$29$	416.488i	0.495229i	−0.968859	+	0.247615i	$0.920353\pi$
$30$	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	$0.0873869\pi$
$37$	742.267i	0.542197i	−0.962552	+	0.271098i	$0.912613\pi$
$38$	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	$-0.291794\pi$
$43$	2250.03	1.21689	0.608444	+	0.793597i	$0.291794\pi$
$44$	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	$0.242409\pi$
$53$	3876.67i	1.38009i	−0.723768	+	0.690044i	$0.757591\pi$
$54$	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	$-0.772323\pi$
$59$	−5255.73	−1.50983	−0.754917	+	0.655821i	$0.772323\pi$
$60$	0	0
$61$	1364.93i	0.366818i	0.983037	+	0.183409i	$0.0587132\pi$
$61$	1364.93i	0.366818i	−0.983037	+	0.183409i	$0.941287\pi$
$62$	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	$-0.488773\pi$
$67$	316.581	0.0705238	0.0352619	+	0.999378i	$0.488773\pi$
$68$	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	$-0.220036\pi$
$83$	10615.1	1.54088	0.770441	+	0.637512i	$0.220036\pi$
$84$	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	$-0.922041\pi$
$101$	− 4946.98i	− 0.484950i	0.970158	−	0.242475i	$-0.0779593\pi$
$102$	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	$-0.440054\pi$
$107$	4286.81	0.374426	0.187213	+	0.982319i	$0.440054\pi$
$108$	0	0
$109$	10601.2i	0.892281i	0.894963	+	0.446141i	$0.147202\pi$
$109$	10601.2i	0.892281i	−0.894963	+	0.446141i	$0.852798\pi$
$110$	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	$-0.436872\pi$
$131$	6762.24	0.394047	0.197023	+	0.980399i	$0.436872\pi$
$132$	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	$-0.699075\pi$
$139$	−22622.3	−1.17086	−0.585432	+	0.810721i	$0.699075\pi$
$140$	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	$0.0391905\pi$
$149$	5453.00i	0.245619i	−0.992430	+	0.122810i	$0.960810\pi$
$150$	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	$-0.899112\pi$
$157$	− 15364.6i	− 0.623337i	0.950191	−	0.311669i	$-0.100888\pi$
$158$	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	$-0.774460\pi$
$163$	−40347.8	−1.51861	−0.759303	+	0.650738i	$0.774460\pi$
$164$	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	$-0.952210\pi$
$173$	− 8953.08i	− 0.299144i	0.988751	−	0.149572i	$-0.0477896\pi$
$174$	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	$-0.514075\pi$
$179$	−2832.69	−0.0884084	−0.0442042	+	0.999023i	$0.514075\pi$
$180$	0	0
$181$	− 57763.3i	− 1.76317i	−0.472023	−	0.881586i	$-0.656476\pi$
$181$	− 57763.3i	− 1.76317i	0.472023	−	0.881586i	$-0.343524\pi$
$182$	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	$-0.677612\pi$
$197$	− 65845.3i	− 1.69665i	0.529477	−	0.848325i	$-0.322388\pi$
$198$	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	$-0.538789\pi$
$211$	−10823.7	−0.243115	−0.121558	+	0.992584i	$0.538789\pi$
$212$	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	$-0.562799\pi$
$227$	−20200.7	−0.392026	−0.196013	+	0.980601i	$0.562799\pi$
$228$	0	0
$229$	75891.6i	1.44718i	0.690230	+	0.723590i	$0.257509\pi$
$229$	75891.6i	1.44718i	−0.690230	+	0.723590i	$0.742491\pi$
$230$	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	$-0.598422\pi$
$251$	−38342.3	−0.608598	−0.304299	+	0.952577i	$0.598422\pi$
$252$	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	$0.0139432\pi$
$269$	6337.36i	0.0875798i	−0.999041	+	0.0437899i	$0.986057\pi$
$270$	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	$-0.891782\pi$
$277$	− 51172.8i	− 0.666930i	0.942763	−	0.333465i	$-0.108218\pi$
$278$	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	$-0.539275\pi$
$283$	−19713.8	−0.246148	−0.123074	+	0.992397i	$0.539275\pi$
$284$	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	$-0.888509\pi$
$293$	− 58916.5i	− 0.686281i	0.939284	−	0.343141i	$-0.111491\pi$
$294$	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	$-0.610310\pi$
$307$	−64023.9	−0.679305	−0.339653	+	0.940551i	$0.610310\pi$
