LMFDB - Embedded newform 32.8.b.a.17.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [32,8,Mod(17,32)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(32, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("32.17");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 32 = 2^{5} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	32.b (of order $2$, degree $1$, not 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:	$9.99632081549$
Analytic rank:	$0$
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} - 3x^{5} - 10x^{4} - 24x^{3} - 320x^{2} - 3072x + 32768 $ x^6 - 3x^5 - 10x^4 - 24x^3 - 320x^2 - 3072*x + 32768
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{30} $
Twist minimal:	no (minimal twist has level 8)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			17.3
Root			$0.776001 - 5.60338i$ of defining polynomial
Character	$\chi$	$=$	32.17
Dual form			32.8.b.a.17.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-21.9408i q^{3} +184.916i q^{5} -1051.96 q^{7} +1705.60 q^{9} +O(q^{10})$	$q-21.9408i q^{3} +184.916i q^{5} -1051.96 q^{7} +1705.60 q^{9} +4324.35i q^{11} +11253.2i q^{13} +4057.20 q^{15} -21746.4 q^{17} +45466.5i q^{19} +23080.8i q^{21} -4414.37 q^{23} +43931.2 q^{25} -85406.8i q^{27} -23687.1i q^{29} -72941.9 q^{31} +94879.8 q^{33} -194523. i q^{35} -483338. i q^{37} +246905. q^{39} -411040. q^{41} -96165.3i q^{43} +315392. i q^{45} +156171. q^{47} +283070. q^{49} +477134. i q^{51} +686962. i q^{53} -799641. q^{55} +997572. q^{57} +1.79961e6i q^{59} +1.36394e6i q^{61} -1.79422e6 q^{63} -2.08090e6 q^{65} -1.08853e6i q^{67} +96854.8i q^{69} +5.60830e6 q^{71} +21698.7 q^{73} -963886. i q^{75} -4.54903e6i q^{77} -2.34010e6 q^{79} +1.85626e6 q^{81} -882169. i q^{83} -4.02125e6i q^{85} -519714. q^{87} -1.34738e6 q^{89} -1.18379e7i q^{91} +1.60040e6i q^{93} -8.40746e6 q^{95} +7.32798e6 q^{97} +7.37562e6i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/32\mathbb{Z}\right)^\times$.

$n$	$5$	$31$
$\chi(n)$	$-1$	$1$

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
$3$	− 21.9408i	− 0.469168i	−0.972096	−	0.234584i	$-0.924627\pi$
$3$	− 21.9408i	− 0.469168i	0.972096	−	0.234584i	$-0.0753728\pi$
$4$	0	0
$5$	184.916i	0.661574i	0.943705	+	0.330787i	$0.107314\pi$
$5$	184.916i	0.661574i	−0.943705	+	0.330787i	$0.892686\pi$
$6$	0	0
$7$	−1051.96	−1.15919	−0.579595	−	0.814905i	$-0.696789\pi$
$7$	−1051.96	−1.15919	−0.579595	+	0.814905i	$0.696789\pi$
$8$	0	0
$9$	1705.60	0.779882
$10$	0	0
$11$	4324.35i	0.979596i	0.871836	+	0.489798i	$0.162929\pi$
$11$	4324.35i	0.979596i	−0.871836	+	0.489798i	$0.837071\pi$
$12$	0	0
$13$	11253.2i	1.42061i	0.703892	+	0.710307i	$0.251444\pi$
$13$	11253.2i	1.42061i	−0.703892	+	0.710307i	$0.748556\pi$
$14$	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
$19$	45466.5i	1.52074i	0.649492	+	0.760368i	$0.274981\pi$
$19$	45466.5i	1.52074i	−0.649492	+	0.760368i	$0.725019\pi$
$20$	0	0
$21$	23080.8i	0.543855i
$22$	0	0
$23$	−4414.37	−0.0756522	−0.0378261	−	0.999284i	$-0.512043\pi$
$23$	−4414.37	−0.0756522	−0.0378261	+	0.999284i	$0.512043\pi$
$24$	0	0
$25$	43931.2	0.562320
$26$	0	0
$27$	− 85406.8i	− 0.835063i
$28$	0	0
$29$	− 23687.1i	− 0.180351i	−0.995926	−	0.0901756i	$-0.971257\pi$
$29$	− 23687.1i	− 0.180351i	0.995926	−	0.0901756i	$-0.0287428\pi$
$30$	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.i	− 0.766890i
$36$	0	0
$37$	− 483338.i	− 1.56872i	−0.620307	−	0.784359i	$-0.712992\pi$
$37$	− 483338.i	− 1.56872i	0.620307	−	0.784359i	$-0.287008\pi$
$38$	0	0
$39$	246905.	0.666506
$40$	0	0
$41$	−411040.	−0.931410	−0.465705	−	0.884940i	$-0.654199\pi$
$41$	−411040.	−0.931410	−0.465705	+	0.884940i	$0.654199\pi$
$42$	0	0
$43$	− 96165.3i	− 0.184450i	−0.995738	−	0.0922250i	$-0.970602\pi$
$43$	− 96165.3i	− 0.184450i	0.995738	−	0.0922250i	$-0.0293979\pi$
$44$	0	0
$45$	315392.i	0.515949i
$46$	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.i	0.503668i
$52$	0	0
$53$	686962.i	0.633822i	0.948455	+	0.316911i	$0.102646\pi$
$53$	686962.i	0.633822i	−0.948455	+	0.316911i	$0.897354\pi$
$54$	0	0
$55$	−799641.	−0.648075
$56$	0	0
$57$	997572.	0.713480
$58$	0	0
$59$	1.79961e6i	1.14077i	0.821378	+	0.570384i	$0.193206\pi$
$59$	1.79961e6i	1.14077i	−0.821378	+	0.570384i	$0.806794\pi$
$60$	0	0
$61$	1.36394e6i	0.769379i	0.923046	+	0.384690i	$0.125692\pi$
$61$	1.36394e6i	0.769379i	−0.923046	+	0.384690i	$0.874308\pi$
