LMFDB - Embedded newform 128.6.b.d.65.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [128,6,Mod(65,128)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(128, 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("128.65");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 128 = 2^{7} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	128.b (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:	$20.5291289361$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-29})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 28x^{2} + 225 $ x^4 + 28*x^2 + 225
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{9} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			65.2
Root			$-0.707107 + 3.80789i$ of defining polynomial
Character	$\chi$	$=$	128.65
Dual form			128.6.b.d.65.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-15.2315i q^{3} +43.0813i q^{5} +22.6274 q^{7} +11.0000 q^{9} +O(q^{10})$	$q-15.2315i q^{3} +43.0813i q^{5} +22.6274 q^{7} +11.0000 q^{9} -258.936i q^{11} +990.870i q^{13} +656.195 q^{15} -218.000 q^{17} +1995.33i q^{19} -344.651i q^{21} +3416.74 q^{23} +1269.00 q^{25} -3868.81i q^{27} -4868.19i q^{29} +8688.93 q^{31} -3944.00 q^{33} +974.819i q^{35} -3231.10i q^{37} +15092.5 q^{39} +8410.00 q^{41} -685.420i q^{43} +473.895i q^{45} +18147.2 q^{47} -16295.0 q^{49} +3320.48i q^{51} +26581.2i q^{53} +11155.3 q^{55} +30392.0 q^{57} +33798.8i q^{59} +27356.6i q^{61} +248.902 q^{63} -42688.0 q^{65} -66729.4i q^{67} -52042.2i q^{69} -21609.2 q^{71} +43006.0 q^{73} -19328.8i q^{75} -5859.06i q^{77} -64759.7 q^{79} -56255.0 q^{81} +70659.1i q^{83} -9391.73i q^{85} -74150.0 q^{87} +93870.0 q^{89} +22420.8i q^{91} -132346. i q^{93} -85961.6 q^{95} -81706.0 q^{97} -2848.30i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 44 q^{9}+O(q^{10}) $ 4 * q + 44 * q^9	$ 4 q + 44 q^{9} - 872 q^{17} + 5076 q^{25} - 15776 q^{33} + 33640 q^{41} - 65180 q^{49} + 121568 q^{57} - 170752 q^{65} + 172024 q^{73} - 225020 q^{81} + 375480 q^{89} - 326824 q^{97}+O(q^{100}) $ 4 * q + 44 * q^9 - 872 * q^17 + 5076 * q^25 - 15776 * q^33 + 33640 * q^41 - 65180 * q^49 + 121568 * q^57 - 170752 * q^65 + 172024 * q^73 - 225020 * q^81 + 375480 * q^89 - 326824 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/128\mathbb{Z}\right)^\times$.

$n$	$5$	$127$
$\chi(n)$	$-1$	$1$

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
$3$	− 15.2315i	− 0.977104i	−0.872535	−	0.488552i	$-0.837525\pi$
$3$	− 15.2315i	− 0.977104i	0.872535	−	0.488552i	$-0.162475\pi$
$4$	0	0
$5$	43.0813i	0.770662i	0.922778	+	0.385331i	$0.125913\pi$
$5$	43.0813i	0.770662i	−0.922778	+	0.385331i	$0.874087\pi$
$6$	0	0
$7$	22.6274	0.174538	0.0872690	−	0.996185i	$-0.472186\pi$
$7$	22.6274	0.174538	0.0872690	+	0.996185i	$0.472186\pi$
$8$	0	0
$9$	11.0000	0.0452675
$10$	0	0
$11$	− 258.936i	− 0.645225i	−0.946531	−	0.322613i	$-0.895439\pi$
$11$	− 258.936i	− 0.645225i	0.946531	−	0.322613i	$-0.104561\pi$
$12$	0	0
$13$	990.870i	1.62614i	0.582165	+	0.813071i	$0.302206\pi$
$13$	990.870i	1.62614i	−0.582165	+	0.813071i	$0.697794\pi$
$14$	0	0
$15$	656.195	0.753017
$16$	0	0
$17$	−218.000	−0.182951	−0.0914754	−	0.995807i	$-0.529158\pi$
$17$	−218.000	−0.182951	−0.0914754	+	0.995807i	$0.529158\pi$
$18$	0	0
$19$	1995.33i	1.26804i	0.773319	+	0.634018i	$0.218595\pi$
$19$	1995.33i	1.26804i	−0.773319	+	0.634018i	$0.781405\pi$
$20$	0	0
$21$	− 344.651i	− 0.170542i
$22$	0	0
$23$	3416.74	1.34677	0.673383	−	0.739294i	$-0.264840\pi$
$23$	3416.74	1.34677	0.673383	+	0.739294i	$0.264840\pi$
$24$	0	0
$25$	1269.00	0.406080
$26$	0	0
$27$	− 3868.81i	− 1.02134i
$28$	0	0
$29$	− 4868.19i	− 1.07491i	−0.843292	−	0.537455i	$-0.819386\pi$
$29$	− 4868.19i	− 1.07491i	0.843292	−	0.537455i	$-0.180614\pi$
$30$	0	0
$31$	8688.93	1.62391	0.811955	−	0.583720i	$-0.198403\pi$
$31$	8688.93	1.62391	0.811955	+	0.583720i	$0.198403\pi$
$32$	0	0
$33$	−3944.00	−0.630452
$34$	0	0
$35$	974.819i	0.134510i
$36$	0	0
$37$	− 3231.10i	− 0.388013i	−0.981000	−	0.194006i	$-0.937852\pi$
$37$	− 3231.10i	− 0.388013i	0.981000	−	0.194006i	$-0.0621482\pi$
$38$	0	0
$39$	15092.5	1.58891
$40$	0	0
$41$	8410.00	0.781333	0.390667	−	0.920532i	$-0.372245\pi$
$41$	8410.00	0.781333	0.390667	+	0.920532i	$0.372245\pi$
$42$	0	0
$43$	− 685.420i	− 0.0565308i	−0.999600	−	0.0282654i	$-0.991002\pi$
