LMFDB - Embedded newform 256.8.a.q.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [256,8,Mod(1,256)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(256, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("256.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 256 = 2^{8} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	256.a (trivial)

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:	yes
Analytic conductor:	$79.9705665239$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 163x^{4} + 4820x^{2} - 15296 $ x^6 - 163x^4 + 4820x^2 - 15296
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{27} $
Twist minimal:	no (minimal twist has level 8)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-11.2068$ of defining polynomial
Character	$\chi$	$=$	256.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+21.9408 q^{3} +184.916 q^{5} +1051.96 q^{7} -1705.60 q^{9} +O(q^{10})$	$q+21.9408 q^{3} +184.916 q^{5} +1051.96 q^{7} -1705.60 q^{9} +4324.35 q^{11} -11253.2 q^{13} +4057.20 q^{15} -21746.4 q^{17} -45466.5 q^{19} +23080.8 q^{21} +4414.37 q^{23} -43931.2 q^{25} -85406.8 q^{27} +23687.1 q^{29} -72941.9 q^{31} +94879.8 q^{33} +194523. q^{35} -483338. q^{37} -246905. q^{39} +411040. q^{41} -96165.3 q^{43} -315392. q^{45} +156171. q^{47} +283070. q^{49} -477134. q^{51} +686962. q^{53} +799641. q^{55} -997572. q^{57} +1.79961e6 q^{59} -1.36394e6 q^{61} -1.79422e6 q^{63} -2.08090e6 q^{65} +1.08853e6 q^{67} +96854.8 q^{69} -5.60830e6 q^{71} -21698.7 q^{73} -963886. q^{75} +4.54903e6 q^{77} -2.34010e6 q^{79} +1.85626e6 q^{81} +882169. q^{83} -4.02125e6 q^{85} +519714. q^{87} +1.34738e6 q^{89} -1.18379e7 q^{91} -1.60040e6 q^{93} -8.40746e6 q^{95} +7.32798e6 q^{97} -7.37562e6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 688 q^{7} + 2918 q^{9}+O(q^{10}) $ 6 * q - 688 * q^7 + 2918 * q^9	$ 6 q - 688 q^{7} + 2918 q^{9} - 17872 q^{15} + 1452 q^{17} - 1296 q^{23} + 39314 q^{25} + 89280 q^{31} + 53880 q^{33} - 328208 q^{39} - 521244 q^{41} - 1566432 q^{47} - 511050 q^{49} - 3270256 q^{55} + 1889896 q^{57} - 5776816 q^{63} + 1416480 q^{65} - 7597104 q^{71} - 2089564 q^{73} - 16015904 q^{79} - 723058 q^{81} - 37453776 q^{87} - 2169084 q^{89} - 48537936 q^{95} - 1088308 q^{97}+O(q^{100}) $ 6 * q - 688 * q^7 + 2918 * q^9 - 17872 * q^15 + 1452 * q^17 - 1296 * q^23 + 39314 * q^25 + 89280 * q^31 + 53880 * q^33 - 328208 * q^39 - 521244 * q^41 - 1566432 * q^47 - 511050 * q^49 - 3270256 * q^55 + 1889896 * q^57 - 5776816 * q^63 + 1416480 * q^65 - 7597104 * q^71 - 2089564 * q^73 - 16015904 * q^79 - 723058 * q^81 - 37453776 * q^87 - 2169084 * q^89 - 48537936 * q^95 - 1088308 * q^97

Display 100 coefficients 10 coefficients

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
$3$	21.9408	0.469168	0.234584	−	0.972096i	$-0.424627\pi$
$3$	21.9408	0.469168	0.234584	+	0.972096i	$0.424627\pi$
$4$	0	0
$5$	184.916	0.661574	0.330787	−	0.943705i	$-0.392686\pi$
$5$	184.916	0.661574	0.330787	+	0.943705i	$0.392686\pi$
$6$	0	0
$7$	1051.96	1.15919	0.579595	−	0.814905i	$-0.303211\pi$
$7$	1051.96	1.15919	0.579595	+	0.814905i	$0.303211\pi$
$8$	0	0
$9$	−1705.60	−0.779882
$10$	0	0
$11$	4324.35	0.979596	0.489798	−	0.871836i	$-0.337071\pi$
$11$	4324.35	0.979596	0.489798	+	0.871836i	$0.337071\pi$
$12$	0	0
$13$	−11253.2	−1.42061	−0.710307	−	0.703892i	$-0.751444\pi$
$13$	−11253.2	−1.42061	−0.710307	+	0.703892i	$0.751444\pi$
$14$	0	0
$15$	4057.20	0.310389
$16$	0	0
$17$	−21746.4	−1.07354	−0.536768	−	0.843730i	$-0.680355\pi$
$17$	−21746.4	−1.07354	−0.536768	+	0.843730i	$0.680355\pi$
$18$	0	0
$19$	−45466.5	−1.52074	−0.760368	−	0.649492i	$-0.774981\pi$
$19$	−45466.5	−1.52074	−0.760368	+	0.649492i	$0.774981\pi$
$20$	0	0
$21$	23080.8	0.543855
$22$	0	0
$23$	4414.37	0.0756522	0.0378261	−	0.999284i	$-0.487957\pi$
$23$	4414.37	0.0756522	0.0378261	+	0.999284i	$0.487957\pi$
$24$	0	0
$25$	−43931.2	−0.562320
$26$	0	0
$27$	−85406.8	−0.835063
$28$	0	0
$29$	23687.1	0.180351	0.0901756	−	0.995926i	$-0.471257\pi$
$29$	23687.1	0.180351	0.0901756	+	0.995926i	$0.471257\pi$
$30$	0	0
$31$	−72941.9	−0.439755	−0.219878	−	0.975527i	$-0.570566\pi$
$31$	−72941.9	−0.439755	−0.219878	+	0.975527i	$0.570566\pi$
$32$	0	0
$33$	94879.8	0.459595
$34$	0	0
$35$	194523.	0.766890
$36$	0	0
$37$	−483338.	−1.56872	−0.784359	−	0.620307i	$-0.787008\pi$
$37$	−483338.	−1.56872	−0.784359	+	0.620307i	$0.787008\pi$
$38$	0	0
$39$	−246905.	−0.666506
$40$	0	0
$41$	411040.	0.931410	0.465705	−	0.884940i	$-0.345801\pi$
$41$	411040.	0.931410	0.465705	+	0.884940i	$0.345801\pi$
$42$	0	0
$43$	−96165.3	−0.184450	−0.0922250	−	0.995738i	$-0.529398\pi$
$43$	−96165.3	−0.184450	−0.0922250	+	0.995738i	$0.529398\pi$
$44$	0	0
$45$	−315392.	−0.515949
$46$	0	0
$47$	156171.	0.219411	0.109705	−	0.993964i	$-0.465009\pi$
$47$	156171.	0.219411	0.109705	+	0.993964i	$0.465009\pi$
$48$	0	0
$49$	283070.	0.343722
$50$	0	0
$51$	−477134.	−0.503668
$52$	0	0
$53$	686962.	0.633822	0.316911	−	0.948455i	$-0.397354\pi$
$53$	686962.	0.633822	0.316911	+	0.948455i	$0.397354\pi$
$54$	0	0