$308$	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	$-0.929247\pi$
$317$	− 44305.6i	− 0.440900i	0.975398	−	0.220450i	$-0.0707525\pi$
$318$	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	$-0.399007\pi$
$331$	68362.0	0.623963	0.311981	+	0.950088i	$0.399007\pi$
$332$	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	$-0.391057\pi$
$347$	80821.6	0.671225	0.335613	+	0.942000i	$0.391057\pi$
$348$	0	0
$349$	29674.7i	0.243633i	0.992553	+	0.121817i	$0.0388719\pi$
$349$	29674.7i	0.243633i	−0.992553	+	0.121817i	$0.961128\pi$
$350$	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	$-0.895823\pi$
$373$	− 89451.2i	− 0.642937i	0.946920	−	0.321469i	$-0.104177\pi$
$374$	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	$-0.748420\pi$
$379$	−202128.	−1.40718	−0.703588	+	0.710609i	$0.748420\pi$
$380$	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	$-0.797083\pi$
$389$	− 180125.i	− 1.19035i	0.803596	−	0.595175i	$-0.202917\pi$
$390$	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	$-0.833784\pi$
$397$	− 157223.i	− 0.997548i	0.866732	−	0.498774i	$-0.166216\pi$
$398$	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	$-0.555671\pi$
$419$	−61097.6	−0.348014	−0.174007	+	0.984744i	$0.555671\pi$
$420$	0	0
$421$	254570.i	1.43629i	0.695893	+	0.718146i	$0.255009\pi$
$421$	254570.i	1.43629i	−0.695893	+	0.718146i	$0.744991\pi$
$422$	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	$-0.470179\pi$
$443$	36718.0	0.187099	0.0935494	+	0.995615i	$0.470179\pi$
$444$	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	$0.167272\pi$
$461$	213221.i	1.00329i	−0.865072	+	0.501647i	$0.832728\pi$
$462$	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	$-0.321335\pi$
$467$	232169.	1.06456	0.532281	+	0.846568i	$0.321335\pi$
$468$	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	$-0.866736\pi$
$491$	−440519.	−1.82727	−0.913633	+	0.406539i	$0.866736\pi$
$492$	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	$-0.329592\pi$
$499$	254053.	1.02029	0.510145	+	0.860088i	$0.329592\pi$
$500$	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	$-0.911846\pi$
$509$	− 141674.i	− 0.546832i	0.961896	−	0.273416i	$-0.0881536\pi$
$510$	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	$-0.815198\pi$
$523$	−457422.	−1.67230	−0.836148	+	0.548503i	$0.815198\pi$
$524$	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	$-0.818495\pi$
$541$	− 315987.i	− 1.07963i	0.841784	−	0.539814i	$-0.181505\pi$
$542$	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	$-0.282941\pi$
$547$	377170.	1.26056	0.630279	+	0.776369i	$0.282941\pi$
$548$	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	$0.148529\pi$
$557$	279142.i	0.899735i	−0.893095	+	0.449867i	$0.851471\pi$
$558$	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	$-0.617517\pi$
$563$	−228763.	−0.721719	−0.360860	+	0.932620i	$0.617517\pi$
$564$	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	$-0.303472\pi$
$571$	377507.	1.15785	0.578925	+	0.815381i	$0.303472\pi$
$572$	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	$-0.653443\pi$
$587$	−319485.	−0.927203	−0.463601	+	0.886044i	$0.653443\pi$
$588$	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	$0.282232\pi$
$613$	582414.i	1.54993i	−0.632007	+	0.774963i	$0.717768\pi$
$614$	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	$-0.220666\pi$
$619$	589437.	1.53835	0.769177	+	0.639035i	$0.220666\pi$
$620$	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	$-0.627538\pi$
$643$	−322522.	−0.780078	−0.390039	+	0.920798i	$0.627538\pi$
$644$	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	$-0.914151\pi$
$653$	− 227229.i	− 0.532890i	0.963850	−	0.266445i	$-0.0858491\pi$
$654$	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	$-0.457675\pi$
$659$	115150.	0.265151	0.132575	+	0.991173i	$0.457675\pi$
$660$	0	0
$661$	− 284004.i	− 0.650012i	−0.945712	−	0.325006i	$-0.894634\pi$