$43$	− 685.420i	− 0.0565308i	0.999600	−	0.0282654i	$-0.00899836\pi$
$44$	0	0
$45$	473.895i	0.0348859i
$46$	0	0
$47$	18147.2	1.19830	0.599149	−	0.800638i	$-0.295506\pi$
$47$	18147.2	1.19830	0.599149	+	0.800638i	$0.295506\pi$
$48$	0	0
$49$	−16295.0	−0.969537
$50$	0	0
$51$	3320.48i	0.178762i
$52$	0	0
$53$	26581.2i	1.29982i	0.760010	+	0.649912i	$0.225194\pi$
$53$	26581.2i	1.29982i	−0.760010	+	0.649912i	$0.774806\pi$
$54$	0	0
$55$	11155.3	0.497250
$56$	0	0
$57$	30392.0	1.23900
$58$	0	0
$59$	33798.8i	1.26407i	0.774940	+	0.632035i	$0.217780\pi$
$59$	33798.8i	1.26407i	−0.774940	+	0.632035i	$0.782220\pi$
$60$	0	0
$61$	27356.6i	0.941322i	0.882314	+	0.470661i	$0.155984\pi$
$61$	27356.6i	0.941322i	−0.882314	+	0.470661i	$0.844016\pi$
$62$	0	0
$63$	248.902	0.00790090
$64$	0	0
$65$	−42688.0	−1.25321
$66$	0	0
$67$	− 66729.4i	− 1.81606i	−0.418905	−	0.908030i	$-0.637586\pi$
$67$	− 66729.4i	− 1.81606i	0.418905	−	0.908030i	$-0.362414\pi$
$68$	0	0
$69$	− 52042.2i	− 1.31593i
$70$	0	0
$71$	−21609.2	−0.508736	−0.254368	−	0.967108i	$-0.581867\pi$
$71$	−21609.2	−0.508736	−0.254368	+	0.967108i	$0.581867\pi$
$72$	0	0
$73$	43006.0	0.944543	0.472272	−	0.881453i	$-0.343434\pi$
$73$	43006.0	0.944543	0.472272	+	0.881453i	$0.343434\pi$
$74$	0	0
$75$	− 19328.8i	− 0.396782i
$76$	0	0
$77$	− 5859.06i	− 0.112616i
$78$	0	0
$79$	−64759.7	−1.16745	−0.583723	−	0.811953i	$-0.698405\pi$
$79$	−64759.7	−1.16745	−0.583723	+	0.811953i	$0.698405\pi$
$80$	0	0
$81$	−56255.0	−0.952683
$82$	0	0
$83$	70659.1i	1.12583i	0.826515	+	0.562915i	$0.190320\pi$
$83$	70659.1i	1.12583i	−0.826515	+	0.562915i	$0.809680\pi$
$84$	0	0
$85$	− 9391.73i	− 0.140993i
$86$	0	0
$87$	−74150.0	−1.05030
$88$	0	0
$89$	93870.0	1.25618	0.628090	−	0.778141i	$-0.283837\pi$
$89$	93870.0	1.25618	0.628090	+	0.778141i	$0.283837\pi$
$90$	0	0
$91$	22420.8i	0.283823i
$92$	0	0
$93$	− 132346.i	− 1.58673i
$94$	0	0
$95$	−85961.6	−0.977227
$96$	0	0
$97$	−81706.0	−0.881708	−0.440854	−	0.897579i	$-0.645324\pi$
$97$	−81706.0	−0.881708	−0.440854	+	0.897579i	$0.645324\pi$
$98$	0	0
$99$	− 2848.30i	− 0.0292077i
$100$	0	0
$101$	87412.0i	0.852644i	0.904572	+	0.426322i	$0.140191\pi$
$101$	87412.0i	0.852644i	−0.904572	+	0.426322i	$0.859809\pi$
$102$	0	0
$103$	−132665.	−1.23214	−0.616072	−	0.787689i	$-0.711277\pi$
$103$	−132665.	−1.23214	−0.616072	+	0.787689i	$0.711277\pi$
$104$	0	0
$105$	14848.0	0.131430
$106$	0	0
$107$	− 87870.8i	− 0.741968i	−0.928639	−	0.370984i	$-0.879020\pi$
$107$	− 87870.8i	− 0.741968i	0.928639	−	0.370984i	$-0.120980\pi$
$108$	0	0
$109$	121791.i	0.981858i	0.871200	+	0.490929i	$0.163343\pi$
$109$	121791.i	0.981858i	−0.871200	+	0.490929i	$0.836657\pi$
$110$	0	0
$111$	−49214.6	−0.379129
$112$	0	0
$113$	−29614.0	−0.218173	−0.109086	−	0.994032i	$-0.534793\pi$
$113$	−29614.0	−0.218173	−0.109086	+	0.994032i	$0.534793\pi$
$114$	0	0
$115$	147198.i	1.03790i
$116$	0	0
$117$	10899.6i	0.0736114i
$118$	0	0
$119$	−4932.78	−0.0319318
$120$	0	0
$121$	94003.0	0.583685
$122$	0	0
$123$	− 128097.i	− 0.763444i
$124$	0	0
$125$	189299.i	1.08361i
$126$	0	0
$127$	−60551.0	−0.333128	−0.166564	−	0.986031i	$-0.553267\pi$
$127$	−60551.0	−0.333128	−0.166564	+	0.986031i	$0.553267\pi$
$128$	0	0
$129$	−10440.0	−0.0552365
$130$	0	0
$131$	− 186754.i	− 0.950805i	−0.879768	−	0.475403i	$-0.842302\pi$
$131$	− 186754.i	− 0.950805i	0.879768	−	0.475403i	$-0.157698\pi$
$132$	0	0
$133$	45149.2i	0.221320i
$134$	0	0
$135$	166674.	0.787104
$136$	0	0
$137$	−182054.	−0.828703	−0.414351	−	0.910117i	$-0.635992\pi$
$137$	−182054.	−0.828703	−0.414351	+	0.910117i	$0.635992\pi$
$138$	0	0
$139$	45131.1i	0.198125i	0.995081	+	0.0990624i	$0.0315843\pi$
$139$	45131.1i	0.198125i	−0.995081	+	0.0990624i	$0.968416\pi$
$140$	0	0
$141$	− 276410.i	− 1.17086i
$142$	0	0
$143$	256572.	1.04923
$144$	0	0
$145$	209728.	0.828393
$146$	0	0
$147$	248198.i	0.947338i
$148$	0	0
$149$	− 262408.i	− 0.968304i	−0.874984	−	0.484152i	$-0.839128\pi$
$149$	− 262408.i	− 0.968304i	0.874984	−	0.484152i	$-0.160872\pi$
$150$	0	0
$151$	113839.	0.406300	0.203150	−	0.979148i	$-0.434882\pi$
$151$	113839.	0.406300	0.203150	+	0.979148i	$0.434882\pi$
$152$	0	0
$153$	−2398.00	−0.00828172
$154$	0	0
$155$	374330.i	1.25149i
$156$	0	0
$157$	114898.i	0.372017i	0.982548	+	0.186009i	$0.0595552\pi$
$157$	114898.i	0.372017i	−0.982548	+	0.186009i	$0.940445\pi$
$158$	0	0
$159$	404872.	1.27006