$55$	799641.	0.648075
$56$	0	0
$57$	−997572.	−0.713480
$58$	0	0
$59$	1.79961e6	1.14077	0.570384	−	0.821378i	$-0.306794\pi$
$59$	1.79961e6	1.14077	0.570384	+	0.821378i	$0.306794\pi$
$60$	0	0
$61$	−1.36394e6	−0.769379	−0.384690	−	0.923046i	$-0.625692\pi$
$61$	−1.36394e6	−0.769379	−0.384690	+	0.923046i	$0.625692\pi$
$62$	0	0
$63$	−1.79422e6	−0.904031
$64$	0	0
$65$	−2.08090e6	−0.939841
$66$	0	0
$67$	1.08853e6	0.442160	0.221080	−	0.975256i	$-0.429042\pi$
$67$	1.08853e6	0.442160	0.221080	+	0.975256i	$0.429042\pi$
$68$	0	0
$69$	96854.8	0.0354936
$70$	0	0
$71$	−5.60830e6	−1.85963	−0.929816	−	0.368025i	$-0.880034\pi$
$71$	−5.60830e6	−1.85963	−0.929816	+	0.368025i	$0.880034\pi$
$72$	0	0
$73$	−21698.7	−0.00652837	−0.00326418	−	0.999995i	$-0.501039\pi$
$73$	−21698.7	−0.00652837	−0.00326418	+	0.999995i	$0.501039\pi$
$74$	0	0
$75$	−963886.	−0.263822
$76$	0	0
$77$	4.54903e6	1.13554
$78$	0	0
$79$	−2.34010e6	−0.533998	−0.266999	−	0.963697i	$-0.586032\pi$
$79$	−2.34010e6	−0.533998	−0.266999	+	0.963697i	$0.586032\pi$
$80$	0	0
$81$	1.85626e6	0.388097
$82$	0	0
$83$	882169.	0.169347	0.0846736	−	0.996409i	$-0.473015\pi$
$83$	882169.	0.169347	0.0846736	+	0.996409i	$0.473015\pi$
$84$	0	0
$85$	−4.02125e6	−0.710223
$86$	0	0
$87$	519714.	0.0846150
$88$	0	0
$89$	1.34738e6	0.202594	0.101297	−	0.994856i	$-0.467701\pi$
$89$	1.34738e6	0.202594	0.101297	+	0.994856i	$0.467701\pi$
$90$	0	0
$91$	−1.18379e7	−1.64676
$92$	0	0
$93$	−1.60040e6	−0.206319
$94$	0	0
$95$	−8.40746e6	−1.00608
$96$	0	0
$97$	7.32798e6	0.815236	0.407618	−	0.913153i	$-0.366360\pi$
$97$	7.32798e6	0.815236	0.407618	+	0.913153i	$0.366360\pi$
$98$	0	0
$99$	−7.37562e6	−0.763969
$100$	0	0
$101$	9.66027e6	0.932963	0.466482	−	0.884531i	$-0.345521\pi$
$101$	9.66027e6	0.932963	0.466482	+	0.884531i	$0.345521\pi$
$102$	0	0
$103$	−1.38659e7	−1.25031	−0.625154	−	0.780501i	$-0.714964\pi$
$103$	−1.38659e7	−1.25031	−0.625154	+	0.780501i	$0.714964\pi$
$104$	0	0
$105$	4.26800e6	0.359800
$106$	0	0
$107$	1.88354e7	1.48638	0.743192	−	0.669078i	$-0.233311\pi$
$107$	1.88354e7	1.48638	0.743192	+	0.669078i	$0.233311\pi$
$108$	0	0
$109$	8.60619e6	0.636529	0.318265	−	0.948002i	$-0.396900\pi$
$109$	8.60619e6	0.636529	0.318265	+	0.948002i	$0.396900\pi$
$110$	0	0
$111$	−1.06048e7	−0.735992
$112$	0	0
$113$	−1.31340e7	−0.856292	−0.428146	−	0.903709i	$-0.640833\pi$
$113$	−1.31340e7	−0.856292	−0.428146	+	0.903709i	$0.640833\pi$
$114$	0	0
$115$	816286.	0.0500495
$116$	0	0
$117$	1.91935e7	1.10791
$118$	0	0
$119$	−2.28763e7	−1.24443
$120$	0	0
$121$	−787128.	−0.0403921
$122$	0	0
$123$	9.01856e6	0.436988
$124$	0	0
$125$	−2.25701e7	−1.03359
$126$	0	0
$127$	−3.18768e7	−1.38090	−0.690449	−	0.723381i	$-0.742587\pi$
$127$	−3.18768e7	−1.38090	−0.690449	+	0.723381i	$0.742587\pi$
$128$	0	0
$129$	−2.10994e6	−0.0865380
$130$	0	0
$131$	1.17050e7	0.454905	0.227453	−	0.973789i	$-0.426960\pi$
$131$	1.17050e7	0.454905	0.227453	+	0.973789i	$0.426960\pi$
$132$	0	0
$133$	−4.78288e7	−1.76282
$134$	0	0
$135$	−1.57931e7	−0.552456
$136$	0	0
$137$	−3.79174e7	−1.25984	−0.629921	−	0.776659i	$-0.716913\pi$
$137$	−3.79174e7	−1.25984	−0.629921	+	0.776659i	$0.716913\pi$
$138$	0	0
$139$	−2.80300e7	−0.885260	−0.442630	−	0.896704i	$-0.645955\pi$
$139$	−2.80300e7	−0.885260	−0.442630	+	0.896704i	$0.645955\pi$
$140$	0	0
$141$	3.42652e6	0.102941
$142$	0	0
$143$	−4.86630e7	−1.39163
$144$	0	0
$145$	4.38012e6	0.119316
$146$	0	0
$147$	6.21078e6	0.161263
$148$	0	0
$149$	−6.82083e6	−0.168922	−0.0844608	−	0.996427i	$-0.526917\pi$
$149$	−6.82083e6	−0.168922	−0.0844608	+	0.996427i	$0.526917\pi$
$150$	0	0
$151$	−7.62068e7	−1.80125	−0.900626	−	0.434595i	$-0.856891\pi$
$151$	−7.62068e7	−1.80125	−0.900626	+	0.434595i	$0.856891\pi$
$152$	0	0
$153$	3.70907e7	0.837230
$154$	0	0
$155$	−1.34881e7	−0.290931
$156$	0	0
$157$	4.28737e7	0.884183	0.442092	−	0.896970i	$-0.354237\pi$
$157$	4.28737e7	0.884183	0.442092	+	0.896970i	$0.354237\pi$
$158$	0	0
$159$	1.50725e7	0.297369
$160$	0	0
$161$	4.64373e6	0.0876952
$162$	0	0
$163$	4.12127e7	0.745374	0.372687	−	0.927957i	$-0.378437\pi$
$163$	4.12127e7	0.745374	0.372687	+	0.927957i	$0.378437\pi$
$164$	0	0
$165$	1.75448e7	0.304056
$166$	0	0
$167$	2.41703e7	0.401582	0.200791	−	0.979634i	$-0.435649\pi$
$167$	2.41703e7	0.401582	0.200791	+	0.979634i	$0.435649\pi$
$168$	0	0
$169$	6.38869e7	1.01814
$170$	0	0
$171$	7.75477e7	1.18599
$172$	0	0
$173$	−5.90454e7	−0.867011	−0.433505	−	0.901151i	$-0.642723\pi$
$173$	−5.90454e7	−0.867011	−0.433505	+	0.901151i	$0.642723\pi$
$174$	0	0
$175$	−4.62137e7	−0.651835
$176$	0	0
$177$	3.94850e7	0.535212
$178$	0	0
$179$	−1.20311e8	−1.56791	−0.783954	−	0.620819i	$-0.786800\pi$
$179$	−1.20311e8	−1.56791	−0.783954	+	0.620819i	$0.786800\pi$
$180$	0	0
$181$	−3.21539e6	−0.0403049	−0.0201525	−	0.999797i	$-0.506415\pi$
$181$	−3.21539e6	−0.0403049	−0.0201525	+	0.999797i	$0.506415\pi$
$182$	0	0
$183$	−2.99259e7	−0.360968
$184$	0	0
$185$	−8.93768e7	−1.03782
$186$	0	0
$187$	−9.40392e7	−1.05163
$188$	0	0
$189$	−8.98443e7	−0.967997
$190$	0	0
$191$	−7.04122e7	−0.731191	−0.365596	−	0.930774i	$-0.619135\pi$
