LMFDB - Embedded newform 32.8.b.a.17.2

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			17.2
Root			$-4.85268 + 2.90715i$ of defining polynomial
Character	$\chi$	$=$	32.17
Dual form			32.8.b.a.17.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-40.2163i q^{3} +324.492i q^{5} +956.960 q^{7} +569.651 q^{9} +O(q^{10})$	$q-40.2163i q^{3} +324.492i q^{5} +956.960 q^{7} +569.651 q^{9} -5452.20i q^{11} -6289.38i q^{13} +13049.8 q^{15} +34587.3 q^{17} +14595.6i q^{19} -38485.4i q^{21} +24667.5 q^{23} -27169.8 q^{25} -110862. i q^{27} +171116. i q^{29} -111688. q^{31} -219267. q^{33} +310526. i q^{35} -103636. i q^{37} -252935. q^{39} +71691.3 q^{41} +328419. i q^{43} +184847. i q^{45} -119043. q^{47} +92230.3 q^{49} -1.39097e6i q^{51} -1.04011e6i q^{53} +1.76919e6 q^{55} +586982. q^{57} +225984. i q^{59} +1.55268e6i q^{61} +545133. q^{63} +2.04085e6 q^{65} -316375. i q^{67} -992033. i q^{69} -538965. q^{71} -2.68512e6 q^{73} +1.09267e6i q^{75} -5.21754e6i q^{77} -8.22632e6 q^{79} -3.21264e6 q^{81} +5.89510e6i q^{83} +1.12233e7i q^{85} +6.88164e6 q^{87} +437005. q^{89} -6.01868e6i q^{91} +4.49169e6i q^{93} -4.73616e6 q^{95} -7.84322e6 q^{97} -3.10585e6i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	− 40.2163i	− 0.859959i	−0.902839	−	0.429979i	$-0.858521\pi$
$3$	− 40.2163i	− 0.859959i	0.902839	−	0.429979i	$-0.141479\pi$
$4$	0	0
$5$	324.492i	1.16094i	0.814283	+	0.580468i	$0.197130\pi$
$5$	324.492i	1.16094i	−0.814283	+	0.580468i	$0.802870\pi$
$6$	0	0
$7$	956.960	1.05451	0.527255	−	0.849707i	$-0.323221\pi$
$7$	956.960	1.05451	0.527255	+	0.849707i	$0.323221\pi$
$8$	0	0
$9$	569.651	0.260471
$10$	0	0
$11$	− 5452.20i	− 1.23509i	−0.786537	−	0.617544i	$-0.788128\pi$
$11$	− 5452.20i	− 1.23509i	0.786537	−	0.617544i	$-0.211872\pi$
$12$	0	0
$13$	− 6289.38i	− 0.793973i	−0.917824	−	0.396987i	$-0.870056\pi$
$13$	− 6289.38i	− 0.793973i	0.917824	−	0.396987i	$-0.129944\pi$
$14$	0	0
$15$	13049.8	0.998357
$16$	0	0
$17$	34587.3	1.70744	0.853720	−	0.520733i	$-0.174341\pi$
$17$	34587.3	1.70744	0.853720	+	0.520733i	$0.174341\pi$
$18$	0	0
$19$	14595.6i	0.488186i	0.969752	+	0.244093i	$0.0784903\pi$
$19$	14595.6i	0.488186i	−0.969752	+	0.244093i	$0.921510\pi$
$20$	0	0
$21$	− 38485.4i	− 0.906835i
$22$	0	0
$23$	24667.5	0.422743	0.211372	−	0.977406i	$-0.432207\pi$
$23$	24667.5	0.422743	0.211372	+	0.977406i	$0.432207\pi$
$24$	0	0
$25$	−27169.8	−0.347774
$26$	0	0
$27$	− 110862.i	− 1.08395i
$28$	0	0
$29$	171116.i	1.30286i	0.758710	+	0.651429i	$0.225830\pi$
$29$	171116.i	1.30286i	−0.758710	+	0.651429i	$0.774170\pi$
$30$	0	0
$31$	−111688.	−0.673352	−0.336676	−	0.941620i	$-0.609303\pi$
$31$	−111688.	−0.673352	−0.336676	+	0.941620i	$0.609303\pi$
$32$	0	0
$33$	−219267.	−1.06212
$34$	0	0
$35$	310526.i	1.22422i
$36$	0	0
$37$	− 103636.i	− 0.336360i	−0.985756	−	0.168180i	$-0.946211\pi$
$37$	− 103636.i	− 0.336360i	0.985756	−	0.168180i	$-0.0537890\pi$
$38$	0	0
$39$	−252935.	−0.682784
$40$	0	0
$41$	71691.3	0.162451	0.0812256	−	0.996696i	$-0.474117\pi$
$41$	71691.3	0.162451	0.0812256	+	0.996696i	$0.474117\pi$
$42$	0	0
$43$	328419.i	0.629925i	0.949104	+	0.314962i	$0.101992\pi$
$43$	328419.i	0.629925i	−0.949104	+	0.314962i	$0.898008\pi$
$44$	0	0
$45$	184847.i	0.302391i
$46$	0	0
$47$	−119043.	−0.167248	−0.0836241	−	0.996497i	$-0.526650\pi$
$47$	−119043.	−0.167248	−0.0836241	+	0.996497i	$0.526650\pi$
$48$	0	0
$49$	92230.3	0.111992
$50$	0	0
$51$	− 1.39097e6i	− 1.46833i
$52$	0	0
$53$	− 1.04011e6i	− 0.959648i	−0.877365	−	0.479824i	$-0.840700\pi$
$53$	− 1.04011e6i	− 0.959648i	0.877365	−	0.479824i	$-0.159300\pi$
$54$	0	0
$55$	1.76919e6	1.43386
$56$	0	0
$57$	586982.	0.419820
$58$	0	0
$59$	225984.i	0.143250i	0.997432	+	0.0716250i	$0.0228185\pi$
$59$	225984.i	0.143250i	−0.997432	+	0.0716250i	$0.977182\pi$
$60$	0	0
$61$	1.55268e6i	0.875843i	0.899013	+	0.437922i	$0.144285\pi$
$61$	1.55268e6i	0.875843i	−0.899013	+	0.437922i	$0.855715\pi$
$62$	0	0
$63$	545133.	0.274670
$64$	0	0
$65$	2.04085e6	0.921753
$66$	0	0
$67$	− 316375.i	− 0.128511i	−0.997933	−	0.0642555i	$-0.979533\pi$
$67$	− 316375.i	− 0.128511i	0.997933	−	0.0642555i	$-0.0204673\pi$
$68$	0	0
$69$	− 992033.i	− 0.363542i
$70$	0	0
$71$	−538965.	−0.178713	−0.0893566	−	0.996000i	$-0.528481\pi$
$71$	−538965.	−0.178713	−0.0893566	+	0.996000i	$0.528481\pi$
$72$	0	0
$73$	−2.68512e6	−0.807856	−0.403928	−	0.914791i	$-0.632355\pi$
$73$	−2.68512e6	−0.807856	−0.403928	+	0.914791i	$0.632355\pi$
$74$	0	0
$75$	1.09267e6i	0.299071i
$76$	0	0
$77$	− 5.21754e6i	− 1.30241i
$78$	0	0
$79$	−8.22632e6	−1.87720	−0.938600	−	0.345007i	$-0.887877\pi$
$79$	−8.22632e6	−1.87720	−0.938600	+	0.345007i	$0.887877\pi$
$80$	0	0
$81$	−3.21264e6	−0.671684
$82$	0	0
$83$	5.89510e6i	1.13167i	0.824520	+	0.565833i	$0.191445\pi$
$83$	5.89510e6i	1.13167i	−0.824520	+	0.565833i	$0.808555\pi$
$84$	0	0
$85$	1.12233e7i	1.98223i
$86$	0	0
$87$	6.88164e6	1.12040