$160$	0	0
$161$	77312.0	0.235062
$162$	0	0
$163$	− 284784.i	− 0.839551i	−0.907628	−	0.419775i	$-0.862109\pi$
$163$	− 284784.i	− 0.839551i	0.907628	−	0.419775i	$-0.137891\pi$
$164$	0	0
$165$	− 169913.i	− 0.485865i
$166$	0	0
$167$	425871.	1.18164	0.590822	−	0.806802i	$-0.298804\pi$
$167$	425871.	1.18164	0.590822	+	0.806802i	$0.298804\pi$
$168$	0	0
$169$	−610531.	−1.64434
$170$	0	0
$171$	21948.7i	0.0574008i
$172$	0	0
$173$	− 165131.i	− 0.419481i	−0.977757	−	0.209741i	$-0.932738\pi$
$173$	− 165131.i	− 0.419481i	0.977757	−	0.209741i	$-0.0672620\pi$
$174$	0	0
$175$	28714.2	0.0708764
$176$	0	0
$177$	514808.	1.23513
$178$	0	0
$179$	− 724671.i	− 1.69047i	−0.534392	−	0.845237i	$-0.679459\pi$
$179$	− 724671.i	− 1.69047i	0.534392	−	0.845237i	$-0.320541\pi$
$180$	0	0
$181$	45709.3i	0.103707i	0.998655	+	0.0518535i	$0.0165129\pi$
$181$	45709.3i	0.103707i	−0.998655	+	0.0518535i	$0.983487\pi$
$182$	0	0
$183$	416684.	0.919769
$184$	0	0
$185$	139200.	0.299027
$186$	0	0
$187$	56448.1i	0.118044i
$188$	0	0
$189$	− 87541.2i	− 0.178262i
$190$	0	0
$191$	434537.	0.861873	0.430936	−	0.902382i	$-0.358183\pi$
$191$	434537.	0.861873	0.430936	+	0.902382i	$0.358183\pi$
$192$	0	0
$193$	277878.	0.536983	0.268492	−	0.963282i	$-0.413475\pi$
$193$	277878.	0.536983	0.268492	+	0.963282i	$0.413475\pi$
$194$	0	0
$195$	650204.i	1.22451i
$196$	0	0
$197$	1.00539e6i	1.84573i	0.385122	+	0.922866i	$0.374159\pi$
$197$	1.00539e6i	1.84573i	−0.385122	+	0.922866i	$0.625841\pi$
$198$	0	0
$199$	−477506.	−0.854765	−0.427382	−	0.904071i	$-0.640564\pi$
$199$	−477506.	−0.854765	−0.427382	+	0.904071i	$0.640564\pi$
$200$	0	0
$201$	−1.01639e6	−1.77448
$202$	0	0
$203$	− 110155.i	− 0.187613i
$204$	0	0
$205$	362314.i	0.602144i
$206$	0	0
$207$	37584.1	0.0609647
$208$	0	0
$209$	516664.	0.818168
$210$	0	0
$211$	18871.9i	0.0291816i	0.999894	+	0.0145908i	$0.00464456\pi$
$211$	18871.9i	0.0291816i	−0.999894	+	0.0145908i	$0.995355\pi$
$212$	0	0
$213$	329141.i	0.497088i
$214$	0	0
$215$	29528.8	0.0435662
$216$	0	0
$217$	196608.	0.283434
$218$	0	0
$219$	− 655048.i	− 0.922917i
$220$	0	0
$221$	− 216010.i	− 0.297504i
$222$	0	0
$223$	−1.23636e6	−1.66488	−0.832441	−	0.554114i	$-0.813057\pi$
$223$	−1.23636e6	−1.66488	−0.832441	+	0.554114i	$0.813057\pi$
$224$	0	0
$225$	13959.0	0.0183822
$226$	0	0
$227$	802261.i	1.03336i	0.856179	+	0.516679i	$0.172832\pi$
$227$	802261.i	1.03336i	−0.856179	+	0.516679i	$0.827168\pi$
$228$	0	0
$229$	− 1.39579e6i	− 1.75886i	−0.476026	−	0.879431i	$-0.657923\pi$
$229$	− 1.39579e6i	− 1.75886i	0.476026	−	0.879431i	$-0.342077\pi$
$230$	0	0
$231$	−89242.5	−0.110038
$232$	0	0
$233$	−812898.	−0.980949	−0.490474	−	0.871456i	$-0.663176\pi$
$233$	−812898.	−0.980949	−0.490474	+	0.871456i	$0.663176\pi$
$234$	0	0
$235$	781805.i	0.923483i
$236$	0	0
$237$	986390.i	1.14072i
$238$	0	0
$239$	−403537.	−0.456971	−0.228486	−	0.973547i	$-0.573377\pi$
$239$	−403537.	−0.456971	−0.228486	+	0.973547i	$0.573377\pi$
$240$	0	0
$241$	560262.	0.621368	0.310684	−	0.950513i	$-0.399442\pi$
$241$	560262.	0.621368	0.310684	+	0.950513i	$0.399442\pi$
$242$	0	0
$243$	− 83270.9i	− 0.0904643i
$244$	0	0
$245$	− 702010.i	− 0.747185i
$246$	0	0
$247$	−1.97712e6	−2.06201
$248$	0	0
$249$	1.07625e6	1.10005
$250$	0	0
$251$	− 901784.i	− 0.903479i	−0.892150	−	0.451740i	$-0.850804\pi$
$251$	− 901784.i	− 0.903479i	0.892150	−	0.451740i	$-0.149196\pi$
$252$	0	0
$253$	− 884718.i	− 0.868968i
$254$	0	0
$255$	−143051.	−0.137765
$256$	0	0
$257$	−870558.	−0.822176	−0.411088	−	0.911596i	$-0.634851\pi$
$257$	−870558.	−0.822176	−0.411088	+	0.911596i	$0.634851\pi$
$258$	0	0
$259$	− 73111.4i	− 0.0677229i
$260$	0	0
$261$	− 53550.1i	− 0.0486585i
$262$	0	0
$263$	−231954.	−0.206782	−0.103391	−	0.994641i	$-0.532969\pi$
$263$	−231954.	−0.206782	−0.103391	+	0.994641i	$0.532969\pi$
$264$	0	0
$265$	−1.14515e6	−1.00172
$266$	0	0
$267$	− 1.42979e6i	− 1.22742i
$268$	0	0
$269$	− 1.43396e6i	− 1.20825i	−0.796890	−	0.604125i	$-0.793523\pi$
$269$	− 1.43396e6i	− 1.20825i	0.796890	−	0.604125i	$-0.206477\pi$
$270$	0	0
$271$	−1.59627e6	−1.32034	−0.660168	−	0.751118i	$-0.729515\pi$
$271$	−1.59627e6	−1.32034	−0.660168	+	0.751118i	$0.729515\pi$
$272$	0	0
$273$	341504.	0.277325
$274$	0	0
$275$	− 328590.i	− 0.262013i
$276$	0	0
$277$	1.38278e6i	1.08281i	0.840761	+	0.541407i	$0.182108\pi$
$277$	1.38278e6i	1.08281i	−0.840761	+	0.541407i	$0.817892\pi$