$191$	−7.04122e7	−0.731191	−0.365596	+	0.930774i	$0.619135\pi$
$192$	0	0
$193$	1.17861e7	0.118010	0.0590050	−	0.998258i	$-0.481207\pi$
$193$	1.17861e7	0.118010	0.0590050	+	0.998258i	$0.481207\pi$
$194$	0	0
$195$	−4.56566e7	−0.440943
$196$	0	0
$197$	8.97008e7	0.835920	0.417960	−	0.908465i	$-0.362745\pi$
$197$	8.97008e7	0.835920	0.417960	+	0.908465i	$0.362745\pi$
$198$	0	0
$199$	−3.98955e7	−0.358871	−0.179436	−	0.983770i	$-0.557427\pi$
$199$	−3.98955e7	−0.358871	−0.179436	+	0.983770i	$0.557427\pi$
$200$	0	0
$201$	2.38833e7	0.207447
$202$	0	0
$203$	2.49178e7	0.209061
$204$	0	0
$205$	7.60078e7	0.616197
$206$	0	0
$207$	−7.52915e6	−0.0589997
$208$	0	0
$209$	−1.96613e8	−1.48971
$210$	0	0
$211$	6.63604e7	0.486318	0.243159	−	0.969986i	$-0.421816\pi$
$211$	6.63604e7	0.486318	0.243159	+	0.969986i	$0.421816\pi$
$212$	0	0
$213$	−1.23051e8	−0.872479
$214$	0	0
$215$	−1.77825e7	−0.122027
$216$	0	0
$217$	−7.67317e7	−0.509760
$218$	0	0
$219$	−476088.	−0.00306290
$220$	0	0
$221$	2.44718e8	1.52508
$222$	0	0
$223$	1.57269e8	0.949675	0.474837	−	0.880074i	$-0.342507\pi$
$223$	1.57269e8	0.949675	0.474837	+	0.880074i	$0.342507\pi$
$224$	0	0
$225$	7.49291e7	0.438543
$226$	0	0
$227$	1.95129e8	1.10721	0.553607	−	0.832778i	$-0.313251\pi$
$227$	1.95129e8	1.10721	0.553607	+	0.832778i	$0.313251\pi$
$228$	0	0
$229$	6.95136e7	0.382512	0.191256	−	0.981540i	$-0.438744\pi$
$229$	6.95136e7	0.382512	0.191256	+	0.981540i	$0.438744\pi$
$230$	0	0
$231$	9.98095e7	0.532758
$232$	0	0
$233$	3.81872e7	0.197776	0.0988878	−	0.995099i	$-0.468472\pi$
$233$	3.81872e7	0.197776	0.0988878	+	0.995099i	$0.468472\pi$
$234$	0	0
$235$	2.88785e7	0.145157
$236$	0	0
$237$	−5.13437e7	−0.250535
$238$	0	0
$239$	2.63549e8	1.24873	0.624366	−	0.781132i	$-0.285357\pi$
$239$	2.63549e8	1.24873	0.624366	+	0.781132i	$0.285357\pi$
$240$	0	0
$241$	1.21337e8	0.558386	0.279193	−	0.960235i	$-0.409933\pi$
$241$	1.21337e8	0.558386	0.279193	+	0.960235i	$0.409933\pi$
$242$	0	0
$243$	2.27512e8	1.01715
$244$	0	0
$245$	5.23441e7	0.227398
$246$	0	0
$247$	5.11645e8	2.16038
$248$	0	0
$249$	1.93555e7	0.0794523
$250$	0	0
$251$	3.62042e7	0.144511	0.0722555	−	0.997386i	$-0.476980\pi$
$251$	3.62042e7	0.144511	0.0722555	+	0.997386i	$0.476980\pi$
$252$	0	0
$253$	1.90893e7	0.0741085
$254$	0	0
$255$	−8.82295e7	−0.333214
$256$	0	0
$257$	1.75550e8	0.645111	0.322555	−	0.946551i	$-0.395458\pi$
$257$	1.75550e8	0.645111	0.322555	+	0.946551i	$0.395458\pi$
$258$	0	0
$259$	−5.08451e8	−1.81844
$260$	0	0
$261$	−4.04008e7	−0.140653
$262$	0	0
$263$	−1.10733e8	−0.375348	−0.187674	−	0.982231i	$-0.560095\pi$
$263$	−1.10733e8	−0.375348	−0.187674	+	0.982231i	$0.560095\pi$
$264$	0	0
$265$	1.27030e8	0.419320
$266$	0	0
$267$	2.95627e7	0.0950505
$268$	0	0
$269$	−3.07561e8	−0.963382	−0.481691	−	0.876341i	$-0.659977\pi$
$269$	−3.07561e8	−0.963382	−0.481691	+	0.876341i	$0.659977\pi$
$270$	0	0
$271$	2.86036e8	0.873027	0.436513	−	0.899698i	$-0.356213\pi$
$271$	2.86036e8	0.873027	0.436513	+	0.899698i	$0.356213\pi$
$272$	0	0
$273$	−2.59734e8	−0.772607
$274$	0	0
$275$	−1.89974e8	−0.550846
$276$	0	0
$277$	3.15642e8	0.892309	0.446155	−	0.894956i	$-0.352793\pi$
$277$	3.15642e8	0.892309	0.446155	+	0.894956i	$0.352793\pi$
$278$	0	0
$279$	1.24410e8	0.342957
$280$	0	0
$281$	2.50451e8	0.673366	0.336683	−	0.941618i	$-0.390695\pi$
$281$	2.50451e8	0.673366	0.336683	+	0.941618i	$0.390695\pi$
$282$	0	0
$283$	3.47387e7	0.0911088	0.0455544	−	0.998962i	$-0.485495\pi$
$283$	3.47387e7	0.0911088	0.0455544	+	0.998962i	$0.485495\pi$
$284$	0	0
$285$	−1.84467e8	−0.472020
$286$	0	0
$287$	4.32397e8	1.07968
$288$	0	0
$289$	6.25676e7	0.152478
$290$	0	0
$291$	1.60782e8	0.382482
$292$	0	0
$293$	−5.34336e8	−1.24102	−0.620508	−	0.784200i	$-0.713074\pi$
$293$	−5.34336e8	−1.24102	−0.620508	+	0.784200i	$0.713074\pi$
$294$	0	0
$295$	3.32777e8	0.754703
$296$	0	0
$297$	−3.69329e8	−0.818024
$298$	0	0
$299$	−4.96760e7	−0.107472
$300$	0	0
$301$	−1.01162e8	−0.213813
$302$	0	0
$303$	2.11954e8	0.437716
$304$	0	0
$305$	−2.52214e8	−0.509002
$306$	0	0
$307$	−1.68851e8	−0.333057	−0.166529	−	0.986037i	$-0.553256\pi$
$307$	−1.68851e8	−0.333057	−0.166529	+	0.986037i	$0.553256\pi$
$308$	0	0
$309$	−3.04229e8	−0.586605
$310$	0	0
$311$	6.99677e8	1.31897	0.659487	−	0.751716i	$-0.270774\pi$
$311$	6.99677e8	1.31897	0.659487	+	0.751716i	$0.270774\pi$
$312$	0	0
$313$	−5.06345e8	−0.933343	−0.466671	−	0.884431i	$-0.654547\pi$
$313$	−5.06345e8	−0.933343	−0.466671	+	0.884431i	$0.654547\pi$
$314$	0	0
$315$	−3.31779e8	−0.598084
$316$	0	0
$317$	4.78996e8	0.844548	0.422274	−	0.906468i	$-0.361232\pi$
$317$	4.78996e8	0.844548	0.422274	+	0.906468i	$0.361232\pi$
$318$	0	0
$319$	1.02431e8	0.176671
$320$	0	0
$321$	4.13263e8	0.697364
$322$	0	0
$323$	9.88733e8	1.63256
$324$	0	0
$325$	4.94369e8	0.798839
$326$	0	0
$327$	1.88827e8	0.298639
$328$	0	0
$329$	1.64285e8	0.254339
$330$	0	0
$331$	−4.20825e8	−0.637828	−0.318914	−	0.947784i	$-0.603318\pi$
$331$	−4.20825e8	−0.637828	−0.318914	+	0.947784i	$0.603318\pi$
$332$	0	0
$333$	8.24382e8	1.22341