$88$	0	0
$89$	437005.	0.0657085	0.0328542	−	0.999460i	$-0.489540\pi$
$89$	437005.	0.0657085	0.0328542	+	0.999460i	$0.489540\pi$
$90$	0	0
$91$	− 6.01868e6i	− 0.837253i
$92$	0	0
$93$	4.49169e6i	0.579055i
$94$	0	0
$95$	−4.73616e6	−0.566753
$96$	0	0
$97$	−7.84322e6	−0.872556	−0.436278	−	0.899812i	$-0.643704\pi$
$97$	−7.84322e6	−0.872556	−0.436278	+	0.899812i	$0.643704\pi$
$98$	0	0
$99$	− 3.10585e6i	− 0.321705i
$100$	0	0
$101$	− 6.19757e6i	− 0.598545i	−0.954168	−	0.299272i	$-0.903256\pi$
$101$	− 6.19757e6i	− 0.598545i	0.954168	−	0.299272i	$-0.0967439\pi$
$102$	0	0
$103$	6.59816e6	0.594966	0.297483	−	0.954727i	$-0.403853\pi$
$103$	6.59816e6	0.594966	0.297483	+	0.954727i	$0.403853\pi$
$104$	0	0
$105$	1.24882e7	1.05278
$106$	0	0
$107$	− 512845.i	− 0.0404709i	−0.999795	−	0.0202354i	$-0.993558\pi$
$107$	− 512845.i	− 0.0404709i	0.999795	−	0.0202354i	$-0.00644158\pi$
$108$	0	0
$109$	− 1.95882e7i	− 1.44878i	−0.689393	−	0.724388i	$-0.742123\pi$
$109$	− 1.95882e7i	− 1.44878i	0.689393	−	0.724388i	$-0.257877\pi$
$110$	0	0
$111$	−4.16785e6	−0.289256
$112$	0	0
$113$	1.88876e7	1.23141	0.615705	−	0.787977i	$-0.288871\pi$
$113$	1.88876e7	1.23141	0.615705	+	0.787977i	$0.288871\pi$
$114$	0	0
$115$	8.00438e6i	0.490778i
$116$	0	0
$117$	− 3.58275e6i	− 0.206807i
$118$	0	0
$119$	3.30987e7	1.80051
$120$	0	0
$121$	−1.02394e7	−0.525441
$122$	0	0
$123$	− 2.88316e6i	− 0.139701i
$124$	0	0
$125$	1.65345e7i	0.757193i
$126$	0	0
$127$	−3.96314e7	−1.71683	−0.858413	−	0.512959i	$-0.828549\pi$
$127$	−3.96314e7	−1.71683	−0.858413	+	0.512959i	$0.828549\pi$
$128$	0	0
$129$	1.32078e7	0.541709
$130$	0	0
$131$	− 3.65337e7i	− 1.41986i	−0.704274	−	0.709928i	$-0.748727\pi$
$131$	− 3.65337e7i	− 1.41986i	0.704274	−	0.709928i	$-0.251273\pi$
$132$	0	0
$133$	1.39675e7i	0.514797i
$134$	0	0
$135$	3.59739e7	1.25840
$136$	0	0
$137$	−2.56967e7	−0.853799	−0.426899	−	0.904299i	$-0.640394\pi$
$137$	−2.56967e7	−0.853799	−0.426899	+	0.904299i	$0.640394\pi$
$138$	0	0
$139$	5.23001e7i	1.65177i	0.563836	+	0.825886i	$0.309325\pi$
$139$	5.23001e7i	1.65177i	−0.563836	+	0.825886i	$0.690675\pi$
$140$	0	0
$141$	4.78747e6i	0.143827i
$142$	0	0
$143$	−3.42910e7	−0.980626
$144$	0	0
$145$	−5.55256e7	−1.51254
$146$	0	0
$147$	− 3.70916e6i	− 0.0963085i
$148$	0	0
$149$	− 1.80406e7i	− 0.446785i	−0.974729	−	0.223392i	$-0.928287\pi$
$149$	− 1.80406e7i	− 0.446785i	0.974729	−	0.223392i	$-0.0717131\pi$
$150$	0	0
$151$	−3.87385e7	−0.915637	−0.457818	−	0.889046i	$-0.651369\pi$
$151$	−3.87385e7	−0.915637	−0.457818	+	0.889046i	$0.651369\pi$
$152$	0	0
$153$	1.97027e7	0.444739
$154$	0	0
$155$	− 3.62420e7i	− 0.781719i
$156$	0	0
$157$	5.12341e7i	1.05660i	0.849058	+	0.528300i	$0.177170\pi$
$157$	5.12341e7i	1.05660i	−0.849058	+	0.528300i	$0.822830\pi$
$158$	0	0
$159$	−4.18292e7	−0.825258
$160$	0	0
$161$	2.36058e7	0.445787
$162$	0	0
$163$	8.57572e7i	1.55101i	0.631343	+	0.775504i	$0.282504\pi$
$163$	8.57572e7i	1.55101i	−0.631343	+	0.775504i	$0.717496\pi$
$164$	0	0
$165$	− 7.11504e7i	− 1.23306i
$166$	0	0
$167$	1.05871e8	1.75901	0.879503	−	0.475893i	$-0.157875\pi$
$167$	1.05871e8	1.75901	0.879503	+	0.475893i	$0.157875\pi$
$168$	0	0
$169$	2.31923e7	0.369606
$170$	0	0
$171$	8.31441e6i	0.127158i
$172$	0	0
$173$	1.98148e7i	0.290956i	0.989361	+	0.145478i	$0.0464720\pi$
$173$	1.98148e7i	0.290956i	−0.989361	+	0.145478i	$0.953528\pi$
$174$	0	0
$175$	−2.60005e7	−0.366731
$176$	0	0
$177$	9.08822e6	0.123189
$178$	0	0
$179$	− 2.97800e7i	− 0.388096i	−0.980992	−	0.194048i	$-0.937838\pi$
$179$	− 2.97800e7i	− 0.388096i	0.980992	−	0.194048i	$-0.0621618\pi$
$180$	0	0
$181$	3.96227e6i	0.0496671i	0.999692	+	0.0248335i	$0.00790558\pi$
$181$	3.96227e6i	0.0496671i	−0.999692	+	0.0248335i	$0.992094\pi$
$182$	0	0
$183$	6.24429e7	0.753189
$184$	0	0
$185$	3.36290e7	0.390493
$186$	0	0
$187$	− 1.88577e8i	− 2.10884i
$188$	0	0
$189$	− 1.06091e8i	− 1.14304i
$190$	0	0
$191$	4.80105e7	0.498562	0.249281	−	0.968431i	$-0.419806\pi$
$191$	4.80105e7	0.498562	0.249281	+	0.968431i	$0.419806\pi$
$192$	0	0
$193$	−4.72502e6	−0.0473100	−0.0236550	−	0.999720i	$-0.507530\pi$
$193$	−4.72502e6	−0.0473100	−0.0236550	+	0.999720i	$0.507530\pi$
$194$	0	0
$195$	− 8.20754e7i	− 0.792669i
$196$	0	0
$197$	1.14882e8i	1.07058i	0.844668	+	0.535290i	$0.179798\pi$
$197$	1.14882e8i	1.07058i	−0.844668	+	0.535290i	$0.820202\pi$
$198$	0	0
$199$	−1.20933e7	−0.108782	−0.0543911	−	0.998520i	$-0.517322\pi$
$199$	−1.20933e7	−0.108782	−0.0543911	+	0.998520i	$0.517322\pi$
$200$	0	0
$201$	−1.27234e7	−0.110514
$202$	0	0
$203$	1.63751e8i	1.37388i
$204$	0	0
$205$	2.32632e7i	0.188596i
$206$	0	0
$207$	1.40518e7	0.110112
$208$	0	0
$209$	7.95784e7	0.602953
$210$	0	0
$211$	1.95850e8i	1.43527i	0.696418	+	0.717636i	$0.254776\pi$
$211$	1.95850e8i	1.43527i	−0.696418	+	0.717636i	$0.745224\pi$
$212$	0	0
$213$	2.16752e7i	0.153686i
$214$	0	0
$215$	−1.06569e8	−0.731302
$216$	0	0
$217$	−1.06881e8	−0.710057
