LMFDB - Embedded newform 320.6.d.b.161.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [320,6,Mod(161,320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(320, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("320.161");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 320 = 2^{6} \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	320.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$51.3228223402$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 384x^{6} + 506x^{5} + 49869x^{4} + 29654x^{3} - 2235516x^{2} - 1528906x + 34180205 $ x^8 - 4x^7 - 384x^6 + 506x^5 + 49869x^4 + 29654x^3 - 2235516x^2 - 1528906*x + 34180205
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.7
Root			$-10.3099 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	320.161
Dual form			320.6.d.b.161.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+21.6198i q^{3} +25.0000i q^{5} +169.466 q^{7} -224.417 q^{9} +O(q^{10})$	$q+21.6198i q^{3} +25.0000i q^{5} +169.466 q^{7} -224.417 q^{9} -512.594i q^{11} +36.4873i q^{13} -540.496 q^{15} -1260.81 q^{17} +790.251i q^{19} +3663.82i q^{21} -4995.19 q^{23} -625.000 q^{25} +401.768i q^{27} -934.435i q^{29} -6690.60 q^{31} +11082.2 q^{33} +4236.64i q^{35} +4305.71i q^{37} -788.849 q^{39} -10359.3 q^{41} +6371.59i q^{43} -5610.42i q^{45} -22120.7 q^{47} +11911.6 q^{49} -27258.6i q^{51} +23246.6i q^{53} +12814.9 q^{55} -17085.1 q^{57} -27758.9i q^{59} -39950.3i q^{61} -38030.9 q^{63} -912.183 q^{65} +55415.0i q^{67} -107995. i q^{69} -39874.7 q^{71} -435.947 q^{73} -13512.4i q^{75} -86867.1i q^{77} +61654.1 q^{79} -63219.4 q^{81} +2943.22i q^{83} -31520.3i q^{85} +20202.3 q^{87} +64783.7 q^{89} +6183.35i q^{91} -144650. i q^{93} -19756.3 q^{95} +159041. q^{97} +115035. i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 320 q^{7} - 1192 q^{9}+O(q^{10}) $ 8 * q + 320 * q^7 - 1192 * q^9	$ 8 q + 320 q^{7} - 1192 q^{9} - 2400 q^{17} - 1760 q^{23} - 5000 q^{25} - 31040 q^{31} + 3760 q^{33} - 4480 q^{39} + 21584 q^{41} - 47680 q^{47} + 82824 q^{49} - 18400 q^{55} - 106640 q^{57} - 322400 q^{63} - 44000 q^{65} - 246720 q^{71} + 46400 q^{73} - 325760 q^{79} - 82328 q^{81} - 636320 q^{87} - 78192 q^{89} + 40800 q^{95} + 24960 q^{97}+O(q^{100}) $ 8 * q + 320 * q^7 - 1192 * q^9 - 2400 * q^17 - 1760 * q^23 - 5000 * q^25 - 31040 * q^31 + 3760 * q^33 - 4480 * q^39 + 21584 * q^41 - 47680 * q^47 + 82824 * q^49 - 18400 * q^55 - 106640 * q^57 - 322400 * q^63 - 44000 * q^65 - 246720 * q^71 + 46400 * q^73 - 325760 * q^79 - 82328 * q^81 - 636320 * q^87 - 78192 * q^89 + 40800 * q^95 + 24960 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/320\mathbb{Z}\right)^\times$.

$n$	$191$	$257$	$261$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	21.6198i	1.38691i	0.720499	+	0.693456i	$0.243913\pi$
$3$	21.6198i	1.38691i	−0.720499	+	0.693456i	$0.756087\pi$
$4$	0	0
$5$	25.0000i	0.447214i
$5$	25.0000i	0.447214i
$6$	0	0
$7$	169.466	1.30718	0.653592	−	0.756847i	$-0.273261\pi$
$7$	169.466	1.30718	0.653592	+	0.756847i	$0.273261\pi$
$8$	0	0
$9$	−224.417	−0.923525
$10$	0	0
$11$	− 512.594i	− 1.27730i	−0.769498	−	0.638649i	$-0.779494\pi$
$11$	− 512.594i	− 1.27730i	0.769498	−	0.638649i	$-0.220506\pi$
$12$	0	0
$13$	36.4873i	0.0598802i	0.999552	+	0.0299401i	$0.00953166\pi$
$13$	36.4873i	0.0598802i	−0.999552	+	0.0299401i	$0.990468\pi$
$14$	0	0
$15$	−540.496	−0.620246
$16$	0	0
$17$	−1260.81	−1.05810	−0.529052	−	0.848589i	$-0.677452\pi$
$17$	−1260.81	−1.05810	−0.529052	+	0.848589i	$0.677452\pi$
$18$	0	0
$19$	790.251i	0.502205i	0.967960	+	0.251103i	$0.0807932\pi$
$19$	790.251i	0.502205i	−0.967960	+	0.251103i	$0.919207\pi$
$20$	0	0
$21$	3663.82i	1.81295i
$22$	0	0
$23$	−4995.19	−1.96894	−0.984470	−	0.175554i	$-0.943828\pi$
$23$	−4995.19	−1.96894	−0.984470	+	0.175554i	$0.943828\pi$
$24$	0	0
$25$	−625.000	−0.200000
$26$	0	0
$27$	401.768i	0.106064i
$28$	0	0
$29$	− 934.435i	− 0.206326i	−0.994664	−	0.103163i	$-0.967104\pi$
$29$	− 934.435i	− 0.206326i	0.994664	−	0.103163i	$-0.0328963\pi$
$30$	0	0
$31$	−6690.60	−1.25043	−0.625217	−	0.780451i	$-0.714989\pi$
$31$	−6690.60	−1.25043	−0.625217	+	0.780451i	$0.714989\pi$
$32$	0	0
$33$	11082.2	1.77150
$34$	0	0
$35$	4236.64i	0.584590i
$36$	0	0
$37$	4305.71i	0.517059i	0.966003	+	0.258530i	$0.0832379\pi$
$37$	4305.71i	0.517059i	−0.966003	+	0.258530i	$0.916762\pi$
$38$	0	0
$39$	−788.849	−0.0830486
$40$	0	0
$41$	−10359.3	−0.962430	−0.481215	−	0.876603i	$-0.659804\pi$
$41$	−10359.3	−0.962430	−0.481215	+	0.876603i	$0.659804\pi$
$42$	0	0
$43$	6371.59i	0.525505i	0.964863	+	0.262752i	$0.0846303\pi$
$43$	6371.59i	0.525505i	−0.964863	+	0.262752i	$0.915370\pi$
$44$	0	0
$45$	− 5610.42i	− 0.413013i
$46$	0	0
$47$	−22120.7	−1.46068	−0.730338	−	0.683085i	$-0.760638\pi$
$47$	−22120.7	−1.46068	−0.730338	+	0.683085i	$0.760638\pi$
$48$	0	0
$49$	11911.6	0.708730
$50$	0	0
$51$	− 27258.6i	− 1.46750i
$52$	0	0
$53$	23246.6i	1.13676i	0.822766	+	0.568380i	$0.192430\pi$
$53$	23246.6i	1.13676i	−0.822766	+	0.568380i	$0.807570\pi$
$54$	0	0
