LMFDB - Embedded newform 256.6.b.d.129.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [256,6,Mod(129,256)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("256.129");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 256 = 2^{8} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	256.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:	$41.0582578721$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 8)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			129.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	256.129
Dual form			256.6.b.d.129.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+20.0000i q^{3} -74.0000i q^{5} -24.0000 q^{7} -157.000 q^{9} +O(q^{10})$	$q+20.0000i q^{3} -74.0000i q^{5} -24.0000 q^{7} -157.000 q^{9} -124.000i q^{11} -478.000i q^{13} +1480.00 q^{15} -1198.00 q^{17} +3044.00i q^{19} -480.000i q^{21} +184.000 q^{23} -2351.00 q^{25} +1720.00i q^{27} +3282.00i q^{29} +5728.00 q^{31} +2480.00 q^{33} +1776.00i q^{35} +10326.0i q^{37} +9560.00 q^{39} +8886.00 q^{41} +9188.00i q^{43} +11618.0i q^{45} -23664.0 q^{47} -16231.0 q^{49} -23960.0i q^{51} +11686.0i q^{53} -9176.00 q^{55} -60880.0 q^{57} -16876.0i q^{59} +18482.0i q^{61} +3768.00 q^{63} -35372.0 q^{65} -15532.0i q^{67} +3680.00i q^{69} -31960.0 q^{71} +4886.00 q^{73} -47020.0i q^{75} +2976.00i q^{77} -44560.0 q^{79} -72551.0 q^{81} +67364.0i q^{83} +88652.0i q^{85} -65640.0 q^{87} -71994.0 q^{89} +11472.0i q^{91} +114560. i q^{93} +225256. q^{95} +48866.0 q^{97} +19468.0i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 48 q^{7} - 314 q^{9}+O(q^{10}) $ 2 * q - 48 * q^7 - 314 * q^9	$ 2 q - 48 q^{7} - 314 q^{9} + 2960 q^{15} - 2396 q^{17} + 368 q^{23} - 4702 q^{25} + 11456 q^{31} + 4960 q^{33} + 19120 q^{39} + 17772 q^{41} - 47328 q^{47} - 32462 q^{49} - 18352 q^{55} - 121760 q^{57} + 7536 q^{63} - 70744 q^{65} - 63920 q^{71} + 9772 q^{73} - 89120 q^{79} - 145102 q^{81} - 131280 q^{87} - 143988 q^{89} + 450512 q^{95} + 97732 q^{97}+O(q^{100}) $ 2 * q - 48 * q^7 - 314 * q^9 + 2960 * q^15 - 2396 * q^17 + 368 * q^23 - 4702 * q^25 + 11456 * q^31 + 4960 * q^33 + 19120 * q^39 + 17772 * q^41 - 47328 * q^47 - 32462 * q^49 - 18352 * q^55 - 121760 * q^57 + 7536 * q^63 - 70744 * q^65 - 63920 * q^71 + 9772 * q^73 - 89120 * q^79 - 145102 * q^81 - 131280 * q^87 - 143988 * q^89 + 450512 * q^95 + 97732 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/256\mathbb{Z}\right)^\times$.

$n$	$5$	$255$
$\chi(n)$	$-1$	$1$

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
$3$	20.0000i	1.28300i	0.767123	+	0.641500i	$0.221688\pi$
$3$	20.0000i	1.28300i	−0.767123	+	0.641500i	$0.778312\pi$
$4$	0	0
$5$	− 74.0000i	− 1.32375i	−0.749613	−	0.661876i	$-0.769760\pi$
$5$	− 74.0000i	− 1.32375i	0.749613	−	0.661876i	$-0.230240\pi$
$6$	0	0
$7$	−24.0000	−0.185125	−0.0925627	−	0.995707i	$-0.529506\pi$
$7$	−24.0000	−0.185125	−0.0925627	+	0.995707i	$0.529506\pi$
$8$	0	0
$9$	−157.000	−0.646091
$10$	0	0
$11$	− 124.000i	− 0.308987i	−0.987994	−	0.154493i	$-0.950625\pi$
$11$	− 124.000i	− 0.308987i	0.987994	−	0.154493i	$-0.0493745\pi$
$12$	0	0
$13$	− 478.000i	− 0.784458i	−0.919868	−	0.392229i	$-0.871704\pi$
$13$	− 478.000i	− 0.784458i	0.919868	−	0.392229i	$-0.128296\pi$
$14$	0	0
$15$	1480.00	1.69837
$16$	0	0
$17$	−1198.00	−1.00539	−0.502695	−	0.864464i	$-0.667658\pi$
$17$	−1198.00	−1.00539	−0.502695	+	0.864464i	$0.667658\pi$
$18$	0	0
$19$	3044.00i	1.93446i	0.253894	+	0.967232i	$0.418288\pi$
$19$	3044.00i	1.93446i	−0.253894	+	0.967232i	$0.581712\pi$
$20$	0	0
$21$	− 480.000i	− 0.237516i
$22$	0	0
$23$	184.000	0.0725268	0.0362634	−	0.999342i	$-0.488454\pi$
$23$	184.000	0.0725268	0.0362634	+	0.999342i	$0.488454\pi$
$24$	0	0
$25$	−2351.00	−0.752320
$26$	0	0
$27$	1720.00i	0.454066i
$28$	0	0
$29$	3282.00i	0.724676i	0.932047	+	0.362338i	$0.118021\pi$
$29$	3282.00i	0.724676i	−0.932047	+	0.362338i	$0.881979\pi$
$30$	0	0
$31$	5728.00	1.07053	0.535265	−	0.844684i	$-0.320212\pi$
$31$	5728.00	1.07053	0.535265	+	0.844684i	$0.320212\pi$
$32$	0	0
$33$	2480.00	0.396430
$34$	0	0
$35$	1776.00i	0.245060i
$36$	0	0
$37$	10326.0i	1.24002i	0.784595	+	0.620009i	$0.212871\pi$
$37$	10326.0i	1.24002i	−0.784595	+	0.620009i	$0.787129\pi$
$38$	0	0
$39$	9560.00	1.00646
$40$	0	0
$41$	8886.00	0.825556	0.412778	−	0.910832i	$-0.364558\pi$
$41$	8886.00	0.825556	0.412778	+	0.910832i	$0.364558\pi$
$42$	0	0
$43$	9188.00i	0.757792i	0.925439	+	0.378896i	$0.123696\pi$
$43$	9188.00i	0.757792i	−0.925439	+	0.378896i	$0.876304\pi$
$44$	0	0
$45$	11618.0i	0.855264i
$46$	0	0
$47$	−23664.0	−1.56258	−0.781292	−	0.624165i	$-0.785439\pi$
$47$	−23664.0	−1.56258	−0.781292	+	0.624165i	$0.785439\pi$
$48$	0	0
$49$	−16231.0	−0.965729
$50$	0	0
$51$	− 23960.0i	− 1.28992i
$52$	0	0
$53$	11686.0i	0.571447i	0.958312	+	0.285724i	$0.0922339\pi$
$53$	11686.0i	0.571447i	−0.958312	+	0.285724i	$0.907766\pi$
$54$	0	0
$55$	−9176.00	−0.409022
$56$	0	0
$57$	−60880.0	−2.48192
$58$	0	0
$59$	− 16876.0i	− 0.631160i	−0.948899	−	0.315580i	$-0.897801\pi$
$59$	− 16876.0i	− 0.631160i	0.948899	−	0.315580i	$-0.102199\pi$
$60$	0	0
$61$	18482.0i	0.635952i	0.948099	+	0.317976i	$0.103003\pi$
$61$	18482.0i	0.635952i	−0.948099	+	0.317976i	$0.896997\pi$