$218$	0	0
$219$	1.07986e8i	0.694723i
$220$	0	0
$221$	− 2.17532e8i	− 1.35566i
$222$	0	0
$223$	−1.08024e8	−0.652311	−0.326156	−	0.945316i	$-0.605753\pi$
$223$	−1.08024e8	−0.652311	−0.326156	+	0.945316i	$0.605753\pi$
$224$	0	0
$225$	−1.54773e7	−0.0905851
$226$	0	0
$227$	− 1.61144e8i	− 0.914374i	−0.889371	−	0.457187i	$-0.848857\pi$
$227$	− 1.61144e8i	− 0.914374i	0.889371	−	0.457187i	$-0.151143\pi$
$228$	0	0
$229$	5.27173e7i	0.290088i	0.989425	+	0.145044i	$0.0463323\pi$
$229$	5.27173e7i	0.290088i	−0.989425	+	0.145044i	$0.953668\pi$
$230$	0	0
$231$	−2.09830e8	−1.12002
$232$	0	0
$233$	−1.79423e8	−0.929249	−0.464625	−	0.885508i	$-0.653811\pi$
$233$	−1.79423e8	−0.929249	−0.464625	+	0.885508i	$0.653811\pi$
$234$	0	0
$235$	− 3.86285e7i	− 0.194165i
$236$	0	0
$237$	3.30832e8i	1.61431i
$238$	0	0
$239$	8.42441e7	0.399160	0.199580	−	0.979882i	$-0.436042\pi$
$239$	8.42441e7	0.399160	0.199580	+	0.979882i	$0.436042\pi$
$240$	0	0
$241$	−2.12302e8	−0.977000	−0.488500	−	0.872564i	$-0.662456\pi$
$241$	−2.12302e8	−0.977000	−0.488500	+	0.872564i	$0.662456\pi$
$242$	0	0
$243$	− 1.13255e8i	− 0.506333i
$244$	0	0
$245$	2.99280e7i	0.130016i
$246$	0	0
$247$	9.17975e7	0.387607
$248$	0	0
$249$	2.37079e8	0.973186
$250$	0	0
$251$	1.18102e8i	0.471411i	0.971825	+	0.235706i	$0.0757401\pi$
$251$	1.18102e8i	0.471411i	−0.971825	+	0.235706i	$0.924260\pi$
$252$	0	0
$253$	− 1.34492e8i	− 0.522125i
$254$	0	0
$255$	4.51359e8	1.70463
$256$	0	0
$257$	1.27463e8	0.468402	0.234201	−	0.972188i	$-0.424753\pi$
$257$	1.27463e8	0.468402	0.234201	+	0.972188i	$0.424753\pi$
$258$	0	0
$259$	− 9.91755e7i	− 0.354695i
$260$	0	0
$261$	9.74762e7i	0.339357i
$262$	0	0
$263$	−4.33125e8	−1.46814	−0.734071	−	0.679073i	$-0.762382\pi$
$263$	−4.33125e8	−1.46814	−0.734071	+	0.679073i	$0.762382\pi$
$264$	0	0
$265$	3.37506e8	1.11409
$266$	0	0
$267$	− 1.75747e7i	− 0.0565066i
$268$	0	0
$269$	− 3.44748e8i	− 1.07986i	−0.841709	−	0.539931i	$-0.818450\pi$
$269$	− 3.44748e8i	− 1.07986i	0.841709	−	0.539931i	$-0.181550\pi$
$270$	0	0
$271$	4.42513e8	1.35062	0.675311	−	0.737533i	$-0.264010\pi$
$271$	4.42513e8	1.35062	0.675311	+	0.737533i	$0.264010\pi$
$272$	0	0
$273$	−2.42049e8	−0.720003
$274$	0	0
$275$	1.48135e8i	0.429531i
$276$	0	0
$277$	− 3.18148e8i	− 0.899395i	−0.893181	−	0.449697i	$-0.851532\pi$
$277$	− 3.18148e8i	− 0.899395i	0.893181	−	0.449697i	$-0.148468\pi$
$278$	0	0
$279$	−6.36234e7	−0.175389
$280$	0	0
$281$	1.28497e8	0.345478	0.172739	−	0.984968i	$-0.444738\pi$
$281$	1.28497e8	0.345478	0.172739	+	0.984968i	$0.444738\pi$
$282$	0	0
$283$	− 3.98970e8i	− 1.04637i	−0.852218	−	0.523187i	$-0.824743\pi$
$283$	− 3.98970e8i	− 1.04637i	0.852218	−	0.523187i	$-0.175257\pi$
$284$	0	0
$285$	1.90471e8i	0.487384i
$286$	0	0
$287$	6.86057e7	0.171306
$288$	0	0
$289$	7.85942e8	1.91535
$290$	0	0
$291$	3.15425e8i	0.750362i
$292$	0	0
$293$	− 2.00958e8i	− 0.466732i	−0.972389	−	0.233366i	$-0.925026\pi$
$293$	− 2.00958e8i	− 0.466732i	0.972389	−	0.233366i	$-0.0749741\pi$
$294$	0	0
$295$	−7.33298e7	−0.166304
$296$	0	0
$297$	−6.04444e8	−1.33878
$298$	0	0
$299$	− 1.55143e8i	− 0.335647i
$300$	0	0
$301$	3.14284e8i	0.664262i
$302$	0	0
$303$	−2.49243e8	−0.514724
$304$	0	0
$305$	−5.03830e8	−1.01680
$306$	0	0
$307$	1.58918e7i	0.0313465i	0.999877	+	0.0156733i	$0.00498916\pi$
$307$	1.58918e7i	0.0313465i	−0.999877	+	0.0156733i	$0.995011\pi$
$308$	0	0
$309$	− 2.65353e8i	− 0.511646i
$310$	0	0
$311$	−4.87710e8	−0.919391	−0.459695	−	0.888077i	$-0.652041\pi$
$311$	−4.87710e8	−0.919391	−0.459695	+	0.888077i	$0.652041\pi$
$312$	0	0
$313$	−3.24731e8	−0.598576	−0.299288	−	0.954163i	$-0.596749\pi$
$313$	−3.24731e8	−0.598576	−0.299288	+	0.954163i	$0.596749\pi$
$314$	0	0
$315$	1.76891e8i	0.318874i
$316$	0	0
$317$	1.06084e9i	1.87043i	0.354086	+	0.935213i	$0.384792\pi$
$317$	1.06084e9i	1.87043i	−0.354086	+	0.935213i	$0.615208\pi$
$318$	0	0
$319$	9.32958e8	1.60914
$320$	0	0
$321$	−2.06247e7	−0.0348033
$322$	0	0
$323$	5.04824e8i	0.833548i
$324$	0	0
$325$	1.70881e8i	0.276123i
$326$	0	0
$327$	−7.87763e8	−1.24589
$328$	0	0
$329$	−1.13919e8	−0.176365
$330$	0	0
$331$	− 2.88487e8i	− 0.437249i	−0.975809	−	0.218624i	$-0.929843\pi$
$331$	− 2.88487e8i	− 0.437249i	0.975809	−	0.218624i	$-0.0701569\pi$
$332$	0	0
$333$	− 5.90363e7i	− 0.0876121i
$334$	0	0
$335$	1.02661e8	0.149193
$336$	0	0
$337$	−1.10595e8	−0.157410	−0.0787051	−	0.996898i	$-0.525079\pi$
$337$	−1.10595e8	−0.157410	−0.0787051	+	0.996898i	$0.525079\pi$
$338$	0	0
$339$	− 7.59590e8i	− 1.05896i
$340$	0	0
$341$	6.08948e8i	0.831649i
$342$	0	0
$343$	−6.99837e8	−0.936414
$344$	0	0
$345$	3.21907e8	0.422049
$346$	0	0
$347$	1.10651e9i	1.42168i	0.703352	+	0.710841i	$0.251686\pi$
$347$	1.10651e9i	1.42168i	−0.703352	+	0.710841i	$0.748314\pi$
$348$	0	0
$349$	− 1.38337e9i	− 1.74201i	−0.491278	−	0.871003i	$-0.663470\pi$
$349$	− 1.38337e9i	− 1.74201i	0.491278	−	0.871003i	$-0.336530\pi$