$55$	12814.9	0.571225
$56$	0	0
$57$	−17085.1	−0.696515
$58$	0	0
$59$	− 27758.9i	− 1.03818i	−0.854720	−	0.519089i	$-0.826271\pi$
$59$	− 27758.9i	− 1.03818i	0.854720	−	0.519089i	$-0.173729\pi$
$60$	0	0
$61$	− 39950.3i	− 1.37466i	−0.726345	−	0.687330i	$-0.758783\pi$
$61$	− 39950.3i	− 1.37466i	0.726345	−	0.687330i	$-0.241217\pi$
$62$	0	0
$63$	−38030.9	−1.20722
$64$	0	0
$65$	−912.183	−0.0267793
$66$	0	0
$67$	55415.0i	1.50814i	0.656797	+	0.754068i	$0.271911\pi$
$67$	55415.0i	1.50814i	−0.656797	+	0.754068i	$0.728089\pi$
$68$	0	0
$69$	− 107995.i	− 2.73075i
$70$	0	0
$71$	−39874.7	−0.938752	−0.469376	−	0.882998i	$-0.655521\pi$
$71$	−39874.7	−0.938752	−0.469376	+	0.882998i	$0.655521\pi$
$72$	0	0
$73$	−435.947	−0.00957473	−0.00478736	−	0.999989i	$-0.501524\pi$
$73$	−435.947	−0.00957473	−0.00478736	+	0.999989i	$0.501524\pi$
$74$	0	0
$75$	− 13512.4i	− 0.277382i
$76$	0	0
$77$	− 86867.1i	− 1.66966i
$78$	0	0
$79$	61654.1	1.11146	0.555730	−	0.831363i	$-0.312439\pi$
$79$	61654.1	1.11146	0.555730	+	0.831363i	$0.312439\pi$
$80$	0	0
$81$	−63219.4	−1.07063
$82$	0	0
$83$	2943.22i	0.0468952i	0.999725	+	0.0234476i	$0.00746428\pi$
$83$	2943.22i	0.0468952i	−0.999725	+	0.0234476i	$0.992536\pi$
$84$	0	0
$85$	− 31520.3i	− 0.473199i
$86$	0	0
$87$	20202.3	0.286156
$88$	0	0
$89$	64783.7	0.866943	0.433472	−	0.901167i	$-0.357288\pi$
$89$	64783.7	0.866943	0.433472	+	0.901167i	$0.357288\pi$
$90$	0	0
$91$	6183.35i	0.0782745i
$92$	0	0
$93$	− 144650.i	− 1.73424i
$94$	0	0
$95$	−19756.3	−0.224593
$96$	0	0
$97$	159041.	1.71625	0.858123	−	0.513445i	$-0.171631\pi$
$97$	159041.	1.71625	0.858123	+	0.513445i	$0.171631\pi$
$98$	0	0
$99$	115035.i	1.17962i
$100$	0	0
$101$	66973.7i	0.653282i	0.945148	+	0.326641i	$0.105917\pi$
$101$	66973.7i	0.653282i	−0.945148	+	0.326641i	$0.894083\pi$
$102$	0	0
$103$	−145452.	−1.35091	−0.675454	−	0.737402i	$-0.736052\pi$
$103$	−145452.	−1.35091	−0.675454	+	0.737402i	$0.736052\pi$
$104$	0	0
$105$	−91595.5	−0.810776
$106$	0	0
$107$	67124.0i	0.566785i	0.959004	+	0.283392i	$0.0914599\pi$
$107$	67124.0i	0.566785i	−0.959004	+	0.283392i	$0.908540\pi$
$108$	0	0
$109$	− 102861.i	− 0.829247i	−0.909993	−	0.414624i	$-0.863913\pi$
$109$	− 102861.i	− 0.829247i	0.909993	−	0.414624i	$-0.136087\pi$
$110$	0	0
$111$	−93088.6	−0.717116
$112$	0	0
$113$	−60698.8	−0.447182	−0.223591	−	0.974683i	$-0.571778\pi$
$113$	−60698.8	−0.447182	−0.223591	+	0.974683i	$0.571778\pi$
$114$	0	0
$115$	− 124880.i	− 0.880537i
$116$	0	0
$117$	− 8188.36i	− 0.0553009i
$118$	0	0
$119$	−213665.	−1.38314
$120$	0	0
$121$	−101702.	−0.631489
$122$	0	0
$123$	− 223965.i	− 1.33481i
$124$	0	0
$125$	− 15625.0i	− 0.0894427i
$126$	0	0
$127$	−318984.	−1.75493	−0.877464	−	0.479642i	$-0.840767\pi$
$127$	−318984.	−1.75493	−0.877464	+	0.479642i	$0.840767\pi$
$128$	0	0
$129$	−137753.	−0.728829
$130$	0	0
$131$	− 359449.i	− 1.83003i	−0.403417	−	0.915016i	$-0.632178\pi$
$131$	− 359449.i	− 1.83003i	0.403417	−	0.915016i	$-0.367822\pi$
$132$	0	0
$133$	133920.i	0.656475i
$134$	0	0
$135$	−10044.2	−0.0474331
$136$	0	0
$137$	289504.	1.31781	0.658906	−	0.752225i	$-0.271019\pi$
$137$	289504.	1.31781	0.658906	+	0.752225i	$0.271019\pi$
$138$	0	0
$139$	318184.i	1.39682i	0.715696	+	0.698412i	$0.246109\pi$
$139$	318184.i	1.39682i	−0.715696	+	0.698412i	$0.753891\pi$
$140$	0	0
$141$	− 478245.i	− 2.02583i
$142$	0	0
$143$	18703.2	0.0764849
$144$	0	0
$145$	23360.9	0.0922718
$146$	0	0
$147$	257527.i	0.982946i
$148$	0	0
$149$	464507.i	1.71406i	0.515266	+	0.857030i	$0.327693\pi$
$149$	464507.i	1.71406i	−0.515266	+	0.857030i	$0.672307\pi$
$150$	0	0
$151$	−13229.4	−0.0472171	−0.0236085	−	0.999721i	$-0.507516\pi$
$151$	−13229.4	−0.0472171	−0.0236085	+	0.999721i	$0.507516\pi$
$152$	0	0
$153$	282948.	0.977186
$154$	0	0
$155$	− 167265.i	− 0.559211i
$156$	0	0
$157$	125487.i	0.406303i	0.979147	+	0.203152i	$0.0651184\pi$
$157$	125487.i	0.406303i	−0.979147	+	0.203152i	$0.934882\pi$
$158$	0	0
$159$	−502587.	−1.57659
$160$	0	0
$161$	−846513.	−2.57377
$162$	0	0
$163$	449649.i	1.32558i	0.748807	+	0.662788i	$0.230627\pi$
$163$	449649.i	1.32558i	−0.748807	+	0.662788i	$0.769373\pi$
$164$	0	0
$165$	277055.i	0.792239i
$166$	0	0
$167$	157771.	0.437760	0.218880	−	0.975752i	$-0.429760\pi$
$167$	157771.	0.437760	0.218880	+	0.975752i	$0.429760\pi$
$168$	0	0
$169$	369962.	0.996414
$170$	0	0
$171$	− 177346.i	− 0.463799i
$172$	0	0
$173$	− 253753.i	− 0.644608i	−0.946636	−	0.322304i	$-0.895543\pi$
$173$	− 253753.i	− 0.644608i	0.946636	−	0.322304i	$-0.104457\pi$
$174$	0	0
$175$	−105916.	−0.261437
$176$	0	0
$177$	600142.	1.43986
$178$	0	0
$179$	− 166500.i	− 0.388403i	−0.980962	−	0.194201i	$-0.937788\pi$
$179$	− 166500.i	− 0.388403i	0.980962	−	0.194201i	$-0.0622115\pi$
$180$	0	0
$181$	639995.i	1.45204i	0.687671	+	0.726022i	$0.258633\pi$
$181$	639995.i	1.45204i	−0.687671	+	0.726022i	$0.741367\pi$
$182$	0	0
$183$	863718.	1.90653
$184$	0	0
$185$	−107643.	−0.231236
$186$	0	0
$187$	646286.i	1.35151i
$188$	0	0
$189$	68086.0i	0.138645i
$190$	0	0
$191$	126967.	0.251829	0.125915	−	0.992041i	$-0.459813\pi$
