LMFDB - Embedded newform 128.5.c.b.127.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [128,5,Mod(127,128)]

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, 0]))

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

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

chi := DirichletCharacter("128.127");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 128 = 2^{7} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	128.c (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:	8.0.205520896.4
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 12x^{6} - 12x^{5} - 8x^{4} + 12x^{3} + 12x^{2} + 4x + 1 $ x^8 - 4x^7 + 12x^6 - 12x^5 - 8x^4 + 12x^3 + 12x^2 + 4*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{39} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.8
Root			$1.55050 + 0.642239i$ of defining polynomial
Character	$\chi$	$=$	128.127
Dual form			128.5.c.b.127.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+16.7135i q^{3} +10.1891 q^{5} +80.4325i q^{7} -198.341 q^{9} +O(q^{10})$	$q+16.7135i q^{3} +10.1891 q^{5} +80.4325i q^{7} -198.341 q^{9} +21.6419i q^{11} +179.095 q^{13} +170.295i q^{15} +148.472 q^{17} -322.535i q^{19} -1344.31 q^{21} +327.187i q^{23} -521.183 q^{25} -1961.18i q^{27} -699.401 q^{29} -1042.58i q^{31} -361.712 q^{33} +819.532i q^{35} +1145.75 q^{37} +2993.31i q^{39} -792.708 q^{41} +231.755i q^{43} -2020.91 q^{45} +2963.98i q^{47} -4068.39 q^{49} +2481.48i q^{51} +4672.55 q^{53} +220.510i q^{55} +5390.68 q^{57} -1055.65i q^{59} +2911.64 q^{61} -15953.1i q^{63} +1824.81 q^{65} +4623.44i q^{67} -5468.45 q^{69} +2548.28i q^{71} +4928.49 q^{73} -8710.79i q^{75} -1740.71 q^{77} +4356.41i q^{79} +16712.5 q^{81} +5655.05i q^{83} +1512.79 q^{85} -11689.4i q^{87} -12202.4 q^{89} +14405.1i q^{91} +17425.2 q^{93} -3286.32i q^{95} +1193.17 q^{97} -4292.47i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 48 q^{5} - 216 q^{9}+O(q^{10}) $ 8 * q + 48 * q^5 - 216 * q^9	$ 8 q + 48 q^{5} - 216 q^{9} + 240 q^{13} + 240 q^{17} - 1216 q^{21} + 664 q^{25} + 432 q^{29} + 992 q^{33} + 2800 q^{37} - 2928 q^{41} - 4880 q^{45} - 5752 q^{49} + 1776 q^{53} + 8608 q^{57} + 12656 q^{61} + 672 q^{65} - 4416 q^{69} + 560 q^{73} - 31296 q^{77} + 14696 q^{81} - 14432 q^{85} - 22992 q^{89} + 56320 q^{93} - 3728 q^{97}+O(q^{100}) $ 8 * q + 48 * q^5 - 216 * q^9 + 240 * q^13 + 240 * q^17 - 1216 * q^21 + 664 * q^25 + 432 * q^29 + 992 * q^33 + 2800 * q^37 - 2928 * q^41 - 4880 * q^45 - 5752 * q^49 + 1776 * q^53 + 8608 * q^57 + 12656 * q^61 + 672 * q^65 - 4416 * q^69 + 560 * q^73 - 31296 * q^77 + 14696 * q^81 - 14432 * q^85 - 22992 * q^89 + 56320 * q^93 - 3728 * 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
$3$	16.7135i	1.85705i	0.371264	+	0.928527i	$0.378925\pi$
$3$	16.7135i	1.85705i	−0.371264	+	0.928527i	$0.621075\pi$
$4$	0	0
$5$	10.1891	0.407562	0.203781	−	0.979016i	$-0.434677\pi$
$5$	10.1891	0.407562	0.203781	+	0.979016i	$0.434677\pi$
$6$	0	0
$7$	80.4325i	1.64148i	0.571302	+	0.820740i	$0.306439\pi$
$7$	80.4325i	1.64148i	−0.571302	+	0.820740i	$0.693561\pi$
$8$	0	0
$9$	−198.341	−2.44865
$10$	0	0
$11$	21.6419i	0.178859i	0.995993	+	0.0894293i	$0.0285043\pi$
$11$	21.6419i	0.178859i	−0.995993	+	0.0894293i	$0.971496\pi$
$12$	0	0
$13$	179.095	1.05974	0.529868	−	0.848080i	$-0.322242\pi$
$13$	179.095	1.05974	0.529868	+	0.848080i	$0.322242\pi$
$14$	0	0
$15$	170.295i	0.756866i
$16$	0	0
$17$	148.472	0.513743	0.256871	−	0.966446i	$-0.417308\pi$
$17$	148.472	0.513743	0.256871	+	0.966446i	$0.417308\pi$
$18$	0	0
$19$	− 322.535i	− 0.893447i	−0.894672	−	0.446724i	$-0.852591\pi$
$19$	− 322.535i	− 0.893447i	0.894672	−	0.446724i	$-0.147409\pi$
$20$	0	0
$21$	−1344.31	−3.04832
$22$	0	0
$23$	327.187i	0.618502i	0.950980	+	0.309251i	$0.100078\pi$
$23$	327.187i	0.618502i	−0.950980	+	0.309251i	$0.899922\pi$
$24$	0	0
$25$	−521.183	−0.833893
$26$	0	0
$27$	− 1961.18i	− 2.69023i
$28$	0	0
$29$	−699.401	−0.831630	−0.415815	−	0.909449i	$-0.636504\pi$
$29$	−699.401	−0.831630	−0.415815	+	0.909449i	$0.636504\pi$
$30$	0	0
$31$	− 1042.58i	− 1.08489i	−0.840090	−	0.542447i	$-0.817498\pi$
$31$	− 1042.58i	− 1.08489i	0.840090	−	0.542447i	$-0.182502\pi$
$32$	0	0
$33$	−361.712	−0.332150
$34$	0	0
$35$	819.532i	0.669005i
$36$	0	0
$37$	1145.75	0.836921	0.418461	−	0.908235i	$-0.362570\pi$
$37$	1145.75	0.836921	0.418461	+	0.908235i	$0.362570\pi$
$38$	0	0
$39$	2993.31i	1.96799i
$40$	0	0
$41$	−792.708	−0.471569	−0.235785	−	0.971805i	$-0.575766\pi$
$41$	−792.708	−0.471569	−0.235785	+	0.971805i	$0.575766\pi$
$42$	0	0
$43$	231.755i	0.125341i	0.998034	+	0.0626704i	$0.0199617\pi$
$43$	231.755i	0.125341i	−0.998034	+	0.0626704i	$0.980038\pi$
$44$	0	0
$45$	−2020.91	−0.997979
$46$	0	0
$47$	2963.98i	1.34177i	0.741560	+	0.670887i	$0.234086\pi$
$47$	2963.98i	1.34177i	−0.741560	+	0.670887i	$0.765914\pi$
$48$	0	0
$49$	−4068.39	−1.69446
$50$	0	0
$51$	2481.48i	0.954048i
$52$	0	0
$53$	4672.55	1.66342	0.831711	−	0.555209i	$-0.187362\pi$
