LMFDB - Embedded newform 128.5.c.a.127.7

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			127.7
Root			$1.12964 + 2.72719i$ of defining polynomial
Character	$\chi$	$=$	128.127
Dual form			128.5.c.a.127.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+7.95199i q^{3} -46.0525 q^{5} +57.5735i q^{7} +17.7658 q^{9} +O(q^{10})$	$q+7.95199i q^{3} -46.0525 q^{5} +57.5735i q^{7} +17.7658 q^{9} -181.609i q^{11} -1.33027 q^{13} -366.209i q^{15} -62.6785 q^{17} -340.238i q^{19} -457.824 q^{21} -447.310i q^{23} +1495.83 q^{25} +785.385i q^{27} -497.710 q^{29} -444.221i q^{31} +1444.15 q^{33} -2651.40i q^{35} -687.281 q^{37} -10.5783i q^{39} -3073.10 q^{41} -307.837i q^{43} -818.160 q^{45} -4046.90i q^{47} -913.711 q^{49} -498.419i q^{51} +1979.75 q^{53} +8363.53i q^{55} +2705.57 q^{57} +2878.33i q^{59} -1532.97 q^{61} +1022.84i q^{63} +61.2623 q^{65} +3968.29i q^{67} +3557.00 q^{69} +910.519i q^{71} -4408.76 q^{73} +11894.8i q^{75} +10455.9 q^{77} -3392.72i q^{79} -4806.34 q^{81} -8386.34i q^{83} +2886.50 q^{85} -3957.79i q^{87} -6526.50 q^{89} -76.5885i q^{91} +3532.44 q^{93} +15668.8i q^{95} -9684.32 q^{97} -3226.43i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	7.95199i	0.883555i	0.897125	+	0.441777i	$0.145652\pi$
$3$	7.95199i	0.883555i	−0.897125	+	0.441777i	$0.854348\pi$
$4$	0	0
$5$	−46.0525	−1.84210	−0.921049	−	0.389446i	$-0.872666\pi$
$5$	−46.0525	−1.84210	−0.921049	+	0.389446i	$0.872666\pi$
$6$	0	0
$7$	57.5735i	1.17497i	0.809235	+	0.587485i	$0.199882\pi$
$7$	57.5735i	1.17497i	−0.809235	+	0.587485i	$0.800118\pi$
$8$	0	0
$9$	17.7658	0.219331
$10$	0	0
$11$	− 181.609i	− 1.50090i	−0.660928	−	0.750449i	$-0.729837\pi$
$11$	− 181.609i	− 1.50090i	0.660928	−	0.750449i	$-0.270163\pi$
$12$	0	0
$13$	−1.33027	−0.00787143	−0.00393572	−	0.999992i	$-0.501253\pi$
$13$	−1.33027	−0.00787143	−0.00393572	+	0.999992i	$0.501253\pi$
$14$	0	0
$15$	− 366.209i	− 1.62759i
$16$	0	0
$17$	−62.6785	−0.216881	−0.108440	−	0.994103i	$-0.534586\pi$
$17$	−62.6785	−0.216881	−0.108440	+	0.994103i	$0.534586\pi$
$18$	0	0
$19$	− 340.238i	− 0.942487i	−0.882003	−	0.471244i	$-0.843805\pi$
$19$	− 340.238i	− 0.942487i	0.882003	−	0.471244i	$-0.156195\pi$
$20$	0	0
$21$	−457.824	−1.03815
$22$	0	0
$23$	− 447.310i	− 0.845576i	−0.906229	−	0.422788i	$-0.861051\pi$
$23$	− 447.310i	− 0.845576i	0.906229	−	0.422788i	$-0.138949\pi$
$24$	0	0
$25$	1495.83	2.39333
$26$	0	0
$27$	785.385i	1.07735i
$28$	0	0
$29$	−497.710	−0.591807	−0.295904	−	0.955218i	$-0.595621\pi$
$29$	−497.710	−0.591807	−0.295904	+	0.955218i	$0.595621\pi$
$30$	0	0
$31$	− 444.221i	− 0.462248i	−0.972924	−	0.231124i	$-0.925760\pi$
$31$	− 444.221i	− 0.462248i	0.972924	−	0.231124i	$-0.0742403\pi$
$32$	0	0
$33$	1444.15	1.32613
$34$	0	0
$35$	− 2651.40i	− 2.16441i
$36$	0	0
$37$	−687.281	−0.502031	−0.251016	−	0.967983i	$-0.580765\pi$
$37$	−687.281	−0.502031	−0.251016	+	0.967983i	$0.580765\pi$
$38$	0	0
$39$	− 10.5783i	− 0.00695484i
$40$	0	0
$41$	−3073.10	−1.82814	−0.914068	−	0.405561i	$-0.867076\pi$
$41$	−3073.10	−1.82814	−0.914068	+	0.405561i	$0.867076\pi$
$42$	0	0
$43$	− 307.837i	− 0.166488i	−0.996529	−	0.0832442i	$-0.973472\pi$
$43$	− 307.837i	− 0.166488i	0.996529	−	0.0832442i	$-0.0265281\pi$
$44$	0	0
$45$	−818.160	−0.404030
$46$	0	0
$47$	− 4046.90i	− 1.83201i	−0.401170	−	0.916004i	$-0.631396\pi$
$47$	− 4046.90i	− 1.83201i	0.401170	−	0.916004i	$-0.368604\pi$
$48$	0	0
$49$	−913.711	−0.380554
$50$	0	0
$51$	− 498.419i	− 0.191626i
$52$	0	0
$53$	1979.75	0.704789	0.352395	−	0.935851i	$-0.385367\pi$
$53$	1979.75	0.704789	0.352395	+	0.935851i	$0.385367\pi$
$54$	0	0
$55$	8363.53i	2.76480i
$56$	0	0
$57$	2705.57	0.832739
$58$	0	0
$59$	2878.33i	0.826869i	0.910534	+	0.413435i	$0.135671\pi$
$59$	2878.33i	0.826869i	−0.910534	+	0.413435i	$0.864329\pi$
$60$	0	0
$61$	−1532.97	−0.411979	−0.205990	−	0.978554i	$-0.566041\pi$
$61$	−1532.97	−0.411979	−0.205990	+	0.978554i	$0.566041\pi$
$62$	0	0
$63$	1022.84i	0.257708i
$64$	0	0
$65$	61.2623	0.0145000
$66$	0	0
$67$	3968.29i	0.884002i	0.897014	+	0.442001i	$0.145731\pi$
$67$	3968.29i	0.884002i	−0.897014	+	0.442001i	$0.854269\pi$
$68$	0	0
$69$	3557.00	0.747113
$70$	0	0
$71$	910.519i	0.180623i	0.995914	+	0.0903114i	$0.0287862\pi$
$71$	910.519i	0.180623i	−0.995914	+	0.0903114i	$0.971214\pi$
$72$	0	0
$73$	−4408.76	−0.827315	−0.413657	−	0.910433i	$-0.635749\pi$
$73$	−4408.76	−0.827315	−0.413657	+	0.910433i	$0.635749\pi$
$74$	0	0
$75$	11894.8i	2.11464i
$76$	0	0
$77$	10455.9	1.76351
$78$	0	0
$79$	− 3392.72i	− 0.543617i	−0.962351	−	0.271809i	$-0.912378\pi$
$79$	− 3392.72i	− 0.543617i	0.962351	−	0.271809i	$-0.0876218\pi$
$80$	0	0