$191$	126967.	0.251829	0.125915	+	0.992041i	$0.459813\pi$
$192$	0	0
$193$	287272.	0.555137	0.277568	−	0.960706i	$-0.410471\pi$
$193$	287272.	0.555137	0.277568	+	0.960706i	$0.410471\pi$
$194$	0	0
$195$	− 19721.2i	− 0.0371405i
$196$	0	0
$197$	− 500809.i	− 0.919405i	−0.888073	−	0.459702i	$-0.847956\pi$
$197$	− 500809.i	− 0.919405i	0.888073	−	0.459702i	$-0.152044\pi$
$198$	0	0
$199$	858127.	1.53610	0.768049	−	0.640392i	$-0.221228\pi$
$199$	858127.	1.53610	0.768049	+	0.640392i	$0.221228\pi$
$200$	0	0
$201$	−1.19806e6	−2.09165
$202$	0	0
$203$	− 158355.i	− 0.269706i
$204$	0	0
$205$	− 258982.i	− 0.430412i
$206$	0	0
$207$	1.12100e6	1.81837
$208$	0	0
$209$	405078.	0.641465
$210$	0	0
$211$	− 982183.i	− 1.51875i	−0.650653	−	0.759375i	$-0.725505\pi$
$211$	− 982183.i	− 1.51875i	0.650653	−	0.759375i	$-0.274495\pi$
$212$	0	0
$213$	− 862083.i	− 1.30197i
$214$	0	0
$215$	−159290.	−0.235013
$216$	0	0
$217$	−1.13383e6	−1.63455
$218$	0	0
$219$	− 9425.09i	− 0.0132793i
$220$	0	0
$221$	− 46003.7i	− 0.0633595i
$222$	0	0
$223$	−188728.	−0.254141	−0.127071	−	0.991894i	$-0.540557\pi$
$223$	−188728.	−0.254141	−0.127071	+	0.991894i	$0.540557\pi$
$224$	0	0
$225$	140260.	0.184705
$226$	0	0
$227$	517079.i	0.666027i	0.942922	+	0.333014i	$0.108066\pi$
$227$	517079.i	0.666027i	−0.942922	+	0.333014i	$0.891934\pi$
$228$	0	0
$229$	160000.i	0.201619i	0.994906	+	0.100809i	$0.0321432\pi$
$229$	160000.i	0.201619i	−0.994906	+	0.100809i	$0.967857\pi$
$230$	0	0
$231$	1.87805e6	2.31568
$232$	0	0
$233$	1.49124e6	1.79953	0.899765	−	0.436375i	$-0.143738\pi$
$233$	1.49124e6	1.79953	0.899765	+	0.436375i	$0.143738\pi$
$234$	0	0
$235$	− 553017.i	− 0.653235i
$236$	0	0
$237$	1.33295e6i	1.54150i
$238$	0	0
$239$	142742.	0.161643	0.0808215	−	0.996729i	$-0.474246\pi$
$239$	142742.	0.161643	0.0808215	+	0.996729i	$0.474246\pi$
$240$	0	0
$241$	−12428.9	−0.0137844	−0.00689222	−	0.999976i	$-0.502194\pi$
$241$	−12428.9	−0.0137844	−0.00689222	+	0.999976i	$0.502194\pi$
$242$	0	0
$243$	− 1.26916e6i	− 1.37880i
$244$	0	0
$245$	297791.i	0.316954i
$246$	0	0
$247$	−28834.2	−0.0300722
$248$	0	0
$249$	−63632.0	−0.0650395
$250$	0	0
$251$	759309.i	0.760736i	0.924835	+	0.380368i	$0.124203\pi$
$251$	759309.i	0.760736i	−0.924835	+	0.380368i	$0.875797\pi$
$252$	0	0
$253$	2.56051e6i	2.51492i
$254$	0	0
$255$	681464.	0.656285
$256$	0	0
$257$	500270.	0.472468	0.236234	−	0.971696i	$-0.424087\pi$
$257$	500270.	0.472468	0.236234	+	0.971696i	$0.424087\pi$
$258$	0	0
$259$	729670.i	0.675891i
$260$	0	0
$261$	209703.i	0.190547i
$262$	0	0
$263$	549782.	0.490119	0.245059	−	0.969508i	$-0.421193\pi$
$263$	549782.	0.490119	0.245059	+	0.969508i	$0.421193\pi$
$264$	0	0
$265$	−581164.	−0.508375
$266$	0	0
$267$	1.40061e6i	1.20237i
$268$	0	0
$269$	1.02271e6i	0.861731i	0.902416	+	0.430865i	$0.141792\pi$
$269$	1.02271e6i	0.861731i	−0.902416	+	0.430865i	$0.858208\pi$
$270$	0	0
$271$	−1.12050e6	−0.926807	−0.463404	−	0.886147i	$-0.653372\pi$
$271$	−1.12050e6	−0.926807	−0.463404	+	0.886147i	$0.653372\pi$
$272$	0	0
$273$	−133683.	−0.108560
$274$	0	0
$275$	320371.i	0.255459i
$276$	0	0
$277$	323208.i	0.253095i	0.991961	+	0.126547i	$0.0403896\pi$
$277$	323208.i	0.253095i	−0.991961	+	0.126547i	$0.959610\pi$
$278$	0	0
$279$	1.50148e6	1.15481
$280$	0	0
$281$	325436.	0.245867	0.122933	−	0.992415i	$-0.460770\pi$
$281$	325436.	0.245867	0.122933	+	0.992415i	$0.460770\pi$
$282$	0	0
$283$	− 2.41107e6i	− 1.78955i	−0.446518	−	0.894775i	$-0.647336\pi$
$283$	− 2.41107e6i	− 1.78955i	0.446518	−	0.894775i	$-0.352664\pi$
$284$	0	0
$285$	− 427127.i	− 0.311491i
$286$	0	0
$287$	−1.75554e6	−1.25807
$288$	0	0
$289$	169793.	0.119585
$290$	0	0
$291$	3.43843e6i	2.38028i
$292$	0	0
$293$	1.89437e6i	1.28913i	0.764550	+	0.644565i	$0.222961\pi$
$293$	1.89437e6i	1.28913i	−0.764550	+	0.644565i	$0.777039\pi$
$294$	0	0
$295$	693972.	0.464287
$296$	0	0
$297$	205944.	0.135475
$298$	0	0
$299$	− 182261.i	− 0.117901i
$300$	0	0
$301$	1.07977e6i	0.686932i
$302$	0	0
$303$	−1.44796e6	−0.906045
$304$	0	0
$305$	998757.	0.614767
$306$	0	0
$307$	− 1.37530e6i	− 0.832823i	−0.909176	−	0.416411i	$-0.863288\pi$
$307$	− 1.37530e6i	− 0.832823i	0.909176	−	0.416411i	$-0.136712\pi$
$308$	0	0
$309$	− 3.14464e6i	− 1.87359i
$310$	0	0
$311$	−2.60469e6	−1.52705	−0.763527	−	0.645776i	$-0.776534\pi$
$311$	−2.60469e6	−1.52705	−0.763527	+	0.645776i	$0.776534\pi$
$312$	0	0
$313$	780226.	0.450152	0.225076	−	0.974341i	$-0.427737\pi$
$313$	780226.	0.450152	0.225076	+	0.974341i	$0.427737\pi$
$314$	0	0
$315$	− 950773.i	− 0.539884i
$316$	0	0
$317$	1.28045e6i	0.715673i	0.933784	+	0.357836i	$0.116485\pi$
$317$	1.28045e6i	0.715673i	−0.933784	+	0.357836i	$0.883515\pi$
$318$	0	0
$319$	−478986.	−0.263540
$320$	0	0
$321$	−1.45121e6	−0.786081
$322$	0	0
$323$	− 996359.i	− 0.531386i
$324$	0	0
$325$	− 22804.6i	− 0.0119760i
$326$	0	0
$327$	2.22383e6	1.15009
$328$	0	0
$329$	−3.74870e6	−1.90937
$330$	0	0
$331$	2.35495e6i	1.18144i	0.806876	+	0.590721i	$0.201157\pi$
$331$	2.35495e6i	1.18144i	−0.806876	+	0.590721i	$0.798843\pi$