$278$	0	0
$279$	95578.2	0.0735103
$280$	0	0
$281$	2.39301e6	1.80791	0.903957	−	0.427622i	$-0.140649\pi$
$281$	2.39301e6	1.80791	0.903957	+	0.427622i	$0.140649\pi$
$282$	0	0
$283$	597671.i	0.443604i	0.975092	+	0.221802i	$0.0711939\pi$
$283$	597671.i	0.443604i	−0.975092	+	0.221802i	$0.928806\pi$
$284$	0	0
$285$	1.30933e6i	0.954852i
$286$	0	0
$287$	190297.	0.136372
$288$	0	0
$289$	−1.37233e6	−0.966529
$290$	0	0
$291$	1.24451e6i	0.861520i
$292$	0	0
$293$	− 874680.i	− 0.595224i	−0.954687	−	0.297612i	$-0.903810\pi$
$293$	− 874680.i	− 0.595224i	0.954687	−	0.297612i	$-0.0961901\pi$
$294$	0	0
$295$	−1.45610e6	−0.974171
$296$	0	0
$297$	−1.00178e6	−0.658991
$298$	0	0
$299$	3.38555e6i	2.19003i
$300$	0	0
$301$	− 15509.3i	− 0.00986678i
$302$	0	0
$303$	1.33142e6	0.833122
$304$	0	0
$305$	−1.17856e6	−0.725441
$306$	0	0
$307$	− 2.70313e6i	− 1.63689i	−0.574582	−	0.818447i	$-0.694835\pi$
$307$	− 2.70313e6i	− 1.63689i	0.574582	−	0.818447i	$-0.305165\pi$
$308$	0	0
$309$	2.02069e6i	1.20393i
$310$	0	0
$311$	2.49642e6	1.46358	0.731789	−	0.681531i	$-0.238686\pi$
$311$	2.49642e6	1.46358	0.731789	+	0.681531i	$0.238686\pi$
$312$	0	0
$313$	2.92331e6	1.68661	0.843303	−	0.537439i	$-0.180608\pi$
$313$	2.92331e6	1.68661	0.843303	+	0.537439i	$0.180608\pi$
$314$	0	0
$315$	10723.0i	0.00608892i
$316$	0	0
$317$	1.61671e6i	0.903618i	0.892115	+	0.451809i	$0.149221\pi$
$317$	1.61671e6i	0.903618i	−0.892115	+	0.451809i	$0.850779\pi$
$318$	0	0
$319$	−1.26055e6	−0.693559
$320$	0	0
$321$	−1.33841e6	−0.724980
$322$	0	0
$323$	− 434982.i	− 0.231988i
$324$	0	0
$325$	1.25741e6i	0.660344i
$326$	0	0
$327$	1.85506e6	0.959377
$328$	0	0
$329$	410624.	0.209148
$330$	0	0
$331$	− 1.14774e6i	− 0.575804i	−0.957660	−	0.287902i	$-0.907042\pi$
$331$	− 1.14774e6i	− 0.575804i	0.957660	−	0.287902i	$-0.0929577\pi$
$332$	0	0
$333$	− 35542.1i	− 0.0175644i
$334$	0	0
$335$	2.87479e6	1.39957
$336$	0	0
$337$	371378.	0.178132	0.0890659	−	0.996026i	$-0.471612\pi$
$337$	371378.	0.178132	0.0890659	+	0.996026i	$0.471612\pi$
$338$	0	0
$339$	451067.i	0.213178i
$340$	0	0
$341$	− 2.24988e6i	− 1.04779i
$342$	0	0
$343$	−749013.	−0.343759
$344$	0	0
$345$	2.24205e6	1.01414
$346$	0	0
$347$	1.13912e6i	0.507863i	0.967222	+	0.253931i	$0.0817238\pi$
$347$	1.13912e6i	0.507863i	−0.967222	+	0.253931i	$0.918276\pi$
$348$	0	0
$349$	− 4.08105e6i	− 1.79353i	−0.442508	−	0.896764i	$-0.645911\pi$
$349$	− 4.08105e6i	− 1.79353i	0.442508	−	0.896764i	$-0.354089\pi$
$350$	0	0
$351$	3.83349e6	1.66084
$352$	0	0
$353$	−3.90953e6	−1.66989	−0.834946	−	0.550332i	$-0.814501\pi$
$353$	−3.90953e6	−1.66989	−0.834946	+	0.550332i	$0.814501\pi$
$354$	0	0
$355$	− 930952.i	− 0.392064i
$356$	0	0
$357$	75133.8i	0.0312007i
$358$	0	0
$359$	2.62028e6	1.07303	0.536514	−	0.843891i	$-0.319741\pi$
$359$	2.62028e6	1.07303	0.536514	+	0.843891i	$0.319741\pi$
$360$	0	0
$361$	−1.50525e6	−0.607913
$362$	0	0
$363$	− 1.43181e6i	− 0.570321i
$364$	0	0
$365$	1.85276e6i	0.727924i
$366$	0	0
$367$	−1.65266e6	−0.640499	−0.320250	−	0.947333i	$-0.603767\pi$
$367$	−1.65266e6	−0.640499	−0.320250	+	0.947333i	$0.603767\pi$
$368$	0	0
$369$	92510.0	0.0353690
$370$	0	0
$371$	601463.i	0.226869i
$372$	0	0
$373$	− 3.09630e6i	− 1.15231i	−0.817339	−	0.576157i	$-0.804552\pi$
$373$	− 3.09630e6i	− 1.15231i	0.817339	−	0.576157i	$-0.195448\pi$
$374$	0	0
$375$	2.88332e6	1.05880
$376$	0	0
$377$	4.82374e6	1.74796
$378$	0	0
$379$	− 647478.i	− 0.231540i	−0.993276	−	0.115770i	$-0.963066\pi$
$379$	− 647478.i	− 0.231540i	0.993276	−	0.115770i	$-0.0369336\pi$
$380$	0	0
$381$	922285.i	0.325501i
$382$	0	0
$383$	2.42168e6	0.843566	0.421783	−	0.906697i	$-0.361404\pi$
$383$	2.42168e6	0.843566	0.421783	+	0.906697i	$0.361404\pi$
$384$	0	0
$385$	252416.	0.0867891
$386$	0	0
$387$	− 7539.62i	− 0.00255901i
$388$	0	0
$389$	− 1.06441e6i	− 0.356644i	−0.983972	−	0.178322i	$-0.942933\pi$
$389$	− 1.06441e6i	− 0.356644i	0.983972	−	0.178322i	$-0.0570669\pi$
$390$	0	0
$391$	−744849.	−0.246392
$392$	0	0
$393$	−2.84455e6	−0.929036
$394$	0	0
$395$	− 2.78993e6i	− 0.899706i
$396$	0	0
$397$	734235.i	0.233808i	0.993143	+	0.116904i	$0.0372969\pi$
$397$	734235.i	0.233808i	−0.993143	+	0.116904i	$0.962703\pi$
$398$	0	0
$399$	687692.	0.216253
$400$	0	0
$401$	−448218.	−0.139197	−0.0695983	−	0.997575i	$-0.522172\pi$
$401$	−448218.	−0.139197	−0.0695983	+	0.997575i	$0.522172\pi$