$81$	−4806.34	−0.732562
$82$	0	0
$83$	− 8386.34i	− 1.21735i	−0.793419	−	0.608676i	$-0.791701\pi$
$83$	− 8386.34i	− 1.21735i	0.793419	−	0.608676i	$-0.208299\pi$
$84$	0	0
$85$	2886.50	0.399516
$86$	0	0
$87$	− 3957.79i	− 0.522894i
$88$	0	0
$89$	−6526.50	−0.823949	−0.411975	−	0.911195i	$-0.635161\pi$
$89$	−6526.50	−0.823949	−0.411975	+	0.911195i	$0.635161\pi$
$90$	0	0
$91$	− 76.5885i	− 0.00924870i
$92$	0	0
$93$	3532.44	0.408422
$94$	0	0
$95$	15668.8i	1.73615i
$96$	0	0
$97$	−9684.32	−1.02926	−0.514631	−	0.857412i	$-0.672071\pi$
$97$	−9684.32	−1.02926	−0.514631	+	0.857412i	$0.672071\pi$
$98$	0	0
$99$	− 3226.43i	− 0.329194i
$100$	0	0
$101$	−15927.9	−1.56140	−0.780701	−	0.624905i	$-0.785138\pi$
$101$	−15927.9	−1.56140	−0.780701	+	0.624905i	$0.785138\pi$
$102$	0	0
$103$	8844.42i	0.833671i	0.908982	+	0.416836i	$0.136861\pi$
$103$	8844.42i	0.833671i	−0.908982	+	0.416836i	$0.863139\pi$
$104$	0	0
$105$	21083.9	1.91237
$106$	0	0
$107$	− 3859.60i	− 0.337112i	−0.985692	−	0.168556i	$-0.946090\pi$
$107$	− 3859.60i	− 0.337112i	0.985692	−	0.168556i	$-0.0539105\pi$
$108$	0	0
$109$	−969.612	−0.0816103	−0.0408052	−	0.999167i	$-0.512992\pi$
$109$	−969.612	−0.0816103	−0.0408052	+	0.999167i	$0.512992\pi$
$110$	0	0
$111$	− 5465.25i	− 0.443572i
$112$	0	0
$113$	12551.5	0.982964	0.491482	−	0.870888i	$-0.336455\pi$
$113$	12551.5	0.982964	0.491482	+	0.870888i	$0.336455\pi$
$114$	0	0
$115$	20599.7i	1.55763i
$116$	0	0
$117$	−23.6334	−0.00172645
$118$	0	0
$119$	− 3608.63i	− 0.254828i
$120$	0	0
$121$	−18340.7	−1.25270
$122$	0	0
$123$	− 24437.2i	− 1.61526i
$124$	0	0
$125$	−40103.8	−2.56665
$126$	0	0
$127$	− 15872.9i	− 0.984123i	−0.870561	−	0.492061i	$-0.836244\pi$
$127$	− 15872.9i	− 0.984123i	0.870561	−	0.492061i	$-0.163756\pi$
$128$	0	0
$129$	2447.92	0.147102
$130$	0	0
$131$	1814.02i	0.105706i	0.998602	+	0.0528529i	$0.0168314\pi$
$131$	1814.02i	0.105706i	−0.998602	+	0.0528529i	$0.983169\pi$
$132$	0	0
$133$	19588.7	1.10739
$134$	0	0
$135$	− 36168.9i	− 1.98458i
$136$	0	0
$137$	22638.7	1.20618	0.603088	−	0.797674i	$-0.293937\pi$
$137$	22638.7	1.20618	0.603088	+	0.797674i	$0.293937\pi$
$138$	0	0
$139$	− 37031.2i	− 1.91663i	−0.285714	−	0.958315i	$-0.592231\pi$
$139$	− 37031.2i	− 1.91663i	0.285714	−	0.958315i	$-0.407769\pi$
$140$	0	0
$141$	32180.9	1.61868
$142$	0	0
$143$	241.589i	0.0118142i
$144$	0	0
$145$	22920.8	1.09017
$146$	0	0
$147$	− 7265.82i	− 0.336241i
$148$	0	0
$149$	1256.13	0.0565797	0.0282899	−	0.999600i	$-0.490994\pi$
$149$	1256.13	0.0565797	0.0282899	+	0.999600i	$0.490994\pi$
$150$	0	0
$151$	− 18288.1i	− 0.802074i	−0.916062	−	0.401037i	$-0.868650\pi$
$151$	− 18288.1i	− 0.802074i	0.916062	−	0.401037i	$-0.131350\pi$
$152$	0	0
$153$	−1113.54	−0.0475687
$154$	0	0
$155$	20457.4i	0.851507i
$156$	0	0
$157$	−15381.9	−0.624038	−0.312019	−	0.950076i	$-0.601005\pi$
$157$	−15381.9	−0.624038	−0.312019	+	0.950076i	$0.601005\pi$
$158$	0	0
$159$	15743.0i	0.622720i
$160$	0	0
$161$	25753.2	0.993526
$162$	0	0
$163$	12787.2i	0.481282i	0.970614	+	0.240641i	$0.0773577\pi$
$163$	12787.2i	0.481282i	−0.970614	+	0.240641i	$0.922642\pi$
$164$	0	0
$165$	−66506.7	−2.44285
$166$	0	0
$167$	− 17486.8i	− 0.627013i	−0.949586	−	0.313507i	$-0.898496\pi$
$167$	− 17486.8i	− 0.627013i	0.949586	−	0.313507i	$-0.101504\pi$
$168$	0	0
$169$	−28559.2	−0.999938
$170$	0	0
$171$	− 6044.61i	− 0.206717i
$172$	0	0
$173$	−12193.5	−0.407415	−0.203708	−	0.979032i	$-0.565299\pi$
$173$	−12193.5	−0.407415	−0.203708	+	0.979032i	$0.565299\pi$
$174$	0	0
$175$	86120.2i	2.81209i
$176$	0	0
$177$	−22888.5	−0.730584
$178$	0	0
$179$	− 2312.77i	− 0.0721814i	−0.999349	−	0.0360907i	$-0.988509\pi$
$179$	− 2312.77i	− 0.0721814i	0.999349	−	0.0360907i	$-0.0114905\pi$
$180$	0	0
$181$	45840.9	1.39925	0.699626	−	0.714510i	$-0.253350\pi$
$181$	45840.9	1.39925	0.699626	+	0.714510i	$0.253350\pi$
$182$	0	0
$183$	− 12190.2i	− 0.364006i
$184$	0	0
$185$	31651.0	0.924791
$186$	0	0
$187$	11383.0i	0.325516i
$188$	0	0
$189$	−45217.4	−1.26585
$190$	0	0
$191$	− 37639.2i	− 1.03175i	−0.856664	−	0.515875i	$-0.827467\pi$
$191$	− 37639.2i	− 1.03175i	0.856664	−	0.515875i	$-0.172533\pi$
$192$	0	0
$193$	−47465.1	−1.27426	−0.637132	−	0.770754i	$-0.719880\pi$
$193$	−47465.1	−1.27426	−0.637132	+	0.770754i	$0.719880\pi$
$194$	0	0
$195$	487.157i	0.0128115i
$196$	0	0
$197$	38867.5	1.00151	0.500754	−	0.865590i	$-0.333056\pi$
$197$	38867.5	1.00151	0.500754	+	0.865590i	$0.333056\pi$
$198$	0	0
$199$	49425.1i	1.24808i	0.781394	+	0.624038i	$0.214509\pi$
$199$	49425.1i	1.24808i	−0.781394	+	0.624038i	$0.785491\pi$
$200$	0	0
$201$	−31555.8	−0.781064
$202$	0	0
$203$	− 28654.9i	− 0.695356i