$332$	0	0
$333$	− 966273.i	− 0.477517i
$334$	0	0
$335$	−1.38537e6	−0.674459
$336$	0	0
$337$	−4.10943e6	−1.97109	−0.985546	−	0.169411i	$-0.945814\pi$
$337$	−4.10943e6	−1.97109	−0.985546	+	0.169411i	$0.945814\pi$
$338$	0	0
$339$	− 1.31230e6i	− 0.620202i
$340$	0	0
$341$	3.42956e6i	1.59718i
$342$	0	0
$343$	−829599.	−0.380744
$344$	0	0
$345$	2.69988e6	1.22123
$346$	0	0
$347$	− 1.41845e6i	− 0.632398i	−0.948693	−	0.316199i	$-0.897593\pi$
$347$	− 1.41845e6i	− 0.632398i	0.948693	−	0.316199i	$-0.102407\pi$
$348$	0	0
$349$	2.03977e6i	0.896434i	0.893925	+	0.448217i	$0.147941\pi$
$349$	2.03977e6i	0.896434i	−0.893925	+	0.448217i	$0.852059\pi$
$350$	0	0
$351$	−14659.5	−0.00635112
$352$	0	0
$353$	129749.	0.0554201	0.0277101	−	0.999616i	$-0.491178\pi$
$353$	129749.	0.0554201	0.0277101	+	0.999616i	$0.491178\pi$
$354$	0	0
$355$	− 996866.i	− 0.419823i
$356$	0	0
$357$	− 4.61939e6i	− 1.91829i
$358$	0	0
$359$	1.54034e6	0.630782	0.315391	−	0.948962i	$-0.397864\pi$
$359$	1.54034e6	0.630782	0.315391	+	0.948962i	$0.397864\pi$
$360$	0	0
$361$	1.85160e6	0.747790
$362$	0	0
$363$	− 2.19878e6i	− 0.875819i
$364$	0	0
$365$	− 10898.7i	− 0.00428195i
$366$	0	0
$367$	−1.89303e6	−0.733655	−0.366828	−	0.930289i	$-0.619556\pi$
$367$	−1.89303e6	−0.733655	−0.366828	+	0.930289i	$0.619556\pi$
$368$	0	0
$369$	2.32479e6	0.888829
$370$	0	0
$371$	3.93950e6i	1.48596i
$372$	0	0
$373$	− 3.29601e6i	− 1.22664i	−0.789836	−	0.613319i	$-0.789834\pi$
$373$	− 3.29601e6i	− 1.22664i	0.789836	−	0.613319i	$-0.210166\pi$
$374$	0	0
$375$	337810.	0.124049
$376$	0	0
$377$	34095.0	0.0123549
$378$	0	0
$379$	1.38042e6i	0.493643i	0.969061	+	0.246821i	$0.0793861\pi$
$379$	1.38042e6i	0.493643i	−0.969061	+	0.246821i	$0.920614\pi$
$380$	0	0
$381$	− 6.89637e6i	− 2.43393i
$382$	0	0
$383$	−3.13515e6	−1.09210	−0.546049	−	0.837753i	$-0.683869\pi$
$383$	−3.13515e6	−1.09210	−0.546049	+	0.837753i	$0.683869\pi$
$384$	0	0
$385$	2.17168e6	0.746696
$386$	0	0
$387$	− 1.42989e6i	− 0.485317i
$388$	0	0
$389$	− 4.57462e6i	− 1.53279i	−0.642372	−	0.766393i	$-0.722050\pi$
$389$	− 4.57462e6i	− 1.53279i	0.642372	−	0.766393i	$-0.277950\pi$
$390$	0	0
$391$	6.29800e6	2.08334
$392$	0	0
$393$	7.77122e6	2.53809
$394$	0	0
$395$	1.54135e6i	0.497060i
$396$	0	0
$397$	2.13071e6i	0.678496i	0.940697	+	0.339248i	$0.110173\pi$
$397$	2.13071e6i	0.678496i	−0.940697	+	0.339248i	$0.889827\pi$
$398$	0	0
$399$	−2.89534e6	−0.910473
$400$	0	0
$401$	−2.03442e6	−0.631799	−0.315900	−	0.948793i	$-0.602306\pi$
$401$	−2.03442e6	−0.631799	−0.315900	+	0.948793i	$0.602306\pi$
$402$	0	0
$403$	− 244122.i	− 0.0748763i
$404$	0	0
$405$	− 1.58049e6i	− 0.478799i
$406$	0	0
$407$	2.20708e6	0.660438
$408$	0	0
$409$	−4.79858e6	−1.41842	−0.709210	−	0.704997i	$-0.750948\pi$
$409$	−4.79858e6	−1.41842	−0.709210	+	0.704997i	$0.750948\pi$
$410$	0	0
$411$	6.25903e6i	1.82769i
$412$	0	0
$413$	− 4.70418e6i	− 1.35709i
$414$	0	0
$415$	−73580.6	−0.0209722
$416$	0	0
$417$	−6.87908e6	−1.93727
$418$	0	0
$419$	− 1.63789e6i	− 0.455773i	−0.973688	−	0.227887i	$-0.926818\pi$
$419$	− 1.63789e6i	− 0.455773i	0.973688	−	0.227887i	$-0.0731816\pi$
$420$	0	0
$421$	− 6.15276e6i	− 1.69186i	−0.533293	−	0.845931i	$-0.679046\pi$
$421$	− 6.15276e6i	− 1.69186i	0.533293	−	0.845931i	$-0.320954\pi$
$422$	0	0
$423$	4.96425e6	1.34897
$424$	0	0
$425$	788008.	0.211621
$426$	0	0
$427$	− 6.77020e6i	− 1.79693i
$428$	0	0
$429$	404360.i	0.106078i
$430$	0	0
$431$	−6.74591e6	−1.74923	−0.874616	−	0.484817i	$-0.838886\pi$
$431$	−6.74591e6	−1.74923	−0.874616	+	0.484817i	$0.838886\pi$
$432$	0	0
$433$	6.05459e6	1.55190	0.775952	−	0.630792i	$-0.217270\pi$
$433$	6.05459e6	1.55190	0.775952	+	0.630792i	$0.217270\pi$
$434$	0	0
$435$	505058.i	0.127973i
$436$	0	0
$437$	− 3.94746e6i	− 0.988812i
$438$	0	0
$439$	−6.84810e6	−1.69593	−0.847967	−	0.530049i	$-0.822174\pi$
$439$	−6.84810e6	−1.69593	−0.847967	+	0.530049i	$0.822174\pi$
$440$	0	0
$441$	−2.67317e6	−0.654530
$442$	0	0
$443$	− 4.22271e6i	− 1.02231i	−0.859489	−	0.511154i	$-0.829218\pi$
$443$	− 4.22271e6i	− 1.02231i	0.859489	−	0.511154i	$-0.170782\pi$
$444$	0	0
$445$	1.61959e6i	0.387709i
$446$	0	0
$447$	−1.00425e7	−2.37725
$448$	0	0
$449$	733420.	0.171687	0.0858434	−	0.996309i	$-0.472642\pi$
$449$	733420.	0.171687	0.0858434	+	0.996309i	$0.472642\pi$
$450$	0	0
$451$	5.31010e6i	1.22931i
$452$	0	0
$453$	− 286018.i	− 0.0654859i
$454$	0	0
$455$	−154584.	−0.0350054
$456$	0	0
$457$	−5.36667e6	−1.20203	−0.601014	−	0.799239i	$-0.705236\pi$
$457$	−5.36667e6	−1.20203	−0.601014	+	0.799239i	$0.705236\pi$
$458$	0	0
$459$	− 506555.i	− 0.112226i
$460$	0	0
$461$	1.80941e6i	0.396538i	0.980148	+	0.198269i	$0.0635320\pi$
$461$	1.80941e6i	0.396538i	−0.980148	+	0.198269i	$0.936468\pi$
$462$	0	0
$463$	4.86270e6	1.05421	0.527103	−	0.849802i	$-0.323278\pi$
$463$	4.86270e6	1.05421	0.527103	+	0.849802i	$0.323278\pi$
$464$	0	0
$465$	3.61624e6	0.775576
$466$	0	0
$467$	8.18481e6i	1.73667i	0.495981	+	0.868334i	$0.334809\pi$
$467$	8.18481e6i	1.73667i	−0.495981	+	0.868334i	$0.665191\pi$
$468$	0	0
$469$	9.39094e6i	1.97141i
$470$	0	0