$62$	0	0
$63$	−1.79422e6	−0.904031
$64$	0	0
$65$	−2.08090e6	−0.939841
$66$	0	0
$67$	− 1.08853e6i	− 0.442160i	−0.975256	−	0.221080i	$-0.929042\pi$
$67$	− 1.08853e6i	− 0.442160i	0.975256	−	0.221080i	$-0.0709582\pi$
$68$	0	0
$69$	96854.8i	0.0354936i
$70$	0	0
$71$	5.60830e6	1.85963	0.929816	−	0.368025i	$-0.119966\pi$
$71$	5.60830e6	1.85963	0.929816	+	0.368025i	$0.119966\pi$
$72$	0	0
$73$	21698.7	0.00652837	0.00326418	−	0.999995i	$-0.498961\pi$
$73$	21698.7	0.00652837	0.00326418	+	0.999995i	$0.498961\pi$
$74$	0	0
$75$	− 963886.i	− 0.263822i
$76$	0	0
$77$	− 4.54903e6i	− 1.13554i
$78$	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.i	− 0.169347i	−0.996409	−	0.0846736i	$-0.973015\pi$
$83$	− 882169.i	− 0.169347i	0.996409	−	0.0846736i	$-0.0269848\pi$
$84$	0	0
$85$	− 4.02125e6i	− 0.710223i
$86$	0	0
$87$	−519714.	−0.0846150
$88$	0	0
$89$	−1.34738e6	−0.202594	−0.101297	−	0.994856i	$-0.532299\pi$
$89$	−1.34738e6	−0.202594	−0.101297	+	0.994856i	$0.532299\pi$
$90$	0	0
$91$	− 1.18379e7i	− 1.64676i
$92$	0	0
$93$	1.60040e6i	0.206319i
$94$	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
$99$	7.37562e6i	0.763969i
$100$	0	0
$101$	9.66027e6i	0.932963i	0.884531	+	0.466482i	$0.154479\pi$
$101$	9.66027e6i	0.932963i	−0.884531	+	0.466482i	$0.845521\pi$
$102$	0	0
$103$	1.38659e7	1.25031	0.625154	−	0.780501i	$-0.285036\pi$
$103$	1.38659e7	1.25031	0.625154	+	0.780501i	$0.285036\pi$
$104$	0	0
$105$	−4.26800e6	−0.359800
$106$	0	0
$107$	1.88354e7i	1.48638i	0.669078	+	0.743192i	$0.266689\pi$
$107$	1.88354e7i	1.48638i	−0.669078	+	0.743192i	$0.733311\pi$
$108$	0	0
$109$	− 8.60619e6i	− 0.636529i	−0.948002	−	0.318265i	$-0.896900\pi$
$109$	− 8.60619e6i	− 0.636529i	0.948002	−	0.318265i	$-0.103100\pi$
$110$	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.i	− 0.0500495i
$116$	0	0
$117$	1.91935e7i	1.10791i
$118$	0	0
$119$	2.28763e7	1.24443
$120$	0	0
$121$	787128.	0.0403921
$122$	0	0
$123$	9.01856e6i	0.436988i
$124$	0	0
$125$	2.25701e7i	1.03359i
$126$	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
$131$	− 1.17050e7i	− 0.454905i	−0.973789	−	0.227453i	$-0.926960\pi$
$131$	− 1.17050e7i	− 0.454905i	0.973789	−	0.227453i	$-0.0730397\pi$
$132$	0	0
$133$	− 4.78288e7i	− 1.76282i
$134$	0	0
$135$	1.57931e7	0.552456
$136$	0	0
$137$	3.79174e7	1.25984	0.629921	−	0.776659i	$-0.283087\pi$
$137$	3.79174e7	1.25984	0.629921	+	0.776659i	$0.283087\pi$
$138$	0	0
$139$	− 2.80300e7i	− 0.885260i	−0.896704	−	0.442630i	$-0.854045\pi$
$139$	− 2.80300e7i	− 0.885260i	0.896704	−	0.442630i	$-0.145955\pi$
$140$	0	0
$141$	− 3.42652e6i	− 0.102941i
$142$	0	0
$143$	−4.86630e7	−1.39163
$144$	0	0
$145$	4.38012e6	0.119316
$146$	0	0
$147$	− 6.21078e6i	− 0.161263i
$148$	0	0
$149$	− 6.82083e6i	− 0.168922i	−0.996427	−	0.0844608i	$-0.973083\pi$
$149$	− 6.82083e6i	− 0.168922i	0.996427	−	0.0844608i	$-0.0269168\pi$
$150$	0	0
$151$	7.62068e7	1.80125	0.900626	−	0.434595i	$-0.143109\pi$
$151$	7.62068e7	1.80125	0.900626	+	0.434595i	$0.143109\pi$
$152$	0	0
$153$	−3.70907e7	−0.837230
$154$	0	0
$155$	− 1.34881e7i	− 0.290931i
$156$	0	0
$157$	− 4.28737e7i	− 0.884183i	−0.896970	−	0.442092i	$-0.854237\pi$
$157$	− 4.28737e7i	− 0.884183i	0.896970	−	0.442092i	$-0.145763\pi$
$158$	0	0
$159$	1.50725e7	0.297369
$160$	0	0
$161$	4.64373e6	0.0876952
$162$	0	0
$163$	− 4.12127e7i	− 0.745374i	−0.927957	−	0.372687i	$-0.878437\pi$
$163$	− 4.12127e7i	− 0.745374i	0.927957	−	0.372687i	$-0.121563\pi$
$164$	0	0
$165$	1.75448e7i	0.304056i
$166$	0	0
$167$	−2.41703e7	−0.401582	−0.200791	−	0.979634i	$-0.564351\pi$
$167$	−2.41703e7	−0.401582	−0.200791	+	0.979634i	$0.564351\pi$
$168$	0	0
$169$	−6.38869e7	−1.01814
$170$	0	0
$171$	7.75477e7i	1.18599i
$172$	0	0
$173$	5.90454e7i	0.867011i	0.901151	+	0.433505i	$0.142723\pi$
$173$	5.90454e7i	0.867011i	−0.901151	+	0.433505i	$0.857277\pi$
$174$	0	0
$175$	−4.62137e7	−0.651835
$176$	0	0
$177$	3.94850e7	0.535212
$178$	0	0
$179$	1.20311e8i	1.56791i	0.620819	+	0.783954i	$0.286800\pi$
$179$	1.20311e8i	1.56791i	−0.620819	+	0.783954i	$0.713200\pi$
$180$	0	0
$181$	− 3.21539e6i	− 0.0403049i	−0.999797	−	0.0201525i	$-0.993585\pi$
$181$	− 3.21539e6i	− 0.0403049i	0.999797	−	0.0201525i	$-0.00641517\pi$
$182$	0	0
$183$	2.99259e7	0.360968
$184$	0	0
$185$	8.93768e7	1.03782
$186$	0	0
$187$	− 9.40392e7i	− 1.05163i
$188$	0	0
$189$	8.98443e7i	0.967997i