$53$	4672.55	1.66342	0.831711	+	0.555209i	$0.187362\pi$
$54$	0	0
$55$	220.510i	0.0728960i
$56$	0	0
$57$	5390.68	1.65918
$58$	0	0
$59$	− 1055.65i	− 0.303262i	−0.988437	−	0.151631i	$-0.951547\pi$
$59$	− 1055.65i	− 0.303262i	0.988437	−	0.151631i	$-0.0484525\pi$
$60$	0	0
$61$	2911.64	0.782488	0.391244	−	0.920287i	$-0.372045\pi$
$61$	2911.64	0.782488	0.391244	+	0.920287i	$0.372045\pi$
$62$	0	0
$63$	− 15953.1i	− 4.01942i
$64$	0	0
$65$	1824.81	0.431908
$66$	0	0
$67$	4623.44i	1.02995i	0.857206	+	0.514974i	$0.172198\pi$
$67$	4623.44i	1.02995i	−0.857206	+	0.514974i	$0.827802\pi$
$68$	0	0
$69$	−5468.45	−1.14859
$70$	0	0
$71$	2548.28i	0.505511i	0.967530	+	0.252756i	$0.0813368\pi$
$71$	2548.28i	0.505511i	−0.967530	+	0.252756i	$0.918663\pi$
$72$	0	0
$73$	4928.49	0.924843	0.462421	−	0.886660i	$-0.346981\pi$
$73$	4928.49	0.924843	0.462421	+	0.886660i	$0.346981\pi$
$74$	0	0
$75$	− 8710.79i	− 1.54859i
$76$	0	0
$77$	−1740.71	−0.293593
$78$	0	0
$79$	4356.41i	0.698031i	0.937117	+	0.349015i	$0.113484\pi$
$79$	4356.41i	0.698031i	−0.937117	+	0.349015i	$0.886516\pi$
$80$	0	0
$81$	16712.5	2.54725
$82$	0	0
$83$	5655.05i	0.820881i	0.911887	+	0.410441i	$0.134625\pi$
$83$	5655.05i	0.820881i	−0.911887	+	0.410441i	$0.865375\pi$
$84$	0	0
$85$	1512.79	0.209382
$86$	0	0
$87$	− 11689.4i	− 1.54438i
$88$	0	0
$89$	−12202.4	−1.54051	−0.770254	−	0.637737i	$-0.779871\pi$
$89$	−12202.4	−1.54051	−0.770254	+	0.637737i	$0.779871\pi$
$90$	0	0
$91$	14405.1i	1.73953i
$92$	0	0
$93$	17425.2	2.01471
$94$	0	0
$95$	− 3286.32i	− 0.364136i
$96$	0	0
$97$	1193.17	0.126811	0.0634057	−	0.997988i	$-0.479804\pi$
$97$	1193.17	0.126811	0.0634057	+	0.997988i	$0.479804\pi$
$98$	0	0
$99$	− 4292.47i	− 0.437963i
$100$	0	0
$101$	10516.4	1.03092	0.515461	−	0.856913i	$-0.327621\pi$
$101$	10516.4	1.03092	0.515461	+	0.856913i	$0.327621\pi$
$102$	0	0
$103$	9535.67i	0.898828i	0.893324	+	0.449414i	$0.148367\pi$
$103$	9535.67i	0.898828i	−0.893324	+	0.449414i	$0.851633\pi$
$104$	0	0
$105$	−13697.2	−1.24238
$106$	0	0
$107$	4262.42i	0.372296i	0.982522	+	0.186148i	$0.0596004\pi$
$107$	4262.42i	0.372296i	−0.982522	+	0.186148i	$0.940400\pi$
$108$	0	0
$109$	1271.33	0.107005	0.0535025	−	0.998568i	$-0.482961\pi$
$109$	1271.33	0.107005	0.0535025	+	0.998568i	$0.482961\pi$
$110$	0	0
$111$	19149.4i	1.55421i
$112$	0	0
$113$	−20599.9	−1.61327	−0.806637	−	0.591047i	$-0.798715\pi$
$113$	−20599.9	−1.61327	−0.806637	+	0.591047i	$0.798715\pi$
$114$	0	0
$115$	3333.73i	0.252078i
$116$	0	0
$117$	−35521.9	−2.59492
$118$	0	0
$119$	11941.9i	0.843298i
$120$	0	0
$121$	14172.6	0.968010
$122$	0	0
$123$	− 13248.9i	− 0.875730i
$124$	0	0
$125$	−11678.5	−0.747426
$126$	0	0
$127$	− 3740.54i	− 0.231914i	−0.993254	−	0.115957i	$-0.963007\pi$
$127$	− 3740.54i	− 0.231914i	0.993254	−	0.115957i	$-0.0369935\pi$
$128$	0	0
$129$	−3873.44	−0.232765
$130$	0	0
$131$	− 19766.1i	− 1.15180i	−0.817519	−	0.575901i	$-0.804651\pi$
$131$	− 19766.1i	− 1.15180i	0.817519	−	0.575901i	$-0.195349\pi$
$132$	0	0
$133$	25942.3	1.46658
$134$	0	0
$135$	− 19982.5i	− 1.09644i
$136$	0	0
$137$	25580.0	1.36288	0.681442	−	0.731872i	$-0.261353\pi$
$137$	25580.0	1.36288	0.681442	+	0.731872i	$0.261353\pi$
$138$	0	0
$139$	6090.33i	0.315218i	0.987502	+	0.157609i	$0.0503786\pi$
$139$	6090.33i	0.315218i	−0.987502	+	0.157609i	$0.949621\pi$
$140$	0	0
$141$	−49538.4	−2.49175
$142$	0	0
$143$	3875.96i	0.189543i
$144$	0	0
$145$	−7126.23	−0.338941
$146$	0	0
$147$	− 67997.0i	− 3.14670i
$148$	0	0
$149$	−1392.16	−0.0627070	−0.0313535	−	0.999508i	$-0.509982\pi$
$149$	−1392.16	−0.0627070	−0.0313535	+	0.999508i	$0.509982\pi$
$150$	0	0
$151$	− 27582.1i	− 1.20969i	−0.796343	−	0.604845i	$-0.793235\pi$
$151$	− 27582.1i	− 1.20969i	0.796343	−	0.604845i	$-0.206765\pi$
$152$	0	0
$153$	−29448.0	−1.25798
$154$	0	0
$155$	− 10622.9i	− 0.442162i
$156$	0	0
$157$	−3752.20	−0.152225	−0.0761126	−	0.997099i	$-0.524251\pi$
$157$	−3752.20	−0.152225	−0.0761126	+	0.997099i	$0.524251\pi$
$158$	0	0
$159$	78094.7i	3.08907i
$160$	0	0
$161$	−26316.5	−1.01526
$162$	0	0
$163$	6009.11i	0.226170i	0.993585	+	0.113085i	$0.0360732\pi$
$163$	6009.11i	0.226170i	−0.993585	+	0.113085i	$0.963927\pi$
$164$	0	0
$165$	−3685.50	−0.135372
$166$	0	0
$167$	− 5470.03i	− 0.196136i	−0.995180	−	0.0980679i	$-0.968734\pi$
$167$	− 5470.03i	− 0.196136i	0.995180	−	0.0980679i	$-0.0312662\pi$
$168$	0	0
$169$	3514.13	0.123039
$170$	0	0
$171$	63971.8i	2.18774i
$172$	0	0
$173$	−33724.9	−1.12683	−0.563415	−	0.826174i	$-0.690513\pi$
$173$	−33724.9	−1.12683	−0.563415	+	0.826174i	$0.690513\pi$
$174$	0	0
$175$	− 41920.1i	− 1.36882i
$176$	0	0
$177$	17643.7	0.563174
$178$	0	0
$179$	53202.0i	1.66043i	0.557441	+	0.830217i	$0.311783\pi$
$179$	53202.0i	1.66043i	−0.557441	+	0.830217i	$0.688217\pi$
$180$	0	0
$181$	41714.5	1.27330	0.636648	−	0.771154i	$-0.280320\pi$
$181$	41714.5	1.27330	0.636648	+	0.771154i	$0.280320\pi$
$182$	0	0
$183$	48663.7i	1.45312i
$184$	0	0
$185$	11674.1	0.341098
$186$	0	0
$187$	3213.20i	0.0918872i
$188$	0	0
$189$	157742.	4.41596
$190$	0	0