$471$	−2.71301e6	−0.563507
$472$	0	0
$473$	3.26604e6	0.671226
$474$	0	0
$475$	− 493907.i	− 0.100441i
$476$	0	0
$477$	− 5.21692e6i	− 1.04983i
$478$	0	0
$479$	−5.20370e6	−1.03627	−0.518136	−	0.855298i	$-0.673374\pi$
$479$	−5.20370e6	−1.03627	−0.518136	+	0.855298i	$0.673374\pi$
$480$	0	0
$481$	−157104.	−0.0309616
$482$	0	0
$483$	− 1.83015e7i	− 3.56959i
$484$	0	0
$485$	3.97602e6i	0.767528i
$486$	0	0
$487$	3.82182e6	0.730211	0.365105	−	0.930966i	$-0.381033\pi$
$487$	3.82182e6	0.730211	0.365105	+	0.930966i	$0.381033\pi$
$488$	0	0
$489$	−9.72133e6	−1.83846
$490$	0	0
$491$	4.68377e6i	0.876782i	0.898784	+	0.438391i	$0.144451\pi$
$491$	4.68377e6i	0.876782i	−0.898784	+	0.438391i	$0.855549\pi$
$492$	0	0
$493$	1.17815e6i	0.218314i
$494$	0	0
$495$	−2.87587e6	−0.527541
$496$	0	0
$497$	−6.75739e6	−1.22712
$498$	0	0
$499$	129527.i	0.0232867i	0.999932	+	0.0116433i	$0.00370628\pi$
$499$	129527.i	0.0232867i	−0.999932	+	0.0116433i	$0.996294\pi$
$500$	0	0
$501$	3.41098e6i	0.607135i
$502$	0	0
$503$	−4.59305e6	−0.809434	−0.404717	−	0.914442i	$-0.632630\pi$
$503$	−4.59305e6	−0.809434	−0.404717	+	0.914442i	$0.632630\pi$
$504$	0	0
$505$	−1.67434e6	−0.292157
$506$	0	0
$507$	7.99851e6i	1.38194i
$508$	0	0
$509$	− 7.25700e6i	− 1.24155i	−0.783991	−	0.620773i	$-0.786819\pi$
$509$	− 7.25700e6i	− 1.24155i	0.783991	−	0.620773i	$-0.213181\pi$
$510$	0	0
$511$	−73878.0	−0.0125159
$512$	0	0
$513$	−317498.	−0.0532657
$514$	0	0
$515$	− 3.63629e6i	− 0.604145i
$516$	0	0
$517$	1.13389e7i	1.86572i
$518$	0	0
$519$	5.48609e6	0.894015
$520$	0	0
$521$	3.82096e6	0.616706	0.308353	−	0.951272i	$-0.400222\pi$
$521$	3.82096e6	0.616706	0.308353	+	0.951272i	$0.400222\pi$
$522$	0	0
$523$	3.14789e6i	0.503228i	0.967828	+	0.251614i	$0.0809613\pi$
$523$	3.14789e6i	0.503228i	−0.967828	+	0.251614i	$0.919039\pi$
$524$	0	0
$525$	− 2.28989e6i	− 0.362590i
$526$	0	0
$527$	8.43559e6	1.32309
$528$	0	0
$529$	1.85156e7	2.87672
$530$	0	0
$531$	6.22956e6i	0.958784i
$532$	0	0
$533$	− 377982.i	− 0.0576306i
$534$	0	0
$535$	−1.67810e6	−0.253474
$536$	0	0
$537$	3.59971e6	0.538681
$538$	0	0
$539$	− 6.10583e6i	− 0.905259i
$540$	0	0
$541$	6.65423e6i	0.977472i	0.872432	+	0.488736i	$0.162542\pi$
$541$	6.65423e6i	0.977472i	−0.872432	+	0.488736i	$0.837458\pi$
$542$	0	0
$543$	−1.38366e7	−2.01386
$544$	0	0
$545$	2.57152e6	0.370851
$546$	0	0
$547$	2.73564e6i	0.390923i	0.980711	+	0.195461i	$0.0626204\pi$
$547$	2.73564e6i	0.390923i	−0.980711	+	0.195461i	$0.937380\pi$
$548$	0	0
$549$	8.96551e6i	1.26953i
$550$	0	0
$551$	738438.	0.103618
$552$	0	0
$553$	1.04483e7	1.45288
$554$	0	0
$555$	− 2.32722e6i	− 0.320704i
$556$	0	0
$557$	− 8.08022e6i	− 1.10353i	−0.833999	−	0.551766i	$-0.813954\pi$
$557$	− 8.08022e6i	− 1.10353i	0.833999	−	0.551766i	$-0.186046\pi$
$558$	0	0
$559$	−232482.	−0.0314674
$560$	0	0
$561$	−1.39726e7	−1.87443
$562$	0	0
$563$	− 8.04656e6i	− 1.06989i	−0.844887	−	0.534945i	$-0.820332\pi$
$563$	− 8.04656e6i	− 1.06989i	0.844887	−	0.534945i	$-0.179668\pi$
$564$	0	0
$565$	− 1.51747e6i	− 0.199986i
$566$	0	0
$567$	−1.07135e7	−1.39951
$568$	0	0
$569$	−5.03321e6	−0.651725	−0.325862	−	0.945417i	$-0.605655\pi$
$569$	−5.03321e6	−0.651725	−0.325862	+	0.945417i	$0.605655\pi$
$570$	0	0
$571$	1.04770e7i	1.34476i	0.740205	+	0.672381i	$0.234728\pi$
$571$	1.04770e7i	1.34476i	−0.740205	+	0.672381i	$0.765272\pi$
$572$	0	0
$573$	2.74499e6i	0.349265i
$574$	0	0
$575$	3.12199e6	0.393788
$576$	0	0
$577$	6.04045e6	0.755318	0.377659	−	0.925945i	$-0.376729\pi$
$577$	6.04045e6	0.755318	0.377659	+	0.925945i	$0.376729\pi$
$578$	0	0
$579$	6.21077e6i	0.769926i
$580$	0	0
$581$	498776.i	0.0613006i
$582$	0	0
$583$	1.19161e7	1.45198
$584$	0	0
$585$	204709.	0.0247313
$586$	0	0
$587$	− 1.24894e6i	− 0.149605i	−0.997198	−	0.0748026i	$-0.976167\pi$
$587$	− 1.24894e6i	− 0.149605i	0.997198	−	0.0748026i	$-0.0238327\pi$
$588$	0	0
$589$	− 5.28725e6i	− 0.627974i
$590$	0	0
$591$	1.08274e7	1.27513
$592$	0	0
$593$	−5.62448e6	−0.656819	−0.328410	−	0.944535i	$-0.606513\pi$
$593$	−5.62448e6	−0.656819	−0.328410	+	0.944535i	$0.606513\pi$
$594$	0	0
$595$	− 5.34162e6i	− 0.618558i
$596$	0	0
$597$	1.85525e7i	2.13043i
$598$	0	0
$599$	1.21046e7	1.37842	0.689211	−	0.724560i	$-0.257957\pi$
$599$	1.21046e7	1.37842	0.689211	+	0.724560i	$0.257957\pi$
$600$	0	0
$601$	−1.35567e7	−1.53098	−0.765488	−	0.643450i	$-0.777502\pi$
$601$	−1.35567e7	−1.53098	−0.765488	+	0.643450i	$0.777502\pi$
$602$	0	0
$603$	− 1.24360e7i	− 1.39280i
$604$	0	0
$605$	− 2.54255e6i	− 0.282410i
$606$	0	0
$607$	9.54199e6	1.05116	0.525578	−	0.850745i	$-0.323849\pi$
$607$	9.54199e6	1.05116	0.525578	+	0.850745i	$0.323849\pi$
$608$	0	0
$609$	3.42360e6	0.374059
$610$	0	0
$611$	− 807125.i	− 0.0874657i
$612$	0	0
$613$	8.07109e6i	0.867523i	0.901028	+	0.433762i	$0.142814\pi$
$613$	8.07109e6i	0.867523i	−0.901028	+	0.433762i	$0.857186\pi$
$614$	0	0
$615$	5.59914e6	0.596943
$616$	0	0
$617$	277031.	0.0292964	0.0146482	−	0.999893i	$-0.495337\pi$
$617$	277031.	0.0292964	0.0146482	+	0.999893i	$0.495337\pi$
$618$	0	0