$190$	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
$195$	4.56566e7i	0.440943i
$196$	0	0
$197$	8.97008e7i	0.835920i	0.908465	+	0.417960i	$0.137255\pi$
$197$	8.97008e7i	0.835920i	−0.908465	+	0.417960i	$0.862745\pi$
$198$	0	0
$199$	3.98955e7	0.358871	0.179436	−	0.983770i	$-0.442573\pi$
$199$	3.98955e7	0.358871	0.179436	+	0.983770i	$0.442573\pi$
$200$	0	0
$201$	−2.38833e7	−0.207447
$202$	0	0
$203$	2.49178e7i	0.209061i
$204$	0	0
$205$	− 7.60078e7i	− 0.616197i
$206$	0	0
$207$	−7.52915e6	−0.0589997
$208$	0	0
$209$	−1.96613e8	−1.48971
$210$	0	0
$211$	− 6.63604e7i	− 0.486318i	−0.969986	−	0.243159i	$-0.921816\pi$
$211$	− 6.63604e7i	− 0.486318i	0.969986	−	0.243159i	$-0.0781837\pi$
$212$	0	0
$213$	− 1.23051e8i	− 0.872479i
$214$	0	0
$215$	1.77825e7	0.122027
$216$	0	0
$217$	7.67317e7	0.509760
$218$	0	0
$219$	− 476088.i	− 0.00306290i
$220$	0	0
$221$	− 2.44718e8i	− 1.52508i
$222$	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
$227$	− 1.95129e8i	− 1.10721i	−0.832778	−	0.553607i	$-0.813251\pi$
$227$	− 1.95129e8i	− 1.10721i	0.832778	−	0.553607i	$-0.186749\pi$
$228$	0	0
$229$	6.95136e7i	0.382512i	0.981540	+	0.191256i	$0.0612561\pi$
$229$	6.95136e7i	0.382512i	−0.981540	+	0.191256i	$0.938744\pi$
$230$	0	0
$231$	−9.98095e7	−0.532758
$232$	0	0
$233$	−3.81872e7	−0.197776	−0.0988878	−	0.995099i	$-0.531528\pi$
$233$	−3.81872e7	−0.197776	−0.0988878	+	0.995099i	$0.531528\pi$
$234$	0	0
$235$	2.88785e7i	0.145157i
$236$	0	0
$237$	5.13437e7i	0.250535i
$238$	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
$243$	− 2.27512e8i	− 1.01715i
$244$	0	0
$245$	5.23441e7i	0.227398i
$246$	0	0
$247$	−5.11645e8	−2.16038
$248$	0	0
$249$	−1.93555e7	−0.0794523
$250$	0	0
$251$	3.62042e7i	0.144511i	0.997386	+	0.0722555i	$0.0230197\pi$
$251$	3.62042e7i	0.144511i	−0.997386	+	0.0722555i	$0.976980\pi$
$252$	0	0
$253$	− 1.90893e7i	− 0.0741085i
$254$	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
$259$	5.08451e8i	1.81844i
$260$	0	0
$261$	− 4.04008e7i	− 0.140653i
$262$	0	0
$263$	1.10733e8	0.375348	0.187674	−	0.982231i	$-0.439905\pi$
$263$	1.10733e8	0.375348	0.187674	+	0.982231i	$0.439905\pi$
$264$	0	0
$265$	−1.27030e8	−0.419320
$266$	0	0
$267$	2.95627e7i	0.0950505i
$268$	0	0
$269$	3.07561e8i	0.963382i	0.876341	+	0.481691i	$0.159977\pi$
$269$	3.07561e8i	0.963382i	−0.876341	+	0.481691i	$0.840023\pi$
$270$	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
$275$	1.89974e8i	0.550846i
$276$	0	0
$277$	3.15642e8i	0.892309i	0.894956	+	0.446155i	$0.147207\pi$
$277$	3.15642e8i	0.892309i	−0.894956	+	0.446155i	$0.852793\pi$
$278$	0	0
$279$	−1.24410e8	−0.342957
$280$	0	0
$281$	−2.50451e8	−0.673366	−0.336683	−	0.941618i	$-0.609305\pi$
$281$	−2.50451e8	−0.673366	−0.336683	+	0.941618i	$0.609305\pi$
$282$	0	0
$283$	3.47387e7i	0.0911088i	0.998962	+	0.0455544i	$0.0145054\pi$
$283$	3.47387e7i	0.0911088i	−0.998962	+	0.0455544i	$0.985495\pi$
$284$	0	0
$285$	1.84467e8i	0.472020i
$286$	0	0
$287$	4.32397e8	1.07968
$288$	0	0
$289$	6.25676e7	0.152478
$290$	0	0
$291$	− 1.60782e8i	− 0.382482i
$292$	0	0
$293$	− 5.34336e8i	− 1.24102i	−0.784200	−	0.620508i	$-0.786926\pi$
$293$	− 5.34336e8i	− 1.24102i	0.784200	−	0.620508i	$-0.213074\pi$
$294$	0	0
$295$	−3.32777e8	−0.754703
$296$	0	0
$297$	3.69329e8	0.818024
$298$	0	0
$299$	− 4.96760e7i	− 0.107472i
$300$	0	0
$301$	1.01162e8i	0.213813i
$302$	0	0
$303$	2.11954e8	0.437716
$304$	0	0
$305$	−2.52214e8	−0.509002
$306$	0	0
$307$	1.68851e8i	0.333057i	0.986037	+	0.166529i	$0.0532558\pi$
$307$	1.68851e8i	0.333057i	−0.986037	+	0.166529i	$0.946744\pi$
$308$	0	0
$309$	− 3.04229e8i	− 0.586605i
$310$	0	0
$311$	−6.99677e8	−1.31897	−0.659487	−	0.751716i	$-0.729226\pi$
$311$	−6.99677e8	−1.31897	−0.659487	+	0.751716i	$0.729226\pi$
$312$	0	0
$313$	5.06345e8	0.933343	0.466671	−	0.884431i	$-0.345453\pi$
$313$	5.06345e8	0.933343	0.466671	+	0.884431i	$0.345453\pi$
$314$	0	0
$315$	− 3.31779e8i	− 0.598084i
$316$	0	0
$317$	− 4.78996e8i	− 0.844548i	−0.906468	−	0.422274i	$-0.861232\pi$
$317$	− 4.78996e8i	− 0.844548i	0.906468	−	0.422274i	$-0.138768\pi$
$318$	0	0
$319$	1.02431e8	0.176671
$320$	0	0
$321$	4.13263e8	0.697364
$322$	0	0
$323$	− 9.88733e8i	− 1.63256i
$324$	0	0
$325$	4.94369e8i	0.798839i
$326$	0	0
$327$	−1.88827e8	−0.298639
$328$	0	0
$329$	−1.64285e8	−0.254339
$330$	0	0
$331$	− 4.20825e8i	− 0.637828i	−0.947784	−	0.318914i	$-0.896682\pi$
$331$	− 4.20825e8i	− 0.637828i	0.947784	−	0.318914i	$-0.103318\pi$