$204$	0	0
$205$	141524.	3.36761
$206$	0	0
$207$	− 7946.83i	− 0.185461i
$208$	0	0
$209$	−61790.2	−1.41458
$210$	0	0
$211$	51920.5i	1.16620i	0.812399	+	0.583101i	$0.198161\pi$
$211$	51920.5i	1.16620i	−0.812399	+	0.583101i	$0.801839\pi$
$212$	0	0
$213$	−7240.44	−0.159590
$214$	0	0
$215$	14176.7i	0.306688i
$216$	0	0
$217$	25575.3	0.543128
$218$	0	0
$219$	− 35058.4i	− 0.730978i
$220$	0	0
$221$	83.3795	0.00170716
$222$	0	0
$223$	− 5552.08i	− 0.111647i	−0.998441	−	0.0558234i	$-0.982222\pi$
$223$	− 5552.08i	− 0.111647i	0.998441	−	0.0558234i	$-0.0177784\pi$
$224$	0	0
$225$	26574.7	0.524932
$226$	0	0
$227$	23448.3i	0.455050i	0.973772	+	0.227525i	$0.0730633\pi$
$227$	23448.3i	0.455050i	−0.973772	+	0.227525i	$0.926937\pi$
$228$	0	0
$229$	−38740.1	−0.738738	−0.369369	−	0.929283i	$-0.620426\pi$
$229$	−38740.1	−0.738738	−0.369369	+	0.929283i	$0.620426\pi$
$230$	0	0
$231$	83144.9i	1.55816i
$232$	0	0
$233$	−48379.7	−0.891152	−0.445576	−	0.895244i	$-0.647001\pi$
$233$	−48379.7	−0.891152	−0.445576	+	0.895244i	$0.647001\pi$
$234$	0	0
$235$	186370.i	3.37474i
$236$	0	0
$237$	26978.8	0.480316
$238$	0	0
$239$	− 72360.9i	− 1.26680i	−0.773824	−	0.633401i	$-0.781659\pi$
$239$	− 72360.9i	− 1.26680i	0.773824	−	0.633401i	$-0.218341\pi$
$240$	0	0
$241$	11716.0	0.201718	0.100859	−	0.994901i	$-0.467841\pi$
$241$	11716.0	0.201718	0.100859	+	0.994901i	$0.467841\pi$
$242$	0	0
$243$	25396.2i	0.430087i
$244$	0	0
$245$	42078.7	0.701019
$246$	0	0
$247$	452.609i	0.00741873i
$248$	0	0
$249$	66688.1	1.07560
$250$	0	0
$251$	1157.11i	0.0183665i	0.999958	+	0.00918325i	$0.00292316\pi$
$251$	1157.11i	0.0183665i	−0.999958	+	0.00918325i	$0.997077\pi$
$252$	0	0
$253$	−81235.3	−1.26912
$254$	0	0
$255$	22953.4i	0.352994i
$256$	0	0
$257$	70976.6	1.07460	0.537302	−	0.843390i	$-0.319443\pi$
$257$	70976.6	1.07460	0.537302	+	0.843390i	$0.319443\pi$
$258$	0	0
$259$	− 39569.2i	− 0.589872i
$260$	0	0
$261$	−8842.24	−0.129802
$262$	0	0
$263$	18456.3i	0.266829i	0.991060	+	0.133414i	$0.0425941\pi$
$263$	18456.3i	0.266829i	−0.991060	+	0.133414i	$0.957406\pi$
$264$	0	0
$265$	−91172.5	−1.29829
$266$	0	0
$267$	− 51898.7i	− 0.728004i
$268$	0	0
$269$	30908.4	0.427142	0.213571	−	0.976928i	$-0.431491\pi$
$269$	30908.4	0.427142	0.213571	+	0.976928i	$0.431491\pi$
$270$	0	0
$271$	105487.i	1.43636i	0.695859	+	0.718178i	$0.255024\pi$
$271$	105487.i	1.43636i	−0.695859	+	0.718178i	$0.744976\pi$
$272$	0	0
$273$	609.031	0.00817173
$274$	0	0
$275$	− 271656.i	− 3.59214i
$276$	0	0
$277$	74390.3	0.969521	0.484760	−	0.874647i	$-0.338907\pi$
$277$	74390.3	0.969521	0.484760	+	0.874647i	$0.338907\pi$
$278$	0	0
$279$	− 7891.95i	− 0.101385i
$280$	0	0
$281$	−72221.4	−0.914646	−0.457323	−	0.889301i	$-0.651192\pi$
$281$	−72221.4	−0.914646	−0.457323	+	0.889301i	$0.651192\pi$
$282$	0	0
$283$	83262.2i	1.03962i	0.854281	+	0.519811i	$0.173998\pi$
$283$	83262.2i	1.03962i	−0.854281	+	0.519811i	$0.826002\pi$
$284$	0	0
$285$	−124598.	−1.53399
$286$	0	0
$287$	− 176929.i	− 2.14800i
$288$	0	0
$289$	−79592.4	−0.952963
$290$	0	0
$291$	− 77009.6i	− 0.909408i
$292$	0	0
$293$	−64847.4	−0.755366	−0.377683	−	0.925935i	$-0.623279\pi$
$293$	−64847.4	−0.755366	−0.377683	+	0.925935i	$0.623279\pi$
$294$	0	0
$295$	− 132554.i	− 1.52317i
$296$	0	0
$297$	142633.	1.61699
$298$	0	0
$299$	595.044i	0.00665590i
$300$	0	0
$301$	17723.3	0.195619
$302$	0	0
$303$	− 126658.i	− 1.37958i
$304$	0	0
$305$	70597.2	0.758906
$306$	0	0
$307$	54615.6i	0.579482i	0.957105	+	0.289741i	$0.0935691\pi$
$307$	54615.6i	0.579482i	−0.957105	+	0.289741i	$0.906431\pi$
$308$	0	0
$309$	−70330.7	−0.736594
$310$	0	0
$311$	142869.i	1.47713i	0.674183	+	0.738564i	$0.264496\pi$
$311$	142869.i	1.47713i	−0.674183	+	0.738564i	$0.735504\pi$
$312$	0	0
$313$	−15326.2	−0.156439	−0.0782195	−	0.996936i	$-0.524923\pi$
$313$	−15326.2	−0.156439	−0.0782195	+	0.996936i	$0.524923\pi$
$314$	0	0
$315$	− 47104.4i	− 0.474723i
$316$	0	0
$317$	−107757.	−1.07233	−0.536165	−	0.844113i	$-0.680127\pi$
$317$	−107757.	−1.07233	−0.536165	+	0.844113i	$0.680127\pi$
$318$	0	0
$319$	90388.5i	0.888243i
$320$	0	0
$321$	30691.5	0.297857
$322$	0	0
$323$	21325.6i	0.204407i
$324$	0	0
$325$	−1989.86	−0.0188389
$326$	0	0
$327$	− 7710.35i	− 0.0721072i
$328$	0	0
$329$	232995.	2.15255
$330$	0	0
$331$	− 72424.3i	− 0.661041i	−0.943799	−	0.330520i	$-0.892776\pi$
$331$	− 72424.3i	− 0.661041i	0.943799	−	0.330520i	$-0.107224\pi$
$332$	0	0
$333$	−12210.1	−0.110111
$334$	0	0
$335$	− 182749.i	− 1.62842i
$336$	0	0
$337$	137409.	1.20992	0.604958	−	0.796257i	$-0.293190\pi$
$337$	137409.	1.20992	0.604958	+	0.796257i	$0.293190\pi$
$338$	0	0
$339$	99809.1i	0.868502i