$62$	0	0
$63$	3768.00	0.119608
$64$	0	0
$65$	−35372.0	−1.03843
$66$	0	0
$67$	− 15532.0i	− 0.422708i	−0.977410	−	0.211354i	$-0.932213\pi$
$67$	− 15532.0i	− 0.422708i	0.977410	−	0.211354i	$-0.0677873\pi$
$68$	0	0
$69$	3680.00i	0.0930519i
$70$	0	0
$71$	−31960.0	−0.752421	−0.376210	−	0.926534i	$-0.622773\pi$
$71$	−31960.0	−0.752421	−0.376210	+	0.926534i	$0.622773\pi$
$72$	0	0
$73$	4886.00	0.107312	0.0536558	−	0.998559i	$-0.482913\pi$
$73$	4886.00	0.107312	0.0536558	+	0.998559i	$0.482913\pi$
$74$	0	0
$75$	− 47020.0i	− 0.965227i
$76$	0	0
$77$	2976.00i	0.0572013i
$78$	0	0
$79$	−44560.0	−0.803299	−0.401650	−	0.915793i	$-0.631563\pi$
$79$	−44560.0	−0.803299	−0.401650	+	0.915793i	$0.631563\pi$
$80$	0	0
$81$	−72551.0	−1.22866
$82$	0	0
$83$	67364.0i	1.07333i	0.843796	+	0.536664i	$0.180316\pi$
$83$	67364.0i	1.07333i	−0.843796	+	0.536664i	$0.819684\pi$
$84$	0	0
$85$	88652.0i	1.33089i
$86$	0	0
$87$	−65640.0	−0.929759
$88$	0	0
$89$	−71994.0	−0.963432	−0.481716	−	0.876327i	$-0.659986\pi$
$89$	−71994.0	−0.963432	−0.481716	+	0.876327i	$0.659986\pi$
$90$	0	0
$91$	11472.0i	0.145223i
$92$	0	0
$93$	114560.i	1.37349i
$94$	0	0
$95$	225256.	2.56075
$96$	0	0
$97$	48866.0	0.527324	0.263662	−	0.964615i	$-0.415070\pi$
$97$	48866.0	0.527324	0.263662	+	0.964615i	$0.415070\pi$
$98$	0	0
$99$	19468.0i	0.199633i
$100$	0	0
$101$	51606.0i	0.503381i	0.967808	+	0.251690i	$0.0809865\pi$
$101$	51606.0i	0.503381i	−0.967808	+	0.251690i	$0.919014\pi$
$102$	0	0
$103$	180424.	1.67572	0.837860	−	0.545886i	$-0.183807\pi$
$103$	180424.	1.67572	0.837860	+	0.545886i	$0.183807\pi$
$104$	0	0
$105$	−35520.0	−0.314412
$106$	0	0
$107$	65700.0i	0.554761i	0.960760	+	0.277381i	$0.0894663\pi$
$107$	65700.0i	0.554761i	−0.960760	+	0.277381i	$0.910534\pi$
$108$	0	0
$109$	112706.i	0.908617i	0.890844	+	0.454308i	$0.150114\pi$
$109$	112706.i	0.908617i	−0.890844	+	0.454308i	$0.849886\pi$
$110$	0	0
$111$	−206520.	−1.59094
$112$	0	0
$113$	−23502.0	−0.173145	−0.0865723	−	0.996246i	$-0.527591\pi$
$113$	−23502.0	−0.173145	−0.0865723	+	0.996246i	$0.527591\pi$
$114$	0	0
$115$	− 13616.0i	− 0.0960075i
$116$	0	0
$117$	75046.0i	0.506831i
$118$	0	0
$119$	28752.0	0.186123
$120$	0	0
$121$	145675.	0.904527
$122$	0	0
$123$	177720.i	1.05919i
$124$	0	0
$125$	− 57276.0i	− 0.327867i
$126$	0	0
$127$	94592.0	0.520409	0.260205	−	0.965553i	$-0.416210\pi$
$127$	94592.0	0.520409	0.260205	+	0.965553i	$0.416210\pi$
$128$	0	0
$129$	−183760.	−0.972247
$130$	0	0
$131$	70292.0i	0.357872i	0.983861	+	0.178936i	$0.0572655\pi$
$131$	70292.0i	0.357872i	−0.983861	+	0.178936i	$0.942735\pi$
$132$	0	0
$133$	− 73056.0i	− 0.358119i
$134$	0	0
$135$	127280.	0.601071
$136$	0	0
$137$	−277290.	−1.26221	−0.631107	−	0.775696i	$-0.717399\pi$
$137$	−277290.	−1.26221	−0.631107	+	0.775696i	$0.717399\pi$
$138$	0	0
$139$	130308.i	0.572050i	0.958222	+	0.286025i	$0.0923341\pi$
$139$	130308.i	0.572050i	−0.958222	+	0.286025i	$0.907666\pi$
$140$	0	0
$141$	− 473280.i	− 2.00480i
$142$	0	0
$143$	−59272.0	−0.242387
$144$	0	0
$145$	242868.	0.959291
$146$	0	0
$147$	− 324620.i	− 1.23903i
$148$	0	0
$149$	− 401530.i	− 1.48167i	−0.671685	−	0.740836i	$-0.734429\pi$
$149$	− 401530.i	− 1.48167i	0.671685	−	0.740836i	$-0.265571\pi$
$150$	0	0
$151$	−75976.0	−0.271165	−0.135583	−	0.990766i	$-0.543291\pi$
$151$	−75976.0	−0.271165	−0.135583	+	0.990766i	$0.543291\pi$
$152$	0	0
$153$	188086.	0.649573
$154$	0	0
$155$	− 423872.i	− 1.41712i
$156$	0	0
$157$	394322.i	1.27674i	0.769730	+	0.638369i	$0.220391\pi$
$157$	394322.i	1.27674i	−0.769730	+	0.638369i	$0.779609\pi$
$158$	0	0
$159$	−233720.	−0.733167
$160$	0	0
$161$	−4416.00	−0.0134265
$162$	0	0
$163$	− 11724.0i	− 0.0345626i	−0.999851	−	0.0172813i	$-0.994499\pi$
$163$	− 11724.0i	− 0.0345626i	0.999851	−	0.0172813i	$-0.00550109\pi$
$164$	0	0
$165$	− 183520.i	− 0.524775i
$166$	0	0
$167$	−551928.	−1.53141	−0.765705	−	0.643192i	$-0.777610\pi$
$167$	−551928.	−1.53141	−0.765705	+	0.643192i	$0.777610\pi$
$168$	0	0
$169$	142809.	0.384626
$170$	0	0
$171$	− 477908.i	− 1.24984i
$172$	0	0
$173$	− 432894.i	− 1.09968i	−0.835270	−	0.549840i	$-0.814689\pi$
$173$	− 432894.i	− 1.09968i	0.835270	−	0.549840i	$-0.185311\pi$
$174$	0	0
$175$	56424.0	0.139274
$176$	0	0
$177$	337520.	0.809779
$178$	0	0
$179$	559620.i	1.30545i	0.757594	+	0.652726i	$0.226375\pi$
$179$	559620.i	1.30545i	−0.757594	+	0.652726i	$0.773625\pi$
$180$	0	0
$181$	604710.i	1.37199i	0.727607	+	0.685995i	$0.240633\pi$
$181$	604710.i	1.37199i	−0.727607	+	0.685995i	$0.759367\pi$
$182$	0	0
$183$	−369640.	−0.815927
$184$	0	0
$185$	764124.	1.64148
$186$	0	0
$187$	148552.i	0.310652i
$188$	0	0
$189$	− 41280.0i	− 0.0840592i
$190$	0	0
$191$	409152.	0.811524	0.405762	−	0.913979i	$-0.367006\pi$
$191$	409152.	0.811524	0.405762	+	0.913979i	$0.367006\pi$
$192$	0	0
$193$	540866.	1.04519	0.522596	−	0.852580i	$-0.324963\pi$
$193$	540866.	1.04519	0.522596	+	0.852580i	$0.324963\pi$
$194$	0	0
$195$	− 707440.i	− 1.33230i
$196$	0	0
$197$	− 629898.i	− 1.15639i	−0.815898	−	0.578195i	$-0.803757\pi$
$197$	− 629898.i	− 1.15639i	0.815898	−	0.578195i	$-0.196243\pi$
$198$	0	0
$199$	283048.	0.506673	0.253336	−	0.967378i	$-0.418472\pi$