$332$	0	0
$333$	− 8.24382e8i	− 1.22341i
$334$	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
$339$	2.88170e8i	0.401745i
$340$	0	0
$341$	− 3.15427e8i	− 0.430782i
$342$	0	0
$343$	5.68554e8	0.760751
$344$	0	0
$345$	−1.79100e7	−0.0234816
$346$	0	0
$347$	5.31496e8i	0.682883i	0.939903	+	0.341442i	$0.110915\pi$
$347$	5.31496e8i	0.682883i	−0.939903	+	0.341442i	$0.889085\pi$
$348$	0	0
$349$	1.31436e9i	1.65511i	0.561385	+	0.827555i	$0.310269\pi$
$349$	1.31436e9i	1.65511i	−0.561385	+	0.827555i	$0.689731\pi$
$350$	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
$355$	1.03706e9i	1.23028i
$356$	0	0
$357$	− 5.01924e8i	− 0.583847i
$358$	0	0
$359$	−7.59524e8	−0.866385	−0.433193	−	0.901301i	$-0.642613\pi$
$359$	−7.59524e8	−0.866385	−0.433193	+	0.901301i	$0.642613\pi$
$360$	0	0
$361$	−1.17333e9	−1.31264
$362$	0	0
$363$	− 1.72702e7i	− 0.0189507i
$364$	0	0
$365$	4.01244e6i	0.00431900i
$366$	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
$371$	− 7.22655e8i	− 0.734720i
$372$	0	0
$373$	4.57130e8i	0.456099i	0.973650	+	0.228049i	$0.0732347\pi$
$373$	4.57130e8i	0.456099i	−0.973650	+	0.228049i	$0.926765\pi$
$374$	0	0
$375$	4.95206e8	0.484927
$376$	0	0
$377$	2.66557e8	0.256209
$378$	0	0
$379$	1.91565e9i	1.80750i	0.428056	+	0.903752i	$0.359199\pi$
$379$	1.91565e9i	1.80750i	−0.428056	+	0.903752i	$0.640801\pi$
$380$	0	0
$381$	6.99402e8i	0.647873i
$382$	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
$387$	− 1.64020e8i	− 0.143849i
$388$	0	0
$389$	2.21214e9i	1.90541i	0.303890	+	0.952707i	$0.401715\pi$
$389$	2.21214e9i	1.90541i	−0.303890	+	0.952707i	$0.598285\pi$
$390$	0	0
$391$	9.59967e7	0.0812153
$392$	0	0
$393$	−2.56817e8	−0.213427
$394$	0	0
$395$	− 4.32721e8i	− 0.353279i
$396$	0	0
$397$	− 1.30879e9i	− 1.04979i	−0.851167	−	0.524896i	$-0.824104\pi$
$397$	− 1.30879e9i	− 1.04979i	0.851167	−	0.524896i	$-0.175896\pi$
$398$	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
$403$	− 8.20832e8i	− 0.624722i
$404$	0	0
$405$	3.43251e8i	0.256755i
$406$	0	0
$407$	2.09013e9	1.53671
$408$	0	0
$409$	1.94107e9	1.40284	0.701421	−	0.712747i	$-0.252549\pi$
$409$	1.94107e9	1.40284	0.701421	+	0.712747i	$0.252549\pi$
$410$	0	0
$411$	− 8.31937e8i	− 0.591077i
$412$	0	0
$413$	− 1.89312e9i	− 1.32237i
$414$	0	0
$415$	1.63127e8	0.112036
$416$	0	0
$417$	−6.15000e8	−0.415336
$418$	0	0
$419$	1.96429e9i	1.30454i	0.757989	+	0.652268i	$0.226182\pi$
$419$	1.96429e9i	1.30454i	−0.757989	+	0.652268i	$0.773818\pi$
$420$	0	0
$421$	− 2.33176e9i	− 1.52299i	−0.648174	−	0.761493i	$-0.724467\pi$
$421$	− 2.33176e9i	− 1.52299i	0.648174	−	0.761493i	$-0.275533\pi$
$422$	0	0
$423$	2.66366e8	0.171115
$424$	0	0
$425$	−9.55346e8	−0.603670
$426$	0	0
$427$	− 1.43480e9i	− 0.891857i
$428$	0	0
$429$	1.06771e9i	0.652907i
$430$	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
$435$	− 9.61033e7i	− 0.0559791i
$436$	0	0
$437$	− 2.00706e8i	− 0.115047i
$438$	0	0
$439$	2.73808e8	0.154461	0.0772307	−	0.997013i	$-0.475392\pi$
$439$	2.73808e8	0.154461	0.0772307	+	0.997013i	$0.475392\pi$
$440$	0	0
$441$	4.82804e8	0.268063
$442$	0	0
$443$	− 2.72048e9i	− 1.48673i	−0.668886	−	0.743365i	$-0.733229\pi$
$443$	− 2.72048e9i	− 1.48673i	0.668886	−	0.743365i	$-0.266771\pi$
$444$	0	0
$445$	− 2.49152e8i	− 0.134031i
$446$	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
$451$	− 1.77748e9i	− 0.912406i
$452$	0	0
$453$	− 1.67204e9i	− 0.845089i
$454$	0	0
$455$	2.18902e9	1.08945
$456$	0	0
$457$	−1.15773e8	−0.0567413	−0.0283706	−	0.999597i	$-0.509032\pi$
$457$	−1.15773e8	−0.0567413	−0.0283706	+	0.999597i	$0.509032\pi$
$458$	0	0
$459$	1.85729e9i	0.896470i
$460$	0	0
$461$	9.34713e8i	0.444350i	0.975007	+	0.222175i	$0.0713156\pi$
$461$	9.34713e8i	0.444350i	−0.975007	+	0.222175i	$0.928684\pi$
$462$	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
$467$	1.65838e9i	0.753485i	0.926318	+	0.376742i	$0.122956\pi$
$467$	1.65838e9i	0.753485i	−0.926318	+	0.376742i	$0.877044\pi$
$468$	0	0
$469$	1.14509e9i	0.512548i
$470$	0	0
$471$	−9.40684e8	−0.414830
$472$	0	0
$473$	4.15853e8	0.180686
$474$	0	0
$475$	1.99740e9i	0.855140i
$476$	0	0
$477$	1.17168e9i	0.494306i
$478$	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
$483$	− 1.01887e8i	− 0.0411438i
$484$	0	0
$485$	1.35506e9i	0.539339i
$486$	0	0
$487$	3.24167e9	1.27180	0.635898	−	0.771773i	$-0.280630\pi$
$487$	3.24167e9	1.27180	0.635898	+	0.771773i	$0.280630\pi$