$340$	0	0
$341$	−80674.3	−0.693788
$342$	0	0
$343$	85628.5i	0.727830i
$344$	0	0
$345$	−163809.	−1.37626
$346$	0	0
$347$	86005.3i	0.714277i	0.934052	+	0.357138i	$0.116248\pi$
$347$	86005.3i	0.714277i	−0.934052	+	0.357138i	$0.883752\pi$
$348$	0	0
$349$	90032.5	0.739177	0.369588	−	0.929196i	$-0.379499\pi$
$349$	90032.5	0.739177	0.369588	+	0.929196i	$0.379499\pi$
$350$	0	0
$351$	− 1044.78i	− 0.00848026i
$352$	0	0
$353$	95205.5	0.764034	0.382017	−	0.924155i	$-0.375230\pi$
$353$	95205.5	0.764034	0.382017	+	0.924155i	$0.375230\pi$
$354$	0	0
$355$	− 41931.7i	− 0.332725i
$356$	0	0
$357$	28695.8	0.225155
$358$	0	0
$359$	10123.7i	0.0785507i	0.999228	+	0.0392754i	$0.0125050\pi$
$359$	10123.7i	0.0785507i	−0.999228	+	0.0392754i	$0.987495\pi$
$360$	0	0
$361$	14559.2	0.111718
$362$	0	0
$363$	− 145845.i	− 1.10683i
$364$	0	0
$365$	203034.	1.52400
$366$	0	0
$367$	− 55995.8i	− 0.415742i	−0.978156	−	0.207871i	$-0.933347\pi$
$367$	− 55995.8i	− 0.415742i	0.978156	−	0.207871i	$-0.0666534\pi$
$368$	0	0
$369$	−54596.1	−0.400967
$370$	0	0
$371$	113981.i	0.828106i
$372$	0	0
$373$	79505.8	0.571454	0.285727	−	0.958311i	$-0.407765\pi$
$373$	79505.8	0.571454	0.285727	+	0.958311i	$0.407765\pi$
$374$	0	0
$375$	− 318905.i	− 2.26777i
$376$	0	0
$377$	662.090	0.00465837
$378$	0	0
$379$	− 138185.i	− 0.962013i	−0.876717	−	0.481007i	$-0.840271\pi$
$379$	− 138185.i	− 0.962013i	0.876717	−	0.481007i	$-0.159729\pi$
$380$	0	0
$381$	126221.	0.869526
$382$	0	0
$383$	230247.i	1.56963i	0.619733	+	0.784813i	$0.287241\pi$
$383$	230247.i	1.56963i	−0.619733	+	0.784813i	$0.712759\pi$
$384$	0	0
$385$	−481518.	−3.24856
$386$	0	0
$387$	− 5468.98i	− 0.0365161i
$388$	0	0
$389$	−257598.	−1.70233	−0.851166	−	0.524897i	$-0.824104\pi$
$389$	−257598.	−1.70233	−0.851166	+	0.524897i	$0.824104\pi$
$390$	0	0
$391$	28036.7i	0.183389i
$392$	0	0
$393$	−14425.0	−0.0933968
$394$	0	0
$395$	156243.i	1.00140i
$396$	0	0
$397$	−158118.	−1.00323	−0.501614	−	0.865091i	$-0.667260\pi$
$397$	−158118.	−1.00323	−0.501614	+	0.865091i	$0.667260\pi$
$398$	0	0
$399$	155769.i	0.978443i
$400$	0	0
$401$	−253655.	−1.57745	−0.788724	−	0.614747i	$-0.789258\pi$
$401$	−253655.	−1.57745	−0.788724	+	0.614747i	$0.789258\pi$
$402$	0	0
$403$	590.934i	0.00363856i
$404$	0	0
$405$	221344.	1.34945
$406$	0	0
$407$	124816.i	0.753498i
$408$	0	0
$409$	−9946.74	−0.0594613	−0.0297306	−	0.999558i	$-0.509465\pi$
$409$	−9946.74	−0.0594613	−0.0297306	+	0.999558i	$0.509465\pi$
$410$	0	0
$411$	180023.i	1.06572i
$412$	0	0
$413$	−165716.	−0.971547
$414$	0	0
$415$	386212.i	2.24248i
$416$	0	0
$417$	294472.	1.69345
$418$	0	0
$419$	− 12318.5i	− 0.0701663i	−0.999384	−	0.0350832i	$-0.988830\pi$
$419$	− 12318.5i	− 0.0701663i	0.999384	−	0.0350832i	$-0.0111696\pi$
$420$	0	0
$421$	306315.	1.72824	0.864120	−	0.503287i	$-0.167876\pi$
$421$	306315.	1.72824	0.864120	+	0.503287i	$0.167876\pi$
$422$	0	0
$423$	− 71896.6i	− 0.401817i
$424$	0	0
$425$	−93756.4	−0.519067
$426$	0	0
$427$	− 88258.7i	− 0.484063i
$428$	0	0
$429$	−1921.11	−0.0104385
$430$	0	0
$431$	266095.i	1.43246i	0.697865	+	0.716229i	$0.254134\pi$
$431$	266095.i	1.43246i	−0.697865	+	0.716229i	$0.745866\pi$
$432$	0	0
$433$	235188.	1.25441	0.627205	−	0.778855i	$-0.284199\pi$
$433$	235188.	1.25441	0.627205	+	0.778855i	$0.284199\pi$
$434$	0	0
$435$	182266.i	0.963223i
$436$	0	0
$437$	−152192.	−0.796945
$438$	0	0
$439$	− 28553.8i	− 0.148161i	−0.997252	−	0.0740807i	$-0.976398\pi$
$439$	− 28553.8i	− 0.148161i	0.997252	−	0.0740807i	$-0.0236022\pi$
$440$	0	0
$441$	−16232.8	−0.0834675
$442$	0	0
$443$	31796.1i	0.162019i	0.996713	+	0.0810096i	$0.0258144\pi$
$443$	31796.1i	0.162019i	−0.996713	+	0.0810096i	$0.974186\pi$
$444$	0	0
$445$	300562.	1.51780
$446$	0	0
$447$	9988.70i	0.0499913i
$448$	0	0
$449$	72803.6	0.361127	0.180564	−	0.983563i	$-0.442208\pi$
$449$	72803.6	0.361127	0.180564	+	0.983563i	$0.442208\pi$
$450$	0	0
$451$	558101.i	2.74385i
$452$	0	0
$453$	145427.	0.708676
$454$	0	0
$455$	3527.09i	0.0170370i
$456$	0	0
$457$	−324179.	−1.55221	−0.776107	−	0.630601i	$-0.782809\pi$
$457$	−324179.	−1.55221	−0.776107	+	0.630601i	$0.782809\pi$
$458$	0	0
$459$	− 49226.8i	− 0.233656i
$460$	0	0
$461$	−129689.	−0.610239	−0.305119	−	0.952314i	$-0.598696\pi$
$461$	−129689.	−0.610239	−0.305119	+	0.952314i	$0.598696\pi$
$462$	0	0
$463$	− 202786.i	− 0.945969i	−0.881071	−	0.472985i	$-0.843177\pi$
$463$	− 202786.i	− 0.945969i	0.881071	−	0.472985i	$-0.156823\pi$
$464$	0	0
$465$	−162677.	−0.752353
$466$	0	0
$467$	− 273594.i	− 1.25451i	−0.778815	−	0.627254i	$-0.784179\pi$
$467$	− 273594.i	− 1.25451i	0.778815	−	0.627254i	$-0.215821\pi$
$468$	0	0
$469$	−228468.	−1.03868