$199$	283048.	0.506673	0.253336	+	0.967378i	$0.418472\pi$
$200$	0	0
$201$	310640.	0.542335
$202$	0	0
$203$	− 78768.0i	− 0.134156i
$204$	0	0
$205$	− 657564.i	− 1.09283i
$206$	0	0
$207$	−28888.0	−0.0468588
$208$	0	0
$209$	377456.	0.597724
$210$	0	0
$211$	142756.i	0.220744i	0.993890	+	0.110372i	$0.0352042\pi$
$211$	142756.i	0.220744i	−0.993890	+	0.110372i	$0.964796\pi$
$212$	0	0
$213$	− 639200.i	− 0.965357i
$214$	0	0
$215$	679912.	1.00313
$216$	0	0
$217$	−137472.	−0.198182
$218$	0	0
$219$	97720.0i	0.137681i
$220$	0	0
$221$	572644.i	0.788686i
$222$	0	0
$223$	−889696.	−1.19806	−0.599031	−	0.800726i	$-0.704447\pi$
$223$	−889696.	−1.19806	−0.599031	+	0.800726i	$0.704447\pi$
$224$	0	0
$225$	369107.	0.486067
$226$	0	0
$227$	1.14316e6i	1.47245i	0.676736	+	0.736226i	$0.263394\pi$
$227$	1.14316e6i	1.47245i	−0.676736	+	0.736226i	$0.736606\pi$
$228$	0	0
$229$	− 695786.i	− 0.876773i	−0.898787	−	0.438386i	$-0.855550\pi$
$229$	− 695786.i	− 0.876773i	0.898787	−	0.438386i	$-0.144450\pi$
$230$	0	0
$231$	−59520.0	−0.0733893
$232$	0	0
$233$	347126.	0.418887	0.209444	−	0.977821i	$-0.432835\pi$
$233$	347126.	0.418887	0.209444	+	0.977821i	$0.432835\pi$
$234$	0	0
$235$	1.75114e6i	2.06847i
$236$	0	0
$237$	− 891200.i	− 1.03063i
$238$	0	0
$239$	1.64296e6	1.86051	0.930255	−	0.366912i	$-0.119585\pi$
$239$	1.64296e6	1.86051	0.930255	+	0.366912i	$0.119585\pi$
$240$	0	0
$241$	−1.16744e6	−1.29477	−0.647383	−	0.762165i	$-0.724137\pi$
$241$	−1.16744e6	−1.29477	−0.647383	+	0.762165i	$0.724137\pi$
$242$	0	0
$243$	− 1.03306e6i	− 1.12230i
$244$	0	0
$245$	1.20109e6i	1.27839i
$246$	0	0
$247$	1.45503e6	1.51751
$248$	0	0
$249$	−1.34728e6	−1.37708
$250$	0	0
$251$	790612.i	0.792098i	0.918229	+	0.396049i	$0.129619\pi$
$251$	790612.i	0.792098i	−0.918229	+	0.396049i	$0.870381\pi$
$252$	0	0
$253$	− 22816.0i	− 0.0224098i
$254$	0	0
$255$	−1.77304e6	−1.70753
$256$	0	0
$257$	−129790.	−0.122577	−0.0612884	−	0.998120i	$-0.519521\pi$
$257$	−129790.	−0.122577	−0.0612884	+	0.998120i	$0.519521\pi$
$258$	0	0
$259$	− 247824.i	− 0.229559i
$260$	0	0
$261$	− 515274.i	− 0.468206i
$262$	0	0
$263$	70888.0	0.0631951	0.0315975	−	0.999501i	$-0.489941\pi$
$263$	70888.0	0.0631951	0.0315975	+	0.999501i	$0.489941\pi$
$264$	0	0
$265$	864764.	0.756455
$266$	0	0
$267$	− 1.43988e6i	− 1.23608i
$268$	0	0
$269$	− 1.79017e6i	− 1.50839i	−0.656649	−	0.754197i	$-0.728027\pi$
$269$	− 1.79017e6i	− 1.50839i	0.656649	−	0.754197i	$-0.271973\pi$
$270$	0	0
$271$	1.77362e6	1.46702	0.733511	−	0.679678i	$-0.237880\pi$
$271$	1.77362e6	1.46702	0.733511	+	0.679678i	$0.237880\pi$
$272$	0	0
$273$	−229440.	−0.186321
$274$	0	0
$275$	291524.i	0.232457i
$276$	0	0
$277$	− 275450.i	− 0.215697i	−0.994167	−	0.107848i	$-0.965604\pi$
$277$	− 275450.i	− 0.215697i	0.994167	−	0.107848i	$-0.0343961\pi$
$278$	0	0
$279$	−899296.	−0.691659
$280$	0	0
$281$	−594170.	−0.448895	−0.224448	−	0.974486i	$-0.572058\pi$
$281$	−594170.	−0.448895	−0.224448	+	0.974486i	$0.572058\pi$
$282$	0	0
$283$	− 1.09243e6i	− 0.810824i	−0.914134	−	0.405412i	$-0.867128\pi$
$283$	− 1.09243e6i	− 0.810824i	0.914134	−	0.405412i	$-0.132872\pi$
$284$	0	0
$285$	4.50512e6i	3.28545i
$286$	0	0
$287$	−213264.	−0.152831
$288$	0	0
$289$	15347.0	0.0108088
$290$	0	0
$291$	977320.i	0.676557i
$292$	0	0
$293$	333654.i	0.227053i	0.993535	+	0.113527i	$0.0362147\pi$
$293$	333654.i	0.227053i	−0.993535	+	0.113527i	$0.963785\pi$
$294$	0	0
$295$	−1.24882e6	−0.835500
$296$	0	0
$297$	213280.	0.140300
$298$	0	0
$299$	− 87952.0i	− 0.0568942i
$300$	0	0
$301$	− 220512.i	− 0.140287i
$302$	0	0
$303$	−1.03212e6	−0.645838
$304$	0	0
$305$	1.36767e6	0.841843
$306$	0	0
$307$	1.05997e6i	0.641872i	0.947101	+	0.320936i	$0.103997\pi$
$307$	1.05997e6i	0.641872i	−0.947101	+	0.320936i	$0.896003\pi$
$308$	0	0
$309$	3.60848e6i	2.14995i
$310$	0	0
$311$	−1.33649e6	−0.783545	−0.391773	−	0.920062i	$-0.628138\pi$
$311$	−1.33649e6	−0.783545	−0.391773	+	0.920062i	$0.628138\pi$
$312$	0	0
$313$	−1.64419e6	−0.948615	−0.474308	−	0.880359i	$-0.657302\pi$
$313$	−1.64419e6	−0.948615	−0.474308	+	0.880359i	$0.657302\pi$
$314$	0	0
$315$	− 278832.i	− 0.158331i
$316$	0	0
$317$	1.72370e6i	0.963414i	0.876332	+	0.481707i	$0.159983\pi$
$317$	1.72370e6i	0.963414i	−0.876332	+	0.481707i	$0.840017\pi$
$318$	0	0
$319$	406968.	0.223915
$320$	0	0
$321$	−1.31400e6	−0.711759
$322$	0	0
$323$	− 3.64671e6i	− 1.94489i
$324$	0	0
$325$	1.12378e6i	0.590163i
$326$	0	0
$327$	−2.25412e6	−1.16576
$328$	0	0
$329$	567936.	0.289274
$330$	0	0
$331$	− 2.74963e6i	− 1.37944i	−0.724074	−	0.689722i	$-0.757733\pi$
$331$	− 2.74963e6i	− 1.37944i	0.724074	−	0.689722i	$-0.242267\pi$
$332$	0	0
$333$	− 1.62118e6i	− 0.801164i
$334$	0	0
$335$	−1.14937e6	−0.559561
$336$	0	0
$337$	−3.41489e6	−1.63796	−0.818978	−	0.573824i	$-0.805459\pi$
$337$	−3.41489e6	−1.63796	−0.818978	+	0.573824i	$0.805459\pi$
$338$	0	0
$339$	− 470040.i	− 0.222145i
$340$	0	0
$341$	− 710272.i	− 0.330780i
$342$	0	0
$343$	792912.	0.363906
$344$	0	0
$345$	272320.	0.123178
$346$	0	0
$347$	− 730764.i	− 0.325802i	−0.986642	−	0.162901i	$-0.947915\pi$
$347$	− 730764.i	− 0.325802i	0.986642	−	0.162901i	$-0.0520851\pi$
$348$	0	0
$349$	2.29749e6i	1.00969i	0.863209	+	0.504847i	$0.168451\pi$