$488$	0	0
$489$	−9.04239e8	−0.349705
$490$	0	0
$491$	2.16351e9i	0.824847i	0.910992	+	0.412423i	$0.135318\pi$
$491$	2.16351e9i	0.824847i	−0.910992	+	0.412423i	$0.864682\pi$
$492$	0	0
$493$	5.15110e8i	0.193613i
$494$	0	0
$495$	−1.36387e9	−0.505422
$496$	0	0
$497$	−5.89969e9	−2.15567
$498$	0	0
$499$	− 2.16713e9i	− 0.780787i	−0.920648	−	0.390394i	$-0.872339\pi$
$499$	− 2.16713e9i	− 0.780787i	0.920648	−	0.390394i	$-0.127661\pi$
$500$	0	0
$501$	5.30316e8i	0.188409i
$502$	0	0
$503$	2.58480e9	0.905606	0.452803	−	0.891611i	$-0.350424\pi$
$503$	2.58480e9	0.905606	0.452803	+	0.891611i	$0.350424\pi$
$504$	0	0
$505$	−1.78633e9	−0.617224
$506$	0	0
$507$	1.40173e9i	0.477680i
$508$	0	0
$509$	2.11747e9i	0.711712i	0.934541	+	0.355856i	$0.115811\pi$
$509$	2.11747e9i	0.711712i	−0.934541	+	0.355856i	$0.884189\pi$
$510$	0	0
$511$	−2.28261e7	−0.00756762
$512$	0	0
$513$	3.88315e9	1.26991
$514$	0	0
$515$	2.56402e9i	0.827172i
$516$	0	0
$517$	6.75339e8i	0.214934i
$518$	0	0
$519$	1.29550e9	0.406773
$520$	0	0
$521$	−2.34354e9	−0.726006	−0.363003	−	0.931788i	$-0.618249\pi$
$521$	−2.34354e9	−0.726006	−0.363003	+	0.931788i	$0.618249\pi$
$522$	0	0
$523$	− 3.26155e9i	− 0.996939i	−0.866907	−	0.498470i	$-0.833896\pi$
$523$	− 3.26155e9i	− 0.996939i	0.866907	−	0.498470i	$-0.166104\pi$
$524$	0	0
$525$	1.01397e9i	0.305820i
$526$	0	0
$527$	1.58622e9	0.472093
$528$	0	0
$529$	−3.38534e9	−0.994277
$530$	0	0
$531$	3.06942e9i	0.889664i
$532$	0	0
$533$	− 4.62554e9i	− 1.32317i
$534$	0	0
$535$	−3.48296e9	−0.983354
$536$	0	0
$537$	2.63973e9	0.735612
$538$	0	0
$539$	1.22409e9i	0.336709i
$540$	0	0
$541$	− 7.24638e9i	− 1.96757i	−0.179344	−	0.983787i	$-0.557397\pi$
$541$	− 7.24638e9i	− 1.96757i	0.179344	−	0.983787i	$-0.442603\pi$
$542$	0	0
$543$	−7.05482e7	−0.0189098
$544$	0	0
$545$	1.59142e9	0.421111
$546$	0	0
$547$	− 6.41460e9i	− 1.67577i	−0.545848	−	0.837884i	$-0.683793\pi$
$547$	− 6.41460e9i	− 1.67577i	0.545848	−	0.837884i	$-0.316207\pi$
$548$	0	0
$549$	2.32634e9i	0.600025i
$550$	0	0
$551$	1.07697e9	0.274267
$552$	0	0
$553$	2.46168e9	0.619005
$554$	0	0
$555$	− 1.96100e9i	− 0.486914i
$556$	0	0
$557$	1.76223e9i	0.432085i	0.976384	+	0.216042i	$0.0693149\pi$
$557$	1.76223e9i	0.432085i	−0.976384	+	0.216042i	$0.930685\pi$
$558$	0	0
$559$	1.08217e9	0.262032
$560$	0	0
$561$	−2.06330e9	−0.493391
$562$	0	0
$563$	− 2.33270e9i	− 0.550909i	−0.961314	−	0.275454i	$-0.911172\pi$
$563$	− 2.33270e9i	− 0.550909i	0.961314	−	0.275454i	$-0.0888283\pi$
$564$	0	0
$565$	− 2.42868e9i	− 0.566501i
$566$	0	0
$567$	−1.95270e9	−0.449878
$568$	0	0
$569$	5.53591e9	1.25978	0.629892	−	0.776683i	$-0.283099\pi$
$569$	5.53591e9	1.25978	0.629892	+	0.776683i	$0.283099\pi$
$570$	0	0
$571$	− 3.70890e9i	− 0.833717i	−0.908971	−	0.416858i	$-0.863131\pi$
$571$	− 3.70890e9i	− 0.833717i	0.908971	−	0.416858i	$-0.136869\pi$
$572$	0	0
$573$	1.54490e9i	0.343051i
$574$	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
$579$	− 2.58596e8i	− 0.0553665i
$580$	0	0
$581$	9.28003e8i	0.196306i
$582$	0	0
$583$	−2.97067e9	−0.620889
$584$	0	0
$585$	−3.54919e9	−0.732965
$586$	0	0
$587$	2.04311e9i	0.416926i	0.978030	+	0.208463i	$0.0668461\pi$
$587$	2.04311e9i	0.416926i	−0.978030	+	0.208463i	$0.933154\pi$
$588$	0	0
$589$	− 3.31641e9i	− 0.668752i
$590$	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
$595$	4.23018e9i	0.823284i
$596$	0	0
$597$	− 8.75341e8i	− 0.168371i
$598$	0	0
$599$	−3.14564e8	−0.0598020	−0.0299010	−	0.999553i	$-0.509519\pi$
$599$	−3.14564e8	−0.0598020	−0.0299010	+	0.999553i	$0.509519\pi$
$600$	0	0
$601$	−2.49586e9	−0.468985	−0.234493	−	0.972118i	$-0.575343\pi$
$601$	−2.49586e9	−0.468985	−0.234493	+	0.972118i	$0.575343\pi$
$602$	0	0
$603$	− 1.85660e9i	− 0.344833i
$604$	0	0
$605$	1.45552e8i	0.0267224i
$606$	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
$611$	1.75743e9i	0.311698i
$612$	0	0
$613$	2.34343e9i	0.410904i	0.978667	+	0.205452i	$0.0658664\pi$
$613$	2.34343e9i	0.410904i	−0.978667	+	0.205452i	$0.934134\pi$
$614$	0	0
$615$	−1.66767e9	−0.289100
$616$	0	0
$617$	−6.26737e8	−0.107420	−0.0537102	−	0.998557i	$-0.517105\pi$
$617$	−6.26737e8	−0.107420	−0.0537102	+	0.998557i	$0.517105\pi$
$618$	0	0
$619$	8.90879e9i	1.50974i	0.655876	+	0.754869i	$0.272300\pi$
$619$	8.90879e9i	1.50974i	−0.655876	+	0.754869i	$0.727700\pi$
$620$	0	0
$621$	3.77017e8i	0.0631743i
$622$	0	0
$623$	1.41739e9	0.234845
