LMFDB - Embedded newform 32.8.b.a.17.6

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			17.6
Root			$5.57668 - 0.949035i$ of defining polynomial
Character	$\chi$	$=$	32.17
Dual form			32.8.b.a.17.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+76.9497i q^{3} +338.443i q^{5} +438.996 q^{7} -3734.25 q^{9} +O(q^{10})$	$q+76.9497i q^{3} +338.443i q^{5} +438.996 q^{7} -3734.25 q^{9} -1966.58i q^{11} -2210.98i q^{13} -26043.0 q^{15} -12114.9 q^{17} +32872.2i q^{19} +33780.6i q^{21} -19605.1 q^{23} -36418.4 q^{25} -119060. i q^{27} -160689. i q^{29} +229270. q^{31} +151328. q^{33} +148575. i q^{35} +496284. i q^{37} +170134. q^{39} +599971. q^{41} +88346.0i q^{43} -1.26383e6i q^{45} -820344. q^{47} -630825. q^{49} -932236. i q^{51} +1.53717e6i q^{53} +665574. q^{55} -2.52950e6 q^{57} +1.82480e6i q^{59} +484582. i q^{61} -1.63932e6 q^{63} +748290. q^{65} -79878.2i q^{67} -1.50860e6i q^{69} -1.27078e6 q^{71} +3.70820e6 q^{73} -2.80238e6i q^{75} -863321. i q^{77} +2.55846e6 q^{79} +994857. q^{81} +1.53414e6i q^{83} -4.10019e6i q^{85} +1.23650e7 q^{87} +1.99492e6 q^{89} -970612. i q^{91} +1.76423e7i q^{93} -1.11253e7 q^{95} -28917.7 q^{97} +7.34370e6i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	76.9497i	1.64544i	0.568446	+	0.822721i	$0.307545\pi$
$3$	76.9497i	1.64544i	−0.568446	+	0.822721i	$0.692455\pi$
$4$	0	0
$5$	338.443i	1.21085i	0.795903	+	0.605425i	$0.206997\pi$
$5$	338.443i	1.21085i	−0.795903	+	0.605425i	$0.793003\pi$
$6$	0	0
$7$	438.996	0.483746	0.241873	−	0.970308i	$-0.422238\pi$
$7$	438.996	0.483746	0.241873	+	0.970308i	$0.422238\pi$
$8$	0	0
$9$	−3734.25	−1.70748
$10$	0	0
$11$	− 1966.58i	− 0.445489i	−0.974877	−	0.222744i	$-0.928498\pi$
$11$	− 1966.58i	− 0.445489i	0.974877	−	0.222744i	$-0.0715015\pi$
$12$	0	0
$13$	− 2210.98i	− 0.279115i	−0.990214	−	0.139557i	$-0.955432\pi$
$13$	− 2210.98i	− 0.279115i	0.990214	−	0.139557i	$-0.0445680\pi$
$14$	0	0
$15$	−26043.0	−1.99238
$16$	0	0
$17$	−12114.9	−0.598064	−0.299032	−	0.954243i	$-0.596664\pi$
$17$	−12114.9	−0.598064	−0.299032	+	0.954243i	$0.596664\pi$
$18$	0	0
$19$	32872.2i	1.09949i	0.835333	+	0.549744i	$0.185275\pi$
$19$	32872.2i	1.09949i	−0.835333	+	0.549744i	$0.814725\pi$
$20$	0	0
$21$	33780.6i	0.795976i
$22$	0	0
$23$	−19605.1	−0.335986	−0.167993	−	0.985788i	$-0.553729\pi$
$23$	−19605.1	−0.335986	−0.167993	+	0.985788i	$0.553729\pi$
$24$	0	0
$25$	−36418.4	−0.466155
$26$	0	0
$27$	− 119060.i	− 1.16411i
$28$	0	0
$29$	− 160689.i	− 1.22347i	−0.791063	−	0.611735i	$-0.790472\pi$
$29$	− 160689.i	− 1.22347i	0.791063	−	0.611735i	$-0.209528\pi$
$30$	0	0
$31$	229270.	1.38224	0.691118	−	0.722742i	$-0.257119\pi$
$31$	229270.	1.38224	0.691118	+	0.722742i	$0.257119\pi$
$32$	0	0
$33$	151328.	0.733026
$34$	0	0
$35$	148575.i	0.585744i
$36$	0	0
$37$	496284.i	1.61074i	0.592775	+	0.805368i	$0.298032\pi$
$37$	496284.i	1.61074i	−0.592775	+	0.805368i	$0.701968\pi$
$38$	0	0
$39$	170134.	0.459267
$40$	0	0
$41$	599971.	1.35952	0.679762	−	0.733433i	$-0.262083\pi$
$41$	599971.	1.35952	0.679762	+	0.733433i	$0.262083\pi$
$42$	0	0
$43$	88346.0i	0.169452i	0.996404	+	0.0847262i	$0.0270015\pi$
$43$	88346.0i	0.169452i	−0.996404	+	0.0847262i	$0.972998\pi$
$44$	0	0
$45$	− 1.26383e6i	− 2.06750i
$46$	0	0
$47$	−820344.	−1.15253	−0.576267	−	0.817262i	$-0.695491\pi$
$47$	−820344.	−1.15253	−0.576267	+	0.817262i	$0.695491\pi$
$48$	0	0
$49$	−630825.	−0.765989
$50$	0	0
$51$	− 932236.i	− 0.984080i
$52$	0	0
$53$	1.53717e6i	1.41826i	0.705077	+	0.709131i	$0.250913\pi$
$53$	1.53717e6i	1.41826i	−0.705077	+	0.709131i	$0.749087\pi$
$54$	0	0
$55$	665574.	0.539420
$56$	0	0
$57$	−2.52950e6	−1.80914
$58$	0	0
$59$	1.82480e6i	1.15673i	0.815776	+	0.578367i	$0.196310\pi$
$59$	1.82480e6i	1.15673i	−0.815776	+	0.578367i	$0.803690\pi$
$60$	0	0
$61$	484582.i	0.273346i	0.990616	+	0.136673i	$0.0436410\pi$
$61$	484582.i	0.273346i	−0.990616	+	0.136673i	$0.956359\pi$
$62$	0	0
$63$	−1.63932e6	−0.825986
$64$	0	0
$65$	748290.	0.337966
$66$	0	0
$67$	− 79878.2i	− 0.0324464i	−0.999868	−	0.0162232i	$-0.994836\pi$
$67$	− 79878.2i	− 0.0324464i	0.999868	−	0.0162232i	$-0.00516423\pi$
$68$	0	0
$69$	− 1.50860e6i	− 0.552845i
$70$	0	0
$71$	−1.27078e6	−0.421373	−0.210686	−	0.977554i	$-0.567570\pi$
$71$	−1.27078e6	−0.421373	−0.210686	+	0.977554i	$0.567570\pi$
$72$	0	0
$73$	3.70820e6	1.11566	0.557832	−	0.829954i	$-0.311633\pi$
$73$	3.70820e6	1.11566	0.557832	+	0.829954i	$0.311633\pi$
$74$	0	0
$75$	− 2.80238e6i	− 0.767031i
$76$	0	0
$77$	− 863321.i	− 0.215504i
$78$	0	0
$79$	2.55846e6	0.583827	0.291914	−	0.956445i	$-0.405708\pi$
$79$	2.55846e6	0.583827	0.291914	+	0.956445i	$0.405708\pi$
$80$	0	0
$81$	994857.	0.208000
$82$	0	0
$83$	1.53414e6i	0.294505i	0.989099	+	0.147252i	$0.0470430\pi$
$83$	1.53414e6i	0.294505i	−0.989099	+	0.147252i	$0.952957\pi$
$84$	0	0
$85$	− 4.10019e6i	− 0.724166i
$86$	0	0
$87$	1.23650e7	2.01315
$88$	0	0