$191$	31457.1i	0.862287i	0.902283	+	0.431143i	$0.141890\pi$
$191$	31457.1i	0.862287i	−0.902283	+	0.431143i	$0.858110\pi$
$192$	0	0
$193$	18486.9	0.496307	0.248153	−	0.968721i	$-0.420176\pi$
$193$	18486.9	0.496307	0.248153	+	0.968721i	$0.420176\pi$
$194$	0	0
$195$	30499.0i	0.802077i
$196$	0	0
$197$	−49310.2	−1.27059	−0.635293	−	0.772271i	$-0.719121\pi$
$197$	−49310.2	−1.27059	−0.635293	+	0.772271i	$0.719121\pi$
$198$	0	0
$199$	− 59391.0i	− 1.49973i	−0.661589	−	0.749867i	$-0.730117\pi$
$199$	− 59391.0i	− 1.49973i	0.661589	−	0.749867i	$-0.269883\pi$
$200$	0	0
$201$	−77273.8	−1.91267
$202$	0	0
$203$	− 56254.6i	− 1.36510i
$204$	0	0
$205$	−8076.94	−0.192194
$206$	0	0
$207$	− 64894.7i	− 1.51450i
$208$	0	0
$209$	6980.25	0.159801
$210$	0	0
$211$	− 33767.5i	− 0.758463i	−0.925302	−	0.379231i	$-0.876188\pi$
$211$	− 33767.5i	− 0.758463i	0.925302	−	0.379231i	$-0.123812\pi$
$212$	0	0
$213$	−42590.7	−0.938762
$214$	0	0
$215$	2361.37i	0.0510842i
$216$	0	0
$217$	83857.6	1.78083
$218$	0	0
$219$	82372.2i	1.71748i
$220$	0	0
$221$	26590.6	0.544431
$222$	0	0
$223$	− 9229.31i	− 0.185592i	−0.995685	−	0.0927961i	$-0.970420\pi$
$223$	− 9229.31i	− 0.185592i	0.995685	−	0.0927961i	$-0.0295805\pi$
$224$	0	0
$225$	103372.	2.04191
$226$	0	0
$227$	− 67230.4i	− 1.30471i	−0.757914	−	0.652355i	$-0.773781\pi$
$227$	− 67230.4i	− 1.30471i	0.757914	−	0.652355i	$-0.226219\pi$
$228$	0	0
$229$	50287.7	0.958939	0.479469	−	0.877559i	$-0.340829\pi$
$229$	50287.7	0.958939	0.479469	+	0.877559i	$0.340829\pi$
$230$	0	0
$231$	− 29093.4i	− 0.545218i
$232$	0	0
$233$	52737.2	0.971416	0.485708	−	0.874121i	$-0.338562\pi$
$233$	52737.2	0.971416	0.485708	+	0.874121i	$0.338562\pi$
$234$	0	0
$235$	30200.1i	0.546856i
$236$	0	0
$237$	−72810.8	−1.29628
$238$	0	0
$239$	− 97480.3i	− 1.70656i	−0.521455	−	0.853279i	$-0.674611\pi$
$239$	− 97480.3i	− 1.70656i	0.521455	−	0.853279i	$-0.325389\pi$
$240$	0	0
$241$	97612.5	1.68063	0.840313	−	0.542101i	$-0.182371\pi$
$241$	97612.5	1.68063	0.840313	+	0.542101i	$0.182371\pi$
$242$	0	0
$243$	120469.i	2.04015i
$244$	0	0
$245$	−41453.1	−0.690597
$246$	0	0
$247$	− 57764.4i	− 0.946818i
$248$	0	0
$249$	−94515.7	−1.52442
$250$	0	0
$251$	− 92490.6i	− 1.46808i	−0.679106	−	0.734041i	$-0.737632\pi$
$251$	− 92490.6i	− 1.46808i	0.679106	−	0.734041i	$-0.262368\pi$
$252$	0	0
$253$	−7080.95	−0.110624
$254$	0	0
$255$	25283.9i	0.388834i
$256$	0	0
$257$	−84387.6	−1.27765	−0.638826	−	0.769351i	$-0.720580\pi$
$257$	−84387.6	−1.27765	−0.638826	+	0.769351i	$0.720580\pi$
$258$	0	0
$259$	92155.2i	1.37379i
$260$	0	0
$261$	138720.	2.03637
$262$	0	0
$263$	136592.i	1.97476i	0.158373	+	0.987379i	$0.449375\pi$
$263$	136592.i	1.97476i	−0.158373	+	0.987379i	$0.550625\pi$
$264$	0	0
$265$	47608.9	0.677948
$266$	0	0
$267$	− 203944.i	− 2.86081i
$268$	0	0
$269$	−54275.8	−0.750070	−0.375035	−	0.927011i	$-0.622369\pi$
$269$	−54275.8	−0.750070	−0.375035	+	0.927011i	$0.622369\pi$
$270$	0	0
$271$	− 14987.4i	− 0.204074i	−0.994781	−	0.102037i	$-0.967464\pi$
$271$	− 14987.4i	− 0.204074i	0.994781	−	0.102037i	$-0.0325359\pi$
$272$	0	0
$273$	−240759.	−3.23041
$274$	0	0
$275$	− 11279.4i	− 0.149149i
$276$	0	0
$277$	−26442.7	−0.344625	−0.172312	−	0.985042i	$-0.555124\pi$
$277$	−26442.7	−0.344625	−0.172312	+	0.985042i	$0.555124\pi$
$278$	0	0
$279$	206787.i	2.65653i
$280$	0	0
$281$	17733.6	0.224587	0.112293	−	0.993675i	$-0.464180\pi$
$281$	17733.6	0.224587	0.112293	+	0.993675i	$0.464180\pi$
$282$	0	0
$283$	106916.i	1.33497i	0.744624	+	0.667484i	$0.232629\pi$
$283$	106916.i	1.33497i	−0.744624	+	0.667484i	$0.767371\pi$
$284$	0	0
$285$	54925.9	0.676220
$286$	0	0
$287$	− 63759.5i	− 0.774071i
$288$	0	0
$289$	−61477.2	−0.736069
$290$	0	0
$291$	19942.0i	0.235496i
$292$	0	0
$293$	99532.1	1.15939	0.579693	−	0.814835i	$-0.303173\pi$
$293$	99532.1	1.15939	0.579693	+	0.814835i	$0.303173\pi$
$294$	0	0
$295$	− 10756.1i	− 0.123598i
$296$	0	0
$297$	42443.6	0.481170
$298$	0	0
$299$	58597.7i	0.655448i
$300$	0	0
$301$	−18640.6	−0.205744
$302$	0	0
$303$	175766.i	1.91448i
$304$	0	0
$305$	29666.9	0.318913
$306$	0	0
$307$	46693.1i	0.495422i	0.968834	+	0.247711i	$0.0796784\pi$
$307$	46693.1i	0.495422i	−0.968834	+	0.247711i	$0.920322\pi$
$308$	0	0
$309$	−159374.	−1.66917
$310$	0	0
$311$	− 21716.8i	− 0.224530i	−0.993678	−	0.112265i	$-0.964189\pi$
$311$	− 21716.8i	− 0.224530i	0.993678	−	0.112265i	$-0.0358106\pi$
$312$	0	0
$313$	55851.2	0.570091	0.285045	−	0.958514i	$-0.407991\pi$
$313$	55851.2	0.570091	0.285045	+	0.958514i	$0.407991\pi$
$314$	0	0
$315$	− 162547.i	− 1.63816i
$316$	0	0
$317$	−105442.	−1.04929	−0.524643	−	0.851322i	$-0.675801\pi$
$317$	−105442.	−1.04929	−0.524643	+	0.851322i	$0.675801\pi$
$318$	0	0
$319$	− 15136.3i	− 0.148744i
$320$	0	0
$321$	−71239.9	−0.691374
$322$	0	0
$323$	− 47887.2i	− 0.459002i
$324$	0	0
$325$	−93341.4	−0.883706
$326$	0	0
$327$	21248.3i	0.198714i
$328$	0	0
$329$	−238400.	−2.20249
$330$	0	0
$331$	− 22761.2i	− 0.207749i	−0.994590	−	0.103874i	$-0.966876\pi$