$624$	0	0
$625$	−7.41437e8	−0.121477
$626$	0	0
$627$	4.31385e9i	0.698922i
$628$	0	0
$629$	1.05109e10i	1.68407i
$630$	0	0
$631$	2.21942e9	0.351671	0.175835	−	0.984420i	$-0.443737\pi$
$631$	2.21942e9	0.351671	0.175835	+	0.984420i	$0.443737\pi$
$632$	0	0
$633$	−1.45600e9	−0.228165
$634$	0	0
$635$	− 5.89451e9i	− 0.913566i
$636$	0	0
$637$	3.18545e9i	0.488296i
$638$	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
$643$	7.71279e9i	1.14412i	0.820210	+	0.572062i	$0.193856\pi$
$643$	7.71279e9i	1.14412i	−0.820210	+	0.572062i	$0.806144\pi$
$644$	0	0
$645$	− 3.90161e8i	− 0.0572513i
$646$	0	0
$647$	−1.18898e10	−1.72587	−0.862935	−	0.505316i	$-0.831376\pi$
$647$	−1.18898e10	−1.72587	−0.862935	+	0.505316i	$0.831376\pi$
$648$	0	0
$649$	−7.78217e9	−1.11749
$650$	0	0
$651$	− 1.68356e9i	− 0.239163i
$652$	0	0
$653$	− 2.00345e9i	− 0.281567i	−0.990040	−	0.140784i	$-0.955038\pi$
$653$	− 2.00345e9i	− 0.281567i	0.990040	−	0.140784i	$-0.0449622\pi$
$654$	0	0
$655$	2.16443e9	0.300954
$656$	0	0
$657$	3.70094e7	0.00509135
$658$	0	0
$659$	− 7.66944e9i	− 1.04391i	−0.852972	−	0.521957i	$-0.825202\pi$
$659$	− 7.66944e9i	− 1.04391i	0.852972	−	0.521957i	$-0.174798\pi$
$660$	0	0
$661$	4.04041e9i	0.544152i	0.962276	+	0.272076i	$0.0877103\pi$
$661$	4.04041e9i	0.544152i	−0.962276	+	0.272076i	$0.912290\pi$
$662$	0	0
$663$	−5.36930e9	−0.715518
$664$	0	0
$665$	8.84429e9	1.16624
$666$	0	0
$667$	1.04564e8i	0.0136440i
$668$	0	0
$669$	− 3.45060e9i	− 0.445557i
$670$	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
$675$	− 3.75203e9i	− 0.469572i
$676$	0	0
$677$	− 1.12671e10i	− 1.39557i	−0.716305	−	0.697787i	$-0.754168\pi$
$677$	− 1.12671e10i	− 1.39557i	0.716305	−	0.697787i	$-0.245832\pi$
$678$	0	0
$679$	−7.70872e9	−0.945014
$680$	0	0
$681$	−4.28129e9	−0.519469
$682$	0	0
$683$	8.00228e9i	0.961039i	0.876984	+	0.480520i	$0.159552\pi$
$683$	8.00228e9i	0.961039i	−0.876984	+	0.480520i	$0.840448\pi$
$684$	0	0
$685$	7.01151e9i	0.833479i
$686$	0	0
$687$	1.52518e9	0.179462
$688$	0	0
$689$	−7.73055e9	−0.900416
$690$	0	0
$691$	− 7.24716e9i	− 0.835593i	−0.908541	−	0.417797i	$-0.862803\pi$
$691$	− 7.24716e9i	− 0.835593i	0.908541	−	0.417797i	$-0.137197\pi$
$692$	0	0
$693$	− 7.75884e9i	− 0.885585i
$694$	0	0
$695$	5.18318e9	0.585665
$696$	0	0
$697$	8.93865e9	0.999902
$698$	0	0
$699$	8.37859e8i	0.0927899i
$700$	0	0
$701$	− 4.68293e8i	− 0.0513458i	−0.999670	−	0.0256729i	$-0.991827\pi$
$701$	− 4.68293e8i	− 0.0513458i	0.999670	−	0.0256729i	$-0.00817283\pi$
$702$	0	0
$703$	2.19757e10	2.38561
$704$	0	0
$705$	6.33617e8	0.0681028
$706$	0	0
$707$	− 1.01622e10i	− 1.08148i
$708$	0	0
$709$	− 1.08926e10i	− 1.14781i	−0.818923	−	0.573903i	$-0.805429\pi$
$709$	− 1.08926e10i	− 1.14781i	0.818923	−	0.573903i	$-0.194571\pi$
$710$	0	0
$711$	−3.99128e9	−0.416455
$712$	0	0
$713$	3.21992e8	0.0332684
$714$	0	0
$715$	− 8.99855e9i	− 0.920664i
$716$	0	0
$717$	− 5.78249e9i	− 0.585865i
$718$	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
$723$	− 2.66224e9i	− 0.261977i
$724$	0	0
$725$	− 1.04060e9i	− 0.101415i
$726$	0	0
$727$	−1.04295e10	−1.00668	−0.503342	−	0.864087i	$-0.667897\pi$
$727$	−1.04295e10	−1.00668	−0.503342	+	0.864087i	$0.667897\pi$
$728$	0	0
$729$	−9.32176e8	−0.0891152
$730$	0	0
$731$	2.09125e9i	0.198014i
$732$	0	0
$733$	1.55538e10i	1.45873i	0.684127	+	0.729363i	$0.260184\pi$
$733$	1.55538e10i	1.45873i	−0.684127	+	0.729363i	$0.739816\pi$
$734$	0	0
$735$	1.14847e9	0.106688
$736$	0	0
$737$	4.70720e9	0.433138
$738$	0	0
$739$	1.07140e10i	0.976554i	0.872689	+	0.488277i	$0.162374\pi$
$739$	1.07140e10i	0.976554i	−0.872689	+	0.488277i	$0.837626\pi$
$740$	0	0
$741$	1.12259e10i	1.01358i
$742$	0	0
$743$	5.46982e9	0.489229	0.244615	−	0.969620i	$-0.421339\pi$
$743$	5.46982e9	0.489229	0.244615	+	0.969620i	$0.421339\pi$
$744$	0	0
$745$	1.26128e9	0.111754
$746$	0	0
$747$	− 1.50463e9i	− 0.132071i
$748$	0	0
$749$	− 1.98140e10i	− 1.72300i
$750$	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
$755$	1.40918e10i	1.19166i
$756$	0	0
$757$	5.38304e9i	0.451016i	0.974241	+	0.225508i	$0.0724043\pi$
$757$	5.38304e9i	0.451016i	−0.974241	+	0.225508i	$0.927596\pi$
$758$	0	0
$759$	−4.18835e8	−0.0347693
$760$	0	0
$761$	−3.28593e9	−0.270279	−0.135139	−	0.990827i	$-0.543148\pi$
$761$	−3.28593e9	−0.270279	−0.135139	+	0.990827i	$0.543148\pi$
$762$	0	0
$763$	9.05334e9i	0.737858i
$764$	0	0
$765$	− 6.85865e9i	− 0.553890i