$470$	0	0
$471$	− 122317.i	− 0.551372i
$472$	0	0
$473$	−55905.9	−0.249882
$474$	0	0
$475$	− 508938.i	− 2.25568i
$476$	0	0
$477$	35172.0	0.154582
$478$	0	0
$479$	− 376594.i	− 1.64136i	−0.571392	−	0.820678i	$-0.693596\pi$
$479$	− 376594.i	− 1.64136i	0.571392	−	0.820678i	$-0.306404\pi$
$480$	0	0
$481$	914.271	0.00395171
$482$	0	0
$483$	204789.i	0.877835i
$484$	0	0
$485$	445987.	1.89600
$486$	0	0
$487$	− 165588.i	− 0.698186i	−0.937088	−	0.349093i	$-0.886490\pi$
$487$	− 165588.i	− 0.698186i	0.937088	−	0.349093i	$-0.113510\pi$
$488$	0	0
$489$	−101684.	−0.425239
$490$	0	0
$491$	216651.i	0.898666i	0.893364	+	0.449333i	$0.148338\pi$
$491$	216651.i	0.898666i	−0.893364	+	0.449333i	$0.851662\pi$
$492$	0	0
$493$	31195.7	0.128352
$494$	0	0
$495$	148585.i	0.606408i
$496$	0	0
$497$	−52421.8	−0.212226
$498$	0	0
$499$	198168.i	0.795850i	0.917418	+	0.397925i	$0.130270\pi$
$499$	198168.i	0.795850i	−0.917418	+	0.397925i	$0.869730\pi$
$500$	0	0
$501$	139055.	0.554001
$502$	0	0
$503$	153994.i	0.608652i	0.952568	+	0.304326i	$0.0984312\pi$
$503$	153994.i	0.608652i	−0.952568	+	0.304326i	$0.901569\pi$
$504$	0	0
$505$	733517.	2.87625
$506$	0	0
$507$	− 227103.i	− 0.883500i
$508$	0	0
$509$	324022.	1.25066	0.625330	−	0.780360i	$-0.284964\pi$
$509$	324022.	1.25066	0.625330	+	0.780360i	$0.284964\pi$
$510$	0	0
$511$	− 253828.i	− 0.972070i
$512$	0	0
$513$	267218.	1.01538
$514$	0	0
$515$	− 407307.i	− 1.53570i
$516$	0	0
$517$	−734953.	−2.74966
$518$	0	0
$519$	− 96962.8i	− 0.359974i
$520$	0	0
$521$	−173876.	−0.640566	−0.320283	−	0.947322i	$-0.603778\pi$
$521$	−173876.	−0.640566	−0.320283	+	0.947322i	$0.603778\pi$
$522$	0	0
$523$	− 210988.i	− 0.771357i	−0.922633	−	0.385678i	$-0.873967\pi$
$523$	− 210988.i	− 0.771357i	0.922633	−	0.385678i	$-0.126033\pi$
$524$	0	0
$525$	−684827.	−2.48463
$526$	0	0
$527$	27843.1i	0.100253i
$528$	0	0
$529$	79755.0	0.285001
$530$	0	0
$531$	51136.0i	0.181358i
$532$	0	0
$533$	4088.05	0.0143900
$534$	0	0
$535$	177744.i	0.620994i
$536$	0	0
$537$	18391.1	0.0637762
$538$	0	0
$539$	165938.i	0.571174i
$540$	0	0
$541$	−212060.	−0.724543	−0.362272	−	0.932073i	$-0.617999\pi$
$541$	−212060.	−0.724543	−0.362272	+	0.932073i	$0.617999\pi$
$542$	0	0
$543$	364526.i	1.23631i
$544$	0	0
$545$	44653.0	0.150334
$546$	0	0
$547$	− 417896.i	− 1.39667i	−0.715771	−	0.698335i	$-0.753925\pi$
$547$	− 417896.i	− 1.39667i	0.715771	−	0.698335i	$-0.246075\pi$
$548$	0	0
$549$	−27234.6	−0.0903599
$550$	0	0
$551$	169340.i	0.557771i
$552$	0	0
$553$	195331.	0.638734
$554$	0	0
$555$	251688.i	0.817104i
$556$	0	0
$557$	−261445.	−0.842693	−0.421346	−	0.906900i	$-0.638442\pi$
$557$	−261445.	−0.842693	−0.421346	+	0.906900i	$0.638442\pi$
$558$	0	0
$559$	409.507i	0.00131050i
$560$	0	0
$561$	−90517.3	−0.287611
$562$	0	0
$563$	− 508256.i	− 1.60349i	−0.597667	−	0.801745i	$-0.703905\pi$
$563$	− 508256.i	− 1.60349i	0.597667	−	0.801745i	$-0.296095\pi$
$564$	0	0
$565$	−578026.	−1.81072
$566$	0	0
$567$	− 276718.i	− 0.860739i
$568$	0	0
$569$	378311.	1.16849	0.584244	−	0.811578i	$-0.301391\pi$
$569$	378311.	1.16849	0.584244	+	0.811578i	$0.301391\pi$
$570$	0	0
$571$	326234.i	1.00059i	0.865855	+	0.500295i	$0.166775\pi$
$571$	326234.i	1.00059i	−0.865855	+	0.500295i	$0.833225\pi$
$572$	0	0
$573$	299307.	0.911607
$574$	0	0
$575$	− 669099.i	− 2.02374i
$576$	0	0
$577$	−239608.	−0.719696	−0.359848	−	0.933011i	$-0.617171\pi$
$577$	−239608.	−0.719696	−0.359848	+	0.933011i	$0.617171\pi$
$578$	0	0
$579$	− 377442.i	− 1.12588i
$580$	0	0
$581$	482831.	1.43035
$582$	0	0
$583$	− 359540.i	− 1.05782i
$584$	0	0
$585$	1088.38	0.00318029
$586$	0	0
$587$	− 3718.92i	− 0.0107929i	−0.999985	−	0.00539647i	$-0.998282\pi$
$587$	− 3718.92i	− 0.0107929i	0.999985	−	0.00539647i	$-0.00171776\pi$
$588$	0	0
$589$	−151141.	−0.435663
$590$	0	0
$591$	309074.i	0.884887i
$592$	0	0
$593$	−358989.	−1.02087	−0.510437	−	0.859915i	$-0.670516\pi$
$593$	−358989.	−1.02087	−0.510437	+	0.859915i	$0.670516\pi$
$594$	0	0
$595$	166186.i	0.469419i
$596$	0	0
$597$	−393028.	−1.10274
$598$	0	0
$599$	− 597544.i	− 1.66539i	−0.553731	−	0.832696i	$-0.686796\pi$
$599$	− 597544.i	− 1.66539i	0.553731	−	0.832696i	$-0.313204\pi$
$600$	0	0
$601$	−515865.	−1.42819	−0.714096	−	0.700047i	$-0.753162\pi$
$601$	−515865.	−1.42819	−0.714096	+	0.700047i	$0.753162\pi$
$602$	0	0
$603$	70499.9i	0.193889i
$604$	0	0
$605$	844636.	2.30759
$606$	0	0
$607$	− 555458.i	− 1.50756i	−0.657128	−	0.753779i	$-0.728229\pi$
$607$	− 555458.i	− 1.50756i	0.657128	−	0.753779i	$-0.271771\pi$
$608$	0	0
$609$	227864.	0.614385
$610$	0	0
$611$	5383.48i	0.0144205i
$612$	0	0
$613$	206906.	0.550621	0.275311	−	0.961355i	$-0.411219\pi$