$619$	6.49138e6i	0.680943i	0.940255	+	0.340471i	$0.110587\pi$
$619$	6.49138e6i	0.680943i	−0.940255	+	0.340471i	$0.889413\pi$
$620$	0	0
$621$	− 2.00691e6i	− 0.208833i
$622$	0	0
$623$	1.09786e7	1.13325
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	8.75772e6i	0.889656i
$628$	0	0
$629$	− 5.42869e6i	− 0.547103i
$630$	0	0
$631$	1.43239e7	1.43215	0.716074	−	0.698025i	$-0.245937\pi$
$631$	1.43239e7	1.43215	0.716074	+	0.698025i	$0.245937\pi$
$632$	0	0
$633$	2.12346e7	2.10637
$634$	0	0
$635$	− 7.97460e6i	− 0.784828i
$636$	0	0
$637$	434623.i	0.0424389i
$638$	0	0
$639$	8.94854e6	0.866962
$640$	0	0
$641$	−1.52078e7	−1.46191	−0.730956	−	0.682425i	$-0.760925\pi$
$641$	−1.52078e7	−1.46191	−0.730956	+	0.682425i	$0.760925\pi$
$642$	0	0
$643$	2.09843e6i	0.200155i	0.994980	+	0.100078i	$0.0319091\pi$
$643$	2.09843e6i	0.200155i	−0.994980	+	0.100078i	$0.968091\pi$
$644$	0	0
$645$	− 3.44382e6i	− 0.325942i
$646$	0	0
$647$	1.04496e7	0.981384	0.490692	−	0.871333i	$-0.336744\pi$
$647$	1.04496e7	0.981384	0.490692	+	0.871333i	$0.336744\pi$
$648$	0	0
$649$	−1.42290e7	−1.32606
$650$	0	0
$651$	− 2.45131e7i	− 2.26697i
$652$	0	0
$653$	5.78680e6i	0.531075i	0.964101	+	0.265537i	$0.0855494\pi$
$653$	5.78680e6i	0.531075i	−0.964101	+	0.265537i	$0.914451\pi$
$654$	0	0
$655$	8.98622e6	0.818415
$656$	0	0
$657$	97833.7	0.00884250
$658$	0	0
$659$	6.05285e6i	0.542933i	0.962448	+	0.271467i	$0.0875087\pi$
$659$	6.05285e6i	0.542933i	−0.962448	+	0.271467i	$0.912491\pi$
$660$	0	0
$661$	− 5.13781e6i	− 0.457377i	−0.973500	−	0.228689i	$-0.926556\pi$
$661$	− 5.13781e6i	− 0.457377i	0.973500	−	0.228689i	$-0.0734438\pi$
$662$	0	0
$663$	994592.	0.0878741
$664$	0	0
$665$	−3.34801e6	−0.293584
$666$	0	0
$667$	4.66768e6i	0.406243i
$668$	0	0
$669$	− 4.08027e6i	− 0.352471i
$670$	0	0
$671$	−2.04783e7	−1.75585
$672$	0	0
$673$	−1.06833e6	−0.0909214	−0.0454607	−	0.998966i	$-0.514476\pi$
$673$	−1.06833e6	−0.0909214	−0.0454607	+	0.998966i	$0.514476\pi$
$674$	0	0
$675$	− 251105.i	− 0.0212127i
$676$	0	0
$677$	− 1.78231e7i	− 1.49456i	−0.664510	−	0.747279i	$-0.731360\pi$
$677$	− 1.78231e7i	− 1.49456i	0.664510	−	0.747279i	$-0.268640\pi$
$678$	0	0
$679$	2.69520e7	2.24345
$680$	0	0
$681$	−1.11792e7	−0.923721
$682$	0	0
$683$	1.21938e7i	1.00020i	0.865968	+	0.500099i	$0.166703\pi$
$683$	1.21938e7i	1.00020i	−0.865968	+	0.500099i	$0.833297\pi$
$684$	0	0
$685$	7.23761e6i	0.589344i
$686$	0	0
$687$	−3.45917e6	−0.279628
$688$	0	0
$689$	−848205.	−0.0680695
$690$	0	0
$691$	− 7.75412e6i	− 0.617785i	−0.951097	−	0.308893i	$-0.900042\pi$
$691$	− 7.75412e6i	− 0.617785i	0.951097	−	0.308893i	$-0.0999583\pi$
$692$	0	0
$693$	1.94944e7i	1.54198i
$694$	0	0
$695$	−7.95460e6	−0.624678
$696$	0	0
$697$	1.30611e7	1.01835
$698$	0	0
$699$	3.22404e7i	2.49579i
$700$	0	0
$701$	2.02808e7i	1.55880i	0.626528	+	0.779399i	$0.284476\pi$
$701$	2.02808e7i	1.55880i	−0.626528	+	0.779399i	$0.715524\pi$
$702$	0	0
$703$	−3.40259e6	−0.259670
$704$	0	0
$705$	1.19561e7	0.905979
$706$	0	0
$707$	1.13497e7i	0.853960i
$708$	0	0
$709$	− 2.36091e6i	− 0.176386i	−0.996103	−	0.0881928i	$-0.971891\pi$
$709$	− 2.36091e6i	− 0.176386i	0.996103	−	0.0881928i	$-0.0281092\pi$
$710$	0	0
$711$	−1.38362e7	−1.02646
$712$	0	0
$713$	3.34208e7	2.46203
$714$	0	0
$715$	467580.i	0.0342051i
$716$	0	0
$717$	3.08606e6i	0.224185i
$718$	0	0
$719$	−4.33349e6	−0.312619	−0.156310	−	0.987708i	$-0.549960\pi$
$719$	−4.33349e6	−0.312619	−0.156310	+	0.987708i	$0.549960\pi$
$720$	0	0
$721$	−2.46491e7	−1.76589
$722$	0	0
$723$	− 268710.i	− 0.0191178i
$724$	0	0
$725$	584022.i	0.0412652i
$726$	0	0
$727$	4.45026e6	0.312284	0.156142	−	0.987735i	$-0.450094\pi$
$727$	4.45026e6	0.312284	0.156142	+	0.987735i	$0.450094\pi$
$728$	0	0
$729$	1.20768e7	0.841650
$730$	0	0
$731$	− 8.03339e6i	− 0.556039i
$732$	0	0
$733$	838509.i	0.0576432i	0.999585	+	0.0288216i	$0.00917547\pi$
$733$	838509.i	0.0576432i	−0.999585	+	0.0288216i	$0.990825\pi$
$734$	0	0
$735$	−6.43818e6	−0.439587
$736$	0	0
$737$	2.84054e7	1.92634
$738$	0	0
$739$	− 2.44547e7i	− 1.64722i	−0.567158	−	0.823609i	$-0.691957\pi$
$739$	− 2.44547e7i	− 1.64722i	0.567158	−	0.823609i	$-0.308043\pi$
$740$	0	0
$741$	− 623389.i	− 0.0417075i
$742$	0	0
$743$	−1.34515e7	−0.893920	−0.446960	−	0.894554i	$-0.647493\pi$
$743$	−1.34515e7	−0.893920	−0.446960	+	0.894554i	$0.647493\pi$
$744$	0	0
$745$	−1.16127e7	−0.766551
$746$	0	0
$747$	− 660509.i	− 0.0433089i
$748$	0	0
$749$	1.13752e7i	0.740892i
$750$	0	0
$751$	3.02598e7	1.95779	0.978896	−	0.204359i	$-0.0655111\pi$
$751$	3.02598e7	1.95779	0.978896	+	0.204359i	$0.0655111\pi$
$752$	0	0
$753$	−1.64161e7	−1.05507
$754$	0	0
$755$	− 330736.i	− 0.0211161i
$756$	0	0
$757$	2.18417e7i	1.38531i	0.721269	+	0.692655i	$0.243559\pi$
$757$	2.18417e7i	1.38531i	−0.721269	+	0.692655i	$0.756441\pi$
$758$	0	0
$759$	−5.53577e7	−3.48798
$760$	0	0
$761$	−2.66377e6	−0.166738	−0.0833692	−	0.996519i	$-0.526568\pi$
$761$	−2.66377e6	−0.166738	−0.0833692	+	0.996519i	$0.526568\pi$
$762$	0	0
$763$	− 1.74314e7i	− 1.08398i
$764$	0	0
$765$	7.07369e6i	0.437011i
$766$	0	0
$767$	1.01285e6	0.0621664