$402$	0	0
$403$	8.60960e6i	2.64071i
$404$	0	0
$405$	− 2.42354e6i	− 0.734197i
$406$	0	0
$407$	−836649.	−0.250356
$408$	0	0
$409$	−84854.0	−0.0250821	−0.0125411	−	0.999921i	$-0.503992\pi$
$409$	−84854.0	−0.0250821	−0.0125411	+	0.999921i	$0.503992\pi$
$410$	0	0
$411$	2.77296e6i	0.809729i
$412$	0	0
$413$	764780.i	0.220628i
$414$	0	0
$415$	−3.04409e6	−0.867635
$416$	0	0
$417$	687416.	0.193588
$418$	0	0
$419$	− 2.75650e6i	− 0.767048i	−0.923531	−	0.383524i	$-0.874710\pi$
$419$	− 2.75650e6i	− 0.767048i	0.923531	−	0.383524i	$-0.125290\pi$
$420$	0	0
$421$	6.94647e6i	1.91011i	0.296423	+	0.955057i	$0.404206\pi$
$421$	6.94647e6i	1.91011i	−0.296423	+	0.955057i	$0.595794\pi$
$422$	0	0
$423$	199619.	0.0542439
$424$	0	0
$425$	−276642.	−0.0742926
$426$	0	0
$427$	619010.i	0.164296i
$428$	0	0
$429$	− 3.90799e6i	− 1.02520i
$430$	0	0
$431$	1.08227e6	0.280635	0.140318	−	0.990107i	$-0.455188\pi$
$431$	1.08227e6	0.280635	0.140318	+	0.990107i	$0.455188\pi$
$432$	0	0
$433$	2.26093e6	0.579520	0.289760	−	0.957099i	$-0.406425\pi$
$433$	2.26093e6	0.579520	0.289760	+	0.957099i	$0.406425\pi$
$434$	0	0
$435$	− 3.19448e6i	− 0.809426i
$436$	0	0
$437$	6.81753e6i	1.70775i
$438$	0	0
$439$	−6.16645e6	−1.52712	−0.763561	−	0.645736i	$-0.776551\pi$
$439$	−6.16645e6	−1.52712	−0.763561	+	0.645736i	$0.776551\pi$
$440$	0	0
$441$	−179245.	−0.0438885
$442$	0	0
$443$	− 6.61587e6i	− 1.60169i	−0.598874	−	0.800843i	$-0.704385\pi$
$443$	− 6.61587e6i	− 1.60169i	0.598874	−	0.800843i	$-0.295615\pi$
$444$	0	0
$445$	4.04404e6i	0.968090i
$446$	0	0
$447$	−3.99688e6	−0.946134
$448$	0	0
$449$	−807882.	−0.189118	−0.0945588	−	0.995519i	$-0.530144\pi$
$449$	−807882.	−0.189118	−0.0945588	+	0.995519i	$0.530144\pi$
$450$	0	0
$451$	− 2.17765e6i	− 0.504136i
$452$	0	0
$453$	− 1.73394e6i	− 0.396998i
$454$	0	0
$455$	−965919.	−0.218732
$456$	0	0
$457$	−6.20353e6	−1.38947	−0.694733	−	0.719267i	$-0.744478\pi$
$457$	−6.20353e6	−1.38947	−0.694733	+	0.719267i	$0.744478\pi$
$458$	0	0
$459$	843401.i	0.186854i
$460$	0	0
$461$	− 1.93853e6i	− 0.424835i	−0.977179	−	0.212417i	$-0.931866\pi$
$461$	− 1.93853e6i	− 0.424835i	0.977179	−	0.212417i	$-0.0681337\pi$
$462$	0	0
$463$	−6.96413e6	−1.50978	−0.754891	−	0.655850i	$-0.772310\pi$
$463$	−6.96413e6	−1.50978	−0.754891	+	0.655850i	$0.772310\pi$
$464$	0	0
$465$	5.70163e6	1.22283
$466$	0	0
$467$	4.65298e6i	0.987277i	0.869667	+	0.493638i	$0.164333\pi$
$467$	4.65298e6i	0.987277i	−0.869667	+	0.493638i	$0.835667\pi$
$468$	0	0
$469$	− 1.50991e6i	− 0.316971i
$470$	0	0
$471$	1.75007e6	0.363499
$472$	0	0
$473$	−177480.	−0.0364751
$474$	0	0
$475$	2.53208e6i	0.514924i
$476$	0	0
$477$	292393.i	0.0588398i
$478$	0	0
$479$	−2.10743e6	−0.419676	−0.209838	−	0.977736i	$-0.567294\pi$
$479$	−2.10743e6	−0.419676	−0.209838	+	0.977736i	$0.567294\pi$
$480$	0	0
$481$	3.20160e6	0.630964
$482$	0	0
$483$	− 1.17758e6i	− 0.229680i
$484$	0	0
$485$	− 3.52000e6i	− 0.679499i
$486$	0	0
$487$	−9.02334e6	−1.72403	−0.862015	−	0.506883i	$-0.830798\pi$
$487$	−9.02334e6	−1.72403	−0.862015	+	0.506883i	$0.830798\pi$
$488$	0	0
$489$	−4.33770e6	−0.820328
$490$	0	0
$491$	2.69262e6i	0.504047i	0.967721	+	0.252023i	$0.0810960\pi$
$491$	2.69262e6i	0.504047i	−0.967721	+	0.252023i	$0.918904\pi$
$492$	0	0
$493$	1.06127e6i	0.196656i
$494$	0	0
$495$	122708.	0.0225093
$496$	0	0
$497$	−488960.	−0.0887937
$498$	0	0
$499$	− 4.32456e6i	− 0.777482i	−0.921347	−	0.388741i	$-0.872910\pi$
$499$	− 4.32456e6i	− 0.777482i	0.921347	−	0.388741i	$-0.127090\pi$
$500$	0	0
$501$	− 6.48667e6i	− 1.15459i
$502$	0	0
$503$	−2.58525e6	−0.455599	−0.227799	−	0.973708i	$-0.573153\pi$
$503$	−2.58525e6	−0.455599	−0.227799	+	0.973708i	$0.573153\pi$
$504$	0	0
$505$	−3.76582e6	−0.657100
$506$	0	0
$507$	9.29933e6i	1.60669i
$508$	0	0
$509$	− 4.02746e6i	− 0.689027i	−0.938781	−	0.344514i	$-0.888044\pi$
$509$	− 4.02746e6i	− 0.689027i	0.938781	−	0.344514i	$-0.111956\pi$
$510$	0	0
$511$	973115.	0.164859
$512$	0	0
$513$	7.71957e6	1.29509
$514$	0	0
$515$	− 5.71536e6i	− 0.949567i
$516$	0	0
$517$	− 4.69897e6i	− 0.773172i
$518$	0	0
$519$	−2.51520e6	−0.409877
$520$	0	0
$521$	2.11324e6	0.341079	0.170539	−	0.985351i	$-0.445449\pi$
$521$	2.11324e6	0.341079	0.170539	+	0.985351i	$0.445449\pi$
$522$	0	0
$523$	− 2.72110e6i	− 0.435001i	−0.976060	−	0.217501i	$-0.930210\pi$
$523$	− 2.72110e6i	− 0.435001i	0.976060	−	0.217501i	$-0.0697904\pi$