$89$	1.99492e6	0.299958	0.149979	−	0.988689i	$-0.452079\pi$
$89$	1.99492e6	0.299958	0.149979	+	0.988689i	$0.452079\pi$
$90$	0	0
$91$	− 970612.i	− 0.135021i
$92$	0	0
$93$	1.76423e7i	2.27439i
$94$	0	0
$95$	−1.11253e7	−1.33131
$96$	0	0
$97$	−28917.7	−0.00321708	−0.00160854	−	0.999999i	$-0.500512\pi$
$97$	−28917.7	−0.00321708	−0.00160854	+	0.999999i	$0.500512\pi$
$98$	0	0
$99$	7.34370e6i	0.760662i
$100$	0	0
$101$	− 1.68077e7i	− 1.62324i	−0.584182	−	0.811622i	$-0.698585\pi$
$101$	− 1.68077e7i	− 1.62324i	0.584182	−	0.811622i	$-0.301415\pi$
$102$	0	0
$103$	1.27746e7	1.15191	0.575953	−	0.817483i	$-0.304631\pi$
$103$	1.27746e7	1.15191	0.575953	+	0.817483i	$0.304631\pi$
$104$	0	0
$105$	−1.14328e7	−0.963807
$106$	0	0
$107$	− 1.35610e7i	− 1.07016i	−0.844803	−	0.535078i	$-0.820282\pi$
$107$	− 1.35610e7i	− 1.07016i	0.844803	−	0.535078i	$-0.179718\pi$
$108$	0	0
$109$	− 4.74206e6i	− 0.350731i	−0.984503	−	0.175366i	$-0.943889\pi$
$109$	− 4.74206e6i	− 0.350731i	0.984503	−	0.175366i	$-0.0561108\pi$
$110$	0	0
$111$	−3.81889e7	−2.65037
$112$	0	0
$113$	−8.06832e6	−0.526028	−0.263014	−	0.964792i	$-0.584717\pi$
$113$	−8.06832e6	−0.526028	−0.263014	+	0.964792i	$0.584717\pi$
$114$	0	0
$115$	− 6.63520e6i	− 0.406828i
$116$	0	0
$117$	8.25635e6i	0.476582i
$118$	0	0
$119$	−5.31839e6	−0.289311
$120$	0	0
$121$	1.56197e7	0.801540
$122$	0	0
$123$	4.61676e7i	2.23702i
$124$	0	0
$125$	1.41153e7i	0.646405i
$126$	0	0
$127$	1.12410e7	0.486960	0.243480	−	0.969906i	$-0.421711\pi$
$127$	1.12410e7	0.486960	0.243480	+	0.969906i	$0.421711\pi$
$128$	0	0
$129$	−6.79820e6	−0.278824
$130$	0	0
$131$	− 8.81527e6i	− 0.342599i	−0.985219	−	0.171299i	$-0.945203\pi$
$131$	− 8.81527e6i	− 0.342599i	0.985219	−	0.171299i	$-0.0547966\pi$
$132$	0	0
$133$	1.44308e7i	0.531874i
$134$	0	0
$135$	4.02951e7	1.40956
$136$	0	0
$137$	−3.33729e7	−1.10885	−0.554424	−	0.832234i	$-0.687061\pi$
$137$	−3.33729e7	−1.10885	−0.554424	+	0.832234i	$0.687061\pi$
$138$	0	0
$139$	− 4.68161e7i	− 1.47857i	−0.673391	−	0.739287i	$-0.735163\pi$
$139$	− 4.68161e7i	− 1.47857i	0.673391	−	0.739287i	$-0.264837\pi$
$140$	0	0
$141$	− 6.31252e7i	− 1.89643i
$142$	0	0
$143$	−4.34806e6	−0.124343
$144$	0	0
$145$	5.43840e7	1.48144
$146$	0	0
$147$	− 4.85418e7i	− 1.26039i
$148$	0	0
$149$	3.83709e7i	0.950277i	0.879911	+	0.475139i	$0.157602\pi$
$149$	3.83709e7i	0.950277i	−0.879911	+	0.475139i	$0.842398\pi$
$150$	0	0
$151$	7.17648e7	1.69626	0.848130	−	0.529788i	$-0.177729\pi$
$151$	7.17648e7	1.69626	0.848130	+	0.529788i	$0.177729\pi$
$152$	0	0
$153$	4.52400e7	1.02118
$154$	0	0
$155$	7.75948e7i	1.67368i
$156$	0	0
$157$	− 4.03778e7i	− 0.832710i	−0.909202	−	0.416355i	$-0.863307\pi$
$157$	− 4.03778e7i	− 0.832710i	0.909202	−	0.416355i	$-0.136693\pi$
$158$	0	0
$159$	−1.18285e8	−2.33367
$160$	0	0
$161$	−8.60656e6	−0.162532
$162$	0	0
$163$	9.84512e7i	1.78059i	0.455383	+	0.890296i	$0.349502\pi$
$163$	9.84512e7i	1.78059i	−0.455383	+	0.890296i	$0.650498\pi$
$164$	0	0
$165$	5.12157e7i	0.887584i
$166$	0	0
$167$	4.24811e7	0.705810	0.352905	−	0.935659i	$-0.385194\pi$
$167$	4.24811e7	0.705810	0.352905	+	0.935659i	$0.385194\pi$
$168$	0	0
$169$	5.78601e7	0.922095
$170$	0	0
$171$	− 1.22753e8i	− 1.87735i
$172$	0	0
$173$	2.65257e7i	0.389498i	0.980853	+	0.194749i	$0.0623893\pi$
$173$	2.65257e7i	0.389498i	−0.980853	+	0.194749i	$0.937611\pi$
$174$	0	0
$175$	−1.59875e7	−0.225501
$176$	0	0
$177$	−1.40418e8	−1.90334
$178$	0	0
$179$	− 2.42148e7i	− 0.315570i	−0.987474	−	0.157785i	$-0.949565\pi$
$179$	− 2.42148e7i	− 0.315570i	0.987474	−	0.157785i	$-0.0504353\pi$
$180$	0	0
$181$	− 2.78961e7i	− 0.349679i	−0.984597	−	0.174839i	$-0.944059\pi$
$181$	− 2.78961e7i	− 0.349679i	0.984597	−	0.174839i	$-0.0559406\pi$
$182$	0	0
$183$	−3.72884e7	−0.449775
$184$	0	0
$185$	−1.67964e8	−1.95036
$186$	0	0
$187$	2.38249e7i	0.266431i
$188$	0	0
$189$	− 5.22671e7i	− 0.563135i
$190$	0	0
$191$	−1.67596e7	−0.174039	−0.0870195	−	0.996207i	$-0.527734\pi$
$191$	−1.67596e7	−0.174039	−0.0870195	+	0.996207i	$0.527734\pi$
$192$	0	0
$193$	8.75008e7	0.876116	0.438058	−	0.898947i	$-0.355667\pi$
$193$	8.75008e7	0.876116	0.438058	+	0.898947i	$0.355667\pi$
$194$	0	0
$195$	5.75806e7i	0.556103i
$196$	0	0
$197$	− 2.56239e7i	− 0.238789i	−0.992847	−	0.119394i	$-0.961905\pi$
$197$	− 2.56239e7i	− 0.238789i	0.992847	−	0.119394i	$-0.0380953\pi$
$198$	0	0
$199$	−5.31884e7	−0.478444	−0.239222	−	0.970965i	$-0.576892\pi$
$199$	−5.31884e7	−0.478444	−0.239222	+	0.970965i	$0.576892\pi$
$200$	0	0
$201$	6.14660e6	0.0533886
$202$	0	0
$203$	− 7.05419e7i	− 0.591849i
$204$	0	0
$205$	2.03056e8i	1.64618i
$206$	0	0
$207$	7.32103e7	0.573688
$208$	0	0
$209$	6.46457e7	0.489810
$210$	0	0
$211$	− 2.01165e7i	− 0.147423i	−0.997280	−	0.0737114i	$-0.976516\pi$
$211$	− 2.01165e7i	− 0.147423i	0.997280	−	0.0737114i	$-0.0234844\pi$
$212$	0	0
$213$	− 9.77862e7i	− 0.693344i
$214$	0	0
$215$	−2.99001e7	−0.205181
$216$	0	0