$349$	2.29749e6i	1.00969i	−0.863209	+	0.504847i	$0.831549\pi$
$350$	0	0
$351$	822160.	0.356196
$352$	0	0
$353$	−1.17072e6	−0.500052	−0.250026	−	0.968239i	$-0.580439\pi$
$353$	−1.17072e6	−0.500052	−0.250026	+	0.968239i	$0.580439\pi$
$354$	0	0
$355$	2.36504e6i	0.996019i
$356$	0	0
$357$	575040.i	0.238796i
$358$	0	0
$359$	3.88654e6	1.59157	0.795787	−	0.605577i	$-0.207058\pi$
$359$	3.88654e6	1.59157	0.795787	+	0.605577i	$0.207058\pi$
$360$	0	0
$361$	−6.78984e6	−2.74215
$362$	0	0
$363$	2.91350e6i	1.16051i
$364$	0	0
$365$	− 361564.i	− 0.142054i
$366$	0	0
$367$	−933040.	−0.361606	−0.180803	−	0.983519i	$-0.557870\pi$
$367$	−933040.	−0.361606	−0.180803	+	0.983519i	$0.557870\pi$
$368$	0	0
$369$	−1.39510e6	−0.533384
$370$	0	0
$371$	− 280464.i	− 0.105789i
$372$	0	0
$373$	− 392218.i	− 0.145967i	−0.997333	−	0.0729836i	$-0.976748\pi$
$373$	− 392218.i	− 0.145967i	0.997333	−	0.0729836i	$-0.0232521\pi$
$374$	0	0
$375$	1.14552e6	0.420653
$376$	0	0
$377$	1.56880e6	0.568477
$378$	0	0
$379$	4.72930e6i	1.69122i	0.533805	+	0.845608i	$0.320762\pi$
$379$	4.72930e6i	1.69122i	−0.533805	+	0.845608i	$0.679238\pi$
$380$	0	0
$381$	1.89184e6i	0.667686i
$382$	0	0
$383$	−1.89734e6	−0.660920	−0.330460	−	0.943820i	$-0.607204\pi$
$383$	−1.89734e6	−0.660920	−0.330460	+	0.943820i	$0.607204\pi$
$384$	0	0
$385$	220224.	0.0757204
$386$	0	0
$387$	− 1.44252e6i	− 0.489602i
$388$	0	0
$389$	− 3.72295e6i	− 1.24742i	−0.781655	−	0.623711i	$-0.785624\pi$
$389$	− 3.72295e6i	− 1.24742i	0.781655	−	0.623711i	$-0.214376\pi$
$390$	0	0
$391$	−220432.	−0.0729177
$392$	0	0
$393$	−1.40584e6	−0.459150
$394$	0	0
$395$	3.29744e6i	1.06337i
$396$	0	0
$397$	− 3.33808e6i	− 1.06297i	−0.847068	−	0.531484i	$-0.821635\pi$
$397$	− 3.33808e6i	− 1.06297i	0.847068	−	0.531484i	$-0.178365\pi$
$398$	0	0
$399$	1.46112e6	0.459466
$400$	0	0
$401$	4.27490e6	1.32759	0.663796	−	0.747913i	$-0.268944\pi$
$401$	4.27490e6	1.32759	0.663796	+	0.747913i	$0.268944\pi$
$402$	0	0
$403$	− 2.73798e6i	− 0.839785i
$404$	0	0
$405$	5.36877e6i	1.62644i
$406$	0	0
$407$	1.28042e6	0.383149
$408$	0	0
$409$	2.57319e6	0.760613	0.380306	−	0.924861i	$-0.375819\pi$
$409$	2.57319e6	0.760613	0.380306	+	0.924861i	$0.375819\pi$
$410$	0	0
$411$	− 5.54580e6i	− 1.61942i
$412$	0	0
$413$	405024.i	0.116844i
$414$	0	0
$415$	4.98494e6	1.42082
$416$	0	0
$417$	−2.60616e6	−0.733941
$418$	0	0
$419$	5.26828e6i	1.46600i	0.680230	+	0.732999i	$0.261880\pi$
$419$	5.26828e6i	1.46600i	−0.680230	+	0.732999i	$0.738120\pi$
$420$	0	0
$421$	− 973354.i	− 0.267649i	−0.991005	−	0.133824i	$-0.957274\pi$
$421$	− 973354.i	− 0.267649i	0.991005	−	0.133824i	$-0.0427258\pi$
$422$	0	0
$423$	3.71525e6	1.00957
$424$	0	0
$425$	2.81650e6	0.756375
$426$	0	0
$427$	− 443568.i	− 0.117731i
$428$	0	0
$429$	− 1.18544e6i	− 0.310983i
$430$	0	0
$431$	−3.55736e6	−0.922433	−0.461216	−	0.887288i	$-0.652587\pi$
$431$	−3.55736e6	−0.922433	−0.461216	+	0.887288i	$0.652587\pi$
$432$	0	0
$433$	−1.95496e6	−0.501092	−0.250546	−	0.968105i	$-0.580610\pi$
$433$	−1.95496e6	−0.501092	−0.250546	+	0.968105i	$0.580610\pi$
$434$	0	0
$435$	4.85736e6i	1.23077i
$436$	0	0
$437$	560096.i	0.140300i
$438$	0	0
$439$	−3.29681e6	−0.816455	−0.408228	−	0.912880i	$-0.633853\pi$
$439$	−3.29681e6	−0.816455	−0.408228	+	0.912880i	$0.633853\pi$
$440$	0	0
$441$	2.54827e6	0.623948
$442$	0	0
$443$	5.05820e6i	1.22458i	0.790634	+	0.612289i	$0.209751\pi$
$443$	5.05820e6i	1.22458i	−0.790634	+	0.612289i	$0.790249\pi$
$444$	0	0
$445$	5.32756e6i	1.27535i
$446$	0	0
$447$	8.03060e6	1.90099
$448$	0	0
$449$	2.12730e6	0.497981	0.248990	−	0.968506i	$-0.419901\pi$
$449$	2.12730e6	0.497981	0.248990	+	0.968506i	$0.419901\pi$
$450$	0	0
$451$	− 1.10186e6i	− 0.255086i
$452$	0	0
$453$	− 1.51952e6i	− 0.347905i
$454$	0	0
$455$	848928.	0.192239
$456$	0	0
$457$	−289130.	−0.0647594	−0.0323797	−	0.999476i	$-0.510309\pi$
$457$	−289130.	−0.0647594	−0.0323797	+	0.999476i	$0.510309\pi$
$458$	0	0
$459$	− 2.06056e6i	− 0.456513i
$460$	0	0
$461$	− 2.66870e6i	− 0.584854i	−0.956288	−	0.292427i	$-0.905537\pi$
$461$	− 2.66870e6i	− 0.584854i	0.956288	−	0.292427i	$-0.0944629\pi$
$462$	0	0
$463$	−7.58619e6	−1.64464	−0.822321	−	0.569024i	$-0.807321\pi$
$463$	−7.58619e6	−1.64464	−0.822321	+	0.569024i	$0.807321\pi$
$464$	0	0
$465$	8.47744e6	1.81816
$466$	0	0
$467$	− 1.41961e6i	− 0.301216i	−0.988594	−	0.150608i	$-0.951877\pi$
$467$	− 1.41961e6i	− 0.301216i	0.988594	−	0.150608i	$-0.0481231\pi$
$468$	0	0
$469$	372768.i	0.0782540i
$470$	0	0
$471$	−7.88644e6	−1.63806
$472$	0	0
$473$	1.13931e6	0.234148
$474$	0	0
$475$	− 7.15644e6i	− 1.45534i
$476$	0	0
$477$	− 1.83470e6i	− 0.369207i
$478$	0	0
$479$	1.88406e6	0.375195	0.187597	−	0.982246i	$-0.439930\pi$
$479$	1.88406e6	0.375195	0.187597	+	0.982246i	$0.439930\pi$
$480$	0	0
$481$	4.93583e6	0.972741
$482$	0	0
$483$	− 88320.0i	− 0.0172263i
$484$	0	0
$485$	− 3.61608e6i	− 0.698046i
$486$	0	0
$487$	−6.01388e6	−1.14903	−0.574516	−	0.818493i	$-0.694810\pi$
$487$	−6.01388e6	−1.14903	−0.574516	+	0.818493i	$0.694810\pi$
$488$	0	0
$489$	234480.	0.0443439
$490$	0	0
$491$	− 4.29232e6i	− 0.803504i	−0.915749	−	0.401752i	$-0.868401\pi$
$491$	− 4.29232e6i	− 0.803504i	0.915749	−	0.401752i	$-0.131599\pi$
$492$	0	0
$493$	− 3.93184e6i	− 0.728581i