$524$	0	0
$525$	− 437362.i	− 0.0692536i
$526$	0	0
$527$	−1.89419e6	−0.297096
$528$	0	0
$529$	5.23777e6	0.813780
$530$	0	0
$531$	371787.i	0.0572213i
$532$	0	0
$533$	8.33322e6i	1.27056i
$534$	0	0
$535$	3.78559e6	0.571806
$536$	0	0
$537$	−1.10379e7	−1.65177
$538$	0	0
$539$	4.21937e6i	0.625569i
$540$	0	0
$541$	1.05683e6i	0.155243i	0.996983	+	0.0776213i	$0.0247325\pi$
$541$	1.05683e6i	0.155243i	−0.996983	+	0.0776213i	$0.975267\pi$
$542$	0	0
$543$	696223.	0.101333
$544$	0	0
$545$	−5.24691e6	−0.756680
$546$	0	0
$547$	3.14920e6i	0.450020i	0.974356	+	0.225010i	$0.0722415\pi$
$547$	3.14920e6i	0.450020i	−0.974356	+	0.225010i	$0.927759\pi$
$548$	0	0
$549$	300923.i	0.0426113i
$550$	0	0
$551$	9.71366e6	1.36302
$552$	0	0
$553$	−1.46534e6	−0.203764
$554$	0	0
$555$	− 2.12023e6i	− 0.292180i
$556$	0	0
$557$	3.91932e6i	0.535270i	0.963520	+	0.267635i	$0.0862422\pi$
$557$	3.91932e6i	0.535270i	−0.963520	+	0.267635i	$0.913758\pi$
$558$	0	0
$559$	679162.	0.0919272
$560$	0	0
$561$	859792.	0.115342
$562$	0	0
$563$	2.62499e6i	0.349025i	0.984655	+	0.174513i	$0.0558349\pi$
$563$	2.62499e6i	0.349025i	−0.984655	+	0.174513i	$0.944165\pi$
$564$	0	0
$565$	− 1.27581e6i	− 0.168138i
$566$	0	0
$567$	−1.27291e6	−0.166279
$568$	0	0
$569$	6.56020e6	0.849448	0.424724	−	0.905323i	$-0.360371\pi$
$569$	6.56020e6	0.849448	0.424724	+	0.905323i	$0.360371\pi$
$570$	0	0
$571$	− 1.14800e7i	− 1.47350i	−0.676165	−	0.736751i	$-0.736359\pi$
$571$	− 1.14800e7i	− 1.47350i	0.676165	−	0.736751i	$-0.263641\pi$
$572$	0	0
$573$	− 6.61867e6i	− 0.842140i
$574$	0	0
$575$	4.33584e6	0.546895
$576$	0	0
$577$	1.04493e7	1.30662	0.653308	−	0.757092i	$-0.273381\pi$
$577$	1.04493e7	1.30662	0.653308	+	0.757092i	$0.273381\pi$
$578$	0	0
$579$	− 4.23251e6i	− 0.524689i
$580$	0	0
$581$	1.59883e6i	0.196500i
$582$	0	0
$583$	6.88283e6	0.838679
$584$	0	0
$585$	−469568.	−0.0567295
$586$	0	0
$587$	− 6.24714e6i	− 0.748318i	−0.927365	−	0.374159i	$-0.877931\pi$
$587$	− 6.24714e6i	− 0.748318i	0.927365	−	0.374159i	$-0.122069\pi$
$588$	0	0
$589$	1.73373e7i	2.05917i
$590$	0	0
$591$	1.53136e7	1.80347
$592$	0	0
$593$	−6.31288e6	−0.737209	−0.368605	−	0.929586i	$-0.620164\pi$
$593$	−6.31288e6	−0.737209	−0.368605	+	0.929586i	$0.620164\pi$
$594$	0	0
$595$	− 212511.i	− 0.0246087i
$596$	0	0
$597$	7.27316e6i	0.835194i
$598$	0	0
$599$	1.21797e6	0.138697	0.0693487	−	0.997592i	$-0.477908\pi$
$599$	1.21797e6	0.138697	0.0693487	+	0.997592i	$0.477908\pi$
$600$	0	0
$601$	−1.05226e7	−1.18833	−0.594167	−	0.804342i	$-0.702518\pi$
$601$	−1.05226e7	−1.18833	−0.594167	+	0.804342i	$0.702518\pi$
$602$	0	0
$603$	− 734023.i	− 0.0822085i
$604$	0	0
$605$	4.04977e6i	0.449824i
$606$	0	0
$607$	2.18789e6	0.241020	0.120510	−	0.992712i	$-0.461547\pi$
$607$	2.18789e6	0.241020	0.120510	+	0.992712i	$0.461547\pi$
$608$	0	0
$609$	−1.67782e6	−0.183317
$610$	0	0
$611$	1.79815e7i	1.94860i
$612$	0	0
$613$	3.98593e6i	0.428428i	0.976787	+	0.214214i	$0.0687190\pi$
$613$	3.98593e6i	0.428428i	−0.976787	+	0.214214i	$0.931281\pi$
$614$	0	0
$615$	5.51860e6	0.588357
$616$	0	0
$617$	1.42965e7	1.51188	0.755938	−	0.654643i	$-0.227181\pi$
$617$	1.42965e7	1.51188	0.755938	+	0.654643i	$0.227181\pi$
$618$	0	0
$619$	1.32078e7i	1.38549i	0.721182	+	0.692746i	$0.243599\pi$
$619$	1.32078e7i	1.38549i	−0.721182	+	0.692746i	$0.756401\pi$
$620$	0	0
$621$	− 1.32187e7i	− 1.37550i
$622$	0	0
$623$	2.12404e6	0.219251
$624$	0	0
$625$	−4.18964e6	−0.429019
$626$	0	0
$627$	− 7.86959e6i	− 0.799435i
$628$	0	0
$629$	704380.i	0.0709872i
$630$	0	0
$631$	1.51047e7	1.51021	0.755107	−	0.655602i	$-0.227585\pi$
$631$	1.51047e7	1.51021	0.755107	+	0.655602i	$0.227585\pi$
$632$	0	0
$633$	287448.	0.0285135
$634$	0	0
$635$	− 2.60862e6i	− 0.256729i
$636$	0	0
$637$	− 1.61462e7i	− 1.57660i
$638$	0	0
$639$	−237701.	−0.0230292
$640$	0	0
$641$	−6.88071e6	−0.661437	−0.330718	−	0.943729i	$-0.607291\pi$
$641$	−6.88071e6	−0.661437	−0.330718	+	0.943729i	$0.607291\pi$
$642$	0	0
$643$	1.44561e7i	1.37887i	0.724346	+	0.689437i	$0.242142\pi$
$643$	1.44561e7i	1.37887i	−0.724346	+	0.689437i	$0.757858\pi$
$644$	0	0
$645$	− 449769.i	− 0.0425687i
$646$	0	0
$647$	−4.83310e6	−0.453905	−0.226953	−	0.973906i	$-0.572876\pi$
$647$	−4.83310e6	−0.453905	−0.226953	+	0.973906i	$0.572876\pi$
$648$	0	0
$649$	8.75174e6	0.815610
$650$	0	0
$651$	− 2.99464e6i	− 0.276944i