$766$	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
$771$	− 3.85170e9i	− 0.302665i
$772$	0	0
$773$	1.08690e10i	0.846370i	0.906043	+	0.423185i	$0.139088\pi$
$773$	1.08690e10i	0.846370i	−0.906043	+	0.423185i	$0.860912\pi$
$774$	0	0
$775$	−3.20443e9	−0.247283
$776$	0	0
$777$	1.11558e10	0.853155
$778$	0	0
$779$	− 1.86886e10i	− 1.41643i
$780$	0	0
$781$	2.42523e10i	1.82169i
$782$	0	0
$783$	−2.02304e9	−0.150605
$784$	0	0
$785$	7.92802e9	0.584953
$786$	0	0
$787$	1.94246e10i	1.42050i	0.703951	+	0.710248i	$0.251417\pi$
$787$	1.94246e10i	1.42050i	−0.703951	+	0.710248i	$0.748583\pi$
$788$	0	0
$789$	− 2.42958e9i	− 0.176101i
$790$	0	0
$791$	1.38164e10	0.992606
$792$	0	0
$793$	−1.53487e10	−1.09299
$794$	0	0
$795$	2.78714e9i	0.196732i
$796$	0	0
$797$	2.30309e10i	1.61141i	0.592314	+	0.805707i	$0.298214\pi$
$797$	2.30309e10i	1.61141i	−0.592314	+	0.805707i	$0.701786\pi$
$798$	0	0
$799$	−3.39616e9	−0.235545
$800$	0	0
$801$	−2.29810e9	−0.157999
$802$	0	0
$803$	9.38331e7i	0.00639516i
$804$	0	0
$805$	8.58697e8i	0.0580169i
$806$	0	0
$807$	6.74814e9	0.451988
$808$	0	0
$809$	−1.50546e10	−0.999656	−0.499828	−	0.866125i	$-0.666603\pi$
$809$	−1.50546e10	−0.999656	−0.499828	+	0.866125i	$0.666603\pi$
$810$	0	0
$811$	1.98821e10i	1.30885i	0.756128	+	0.654424i	$0.227089\pi$
$811$	1.98821e10i	1.30885i	−0.756128	+	0.654424i	$0.772911\pi$
$812$	0	0
$813$	− 6.27585e9i	− 0.409596i
$814$	0	0
$815$	7.62086e9	0.493120
$816$	0	0
$817$	4.37230e9	0.280500
$818$	0	0
$819$	− 2.01908e10i	− 1.28428i
$820$	0	0
$821$	− 2.29870e10i	− 1.44971i	−0.688901	−	0.724855i	$-0.741907\pi$
$821$	− 2.29870e10i	− 1.44971i	0.688901	−	0.724855i	$-0.258093\pi$
$822$	0	0
$823$	−1.46696e10	−0.917316	−0.458658	−	0.888613i	$-0.651670\pi$
$823$	−1.46696e10	−0.917316	−0.458658	+	0.888613i	$0.651670\pi$
$824$	0	0
$825$	4.16819e9	0.258439
$826$	0	0
$827$	− 1.26591e9i	− 0.0778277i	−0.999243	−	0.0389138i	$-0.987610\pi$
$827$	− 1.26591e9i	− 0.0778277i	0.999243	−	0.0389138i	$-0.0123898\pi$
$828$	0	0
$829$	− 2.49778e10i	− 1.52270i	−0.648342	−	0.761349i	$-0.724537\pi$
$829$	− 2.49778e10i	− 1.52270i	0.648342	−	0.761349i	$-0.275463\pi$
$830$	0	0
$831$	6.92544e9	0.418643
$832$	0	0
$833$	−6.15576e9	−0.368998
$834$	0	0
$835$	− 4.46947e9i	− 0.265676i
$836$	0	0
$837$	6.22973e9i	0.367223i
$838$	0	0
$839$	3.09477e10	1.80909	0.904547	−	0.426373i	$-0.140209\pi$
$839$	3.09477e10	1.80909	0.904547	+	0.426373i	$0.140209\pi$
$840$	0	0
$841$	1.66888e10	0.967473
$842$	0	0
$843$	5.49510e9i	0.315921i
$844$	0	0
$845$	− 1.18137e10i	− 0.673577i
$846$	0	0
$847$	−8.28024e8	−0.0468221
$848$	0	0
$849$	7.62194e8	0.0427453
$850$	0	0
$851$	2.13363e9i	0.118677i
$852$	0	0
$853$	1.86385e10i	1.02823i	0.857723	+	0.514113i	$0.171879\pi$
$853$	1.86385e10i	1.02823i	−0.857723	+	0.514113i	$0.828121\pi$
$854$	0	0
$855$	−1.43398e10	−0.784623
$856$	0	0
$857$	−2.37366e10	−1.28821	−0.644103	−	0.764939i	$-0.722769\pi$
$857$	−2.37366e10	−1.28821	−0.644103	+	0.764939i	$0.722769\pi$
$858$	0	0
$859$	− 2.46268e10i	− 1.32566i	−0.748770	−	0.662830i	$-0.769355\pi$
$859$	− 2.46268e10i	− 1.32566i	0.748770	−	0.662830i	$-0.230645\pi$
$860$	0	0
$861$	− 9.48713e9i	− 0.506552i
$862$	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
$867$	− 1.37278e9i	− 0.0715378i
$868$	0	0
$869$	− 1.01194e10i	− 0.523102i
$870$	0	0
$871$	1.22495e10	0.628139
$872$	0	0
$873$	1.24986e10	0.635788
$874$	0	0
$875$	− 2.37428e10i	− 1.19813i
$876$	0	0
$877$	− 1.07280e10i	− 0.537057i	−0.963272	−	0.268528i	$-0.913463\pi$
$877$	− 1.07280e10i	− 0.537057i	0.963272	−	0.268528i	$-0.0865373\pi$
$878$	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
$883$	− 1.60870e10i	− 0.786342i	−0.919465	−	0.393171i	$-0.871378\pi$
$883$	− 1.60870e10i	− 0.786342i	0.919465	−	0.393171i	$-0.128622\pi$
$884$	0	0
$885$	7.30139e9i	0.354082i
$886$	0	0
$887$	2.61884e10	1.26001	0.630007	−	0.776589i	$-0.283052\pi$
$887$	2.61884e10	1.26001	0.630007	+	0.776589i	$0.283052\pi$
$888$	0	0
$889$	3.35330e10	1.60072
$890$	0	0
$891$	8.02711e9i	0.380178i
$892$	0	0
$893$	7.10056e9i	0.333666i
$894$	0	0
$895$	−2.22474e10	−1.03729
$896$	0	0
$897$	−1.08993e9	−0.0504226
$898$	0	0
$899$	1.72778e9i	0.0793104i
$900$	0	0
$901$	− 1.49390e10i	− 0.680430i
$902$	0	0
$903$	2.21957e9	0.100314
$904$	0	0
$905$	5.94575e8	0.0266647
$906$	0	0
$907$	1.79753e9i	0.0799929i	0.999200	+	0.0399964i	$0.0127347\pi$