$494$	0	0
$495$	1.44063e6	0.264265
$496$	0	0
$497$	767040.	0.139292
$498$	0	0
$499$	1.34509e6i	0.241825i	0.992663	+	0.120912i	$0.0385820\pi$
$499$	1.34509e6i	0.241825i	−0.992663	+	0.120912i	$0.961418\pi$
$500$	0	0
$501$	− 1.10386e7i	− 1.96480i
$502$	0	0
$503$	202008.	0.0355999	0.0177999	−	0.999842i	$-0.494334\pi$
$503$	202008.	0.0355999	0.0177999	+	0.999842i	$0.494334\pi$
$504$	0	0
$505$	3.81884e6	0.666352
$506$	0	0
$507$	2.85618e6i	0.493476i
$508$	0	0
$509$	− 9.78344e6i	− 1.67377i	−0.547375	−	0.836887i	$-0.684373\pi$
$509$	− 9.78344e6i	− 1.67377i	0.547375	−	0.836887i	$-0.315627\pi$
$510$	0	0
$511$	−117264.	−0.0198661
$512$	0	0
$513$	−5.23568e6	−0.878374
$514$	0	0
$515$	− 1.33514e7i	− 2.21824i
$516$	0	0
$517$	2.93434e6i	0.482818i
$518$	0	0
$519$	8.65788e6	1.41089
$520$	0	0
$521$	1.04830e7	1.69197	0.845985	−	0.533207i	$-0.179013\pi$
$521$	1.04830e7	1.69197	0.845985	+	0.533207i	$0.179013\pi$
$522$	0	0
$523$	− 6.21017e6i	− 0.992772i	−0.868102	−	0.496386i	$-0.834660\pi$
$523$	− 6.21017e6i	− 0.992772i	0.868102	−	0.496386i	$-0.165340\pi$
$524$	0	0
$525$	1.12848e6i	0.178688i
$526$	0	0
$527$	−6.86214e6	−1.07630
$528$	0	0
$529$	−6.40249e6	−0.994740
$530$	0	0
$531$	2.64953e6i	0.407787i
$532$	0	0
$533$	− 4.24751e6i	− 0.647614i
$534$	0	0
$535$	4.86180e6	0.734366
$536$	0	0
$537$	−1.11924e7	−1.67489
$538$	0	0
$539$	2.01264e6i	0.298397i
$540$	0	0
$541$	− 5.08088e6i	− 0.746355i	−0.927760	−	0.373178i	$-0.878268\pi$
$541$	− 5.08088e6i	− 0.746355i	0.927760	−	0.373178i	$-0.121732\pi$
$542$	0	0
$543$	−1.20942e7	−1.76026
$544$	0	0
$545$	8.34024e6	1.20278
$546$	0	0
$547$	3.34687e6i	0.478267i	0.970987	+	0.239133i	$0.0768633\pi$
$547$	3.34687e6i	0.478267i	−0.970987	+	0.239133i	$0.923137\pi$
$548$	0	0
$549$	− 2.90167e6i	− 0.410883i
$550$	0	0
$551$	−9.99041e6	−1.40186
$552$	0	0
$553$	1.06944e6	0.148711
$554$	0	0
$555$	1.52825e7i	2.10601i
$556$	0	0
$557$	− 7.00377e6i	− 0.956520i	−0.878218	−	0.478260i	$-0.841268\pi$
$557$	− 7.00377e6i	− 0.956520i	0.878218	−	0.478260i	$-0.158732\pi$
$558$	0	0
$559$	4.39186e6	0.594456
$560$	0	0
$561$	−2.97104e6	−0.398567
$562$	0	0
$563$	− 1.29819e7i	− 1.72610i	−0.505116	−	0.863052i	$-0.668550\pi$
$563$	− 1.29819e7i	− 1.72610i	0.505116	−	0.863052i	$-0.331450\pi$
$564$	0	0
$565$	1.73915e6i	0.229200i
$566$	0	0
$567$	1.74122e6	0.227456
$568$	0	0
$569$	−1.89942e6	−0.245946	−0.122973	−	0.992410i	$-0.539243\pi$
$569$	−1.89942e6	−0.245946	−0.122973	+	0.992410i	$0.539243\pi$
$570$	0	0
$571$	1.66300e6i	0.213452i	0.994288	+	0.106726i	$0.0340368\pi$
$571$	1.66300e6i	0.213452i	−0.994288	+	0.106726i	$0.965963\pi$
$572$	0	0
$573$	8.18304e6i	1.04119i
$574$	0	0
$575$	−432584.	−0.0545633
$576$	0	0
$577$	8.77344e6	1.09706	0.548530	−	0.836131i	$-0.315188\pi$
$577$	8.77344e6	1.09706	0.548530	+	0.836131i	$0.315188\pi$
$578$	0	0
$579$	1.08173e7i	1.34098i
$580$	0	0
$581$	− 1.61674e6i	− 0.198700i
$582$	0	0
$583$	1.44906e6	0.176570
$584$	0	0
$585$	5.55340e6	0.670918
$586$	0	0
$587$	− 5.18393e6i	− 0.620961i	−0.950580	−	0.310480i	$-0.899510\pi$
$587$	− 5.18393e6i	− 0.620961i	0.950580	−	0.310480i	$-0.100490\pi$
$588$	0	0
$589$	1.74360e7i	2.07090i
$590$	0	0
$591$	1.25980e7	1.48365
$592$	0	0
$593$	8.49858e6	0.992452	0.496226	−	0.868193i	$-0.334719\pi$
$593$	8.49858e6	0.992452	0.496226	+	0.868193i	$0.334719\pi$
$594$	0	0
$595$	− 2.12765e6i	− 0.246381i
$596$	0	0
$597$	5.66096e6i	0.650061i
$598$	0	0
$599$	1.12471e7	1.28078	0.640388	−	0.768051i	$-0.278773\pi$
$599$	1.12471e7	1.28078	0.640388	+	0.768051i	$0.278773\pi$
$600$	0	0
$601$	3.46439e6	0.391238	0.195619	−	0.980680i	$-0.437328\pi$
$601$	3.46439e6	0.391238	0.195619	+	0.980680i	$0.437328\pi$
$602$	0	0
$603$	2.43852e6i	0.273108i
$604$	0	0
$605$	− 1.07799e7i	− 1.19737i
$606$	0	0
$607$	999712.	0.110129	0.0550647	−	0.998483i	$-0.482463\pi$
$607$	999712.	0.110129	0.0550647	+	0.998483i	$0.482463\pi$
$608$	0	0
$609$	1.57536e6	0.172122
$610$	0	0
$611$	1.13114e7i	1.22578i
$612$	0	0
$613$	9.81340e6i	1.05480i	0.849619	+	0.527398i	$0.176832\pi$
$613$	9.81340e6i	1.05480i	−0.849619	+	0.527398i	$0.823168\pi$
$614$	0	0
$615$	1.31513e7	1.40210
$616$	0	0
$617$	5.34745e6	0.565501	0.282751	−	0.959193i	$-0.408753\pi$
$617$	5.34745e6	0.565501	0.282751	+	0.959193i	$0.408753\pi$
$618$	0	0
$619$	6.82768e6i	0.716221i	0.933679	+	0.358110i	$0.116579\pi$
$619$	6.82768e6i	0.716221i	−0.933679	+	0.358110i	$0.883421\pi$
$620$	0	0
$621$	316480.i	0.0329319i
$622$	0	0
$623$	1.72786e6	0.178356
$624$	0	0
$625$	−1.15853e7	−1.18633
$626$	0	0
$627$	7.54912e6i	0.766880i
$628$	0	0
$629$	− 1.23705e7i	− 1.24670i
$630$	0	0
$631$	−3.60970e6	−0.360909	−0.180455	−	0.983583i	$-0.557757\pi$
$631$	−3.60970e6	−0.360909	−0.180455	+	0.983583i	$0.557757\pi$
$632$	0	0
$633$	−2.85512e6	−0.283214
$634$	0	0
$635$	− 6.99981e6i	− 0.688893i
$636$	0	0
$637$	7.75842e6i	0.757573i
$638$	0	0
$639$	5.01772e6	0.486132
$640$	0	0
$641$	−1.33853e7	−1.28672	−0.643361	−	0.765563i	$-0.722460\pi$
$641$	−1.33853e7	−1.28672	−0.643361	+	0.765563i	$0.722460\pi$
$642$	0	0
$643$	− 9.91115e6i	− 0.945358i	−0.881235	−	0.472679i	$-0.843287\pi$
$643$	− 9.91115e6i	− 0.945358i	0.881235	−	0.472679i	$-0.156713\pi$
$644$	0	0
$645$	1.35982e7i	1.28701i
$646$	0	0