$350$	0	0
$351$	−6.97254e8	−0.860630
$352$	0	0
$353$	−2.47617e8	−0.299618	−0.149809	−	0.988715i	$-0.547866\pi$
$353$	−2.47617e8	−0.299618	−0.149809	+	0.988715i	$0.547866\pi$
$354$	0	0
$355$	− 1.74890e8i	− 0.207475i
$356$	0	0
$357$	− 1.33111e9i	− 1.54837i
$358$	0	0
$359$	1.38641e9	1.58148	0.790738	−	0.612155i	$-0.209697\pi$
$359$	1.38641e9	1.58148	0.790738	+	0.612155i	$0.209697\pi$
$360$	0	0
$361$	6.80839e8	0.761674
$362$	0	0
$363$	4.11789e8i	0.451858i
$364$	0	0
$365$	− 8.71299e8i	− 0.937869i
$366$	0	0
$367$	7.49367e8	0.791341	0.395670	−	0.918393i	$-0.370512\pi$
$367$	7.49367e8	0.791341	0.395670	+	0.918393i	$0.370512\pi$
$368$	0	0
$369$	4.08390e7	0.0423139
$370$	0	0
$371$	− 9.95340e8i	− 1.01196i
$372$	0	0
$373$	1.49519e9i	1.49181i	0.666051	+	0.745906i	$0.267983\pi$
$373$	1.49519e9i	1.49181i	−0.666051	+	0.745906i	$0.732017\pi$
$374$	0	0
$375$	6.64957e8	0.651155
$376$	0	0
$377$	1.07621e9	1.03443
$378$	0	0
$379$	7.92096e7i	0.0747379i	0.999302	+	0.0373689i	$0.0118977\pi$
$379$	7.92096e7i	0.0747379i	−0.999302	+	0.0373689i	$0.988102\pi$
$380$	0	0
$381$	1.59383e9i	1.47640i
$382$	0	0
$383$	−4.80285e8	−0.436820	−0.218410	−	0.975857i	$-0.570087\pi$
$383$	−4.80285e8	−0.436820	−0.218410	+	0.975857i	$0.570087\pi$
$384$	0	0
$385$	1.69305e9	1.51202
$386$	0	0
$387$	1.87084e8i	0.164077i
$388$	0	0
$389$	1.07150e9i	0.922928i	0.887159	+	0.461464i	$0.152676\pi$
$389$	1.07150e9i	0.922928i	−0.887159	+	0.461464i	$0.847324\pi$
$390$	0	0
$391$	8.53180e8	0.721809
$392$	0	0
$393$	−1.46925e9	−1.22102
$394$	0	0
$395$	− 2.66937e9i	− 2.17931i
$396$	0	0
$397$	− 2.03185e9i	− 1.62976i	−0.579627	−	0.814882i	$-0.696802\pi$
$397$	− 2.03185e9i	− 1.62976i	0.579627	−	0.814882i	$-0.303198\pi$
$398$	0	0
$399$	5.61719e8	0.442705
$400$	0	0
$401$	−2.57759e9	−1.99622	−0.998111	−	0.0614301i	$-0.980434\pi$
$401$	−2.57759e9	−1.99622	−0.998111	+	0.0614301i	$0.980434\pi$
$402$	0	0
$403$	7.02451e8i	0.534624i
$404$	0	0
$405$	− 1.04248e9i	− 0.779782i
$406$	0	0
$407$	−5.65044e8	−0.415434
$408$	0	0
$409$	3.30242e8	0.238672	0.119336	−	0.992854i	$-0.461923\pi$
$409$	3.30242e8	0.238672	0.119336	+	0.992854i	$0.461923\pi$
$410$	0	0
$411$	1.03343e9i	0.734232i
$412$	0	0
$413$	2.16257e8i	0.151059i
$414$	0	0
$415$	−1.91291e9	−1.31379
$416$	0	0
$417$	2.10331e9	1.42046
$418$	0	0
$419$	5.80021e7i	0.0385207i	0.999815	+	0.0192604i	$0.00613114\pi$
$419$	5.80021e7i	0.0385207i	−0.999815	+	0.0192604i	$0.993869\pi$
$420$	0	0
$421$	− 1.90609e8i	− 0.124496i	−0.998061	−	0.0622480i	$-0.980173\pi$
$421$	− 1.90609e8i	− 0.124496i	0.998061	−	0.0622480i	$-0.0198270\pi$
$422$	0	0
$423$	−6.78129e7	−0.0435633
$424$	0	0
$425$	−9.39731e8	−0.593803
$426$	0	0
$427$	1.48585e9i	0.923586i
$428$	0	0
$429$	1.37906e9i	0.843298i
$430$	0	0
$431$	−2.42923e9	−1.46150	−0.730749	−	0.682646i	$-0.760829\pi$
$431$	−2.42923e9	−1.46150	−0.730749	+	0.682646i	$0.760829\pi$
$432$	0	0
$433$	−2.37902e9	−1.40828	−0.704141	−	0.710060i	$-0.748668\pi$
$433$	−2.37902e9	−1.40828	−0.704141	+	0.710060i	$0.748668\pi$
$434$	0	0
$435$	2.23303e9i	1.30072i
$436$	0	0
$437$	3.60037e8i	0.206378i
$438$	0	0
$439$	1.33161e9	0.751194	0.375597	−	0.926783i	$-0.377438\pi$
$439$	1.33161e9	0.751194	0.375597	+	0.926783i	$0.377438\pi$
$440$	0	0
$441$	5.25390e7	0.0291707
$442$	0	0
$443$	5.02643e8i	0.274692i	0.990523	+	0.137346i	$0.0438573\pi$
$443$	5.02643e8i	0.274692i	−0.990523	+	0.137346i	$0.956143\pi$
$444$	0	0
$445$	1.41805e8i	0.0762834i
$446$	0	0
$447$	−7.25525e8	−0.384217
$448$	0	0
$449$	3.14785e9	1.64116	0.820580	−	0.571531i	$-0.193650\pi$
$449$	3.14785e9	1.64116	0.820580	+	0.571531i	$0.193650\pi$
$450$	0	0
$451$	− 3.90876e8i	− 0.200641i
$452$	0	0
$453$	1.55792e9i	0.787410i
$454$	0	0
$455$	1.95301e9	0.971998
$456$	0	0
$457$	−2.68422e9	−1.31556	−0.657782	−	0.753209i	$-0.728505\pi$
$457$	−2.68422e9	−1.31556	−0.657782	+	0.753209i	$0.728505\pi$
$458$	0	0
$459$	− 3.83442e9i	− 1.85078i
$460$	0	0
$461$	− 1.30434e9i	− 0.620065i	−0.950726	−	0.310033i	$-0.899660\pi$
$461$	− 1.30434e9i	− 0.620065i	0.950726	−	0.310033i	$-0.100340\pi$
$462$	0	0
$463$	−2.86853e9	−1.34315	−0.671577	−	0.740934i	$-0.734383\pi$
$463$	−2.86853e9	−1.34315	−0.671577	+	0.740934i	$0.734383\pi$
$464$	0	0
$465$	−1.45752e9	−0.672246
$466$	0	0
$467$	5.96519e8i	0.271029i	0.990775	+	0.135514i	$0.0432687\pi$
$467$	5.96519e8i	0.271029i	−0.990775	+	0.135514i	$0.956731\pi$
$468$	0	0
$469$	− 3.02758e8i	− 0.135516i
$470$	0	0
$471$	2.06045e9	0.908632
$472$	0	0
$473$	1.79061e9	0.778012
$474$	0	0
$475$	− 3.96561e8i	− 0.169778i
$476$	0	0
$477$	− 5.92497e8i	− 0.249961i
$478$	0	0
$479$	2.16068e9	0.898289	0.449144	−	0.893459i	$-0.351729\pi$
$479$	2.16068e9	0.898289	0.449144	+	0.893459i	$0.351729\pi$
$480$	0	0
$481$	−6.51805e8	−0.267061
$482$	0	0
$483$	− 9.49337e8i	− 0.383359i
$484$	0	0
$485$	− 2.54506e9i	− 1.01298i
$486$	0	0
$487$	−1.41934e8	−0.0556847	−0.0278424	−	0.999612i	$-0.508864\pi$