$613$	206906.	0.550621	0.275311	+	0.961355i	$0.411219\pi$
$614$	0	0
$615$	1.12539e6i	2.97546i
$616$	0	0
$617$	405517.	1.06522	0.532609	−	0.846362i	$-0.321212\pi$
$617$	405517.	1.06522	0.532609	+	0.846362i	$0.321212\pi$
$618$	0	0
$619$	− 430349.i	− 1.12315i	−0.827424	−	0.561577i	$-0.810195\pi$
$619$	− 430349.i	− 1.12315i	0.827424	−	0.561577i	$-0.189805\pi$
$620$	0	0
$621$	351310.	0.910978
$622$	0	0
$623$	− 375754.i	− 0.968116i
$624$	0	0
$625$	911987.	2.33469
$626$	0	0
$627$	− 491355.i	− 1.24986i
$628$	0	0
$629$	43077.8	0.108881
$630$	0	0
$631$	329465.i	0.827467i	0.910398	+	0.413733i	$0.135775\pi$
$631$	329465.i	0.827467i	−0.910398	+	0.413733i	$0.864225\pi$
$632$	0	0
$633$	−412871.	−1.03040
$634$	0	0
$635$	730987.i	1.81285i
$636$	0	0
$637$	1215.48	0.00299551
$638$	0	0
$639$	16176.1i	0.0396162i
$640$	0	0
$641$	−561428.	−1.36640	−0.683200	−	0.730231i	$-0.739412\pi$
$641$	−561428.	−1.36640	−0.683200	+	0.730231i	$0.739412\pi$
$642$	0	0
$643$	151040.i	0.365317i	0.983176	+	0.182658i	$0.0584702\pi$
$643$	151040.i	0.365317i	−0.983176	+	0.182658i	$0.941530\pi$
$644$	0	0
$645$	−112733.	−0.270976
$646$	0	0
$647$	781486.i	1.86686i	0.358755	+	0.933432i	$0.383201\pi$
$647$	781486.i	1.86686i	−0.358755	+	0.933432i	$0.616799\pi$
$648$	0	0
$649$	522730.	1.24105
$650$	0	0
$651$	203375.i	0.479883i
$652$	0	0
$653$	−652623.	−1.53051	−0.765255	−	0.643727i	$-0.777387\pi$
$653$	−652623.	−1.53051	−0.765255	+	0.643727i	$0.777387\pi$
$654$	0	0
$655$	− 83539.9i	− 0.194720i
$656$	0	0
$657$	−78325.3	−0.181456
$658$	0	0
$659$	807498.i	1.85939i	0.368330	+	0.929695i	$0.379930\pi$
$659$	807498.i	1.85939i	−0.368330	+	0.929695i	$0.620070\pi$
$660$	0	0
$661$	830419.	1.90062	0.950308	−	0.311312i	$-0.100768\pi$
$661$	830419.	1.90062	0.950308	+	0.311312i	$0.100768\pi$
$662$	0	0
$663$	663.033i	0.00150837i
$664$	0	0
$665$	−902108.	−2.03993
$666$	0	0
$667$	222631.i	0.500418i
$668$	0	0
$669$	44150.1	0.0986460
$670$	0	0
$671$	278402.i	0.618339i
$672$	0	0
$673$	548179.	1.21030	0.605148	−	0.796113i	$-0.293114\pi$
$673$	548179.	1.21030	0.605148	+	0.796113i	$0.293114\pi$
$674$	0	0
$675$	1.17480e6i	2.57844i
$676$	0	0
$677$	144840.	0.316018	0.158009	−	0.987438i	$-0.449492\pi$
$677$	144840.	0.316018	0.158009	+	0.987438i	$0.449492\pi$
$678$	0	0
$679$	− 557560.i	− 1.20935i
$680$	0	0
$681$	−186460.	−0.402061
$682$	0	0
$683$	361065.i	0.774004i	0.922079	+	0.387002i	$0.126489\pi$
$683$	361065.i	0.774004i	−0.922079	+	0.387002i	$0.873511\pi$
$684$	0	0
$685$	−1.04257e6	−2.22190
$686$	0	0
$687$	− 308061.i	− 0.652715i
$688$	0	0
$689$	−2633.61	−0.00554770
$690$	0	0
$691$	− 630523.i	− 1.32052i	−0.751037	−	0.660260i	$-0.770446\pi$
$691$	− 630523.i	− 1.32052i	0.751037	−	0.660260i	$-0.229554\pi$
$692$	0	0
$693$	185757.	0.386793
$694$	0	0
$695$	1.70538e6i	3.53062i
$696$	0	0
$697$	192617.	0.396487
$698$	0	0
$699$	− 384715.i	− 0.787381i
$700$	0	0
$701$	306464.	0.623653	0.311827	−	0.950139i	$-0.399059\pi$
$701$	306464.	0.623653	0.311827	+	0.950139i	$0.399059\pi$
$702$	0	0
$703$	233839.i	0.473158i
$704$	0	0
$705$	−1.48201e6	−2.98177
$706$	0	0
$707$	− 917023.i	− 1.83460i
$708$	0	0
$709$	675872.	1.34453	0.672267	−	0.740309i	$-0.265321\pi$
$709$	675872.	1.34453	0.672267	+	0.740309i	$0.265321\pi$
$710$	0	0
$711$	− 60274.4i	− 0.119232i
$712$	0	0
$713$	−198704.	−0.390866
$714$	0	0
$715$	− 11125.8i	− 0.0217630i
$716$	0	0
$717$	575414.	1.11929
$718$	0	0
$719$	334370.i	0.646800i	0.946262	+	0.323400i	$0.104826\pi$
$719$	334370.i	0.646800i	−0.946262	+	0.323400i	$0.895174\pi$
$720$	0	0
$721$	−509204.	−0.979539
$722$	0	0
$723$	93165.6i	0.178229i
$724$	0	0
$725$	−744489.	−1.41639
$726$	0	0
$727$	− 651244.i	− 1.23218i	−0.787675	−	0.616091i	$-0.788715\pi$
$727$	− 651244.i	− 1.23218i	0.787675	−	0.616091i	$-0.211285\pi$
$728$	0	0
$729$	−591264.	−1.11257
$730$	0	0
$731$	19294.8i	0.0361081i
$732$	0	0
$733$	712258.	1.32565	0.662825	−	0.748774i	$-0.269357\pi$
$733$	712258.	1.32565	0.662825	+	0.748774i	$0.269357\pi$
$734$	0	0
$735$	334609.i	0.619388i
$736$	0	0
$737$	720675.	1.32680
$738$	0	0
$739$	− 894022.i	− 1.63704i	−0.574478	−	0.818520i	$-0.694795\pi$
$739$	− 894022.i	− 1.63704i	0.574478	−	0.818520i	$-0.305205\pi$
$740$	0	0
$741$	−3599.14	−0.00655485
$742$	0	0
$743$	− 746574.i	− 1.35237i	−0.736732	−	0.676184i	$-0.763632\pi$
$743$	− 746574.i	− 1.35237i	0.736732	−	0.676184i	$-0.236368\pi$
$744$	0	0
$745$	−57847.7	−0.104225
$746$	0	0
$747$	− 148990.i	− 0.267004i
$748$	0	0
$749$	222211.	0.396097
$750$	0	0
$751$	941562.i	1.66943i	0.550680	+	0.834716i	$0.314368\pi$
$751$	941562.i	1.66943i	−0.550680	+	0.834716i	$0.685632\pi$
$752$	0	0