$647$	−1.78359e7	−1.67508	−0.837539	−	0.546378i	$-0.816006\pi$
$647$	−1.78359e7	−1.67508	−0.837539	+	0.546378i	$0.816006\pi$
$648$	0	0
$649$	−2.09262e6	−0.195020
$650$	0	0
$651$	− 2.74944e6i	− 0.254268i
$652$	0	0
$653$	4.32323e6i	0.396758i	0.980125	+	0.198379i	$0.0635677\pi$
$653$	4.32323e6i	0.396758i	−0.980125	+	0.198379i	$0.936432\pi$
$654$	0	0
$655$	5.20161e6	0.473734
$656$	0	0
$657$	−767102.	−0.0693330
$658$	0	0
$659$	1.97858e7i	1.77476i	0.461035	+	0.887382i	$0.347478\pi$
$659$	1.97858e7i	1.77476i	−0.461035	+	0.887382i	$0.652522\pi$
$660$	0	0
$661$	1.57772e7i	1.40451i	0.711925	+	0.702255i	$0.247824\pi$
$661$	1.57772e7i	1.40451i	−0.711925	+	0.702255i	$0.752176\pi$
$662$	0	0
$663$	−1.14529e7	−1.01188
$664$	0	0
$665$	−5.40614e6	−0.474060
$666$	0	0
$667$	603888.i	0.0525584i
$668$	0	0
$669$	− 1.77939e7i	− 1.53711i
$670$	0	0
$671$	2.29177e6	0.196501
$672$	0	0
$673$	6.78762e6	0.577670	0.288835	−	0.957379i	$-0.406732\pi$
$673$	6.78762e6	0.577670	0.288835	+	0.957379i	$0.406732\pi$
$674$	0	0
$675$	− 4.04372e6i	− 0.341603i
$676$	0	0
$677$	− 1.49942e7i	− 1.25734i	−0.777673	−	0.628669i	$-0.783600\pi$
$677$	− 1.49942e7i	− 1.25734i	0.777673	−	0.628669i	$-0.216400\pi$
$678$	0	0
$679$	−1.17278e6	−0.0976211
$680$	0	0
$681$	−2.28631e7	−1.88916
$682$	0	0
$683$	− 1.15580e7i	− 0.948053i	−0.880511	−	0.474026i	$-0.842800\pi$
$683$	− 1.15580e7i	− 0.948053i	0.880511	−	0.474026i	$-0.157200\pi$
$684$	0	0
$685$	2.05195e7i	1.67086i
$686$	0	0
$687$	1.39157e7	1.12490
$688$	0	0
$689$	5.58591e6	0.448276
$690$	0	0
$691$	− 220156.i	− 0.0175402i	−0.999962	−	0.00877012i	$-0.997208\pi$
$691$	− 220156.i	− 0.0175402i	0.999962	−	0.00877012i	$-0.00279165\pi$
$692$	0	0
$693$	− 467232.i	− 0.0369572i
$694$	0	0
$695$	9.64279e6	0.757253
$696$	0	0
$697$	−1.06454e7	−0.830006
$698$	0	0
$699$	6.94252e6i	0.537433i
$700$	0	0
$701$	− 4.78933e6i	− 0.368111i	−0.982916	−	0.184056i	$-0.941077\pi$
$701$	− 4.78933e6i	− 0.368111i	0.982916	−	0.184056i	$-0.0589227\pi$
$702$	0	0
$703$	−3.14323e7	−2.39877
$704$	0	0
$705$	−3.50227e7	−2.65385
$706$	0	0
$707$	− 1.23854e6i	− 0.0931886i
$708$	0	0
$709$	4.26892e6i	0.318935i	0.987203	+	0.159468i	$0.0509777\pi$
$709$	4.26892e6i	0.318935i	−0.987203	+	0.159468i	$0.949022\pi$
$710$	0	0
$711$	6.99592e6	0.519004
$712$	0	0
$713$	1.05395e6	0.0776421
$714$	0	0
$715$	4.38613e6i	0.320860i
$716$	0	0
$717$	3.28592e7i	2.38704i
$718$	0	0
$719$	1.61960e7	1.16838	0.584190	−	0.811617i	$-0.301412\pi$
$719$	1.61960e7	1.16838	0.584190	+	0.811617i	$0.301412\pi$
$720$	0	0
$721$	−4.33018e6	−0.310218
$722$	0	0
$723$	− 2.33488e7i	− 1.66119i
$724$	0	0
$725$	− 7.71598e6i	− 0.545188i
$726$	0	0
$727$	6.53426e6	0.458522	0.229261	−	0.973365i	$-0.426369\pi$
$727$	6.53426e6	0.458522	0.229261	+	0.973365i	$0.426369\pi$
$728$	0	0
$729$	3.03131e6	0.211257
$730$	0	0
$731$	− 1.10072e7i	− 0.761876i
$732$	0	0
$733$	− 1.31617e7i	− 0.904800i	−0.891815	−	0.452400i	$-0.850568\pi$
$733$	− 1.31617e7i	− 0.904800i	0.891815	−	0.452400i	$-0.149432\pi$
$734$	0	0
$735$	−2.40219e7	−1.64017
$736$	0	0
$737$	−1.92597e6	−0.130611
$738$	0	0
$739$	− 1.42348e7i	− 0.958825i	−0.877590	−	0.479412i	$-0.840850\pi$
$739$	− 1.42348e7i	− 0.958825i	0.877590	−	0.479412i	$-0.159150\pi$
$740$	0	0
$741$	2.91006e7i	1.94696i
$742$	0	0
$743$	−2.15835e7	−1.43434	−0.717168	−	0.696901i	$-0.754562\pi$
$743$	−2.15835e7	−1.43434	−0.717168	+	0.696901i	$0.754562\pi$
$744$	0	0
$745$	−2.97132e7	−1.96137
$746$	0	0
$747$	− 1.05761e7i	− 0.693467i
$748$	0	0
$749$	− 1.57680e6i	− 0.102700i
$750$	0	0
$751$	−1.86594e7	−1.20725	−0.603625	−	0.797268i	$-0.706278\pi$
$751$	−1.86594e7	−1.20725	−0.603625	+	0.797268i	$0.706278\pi$
$752$	0	0
$753$	−1.58122e7	−1.01626
$754$	0	0
$755$	5.62222e6i	0.358956i
$756$	0	0
$757$	− 2.56681e6i	− 0.162800i	−0.996682	−	0.0813999i	$-0.974061\pi$
$757$	− 2.56681e6i	− 0.162800i	0.996682	−	0.0813999i	$-0.0259391\pi$
$758$	0	0
$759$	456320.	0.0287518
$760$	0	0
$761$	2.59586e7	1.62487	0.812436	−	0.583051i	$-0.198141\pi$
$761$	2.59586e7	1.62487	0.812436	+	0.583051i	$0.198141\pi$
$762$	0	0
$763$	− 2.70494e6i	− 0.168208i
$764$	0	0
$765$	− 1.39184e7i	− 0.859874i
$766$	0	0
$767$	−8.06673e6	−0.495118
$768$	0	0
$769$	5.53267e6	0.337380	0.168690	−	0.985669i	$-0.446046\pi$
$769$	5.53267e6	0.337380	0.168690	+	0.985669i	$0.446046\pi$
$770$	0	0
$771$	− 2.59580e6i	− 0.157266i
$772$	0	0
$773$	8.32940e6i	0.501378i	0.968068	+	0.250689i	$0.0806571\pi$
$773$	8.32940e6i	0.501378i	−0.968068	+	0.250689i	$0.919343\pi$
$774$	0	0
$775$	−1.34665e7	−0.805381
$776$	0	0
$777$	4.95648e6	0.294524
$778$	0	0
$779$	2.70490e7i	1.59701i
$780$	0	0
$781$	3.96304e6i	0.232488i
$782$	0	0
$783$	−5.64504e6	−0.329051
$784$	0	0
$785$	2.91798e7	1.69009
$786$	0	0
$787$	− 1.36523e7i	− 0.785719i	−0.919598	−	0.392860i	$-0.871486\pi$
$787$	− 1.36523e7i	− 0.785719i	0.919598	−	0.392860i	$-0.128514\pi$
$788$	0	0
$789$	1.41776e6i	0.0810793i
$790$	0	0
$791$	564048.	0.0320535
$792$	0	0
$793$	8.83440e6	0.498877
$794$	0	0
$795$	1.72953e7i	0.970532i
$796$	0	0
$797$	8.54626e6i	0.476574i	0.971195	+	0.238287i	$0.0765859\pi$
$797$	8.54626e6i	0.476574i	−0.971195	+	0.238287i	$0.923414\pi$
$798$	0	0
$799$	2.83495e7	1.57101
$800$	0	0
$801$	1.13031e7	0.622465