$331$	− 22761.2i	− 0.207749i	0.994590	−	0.103874i	$-0.0331240\pi$
$332$	0	0
$333$	−227248.	−2.04933
$334$	0	0
$335$	47108.5i	0.419768i
$336$	0	0
$337$	134406.	1.18348	0.591739	−	0.806130i	$-0.298442\pi$
$337$	134406.	1.18348	0.591739	+	0.806130i	$0.298442\pi$
$338$	0	0
$339$	− 344296.i	− 2.99594i
$340$	0	0
$341$	22563.5	0.194043
$342$	0	0
$343$	− 134112.i	− 1.13994i
$344$	0	0
$345$	−55718.3	−0.468123
$346$	0	0
$347$	− 211717.i	− 1.75832i	−0.476527	−	0.879160i	$-0.658105\pi$
$347$	− 211717.i	− 1.75832i	0.476527	−	0.879160i	$-0.341895\pi$
$348$	0	0
$349$	241146.	1.97984	0.989918	−	0.141638i	$-0.0452370\pi$
$349$	241146.	1.97984	0.989918	+	0.141638i	$0.0452370\pi$
$350$	0	0
$351$	− 351238.i	− 2.85093i
$352$	0	0
$353$	172104.	1.38115	0.690577	−	0.723259i	$-0.257357\pi$
$353$	172104.	1.38115	0.690577	+	0.723259i	$0.257357\pi$
$354$	0	0
$355$	25964.6i	0.206027i
$356$	0	0
$357$	−199592.	−1.56605
$358$	0	0
$359$	210237.i	1.63125i	0.578584	+	0.815623i	$0.303606\pi$
$359$	210237.i	1.63125i	−0.578584	+	0.815623i	$0.696394\pi$
$360$	0	0
$361$	26292.5	0.201752
$362$	0	0
$363$	236874.i	1.79765i
$364$	0	0
$365$	50216.6	0.376931
$366$	0	0
$367$	111614.i	0.828681i	0.910122	+	0.414341i	$0.135988\pi$
$367$	111614.i	0.828681i	−0.910122	+	0.414341i	$0.864012\pi$
$368$	0	0
$369$	157226.	1.15471
$370$	0	0
$371$	375825.i	2.73047i
$372$	0	0
$373$	173970.	1.25042	0.625211	−	0.780455i	$-0.285013\pi$
$373$	173970.	1.25042	0.625211	+	0.780455i	$0.285013\pi$
$374$	0	0
$375$	− 195189.i	− 1.38801i
$376$	0	0
$377$	−125259.	−0.881308
$378$	0	0
$379$	28736.9i	0.200060i	0.994984	+	0.100030i	$0.0318939\pi$
$379$	28736.9i	0.200060i	−0.994984	+	0.100030i	$0.968106\pi$
$380$	0	0
$381$	62517.5	0.430677
$382$	0	0
$383$	470.577i	0.00320799i	0.999999	+	0.00160399i	$0.000510568\pi$
$383$	470.577i	0.00320799i	−0.999999	+	0.00160399i	$0.999489\pi$
$384$	0	0
$385$	−17736.2	−0.119657
$386$	0	0
$387$	− 45966.5i	− 0.306916i
$388$	0	0
$389$	−185468.	−1.22566	−0.612831	−	0.790214i	$-0.709969\pi$
$389$	−185468.	−1.22566	−0.612831	+	0.790214i	$0.709969\pi$
$390$	0	0
$391$	48578.0i	0.317751i
$392$	0	0
$393$	330360.	2.13896
$394$	0	0
$395$	44387.7i	0.284491i
$396$	0	0
$397$	292155.	1.85367	0.926835	−	0.375469i	$-0.122518\pi$
$397$	292155.	1.85367	0.926835	+	0.375469i	$0.122518\pi$
$398$	0	0
$399$	433586.i	2.72351i
$400$	0	0
$401$	−113040.	−0.702982	−0.351491	−	0.936191i	$-0.614325\pi$
$401$	−113040.	−0.702982	−0.351491	+	0.936191i	$0.614325\pi$
$402$	0	0
$403$	− 186722.i	− 1.14970i
$404$	0	0
$405$	170285.	1.03816
$406$	0	0
$407$	24796.1i	0.149691i
$408$	0	0
$409$	−318260.	−1.90255	−0.951273	−	0.308350i	$-0.900223\pi$
$409$	−318260.	−1.90255	−0.951273	+	0.308350i	$0.900223\pi$
$410$	0	0
$411$	427531.i	2.53095i
$412$	0	0
$413$	84909.0	0.497798
$414$	0	0
$415$	57619.7i	0.334560i
$416$	0	0
$417$	−101791.	−0.585377
$418$	0	0
$419$	86121.4i	0.490549i	0.969454	+	0.245275i	$0.0788782\pi$
$419$	86121.4i	0.490549i	−0.969454	+	0.245275i	$0.921122\pi$
$420$	0	0
$421$	151053.	0.852244	0.426122	−	0.904666i	$-0.359879\pi$
$421$	151053.	0.852244	0.426122	+	0.904666i	$0.359879\pi$
$422$	0	0
$423$	− 587878.i	− 3.28554i
$424$	0	0
$425$	−77380.9	−0.428406
$426$	0	0
$427$	234191.i	1.28444i
$428$	0	0
$429$	−64780.8	−0.351991
$430$	0	0
$431$	− 122934.i	− 0.661785i	−0.943668	−	0.330893i	$-0.892650\pi$
$431$	− 122934.i	− 0.661785i	0.943668	−	0.330893i	$-0.107350\pi$
$432$	0	0
$433$	−204083.	−1.08851	−0.544255	−	0.838920i	$-0.683187\pi$
$433$	−204083.	−1.08851	−0.544255	+	0.838920i	$0.683187\pi$
$434$	0	0
$435$	− 119104.i	− 0.629432i
$436$	0	0
$437$	105529.	0.552599
$438$	0	0
$439$	− 219346.i	− 1.13815i	−0.822284	−	0.569077i	$-0.807301\pi$
$439$	− 219346.i	− 1.13815i	0.822284	−	0.569077i	$-0.192699\pi$
$440$	0	0
$441$	806928.	4.14914
$442$	0	0
$443$	39511.3i	0.201332i	0.994920	+	0.100666i	$0.0320974\pi$
$443$	39511.3i	0.201332i	−0.994920	+	0.100666i	$0.967903\pi$
$444$	0	0
$445$	−124331.	−0.627853
$446$	0	0
$447$	− 23267.8i	− 0.116450i
$448$	0	0
$449$	204669.	1.01522	0.507608	−	0.861588i	$-0.330530\pi$
$449$	204669.	1.01522	0.507608	+	0.861588i	$0.330530\pi$
$450$	0	0
$451$	− 17155.7i	− 0.0843442i
$452$	0	0
$453$	460994.	2.24646
$454$	0	0
$455$	146774.i	0.708969i
$456$	0	0
$457$	−62590.3	−0.299691	−0.149846	−	0.988709i	$-0.547878\pi$
$457$	−62590.3	−0.299691	−0.149846	+	0.988709i	$0.547878\pi$
$458$	0	0
$459$	− 291179.i	− 1.38208i
$460$	0	0
$461$	73091.9	0.343928	0.171964	−	0.985103i	$-0.444989\pi$
$461$	73091.9	0.343928	0.171964	+	0.985103i	$0.444989\pi$
$462$	0	0
$463$	− 143210.i	− 0.668055i	−0.942563	−	0.334028i	$-0.891592\pi$
$463$	− 143210.i	− 0.668055i	0.942563	−	0.334028i	$-0.108408\pi$
$464$	0	0
$465$	177547.	0.821119
$466$	0	0
$467$	149694.i	0.686389i	0.939264	+	0.343194i	$0.111509\pi$
$467$	149694.i	0.686389i	−0.939264	+	0.343194i	$0.888491\pi$
$468$	0	0
$469$	−371875.	−1.69064
$470$	0	0
$471$	− 62712.4i	− 0.282691i
$472$	0	0
$473$	−5015.62	−0.0224183
$474$	0	0
$475$	168100.i	0.745040i
$476$	0	0