$217$	1.00649e8	0.668651
$218$	0	0
$219$	2.85345e8i	1.83576i
$220$	0	0
$221$	2.67858e7i	0.166929i
$222$	0	0
$223$	1.67012e8	1.00851	0.504254	−	0.863555i	$-0.331768\pi$
$223$	1.67012e8	1.00851	0.504254	+	0.863555i	$0.331768\pi$
$224$	0	0
$225$	1.35995e8	0.795949
$226$	0	0
$227$	− 1.37308e8i	− 0.779122i	−0.921001	−	0.389561i	$-0.872627\pi$
$227$	− 1.37308e8i	− 0.779122i	0.921001	−	0.389561i	$-0.127373\pi$
$228$	0	0
$229$	− 2.67935e8i	− 1.47436i	−0.675694	−	0.737182i	$-0.736156\pi$
$229$	− 2.67935e8i	− 1.47436i	0.675694	−	0.737182i	$-0.263844\pi$
$230$	0	0
$231$	6.64322e7	0.354599
$232$	0	0
$233$	1.98032e8	1.02563	0.512815	−	0.858499i	$-0.328603\pi$
$233$	1.98032e8	1.02563	0.512815	+	0.858499i	$0.328603\pi$
$234$	0	0
$235$	− 2.77639e8i	− 1.39554i
$236$	0	0
$237$	1.96873e8i	0.960654i
$238$	0	0
$239$	−8.22277e7	−0.389606	−0.194803	−	0.980842i	$-0.562407\pi$
$239$	−8.22277e7	−0.389606	−0.194803	+	0.980842i	$0.562407\pi$
$240$	0	0
$241$	−2.70650e8	−1.24551	−0.622757	−	0.782415i	$-0.713988\pi$
$241$	−2.70650e8	−1.24551	−0.622757	+	0.782415i	$0.713988\pi$
$242$	0	0
$243$	− 1.83831e8i	− 0.821860i
$244$	0	0
$245$	− 2.13498e8i	− 0.927498i
$246$	0	0
$247$	7.26797e7	0.306884
$248$	0	0
$249$	−1.18052e8	−0.484591
$250$	0	0
$251$	2.90747e8i	1.16053i	0.814427	+	0.580266i	$0.197051\pi$
$251$	2.90747e8i	1.16053i	−0.814427	+	0.580266i	$0.802949\pi$
$252$	0	0
$253$	3.85549e7i	0.149678i
$254$	0	0
$255$	3.15508e8	1.19157
$256$	0	0
$257$	4.36047e7	0.160239	0.0801193	−	0.996785i	$-0.474470\pi$
$257$	4.36047e7	0.160239	0.0801193	+	0.996785i	$0.474470\pi$
$258$	0	0
$259$	2.17867e8i	0.779188i
$260$	0	0
$261$	6.00053e8i	2.08905i
$262$	0	0
$263$	−4.27678e8	−1.44968	−0.724840	−	0.688917i	$-0.758086\pi$
$263$	−4.27678e8	−1.44968	−0.724840	+	0.688917i	$0.758086\pi$
$264$	0	0
$265$	−5.20244e8	−1.71730
$266$	0	0
$267$	1.53509e8i	0.493564i
$268$	0	0
$269$	− 8.26134e7i	− 0.258772i	−0.991594	−	0.129386i	$-0.958699\pi$
$269$	− 8.26134e7i	− 0.258772i	0.991594	−	0.129386i	$-0.0413007\pi$
$270$	0	0
$271$	−6.15189e7	−0.187766	−0.0938829	−	0.995583i	$-0.529928\pi$
$271$	−6.15189e7	−0.187766	−0.0938829	+	0.995583i	$0.529928\pi$
$272$	0	0
$273$	7.46883e7	0.222169
$274$	0	0
$275$	7.16196e7i	0.207667i
$276$	0	0
$277$	4.39237e7i	0.124171i	0.998071	+	0.0620854i	$0.0197751\pi$
$277$	4.39237e7i	0.124171i	−0.998071	+	0.0620854i	$0.980225\pi$
$278$	0	0
$279$	−8.56153e8	−2.36013
$280$	0	0
$281$	5.80931e8	1.56190	0.780948	−	0.624596i	$-0.214736\pi$
$281$	5.80931e8	1.56190	0.780948	+	0.624596i	$0.214736\pi$
$282$	0	0
$283$	− 6.03790e8i	− 1.58356i	−0.610810	−	0.791778i	$-0.709156\pi$
$283$	− 6.03790e8i	− 1.58356i	0.610810	−	0.791778i	$-0.290844\pi$
$284$	0	0
$285$	− 8.56091e8i	− 2.19060i
$286$	0	0
$287$	2.63385e8	0.657665
$288$	0	0
$289$	−2.63568e8	−0.642319
$290$	0	0
$291$	− 2.22520e6i	− 0.00529352i
$292$	0	0
$293$	− 1.10504e8i	− 0.256649i	−0.991732	−	0.128325i	$-0.959040\pi$
$293$	− 1.10504e8i	− 0.256649i	0.991732	−	0.128325i	$-0.0409600\pi$
$294$	0	0
$295$	−6.17591e8	−1.40063
$296$	0	0
$297$	−2.34142e8	−0.518599
$298$	0	0
$299$	4.33464e7i	0.0937787i
$300$	0	0
$301$	3.87836e7i	0.0819719i
$302$	0	0
$303$	1.29335e9	2.67095
$304$	0	0
$305$	−1.64003e8	−0.330981
$306$	0	0
$307$	7.03386e8i	1.38742i	0.720252	+	0.693712i	$0.244026\pi$
$307$	7.03386e8i	1.38742i	−0.720252	+	0.693712i	$0.755974\pi$
$308$	0	0
$309$	9.83002e8i	1.89539i
$310$	0	0
$311$	8.61240e8	1.62354	0.811769	−	0.583978i	$-0.198505\pi$
$311$	8.61240e8	1.62354	0.811769	+	0.583978i	$0.198505\pi$
$312$	0	0
$313$	2.42056e8	0.446181	0.223090	−	0.974798i	$-0.428385\pi$
$313$	2.42056e8	0.446181	0.223090	+	0.974798i	$0.428385\pi$
$314$	0	0
$315$	− 5.54817e8i	− 1.00014i
$316$	0	0
$317$	5.38362e8i	0.949221i	0.880196	+	0.474610i	$0.157411\pi$
$317$	5.38362e8i	0.949221i	−0.880196	+	0.474610i	$0.842589\pi$
$318$	0	0
$319$	−3.16007e8	−0.545042
$320$	0	0
$321$	1.04351e9	1.76088
$322$	0	0
$323$	− 3.98242e8i	− 0.657565i
$324$	0	0
$325$	8.05203e7i	0.130111i
$326$	0	0
$327$	3.64900e8	0.577108
$328$	0	0
$329$	−3.60128e8	−0.557534
$330$	0	0
$331$	1.05054e9i	1.59226i	0.605124	+	0.796131i	$0.293123\pi$
$331$	1.05054e9i	1.59226i	−0.605124	+	0.796131i	$0.706877\pi$
$332$	0	0
$333$	− 1.85325e9i	− 2.75029i
$334$	0	0
$335$	2.70342e7	0.0392877
$336$	0	0
$337$	−2.04579e8	−0.291177	−0.145589	−	0.989345i	$-0.546508\pi$
$337$	−2.04579e8	−0.291177	−0.145589	+	0.989345i	$0.546508\pi$
$338$	0	0
$339$	− 6.20855e8i	− 0.865548i
$340$	0	0
$341$	− 4.50878e8i	− 0.615770i
$342$	0	0
$343$	−6.38462e8	−0.854291
$344$	0	0
$345$	5.10576e8	0.669412
$346$	0	0
$347$	− 6.28852e8i	− 0.807970i	−0.914766	−	0.403985i	$-0.867625\pi$
$347$	− 6.28852e8i	− 0.807970i	0.914766	−	0.403985i	$-0.132375\pi$
$348$	0	0
$349$	9.86717e8i	1.24252i	0.783604	+	0.621260i	$0.213379\pi$
$349$	9.86717e8i	1.24252i	−0.783604	+	0.621260i	$0.786621\pi$
$350$	0	0
$351$	−2.63240e8	−0.324921