$334$	0	0
$335$	2.01287e8	0.292522
$336$	0	0
$337$	−1.02987e9	−1.46580	−0.732902	−	0.680334i	$-0.761835\pi$
$337$	−1.02987e9	−1.46580	−0.732902	+	0.680334i	$0.761835\pi$
$338$	0	0
$339$	−2.88170e8	−0.401745
$340$	0	0
$341$	−3.15427e8	−0.430782
$342$	0	0
$343$	−5.68554e8	−0.760751
$344$	0	0
$345$	1.79100e7	0.0234816
$346$	0	0
$347$	5.31496e8	0.682883	0.341442	−	0.939903i	$-0.389085\pi$
$347$	5.31496e8	0.682883	0.341442	+	0.939903i	$0.389085\pi$
$348$	0	0
$349$	−1.31436e9	−1.65511	−0.827555	−	0.561385i	$-0.810269\pi$
$349$	−1.31436e9	−1.65511	−0.827555	+	0.561385i	$0.810269\pi$
$350$	0	0
$351$	9.61103e8	1.18630
$352$	0	0
$353$	1.42031e9	1.71858	0.859291	−	0.511486i	$-0.170905\pi$
$353$	1.42031e9	1.71858	0.859291	+	0.511486i	$0.170905\pi$
$354$	0	0
$355$	−1.03706e9	−1.23028
$356$	0	0
$357$	−5.01924e8	−0.583847
$358$	0	0
$359$	7.59524e8	0.866385	0.433193	−	0.901301i	$-0.357387\pi$
$359$	7.59524e8	0.866385	0.433193	+	0.901301i	$0.357387\pi$
$360$	0	0
$361$	1.17333e9	1.31264
$362$	0	0
$363$	−1.72702e7	−0.0189507
$364$	0	0
$365$	−4.01244e6	−0.00431900
$366$	0	0
$367$	−3.76159e8	−0.397228	−0.198614	−	0.980078i	$-0.563644\pi$
$367$	−3.76159e8	−0.397228	−0.198614	+	0.980078i	$0.563644\pi$
$368$	0	0
$369$	−7.01071e8	−0.726390
$370$	0	0
$371$	7.22655e8	0.734720
$372$	0	0
$373$	4.57130e8	0.456099	0.228049	−	0.973650i	$-0.426765\pi$
$373$	4.57130e8	0.456099	0.228049	+	0.973650i	$0.426765\pi$
$374$	0	0
$375$	−4.95206e8	−0.484927
$376$	0	0
$377$	−2.66557e8	−0.256209
$378$	0	0
$379$	1.91565e9	1.80750	0.903752	−	0.428056i	$-0.140801\pi$
$379$	1.91565e9	1.80750	0.903752	+	0.428056i	$0.140801\pi$
$380$	0	0
$381$	−6.99402e8	−0.647873
$382$	0	0
$383$	−1.57853e9	−1.43568	−0.717839	−	0.696209i	$-0.754869\pi$
$383$	−1.57853e9	−1.43568	−0.717839	+	0.696209i	$0.754869\pi$
$384$	0	0
$385$	8.41187e8	0.751243
$386$	0	0
$387$	1.64020e8	0.143849
$388$	0	0
$389$	2.21214e9	1.90541	0.952707	−	0.303890i	$-0.0982854\pi$
$389$	2.21214e9	1.90541	0.952707	+	0.303890i	$0.0982854\pi$
$390$	0	0
$391$	−9.59967e7	−0.0812153
$392$	0	0
$393$	2.56817e8	0.213427
$394$	0	0
$395$	−4.32721e8	−0.353279
$396$	0	0
$397$	1.30879e9	1.04979	0.524896	−	0.851167i	$-0.324104\pi$
$397$	1.30879e9	1.04979	0.524896	+	0.851167i	$0.324104\pi$
$398$	0	0
$399$	−1.04940e9	−0.827060
$400$	0	0
$401$	1.04514e8	0.0809413	0.0404706	−	0.999181i	$-0.487114\pi$
$401$	1.04514e8	0.0809413	0.0404706	+	0.999181i	$0.487114\pi$
$402$	0	0
$403$	8.20832e8	0.624722
$404$	0	0
$405$	3.43251e8	0.256755
$406$	0	0
$407$	−2.09013e9	−1.53671
$408$	0	0
$409$	−1.94107e9	−1.40284	−0.701421	−	0.712747i	$-0.747451\pi$
$409$	−1.94107e9	−1.40284	−0.701421	+	0.712747i	$0.747451\pi$
$410$	0	0
$411$	−8.31937e8	−0.591077
$412$	0	0
$413$	1.89312e9	1.32237
$414$	0	0
$415$	1.63127e8	0.112036
$416$	0	0
$417$	−6.15000e8	−0.415336
$418$	0	0
$419$	−1.96429e9	−1.30454	−0.652268	−	0.757989i	$-0.726182\pi$
$419$	−1.96429e9	−1.30454	−0.652268	+	0.757989i	$0.726182\pi$
$420$	0	0
$421$	−2.33176e9	−1.52299	−0.761493	−	0.648174i	$-0.775533\pi$
$421$	−2.33176e9	−1.52299	−0.761493	+	0.648174i	$0.775533\pi$
$422$	0	0
$423$	−2.66366e8	−0.171115
$424$	0	0
$425$	9.55346e8	0.603670
$426$	0	0
$427$	−1.43480e9	−0.891857
$428$	0	0
$429$	−1.06771e9	−0.652907
$430$	0	0
$431$	−1.72745e8	−0.103929	−0.0519643	−	0.998649i	$-0.516548\pi$
$431$	−1.72745e8	−0.103929	−0.0519643	+	0.998649i	$0.516548\pi$
$432$	0	0
$433$	−2.71381e9	−1.60647	−0.803233	−	0.595665i	$-0.796889\pi$
$433$	−2.71381e9	−1.60647	−0.803233	+	0.595665i	$0.796889\pi$
$434$	0	0
$435$	9.61033e7	0.0559791
$436$	0	0
$437$	−2.00706e8	−0.115047
$438$	0	0
$439$	−2.73808e8	−0.154461	−0.0772307	−	0.997013i	$-0.524608\pi$
$439$	−2.73808e8	−0.154461	−0.0772307	+	0.997013i	$0.524608\pi$
$440$	0	0
$441$	−4.82804e8	−0.268063
$442$	0	0
$443$	−2.72048e9	−1.48673	−0.743365	−	0.668886i	$-0.766771\pi$
$443$	−2.72048e9	−1.48673	−0.743365	+	0.668886i	$0.766771\pi$
$444$	0	0
$445$	2.49152e8	0.134031
$446$	0	0
$447$	−1.49654e8	−0.0792526
$448$	0	0
$449$	1.69111e9	0.881675	0.440837	−	0.897587i	$-0.354682\pi$
$449$	1.69111e9	0.881675	0.440837	+	0.897587i	$0.354682\pi$
$450$	0	0
$451$	1.77748e9	0.912406
$452$	0	0
$453$	−1.67204e9	−0.845089
$454$	0	0
$455$	−2.18902e9	−1.08945
$456$	0	0
$457$	1.15773e8	0.0567413	0.0283706	−	0.999597i	$-0.490968\pi$
$457$	1.15773e8	0.0567413	0.0283706	+	0.999597i	$0.490968\pi$
$458$	0	0
$459$	1.85729e9	0.896470
$460$	0	0
$461$	−9.34713e8	−0.444350	−0.222175	−	0.975007i	$-0.571316\pi$
$461$	−9.34713e8	−0.444350	−0.222175	+	0.975007i	$0.571316\pi$
$462$	0	0
$463$	−1.91738e9	−0.897790	−0.448895	−	0.893584i	$-0.648182\pi$
$463$	−1.91738e9	−0.897790	−0.448895	+	0.893584i	$0.648182\pi$
$464$	0	0
$465$	−2.95940e8	−0.136495
$466$	0	0
$467$	−1.65838e9	−0.753485	−0.376742	−	0.926318i	$-0.622956\pi$
$467$	−1.65838e9	−0.753485	−0.376742	+	0.926318i	$0.622956\pi$
$468$	0	0
$469$	1.14509e9	0.512548
$470$	0	0
$471$	9.40684e8	0.414830
$472$	0	0
$473$	−4.15853e8	−0.180686
$474$	0	0
$475$	1.99740e9	0.855140
$476$	0	0
$477$	−1.17168e9	−0.494306