$768$	0	0
$769$	1.79300e6	0.109336	0.0546681	−	0.998505i	$-0.482590\pi$
$769$	1.79300e6	0.109336	0.0546681	+	0.998505i	$0.482590\pi$
$770$	0	0
$771$	1.08158e7i	0.655271i
$772$	0	0
$773$	− 3.03672e7i	− 1.82792i	−0.405810	−	0.913958i	$-0.633010\pi$
$773$	− 3.03672e7i	− 1.82792i	0.405810	−	0.913958i	$-0.366990\pi$
$774$	0	0
$775$	4.18162e6	0.250087
$776$	0	0
$777$	−1.57753e7	−0.937402
$778$	0	0
$779$	− 8.18642e6i	− 0.483337i
$780$	0	0
$781$	2.04395e7i	1.19907i
$782$	0	0
$783$	375426.	0.0218837
$784$	0	0
$785$	−3.13718e6	−0.181704
$786$	0	0
$787$	1.37601e7i	0.791925i	0.918267	+	0.395962i	$0.129589\pi$
$787$	1.37601e7i	0.791925i	−0.918267	+	0.395962i	$0.870411\pi$
$788$	0	0
$789$	1.18862e7i	0.679752i
$790$	0	0
$791$	−1.02864e7	−0.584549
$792$	0	0
$793$	1.45768e6	0.0823150
$794$	0	0
$795$	− 1.25647e7i	− 0.705071i
$796$	0	0
$797$	− 1.94451e6i	− 0.108434i	−0.998529	−	0.0542169i	$-0.982734\pi$
$797$	− 1.94451e6i	− 0.108434i	0.998529	−	0.0542169i	$-0.0172662\pi$
$798$	0	0
$799$	2.78901e7	1.54555
$800$	0	0
$801$	−1.45385e7	−0.800644
$802$	0	0
$803$	223464.i	0.0122298i
$804$	0	0
$805$	− 2.11628e7i	− 1.15102i
$806$	0	0
$807$	−2.21108e7	−1.19514
$808$	0	0
$809$	4.71536e6	0.253305	0.126652	−	0.991947i	$-0.459577\pi$
$809$	4.71536e6	0.253305	0.126652	+	0.991947i	$0.459577\pi$
$810$	0	0
$811$	− 7.50002e6i	− 0.400415i	−0.979754	−	0.200207i	$-0.935838\pi$
$811$	− 7.50002e6i	− 0.400415i	0.979754	−	0.200207i	$-0.0641616\pi$
$812$	0	0
$813$	− 2.42251e7i	− 1.28540i
$814$	0	0
$815$	−1.12412e7	−0.592816
$816$	0	0
$817$	−5.03516e6	−0.263911
$818$	0	0
$819$	− 1.38765e6i	− 0.0722885i
$820$	0	0
$821$	− 2.84972e7i	− 1.47552i	−0.675063	−	0.737760i	$-0.735884\pi$
$821$	− 2.84972e7i	− 1.47552i	0.675063	−	0.737760i	$-0.264116\pi$
$822$	0	0
$823$	−6.20948e6	−0.319562	−0.159781	−	0.987152i	$-0.551079\pi$
$823$	−6.20948e6	−0.319562	−0.159781	+	0.987152i	$0.551079\pi$
$824$	0	0
$825$	−6.92637e6	−0.354300
$826$	0	0
$827$	1.27490e7i	0.648207i	0.946022	+	0.324104i	$0.105063\pi$
$827$	1.27490e7i	0.648207i	−0.946022	+	0.324104i	$0.894937\pi$
$828$	0	0
$829$	2.60957e7i	1.31881i	0.751787	+	0.659406i	$0.229192\pi$
$829$	2.60957e7i	1.31881i	−0.751787	+	0.659406i	$0.770808\pi$
$830$	0	0
$831$	−6.98771e6	−0.351020
$832$	0	0
$833$	−1.50183e7	−0.749910
$834$	0	0
$835$	3.94428e6i	0.195772i
$836$	0	0
$837$	− 2.68807e6i	− 0.132626i
$838$	0	0
$839$	−2.88675e7	−1.41581	−0.707903	−	0.706310i	$-0.750359\pi$
$839$	−2.88675e7	−1.41581	−0.707903	+	0.706310i	$0.750359\pi$
$840$	0	0
$841$	1.96380e7	0.957430
$842$	0	0
$843$	7.03587e6i	0.340996i
$844$	0	0
$845$	9.24904e6i	0.445610i
$846$	0	0
$847$	−1.72350e7	−0.825472
$848$	0	0
$849$	5.21269e7	2.48195
$850$	0	0
$851$	− 2.15078e7i	− 1.01806i
$852$	0	0
$853$	1.83501e7i	0.863505i	0.901992	+	0.431753i	$0.142105\pi$
$853$	1.83501e7i	0.863505i	−0.901992	+	0.431753i	$0.857895\pi$
$854$	0	0
$855$	4.43364e6	0.207417
$856$	0	0
$857$	−2.16867e7	−1.00865	−0.504327	−	0.863513i	$-0.668259\pi$
$857$	−2.16867e7	−1.00865	−0.504327	+	0.863513i	$0.668259\pi$
$858$	0	0
$859$	− 7.71806e6i	− 0.356883i	−0.983951	−	0.178441i	$-0.942895\pi$
$859$	− 7.71806e6i	− 0.356883i	0.983951	−	0.178441i	$-0.0571055\pi$
$860$	0	0
$861$	− 3.79545e7i	− 1.74484i
$862$	0	0
$863$	2.51914e7	1.15140	0.575700	−	0.817661i	$-0.304730\pi$
$863$	2.51914e7	1.15140	0.575700	+	0.817661i	$0.304730\pi$
$864$	0	0
$865$	6.34382e6	0.288277
$866$	0	0
$867$	3.67090e6i	0.165854i
$868$	0	0
$869$	− 3.16035e7i	− 1.41967i
$870$	0	0
$871$	−2.02194e6	−0.0903075
$872$	0	0
$873$	−3.56914e7	−1.58500
$874$	0	0
$875$	− 2.64790e6i	− 0.116918i
$876$	0	0
$877$	− 1.45084e7i	− 0.636971i	−0.947928	−	0.318485i	$-0.896826\pi$
$877$	− 1.45084e7i	− 0.636971i	0.947928	−	0.318485i	$-0.103174\pi$
$878$	0	0
$879$	−4.09560e7	−1.78791
$880$	0	0
$881$	4.13732e7	1.79589	0.897943	−	0.440111i	$-0.145061\pi$
$881$	4.13732e7	1.79589	0.897943	+	0.440111i	$0.145061\pi$
$882$	0	0
$883$	5.77268e6i	0.249158i	0.992210	+	0.124579i	$0.0397581\pi$
$883$	5.77268e6i	0.249158i	−0.992210	+	0.124579i	$0.960242\pi$
$884$	0	0
$885$	1.50036e7i	0.643926i
$886$	0	0
$887$	−9.78721e6	−0.417686	−0.208843	−	0.977949i	$-0.566970\pi$
$887$	−9.78721e6	−0.417686	−0.208843	+	0.977949i	$0.566970\pi$
$888$	0	0
$889$	−5.40568e7	−2.29401
$890$	0	0
$891$	3.24059e7i	1.36751i
$892$	0	0
$893$	− 1.74809e7i	− 0.733560i
$894$	0	0
$895$	4.16251e6	0.173699
$896$	0	0
$897$	3.94045e6	0.163518
$898$	0	0
$899$	6.25193e6i	0.257997i
$900$	0	0
$901$	− 2.93096e7i	− 1.20281i
$902$	0	0
$903$	−2.33444e7	−0.952714
$904$	0	0
$905$	−1.59999e7	−0.649374
$906$	0	0
$907$	1.90645e6i	0.0769497i	0.999260	+	0.0384749i	$0.0122500\pi$
$907$	1.90645e6i	0.0769497i	−0.999260	+	0.0384749i	$0.987750\pi$
$908$	0	0
$909$	− 1.50300e7i	− 0.603323i
$910$	0	0
$911$	6.24825e6	0.249438	0.124719	−	0.992192i	$-0.460197\pi$
$911$	6.24825e6	0.249438	0.124719	+	0.992192i	$0.460197\pi$
$912$	0	0
$913$	1.50868e6	0.0598991
$914$	0	0
$915$	2.15929e7i	0.852627i
$916$	0	0
$917$	− 6.09142e7i	− 2.39219i
$918$	0	0