$487$	−1.41934e8	−0.0556847	−0.0278424	+	0.999612i	$0.508864\pi$
$488$	0	0
$489$	3.44884e9	1.33380
$490$	0	0
$491$	2.38677e9i	0.909966i	0.890500	+	0.454983i	$0.150355\pi$
$491$	2.38677e9i	0.909966i	−0.890500	+	0.454983i	$0.849645\pi$
$492$	0	0
$493$	5.91843e9i	2.22455i
$494$	0	0
$495$	1.00782e9	0.373479
$496$	0	0
$497$	−5.15769e8	−0.188455
$498$	0	0
$499$	− 5.23900e9i	− 1.88754i	−0.330601	−	0.943771i	$-0.607251\pi$
$499$	− 5.23900e9i	− 1.88754i	0.330601	−	0.943771i	$-0.392749\pi$
$500$	0	0
$501$	− 4.25772e9i	− 1.51267i
$502$	0	0
$503$	3.63292e9	1.27282	0.636411	−	0.771350i	$-0.280418\pi$
$503$	3.63292e9	1.27282	0.636411	+	0.771350i	$0.280418\pi$
$504$	0	0
$505$	2.01106e9	0.694872
$506$	0	0
$507$	− 9.32706e8i	− 0.317846i
$508$	0	0
$509$	− 2.58693e9i	− 0.869505i	−0.900550	−	0.434753i	$-0.856836\pi$
$509$	− 2.58693e9i	− 0.869505i	0.900550	−	0.434753i	$-0.143164\pi$
$510$	0	0
$511$	−2.56955e9	−0.851892
$512$	0	0
$513$	1.61811e9	0.529171
$514$	0	0
$515$	2.14105e9i	0.690718i
$516$	0	0
$517$	6.49047e8i	0.206566i
$518$	0	0
$519$	7.96876e8	0.250210
$520$	0	0
$521$	1.08542e8	0.0336253	0.0168127	−	0.999859i	$-0.494648\pi$
$521$	1.08542e8	0.0336253	0.0168127	+	0.999859i	$0.494648\pi$
$522$	0	0
$523$	6.10725e9i	1.86676i	0.358884	+	0.933382i	$0.383157\pi$
$523$	6.10725e9i	1.86676i	−0.358884	+	0.933382i	$0.616843\pi$
$524$	0	0
$525$	1.04564e9i	0.315374i
$526$	0	0
$527$	−3.86300e9	−1.14971
$528$	0	0
$529$	−2.79634e9	−0.821288
$530$	0	0
$531$	1.28732e8i	0.0373125i
$532$	0	0
$533$	− 4.50894e8i	− 0.128982i
$534$	0	0
$535$	1.66414e8	0.0469841
$536$	0	0
$537$	−1.19764e9	−0.333747
$538$	0	0
$539$	− 5.02858e8i	− 0.138320i
$540$	0	0
$541$	− 5.39345e8i	− 0.146445i	−0.997316	−	0.0732227i	$-0.976672\pi$
$541$	− 5.39345e8i	− 0.146445i	0.997316	−	0.0732227i	$-0.0233284\pi$
$542$	0	0
$543$	1.59348e8	0.0427116
$544$	0	0
$545$	6.35620e9	1.68194
$546$	0	0
$547$	8.82287e7i	0.0230491i	0.999934	+	0.0115246i	$0.00366846\pi$
$547$	8.82287e7i	0.0230491i	−0.999934	+	0.0115246i	$0.996332\pi$
$548$	0	0
$549$	8.84483e8i	0.228132i
$550$	0	0
$551$	−2.49754e9	−0.636037
$552$	0	0
$553$	−7.87226e9	−1.97953
$554$	0	0
$555$	− 1.35243e9i	− 0.335807i
$556$	0	0
$557$	5.57233e8i	0.136629i	0.997664	+	0.0683147i	$0.0217622\pi$
$557$	5.57233e8i	0.136629i	−0.997664	+	0.0683147i	$0.978238\pi$
$558$	0	0
$559$	2.06555e9	0.500143
$560$	0	0
$561$	−7.58386e9	−1.81351
$562$	0	0
$563$	− 1.17012e9i	− 0.276344i	−0.990408	−	0.138172i	$-0.955877\pi$
$563$	− 1.17012e9i	− 0.276344i	0.990408	−	0.138172i	$-0.0441227\pi$
$564$	0	0
$565$	6.12887e9i	1.42959i
$566$	0	0
$567$	−3.07437e9	−0.708297
$568$	0	0
$569$	−2.39181e9	−0.544295	−0.272147	−	0.962256i	$-0.587734\pi$
$569$	−2.39181e9	−0.544295	−0.272147	+	0.962256i	$0.587734\pi$
$570$	0	0
$571$	− 3.15823e9i	− 0.709933i	−0.934879	−	0.354966i	$-0.884492\pi$
$571$	− 3.15823e9i	− 0.709933i	0.934879	−	0.354966i	$-0.115508\pi$
$572$	0	0
$573$	− 1.93080e9i	− 0.428743i
$574$	0	0
$575$	−6.70211e8	−0.147019
$576$	0	0
$577$	4.03435e9	0.874296	0.437148	−	0.899390i	$-0.355989\pi$
$577$	4.03435e9	0.874296	0.437148	+	0.899390i	$0.355989\pi$
$578$	0	0
$579$	1.90023e8i	0.0406846i
$580$	0	0
$581$	5.64138e9i	1.19335i
$582$	0	0
$583$	−5.67087e9	−1.18525
$584$	0	0
$585$	1.16257e9	0.240090
$586$	0	0
$587$	− 2.72240e9i	− 0.555544i	−0.960647	−	0.277772i	$-0.910404\pi$
$587$	− 2.72240e9i	− 0.555544i	0.960647	−	0.277772i	$-0.0895960\pi$
$588$	0	0
$589$	− 1.63016e9i	− 0.328721i
$590$	0	0
$591$	4.62012e9	0.920655
$592$	0	0
$593$	−2.29251e9	−0.451460	−0.225730	−	0.974190i	$-0.572477\pi$
$593$	−2.29251e9	−0.451460	−0.225730	+	0.974190i	$0.572477\pi$
$594$	0	0
$595$	1.07402e10i	2.09028i
$596$	0	0
$597$	4.86346e8i	0.0935482i
$598$	0	0
$599$	3.58734e9	0.681991	0.340995	−	0.940065i	$-0.389236\pi$
$599$	3.58734e9	0.681991	0.340995	+	0.940065i	$0.389236\pi$
$600$	0	0
$601$	8.20369e9	1.54152	0.770759	−	0.637127i	$-0.219877\pi$
$601$	8.20369e9	1.54152	0.770759	+	0.637127i	$0.219877\pi$
$602$	0	0
$603$	− 1.80223e8i	− 0.0334734i
$604$	0	0
$605$	− 3.32259e9i	− 0.610004i
$606$	0	0
$607$	4.60087e9	0.834986	0.417493	−	0.908680i	$-0.362909\pi$
$607$	4.60087e9	0.834986	0.417493	+	0.908680i	$0.362909\pi$
$608$	0	0
$609$	6.58546e9	1.18148
$610$	0	0
$611$	7.48707e8i	0.132791i
$612$	0	0
$613$	− 8.55728e9i	− 1.50046i	−0.661178	−	0.750229i	$-0.729943\pi$
$613$	− 8.55728e9i	− 1.50046i	0.661178	−	0.750229i	$-0.270057\pi$
$614$	0	0
$615$	9.35561e8	0.162184
$616$	0	0
$617$	−2.58089e9	−0.442355	−0.221178	−	0.975234i	$-0.570990\pi$
$617$	−2.58089e9	−0.442355	−0.221178	+	0.975234i	$0.570990\pi$
$618$	0	0
$619$	5.26641e9i	0.892478i	0.894914	+	0.446239i	$0.147237\pi$
$619$	5.26641e9i	0.892478i	−0.894914	+	0.446239i	$0.852763\pi$
$620$	0	0
$621$	− 2.73469e9i	− 0.458234i
$622$	0	0
$623$	4.18197e8	0.0692903
$624$	0	0
$625$	−7.48796e9	−1.22683
$626$	0	0
$627$	− 3.20035e9i	− 0.518514i
$628$	0	0