$477$	−926758.	−4.07314
$478$	0	0
$479$	− 79876.1i	− 0.348133i	−0.984734	−	0.174067i	$-0.944309\pi$
$479$	− 79876.1i	− 0.348133i	0.984734	−	0.174067i	$-0.0556908\pi$
$480$	0	0
$481$	205198.	0.886915
$482$	0	0
$483$	− 439841.i	− 1.88539i
$484$	0	0
$485$	12157.3	0.0516835
$486$	0	0
$487$	345681.i	1.45753i	0.684763	+	0.728766i	$0.259906\pi$
$487$	345681.i	1.45753i	−0.684763	+	0.728766i	$0.740094\pi$
$488$	0	0
$489$	−100433.	−0.420010
$490$	0	0
$491$	− 319173.i	− 1.32392i	−0.749538	−	0.661961i	$-0.769724\pi$
$491$	− 319173.i	− 1.32392i	0.749538	−	0.661961i	$-0.230276\pi$
$492$	0	0
$493$	−103841.	−0.427244
$494$	0	0
$495$	− 43736.2i	− 0.178497i
$496$	0	0
$497$	−204965.	−0.829786
$498$	0	0
$499$	− 186219.i	− 0.747866i	−0.927456	−	0.373933i	$-0.878009\pi$
$499$	− 186219.i	− 0.747866i	0.927456	−	0.373933i	$-0.121991\pi$
$500$	0	0
$501$	91423.3	0.364235
$502$	0	0
$503$	109815.i	0.434038i	0.976167	+	0.217019i	$0.0696333\pi$
$503$	109815.i	0.434038i	−0.976167	+	0.217019i	$0.930367\pi$
$504$	0	0
$505$	107153.	0.420165
$506$	0	0
$507$	58733.4i	0.228491i
$508$	0	0
$509$	248990.	0.961052	0.480526	−	0.876981i	$-0.340446\pi$
$509$	248990.	0.961052	0.480526	+	0.876981i	$0.340446\pi$
$510$	0	0
$511$	396411.i	1.51811i
$512$	0	0
$513$	−632547.	−2.40358
$514$	0	0
$515$	97159.5i	0.366328i
$516$	0	0
$517$	−64146.0	−0.239988
$518$	0	0
$519$	− 563661.i	− 2.09258i
$520$	0	0
$521$	163712.	0.603123	0.301562	−	0.953447i	$-0.402492\pi$
$521$	163712.	0.603123	0.301562	+	0.953447i	$0.402492\pi$
$522$	0	0
$523$	106787.i	0.390406i	0.980763	+	0.195203i	$0.0625366\pi$
$523$	106787.i	0.390406i	−0.980763	+	0.195203i	$0.937463\pi$
$524$	0	0
$525$	700631.	2.54197
$526$	0	0
$527$	− 154794.i	− 0.557356i
$528$	0	0
$529$	172789.	0.617456
$530$	0	0
$531$	209380.i	0.742583i
$532$	0	0
$533$	−141970.	−0.499739
$534$	0	0
$535$	43430.0i	0.151734i
$536$	0	0
$537$	−889191.	−3.08352
$538$	0	0
$539$	− 88047.6i	− 0.303068i
$540$	0	0
$541$	23108.9	0.0789558	0.0394779	−	0.999220i	$-0.487431\pi$
$541$	23108.9	0.0789558	0.0394779	+	0.999220i	$0.487431\pi$
$542$	0	0
$543$	697194.i	2.36458i
$544$	0	0
$545$	12953.6	0.0436112
$546$	0	0
$547$	− 99858.8i	− 0.333743i	−0.985979	−	0.166871i	$-0.946634\pi$
$547$	− 99858.8i	− 0.333743i	0.985979	−	0.166871i	$-0.0533664\pi$
$548$	0	0
$549$	−577497.	−1.91604
$550$	0	0
$551$	225581.i	0.743018i
$552$	0	0
$553$	−350397.	−1.14580
$554$	0	0
$555$	195114.i	0.633437i
$556$	0	0
$557$	404414.	1.30352	0.651758	−	0.758427i	$-0.274032\pi$
$557$	404414.	1.30352	0.651758	+	0.758427i	$0.274032\pi$
$558$	0	0
$559$	41506.2i	0.132828i
$560$	0	0
$561$	−53703.9	−0.170640
$562$	0	0
$563$	− 69169.2i	− 0.218221i	−0.994030	−	0.109110i	$-0.965200\pi$
$563$	− 69169.2i	− 0.218221i	0.994030	−	0.109110i	$-0.0348002\pi$
$564$	0	0
$565$	−209893.	−0.657510
$566$	0	0
$567$	1.34423e6i	4.18126i
$568$	0	0
$569$	123521.	0.381519	0.190759	−	0.981637i	$-0.438905\pi$
$569$	123521.	0.381519	0.190759	+	0.981637i	$0.438905\pi$
$570$	0	0
$571$	− 192876.i	− 0.591571i	−0.955254	−	0.295785i	$-0.904419\pi$
$571$	− 192876.i	− 0.591571i	0.955254	−	0.295785i	$-0.0955814\pi$
$572$	0	0
$573$	−525758.	−1.60131
$574$	0	0
$575$	− 170525.i	− 0.515764i
$576$	0	0
$577$	−94086.0	−0.282601	−0.141300	−	0.989967i	$-0.545128\pi$
$577$	−94086.0	−0.282601	−0.141300	+	0.989967i	$0.545128\pi$
$578$	0	0
$579$	308981.i	0.921669i
$580$	0	0
$581$	−454850.	−1.34746
$582$	0	0
$583$	101123.i	0.297517i
$584$	0	0
$585$	−361935.	−1.05759
$586$	0	0
$587$	− 304527.i	− 0.883792i	−0.897066	−	0.441896i	$-0.854306\pi$
$587$	− 304527.i	− 0.883792i	0.897066	−	0.441896i	$-0.145694\pi$
$588$	0	0
$589$	−336269.	−0.969296
$590$	0	0
$591$	− 824146.i	− 2.35955i
$592$	0	0
$593$	122010.	0.346966	0.173483	−	0.984837i	$-0.444498\pi$
$593$	122010.	0.346966	0.173483	+	0.984837i	$0.444498\pi$
$594$	0	0
$595$	121677.i	0.343697i
$596$	0	0
$597$	992631.	2.78509
$598$	0	0
$599$	− 329463.i	− 0.918232i	−0.888376	−	0.459116i	$-0.848166\pi$
$599$	− 329463.i	− 0.918232i	0.888376	−	0.459116i	$-0.151834\pi$
$600$	0	0
$601$	−102795.	−0.284593	−0.142297	−	0.989824i	$-0.545449\pi$
$601$	−102795.	−0.284593	−0.142297	+	0.989824i	$0.545449\pi$
$602$	0	0
$603$	− 917017.i	− 2.52199i
$604$	0	0
$605$	144406.	0.394524
$606$	0	0
$607$	560628.i	1.52159i	0.648993	+	0.760794i	$0.275190\pi$
$607$	560628.i	1.52159i	−0.648993	+	0.760794i	$0.724810\pi$
$608$	0	0
$609$	940210.	2.53507
$610$	0	0
$611$	530834.i	1.42192i
$612$	0	0
$613$	−454035.	−1.20828	−0.604140	−	0.796878i	$-0.706483\pi$
$613$	−454035.	−1.20828	−0.604140	+	0.796878i	$0.706483\pi$
$614$	0	0
$615$	− 134994.i	− 0.356914i
$616$	0	0
$617$	245819.	0.645721	0.322860	−	0.946447i	$-0.395356\pi$
$617$	245819.	0.645721	0.322860	+	0.946447i	$0.395356\pi$
$618$	0	0
$619$	− 577432.i	− 1.50702i	−0.657435	−	0.753511i	$-0.728359\pi$
$619$	− 577432.i	− 1.50702i	0.657435	−	0.753511i	$-0.271641\pi$
$620$	0	0
$621$	641672.	1.66391
$622$	0	0
$623$	− 981467.i	− 2.52871i
$624$	0	0
$625$	206746.	0.529270
$626$	0	0