$753$	−9201.32	−0.0162278
$754$	0	0
$755$	842212.i	1.47750i
$756$	0	0
$757$	−266806.	−0.465591	−0.232796	−	0.972526i	$-0.574787\pi$
$757$	−266806.	−0.465591	−0.232796	+	0.972526i	$0.574787\pi$
$758$	0	0
$759$	− 645983.i	− 1.12134i
$760$	0	0
$761$	208021.	0.359202	0.179601	−	0.983740i	$-0.442519\pi$
$761$	208021.	0.359202	0.179601	+	0.983740i	$0.442519\pi$
$762$	0	0
$763$	− 55824.0i	− 0.0958897i
$764$	0	0
$765$	51281.1	0.0876263
$766$	0	0
$767$	− 3828.97i	− 0.00650865i
$768$	0	0
$769$	385383.	0.651688	0.325844	−	0.945424i	$-0.394352\pi$
$769$	385383.	0.651688	0.325844	+	0.945424i	$0.394352\pi$
$770$	0	0
$771$	564405.i	0.949472i
$772$	0	0
$773$	312746.	0.523399	0.261699	−	0.965149i	$-0.415717\pi$
$773$	312746.	0.523399	0.261699	+	0.965149i	$0.415717\pi$
$774$	0	0
$775$	− 664478.i	− 1.10631i
$776$	0	0
$777$	314654.	0.521184
$778$	0	0
$779$	1.04558e6i	1.72299i
$780$	0	0
$781$	165358.	0.271096
$782$	0	0
$783$	− 390894.i	− 0.637581i
$784$	0	0
$785$	708375.	1.14954
$786$	0	0
$787$	− 473462.i	− 0.764426i	−0.924074	−	0.382213i	$-0.875162\pi$
$787$	− 473462.i	− 0.764426i	0.924074	−	0.382213i	$-0.124838\pi$
$788$	0	0
$789$	−146764.	−0.235758
$790$	0	0
$791$	722632.i	1.15495i
$792$	0	0
$793$	2039.27	0.00324287
$794$	0	0
$795$	− 725003.i	− 1.14711i
$796$	0	0
$797$	−861999.	−1.35703	−0.678516	−	0.734586i	$-0.737377\pi$
$797$	−861999.	−1.35703	−0.678516	+	0.734586i	$0.737377\pi$
$798$	0	0
$799$	253654.i	0.397327i
$800$	0	0
$801$	−115949.	−0.180718
$802$	0	0
$803$	800670.i	1.24172i
$804$	0	0
$805$	−1.18600e6	−1.83017
$806$	0	0
$807$	245784.i	0.377403i
$808$	0	0
$809$	303009.	0.462976	0.231488	−	0.972838i	$-0.425641\pi$
$809$	303009.	0.462976	0.231488	+	0.972838i	$0.425641\pi$
$810$	0	0
$811$	− 223545.i	− 0.339878i	−0.985455	−	0.169939i	$-0.945643\pi$
$811$	− 223545.i	− 0.339878i	0.985455	−	0.169939i	$-0.0543571\pi$
$812$	0	0
$813$	−838835.	−1.26910
$814$	0	0
$815$	− 588881.i	− 0.886569i
$816$	0	0
$817$	−104738.	−0.156913
$818$	0	0
$819$	− 1360.66i	− 0.00202853i
$820$	0	0
$821$	−29800.5	−0.0442117	−0.0221058	−	0.999756i	$-0.507037\pi$
$821$	−29800.5	−0.0442117	−0.0221058	+	0.999756i	$0.507037\pi$
$822$	0	0
$823$	− 1.31384e6i	− 1.93974i	−0.243629	−	0.969868i	$-0.578338\pi$
$823$	− 1.31384e6i	− 1.93974i	0.243629	−	0.969868i	$-0.421662\pi$
$824$	0	0
$825$	2.16020e6	3.17385
$826$	0	0
$827$	− 809709.i	− 1.18391i	−0.805972	−	0.591954i	$-0.798357\pi$
$827$	− 809709.i	− 1.18391i	0.805972	−	0.591954i	$-0.201643\pi$
$828$	0	0
$829$	−367981.	−0.535447	−0.267723	−	0.963496i	$-0.586271\pi$
$829$	−367981.	−0.535447	−0.267723	+	0.963496i	$0.586271\pi$
$830$	0	0
$831$	591551.i	0.856624i
$832$	0	0
$833$	57270.1	0.0825350
$834$	0	0
$835$	805309.i	1.15502i
$836$	0	0
$837$	348884.	0.498001
$838$	0	0
$839$	− 528646.i	− 0.751002i	−0.926822	−	0.375501i	$-0.877471\pi$
$839$	− 528646.i	− 0.751002i	0.926822	−	0.375501i	$-0.122529\pi$
$840$	0	0
$841$	−459566.	−0.649764
$842$	0	0
$843$	− 574304.i	− 0.808140i
$844$	0	0
$845$	1.31522e6	1.84198
$846$	0	0
$847$	− 1.05594e6i	− 1.47188i
$848$	0	0
$849$	−662101.	−0.918562
$850$	0	0
$851$	307427.i	0.424506i
$852$	0	0
$853$	−342907.	−0.471279	−0.235640	−	0.971840i	$-0.575719\pi$
$853$	−342907.	−0.471279	−0.235640	+	0.971840i	$0.575719\pi$
$854$	0	0
$855$	278369.i	0.380793i
$856$	0	0
$857$	−35257.6	−0.0480055	−0.0240028	−	0.999712i	$-0.507641\pi$
$857$	−35257.6	−0.0480055	−0.0240028	+	0.999712i	$0.507641\pi$
$858$	0	0
$859$	− 1.36501e6i	− 1.84990i	−0.380086	−	0.924951i	$-0.624105\pi$
$859$	− 1.36501e6i	− 1.84990i	0.380086	−	0.924951i	$-0.375895\pi$
$860$	0	0
$861$	1.40694e6	1.89788
$862$	0	0
$863$	410558.i	0.551255i	0.961264	+	0.275628i	$0.0888857\pi$
$863$	410558.i	0.551255i	−0.961264	+	0.275628i	$0.911114\pi$
$864$	0	0
$865$	561542.	0.750499
$866$	0	0
$867$	− 632918.i	− 0.841995i
$868$	0	0
$869$	−616147.	−0.815914
$870$	0	0
$871$	− 5278.90i	− 0.00695837i
$872$	0	0
$873$	−172050.	−0.225749
$874$	0	0
$875$	− 2.30892e6i	− 3.01573i
$876$	0	0
$877$	−1.21087e6	−1.57434	−0.787168	−	0.616739i	$-0.788453\pi$
$877$	−1.21087e6	−1.57434	−0.787168	+	0.616739i	$0.788453\pi$
$878$	0	0
$879$	− 515666.i	− 0.667407i
$880$	0	0
$881$	132858.	0.171173	0.0855867	−	0.996331i	$-0.472724\pi$
$881$	132858.	0.171173	0.0855867	+	0.996331i	$0.472724\pi$
$882$	0	0
$883$	− 687910.i	− 0.882287i	−0.897437	−	0.441144i	$-0.854573\pi$
$883$	− 687910.i	− 0.882287i	0.897437	−	0.441144i	$-0.145427\pi$
$884$	0	0
$885$	1.05407e6	1.34581
$886$	0	0
$887$	− 617026.i	− 0.784253i	−0.919911	−	0.392126i	$-0.871740\pi$
$887$	− 617026.i	− 0.784253i	0.919911	−	0.392126i	$-0.128260\pi$
$888$	0	0
$889$	913860.	1.15631