$629$	− 3.58449e9i	− 0.574314i
$630$	0	0
$631$	−8.32515e9	−1.31914	−0.659568	−	0.751645i	$-0.729261\pi$
$631$	−8.32515e9	−1.31914	−0.659568	+	0.751645i	$0.729261\pi$
$632$	0	0
$633$	7.87635e9	1.23428
$634$	0	0
$635$	− 1.28601e10i	− 1.99313i
$636$	0	0
$637$	− 5.80071e8i	− 0.0889187i
$638$	0	0
$639$	−3.07022e8	−0.0465496
$640$	0	0
$641$	4.26190e9	0.639146	0.319573	−	0.947562i	$-0.396460\pi$
$641$	4.26190e9	0.639146	0.319573	+	0.947562i	$0.396460\pi$
$642$	0	0
$643$	− 1.26588e10i	− 1.87782i	−0.344167	−	0.938908i	$-0.611839\pi$
$643$	− 1.26588e10i	− 1.87782i	0.344167	−	0.938908i	$-0.388161\pi$
$644$	0	0
$645$	4.28582e9i	0.628890i
$646$	0	0
$647$	7.38061e9	1.07134	0.535670	−	0.844427i	$-0.320059\pi$
$647$	7.38061e9	1.07134	0.535670	+	0.844427i	$0.320059\pi$
$648$	0	0
$649$	1.23211e9	0.176926
$650$	0	0
$651$	4.29837e9i	0.610620i
$652$	0	0
$653$	3.21579e9i	0.451951i	0.974133	+	0.225975i	$0.0725569\pi$
$653$	3.21579e9i	0.451951i	−0.974133	+	0.225975i	$0.927443\pi$
$654$	0	0
$655$	1.18549e10	1.64836
$656$	0	0
$657$	−1.52958e9	−0.210423
$658$	0	0
$659$	5.17004e9i	0.703711i	0.936054	+	0.351856i	$0.114449\pi$
$659$	5.17004e9i	0.703711i	−0.936054	+	0.351856i	$0.885551\pi$
$660$	0	0
$661$	1.95604e9i	0.263435i	0.991287	+	0.131717i	$0.0420491\pi$
$661$	1.95604e9i	0.263435i	−0.991287	+	0.131717i	$0.957951\pi$
$662$	0	0
$663$	−8.74835e9	−1.16581
$664$	0	0
$665$	−4.53232e9	−0.597647
$666$	0	0
$667$	4.22099e9i	0.550775i
$668$	0	0
$669$	4.34434e9i	0.560961i
$670$	0	0
$671$	8.46551e9	1.08174
$672$	0	0
$673$	1.54679e9	0.195605	0.0978024	−	0.995206i	$-0.468819\pi$
$673$	1.54679e9	0.195605	0.0978024	+	0.995206i	$0.468819\pi$
$674$	0	0
$675$	3.01211e9i	0.376971i
$676$	0	0
$677$	8.55209e9i	1.05928i	0.848222	+	0.529642i	$0.177674\pi$
$677$	8.55209e9i	1.05928i	−0.848222	+	0.529642i	$0.822326\pi$
$678$	0	0
$679$	−7.50565e9	−0.920119
$680$	0	0
$681$	−6.48061e9	−0.786323
$682$	0	0
$683$	7.26976e9i	0.873067i	0.899688	+	0.436534i	$0.143794\pi$
$683$	7.26976e9i	0.873067i	−0.899688	+	0.436534i	$0.856206\pi$
$684$	0	0
$685$	− 8.33837e9i	− 0.991206i
$686$	0	0
$687$	2.12010e9	0.249463
$688$	0	0
$689$	−6.54162e9	−0.761935
$690$	0	0
$691$	− 5.19893e9i	− 0.599434i	−0.954028	−	0.299717i	$-0.903108\pi$
$691$	− 5.19893e9i	− 0.599434i	0.954028	−	0.299717i	$-0.0968922\pi$
$692$	0	0
$693$	− 2.97218e9i	− 0.339241i
$694$	0	0
$695$	−1.69709e10	−1.91760
$696$	0	0
$697$	2.47961e9	0.277376
$698$	0	0
$699$	7.21572e9i	0.799116i
$700$	0	0
$701$	1.35221e10i	1.48262i	0.671163	+	0.741310i	$0.265795\pi$
$701$	1.35221e10i	1.48262i	−0.671163	+	0.741310i	$0.734205\pi$
$702$	0	0
$703$	1.51263e9	0.164206
$704$	0	0
$705$	−1.55349e9	−0.166974
$706$	0	0
$707$	− 5.93083e9i	− 0.631172i
$708$	0	0
$709$	− 1.05901e10i	− 1.11594i	−0.829862	−	0.557969i	$-0.811581\pi$
$709$	− 1.05901e10i	− 1.11594i	0.829862	−	0.557969i	$-0.188419\pi$
$710$	0	0
$711$	−4.68613e9	−0.488957
$712$	0	0
$713$	−2.75507e9	−0.284655
$714$	0	0
$715$	− 1.11271e10i	− 1.13845i
$716$	0	0
$717$	− 3.38799e9i	− 0.343261i
$718$	0	0
$719$	−1.49161e10	−1.49659	−0.748297	−	0.663363i	$-0.769128\pi$
$719$	−1.49161e10	−1.49659	−0.748297	+	0.663363i	$0.769128\pi$
$720$	0	0
$721$	6.31417e9	0.627398
$722$	0	0
$723$	8.53800e9i	0.840180i
$724$	0	0
$725$	− 4.64919e9i	− 0.453100i
$726$	0	0
$727$	8.90159e9	0.859206	0.429603	−	0.903018i	$-0.358654\pi$
$727$	8.90159e9	0.859206	0.429603	+	0.903018i	$0.358654\pi$
$728$	0	0
$729$	−1.15808e10	−1.10711
$730$	0	0
$731$	1.13591e10i	1.07556i
$732$	0	0
$733$	7.99792e9i	0.750090i	0.927007	+	0.375045i	$0.122373\pi$
$733$	7.99792e9i	0.750090i	−0.927007	+	0.375045i	$0.877627\pi$
$734$	0	0
$735$	1.20359e9	0.111808
$736$	0	0
$737$	−1.72494e9	−0.158722
$738$	0	0
$739$	1.03852e10i	0.946588i	0.880905	+	0.473294i	$0.156935\pi$
$739$	1.03852e10i	0.946588i	−0.880905	+	0.473294i	$0.843065\pi$
$740$	0	0
$741$	− 3.69175e9i	− 0.333326i
$742$	0	0
$743$	3.73477e9	0.334044	0.167022	−	0.985953i	$-0.446585\pi$
$743$	3.73477e9	0.334044	0.167022	+	0.985953i	$0.446585\pi$
$744$	0	0
$745$	5.85402e9	0.518689
$746$	0	0
$747$	3.35815e9i	0.294766i
$748$	0	0
$749$	− 4.90772e8i	− 0.0426770i
$750$	0	0
$751$	1.34330e10	1.15726	0.578631	−	0.815589i	$-0.303587\pi$
$751$	1.34330e10	1.15726	0.578631	+	0.815589i	$0.303587\pi$
$752$	0	0
$753$	4.74963e9	0.405394
$754$	0	0
$755$	− 1.25703e10i	− 1.06300i
$756$	0	0
$757$	6.78007e9i	0.568065i	0.958815	+	0.284033i	$0.0916724\pi$
$757$	6.78007e9i	0.568065i	−0.958815	+	0.284033i	$0.908328\pi$
$758$	0	0
$759$	−5.40877e9	−0.449006
$760$	0	0
$761$	8.01137e9	0.658962	0.329481	−	0.944162i	$-0.393126\pi$
$761$	8.01137e9	0.658962	0.329481	+	0.944162i	$0.393126\pi$
$762$	0	0
$763$	− 1.87451e10i	− 1.52775i
$764$	0	0
$765$	6.39335e9i	0.516314i
$766$	0	0
$767$	1.42130e9	0.113737
$768$	0	0
$769$	1.46553e10	1.16213	0.581063	−	0.813858i	$-0.302637\pi$
$769$	1.46553e10	1.16213	0.581063	+	0.813858i	$0.302637\pi$