$478$	0	0
$479$	−4.09755e9	−1.70353	−0.851766	−	0.523923i	$-0.824468\pi$
$479$	−4.09755e9	−1.70353	−0.851766	+	0.523923i	$0.824468\pi$
$480$	0	0
$481$	5.43912e9	2.22854
$482$	0	0
$483$	1.01887e8	0.0411438
$484$	0	0
$485$	1.35506e9	0.539339
$486$	0	0
$487$	−3.24167e9	−1.27180	−0.635898	−	0.771773i	$-0.719370\pi$
$487$	−3.24167e9	−1.27180	−0.635898	+	0.771773i	$0.719370\pi$
$488$	0	0
$489$	9.04239e8	0.349705
$490$	0	0
$491$	2.16351e9	0.824847	0.412423	−	0.910992i	$-0.364682\pi$
$491$	2.16351e9	0.824847	0.412423	+	0.910992i	$0.364682\pi$
$492$	0	0
$493$	−5.15110e8	−0.193613
$494$	0	0
$495$	−1.36387e9	−0.505422
$496$	0	0
$497$	−5.89969e9	−2.15567
$498$	0	0
$499$	2.16713e9	0.780787	0.390394	−	0.920648i	$-0.372339\pi$
$499$	2.16713e9	0.780787	0.390394	+	0.920648i	$0.372339\pi$
$500$	0	0
$501$	5.30316e8	0.188409
$502$	0	0
$503$	−2.58480e9	−0.905606	−0.452803	−	0.891611i	$-0.649576\pi$
$503$	−2.58480e9	−0.905606	−0.452803	+	0.891611i	$0.649576\pi$
$504$	0	0
$505$	1.78633e9	0.617224
$506$	0	0
$507$	1.40173e9	0.477680
$508$	0	0
$509$	−2.11747e9	−0.711712	−0.355856	−	0.934541i	$-0.615811\pi$
$509$	−2.11747e9	−0.711712	−0.355856	+	0.934541i	$0.615811\pi$
$510$	0	0
$511$	−2.28261e7	−0.00756762
$512$	0	0
$513$	3.88315e9	1.26991
$514$	0	0
$515$	−2.56402e9	−0.827172
$516$	0	0
$517$	6.75339e8	0.214934
$518$	0	0
$519$	−1.29550e9	−0.406773
$520$	0	0
$521$	2.34354e9	0.726006	0.363003	−	0.931788i	$-0.381751\pi$
$521$	2.34354e9	0.726006	0.363003	+	0.931788i	$0.381751\pi$
$522$	0	0
$523$	−3.26155e9	−0.996939	−0.498470	−	0.866907i	$-0.666104\pi$
$523$	−3.26155e9	−0.996939	−0.498470	+	0.866907i	$0.666104\pi$
$524$	0	0
$525$	−1.01397e9	−0.305820
$526$	0	0
$527$	1.58622e9	0.472093
$528$	0	0
$529$	−3.38534e9	−0.994277
$530$	0	0
$531$	−3.06942e9	−0.889664
$532$	0	0
$533$	−4.62554e9	−1.32317
$534$	0	0
$535$	3.48296e9	0.983354
$536$	0	0
$537$	−2.63973e9	−0.735612
$538$	0	0
$539$	1.22409e9	0.336709
$540$	0	0
$541$	7.24638e9	1.96757	0.983787	−	0.179344i	$-0.0573974\pi$
$541$	7.24638e9	1.96757	0.983787	+	0.179344i	$0.0573974\pi$
$542$	0	0
$543$	−7.05482e7	−0.0189098
$544$	0	0
$545$	1.59142e9	0.421111
$546$	0	0
$547$	6.41460e9	1.67577	0.837884	−	0.545848i	$-0.183793\pi$
$547$	6.41460e9	1.67577	0.837884	+	0.545848i	$0.183793\pi$
$548$	0	0
$549$	2.32634e9	0.600025
$550$	0	0
$551$	−1.07697e9	−0.274267
$552$	0	0
$553$	−2.46168e9	−0.619005
$554$	0	0
$555$	−1.96100e9	−0.486914
$556$	0	0
$557$	−1.76223e9	−0.432085	−0.216042	−	0.976384i	$-0.569315\pi$
$557$	−1.76223e9	−0.432085	−0.216042	+	0.976384i	$0.569315\pi$
$558$	0	0
$559$	1.08217e9	0.262032
$560$	0	0
$561$	−2.06330e9	−0.493391
$562$	0	0
$563$	2.33270e9	0.550909	0.275454	−	0.961314i	$-0.411172\pi$
$563$	2.33270e9	0.550909	0.275454	+	0.961314i	$0.411172\pi$
$564$	0	0
$565$	−2.42868e9	−0.566501
$566$	0	0
$567$	1.95270e9	0.449878
$568$	0	0
$569$	−5.53591e9	−1.25978	−0.629892	−	0.776683i	$-0.716901\pi$
$569$	−5.53591e9	−1.25978	−0.629892	+	0.776683i	$0.716901\pi$
$570$	0	0
$571$	−3.70890e9	−0.833717	−0.416858	−	0.908971i	$-0.636869\pi$
$571$	−3.70890e9	−0.833717	−0.416858	+	0.908971i	$0.636869\pi$
$572$	0	0
$573$	−1.54490e9	−0.343051
$574$	0	0
$575$	−1.93929e8	−0.0425407
$576$	0	0
$577$	3.63713e9	0.788212	0.394106	−	0.919065i	$-0.371054\pi$
$577$	3.63713e9	0.788212	0.394106	+	0.919065i	$0.371054\pi$
$578$	0	0
$579$	2.58596e8	0.0553665
$580$	0	0
$581$	9.28003e8	0.196306
$582$	0	0
$583$	2.97067e9	0.620889
$584$	0	0
$585$	3.54919e9	0.732965
$586$	0	0
$587$	2.04311e9	0.416926	0.208463	−	0.978030i	$-0.433154\pi$
$587$	2.04311e9	0.416926	0.208463	+	0.978030i	$0.433154\pi$
$588$	0	0
$589$	3.31641e9	0.668752
$590$	0	0
$591$	1.96811e9	0.392187
$592$	0	0
$593$	−5.74764e9	−1.13187	−0.565937	−	0.824448i	$-0.691486\pi$
$593$	−5.74764e9	−1.13187	−0.565937	+	0.824448i	$0.691486\pi$
$594$	0	0
$595$	−4.23018e9	−0.823284
$596$	0	0
$597$	−8.75341e8	−0.168371
$598$	0	0
$599$	3.14564e8	0.0598020	0.0299010	−	0.999553i	$-0.490481\pi$
$599$	3.14564e8	0.0598020	0.0299010	+	0.999553i	$0.490481\pi$
$600$	0	0
$601$	2.49586e9	0.468985	0.234493	−	0.972118i	$-0.424657\pi$
$601$	2.49586e9	0.468985	0.234493	+	0.972118i	$0.424657\pi$
$602$	0	0
$603$	−1.85660e9	−0.344833
$604$	0	0
$605$	−1.45552e8	−0.0267224
$606$	0	0
$607$	2.81126e9	0.510200	0.255100	−	0.966915i	$-0.417892\pi$
$607$	2.81126e9	0.510200	0.255100	+	0.966915i	$0.417892\pi$
$608$	0	0
$609$	5.46717e8	0.0980849
$610$	0	0
$611$	−1.75743e9	−0.311698
$612$	0	0
$613$	2.34343e9	0.410904	0.205452	−	0.978667i	$-0.434134\pi$
$613$	2.34343e9	0.410904	0.205452	+	0.978667i	$0.434134\pi$
$614$	0	0
$615$	1.66767e9	0.289100
$616$	0	0
$617$	6.26737e8	0.107420	0.0537102	−	0.998557i	$-0.482895\pi$
$617$	6.26737e8	0.107420	0.0537102	+	0.998557i	$0.482895\pi$
$618$	0	0
$619$	8.90879e9	1.50974	0.754869	−	0.655876i	$-0.227700\pi$
$619$	8.90879e9	1.50974	0.754869	+	0.655876i	$0.227700\pi$
$620$	0	0
$621$	−3.77017e8	−0.0631743
$622$	0	0
$623$	1.41739e9	0.234845
$624$	0	0
$625$	−7.41437e8	−0.121477
$626$	0	0
$627$	−4.31385e9	−0.698922
$628$	0	0