$352$	0	0
$353$	5.59732e8	0.677281	0.338641	−	0.940916i	$-0.390033\pi$
$353$	5.59732e8	0.677281	0.338641	+	0.940916i	$0.390033\pi$
$354$	0	0
$355$	− 4.30087e8i	− 0.510219i
$356$	0	0
$357$	− 4.09248e8i	− 0.476045i
$358$	0	0
$359$	−1.42390e8	−0.162424	−0.0812119	−	0.996697i	$-0.525879\pi$
$359$	−1.42390e8	−0.162424	−0.0812119	+	0.996697i	$0.525879\pi$
$360$	0	0
$361$	−1.86707e8	−0.208875
$362$	0	0
$363$	1.20193e9i	1.31889i
$364$	0	0
$365$	1.25501e9i	1.35090i
$366$	0	0
$367$	−7.13452e8	−0.753414	−0.376707	−	0.926333i	$-0.622944\pi$
$367$	−7.13452e8	−0.753414	−0.376707	+	0.926333i	$0.622944\pi$
$368$	0	0
$369$	−2.24044e9	−2.32136
$370$	0	0
$371$	6.74812e8i	0.686079i
$372$	0	0
$373$	− 4.14729e8i	− 0.413794i	−0.978363	−	0.206897i	$-0.933664\pi$
$373$	− 4.14729e8i	− 0.413794i	0.978363	−	0.206897i	$-0.0663364\pi$
$374$	0	0
$375$	−1.08617e9	−1.06362
$376$	0	0
$377$	−3.55280e8	−0.341489
$378$	0	0
$379$	− 1.23625e8i	− 0.116646i	−0.998298	−	0.0583229i	$-0.981425\pi$
$379$	− 1.23625e8i	− 0.116646i	0.998298	−	0.0583229i	$-0.0185753\pi$
$380$	0	0
$381$	8.64994e8i	0.801264i
$382$	0	0
$383$	1.35784e9	1.23496	0.617480	−	0.786586i	$-0.288154\pi$
$383$	1.35784e9	1.23496	0.617480	+	0.786586i	$0.288154\pi$
$384$	0	0
$385$	2.92184e8	0.260942
$386$	0	0
$387$	− 3.29906e8i	− 0.289336i
$388$	0	0
$389$	− 1.00573e9i	− 0.866281i	−0.901326	−	0.433141i	$-0.857405\pi$
$389$	− 1.00573e9i	− 0.866281i	0.901326	−	0.433141i	$-0.142595\pi$
$390$	0	0
$391$	2.37513e8	0.200941
$392$	0	0
$393$	6.78332e8	0.563726
$394$	0	0
$395$	8.65893e8i	0.706927i
$396$	0	0
$397$	2.30080e8i	0.184549i	0.995734	+	0.0922747i	$0.0294138\pi$
$397$	2.30080e8i	0.184549i	−0.995734	+	0.0922747i	$0.970586\pi$
$398$	0	0
$399$	−1.11044e9	−0.875167
$400$	0	0
$401$	−1.24791e9	−0.966446	−0.483223	−	0.875497i	$-0.660534\pi$
$401$	−1.24791e9	−0.966446	−0.483223	+	0.875497i	$0.660534\pi$
$402$	0	0
$403$	− 5.06912e8i	− 0.385802i
$404$	0	0
$405$	3.36702e8i	0.251857i
$406$	0	0
$407$	9.75982e8	0.717565
$408$	0	0
$409$	−1.79923e9	−1.30033	−0.650166	−	0.759792i	$-0.725301\pi$
$409$	−1.79923e9	−1.30033	−0.650166	+	0.759792i	$0.725301\pi$
$410$	0	0
$411$	− 2.56803e9i	− 1.82454i
$412$	0	0
$413$	8.01081e8i	0.559566i
$414$	0	0
$415$	−5.19219e8	−0.356601
$416$	0	0
$417$	3.60248e9	2.43291
$418$	0	0
$419$	− 2.66870e8i	− 0.177235i	−0.996066	−	0.0886177i	$-0.971755\pi$
$419$	− 2.66870e8i	− 0.177235i	0.996066	−	0.0886177i	$-0.0282449\pi$
$420$	0	0
$421$	− 2.70575e9i	− 1.76726i	−0.468185	−	0.883630i	$-0.655092\pi$
$421$	− 2.70575e9i	− 1.76726i	0.468185	−	0.883630i	$-0.344908\pi$
$422$	0	0
$423$	3.06337e9	1.96792
$424$	0	0
$425$	4.41204e8	0.278791
$426$	0	0
$427$	2.12730e8i	0.132230i
$428$	0	0
$429$	− 3.34582e8i	− 0.204598i
$430$	0	0
$431$	−1.78455e9	−1.07364	−0.536820	−	0.843697i	$-0.680374\pi$
$431$	−1.78455e9	−1.07364	−0.536820	+	0.843697i	$0.680374\pi$
$432$	0	0
$433$	1.21276e9	0.717905	0.358953	−	0.933356i	$-0.383134\pi$
$433$	1.21276e9	0.717905	0.358953	+	0.933356i	$0.383134\pi$
$434$	0	0
$435$	4.18483e9i	2.43762i
$436$	0	0
$437$	− 6.44461e8i	− 0.369413i
$438$	0	0
$439$	1.76141e9	0.993654	0.496827	−	0.867850i	$-0.334498\pi$
$439$	1.76141e9	0.993654	0.496827	+	0.867850i	$0.334498\pi$
$440$	0	0
$441$	2.35566e9	1.30791
$442$	0	0
$443$	− 8.06208e8i	− 0.440590i	−0.975433	−	0.220295i	$-0.929298\pi$
$443$	− 8.06208e8i	− 0.440590i	0.975433	−	0.220295i	$-0.0707019\pi$
$444$	0	0
$445$	6.75166e8i	0.363204i
$446$	0	0
$447$	−2.95263e9	−1.56363
$448$	0	0
$449$	5.85913e8	0.305472	0.152736	−	0.988267i	$-0.451192\pi$
$449$	5.85913e8	0.305472	0.152736	+	0.988267i	$0.451192\pi$
$450$	0	0
$451$	− 1.17989e9i	− 0.605653i
$452$	0	0
$453$	5.52228e9i	2.79110i
$454$	0	0
$455$	3.28496e8	0.163490
$456$	0	0
$457$	5.52640e8	0.270854	0.135427	−	0.990787i	$-0.456759\pi$
$457$	5.52640e8	0.270854	0.135427	+	0.990787i	$0.456759\pi$
$458$	0	0
$459$	1.44240e9i	0.696213i
$460$	0	0
$461$	− 1.74101e9i	− 0.827652i	−0.910356	−	0.413826i	$-0.864192\pi$
$461$	− 1.74101e9i	− 0.827652i	0.910356	−	0.413826i	$-0.135808\pi$
$462$	0	0
$463$	2.84431e9	1.33181	0.665906	−	0.746035i	$-0.268045\pi$
$463$	2.84431e9	1.33181	0.665906	+	0.746035i	$0.268045\pi$
$464$	0	0
$465$	−5.97090e9	−2.75394
$466$	0	0
$467$	− 1.63130e9i	− 0.741183i	−0.928796	−	0.370592i	$-0.879155\pi$
$467$	− 1.63130e9i	− 0.741183i	0.928796	−	0.370592i	$-0.120845\pi$
$468$	0	0
$469$	− 3.50662e7i	− 0.0156958i
$470$	0	0
$471$	3.10706e9	1.37018
$472$	0	0
$473$	1.73739e8	0.0754891
$474$	0	0
$475$	− 1.19715e9i	− 0.512532i
$476$	0	0
$477$	− 5.74018e9i	− 2.42165i
$478$	0	0
$479$	−3.69345e9	−1.53553	−0.767765	−	0.640732i	$-0.778631\pi$
$479$	−3.69345e9	−1.53553	−0.767765	+	0.640732i	$0.778631\pi$
$480$	0	0
$481$	1.09727e9	0.449580
$482$	0	0
$483$	− 6.62272e8i	− 0.267437i
$484$	0	0
$485$	− 9.78697e6i	− 0.00389540i
$486$	0	0
$487$	−1.57153e9	−0.616554	−0.308277	−	0.951297i	$-0.599752\pi$