$652$	0	0
$653$	− 8.22289e6i	− 0.754643i	−0.926082	−	0.377321i	$-0.876845\pi$
$653$	− 8.22289e6i	− 0.754643i	0.926082	−	0.377321i	$-0.123155\pi$
$654$	0	0
$655$	8.04561e6	0.732750
$656$	0	0
$657$	473066.	0.0427571
$658$	0	0
$659$	− 5.93447e6i	− 0.532315i	−0.963930	−	0.266157i	$-0.914246\pi$
$659$	− 5.93447e6i	− 0.532315i	0.963930	−	0.266157i	$-0.0857541\pi$
$660$	0	0
$661$	− 6.07425e6i	− 0.540741i	−0.962756	−	0.270370i	$-0.912854\pi$
$661$	− 6.07425e6i	− 0.540741i	0.962756	−	0.270370i	$-0.0871461\pi$
$662$	0	0
$663$	−3.29016e6	−0.290692
$664$	0	0
$665$	−1.94509e6	−0.170563
$666$	0	0
$667$	− 1.66333e7i	− 1.44765i
$668$	0	0
$669$	1.88317e7i	1.62676i
$670$	0	0
$671$	7.08363e6	0.607364
$672$	0	0
$673$	−4.01847e6	−0.341998	−0.170999	−	0.985271i	$-0.554699\pi$
$673$	−4.01847e6	−0.341998	−0.170999	+	0.985271i	$0.554699\pi$
$674$	0	0
$675$	− 4.90952e6i	− 0.414744i
$676$	0	0
$677$	− 1.84827e7i	− 1.54986i	−0.632044	−	0.774932i	$-0.717784\pi$
$677$	− 1.84827e7i	− 1.54986i	0.632044	−	0.774932i	$-0.282216\pi$
$678$	0	0
$679$	−1.84880e6	−0.153891
$680$	0	0
$681$	1.22197e7	1.00970
$682$	0	0
$683$	1.40610e6i	0.115336i	0.998336	+	0.0576679i	$0.0183665\pi$
$683$	1.40610e6i	0.115336i	−0.998336	+	0.0576679i	$0.981634\pi$
$684$	0	0
$685$	− 7.84313e6i	− 0.638650i
$686$	0	0
$687$	−2.12601e7	−1.71859
$688$	0	0
$689$	−2.63385e7	−2.11370
$690$	0	0
$691$	− 7.01640e6i	− 0.559009i	−0.960144	−	0.279505i	$-0.909830\pi$
$691$	− 7.01640e6i	− 0.559009i	0.960144	−	0.279505i	$-0.0901702\pi$
$692$	0	0
$693$	− 64449.7i	− 0.00509786i
$694$	0	0
$695$	−1.94431e6	−0.152687
$696$	0	0
$697$	−1.83338e6	−0.142946
$698$	0	0
$699$	1.23817e7i	0.958489i
$700$	0	0
$701$	1.28008e7i	0.983879i	0.870629	+	0.491940i	$0.163712\pi$
$701$	1.28008e7i	0.983879i	−0.870629	+	0.491940i	$0.836288\pi$
$702$	0	0
$703$	6.44712e6	0.492014
$704$	0	0
$705$	1.19081e7	0.902339
$706$	0	0
$707$	1.97791e6i	0.148819i
$708$	0	0
$709$	− 3.05442e6i	− 0.228199i	−0.993469	−	0.114099i	$-0.963602\pi$
$709$	− 3.05442e6i	− 0.228199i	0.993469	−	0.114099i	$-0.0363982\pi$
$710$	0	0
$711$	−712356.	−0.0528474
$712$	0	0
$713$	2.96878e7	2.18703
$714$	0	0
$715$	1.10535e7i	0.808600i
$716$	0	0
$717$	6.14650e6i	0.446509i
$718$	0	0
$719$	−1.99890e7	−1.44201	−0.721006	−	0.692928i	$-0.756320\pi$
$719$	−1.99890e7	−1.44201	−0.721006	+	0.692928i	$0.756320\pi$
$720$	0	0
$721$	−3.00186e6	−0.215056
$722$	0	0
$723$	− 8.53366e6i	− 0.607141i
$724$	0	0
$725$	− 6.17773e6i	− 0.436500i
$726$	0	0
$727$	−1.34254e7	−0.942088	−0.471044	−	0.882110i	$-0.656123\pi$
$727$	−1.34254e7	−0.942088	−0.471044	+	0.882110i	$0.656123\pi$
$728$	0	0
$729$	−1.49383e7	−1.04108
$730$	0	0
$731$	149421.i	0.0103424i
$732$	0	0
$733$	1.89485e6i	0.130261i	0.997877	+	0.0651304i	$0.0207463\pi$
$733$	1.89485e6i	0.130261i	−0.997877	+	0.0651304i	$0.979254\pi$
$734$	0	0
$735$	−1.06927e7	−0.730078
$736$	0	0
$737$	−1.72787e7	−1.17177
$738$	0	0
$739$	− 1.52333e7i	− 1.02608i	−0.858364	−	0.513041i	$-0.828519\pi$
$739$	− 1.52333e7i	− 1.02608i	0.858364	−	0.513041i	$-0.171481\pi$
$740$	0	0
$741$	3.01145e7i	2.01479i
$742$	0	0
$743$	−1.79078e7	−1.19006	−0.595031	−	0.803703i	$-0.702860\pi$
$743$	−1.79078e7	−1.19006	−0.595031	+	0.803703i	$0.702860\pi$
$744$	0	0
$745$	1.13049e7	0.746235
$746$	0	0
$747$	777251.i	0.0509635i
$748$	0	0
$749$	− 1.98829e6i	− 0.129502i
$750$	0	0
$751$	2.08382e7	1.34822	0.674109	−	0.738632i	$-0.264528\pi$
$751$	2.08382e7	1.34822	0.674109	+	0.738632i	$0.264528\pi$
$752$	0	0
$753$	−1.37356e7	−0.882793
$754$	0	0
$755$	4.90431e6i	0.313120i
$756$	0	0
$757$	− 3.00548e6i	− 0.190623i	−0.995448	−	0.0953113i	$-0.969615\pi$
$757$	− 3.00548e6i	− 0.190623i	0.995448	−	0.0953113i	$-0.0303847\pi$
$758$	0	0
$759$	−1.34756e7	−0.849072
$760$	0	0
$761$	1.28926e7	0.807013	0.403507	−	0.914977i	$-0.367791\pi$
$761$	1.28926e7	0.807013	0.403507	+	0.914977i	$0.367791\pi$
$762$	0	0
$763$	2.75581e6i	0.171371i
$764$	0	0
$765$	− 103309.i	− 0.00638241i
$766$	0	0
$767$	−3.34902e7	−2.05556
$768$	0	0
$769$	1.07309e7	0.654364	0.327182	−	0.944961i	$-0.393901\pi$
$769$	1.07309e7	0.654364	0.327182	+	0.944961i	$0.393901\pi$
$770$	0	0
$771$	1.32599e7i	0.803352i
$772$	0	0
$773$	1.83952e7i	1.10728i	0.832757	+	0.553639i	$0.186761\pi$
$773$	1.83952e7i	1.10728i	−0.832757	+	0.553639i	$0.813239\pi$
$774$	0	0
$775$	1.10262e7	0.659437
$776$	0	0