$627$	116664.i	0.296759i
$628$	0	0
$629$	170111.	0.429962
$630$	0	0
$631$	− 588831.i	− 1.47888i	−0.673224	−	0.739438i	$-0.735091\pi$
$631$	− 588831.i	− 1.47888i	0.673224	−	0.739438i	$-0.264909\pi$
$632$	0	0
$633$	564373.	1.40851
$634$	0	0
$635$	− 38112.6i	− 0.0945193i
$636$	0	0
$637$	−728630.	−1.79568
$638$	0	0
$639$	− 505428.i	− 1.23782i
$640$	0	0
$641$	−86370.6	−0.210208	−0.105104	−	0.994461i	$-0.533518\pi$
$641$	−86370.6	−0.210208	−0.105104	+	0.994461i	$0.533518\pi$
$642$	0	0
$643$	− 671327.i	− 1.62372i	−0.583849	−	0.811862i	$-0.698454\pi$
$643$	− 671327.i	− 1.62372i	0.583849	−	0.811862i	$-0.301546\pi$
$644$	0	0
$645$	−39466.7	−0.0948661
$646$	0	0
$647$	208533.i	0.498158i	0.968483	+	0.249079i	$0.0801278\pi$
$647$	208533.i	0.498158i	−0.968483	+	0.249079i	$0.919872\pi$
$648$	0	0
$649$	22846.4	0.0542410
$650$	0	0
$651$	1.40155e6i	3.30710i
$652$	0	0
$653$	−556984.	−1.30622	−0.653110	−	0.757263i	$-0.726536\pi$
$653$	−556984.	−1.30622	−0.653110	+	0.757263i	$0.726536\pi$
$654$	0	0
$655$	− 201398.i	− 0.469431i
$656$	0	0
$657$	−977521.	−2.26462
$658$	0	0
$659$	− 295904.i	− 0.681364i	−0.940179	−	0.340682i	$-0.889342\pi$
$659$	− 295904.i	− 0.681364i	0.940179	−	0.340682i	$-0.110658\pi$
$660$	0	0
$661$	280826.	0.642739	0.321370	−	0.946954i	$-0.395857\pi$
$661$	280826.	0.642739	0.321370	+	0.946954i	$0.395857\pi$
$662$	0	0
$663$	444421.i	1.01104i
$664$	0	0
$665$	264327.	0.597721
$666$	0	0
$667$	− 228835.i	− 0.514365i
$668$	0	0
$669$	154254.	0.344655
$670$	0	0
$671$	63013.4i	0.139955i
$672$	0	0
$673$	−395861.	−0.874003	−0.437001	−	0.899461i	$-0.643960\pi$
$673$	−395861.	−0.874003	−0.437001	+	0.899461i	$0.643960\pi$
$674$	0	0
$675$	1.02213e6i	2.24336i
$676$	0	0
$677$	296569.	0.647065	0.323532	−	0.946217i	$-0.395130\pi$
$677$	296569.	0.647065	0.323532	+	0.946217i	$0.395130\pi$
$678$	0	0
$679$	95969.5i	0.208158i
$680$	0	0
$681$	1.12365e6	2.42292
$682$	0	0
$683$	− 144692.i	− 0.310172i	−0.987901	−	0.155086i	$-0.950435\pi$
$683$	− 144692.i	− 0.310172i	0.987901	−	0.155086i	$-0.0495654\pi$
$684$	0	0
$685$	260636.	0.555461
$686$	0	0
$687$	840483.i	1.78080i
$688$	0	0
$689$	836832.	1.76279
$690$	0	0
$691$	− 359909.i	− 0.753766i	−0.926261	−	0.376883i	$-0.876996\pi$
$691$	− 359909.i	− 0.753766i	0.926261	−	0.376883i	$-0.123004\pi$
$692$	0	0
$693$	345254.	0.718907
$694$	0	0
$695$	62054.7i	0.128471i
$696$	0	0
$697$	−117695.	−0.242265
$698$	0	0
$699$	881423.i	1.80397i
$700$	0	0
$701$	−75448.9	−0.153538	−0.0767692	−	0.997049i	$-0.524460\pi$
$701$	−75448.9	−0.153538	−0.0767692	+	0.997049i	$0.524460\pi$
$702$	0	0
$703$	− 369542.i	− 0.747745i
$704$	0	0
$705$	−504750.	−1.01554
$706$	0	0
$707$	845863.i	1.69224i
$708$	0	0
$709$	484369.	0.963571	0.481785	−	0.876289i	$-0.339988\pi$
$709$	484369.	0.963571	0.481785	+	0.876289i	$0.339988\pi$
$710$	0	0
$711$	− 864054.i	− 1.70923i
$712$	0	0
$713$	341120.	0.671009
$714$	0	0
$715$	39492.4i	0.0772505i
$716$	0	0
$717$	1.62924e6	3.16917
$718$	0	0
$719$	− 47114.4i	− 0.0911373i	−0.998961	−	0.0455687i	$-0.985490\pi$
$719$	− 47114.4i	− 0.0911373i	0.998961	−	0.0455687i	$-0.0145100\pi$
$720$	0	0
$721$	−766978.	−1.47541
$722$	0	0
$723$	1.63145e6i	3.12102i
$724$	0	0
$725$	364516.	0.693490
$726$	0	0
$727$	134342.i	0.254180i	0.991891	+	0.127090i	$0.0405637\pi$
$727$	134342.i	0.254180i	−0.991891	+	0.127090i	$0.959436\pi$
$728$	0	0
$729$	−659746.	−1.24143
$730$	0	0
$731$	34409.0i	0.0643929i
$732$	0	0
$733$	−429578.	−0.799529	−0.399764	−	0.916618i	$-0.630908\pi$
$733$	−429578.	−0.799529	−0.399764	+	0.916618i	$0.630908\pi$
$734$	0	0
$735$	− 692826.i	− 1.28248i
$736$	0	0
$737$	−100060.	−0.184215
$738$	0	0
$739$	− 975871.i	− 1.78691i	−0.449148	−	0.893457i	$-0.648273\pi$
$739$	− 975871.i	− 1.78691i	0.449148	−	0.893457i	$-0.351727\pi$
$740$	0	0
$741$	965445.	1.75829
$742$	0	0
$743$	671044.i	1.21555i	0.794109	+	0.607776i	$0.207938\pi$
$743$	671044.i	1.21555i	−0.794109	+	0.607776i	$0.792062\pi$
$744$	0	0
$745$	−14184.8	−0.0255570
$746$	0	0
$747$	− 1.12163e6i	− 2.01005i
$748$	0	0
$749$	−342837.	−0.611116
$750$	0	0
$751$	272637.i	0.483399i	0.970351	+	0.241699i	$0.0777048\pi$
$751$	272637.i	0.483399i	−0.970351	+	0.241699i	$0.922295\pi$
$752$	0	0
$753$	1.54584e6	2.72631
$754$	0	0
$755$	− 281036.i	− 0.493024i
$756$	0	0
$757$	−850551.	−1.48426	−0.742128	−	0.670259i	$-0.766183\pi$
$757$	−850551.	−1.48426	−0.742128	+	0.670259i	$0.766183\pi$
$758$	0	0
$759$	− 118347.i	− 0.205435i
$760$	0	0
$761$	65767.8	0.113565	0.0567824	−	0.998387i	$-0.481916\pi$
$761$	65767.8	0.113565	0.0567824	+	0.998387i	$0.481916\pi$
$762$	0	0
$763$	102256.i	0.175647i
$764$	0	0
$765$	−300047.	−0.512704
$766$	0	0
$767$	− 189063.i	− 0.321378i
$768$	0	0
$769$	−542462.	−0.917310	−0.458655	−	0.888614i	$-0.651669\pi$
$769$	−542462.	−0.917310	−0.458655	+	0.888614i	$0.651669\pi$
$770$	0	0
$771$	− 1.41041e6i	− 2.37267i
$772$	0	0
$773$	−91838.1	−0.153697	−0.0768483	−	0.997043i	$-0.524486\pi$
$773$	−91838.1	−0.153697	−0.0768483	+	0.997043i	$0.524486\pi$
$774$	0	0
$775$	543377.i	0.904686i