$770$	0	0
$771$	− 5.12609e9i	− 0.402806i
$772$	0	0
$773$	− 2.33296e10i	− 1.81668i	−0.418233	−	0.908340i	$-0.637350\pi$
$773$	− 2.33296e10i	− 1.81668i	0.418233	−	0.908340i	$-0.362650\pi$
$774$	0	0
$775$	3.03456e9	0.234174
$776$	0	0
$777$	−3.98847e9	−0.305023
$778$	0	0
$779$	1.04638e9i	0.0793064i
$780$	0	0
$781$	2.93855e9i	0.220726i
$782$	0	0
$783$	1.89703e10	1.41224
$784$	0	0
$785$	−1.66250e10	−1.22664
$786$	0	0
$787$	− 5.27740e8i	− 0.0385930i	−0.999814	−	0.0192965i	$-0.993857\pi$
$787$	− 5.27740e8i	− 0.0385930i	0.999814	−	0.0192965i	$-0.00614264\pi$
$788$	0	0
$789$	1.74187e10i	1.26254i
$790$	0	0
$791$	1.80747e10	1.29853
$792$	0	0
$793$	9.76536e9	0.695396
$794$	0	0
$795$	− 1.35732e10i	− 0.958072i
$796$	0	0
$797$	− 2.41514e9i	− 0.168981i	−0.996424	−	0.0844907i	$-0.973074\pi$
$797$	− 2.41514e9i	− 0.168981i	0.996424	−	0.0844907i	$-0.0269263\pi$
$798$	0	0
$799$	−4.11738e9	−0.285566
$800$	0	0
$801$	2.48940e8	0.0171152
$802$	0	0
$803$	1.46398e10i	0.997773i
$804$	0	0
$805$	7.65988e9i	0.517531i
$806$	0	0
$807$	−1.38645e10	−0.928636
$808$	0	0
$809$	1.16364e10	0.772679	0.386340	−	0.922357i	$-0.373739\pi$
$809$	1.16364e10	0.772679	0.386340	+	0.922357i	$0.373739\pi$
$810$	0	0
$811$	− 1.44359e10i	− 0.950320i	−0.879899	−	0.475160i	$-0.842390\pi$
$811$	− 1.44359e10i	− 0.950320i	0.879899	−	0.475160i	$-0.157610\pi$
$812$	0	0
$813$	− 1.77962e10i	− 1.16148i
$814$	0	0
$815$	−2.78275e10	−1.80062
$816$	0	0
$817$	−4.79348e9	−0.307521
$818$	0	0
$819$	− 3.42855e9i	− 0.218080i
$820$	0	0
$821$	− 7.63805e9i	− 0.481706i	−0.970562	−	0.240853i	$-0.922573\pi$
$821$	− 7.63805e9i	− 0.481706i	0.970562	−	0.240853i	$-0.0774271\pi$
$822$	0	0
$823$	2.16446e10	1.35348	0.676738	−	0.736224i	$-0.263393\pi$
$823$	2.16446e10	1.35348	0.676738	+	0.736224i	$0.263393\pi$
$824$	0	0
$825$	5.95746e9	0.369379
$826$	0	0
$827$	1.57823e10i	0.970288i	0.874434	+	0.485144i	$0.161233\pi$
$827$	1.57823e10i	0.970288i	−0.874434	+	0.485144i	$0.838767\pi$
$828$	0	0
$829$	− 2.63296e10i	− 1.60510i	−0.596582	−	0.802552i	$-0.703475\pi$
$829$	− 2.63296e10i	− 1.60510i	0.596582	−	0.802552i	$-0.296525\pi$
$830$	0	0
$831$	−1.27947e10	−0.773442
$832$	0	0
$833$	3.19000e9	0.191220
$834$	0	0
$835$	3.43541e10i	2.04210i
$836$	0	0
$837$	1.23820e10i	0.729882i
$838$	0	0
$839$	1.84995e9	0.108142	0.0540710	−	0.998537i	$-0.482780\pi$
$839$	1.84995e9	0.108142	0.0540710	+	0.998537i	$0.482780\pi$
$840$	0	0
$841$	−1.20307e10	−0.697438
$842$	0	0
$843$	− 5.16767e9i	− 0.297097i
$844$	0	0
$845$	7.52569e9i	0.429090i
$846$	0	0
$847$	−9.79866e9	−0.554083
$848$	0	0
$849$	−1.60451e10	−0.899839
$850$	0	0
$851$	− 2.55643e9i	− 0.142194i
$852$	0	0
$853$	− 6.28088e7i	− 0.00346496i	−0.999998	−	0.00173248i	$-0.999449\pi$
$853$	− 6.28088e7i	− 0.00346496i	0.999998	−	0.00173248i	$-0.000551466\pi$
$854$	0	0
$855$	−2.69796e9	−0.147623
$856$	0	0
$857$	−6.25595e9	−0.339516	−0.169758	−	0.985486i	$-0.554299\pi$
$857$	−6.25595e9	−0.339516	−0.169758	+	0.985486i	$0.554299\pi$
$858$	0	0
$859$	2.48119e10i	1.33562i	0.744331	+	0.667811i	$0.232769\pi$
$859$	2.48119e10i	1.33562i	−0.744331	+	0.667811i	$0.767231\pi$
$860$	0	0
$861$	− 2.75907e9i	− 0.147316i
$862$	0	0
$863$	−1.28944e10	−0.682909	−0.341455	−	0.939898i	$-0.610920\pi$
$863$	−1.28944e10	−0.682909	−0.341455	+	0.939898i	$0.610920\pi$
$864$	0	0
$865$	−6.42973e9	−0.337782
$866$	0	0
$867$	− 3.16077e10i	− 1.64712i
$868$	0	0
$869$	4.48516e10i	2.31851i
$870$	0	0
$871$	−1.98980e9	−0.102034
$872$	0	0
$873$	−4.46789e9	−0.227276
$874$	0	0
$875$	1.58229e10i	0.798468i
$876$	0	0
$877$	2.75670e10i	1.38004i	0.723792	+	0.690018i	$0.242397\pi$
$877$	2.75670e10i	1.38004i	−0.723792	+	0.690018i	$0.757603\pi$
$878$	0	0
$879$	−8.08177e9	−0.401371
$880$	0	0
$881$	2.74343e10	1.35169	0.675847	−	0.737042i	$-0.263778\pi$
$881$	2.74343e10	1.35169	0.675847	+	0.737042i	$0.263778\pi$
$882$	0	0
$883$	− 2.09906e10i	− 1.02604i	−0.858378	−	0.513018i	$-0.828527\pi$
$883$	− 2.09906e10i	− 1.02604i	0.858378	−	0.513018i	$-0.171473\pi$
$884$	0	0
$885$	2.94905e9i	0.143015i
$886$	0	0
$887$	2.39134e10	1.15056	0.575280	−	0.817957i	$-0.304893\pi$
$887$	2.39134e10	1.15056	0.575280	+	0.817957i	$0.304893\pi$
$888$	0	0
$889$	−3.79257e10	−1.81041
$890$	0	0
$891$	1.75160e10i	0.829588i
$892$	0	0
$893$	− 1.73751e9i	− 0.0816483i
$894$	0	0
$895$	9.66337e9	0.450555
$896$	0	0
$897$	−6.23927e9	−0.288643
$898$	0	0
$899$	− 1.91117e10i	− 0.877282i
$900$	0	0
$901$	− 3.59744e10i	− 1.63854i
$902$	0	0
$903$	1.26393e10	0.571238
$904$	0	0
$905$	−1.28572e9	−0.0576603
$906$	0	0
$907$	− 3.84425e10i	− 1.71075i	−0.518012	−	0.855373i	$-0.673328\pi$
$907$	− 3.84425e10i	− 1.71075i	0.518012	−	0.855373i	$-0.326672\pi$
$908$	0	0
$909$	− 3.53045e9i	− 0.155904i
$910$	0	0
$911$	−3.64507e10	−1.59732	−0.798660	−	0.601783i	$-0.794457\pi$
$911$	−3.64507e10	−1.59732	−0.798660	+	0.601783i	$0.794457\pi$
$912$	0	0
$913$	3.21413e10	1.39771
$914$	0	0