$487$	−1.57153e9	−0.616554	−0.308277	+	0.951297i	$0.599752\pi$
$488$	0	0
$489$	−7.57578e9	−2.92986
$490$	0	0
$491$	− 1.63493e9i	− 0.623323i	−0.950193	−	0.311662i	$-0.899114\pi$
$491$	− 1.63493e9i	− 0.623323i	0.950193	−	0.311662i	$-0.100886\pi$
$492$	0	0
$493$	1.94673e9i	0.731713i
$494$	0	0
$495$	−2.48542e9	−0.921047
$496$	0	0
$497$	−5.57868e8	−0.203838
$498$	0	0
$499$	− 6.86870e8i	− 0.247470i	−0.992315	−	0.123735i	$-0.960513\pi$
$499$	− 6.86870e8i	− 0.247470i	0.992315	−	0.123735i	$-0.0394873\pi$
$500$	0	0
$501$	3.26890e9i	1.16137i
$502$	0	0
$503$	−2.33472e9	−0.817990	−0.408995	−	0.912537i	$-0.634121\pi$
$503$	−2.33472e9	−0.817990	−0.408995	+	0.912537i	$0.634121\pi$
$504$	0	0
$505$	5.68845e9	1.96550
$506$	0	0
$507$	4.45231e9i	1.51725i
$508$	0	0
$509$	6.21342e8i	0.208842i	0.994533	+	0.104421i	$0.0332990\pi$
$509$	6.21342e8i	0.208842i	−0.994533	+	0.104421i	$0.966701\pi$
$510$	0	0
$511$	1.62789e9	0.539699
$512$	0	0
$513$	3.91378e9	1.27993
$514$	0	0
$515$	4.32347e9i	1.39478i
$516$	0	0
$517$	1.61327e9i	0.513441i
$518$	0	0
$519$	−2.04114e9	−0.640896
$520$	0	0
$521$	−1.65562e9	−0.512897	−0.256448	−	0.966558i	$-0.582552\pi$
$521$	−1.65562e9	−0.512897	−0.256448	+	0.966558i	$0.582552\pi$
$522$	0	0
$523$	4.42671e9i	1.35309i	0.736403	+	0.676543i	$0.236523\pi$
$523$	4.42671e9i	1.35309i	−0.736403	+	0.676543i	$0.763477\pi$
$524$	0	0
$525$	− 1.23024e9i	− 0.371049i
$526$	0	0
$527$	−2.77758e9	−0.826665
$528$	0	0
$529$	−3.02047e9	−0.887113
$530$	0	0
$531$	− 6.81427e9i	− 1.97510i
$532$	0	0
$533$	− 1.32652e9i	− 0.379463i
$534$	0	0
$535$	4.58961e9	1.29580
$536$	0	0
$537$	1.86332e9	0.519252
$538$	0	0
$539$	1.24057e9i	0.341240i
$540$	0	0
$541$	− 3.71878e9i	− 1.00974i	−0.863195	−	0.504871i	$-0.831540\pi$
$541$	− 3.71878e9i	− 1.00974i	0.863195	−	0.504871i	$-0.168460\pi$
$542$	0	0
$543$	2.14660e9	0.575376
$544$	0	0
$545$	1.60492e9	0.424683
$546$	0	0
$547$	− 3.20749e9i	− 0.837934i	−0.908002	−	0.418967i	$-0.862392\pi$
$547$	− 3.20749e9i	− 0.837934i	0.908002	−	0.418967i	$-0.137608\pi$
$548$	0	0
$549$	− 1.80955e9i	− 0.466732i
$550$	0	0
$551$	5.28219e9	1.34519
$552$	0	0
$553$	1.12316e9	0.282424
$554$	0	0
$555$	− 1.29247e10i	− 3.20920i
$556$	0	0
$557$	4.80739e9i	1.17873i	0.807865	+	0.589367i	$0.200623\pi$
$557$	4.80739e9i	1.17873i	−0.807865	+	0.589367i	$0.799377\pi$
$558$	0	0
$559$	1.95331e8	0.0472967
$560$	0	0
$561$	−1.83332e9	−0.438397
$562$	0	0
$563$	3.77127e9i	0.890653i	0.895368	+	0.445326i	$0.146912\pi$
$563$	3.77127e9i	0.890653i	−0.895368	+	0.445326i	$0.853088\pi$
$564$	0	0
$565$	− 2.73066e9i	− 0.636940i
$566$	0	0
$567$	4.36739e8	0.100619
$568$	0	0
$569$	2.09341e9	0.476388	0.238194	−	0.971218i	$-0.423445\pi$
$569$	2.09341e9	0.476388	0.238194	+	0.971218i	$0.423445\pi$
$570$	0	0
$571$	6.86085e9i	1.54224i	0.636691	+	0.771119i	$0.280303\pi$
$571$	6.86085e9i	1.54224i	−0.636691	+	0.771119i	$0.719697\pi$
$572$	0	0
$573$	− 1.28965e9i	− 0.286371i
$574$	0	0
$575$	7.13986e8	0.156622
$576$	0	0
$577$	5.70742e9	1.23687	0.618435	−	0.785836i	$-0.287767\pi$
$577$	5.70742e9	1.23687	0.618435	+	0.785836i	$0.287767\pi$
$578$	0	0
$579$	6.73316e9i	1.44160i
$580$	0	0
$581$	6.73483e8i	0.142466i
$582$	0	0
$583$	3.02297e9	0.631820
$584$	0	0
$585$	−2.79430e9	−0.577069
$586$	0	0
$587$	− 1.56510e9i	− 0.319380i	−0.987167	−	0.159690i	$-0.948951\pi$
$587$	− 1.56510e9i	− 0.319380i	0.987167	−	0.159690i	$-0.0510495\pi$
$588$	0	0
$589$	7.53661e9i	1.51975i
$590$	0	0
$591$	1.97175e9	0.392913
$592$	0	0
$593$	−2.78410e9	−0.548268	−0.274134	−	0.961692i	$-0.588391\pi$
$593$	−2.78410e9	−0.548268	−0.274134	+	0.961692i	$0.588391\pi$
$594$	0	0
$595$	− 1.79997e9i	− 0.350312i
$596$	0	0
$597$	− 4.09283e9i	− 0.787251i
$598$	0	0
$599$	−6.18885e9	−1.17657	−0.588283	−	0.808655i	$-0.700196\pi$
$599$	−6.18885e9	−1.17657	−0.588283	+	0.808655i	$0.700196\pi$
$600$	0	0
$601$	−2.42206e9	−0.455118	−0.227559	−	0.973764i	$-0.573074\pi$
$601$	−2.42206e9	−0.455118	−0.227559	+	0.973764i	$0.573074\pi$
$602$	0	0
$603$	2.98285e8i	0.0554014i
$604$	0	0
$605$	5.28639e9i	0.970544i
$606$	0	0
$607$	4.62465e9	0.839302	0.419651	−	0.907686i	$-0.362152\pi$
$607$	4.62465e9	0.839302	0.419651	+	0.907686i	$0.362152\pi$
$608$	0	0
$609$	5.42817e9	0.973852
$610$	0	0
$611$	1.81376e9i	0.321689i
$612$	0	0
$613$	− 5.45433e9i	− 0.956378i	−0.878257	−	0.478189i	$-0.841293\pi$
$613$	− 5.45433e9i	− 0.956378i	0.878257	−	0.478189i	$-0.158707\pi$
$614$	0	0
$615$	−1.56251e10	−2.70869
$616$	0	0
$617$	−2.32837e9	−0.399074	−0.199537	−	0.979890i	$-0.563944\pi$
$617$	−2.32837e9	−0.399074	−0.199537	+	0.979890i	$0.563944\pi$
$618$	0	0
$619$	− 9.58626e9i	− 1.62455i	−0.583278	−	0.812273i	$-0.698230\pi$
$619$	− 9.58626e9i	− 1.62455i	0.583278	−	0.812273i	$-0.301770\pi$
$620$	0	0
$621$	2.33419e9i	0.391125i
$622$	0	0
$623$	8.75763e8	0.145104
$624$	0	0
$625$	−7.62240e9	−1.24885
$626$	0	0
$627$	4.97446e9i	0.805953i
$628$	0	0