$776$	0	0
$777$	−1.54024e6	−2.55120
$778$	0	0
$779$	255676.i	0.421322i
$780$	0	0
$781$	−55149.6	−0.0904150
$782$	0	0
$783$	1.37165e6i	2.23727i
$784$	0	0
$785$	−38231.4	−0.0620412
$786$	0	0
$787$	546617.i	0.882539i	0.897375	+	0.441270i	$0.145472\pi$
$787$	546617.i	0.882539i	−0.897375	+	0.441270i	$0.854528\pi$
$788$	0	0
$789$	−2.28293e6	−3.66724
$790$	0	0
$791$	− 1.65690e6i	− 2.64816i
$792$	0	0
$793$	521461.	0.829231
$794$	0	0
$795$	795711.i	1.25899i
$796$	0	0
$797$	−822216.	−1.29440	−0.647201	−	0.762320i	$-0.724060\pi$
$797$	−822216.	−1.29440	−0.647201	+	0.762320i	$0.724060\pi$
$798$	0	0
$799$	440066.i	0.689326i
$800$	0	0
$801$	2.42023e6	3.77217
$802$	0	0
$803$	106662.i	0.165416i
$804$	0	0
$805$	−268140.	−0.413781
$806$	0	0
$807$	− 907138.i	− 1.39292i
$808$	0	0
$809$	−216481.	−0.330767	−0.165384	−	0.986229i	$-0.552886\pi$
$809$	−216481.	−0.330767	−0.165384	+	0.986229i	$0.552886\pi$
$810$	0	0
$811$	− 64418.9i	− 0.0979427i	−0.998800	−	0.0489713i	$-0.984406\pi$
$811$	− 64418.9i	− 0.0979427i	0.998800	−	0.0489713i	$-0.0155943\pi$
$812$	0	0
$813$	250491.	0.378976
$814$	0	0
$815$	61227.2i	0.0921784i
$816$	0	0
$817$	74749.0	0.111985
$818$	0	0
$819$	− 2.85712e6i	− 4.25952i
$820$	0	0
$821$	1.13414e6	1.68260	0.841300	−	0.540569i	$-0.181791\pi$
$821$	1.13414e6	1.68260	0.841300	+	0.540569i	$0.181791\pi$
$822$	0	0
$823$	− 686626.i	− 1.01373i	−0.862026	−	0.506863i	$-0.830805\pi$
$823$	− 686626.i	− 1.01373i	0.862026	−	0.506863i	$-0.169195\pi$
$824$	0	0
$825$	188518.	0.276978
$826$	0	0
$827$	453622.i	0.663259i	0.943410	+	0.331630i	$0.107598\pi$
$827$	453622.i	0.663259i	−0.943410	+	0.331630i	$0.892402\pi$
$828$	0	0
$829$	649128.	0.944542	0.472271	−	0.881453i	$-0.343434\pi$
$829$	649128.	0.944542	0.472271	+	0.881453i	$0.343434\pi$
$830$	0	0
$831$	− 441950.i	− 0.639987i
$832$	0	0
$833$	−604040.	−0.870515
$834$	0	0
$835$	− 55734.5i	− 0.0799376i
$836$	0	0
$837$	−2.04469e6	−2.91861
$838$	0	0
$839$	564460.i	0.801879i	0.916104	+	0.400940i	$0.131316\pi$
$839$	564460.i	0.801879i	−0.916104	+	0.400940i	$0.868684\pi$
$840$	0	0
$841$	−218120.	−0.308392
$842$	0	0
$843$	296390.i	0.417070i
$844$	0	0
$845$	35805.7	0.0501462
$846$	0	0
$847$	1.13994e6i	1.58897i
$848$	0	0
$849$	−1.78694e6	−2.47911
$850$	0	0
$851$	374873.i	0.517637i
$852$	0	0
$853$	−6797.28	−0.00934194	−0.00467097	−	0.999989i	$-0.501487\pi$
$853$	−6797.28	−0.00934194	−0.00467097	+	0.999989i	$0.501487\pi$
$854$	0	0
$855$	651812.i	0.891642i
$856$	0	0
$857$	−936730.	−1.27542	−0.637709	−	0.770277i	$-0.720118\pi$
$857$	−936730.	−1.27542	−0.637709	+	0.770277i	$0.720118\pi$
$858$	0	0
$859$	− 850774.i	− 1.15300i	−0.817099	−	0.576498i	$-0.804419\pi$
$859$	− 850774.i	− 1.15300i	0.817099	−	0.576498i	$-0.195581\pi$
$860$	0	0
$861$	1.06564e6	1.43749
$862$	0	0
$863$	− 657197.i	− 0.882417i	−0.897405	−	0.441209i	$-0.854550\pi$
$863$	− 657197.i	− 0.882417i	0.897405	−	0.441209i	$-0.145450\pi$
$864$	0	0
$865$	−343625.	−0.459253
$866$	0	0
$867$	− 1.02750e6i	− 1.36692i
$868$	0	0
$869$	−94280.9	−0.124849
$870$	0	0
$871$	828036.i	1.09147i
$872$	0	0
$873$	−236654.	−0.310517
$874$	0	0
$875$	− 939333.i	− 1.22688i
$876$	0	0
$877$	−751857.	−0.977544	−0.488772	−	0.872412i	$-0.662555\pi$
$877$	−751857.	−0.977544	−0.488772	+	0.872412i	$0.662555\pi$
$878$	0	0
$879$	1.66353e6i	2.15304i
$880$	0	0
$881$	−277963.	−0.358125	−0.179062	−	0.983838i	$-0.557306\pi$
$881$	−277963.	−0.358125	−0.179062	+	0.983838i	$0.557306\pi$
$882$	0	0
$883$	481184.i	0.617149i	0.951200	+	0.308574i	$0.0998519\pi$
$883$	481184.i	0.617149i	−0.951200	+	0.308574i	$0.900148\pi$
$884$	0	0
$885$	179773.	0.229529
$886$	0	0
$887$	772804.i	0.982250i	0.871089	+	0.491125i	$0.163414\pi$
$887$	772804.i	0.982250i	−0.871089	+	0.491125i	$0.836586\pi$
$888$	0	0
$889$	300861.	0.380682
$890$	0	0
$891$	361690.i	0.455597i
$892$	0	0
$893$	955985.	1.19880
$894$	0	0
$895$	542078.i	0.676730i
$896$	0	0
$897$	−979373.	−1.21720
$898$	0	0
$899$	729184.i	0.902231i
$900$	0	0
$901$	693741.	0.854571
$902$	0	0
$903$	− 311550.i	− 0.382079i
$904$	0	0
$905$	425031.	0.518948
$906$	0	0
$907$	1.03068e6i	1.25288i	0.779469	+	0.626440i	$0.215489\pi$
$907$	1.03068e6i	1.25288i	−0.779469	+	0.626440i	$0.784511\pi$
$908$	0	0
$909$	−2.08584e6	−2.52437
$910$	0	0
$911$	716922.i	0.863843i	0.901911	+	0.431922i	$0.142164\pi$
$911$	716922.i	0.863843i	−0.901911	+	0.431922i	$0.857836\pi$
$912$	0	0
$913$	−122386.	−0.146822
$914$	0	0
$915$	495837.i	0.592239i
$916$	0	0
$917$	1.58984e6	1.89066
$918$	0	0
$919$	223092.i	0.264152i	0.991240	+	0.132076i	$0.0421642\pi$
$919$	223092.i	0.264152i	−0.991240	+	0.132076i	$0.957836\pi$
$920$	0	0
$921$	−780404.	−0.920027
$922$	0	0
$923$	456385.i	0.535708i
$924$	0	0
$925$	−597143.	−0.697903
$926$	0	0
$927$	− 1.89131e6i	− 2.20092i
$928$	0	0
$929$	−28837.6	−0.0334140	−0.0167070	−	0.999860i	$-0.505318\pi$
$929$	−28837.6	−0.0334140	−0.0167070	+	0.999860i	$0.505318\pi$
$930$	0	0
$931$	1.31220e6i	1.51391i
$932$	0	0
$933$	362963.	0.416965
$934$	0	0
$935$	32739.5i	0.0374498i