$629$	1.05109e10	1.68407
$630$	0	0
$631$	−2.21942e9	−0.351671	−0.175835	−	0.984420i	$-0.556263\pi$
$631$	−2.21942e9	−0.351671	−0.175835	+	0.984420i	$0.556263\pi$
$632$	0	0
$633$	1.45600e9	0.228165
$634$	0	0
$635$	−5.89451e9	−0.913566
$636$	0	0
$637$	−3.18545e9	−0.488296
$638$	0	0
$639$	9.56552e9	1.45029
$640$	0	0
$641$	8.58104e9	1.28688	0.643438	−	0.765498i	$-0.277507\pi$
$641$	8.58104e9	1.28688	0.643438	+	0.765498i	$0.277507\pi$
$642$	0	0
$643$	−7.71279e9	−1.14412	−0.572062	−	0.820210i	$-0.693856\pi$
$643$	−7.71279e9	−1.14412	−0.572062	+	0.820210i	$0.693856\pi$
$644$	0	0
$645$	−3.90161e8	−0.0572513
$646$	0	0
$647$	1.18898e10	1.72587	0.862935	−	0.505316i	$-0.168624\pi$
$647$	1.18898e10	1.72587	0.862935	+	0.505316i	$0.168624\pi$
$648$	0	0
$649$	7.78217e9	1.11749
$650$	0	0
$651$	−1.68356e9	−0.239163
$652$	0	0
$653$	2.00345e9	0.281567	0.140784	−	0.990040i	$-0.455038\pi$
$653$	2.00345e9	0.281567	0.140784	+	0.990040i	$0.455038\pi$
$654$	0	0
$655$	2.16443e9	0.300954
$656$	0	0
$657$	3.70094e7	0.00509135
$658$	0	0
$659$	7.66944e9	1.04391	0.521957	−	0.852972i	$-0.325202\pi$
$659$	7.66944e9	1.04391	0.521957	+	0.852972i	$0.325202\pi$
$660$	0	0
$661$	4.04041e9	0.544152	0.272076	−	0.962276i	$-0.412290\pi$
$661$	4.04041e9	0.544152	0.272076	+	0.962276i	$0.412290\pi$
$662$	0	0
$663$	5.36930e9	0.715518
$664$	0	0
$665$	−8.84429e9	−1.16624
$666$	0	0
$667$	1.04564e8	0.0136440
$668$	0	0
$669$	3.45060e9	0.445557
$670$	0	0
$671$	−5.89816e9	−0.753681
$672$	0	0
$673$	4.11138e9	0.519918	0.259959	−	0.965620i	$-0.416291\pi$
$673$	4.11138e9	0.519918	0.259959	+	0.965620i	$0.416291\pi$
$674$	0	0
$675$	3.75203e9	0.469572
$676$	0	0
$677$	−1.12671e10	−1.39557	−0.697787	−	0.716305i	$-0.745832\pi$
$677$	−1.12671e10	−1.39557	−0.697787	+	0.716305i	$0.745832\pi$
$678$	0	0
$679$	7.70872e9	0.945014
$680$	0	0
$681$	4.28129e9	0.519469
$682$	0	0
$683$	8.00228e9	0.961039	0.480520	−	0.876984i	$-0.340448\pi$
$683$	8.00228e9	0.961039	0.480520	+	0.876984i	$0.340448\pi$
$684$	0	0
$685$	−7.01151e9	−0.833479
$686$	0	0
$687$	1.52518e9	0.179462
$688$	0	0
$689$	−7.73055e9	−0.900416
$690$	0	0
$691$	7.24716e9	0.835593	0.417797	−	0.908541i	$-0.362803\pi$
$691$	7.24716e9	0.835593	0.417797	+	0.908541i	$0.362803\pi$
$692$	0	0
$693$	−7.75884e9	−0.885585
$694$	0	0
$695$	−5.18318e9	−0.585665
$696$	0	0
$697$	−8.93865e9	−0.999902
$698$	0	0
$699$	8.37859e8	0.0927899
$700$	0	0
$701$	4.68293e8	0.0513458	0.0256729	−	0.999670i	$-0.491827\pi$
$701$	4.68293e8	0.0513458	0.0256729	+	0.999670i	$0.491827\pi$
$702$	0	0
$703$	2.19757e10	2.38561
$704$	0	0
$705$	6.33617e8	0.0681028
$706$	0	0
$707$	1.01622e10	1.08148
$708$	0	0
$709$	−1.08926e10	−1.14781	−0.573903	−	0.818923i	$-0.694571\pi$
$709$	−1.08926e10	−1.14781	−0.573903	+	0.818923i	$0.694571\pi$
$710$	0	0
$711$	3.99128e9	0.416455
$712$	0	0
$713$	−3.21992e8	−0.0332684
$714$	0	0
$715$	−8.99855e9	−0.920664
$716$	0	0
$717$	5.78249e9	0.585865
$718$	0	0
$719$	7.40284e9	0.742758	0.371379	−	0.928481i	$-0.378885\pi$
$719$	7.40284e9	0.742758	0.371379	+	0.928481i	$0.378885\pi$
$720$	0	0
$721$	−1.45863e10	−1.44935
$722$	0	0
$723$	2.66224e9	0.261977
$724$	0	0
$725$	−1.04060e9	−0.101415
$726$	0	0
$727$	1.04295e10	1.00668	0.503342	−	0.864087i	$-0.332103\pi$
$727$	1.04295e10	1.00668	0.503342	+	0.864087i	$0.332103\pi$
$728$	0	0
$729$	9.32176e8	0.0891152
$730$	0	0
$731$	2.09125e9	0.198014
$732$	0	0
$733$	−1.55538e10	−1.45873	−0.729363	−	0.684127i	$-0.760184\pi$
$733$	−1.55538e10	−1.45873	−0.729363	+	0.684127i	$0.760184\pi$
$734$	0	0
$735$	1.14847e9	0.106688
$736$	0	0
$737$	4.70720e9	0.433138
$738$	0	0
$739$	−1.07140e10	−0.976554	−0.488277	−	0.872689i	$-0.662374\pi$
$739$	−1.07140e10	−0.976554	−0.488277	+	0.872689i	$0.662374\pi$
$740$	0	0
$741$	1.12259e10	1.01358
$742$	0	0
$743$	−5.46982e9	−0.489229	−0.244615	−	0.969620i	$-0.578661\pi$
$743$	−5.46982e9	−0.489229	−0.244615	+	0.969620i	$0.578661\pi$
$744$	0	0
$745$	−1.26128e9	−0.111754
$746$	0	0
$747$	−1.50463e9	−0.132071
$748$	0	0
$749$	1.98140e10	1.72300
$750$	0	0
$751$	−5.68185e9	−0.489497	−0.244749	−	0.969587i	$-0.578705\pi$
$751$	−5.68185e9	−0.489497	−0.244749	+	0.969587i	$0.578705\pi$
$752$	0	0
$753$	7.94349e8	0.0677999
$754$	0	0
$755$	−1.40918e10	−1.19166
$756$	0	0
$757$	5.38304e9	0.451016	0.225508	−	0.974241i	$-0.427596\pi$
$757$	5.38304e9	0.451016	0.225508	+	0.974241i	$0.427596\pi$
$758$	0	0
$759$	4.18835e8	0.0347693
$760$	0	0
$761$	3.28593e9	0.270279	0.135139	−	0.990827i	$-0.456852\pi$
$761$	3.28593e9	0.270279	0.135139	+	0.990827i	$0.456852\pi$
$762$	0	0
$763$	9.05334e9	0.737858
$764$	0	0
$765$	6.85865e9	0.553890
$766$	0	0
$767$	−2.02515e10	−1.62059
$768$	0	0
$769$	7.10914e9	0.563735	0.281868	−	0.959453i	$-0.409046\pi$
$769$	7.10914e9	0.563735	0.281868	+	0.959453i	$0.409046\pi$
$770$	0	0
$771$	3.85170e9	0.302665
$772$	0	0
$773$	1.08690e10	0.846370	0.423185	−	0.906043i	$-0.360912\pi$
$773$	1.08690e10	0.846370	0.423185	+	0.906043i	$0.360912\pi$
$774$	0	0
$775$	3.20443e9	0.247283
$776$	0	0
$777$	−1.11558e10	−0.853155
$778$	0	0
$779$	−1.86886e10	−1.41643
$780$	0	0