$907$	1.79753e9i	0.0799929i	−0.999200	+	0.0399964i	$0.987265\pi$
$908$	0	0
$909$	1.64766e10i	0.727601i
$910$	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
$915$	5.53377e9i	0.238807i
$916$	0	0
$917$	1.23131e10i	0.527322i
$918$	0	0
$919$	2.44812e10	1.04047	0.520234	−	0.854024i	$-0.325845\pi$
$919$	2.44812e10	1.04047	0.520234	+	0.854024i	$0.325845\pi$
$920$	0	0
$921$	3.70472e9	0.156260
$922$	0	0
$923$	6.31115e10i	2.64182i
$924$	0	0
$925$	− 2.12336e10i	− 0.882121i
$926$	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
$931$	1.28702e10i	0.522711i
$932$	0	0
$933$	1.53515e10i	0.618820i
$934$	0	0
$935$	1.73893e10	0.695732
$936$	0	0
$937$	−2.78747e10	−1.10693	−0.553466	−	0.832871i	$-0.686695\pi$
$937$	−2.78747e10	−1.10693	−0.553466	+	0.832871i	$0.686695\pi$
$938$	0	0
$939$	− 1.11096e10i	− 0.437894i
$940$	0	0
$941$	3.45921e10i	1.35336i	0.736277	+	0.676680i	$0.236582\pi$
$941$	3.45921e10i	1.35336i	−0.736277	+	0.676680i	$0.763418\pi$
$942$	0	0
$943$	1.81448e9	0.0704632
$944$	0	0
$945$	−1.66136e10	−0.640402
$946$	0	0
$947$	− 2.20475e10i	− 0.843598i	−0.906689	−	0.421799i	$-0.861399\pi$
$947$	− 2.20475e10i	− 0.843598i	0.906689	−	0.421799i	$-0.138601\pi$
$948$	0	0
$949$	2.44181e8i	0.00927429i
$950$	0	0
$951$	−1.05096e10	−0.396235
$952$	0	0
$953$	−3.95876e9	−0.148161	−0.0740805	−	0.997252i	$-0.523602\pi$
$953$	−3.95876e9	−0.148161	−0.0740805	+	0.997252i	$0.523602\pi$
$954$	0	0
$955$	− 1.30203e10i	− 0.483737i
$956$	0	0
$957$	− 2.24743e9i	− 0.0828885i
$958$	0	0
$959$	−3.98874e10	−1.46040
$960$	0	0
$961$	−2.21921e10	−0.806615
$962$	0	0
$963$	3.21256e10i	1.15920i
$964$	0	0
$965$	2.17943e9i	0.0780724i
$966$	0	0
$967$	9.76598e9	0.347315	0.173657	−	0.984806i	$-0.444441\pi$
$967$	9.76598e9	0.347315	0.173657	+	0.984806i	$0.444441\pi$
$968$	0	0
$969$	−2.16936e10	−0.765946
$970$	0	0
$971$	2.15553e10i	0.755589i	0.925889	+	0.377795i	$0.123317\pi$
$971$	2.15553e10i	0.755589i	−0.925889	+	0.377795i	$0.876683\pi$
$972$	0	0
$973$	2.94863e10i	1.02619i
$974$	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
$979$	− 5.82657e9i	− 0.198460i
$980$	0	0
$981$	− 1.46787e10i	− 0.496417i
$982$	0	0
$983$	2.85116e10	0.957380	0.478690	−	0.877984i	$-0.341112\pi$
$983$	2.85116e10	0.957380	0.478690	+	0.877984i	$0.341112\pi$
$984$	0	0
$985$	−1.65871e10	−0.553023
$986$	0	0
$987$	3.60455e9i	0.119328i
$988$	0	0
$989$	4.24509e8i	0.0139540i
$990$	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
$995$	7.37731e9i	0.237420i
$996$	0	0
$997$	− 1.30057e10i	− 0.415624i	−0.978169	−	0.207812i	$-0.933366\pi$
$997$	− 1.30057e10i	− 0.415624i	0.978169	−	0.207812i	$-0.0666343\pi$
$998$	0	0
$999$	−4.12804e10	−1.30998

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.8.b.a.5.3	✓	6	8.3	odd	2
8.8.b.a.5.4	yes	6	4.3	odd	2
32.8.b.a.17.3		6	1.1	even	1	trivial
32.8.b.a.17.4		6	8.5	even	2	inner
72.8.d.b.37.3		6	12.11	even	2
72.8.d.b.37.4		6	24.11	even	2
256.8.a.q.1.3		6	16.5	even	4
256.8.a.q.1.4		6	16.13	even	4
256.8.a.r.1.3		6	16.3	odd	4
256.8.a.r.1.4		6	16.11	odd	4
288.8.d.b.145.3		6	3.2	odd	2
288.8.d.b.145.4		6	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:	\( N \)	\(=\)	\( 32 = 2^{5} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	32.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-21.9408i q^{3} +184.916i q^{5} -1051.96 q^{7} +1705.60 q^{9} +O(q^{10})\)	\(q-21.9408i q^{3} +184.916i q^{5} -1051.96 q^{7} +1705.60 q^{9} +4324.35i q^{11} +11253.2i q^{13} +4057.20 q^{15} -21746.4 q^{17} +45466.5i q^{19} +23080.8i q^{21} -4414.37 q^{23} +43931.2 q^{25} -85406.8i q^{27} -23687.1i q^{29} -72941.9 q^{31} +94879.8 q^{33} -194523. i q^{35} -483338. i q^{37} +246905. q^{39} -411040. q^{41} -96165.3i q^{43} +315392. i q^{45} +156171. q^{47} +283070. q^{49} +477134. i q^{51} +686962. i q^{53} -799641. q^{55} +997572. q^{57} +1.79961e6i q^{59} +1.36394e6i q^{61} -1.79422e6 q^{63} -2.08090e6 q^{65} -1.08853e6i q^{67} +96854.8i q^{69} +5.60830e6 q^{71} +21698.7 q^{73} -963886. i q^{75} -4.54903e6i q^{77} -2.34010e6 q^{79} +1.85626e6 q^{81} -882169. i q^{83} -4.02125e6i q^{85} -519714. q^{87} -1.34738e6 q^{89} -1.18379e7i q^{91} +1.60040e6i q^{93} -8.40746e6 q^{95} +7.32798e6 q^{97} +7.37562e6i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 688 q^{7} - 2918 q^{9}+O(q^{10}) \) 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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