$802$	0	0
$803$	− 605864.i	− 0.0331578i
$804$	0	0
$805$	326784.i	0.0177734i
$806$	0	0
$807$	3.58035e7	1.93527
$808$	0	0
$809$	−7.58484e6	−0.407451	−0.203725	−	0.979028i	$-0.565305\pi$
$809$	−7.58484e6	−0.407451	−0.203725	+	0.979028i	$0.565305\pi$
$810$	0	0
$811$	− 6.18473e6i	− 0.330194i	−0.986277	−	0.165097i	$-0.947206\pi$
$811$	− 6.18473e6i	− 0.330194i	0.986277	−	0.165097i	$-0.0527937\pi$
$812$	0	0
$813$	3.54723e7i	1.88219i
$814$	0	0
$815$	−867576.	−0.0457524
$816$	0	0
$817$	−2.79683e7	−1.46592
$818$	0	0
$819$	− 1.80110e6i	− 0.0938273i
$820$	0	0
$821$	− 2.78102e6i	− 0.143995i	−0.997405	−	0.0719973i	$-0.977063\pi$
$821$	− 2.78102e6i	− 0.143995i	0.997405	−	0.0719973i	$-0.0229373\pi$
$822$	0	0
$823$	1.63895e7	0.843461	0.421731	−	0.906721i	$-0.361423\pi$
$823$	1.63895e7	0.843461	0.421731	+	0.906721i	$0.361423\pi$
$824$	0	0
$825$	−5.83048e6	−0.298242
$826$	0	0
$827$	− 2.29511e7i	− 1.16692i	−0.812142	−	0.583459i	$-0.801699\pi$
$827$	− 2.29511e7i	− 1.16692i	0.812142	−	0.583459i	$-0.198301\pi$
$828$	0	0
$829$	3.50136e6i	0.176950i	0.996078	+	0.0884750i	$0.0281993\pi$
$829$	3.50136e6i	0.176950i	−0.996078	+	0.0884750i	$0.971801\pi$
$830$	0	0
$831$	5.50900e6	0.276739
$832$	0	0
$833$	1.94447e7	0.970934
$834$	0	0
$835$	4.08427e7i	2.02721i
$836$	0	0
$837$	9.85216e6i	0.486091i
$838$	0	0
$839$	5.29668e6	0.259776	0.129888	−	0.991529i	$-0.458538\pi$
$839$	5.29668e6	0.259776	0.129888	+	0.991529i	$0.458538\pi$
$840$	0	0
$841$	9.73962e6	0.474845
$842$	0	0
$843$	− 1.18834e7i	− 0.575933i
$844$	0	0
$845$	− 1.05679e7i	− 0.509150i
$846$	0	0
$847$	−3.49620e6	−0.167451
$848$	0	0
$849$	2.18486e7	1.04029
$850$	0	0
$851$	1.89998e6i	0.0899344i
$852$	0	0
$853$	− 2.02948e7i	− 0.955021i	−0.878626	−	0.477511i	$-0.841539\pi$
$853$	− 2.02948e7i	− 0.955021i	0.878626	−	0.477511i	$-0.158461\pi$
$854$	0	0
$855$	−3.53652e7	−1.65448
$856$	0	0
$857$	4.82785e6	0.224544	0.112272	−	0.993678i	$-0.464187\pi$
$857$	4.82785e6	0.224544	0.112272	+	0.993678i	$0.464187\pi$
$858$	0	0
$859$	1.30210e7i	0.602092i	0.953610	+	0.301046i	$0.0973358\pi$
$859$	1.30210e7i	0.602092i	−0.953610	+	0.301046i	$0.902664\pi$
$860$	0	0
$861$	− 4.26528e6i	− 0.196083i
$862$	0	0
$863$	3.92387e7	1.79344	0.896721	−	0.442596i	$-0.145942\pi$
$863$	3.92387e7	1.79344	0.896721	+	0.442596i	$0.145942\pi$
$864$	0	0
$865$	−3.20342e7	−1.45570
$866$	0	0
$867$	306940.i	0.0138677i
$868$	0	0
$869$	5.52544e6i	0.248209i
$870$	0	0
$871$	−7.42430e6	−0.331596
$872$	0	0
$873$	−7.67196e6	−0.340699
$874$	0	0
$875$	1.37462e6i	0.0606965i
$876$	0	0
$877$	− 1.34622e7i	− 0.591041i	−0.955336	−	0.295520i	$-0.904507\pi$
$877$	− 1.34622e7i	− 0.591041i	0.955336	−	0.295520i	$-0.0954930\pi$
$878$	0	0
$879$	−6.67308e6	−0.291309
$880$	0	0
$881$	−917710.	−0.0398351	−0.0199175	−	0.999802i	$-0.506340\pi$
$881$	−917710.	−0.0398351	−0.0199175	+	0.999802i	$0.506340\pi$
$882$	0	0
$883$	2.45488e7i	1.05957i	0.848133	+	0.529784i	$0.177727\pi$
$883$	2.45488e7i	1.05957i	−0.848133	+	0.529784i	$0.822273\pi$
$884$	0	0
$885$	− 2.49765e7i	− 1.07195i
$886$	0	0
$887$	−1.61463e7	−0.689070	−0.344535	−	0.938773i	$-0.611963\pi$
$887$	−1.61463e7	−0.689070	−0.344535	+	0.938773i	$0.611963\pi$
$888$	0	0
$889$	−2.27021e6	−0.0963410
$890$	0	0
$891$	8.99632e6i	0.379639i
$892$	0	0
$893$	− 7.20332e7i	− 3.02276i
$894$	0	0
$895$	4.14119e7	1.72809
$896$	0	0
$897$	1.75904e6	0.0729953
$898$	0	0
$899$	1.87993e7i	0.775787i
$900$	0	0
$901$	− 1.39998e7i	− 0.574527i
$902$	0	0
$903$	4.41024e6	0.179988
$904$	0	0
$905$	4.47485e7	1.81617
$906$	0	0
$907$	2.03361e7i	0.820824i	0.911900	+	0.410412i	$0.134615\pi$
$907$	2.03361e7i	0.820824i	−0.911900	+	0.410412i	$0.865385\pi$
$908$	0	0
$909$	− 8.10214e6i	− 0.325230i
$910$	0	0
$911$	−1.07726e7	−0.430054	−0.215027	−	0.976608i	$-0.568984\pi$
$911$	−1.07726e7	−0.430054	−0.215027	+	0.976608i	$0.568984\pi$
$912$	0	0
$913$	8.35314e6	0.331644
$914$	0	0
$915$	2.73534e7i	1.08009i
$916$	0	0
$917$	− 1.68701e6i	− 0.0662512i
$918$	0	0
$919$	4.18566e7	1.63484	0.817419	−	0.576043i	$-0.195404\pi$
$919$	4.18566e7	1.63484	0.817419	+	0.576043i	$0.195404\pi$
$920$	0	0
$921$	−2.11994e7	−0.823522
$922$	0	0
$923$	1.52769e7i	0.590242i
$924$	0	0
$925$	− 2.42764e7i	− 0.932890i
$926$	0	0
$927$	−2.83266e7	−1.08267
$928$	0	0
$929$	2.99845e7	1.13988	0.569939	−	0.821687i	$-0.306967\pi$
$929$	2.99845e7	1.13988	0.569939	+	0.821687i	$0.306967\pi$
$930$	0	0
$931$	− 4.94072e7i	− 1.86817i
$932$	0	0
$933$	− 2.67298e7i	− 1.00529i
$934$	0	0
$935$	1.09928e7	0.411227
$936$	0	0
$937$	−1.42402e7	−0.529867	−0.264934	−	0.964267i	$-0.585350\pi$
$937$	−1.42402e7	−0.529867	−0.264934	+	0.964267i	$0.585350\pi$
$938$	0	0
$939$	− 3.28837e7i	− 1.21707i
$940$	0	0
$941$	4.14546e7i	1.52615i	0.646307	+	0.763077i	$0.276313\pi$
$941$	4.14546e7i	1.52615i	−0.646307	+	0.763077i	$0.723687\pi$
$942$	0	0
$943$	1.63502e6	0.0598749
$944$	0	0
$945$	−3.05472e6	−0.111274
$946$	0	0
$947$	− 1.54079e7i	− 0.558300i	−0.960248	−	0.279150i	$-0.909947\pi$
$947$	− 1.54079e7i	− 0.558300i	0.960248	−	0.279150i	$-0.0900527\pi$
$948$	0	0
$949$	− 2.33551e6i	− 0.0841813i
$950$	0	0
$951$	−3.44740e7	−1.23606
$952$	0	0
$953$	2.06328e7	0.735912	0.367956	−	0.929843i	$-0.380058\pi$