$629$	− 6.01242e9i	− 0.963324i
$630$	0	0
$631$	−1.18616e10	−1.87949	−0.939747	−	0.341870i	$-0.888940\pi$
$631$	−1.18616e10	−1.87949	−0.939747	+	0.341870i	$0.888940\pi$
$632$	0	0
$633$	1.54796e9	0.242576
$634$	0	0
$635$	3.80445e9i	0.589635i
$636$	0	0
$637$	1.39474e9i	0.213799i
$638$	0	0
$639$	4.74542e9	0.719484
$640$	0	0
$641$	−1.06130e8	−0.0159161	−0.00795805	−	0.999968i	$-0.502533\pi$
$641$	−1.06130e8	−0.0159161	−0.00795805	+	0.999968i	$0.502533\pi$
$642$	0	0
$643$	− 2.19289e9i	− 0.325296i	−0.986684	−	0.162648i	$-0.947996\pi$
$643$	− 2.19289e9i	− 0.325296i	0.986684	−	0.162648i	$-0.0520035\pi$
$644$	0	0
$645$	− 2.30080e9i	− 0.337614i
$646$	0	0
$647$	4.23914e9	0.615337	0.307668	−	0.951494i	$-0.400451\pi$
$647$	4.23914e9	0.615337	0.307668	+	0.951494i	$0.400451\pi$
$648$	0	0
$649$	3.58862e9	0.515312
$650$	0	0
$651$	7.74489e9i	1.10023i
$652$	0	0
$653$	− 9.93257e9i	− 1.39594i	−0.716129	−	0.697968i	$-0.754088\pi$
$653$	− 9.93257e9i	− 1.39594i	0.716129	−	0.697968i	$-0.245912\pi$
$654$	0	0
$655$	2.98346e9	0.414836
$656$	0	0
$657$	−1.38474e10	−1.90497
$658$	0	0
$659$	− 1.36634e10i	− 1.85977i	−0.367852	−	0.929884i	$-0.619907\pi$
$659$	− 1.36634e10i	− 1.85977i	0.367852	−	0.929884i	$-0.380093\pi$
$660$	0	0
$661$	1.03765e10i	1.39747i	0.715378	+	0.698737i	$0.246254\pi$
$661$	1.03765e10i	1.39747i	−0.715378	+	0.698737i	$0.753746\pi$
$662$	0	0
$663$	−2.06115e9	−0.274671
$664$	0	0
$665$	−4.88398e9	−0.644019
$666$	0	0
$667$	3.15032e9i	0.411069i
$668$	0	0
$669$	1.28515e10i	1.65944i
$670$	0	0
$671$	9.52968e8	0.121773
$672$	0	0
$673$	4.70776e9	0.595336	0.297668	−	0.954669i	$-0.403791\pi$
$673$	4.70776e9	0.595336	0.297668	+	0.954669i	$0.403791\pi$
$674$	0	0
$675$	4.33599e9i	0.542657i
$676$	0	0
$677$	9.55050e9i	1.18295i	0.806324	+	0.591474i	$0.201454\pi$
$677$	9.55050e9i	1.18295i	−0.806324	+	0.591474i	$0.798546\pi$
$678$	0	0
$679$	−1.26947e7	−0.00155625
$680$	0	0
$681$	1.05658e10	1.28200
$682$	0	0
$683$	− 1.06442e10i	− 1.27832i	−0.769073	−	0.639161i	$-0.779282\pi$
$683$	− 1.06442e10i	− 1.27832i	0.769073	−	0.639161i	$-0.220718\pi$
$684$	0	0
$685$	− 1.12948e10i	− 1.34265i
$686$	0	0
$687$	2.06175e10	2.42598
$688$	0	0
$689$	3.39865e9	0.395858
$690$	0	0
$691$	8.41537e9i	0.970287i	0.874435	+	0.485143i	$0.161233\pi$
$691$	8.41537e9i	0.970287i	−0.874435	+	0.485143i	$0.838767\pi$
$692$	0	0
$693$	3.22386e9i	0.367967i
$694$	0	0
$695$	1.58445e10	1.79033
$696$	0	0
$697$	−7.26858e9	−0.813083
$698$	0	0
$699$	1.52385e10i	1.68761i
$700$	0	0
$701$	− 8.95355e9i	− 0.981707i	−0.871242	−	0.490854i	$-0.836685\pi$
$701$	− 8.95355e9i	− 0.981707i	0.871242	−	0.490854i	$-0.163315\pi$
$702$	0	0
$703$	−1.63139e10	−1.77099
$704$	0	0
$705$	2.13643e10	2.29629
$706$	0	0
$707$	− 7.37853e9i	− 0.785239i
$708$	0	0
$709$	− 8.11796e9i	− 0.855431i	−0.903913	−	0.427716i	$-0.859318\pi$
$709$	− 8.11796e9i	− 0.855431i	0.903913	−	0.427716i	$-0.140682\pi$
$710$	0	0
$711$	−9.55395e9	−0.996872
$712$	0	0
$713$	−4.49486e9	−0.464412
$714$	0	0
$715$	− 1.47157e9i	− 0.150560i
$716$	0	0
$717$	− 6.32739e9i	− 0.641073i
$718$	0	0
$719$	−9.54339e8	−0.0957528	−0.0478764	−	0.998853i	$-0.515245\pi$
$719$	−9.54339e8	−0.0957528	−0.0478764	+	0.998853i	$0.515245\pi$
$720$	0	0
$721$	5.60800e9	0.557231
$722$	0	0
$723$	− 2.08264e10i	− 2.04942i
$724$	0	0
$725$	5.85203e9i	0.570327i
$726$	0	0
$727$	−1.75084e10	−1.68996	−0.844979	−	0.534799i	$-0.820387\pi$
$727$	−1.75084e10	−1.68996	−0.844979	+	0.534799i	$0.820387\pi$
$728$	0	0
$729$	1.63215e10	1.56032
$730$	0	0
$731$	− 1.07030e9i	− 0.101343i
$732$	0	0
$733$	− 1.16062e10i	− 1.08849i	−0.838925	−	0.544247i	$-0.816815\pi$
$733$	− 1.16062e10i	− 1.08849i	0.838925	−	0.544247i	$-0.183185\pi$
$734$	0	0
$735$	1.64286e10	1.52614
$736$	0	0
$737$	−1.57087e8	−0.0144545
$738$	0	0
$739$	4.74800e9i	0.432768i	0.976308	+	0.216384i	$0.0694263\pi$
$739$	4.74800e9i	0.432768i	−0.976308	+	0.216384i	$0.930574\pi$
$740$	0	0
$741$	5.59268e9i	0.504959i
$742$	0	0
$743$	4.35857e9	0.389837	0.194918	−	0.980819i	$-0.437556\pi$
$743$	4.35857e9	0.389837	0.194918	+	0.980819i	$0.437556\pi$
$744$	0	0
$745$	−1.29864e10	−1.15064
$746$	0	0
$747$	− 5.72888e9i	− 0.502860i
$748$	0	0
$749$	− 5.95321e9i	− 0.517684i
$750$	0	0
$751$	6.10987e9	0.526371	0.263186	−	0.964745i	$-0.415227\pi$
$751$	6.10987e9	0.526371	0.263186	+	0.964745i	$0.415227\pi$
$752$	0	0
$753$	−2.23729e10	−1.90959
$754$	0	0
$755$	2.42883e10i	2.05391i
$756$	0	0
$757$	2.29472e10i	1.92262i	0.275460	+	0.961312i	$0.411170\pi$
$757$	2.29472e10i	1.92262i	−0.275460	+	0.961312i	$0.588830\pi$
$758$	0	0
$759$	−2.96679e9	−0.246286
$760$	0	0
$761$	7.15151e9	0.588236	0.294118	−	0.955769i	$-0.404974\pi$
$761$	7.15151e9	0.588236	0.294118	+	0.955769i	$0.404974\pi$
$762$	0	0
$763$	− 2.08175e9i	− 0.169665i
$764$	0	0
$765$	1.53111e10i	1.23650i
$766$	0	0
$767$	4.03460e9	0.322862
$768$	0	0
$769$	−1.85381e10	−1.47002	−0.735011	−	0.678055i	$-0.762823\pi$