$781$	−2.42523e10	−1.82169
$782$	0	0
$783$	−2.02304e9	−0.150605
$784$	0	0
$785$	7.92802e9	0.584953
$786$	0	0
$787$	−1.94246e10	−1.42050	−0.710248	−	0.703951i	$-0.751417\pi$
$787$	−1.94246e10	−1.42050	−0.710248	+	0.703951i	$0.751417\pi$
$788$	0	0
$789$	−2.42958e9	−0.176101
$790$	0	0
$791$	−1.38164e10	−0.992606
$792$	0	0
$793$	1.53487e10	1.09299
$794$	0	0
$795$	2.78714e9	0.196732
$796$	0	0
$797$	−2.30309e10	−1.61141	−0.805707	−	0.592314i	$-0.798214\pi$
$797$	−2.30309e10	−1.61141	−0.805707	+	0.592314i	$0.798214\pi$
$798$	0	0
$799$	−3.39616e9	−0.235545
$800$	0	0
$801$	−2.29810e9	−0.157999
$802$	0	0
$803$	−9.38331e7	−0.00639516
$804$	0	0
$805$	8.58697e8	0.0580169
$806$	0	0
$807$	−6.74814e9	−0.451988
$808$	0	0
$809$	1.50546e10	0.999656	0.499828	−	0.866125i	$-0.333397\pi$
$809$	1.50546e10	0.999656	0.499828	+	0.866125i	$0.333397\pi$
$810$	0	0
$811$	1.98821e10	1.30885	0.654424	−	0.756128i	$-0.272911\pi$
$811$	1.98821e10	1.30885	0.654424	+	0.756128i	$0.272911\pi$
$812$	0	0
$813$	6.27585e9	0.409596
$814$	0	0
$815$	7.62086e9	0.493120
$816$	0	0
$817$	4.37230e9	0.280500
$818$	0	0
$819$	2.01908e10	1.28428
$820$	0	0
$821$	−2.29870e10	−1.44971	−0.724855	−	0.688901i	$-0.758093\pi$
$821$	−2.29870e10	−1.44971	−0.724855	+	0.688901i	$0.758093\pi$
$822$	0	0
$823$	1.46696e10	0.917316	0.458658	−	0.888613i	$-0.348330\pi$
$823$	1.46696e10	0.917316	0.458658	+	0.888613i	$0.348330\pi$
$824$	0	0
$825$	−4.16819e9	−0.258439
$826$	0	0
$827$	−1.26591e9	−0.0778277	−0.0389138	−	0.999243i	$-0.512390\pi$
$827$	−1.26591e9	−0.0778277	−0.0389138	+	0.999243i	$0.512390\pi$
$828$	0	0
$829$	2.49778e10	1.52270	0.761349	−	0.648342i	$-0.224537\pi$
$829$	2.49778e10	1.52270	0.761349	+	0.648342i	$0.224537\pi$
$830$	0	0
$831$	6.92544e9	0.418643
$832$	0	0
$833$	−6.15576e9	−0.368998
$834$	0	0
$835$	4.46947e9	0.265676
$836$	0	0
$837$	6.22973e9	0.367223
$838$	0	0
$839$	−3.09477e10	−1.80909	−0.904547	−	0.426373i	$-0.859791\pi$
$839$	−3.09477e10	−1.80909	−0.904547	+	0.426373i	$0.859791\pi$
$840$	0	0
$841$	−1.66888e10	−0.967473
$842$	0	0
$843$	5.49510e9	0.315921
$844$	0	0
$845$	1.18137e10	0.673577
$846$	0	0
$847$	−8.28024e8	−0.0468221
$848$	0	0
$849$	7.62194e8	0.0427453
$850$	0	0
$851$	−2.13363e9	−0.118677
$852$	0	0
$853$	1.86385e10	1.02823	0.514113	−	0.857723i	$-0.328121\pi$
$853$	1.86385e10	1.02823	0.514113	+	0.857723i	$0.328121\pi$
$854$	0	0
$855$	1.43398e10	0.784623
$856$	0	0
$857$	2.37366e10	1.28821	0.644103	−	0.764939i	$-0.277231\pi$
$857$	2.37366e10	1.28821	0.644103	+	0.764939i	$0.277231\pi$
$858$	0	0
$859$	−2.46268e10	−1.32566	−0.662830	−	0.748770i	$-0.730645\pi$
$859$	−2.46268e10	−1.32566	−0.662830	+	0.748770i	$0.730645\pi$
$860$	0	0
$861$	9.48713e9	0.506552
$862$	0	0
$863$	−2.50051e10	−1.32431	−0.662157	−	0.749365i	$-0.730359\pi$
$863$	−2.50051e10	−1.32431	−0.662157	+	0.749365i	$0.730359\pi$
$864$	0	0
$865$	−1.09184e10	−0.573592
$866$	0	0
$867$	1.37278e9	0.0715378
$868$	0	0
$869$	−1.01194e10	−0.523102
$870$	0	0
$871$	−1.22495e10	−0.628139
$872$	0	0
$873$	−1.24986e10	−0.635788
$874$	0	0
$875$	−2.37428e10	−1.19813
$876$	0	0
$877$	1.07280e10	0.537057	0.268528	−	0.963272i	$-0.413463\pi$
$877$	1.07280e10	0.537057	0.268528	+	0.963272i	$0.413463\pi$
$878$	0	0
$879$	−1.17238e10	−0.582245
$880$	0	0
$881$	5.94499e9	0.292911	0.146455	−	0.989217i	$-0.453214\pi$
$881$	5.94499e9	0.292911	0.146455	+	0.989217i	$0.453214\pi$
$882$	0	0
$883$	1.60870e10	0.786342	0.393171	−	0.919465i	$-0.371378\pi$
$883$	1.60870e10	0.786342	0.393171	+	0.919465i	$0.371378\pi$
$884$	0	0
$885$	7.30139e9	0.354082
$886$	0	0
$887$	−2.61884e10	−1.26001	−0.630007	−	0.776589i	$-0.716948\pi$
$887$	−2.61884e10	−1.26001	−0.630007	+	0.776589i	$0.716948\pi$
$888$	0	0
$889$	−3.35330e10	−1.60072
$890$	0	0
$891$	8.02711e9	0.380178
$892$	0	0
$893$	−7.10056e9	−0.333666
$894$	0	0
$895$	−2.22474e10	−1.03729
$896$	0	0
$897$	−1.08993e9	−0.0504226
$898$	0	0
$899$	−1.72778e9	−0.0793104
$900$	0	0
$901$	−1.49390e10	−0.680430
$902$	0	0
$903$	−2.21957e9	−0.100314
$904$	0	0
$905$	−5.94575e8	−0.0266647
$906$	0	0
$907$	1.79753e9	0.0799929	0.0399964	−	0.999200i	$-0.487265\pi$
$907$	1.79753e9	0.0799929	0.0399964	+	0.999200i	$0.487265\pi$
$908$	0	0
$909$	−1.64766e10	−0.727601
$910$	0	0
$911$	−3.70021e10	−1.62148	−0.810740	−	0.585406i	$-0.800935\pi$
$911$	−3.70021e10	−1.62148	−0.810740	+	0.585406i	$0.800935\pi$
$912$	0	0
$913$	3.81481e9	0.165892
$914$	0	0
$915$	−5.53377e9	−0.238807
$916$	0	0
$917$	1.23131e10	0.527322
$918$	0	0
$919$	−2.44812e10	−1.04047	−0.520234	−	0.854024i	$-0.674155\pi$
$919$	−2.44812e10	−1.04047	−0.520234	+	0.854024i	$0.674155\pi$
$920$	0	0
$921$	−3.70472e9	−0.156260
$922$	0	0
$923$	6.31115e10	2.64182
$924$	0	0
$925$	2.12336e10	0.882121
$926$	0	0
$927$	2.36497e10	0.975093
$928$	0	0
$929$	−6.19981e9	−0.253702	−0.126851	−	0.991922i	$-0.540487\pi$
$929$	−6.19981e9	−0.253702	−0.126851	+	0.991922i	$0.540487\pi$
$930$	0	0
$931$	−1.28702e10	−0.522711
$932$	0	0
$933$	1.53515e10	0.618820
$934$	0	0
$935$	−1.73893e10	−0.695732
$936$	0	0
$937$	2.78747e10	1.10693	0.553466	−	0.832871i	$-0.313305\pi$