$936$	0	0
$937$	887467.	1.01082	0.505409	−	0.862880i	$-0.331342\pi$
$937$	887467.	1.01082	0.505409	+	0.862880i	$0.331342\pi$
$938$	0	0
$939$	933469.i	1.05869i
$940$	0	0
$941$	−186343.	−0.210442	−0.105221	−	0.994449i	$-0.533555\pi$
$941$	−186343.	−0.210442	−0.105221	+	0.994449i	$0.533555\pi$
$942$	0	0
$943$	− 259364.i	− 0.291666i
$944$	0	0
$945$	1.60725e6	1.79978
$946$	0	0
$947$	− 1.50276e6i	− 1.67567i	−0.545920	−	0.837837i	$-0.683820\pi$
$947$	− 1.50276e6i	− 1.67567i	0.545920	−	0.837837i	$-0.316180\pi$
$948$	0	0
$949$	882669.	0.980089
$950$	0	0
$951$	− 1.76230e6i	− 1.94858i
$952$	0	0
$953$	740506.	0.815347	0.407674	−	0.913128i	$-0.366340\pi$
$953$	740506.	0.815347	0.407674	+	0.913128i	$0.366340\pi$
$954$	0	0
$955$	320518.i	0.351436i
$956$	0	0
$957$	252981.	0.276226
$958$	0	0
$959$	2.05746e6i	2.23715i
$960$	0	0
$961$	−163459.	−0.176996
$962$	0	0
$963$	− 845411.i	− 0.911624i
$964$	0	0
$965$	188364.	0.202276
$966$	0	0
$967$	401772.i	0.429662i	0.976651	+	0.214831i	$0.0689201\pi$
$967$	401772.i	0.429662i	−0.976651	+	0.214831i	$0.931080\pi$
$968$	0	0
$969$	800363.	0.852392
$970$	0	0
$971$	1.06855e6i	1.13333i	0.823947	+	0.566667i	$0.191768\pi$
$971$	1.06855e6i	1.13333i	−0.823947	+	0.566667i	$0.808232\pi$
$972$	0	0
$973$	−489860.	−0.517424
$974$	0	0
$975$	− 1.56006e6i	− 1.64109i
$976$	0	0
$977$	−674685.	−0.706825	−0.353412	−	0.935468i	$-0.614979\pi$
$977$	−674685.	−0.706825	−0.353412	+	0.935468i	$0.614979\pi$
$978$	0	0
$979$	− 264082.i	− 0.275533i
$980$	0	0
$981$	−252156.	−0.262018
$982$	0	0
$983$	− 751517.i	− 0.777735i	−0.921294	−	0.388868i	$-0.872866\pi$
$983$	− 751517.i	− 0.777735i	0.921294	−	0.388868i	$-0.127134\pi$
$984$	0	0
$985$	−502424.	−0.517843
$986$	0	0
$987$	− 3.98450e6i	− 4.09015i
$988$	0	0
$989$	−75827.3	−0.0775235
$990$	0	0
$991$	− 328889.i	− 0.334890i	−0.985881	−	0.167445i	$-0.946448\pi$
$991$	− 328889.i	− 0.334890i	0.985881	−	0.167445i	$-0.0535516\pi$
$992$	0	0
$993$	380418.	0.385801
$994$	0	0
$995$	− 605138.i	− 0.611235i
$996$	0	0
$997$	105510.	0.106146	0.0530729	−	0.998591i	$-0.483098\pi$
$997$	105510.	0.106146	0.0530729	+	0.998591i	$0.483098\pi$
$998$	0	0
$999$	− 2.24701e6i	− 2.25151i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	128.5.c.b.127.8	yes	8
3.2	odd	2		1152.5.g.a.127.4		8
4.3	odd	2	inner	128.5.c.b.127.1	yes	8
8.3	odd	2		128.5.c.a.127.8	yes	8
8.5	even	2		128.5.c.a.127.1	✓	8
12.11	even	2		1152.5.g.a.127.3		8
16.3	odd	4		256.5.d.h.127.7		8
16.5	even	4		256.5.d.h.127.8		8
16.11	odd	4		256.5.d.g.127.2		8
16.13	even	4		256.5.d.g.127.1		8
24.5	odd	2		1152.5.g.b.127.6		8
24.11	even	2		1152.5.g.b.127.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.5.c.a.127.1	✓	8	8.5	even	2
128.5.c.a.127.8	yes	8	8.3	odd	2
128.5.c.b.127.1	yes	8	4.3	odd	2	inner
128.5.c.b.127.8	yes	8	1.1	even	1	trivial
256.5.d.g.127.1		8	16.13	even	4
256.5.d.g.127.2		8	16.11	odd	4
256.5.d.h.127.7		8	16.3	odd	4
256.5.d.h.127.8		8	16.5	even	4
1152.5.g.a.127.3		8	12.11	even	2
1152.5.g.a.127.4		8	3.2	odd	2
1152.5.g.b.127.5		8	24.11	even	2
1152.5.g.b.127.6		8	24.5	odd	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.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+16.7135i q^{3} +10.1891 q^{5} +80.4325i q^{7} -198.341 q^{9} +O(q^{10})\)	\(q+16.7135i q^{3} +10.1891 q^{5} +80.4325i q^{7} -198.341 q^{9} +21.6419i q^{11} +179.095 q^{13} +170.295i q^{15} +148.472 q^{17} -322.535i q^{19} -1344.31 q^{21} +327.187i q^{23} -521.183 q^{25} -1961.18i q^{27} -699.401 q^{29} -1042.58i q^{31} -361.712 q^{33} +819.532i q^{35} +1145.75 q^{37} +2993.31i q^{39} -792.708 q^{41} +231.755i q^{43} -2020.91 q^{45} +2963.98i q^{47} -4068.39 q^{49} +2481.48i q^{51} +4672.55 q^{53} +220.510i q^{55} +5390.68 q^{57} -1055.65i q^{59} +2911.64 q^{61} -15953.1i q^{63} +1824.81 q^{65} +4623.44i q^{67} -5468.45 q^{69} +2548.28i q^{71} +4928.49 q^{73} -8710.79i q^{75} -1740.71 q^{77} +4356.41i q^{79} +16712.5 q^{81} +5655.05i q^{83} +1512.79 q^{85} -11689.4i q^{87} -12202.4 q^{89} +14405.1i q^{91} +17425.2 q^{93} -3286.32i q^{95} +1193.17 q^{97} -4292.47i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 48 q^{5} - 216 q^{9}+O(q^{10}) \) 8 * q + 48 * q^5 - 216 * q^9	\( 8 q + 48 q^{5} - 216 q^{9} + 240 q^{13} + 240 q^{17} - 1216 q^{21} + 664 q^{25} + 432 q^{29} + 992 q^{33} + 2800 q^{37} - 2928 q^{41} - 4880 q^{45} - 5752 q^{49} + 1776 q^{53} + 8608 q^{57} + 12656 q^{61} + 672 q^{65} - 4416 q^{69} + 560 q^{73} - 31296 q^{77} + 14696 q^{81} - 14432 q^{85} - 22992 q^{89} + 56320 q^{93} - 3728 q^{97}+O(q^{100}) \) 8 * q + 48 * q^5 - 216 * q^9 + 240 * q^13 + 240 * q^17 - 1216 * q^21 + 664 * q^25 + 432 * q^29 + 992 * q^33 + 2800 * q^37 - 2928 * q^41 - 4880 * q^45 - 5752 * q^49 + 1776 * q^53 + 8608 * q^57 + 12656 * q^61 + 672 * q^65 - 4416 * q^69 + 560 * q^73 - 31296 * q^77 + 14696 * q^81 - 14432 * q^85 - 22992 * q^89 + 56320 * q^93 - 3728 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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