$769$	−1.85381e10	−1.47002	−0.735011	+	0.678055i	$0.762823\pi$
$770$	0	0
$771$	3.35537e9i	0.263663i
$772$	0	0
$773$	8.25535e9i	0.642846i	0.946936	+	0.321423i	$0.104161\pi$
$773$	8.25535e9i	0.642846i	−0.946936	+	0.321423i	$0.895839\pi$
$774$	0	0
$775$	−8.34966e9	−0.644336
$776$	0	0
$777$	−1.67648e10	−1.28211
$778$	0	0
$779$	1.97223e10i	1.49478i
$780$	0	0
$781$	2.49909e9i	0.187717i
$782$	0	0
$783$	−1.91317e10	−1.42425
$784$	0	0
$785$	1.36656e10	1.00829
$786$	0	0
$787$	1.53185e9i	0.112023i	0.998430	+	0.0560113i	$0.0178383\pi$
$787$	1.53185e9i	0.112023i	−0.998430	+	0.0560113i	$0.982162\pi$
$788$	0	0
$789$	− 3.29097e10i	− 2.38536i
$790$	0	0
$791$	−3.54196e9	−0.254464
$792$	0	0
$793$	1.07140e9	0.0762950
$794$	0	0
$795$	− 4.00326e10i	− 2.82572i
$796$	0	0
$797$	2.78553e9i	0.194896i	0.995241	+	0.0974480i	$0.0310680\pi$
$797$	2.78553e9i	0.194896i	−0.995241	+	0.0974480i	$0.968932\pi$
$798$	0	0
$799$	9.93837e9	0.689289
$800$	0	0
$801$	−7.44954e9	−0.512172
$802$	0	0
$803$	− 7.29247e9i	− 0.497016i
$804$	0	0
$805$	− 2.91283e9i	− 0.196802i
$806$	0	0
$807$	6.35708e9	0.425794
$808$	0	0
$809$	1.17657e10	0.781267	0.390634	−	0.920546i	$-0.372256\pi$
$809$	1.17657e10	0.781267	0.390634	+	0.920546i	$0.372256\pi$
$810$	0	0
$811$	− 6.29491e9i	− 0.414397i	−0.978299	−	0.207198i	$-0.933565\pi$
$811$	− 6.29491e9i	− 0.414397i	0.978299	−	0.207198i	$-0.0664346\pi$
$812$	0	0
$813$	− 4.73386e9i	− 0.308957i
$814$	0	0
$815$	−3.33201e10	−2.15603
$816$	0	0
$817$	−2.90413e9	−0.186311
$818$	0	0
$819$	3.62451e9i	0.230545i
$820$	0	0
$821$	− 4.27400e9i	− 0.269546i	−0.990876	−	0.134773i	$-0.956969\pi$
$821$	− 4.27400e9i	− 0.269546i	0.990876	−	0.134773i	$-0.0430306\pi$
$822$	0	0
$823$	3.16411e9	0.197858	0.0989288	−	0.995095i	$-0.468458\pi$
$823$	3.16411e9	0.197858	0.0989288	+	0.995095i	$0.468458\pi$
$824$	0	0
$825$	−5.51111e9	−0.341704
$826$	0	0
$827$	3.47251e9i	0.213488i	0.994287	+	0.106744i	$0.0340426\pi$
$827$	3.47251e9i	0.213488i	−0.994287	+	0.106744i	$0.965957\pi$
$828$	0	0
$829$	5.21388e9i	0.317848i	0.987291	+	0.158924i	$0.0508026\pi$
$829$	5.21388e9i	0.317848i	−0.987291	+	0.158924i	$0.949197\pi$
$830$	0	0
$831$	−3.37991e9	−0.204316
$832$	0	0
$833$	7.64237e9	0.458111
$834$	0	0
$835$	1.43774e10i	0.854629i
$836$	0	0
$837$	− 2.72970e10i	− 1.60908i
$838$	0	0
$839$	2.48652e10	1.45353	0.726766	−	0.686885i	$-0.241023\pi$
$839$	2.48652e10	1.45353	0.726766	+	0.686885i	$0.241023\pi$
$840$	0	0
$841$	−8.57107e9	−0.496877
$842$	0	0
$843$	4.47024e10i	2.57001i
$844$	0	0
$845$	1.95823e10i	1.11652i
$846$	0	0
$847$	6.85701e9	0.387742
$848$	0	0
$849$	4.64614e10	2.60565
$850$	0	0
$851$	− 9.72969e9i	− 0.541185i
$852$	0	0
$853$	3.01930e10i	1.66565i	0.553536	+	0.832826i	$0.313278\pi$
$853$	3.01930e10i	1.66565i	−0.553536	+	0.832826i	$0.686722\pi$
$854$	0	0
$855$	4.15448e10	2.27319
$856$	0	0
$857$	2.57761e10	1.39889	0.699446	−	0.714685i	$-0.253430\pi$
$857$	2.57761e10	1.39889	0.699446	+	0.714685i	$0.253430\pi$
$858$	0	0
$859$	− 2.77848e10i	− 1.49565i	−0.663894	−	0.747827i	$-0.731097\pi$
$859$	− 2.77848e10i	− 1.49565i	0.663894	−	0.747827i	$-0.268903\pi$
$860$	0	0
$861$	2.02674e10i	1.08215i
$862$	0	0
$863$	−3.02990e10	−1.60469	−0.802344	−	0.596862i	$-0.796414\pi$
$863$	−3.02990e10	−1.60469	−0.802344	+	0.596862i	$0.796414\pi$
$864$	0	0
$865$	−8.97743e9	−0.471624
$866$	0	0
$867$	− 2.02815e10i	− 1.05690i
$868$	0	0
$869$	− 5.03142e9i	− 0.260089i
$870$	0	0
$871$	−1.76609e8	−0.00905627
$872$	0	0
$873$	1.07986e8	0.00549309
$874$	0	0
$875$	6.19656e9i	0.312696i
$876$	0	0
$877$	2.37410e9i	0.118850i	0.998233	+	0.0594252i	$0.0189268\pi$
$877$	2.37410e9i	0.118850i	−0.998233	+	0.0594252i	$0.981073\pi$
$878$	0	0
$879$	8.50323e9	0.422302
$880$	0	0
$881$	−3.28058e10	−1.61635	−0.808175	−	0.588942i	$-0.799544\pi$
$881$	−3.28058e10	−1.61635	−0.808175	+	0.588942i	$0.799544\pi$
$882$	0	0
$883$	2.14967e10i	1.05078i	0.850863	+	0.525388i	$0.176080\pi$
$883$	2.14967e10i	1.05078i	−0.850863	+	0.525388i	$0.823920\pi$
$884$	0	0
$885$	− 4.75234e10i	− 2.30466i
$886$	0	0
$887$	1.50080e10	0.722089	0.361045	−	0.932549i	$-0.382420\pi$
$887$	1.50080e10	0.722089	0.361045	+	0.932549i	$0.382420\pi$
$888$	0	0
$889$	4.93477e9	0.235565
$890$	0	0
$891$	− 1.95647e9i	− 0.0926617i
$892$	0	0
$893$	− 2.69665e10i	− 1.26720i
$894$	0	0
$895$	8.19532e9	0.382107
$896$	0	0
$897$	−3.33549e9	−0.154307
$898$	0	0
$899$	− 3.68412e10i	− 1.69112i
$900$	0	0
$901$	− 1.86226e10i	− 0.848212i
$902$	0	0
$903$	−2.98438e9	−0.134880
$904$	0	0
$905$	9.44124e9	0.423408
$906$	0	0
$907$	1.57906e10i	0.702704i	0.936243	+	0.351352i	$0.114278\pi$
$907$	1.57906e10i	0.702704i	−0.936243	+	0.351352i	$0.885722\pi$
$908$	0	0
$909$	6.27643e10i	2.77165i
$910$	0	0
$911$	6.82128e9	0.298917	0.149459	−	0.988768i	$-0.452247\pi$
$911$	6.82128e9	0.298917	0.149459	+	0.988768i	$0.452247\pi$
$912$	0	0
$913$	3.01701e9	0.131199
$914$	0	0
$915$	− 1.26200e10i	− 0.544609i
$916$	0	0