$937$	2.78747e10	1.10693	0.553466	+	0.832871i	$0.313305\pi$
$938$	0	0
$939$	−1.11096e10	−0.437894
$940$	0	0
$941$	−3.45921e10	−1.35336	−0.676680	−	0.736277i	$-0.736582\pi$
$941$	−3.45921e10	−1.35336	−0.676680	+	0.736277i	$0.736582\pi$
$942$	0	0
$943$	1.81448e9	0.0704632
$944$	0	0
$945$	−1.66136e10	−0.640402
$946$	0	0
$947$	2.20475e10	0.843598	0.421799	−	0.906689i	$-0.361399\pi$
$947$	2.20475e10	0.843598	0.421799	+	0.906689i	$0.361399\pi$
$948$	0	0
$949$	2.44181e8	0.00927429
$950$	0	0
$951$	1.05096e10	0.396235
$952$	0	0
$953$	3.95876e9	0.148161	0.0740805	−	0.997252i	$-0.476398\pi$
$953$	3.95876e9	0.148161	0.0740805	+	0.997252i	$0.476398\pi$
$954$	0	0
$955$	−1.30203e10	−0.483737
$956$	0	0
$957$	2.24743e9	0.0828885
$958$	0	0
$959$	−3.98874e10	−1.46040
$960$	0	0
$961$	−2.21921e10	−0.806615
$962$	0	0
$963$	−3.21256e10	−1.15920
$964$	0	0
$965$	2.17943e9	0.0780724
$966$	0	0
$967$	−9.76598e9	−0.347315	−0.173657	−	0.984806i	$-0.555559\pi$
$967$	−9.76598e9	−0.347315	−0.173657	+	0.984806i	$0.555559\pi$
$968$	0	0
$969$	2.16936e10	0.765946
$970$	0	0
$971$	2.15553e10	0.755589	0.377795	−	0.925889i	$-0.376683\pi$
$971$	2.15553e10	0.755589	0.377795	+	0.925889i	$0.376683\pi$
$972$	0	0
$973$	−2.94863e10	−1.02619
$974$	0	0
$975$	1.08468e10	0.374789
$976$	0	0
$977$	5.24340e10	1.79880	0.899398	−	0.437131i	$-0.144005\pi$
$977$	5.24340e10	1.79880	0.899398	+	0.437131i	$0.144005\pi$
$978$	0	0
$979$	5.82657e9	0.198460
$980$	0	0
$981$	−1.46787e10	−0.496417
$982$	0	0
$983$	−2.85116e10	−0.957380	−0.478690	−	0.877984i	$-0.658888\pi$
$983$	−2.85116e10	−0.957380	−0.478690	+	0.877984i	$0.658888\pi$
$984$	0	0
$985$	1.65871e10	0.553023
$986$	0	0
$987$	3.60455e9	0.119328
$988$	0	0
$989$	−4.24509e8	−0.0139540
$990$	0	0
$991$	2.72290e10	0.888738	0.444369	−	0.895844i	$-0.353428\pi$
$991$	2.72290e10	0.888738	0.444369	+	0.895844i	$0.353428\pi$
$992$	0	0
$993$	−9.23324e9	−0.299248
$994$	0	0
$995$	−7.37731e9	−0.237420
$996$	0	0
$997$	−1.30057e10	−0.415624	−0.207812	−	0.978169i	$-0.566634\pi$
$997$	−1.30057e10	−0.415624	−0.207812	+	0.978169i	$0.566634\pi$
$998$	0	0
$999$	4.12804e10	1.30998

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	256.8.a.q.1.4		6
4.3	odd	2		256.8.a.r.1.3		6
8.3	odd	2		256.8.a.r.1.4		6
8.5	even	2	inner	256.8.a.q.1.3		6
16.3	odd	4		8.8.b.a.5.3	✓	6
16.5	even	4		32.8.b.a.17.3		6
16.11	odd	4		8.8.b.a.5.4	yes	6
16.13	even	4		32.8.b.a.17.4		6
48.5	odd	4		288.8.d.b.145.3		6
48.11	even	4		72.8.d.b.37.3		6
48.29	odd	4		288.8.d.b.145.4		6
48.35	even	4		72.8.d.b.37.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.8.b.a.5.3	✓	6	16.3	odd	4
8.8.b.a.5.4	yes	6	16.11	odd	4
32.8.b.a.17.3		6	16.5	even	4
32.8.b.a.17.4		6	16.13	even	4
72.8.d.b.37.3		6	48.11	even	4
72.8.d.b.37.4		6	48.35	even	4
256.8.a.q.1.3		6	8.5	even	2	inner
256.8.a.q.1.4		6	1.1	even	1	trivial
256.8.a.r.1.3		6	4.3	odd	2
256.8.a.r.1.4		6	8.3	odd	2
288.8.d.b.145.3		6	48.5	odd	4
288.8.d.b.145.4		6	48.29	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 256 = 2^{8} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	256.a (trivial)

\(f(q)\)	\(=\)	\(q+21.9408 q^{3} +184.916 q^{5} +1051.96 q^{7} -1705.60 q^{9} +O(q^{10})\)	\(q+21.9408 q^{3} +184.916 q^{5} +1051.96 q^{7} -1705.60 q^{9} +4324.35 q^{11} -11253.2 q^{13} +4057.20 q^{15} -21746.4 q^{17} -45466.5 q^{19} +23080.8 q^{21} +4414.37 q^{23} -43931.2 q^{25} -85406.8 q^{27} +23687.1 q^{29} -72941.9 q^{31} +94879.8 q^{33} +194523. q^{35} -483338. q^{37} -246905. q^{39} +411040. q^{41} -96165.3 q^{43} -315392. q^{45} +156171. q^{47} +283070. q^{49} -477134. q^{51} +686962. q^{53} +799641. q^{55} -997572. q^{57} +1.79961e6 q^{59} -1.36394e6 q^{61} -1.79422e6 q^{63} -2.08090e6 q^{65} +1.08853e6 q^{67} +96854.8 q^{69} -5.60830e6 q^{71} -21698.7 q^{73} -963886. q^{75} +4.54903e6 q^{77} -2.34010e6 q^{79} +1.85626e6 q^{81} +882169. q^{83} -4.02125e6 q^{85} +519714. q^{87} +1.34738e6 q^{89} -1.18379e7 q^{91} -1.60040e6 q^{93} -8.40746e6 q^{95} +7.32798e6 q^{97} -7.37562e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 688 q^{7} + 2918 q^{9}+O(q^{10}) \) 6 * q - 688 * q^7 + 2918 * q^9	\( 6 q - 688 q^{7} + 2918 q^{9} - 17872 q^{15} + 1452 q^{17} - 1296 q^{23} + 39314 q^{25} + 89280 q^{31} + 53880 q^{33} - 328208 q^{39} - 521244 q^{41} - 1566432 q^{47} - 511050 q^{49} - 3270256 q^{55} + 1889896 q^{57} - 5776816 q^{63} + 1416480 q^{65} - 7597104 q^{71} - 2089564 q^{73} - 16015904 q^{79} - 723058 q^{81} - 37453776 q^{87} - 2169084 q^{89} - 48537936 q^{95} - 1088308 q^{97}+O(q^{100}) \) 6 * q - 688 * q^7 + 2918 * q^9 - 17872 * q^15 + 1452 * q^17 - 1296 * q^23 + 39314 * q^25 + 89280 * q^31 + 53880 * q^33 - 328208 * q^39 - 521244 * q^41 - 1566432 * q^47 - 511050 * q^49 - 3270256 * q^55 + 1889896 * q^57 - 5776816 * q^63 + 1416480 * q^65 - 7597104 * q^71 - 2089564 * q^73 - 16015904 * q^79 - 723058 * q^81 - 37453776 * q^87 - 2169084 * q^89 - 48537936 * q^95 - 1088308 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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