$661$	− 284004.i	− 0.650012i	0.945712	−	0.325006i	$-0.105366\pi$
$662$	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	$-0.882890\pi$
$677$	− 329692.i	− 0.719335i	0.933081	−	0.359667i	$-0.117110\pi$
$678$	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	$-0.502410\pi$
$683$	−7064.69	−0.0151444	−0.00757220	+	0.999971i	$0.502410\pi$
$684$	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	$-0.345845\pi$
$691$	444613.	0.931165	0.465582	+	0.885005i	$0.345845\pi$
$692$	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	$-0.919732\pi$
$701$	− 245215.i	− 0.499011i	0.968373	−	0.249506i	$-0.0802681\pi$
$702$	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	$-0.823078\pi$
$709$	− 530467.i	− 1.05527i	0.849470	−	0.527637i	$-0.176922\pi$
$710$	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	$-0.807214\pi$
$733$	− 611757.i	− 1.13860i	0.822130	−	0.569300i	$-0.192786\pi$
$734$	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	$-0.482357\pi$
$739$	60509.1	0.110798	0.0553990	+	0.998464i	$0.482357\pi$
$740$	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	$-0.912999\pi$
$757$	− 309368.i	− 0.539862i	0.962880	−	0.269931i	$-0.0870010\pi$
$758$	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	$0.126020\pi$
$773$	460864.i	0.771282i	−0.922649	+	0.385641i	$0.873980\pi$
$774$	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	$-0.320106\pi$
$787$	663401.	1.07109	0.535546	+	0.844506i	$0.320106\pi$
$788$	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	$0.267726\pi$
$797$	946927.i	1.49073i	−0.666655	+	0.745367i	$0.732274\pi$
$798$	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	$-0.642593\pi$
$811$	−569764.	−0.866270	−0.433135	+	0.901329i	$0.642593\pi$
$812$	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	$-0.980541\pi$
$821$	− 82359.5i	− 0.122188i	0.998132	−	0.0610938i	$-0.0194589\pi$
$822$	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	$-0.562455\pi$
$827$	−266668.	−0.389905	−0.194953	+	0.980813i	$0.562455\pi$
$828$	0	0
$829$	912028.i	1.32709i	0.748138	+	0.663543i	$0.230948\pi$
$829$	912028.i	1.32709i	−0.748138	+	0.663543i	$0.769052\pi$
$830$	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	$0.264501\pi$
$853$	1.07479e6i	1.47715i	−0.674171	+	0.738576i	$0.735499\pi$
$854$	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	$-0.544132\pi$
$859$	−203953.	−0.276403	−0.138202	+	0.990404i	$0.544132\pi$
$860$	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	$0.0773912\pi$
$877$	370325.i	0.481486i	−0.970589	+	0.240743i	$0.922609\pi$
$878$	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	$-0.307964\pi$
$883$	884732.	1.13472	0.567362	+	0.823469i	$0.307964\pi$
$884$	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	$-0.578638\pi$
$907$	−402345.	−0.489085	−0.244542	+	0.969639i	$0.578638\pi$
$908$	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	$-0.865437\pi$
$941$	− 726562.i	− 0.820528i	0.911967	−	0.410264i	$-0.134563\pi$
$942$	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	$-0.602623\pi$
$947$	−568295.	−0.633686	−0.316843	+	0.948478i	$0.602623\pi$
$948$	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	$-0.512466\pi$
$971$	−73831.0	−0.0783069	−0.0391535	+	0.999233i	$0.512466\pi$
$972$	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	$-0.689212\pi$
$997$	− 1.64701e6i	− 1.65694i	0.560033	−	0.828470i	$-0.310788\pi$
$998$	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.63.1	✓	8
3.2	odd	2		1152.5.b.l.703.7		8
4.3	odd	2	inner	128.5.d.d.63.7	yes	8
8.3	odd	2	inner	128.5.d.d.63.2	yes	8
8.5	even	2	inner	128.5.d.d.63.8	yes	8
12.11	even	2		1152.5.b.l.703.8		8
16.3	odd	4		256.5.c.g.255.4		4
16.5	even	4		256.5.c.k.255.4		4
16.11	odd	4		256.5.c.k.255.1		4
16.13	even	4		256.5.c.g.255.1		4
24.5	odd	2		1152.5.b.l.703.1		8
24.11	even	2		1152.5.b.l.703.2		8

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