$917$	− 3.86987e9i	− 0.165731i
$918$	0	0
$919$	1.08588e10	0.461506	0.230753	−	0.973012i	$-0.425881\pi$
$919$	1.08588e10	0.461506	0.230753	+	0.973012i	$0.425881\pi$
$920$	0	0
$921$	−5.41253e10	−2.28293
$922$	0	0
$923$	2.80967e9i	0.117611i
$924$	0	0
$925$	− 1.80739e10i	− 0.750853i
$926$	0	0
$927$	−4.77036e10	−1.96685
$928$	0	0
$929$	−2.99248e9	−0.122455	−0.0612275	−	0.998124i	$-0.519502\pi$
$929$	−2.99248e9	−0.122455	−0.0612275	+	0.998124i	$0.519502\pi$
$930$	0	0
$931$	− 2.07366e10i	− 0.842197i
$932$	0	0
$933$	6.62721e10i	2.67144i
$934$	0	0
$935$	−8.06335e9	−0.322608
$936$	0	0
$937$	2.39333e10	0.950417	0.475208	−	0.879873i	$-0.342373\pi$
$937$	2.39333e10	0.950417	0.475208	+	0.879873i	$0.342373\pi$
$938$	0	0
$939$	1.86261e10i	0.734164i
$940$	0	0
$941$	− 3.17291e10i	− 1.24135i	−0.784068	−	0.620675i	$-0.786859\pi$
$941$	− 3.17291e10i	− 1.24135i	0.784068	−	0.620675i	$-0.213141\pi$
$942$	0	0
$943$	−1.17625e10	−0.456781
$944$	0	0
$945$	1.76894e10	0.681871
$946$	0	0
$947$	4.13161e10i	1.58086i	0.612551	+	0.790431i	$0.290143\pi$
$947$	4.13161e10i	1.58086i	−0.612551	+	0.790431i	$0.709857\pi$
$948$	0	0
$949$	− 8.19876e9i	− 0.311399i
$950$	0	0
$951$	−4.14268e10	−1.56189
$952$	0	0
$953$	−4.31993e10	−1.61678	−0.808390	−	0.588647i	$-0.799661\pi$
$953$	−4.31993e10	−1.61678	−0.808390	+	0.588647i	$0.799661\pi$
$954$	0	0
$955$	− 5.67216e9i	− 0.210735i
$956$	0	0
$957$	− 2.43167e10i	− 0.896835i
$958$	0	0
$959$	−1.46506e10	−0.536401
$960$	0	0
$961$	2.50523e10	0.910574
$962$	0	0
$963$	5.06400e10i	1.82727i
$964$	0	0
$965$	2.96140e10i	1.06084i
$966$	0	0
$967$	−6.99318e9	−0.248703	−0.124352	−	0.992238i	$-0.539685\pi$
$967$	−6.99318e9	−0.248703	−0.124352	+	0.992238i	$0.539685\pi$
$968$	0	0
$969$	3.06446e10	1.08198
$970$	0	0
$971$	− 8.48357e9i	− 0.297380i	−0.988884	−	0.148690i	$-0.952494\pi$
$971$	− 8.48357e9i	− 0.297380i	0.988884	−	0.148690i	$-0.0475056\pi$
$972$	0	0
$973$	− 2.05521e10i	− 0.715255i
$974$	0	0
$975$	−6.19601e9	−0.214090
$976$	0	0
$977$	−1.72197e10	−0.590739	−0.295370	−	0.955383i	$-0.595443\pi$
$977$	−1.72197e10	−0.590739	−0.295370	+	0.955383i	$0.595443\pi$
$978$	0	0
$979$	− 3.92317e9i	− 0.133628i
$980$	0	0
$981$	1.77081e10i	0.598866i
$982$	0	0
$983$	3.76267e10	1.26345	0.631725	−	0.775192i	$-0.282347\pi$
$983$	3.76267e10	1.26345	0.631725	+	0.775192i	$0.282347\pi$
$984$	0	0
$985$	8.67222e9	0.289137
$986$	0	0
$987$	− 2.77117e10i	− 0.917389i
$988$	0	0
$989$	− 1.73203e9i	− 0.0569336i
$990$	0	0
$991$	3.13036e9	0.102173	0.0510866	−	0.998694i	$-0.483732\pi$
$991$	3.13036e9	0.102173	0.0510866	+	0.998694i	$0.483732\pi$
$992$	0	0
$993$	−8.08387e10	−2.61997
$994$	0	0
$995$	− 1.80012e10i	− 0.579323i
$996$	0	0
$997$	− 1.67605e9i	− 0.0535617i	−0.999641	−	0.0267808i	$-0.991474\pi$
$997$	− 1.67605e9i	− 0.0535617i	0.999641	−	0.0267808i	$-0.00852563\pi$
$998$	0	0
$999$	5.90878e10	1.87508

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	32.8.b.a.17.6		6
3.2	odd	2		288.8.d.b.145.1		6
4.3	odd	2		8.8.b.a.5.6	yes	6
8.3	odd	2		8.8.b.a.5.5	✓	6
8.5	even	2	inner	32.8.b.a.17.1		6
12.11	even	2		72.8.d.b.37.1		6
16.3	odd	4		256.8.a.r.1.6		6
16.5	even	4		256.8.a.q.1.6		6
16.11	odd	4		256.8.a.r.1.1		6
16.13	even	4		256.8.a.q.1.1		6
24.5	odd	2		288.8.d.b.145.6		6
24.11	even	2		72.8.d.b.37.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.8.b.a.5.5	✓	6	8.3	odd	2
8.8.b.a.5.6	yes	6	4.3	odd	2
32.8.b.a.17.1		6	8.5	even	2	inner
32.8.b.a.17.6		6	1.1	even	1	trivial
72.8.d.b.37.1		6	12.11	even	2
72.8.d.b.37.2		6	24.11	even	2
256.8.a.q.1.1		6	16.13	even	4
256.8.a.q.1.6		6	16.5	even	4
256.8.a.r.1.1		6	16.11	odd	4
256.8.a.r.1.6		6	16.3	odd	4
288.8.d.b.145.1		6	3.2	odd	2
288.8.d.b.145.6		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)

\(f(q)\)	\(=\)	\(q+76.9497i q^{3} +338.443i q^{5} +438.996 q^{7} -3734.25 q^{9} +O(q^{10})\)	\(q+76.9497i q^{3} +338.443i q^{5} +438.996 q^{7} -3734.25 q^{9} -1966.58i q^{11} -2210.98i q^{13} -26043.0 q^{15} -12114.9 q^{17} +32872.2i q^{19} +33780.6i q^{21} -19605.1 q^{23} -36418.4 q^{25} -119060. i q^{27} -160689. i q^{29} +229270. q^{31} +151328. q^{33} +148575. i q^{35} +496284. i q^{37} +170134. q^{39} +599971. q^{41} +88346.0i q^{43} -1.26383e6i q^{45} -820344. q^{47} -630825. q^{49} -932236. i q^{51} +1.53717e6i q^{53} +665574. q^{55} -2.52950e6 q^{57} +1.82480e6i q^{59} +484582. i q^{61} -1.63932e6 q^{63} +748290. q^{65} -79878.2i q^{67} -1.50860e6i q^{69} -1.27078e6 q^{71} +3.70820e6 q^{73} -2.80238e6i q^{75} -863321. i q^{77} +2.55846e6 q^{79} +994857. q^{81} +1.53414e6i q^{83} -4.10019e6i q^{85} +1.23650e7 q^{87} +1.99492e6 q^{89} -970612. i q^{91} +1.76423e7i q^{93} -1.11253e7 q^{95} -28917.7 q^{97} +7.34370e6i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 688 q^{7} - 2918 q^{9}+O(q^{10}) \) 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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