$890$	0	0
$891$	872874.i	1.09950i
$892$	0	0
$893$	−1.37691e6	−1.72664
$894$	0	0
$895$	106509.i	0.132965i
$896$	0	0
$897$	−4731.78	−0.00588085
$898$	0	0
$899$	221093.i	0.273562i
$900$	0	0
$901$	−124088.	−0.152855
$902$	0	0
$903$	140935.i	0.172840i
$904$	0	0
$905$	−2.11109e6	−2.57756
$906$	0	0
$907$	586008.i	0.712343i	0.934421	+	0.356172i	$0.115918\pi$
$907$	586008.i	0.712343i	−0.934421	+	0.356172i	$0.884082\pi$
$908$	0	0
$909$	−282972.	−0.342464
$910$	0	0
$911$	318002.i	0.383172i	0.981476	+	0.191586i	$0.0613630\pi$
$911$	318002.i	0.383172i	−0.981476	+	0.191586i	$0.938637\pi$
$912$	0	0
$913$	−1.52303e6	−1.82712
$914$	0	0
$915$	561389.i	0.670535i
$916$	0	0
$917$	−104439.	−0.124201
$918$	0	0
$919$	241621.i	0.286091i	0.989716	+	0.143045i	$0.0456895\pi$
$919$	241621.i	0.286091i	−0.989716	+	0.143045i	$0.954311\pi$
$920$	0	0
$921$	−434302.	−0.512004
$922$	0	0
$923$	− 1211.24i	− 0.00142176i
$924$	0	0
$925$	−1.02806e6	−1.20153
$926$	0	0
$927$	157128.i	0.182850i
$928$	0	0
$929$	669730.	0.776012	0.388006	−	0.921657i	$-0.373164\pi$
$929$	669730.	0.776012	0.388006	+	0.921657i	$0.373164\pi$
$930$	0	0
$931$	310879.i	0.358668i
$932$	0	0
$933$	−1.13610e6	−1.30512
$934$	0	0
$935$	− 524214.i	− 0.599633i
$936$	0	0
$937$	228556.	0.260324	0.130162	−	0.991493i	$-0.458450\pi$
$937$	228556.	0.260324	0.130162	+	0.991493i	$0.458450\pi$
$938$	0	0
$939$	− 121874.i	− 0.138222i
$940$	0	0
$941$	637202.	0.719611	0.359806	−	0.933027i	$-0.382843\pi$
$941$	637202.	0.719611	0.359806	+	0.933027i	$0.382843\pi$
$942$	0	0
$943$	1.37463e6i	1.54583i
$944$	0	0
$945$	2.08237e6	2.33182
$946$	0	0
$947$	− 714073.i	− 0.796237i	−0.917334	−	0.398119i	$-0.869663\pi$
$947$	− 714073.i	− 0.796237i	0.917334	−	0.398119i	$-0.130337\pi$
$948$	0	0
$949$	5864.85	0.00651216
$950$	0	0
$951$	− 856885.i	− 0.947462i
$952$	0	0
$953$	672082.	0.740009	0.370004	−	0.929030i	$-0.379356\pi$
$953$	672082.	0.740009	0.370004	+	0.929030i	$0.379356\pi$
$954$	0	0
$955$	1.73338e6i	1.90058i
$956$	0	0
$957$	−718768.	−0.784811
$958$	0	0
$959$	1.30339e6i	1.41722i
$960$	0	0
$961$	726189.	0.786327
$962$	0	0
$963$	− 68569.0i	− 0.0739393i
$964$	0	0
$965$	2.18588e6	2.34732
$966$	0	0
$967$	1.01554e6i	1.08603i	0.839722	+	0.543017i	$0.182718\pi$
$967$	1.01554e6i	1.08603i	−0.839722	+	0.543017i	$0.817282\pi$
$968$	0	0
$969$	−169581.	−0.180605
$970$	0	0
$971$	1.00456e6i	1.06546i	0.846287	+	0.532728i	$0.178833\pi$
$971$	1.00456e6i	1.06546i	−0.846287	+	0.532728i	$0.821167\pi$
$972$	0	0
$973$	2.13202e6	2.25198
$974$	0	0
$975$	− 15823.4i	− 0.0166452i
$976$	0	0
$977$	2168.27	0.00227156	0.00113578	−	0.999999i	$-0.499638\pi$
$977$	2168.27	0.00227156	0.00113578	+	0.999999i	$0.499638\pi$
$978$	0	0
$979$	1.18527e6i	1.23666i
$980$	0	0
$981$	−17226.0	−0.0178997
$982$	0	0
$983$	626048.i	0.647889i	0.946076	+	0.323944i	$0.105009\pi$
$983$	626048.i	0.647889i	−0.946076	+	0.323944i	$0.894991\pi$
$984$	0	0
$985$	−1.78994e6	−1.84488
$986$	0	0
$987$	1.85277e6i	1.90190i
$988$	0	0
$989$	−137698.	−0.140779
$990$	0	0
$991$	− 841971.i	− 0.857334i	−0.903463	−	0.428667i	$-0.858983\pi$
$991$	− 841971.i	− 0.857334i	0.903463	−	0.428667i	$-0.141017\pi$
$992$	0	0
$993$	575917.	0.584065
$994$	0	0
$995$	− 2.27615e6i	− 2.29908i
$996$	0	0
$997$	29692.0	0.0298710	0.0149355	−	0.999888i	$-0.495246\pi$
$997$	29692.0	0.0298710	0.0149355	+	0.999888i	$0.495246\pi$
$998$	0	0
$999$	− 539780.i	− 0.540861i

Twists

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

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

\(f(q)\)	\(=\)	\(q+7.95199i q^{3} -46.0525 q^{5} +57.5735i q^{7} +17.7658 q^{9} +O(q^{10})\)	\(q+7.95199i q^{3} -46.0525 q^{5} +57.5735i q^{7} +17.7658 q^{9} -181.609i q^{11} -1.33027 q^{13} -366.209i q^{15} -62.6785 q^{17} -340.238i q^{19} -457.824 q^{21} -447.310i q^{23} +1495.83 q^{25} +785.385i q^{27} -497.710 q^{29} -444.221i q^{31} +1444.15 q^{33} -2651.40i q^{35} -687.281 q^{37} -10.5783i q^{39} -3073.10 q^{41} -307.837i q^{43} -818.160 q^{45} -4046.90i q^{47} -913.711 q^{49} -498.419i q^{51} +1979.75 q^{53} +8363.53i q^{55} +2705.57 q^{57} +2878.33i q^{59} -1532.97 q^{61} +1022.84i q^{63} +61.2623 q^{65} +3968.29i q^{67} +3557.00 q^{69} +910.519i q^{71} -4408.76 q^{73} +11894.8i q^{75} +10455.9 q^{77} -3392.72i q^{79} -4806.34 q^{81} -8386.34i q^{83} +2886.50 q^{85} -3957.79i q^{87} -6526.50 q^{89} -76.5885i q^{91} +3532.44 q^{93} +15668.8i q^{95} -9684.32 q^{97} -3226.43i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 48 q^{5} - 216 q^{9}+O(q^{10}) \) 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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