$915$	2.02622e10i	0.874405i
$916$	0	0
$917$	− 3.49613e10i	− 1.49725i
$918$	0	0
$919$	−2.37343e10	−1.00872	−0.504361	−	0.863493i	$-0.668272\pi$
$919$	−2.37343e10	−1.00872	−0.504361	+	0.863493i	$0.668272\pi$
$920$	0	0
$921$	6.39111e8	0.0269567
$922$	0	0
$923$	3.38976e9i	0.141894i
$924$	0	0
$925$	2.81577e9i	0.116977i
$926$	0	0
$927$	3.75864e9	0.154972
$928$	0	0
$929$	−4.23952e10	−1.73485	−0.867424	−	0.497570i	$-0.834226\pi$
$929$	−4.23952e10	−1.73485	−0.867424	+	0.497570i	$0.834226\pi$
$930$	0	0
$931$	1.34616e9i	0.0546730i
$932$	0	0
$933$	1.96139e10i	0.790638i
$934$	0	0
$935$	6.11916e10	2.44823
$936$	0	0
$937$	3.70014e10	1.46937	0.734683	−	0.678411i	$-0.237331\pi$
$937$	3.70014e10	1.46937	0.734683	+	0.678411i	$0.237331\pi$
$938$	0	0
$939$	1.30595e10i	0.514750i
$940$	0	0
$941$	− 2.10617e10i	− 0.824005i	−0.911183	−	0.412003i	$-0.864829\pi$
$941$	− 2.10617e10i	− 0.824005i	0.911183	−	0.412003i	$-0.135171\pi$
$942$	0	0
$943$	1.76844e9	0.0686752
$944$	0	0
$945$	3.44256e10	1.32700
$946$	0	0
$947$	1.35146e10i	0.517104i	0.965997	+	0.258552i	$0.0832454\pi$
$947$	1.35146e10i	0.517104i	−0.965997	+	0.258552i	$0.916755\pi$
$948$	0	0
$949$	1.68877e10i	0.641416i
$950$	0	0
$951$	4.26628e10	1.60849
$952$	0	0
$953$	4.26831e10	1.59746	0.798731	−	0.601688i	$-0.205505\pi$
$953$	4.26831e10	1.59746	0.798731	+	0.601688i	$0.205505\pi$
$954$	0	0
$955$	1.55790e10i	0.578799i
$956$	0	0
$957$	− 3.75201e10i	− 1.38380i
$958$	0	0
$959$	−2.45907e10	−0.900340
$960$	0	0
$961$	−1.50383e10	−0.546597
$962$	0	0
$963$	− 2.92142e8i	− 0.0105415i
$964$	0	0
$965$	− 1.53323e9i	− 0.0549239i
$966$	0	0
$967$	−2.32274e10	−0.826053	−0.413026	−	0.910719i	$-0.635528\pi$
$967$	−2.32274e10	−0.826053	−0.413026	+	0.910719i	$0.635528\pi$
$968$	0	0
$969$	2.03021e10	0.716817
$970$	0	0
$971$	− 4.85678e10i	− 1.70248i	−0.524780	−	0.851238i	$-0.675852\pi$
$971$	− 4.85678e10i	− 1.70248i	0.524780	−	0.851238i	$-0.324148\pi$
$972$	0	0
$973$	5.00491e10i	1.74181i
$974$	0	0
$975$	6.87221e9	0.237454
$976$	0	0
$977$	1.54549e10	0.530193	0.265096	−	0.964222i	$-0.414596\pi$
$977$	1.54549e10	0.530193	0.265096	+	0.964222i	$0.414596\pi$
$978$	0	0
$979$	− 2.38264e9i	− 0.0811557i
$980$	0	0
$981$	− 1.11584e10i	− 0.377364i
$982$	0	0
$983$	1.59192e10	0.534546	0.267273	−	0.963621i	$-0.413877\pi$
$983$	1.59192e10	0.534546	0.267273	+	0.963621i	$0.413877\pi$
$984$	0	0
$985$	−3.72782e10	−1.24288
$986$	0	0
$987$	4.58142e9i	0.151667i
$988$	0	0
$989$	8.10126e9i	0.266296i
$990$	0	0
$991$	−1.49883e10	−0.489208	−0.244604	−	0.969623i	$-0.578658\pi$
$991$	−1.49883e10	−0.489208	−0.244604	+	0.969623i	$0.578658\pi$
$992$	0	0
$993$	−1.16019e10	−0.376016
$994$	0	0
$995$	− 3.92416e9i	− 0.126289i
$996$	0	0
$997$	− 3.44101e10i	− 1.09965i	−0.835281	−	0.549824i	$-0.814695\pi$
$997$	− 3.44101e10i	− 1.09965i	0.835281	−	0.549824i	$-0.185305\pi$
$998$	0	0
$999$	−1.14893e10	−0.364598

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	32.8.b.a.17.2		6
3.2	odd	2		288.8.d.b.145.2		6
4.3	odd	2		8.8.b.a.5.1	✓	6
8.3	odd	2		8.8.b.a.5.2	yes	6
8.5	even	2	inner	32.8.b.a.17.5		6
12.11	even	2		72.8.d.b.37.6		6
16.3	odd	4		256.8.a.r.1.2		6
16.5	even	4		256.8.a.q.1.2		6
16.11	odd	4		256.8.a.r.1.5		6
16.13	even	4		256.8.a.q.1.5		6
24.5	odd	2		288.8.d.b.145.5		6
24.11	even	2		72.8.d.b.37.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.8.b.a.5.1	✓	6	4.3	odd	2
8.8.b.a.5.2	yes	6	8.3	odd	2
32.8.b.a.17.2		6	1.1	even	1	trivial
32.8.b.a.17.5		6	8.5	even	2	inner
72.8.d.b.37.5		6	24.11	even	2
72.8.d.b.37.6		6	12.11	even	2
256.8.a.q.1.2		6	16.5	even	4
256.8.a.q.1.5		6	16.13	even	4
256.8.a.r.1.2		6	16.3	odd	4
256.8.a.r.1.5		6	16.11	odd	4
288.8.d.b.145.2		6	3.2	odd	2
288.8.d.b.145.5		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)

\(f(q)\)	\(=\)	\(q-40.2163i q^{3} +324.492i q^{5} +956.960 q^{7} +569.651 q^{9} +O(q^{10})\)	\(q-40.2163i q^{3} +324.492i q^{5} +956.960 q^{7} +569.651 q^{9} -5452.20i q^{11} -6289.38i q^{13} +13049.8 q^{15} +34587.3 q^{17} +14595.6i q^{19} -38485.4i q^{21} +24667.5 q^{23} -27169.8 q^{25} -110862. i q^{27} +171116. i q^{29} -111688. q^{31} -219267. q^{33} +310526. i q^{35} -103636. i q^{37} -252935. q^{39} +71691.3 q^{41} +328419. i q^{43} +184847. i q^{45} -119043. q^{47} +92230.3 q^{49} -1.39097e6i q^{51} -1.04011e6i q^{53} +1.76919e6 q^{55} +586982. q^{57} +225984. i q^{59} +1.55268e6i q^{61} +545133. q^{63} +2.04085e6 q^{65} -316375. i q^{67} -992033. i q^{69} -538965. q^{71} -2.68512e6 q^{73} +1.09267e6i q^{75} -5.21754e6i q^{77} -8.22632e6 q^{79} -3.21264e6 q^{81} +5.89510e6i q^{83} +1.12233e7i q^{85} +6.88164e6 q^{87} +437005. q^{89} -6.01868e6i q^{91} +4.49169e6i q^{93} -4.73616e6 q^{95} -7.84322e6 q^{97} -3.10585e6i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 688 q^{7} - 2918 q^{9}+O(q^{10}) \) 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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