$953$	2.06328e7	0.735912	0.367956	+	0.929843i	$0.380058\pi$
$954$	0	0
$955$	− 3.02772e7i	− 1.07426i
$956$	0	0
$957$	8.13936e6i	0.287283i
$958$	0	0
$959$	6.65496e6	0.233668
$960$	0	0
$961$	4.18083e6	0.146034
$962$	0	0
$963$	− 1.03149e7i	− 0.358426i
$964$	0	0
$965$	− 4.00241e7i	− 1.38358i
$966$	0	0
$967$	1.18724e7	0.408294	0.204147	−	0.978940i	$-0.434558\pi$
$967$	1.18724e7	0.408294	0.204147	+	0.978940i	$0.434558\pi$
$968$	0	0
$969$	7.29342e7	2.49530
$970$	0	0
$971$	− 1.53222e6i	− 0.0521523i	−0.999660	−	0.0260761i	$-0.991699\pi$
$971$	− 1.53222e6i	− 0.0521523i	0.999660	−	0.0260761i	$-0.00830123\pi$
$972$	0	0
$973$	− 3.12739e6i	− 0.105901i
$974$	0	0
$975$	−2.24756e7	−0.757180
$976$	0	0
$977$	1.74321e7	0.584269	0.292135	−	0.956377i	$-0.405635\pi$
$977$	1.74321e7	0.584269	0.292135	+	0.956377i	$0.405635\pi$
$978$	0	0
$979$	8.92726e6i	0.297688i
$980$	0	0
$981$	− 1.76948e7i	− 0.587049i
$982$	0	0
$983$	2.23270e6	0.0736963	0.0368482	−	0.999321i	$-0.488268\pi$
$983$	2.23270e6	0.0736963	0.0368482	+	0.999321i	$0.488268\pi$
$984$	0	0
$985$	−4.66125e7	−1.53078
$986$	0	0
$987$	1.13587e7i	0.371139i
$988$	0	0
$989$	1.69059e6i	0.0549602i
$990$	0	0
$991$	−2.22501e7	−0.719693	−0.359847	−	0.933011i	$-0.617171\pi$
$991$	−2.22501e7	−0.719693	−0.359847	+	0.933011i	$0.617171\pi$
$992$	0	0
$993$	5.49926e7	1.76983
$994$	0	0
$995$	− 2.09456e7i	− 0.670709i
$996$	0	0
$997$	5.32662e7i	1.69712i	0.529095	+	0.848562i	$0.322531\pi$
$997$	5.32662e7i	1.69712i	−0.529095	+	0.848562i	$0.677469\pi$
$998$	0	0
$999$	−1.77607e7	−0.563050

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	256.6.b.d.129.2		2
4.3	odd	2		256.6.b.f.129.1		2
8.3	odd	2		256.6.b.f.129.2		2
8.5	even	2	inner	256.6.b.d.129.1		2
16.3	odd	4		8.6.a.a.1.1	✓	1
16.5	even	4		64.6.a.g.1.1		1
16.11	odd	4		64.6.a.a.1.1		1
16.13	even	4		16.6.a.a.1.1		1
48.5	odd	4		576.6.a.h.1.1		1
48.11	even	4		576.6.a.g.1.1		1
48.29	odd	4		144.6.a.k.1.1		1
48.35	even	4		72.6.a.f.1.1		1
80.3	even	4		200.6.c.a.49.2		2
80.13	odd	4		400.6.c.d.49.1		2
80.19	odd	4		200.6.a.a.1.1		1
80.29	even	4		400.6.a.l.1.1		1
80.67	even	4		200.6.c.a.49.1		2
80.77	odd	4		400.6.c.d.49.2		2
112.3	even	12		392.6.i.e.177.1		2
112.13	odd	4		784.6.a.l.1.1		1
112.19	even	12		392.6.i.e.361.1		2
112.51	odd	12		392.6.i.b.361.1		2
112.67	odd	12		392.6.i.b.177.1		2
112.83	even	4		392.6.a.b.1.1		1
176.131	even	4		968.6.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.6.a.a.1.1	✓	1	16.3	odd	4
16.6.a.a.1.1		1	16.13	even	4
64.6.a.a.1.1		1	16.11	odd	4
64.6.a.g.1.1		1	16.5	even	4
72.6.a.f.1.1		1	48.35	even	4
144.6.a.k.1.1		1	48.29	odd	4
200.6.a.a.1.1		1	80.19	odd	4
200.6.c.a.49.1		2	80.67	even	4
200.6.c.a.49.2		2	80.3	even	4
256.6.b.d.129.1		2	8.5	even	2	inner
256.6.b.d.129.2		2	1.1	even	1	trivial
256.6.b.f.129.1		2	4.3	odd	2
256.6.b.f.129.2		2	8.3	odd	2
392.6.a.b.1.1		1	112.83	even	4
392.6.i.b.177.1		2	112.67	odd	12
392.6.i.b.361.1		2	112.51	odd	12
392.6.i.e.177.1		2	112.3	even	12
392.6.i.e.361.1		2	112.19	even	12
400.6.a.l.1.1		1	80.29	even	4
400.6.c.d.49.1		2	80.13	odd	4
400.6.c.d.49.2		2	80.77	odd	4
576.6.a.g.1.1		1	48.11	even	4
576.6.a.h.1.1		1	48.5	odd	4
784.6.a.l.1.1		1	112.13	odd	4
968.6.a.a.1.1		1	176.131	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 256 = 2^{8} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	256.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+20.0000i q^{3} -74.0000i q^{5} -24.0000 q^{7} -157.000 q^{9} +O(q^{10})\)	\(q+20.0000i q^{3} -74.0000i q^{5} -24.0000 q^{7} -157.000 q^{9} -124.000i q^{11} -478.000i q^{13} +1480.00 q^{15} -1198.00 q^{17} +3044.00i q^{19} -480.000i q^{21} +184.000 q^{23} -2351.00 q^{25} +1720.00i q^{27} +3282.00i q^{29} +5728.00 q^{31} +2480.00 q^{33} +1776.00i q^{35} +10326.0i q^{37} +9560.00 q^{39} +8886.00 q^{41} +9188.00i q^{43} +11618.0i q^{45} -23664.0 q^{47} -16231.0 q^{49} -23960.0i q^{51} +11686.0i q^{53} -9176.00 q^{55} -60880.0 q^{57} -16876.0i q^{59} +18482.0i q^{61} +3768.00 q^{63} -35372.0 q^{65} -15532.0i q^{67} +3680.00i q^{69} -31960.0 q^{71} +4886.00 q^{73} -47020.0i q^{75} +2976.00i q^{77} -44560.0 q^{79} -72551.0 q^{81} +67364.0i q^{83} +88652.0i q^{85} -65640.0 q^{87} -71994.0 q^{89} +11472.0i q^{91} +114560. i q^{93} +225256. q^{95} +48866.0 q^{97} +19468.0i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 48 q^{7} - 314 q^{9}+O(q^{10}) \) 2 * q - 48 * q^7 - 314 * q^9	\( 2 q - 48 q^{7} - 314 q^{9} + 2960 q^{15} - 2396 q^{17} + 368 q^{23} - 4702 q^{25} + 11456 q^{31} + 4960 q^{33} + 19120 q^{39} + 17772 q^{41} - 47328 q^{47} - 32462 q^{49} - 18352 q^{55} - 121760 q^{57} + 7536 q^{63} - 70744 q^{65} - 63920 q^{71} + 9772 q^{73} - 89120 q^{79} - 145102 q^{81} - 131280 q^{87} - 143988 q^{89} + 450512 q^{95} + 97732 q^{97}+O(q^{100}) \) 2 * q - 48 * q^7 - 314 * q^9 + 2960 * q^15 - 2396 * q^17 + 368 * q^23 - 4702 * q^25 + 11456 * q^31 + 4960 * q^33 + 19120 * q^39 + 17772 * q^41 - 47328 * q^47 - 32462 * q^49 - 18352 * q^55 - 121760 * q^57 + 7536 * q^63 - 70744 * q^65 - 63920 * q^71 + 9772 * q^73 - 89120 * q^79 - 145102 * q^81 - 131280 * q^87 - 143988 * q^89 + 450512 * q^95 + 97732 * q^97

Introduction

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