$919$	3.90166e6	0.152391	0.0761957	−	0.997093i	$-0.475723\pi$
$919$	3.90166e6	0.152391	0.0761957	+	0.997093i	$0.475723\pi$
$920$	0	0
$921$	2.97338e7	1.15505
$922$	0	0
$923$	− 1.45492e6i	− 0.0562127i
$924$	0	0
$925$	− 2.69107e6i	− 0.103412i
$926$	0	0
$927$	3.26418e7	1.24760
$928$	0	0
$929$	2.94138e6	0.111818	0.0559090	−	0.998436i	$-0.482194\pi$
$929$	2.94138e6	0.111818	0.0559090	+	0.998436i	$0.482194\pi$
$930$	0	0
$931$	9.41318e6i	0.355928i
$932$	0	0
$933$	− 5.63129e7i	− 2.11789i
$934$	0	0
$935$	−1.61571e7	−0.604415
$936$	0	0
$937$	2.73011e7	1.01585	0.507926	−	0.861401i	$-0.330412\pi$
$937$	2.73011e7	1.01585	0.507926	+	0.861401i	$0.330412\pi$
$938$	0	0
$939$	1.68683e7i	0.624322i
$940$	0	0
$941$	− 2.99537e6i	− 0.110275i	−0.998479	−	0.0551374i	$-0.982440\pi$
$941$	− 2.99537e6i	− 0.110275i	0.998479	−	0.0551374i	$-0.0175597\pi$
$942$	0	0
$943$	5.17465e7	1.89497
$944$	0	0
$945$	−1.70215e6	−0.0620038
$946$	0	0
$947$	− 2.96401e7i	− 1.07400i	−0.843582	−	0.537000i	$-0.819558\pi$
$947$	− 2.96401e7i	− 1.07400i	0.843582	−	0.537000i	$-0.180442\pi$
$948$	0	0
$949$	− 15906.5i	0 0.000573337i
$950$	0	0
$951$	−2.76831e7	−0.992575
$952$	0	0
$953$	2.57409e7	0.918101	0.459051	−	0.888410i	$-0.348190\pi$
$953$	2.57409e7	0.918101	0.459051	+	0.888410i	$0.348190\pi$
$954$	0	0
$955$	3.17416e6i	0.112621i
$956$	0	0
$957$	− 1.03556e7i	− 0.365506i
$958$	0	0
$959$	4.90611e7	1.72262
$960$	0	0
$961$	1.61349e7	0.563584
$962$	0	0
$963$	− 1.50637e7i	− 0.523440i
$964$	0	0
$965$	7.18180e6i	0.248265i
$966$	0	0
$967$	2.11539e7	0.727487	0.363743	−	0.931499i	$-0.381499\pi$
$967$	2.11539e7	0.727487	0.363743	+	0.931499i	$0.381499\pi$
$968$	0	0
$969$	2.15411e7	0.736985
$970$	0	0
$971$	4.18119e7i	1.42315i	0.702608	+	0.711577i	$0.252019\pi$
$971$	4.18119e7i	1.42315i	−0.702608	+	0.711577i	$0.747981\pi$
$972$	0	0
$973$	5.39213e7i	1.82590i
$974$	0	0
$975$	493031.	0.0166097
$976$	0	0
$977$	−3.93721e7	−1.31963	−0.659815	−	0.751428i	$-0.729365\pi$
$977$	−3.93721e7	−1.31963	−0.659815	+	0.751428i	$0.729365\pi$
$978$	0	0
$979$	− 3.32078e7i	− 1.10734i
$980$	0	0
$981$	2.30837e7i	0.765831i
$982$	0	0
$983$	−7.10636e6	−0.234565	−0.117283	−	0.993099i	$-0.537418\pi$
$983$	−7.10636e6	−0.234565	−0.117283	+	0.993099i	$0.537418\pi$
$984$	0	0
$985$	1.25202e7	0.411170
$986$	0	0
$987$	− 8.10462e7i	− 2.64813i
$988$	0	0
$989$	− 3.18273e7i	− 1.03469i
$990$	0	0
$991$	−2.70666e6	−0.0875487	−0.0437744	−	0.999041i	$-0.513938\pi$
$991$	−2.70666e6	−0.0875487	−0.0437744	+	0.999041i	$0.513938\pi$
$992$	0	0
$993$	−5.09137e7	−1.63856
$994$	0	0
$995$	2.14532e7i	0.686964i
$996$	0	0
$997$	1.58535e7i	0.505112i	0.967582	+	0.252556i	$0.0812713\pi$
$997$	1.58535e7i	0.505112i	−0.967582	+	0.252556i	$0.918729\pi$
$998$	0	0
$999$	−1.72990e6	−0.0548412

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.6.d.b.161.7	yes	8
4.3	odd	2		320.6.d.a.161.2	✓	8
8.3	odd	2		320.6.d.a.161.7	yes	8
8.5	even	2	inner	320.6.d.b.161.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.6.d.a.161.2	✓	8	4.3	odd	2
320.6.d.a.161.7	yes	8	8.3	odd	2
320.6.d.b.161.2	yes	8	8.5	even	2	inner
320.6.d.b.161.7	yes	8	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	320.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+21.6198i q^{3} +25.0000i q^{5} +169.466 q^{7} -224.417 q^{9} +O(q^{10})\)	\(q+21.6198i q^{3} +25.0000i q^{5} +169.466 q^{7} -224.417 q^{9} -512.594i q^{11} +36.4873i q^{13} -540.496 q^{15} -1260.81 q^{17} +790.251i q^{19} +3663.82i q^{21} -4995.19 q^{23} -625.000 q^{25} +401.768i q^{27} -934.435i q^{29} -6690.60 q^{31} +11082.2 q^{33} +4236.64i q^{35} +4305.71i q^{37} -788.849 q^{39} -10359.3 q^{41} +6371.59i q^{43} -5610.42i q^{45} -22120.7 q^{47} +11911.6 q^{49} -27258.6i q^{51} +23246.6i q^{53} +12814.9 q^{55} -17085.1 q^{57} -27758.9i q^{59} -39950.3i q^{61} -38030.9 q^{63} -912.183 q^{65} +55415.0i q^{67} -107995. i q^{69} -39874.7 q^{71} -435.947 q^{73} -13512.4i q^{75} -86867.1i q^{77} +61654.1 q^{79} -63219.4 q^{81} +2943.22i q^{83} -31520.3i q^{85} +20202.3 q^{87} +64783.7 q^{89} +6183.35i q^{91} -144650. i q^{93} -19756.3 q^{95} +159041. q^{97} +115035. i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 320 q^{7} - 1192 q^{9}+O(q^{10}) \) 8 * q + 320 * q^7 - 1192 * q^9	\( 8 q + 320 q^{7} - 1192 q^{9} - 2400 q^{17} - 1760 q^{23} - 5000 q^{25} - 31040 q^{31} + 3760 q^{33} - 4480 q^{39} + 21584 q^{41} - 47680 q^{47} + 82824 q^{49} - 18400 q^{55} - 106640 q^{57} - 322400 q^{63} - 44000 q^{65} - 246720 q^{71} + 46400 q^{73} - 325760 q^{79} - 82328 q^{81} - 636320 q^{87} - 78192 q^{89} + 40800 q^{95} + 24960 q^{97}+O(q^{100}) \) 8 * q + 320 * q^7 - 1192 * q^9 - 2400 * q^17 - 1760 * q^23 - 5000 * q^25 - 31040 * q^31 + 3760 * q^33 - 4480 * q^39 + 21584 * q^41 - 47680 * q^47 + 82824 * q^49 - 18400 * q^55 - 106640 * q^57 - 322400 * q^63 - 44000 * q^65 - 246720 * q^71 + 46400 * q^73 - 325760 * q^79 - 82328 * q^81 - 636320 * q^87 - 78192 * q^89 + 40800 * q^95 + 24960 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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