$777$	−1.11360e6	−0.0661724
$778$	0	0
$779$	1.67807e7i	0.990758i
$780$	0	0
$781$	5.59540e6i	0.328249i
$782$	0	0
$783$	−1.88341e7	−1.09784
$784$	0	0
$785$	−4.94995e6	−0.286699
$786$	0	0
$787$	− 4.26448e6i	− 0.245431i	−0.992442	−	0.122716i	$-0.960840\pi$
$787$	− 4.26448e6i	− 0.245431i	0.992442	−	0.122716i	$-0.0391603\pi$
$788$	0	0
$789$	3.53301e6i	0.202047i
$790$	0	0
$791$	−670088.	−0.0380795
$792$	0	0
$793$	−2.71069e7	−1.53072
$794$	0	0
$795$	1.74424e7i	0.978790i
$796$	0	0
$797$	− 2.96002e7i	− 1.65063i	−0.564676	−	0.825313i	$-0.690999\pi$
$797$	− 2.96002e7i	− 1.65063i	0.564676	−	0.825313i	$-0.309001\pi$
$798$	0	0
$799$	−3.95609e6	−0.219229
$800$	0	0
$801$	1.03257e6	0.0568641
$802$	0	0
$803$	− 1.11358e7i	− 0.609443i
$804$	0	0
$805$	3.33070e6i	0.181153i
$806$	0	0
$807$	−2.18415e7	−1.18059
$808$	0	0
$809$	3.37728e7	1.81424	0.907121	−	0.420870i	$-0.138275\pi$
$809$	3.37728e7	1.81424	0.907121	+	0.420870i	$0.138275\pi$
$810$	0	0
$811$	− 2.05658e7i	− 1.09798i	−0.835830	−	0.548988i	$-0.815013\pi$
$811$	− 2.05658e7i	− 1.09798i	0.835830	−	0.548988i	$-0.184987\pi$
$812$	0	0
$813$	2.43137e7i	1.29010i
$814$	0	0
$815$	1.22689e7	0.647010
$816$	0	0
$817$	1.36764e6	0.0716831
$818$	0	0
$819$	246629.i	0.0128480i
$820$	0	0
$821$	− 1.78535e7i	− 0.924414i	−0.886772	−	0.462207i	$-0.847058\pi$
$821$	− 1.78535e7i	− 0.924414i	0.886772	−	0.462207i	$-0.152942\pi$
$822$	0	0
$823$	8.13336e6	0.418572	0.209286	−	0.977854i	$-0.432886\pi$
$823$	8.13336e6	0.418572	0.209286	+	0.977854i	$0.432886\pi$
$824$	0	0
$825$	−5.00494e6	−0.256014
$826$	0	0
$827$	− 2.12039e7i	− 1.07808i	−0.842279	−	0.539042i	$-0.818787\pi$
$827$	− 2.12039e7i	− 1.07808i	0.842279	−	0.539042i	$-0.181213\pi$
$828$	0	0
$829$	− 3.73037e6i	− 0.188523i	−0.995547	−	0.0942617i	$-0.969951\pi$
$829$	− 3.73037e6i	− 0.188523i	0.995547	−	0.0942617i	$-0.0300491\pi$
$830$	0	0
$831$	2.10619e7	1.05802
$832$	0	0
$833$	3.55231e6	0.177377
$834$	0	0
$835$	1.83471e7i	0.910648i
$836$	0	0
$837$	− 3.36158e7i	− 1.65856i
$838$	0	0
$839$	−1.64106e7	−0.804861	−0.402430	−	0.915451i	$-0.631834\pi$
$839$	−1.64106e7	−0.804861	−0.402430	+	0.915451i	$0.631834\pi$
$840$	0	0
$841$	−3.18812e6	−0.155433
$842$	0	0
$843$	− 3.64492e7i	− 1.76652i
$844$	0	0
$845$	− 2.63025e7i	− 1.26723i
$846$	0	0
$847$	2.12705e6	0.101875
$848$	0	0
$849$	9.10345e6	0.433448
$850$	0	0
$851$	− 1.10398e7i	− 0.522563i
$852$	0	0
$853$	− 8.74236e6i	− 0.411392i	−0.978616	−	0.205696i	$-0.934054\pi$
$853$	− 8.74236e6i	− 0.411392i	0.978616	−	0.205696i	$-0.0659459\pi$
$854$	0	0
$855$	−945577.	−0.0442366
$856$	0	0
$857$	−3.31459e7	−1.54162	−0.770810	−	0.637065i	$-0.780148\pi$
$857$	−3.31459e7	−1.54162	−0.770810	+	0.637065i	$0.780148\pi$
$858$	0	0
$859$	− 898006.i	− 0.0415237i	−0.999784	−	0.0207619i	$-0.993391\pi$
$859$	− 898006.i	− 0.0415237i	0.999784	−	0.0207619i	$-0.00660918\pi$
$860$	0	0
$861$	− 2.89851e6i	− 0.133250i
$862$	0	0
$863$	−1.63551e6	−0.0747526	−0.0373763	−	0.999301i	$-0.511900\pi$
$863$	−1.63551e6	−0.0747526	−0.0373763	+	0.999301i	$0.511900\pi$
$864$	0	0
$865$	7.11405e6	0.323278
$866$	0	0
$867$	2.09028e7i	0.944400i
$868$	0	0
$869$	1.67686e7i	0.753266i
$870$	0	0
$871$	6.61202e7	2.95317
$872$	0	0
$873$	−898766.	−0.0399127
$874$	0	0
$875$	4.28335e6i	0.189132i
$876$	0	0
$877$	− 1.42532e7i	− 0.625770i	−0.949791	−	0.312885i	$-0.898705\pi$
$877$	− 1.42532e7i	− 0.625770i	0.949791	−	0.312885i	$-0.101295\pi$
$878$	0	0
$879$	−1.33227e7	−0.581595
$880$	0	0
$881$	−1.24428e7	−0.540105	−0.270053	−	0.962846i	$-0.587041\pi$
$881$	−1.24428e7	−0.540105	−0.270053	+	0.962846i	$0.587041\pi$
$882$	0	0
$883$	4.34630e7i	1.87593i	0.346725	+	0.937967i	$0.387294\pi$
$883$	4.34630e7i	1.87593i	−0.346725	+	0.937967i	$0.612706\pi$
$884$	0	0
$885$	2.21786e7i	0.951867i
$886$	0	0
$887$	−4.26973e6	−0.182218	−0.0911089	−	0.995841i	$-0.529041\pi$
$887$	−4.26973e6	−0.182218	−0.0911089	+	0.995841i	$0.529041\pi$
$888$	0	0
$889$	−1.37011e6	−0.0581436
$890$	0	0
$891$	1.45665e7i	0.614695i
$892$	0	0
$893$	3.62097e7i	1.51948i
$894$	0	0
$895$	3.12198e7	1.30278
$896$	0	0
$897$	5.15671e7	2.13989
$898$	0	0
$899$	− 4.22993e7i	− 1.74556i
$900$	0	0
$901$	− 5.79470e6i	− 0.237804i
$902$	0	0
$903$	−236230.	−0.00964087
$904$	0	0
$905$	−1.96922e6	−0.0799230
$906$	0	0
$907$	3.88728e7i	1.56902i	0.620119	+	0.784508i	$0.287084\pi$
$907$	3.88728e7i	1.56902i	−0.620119	+	0.784508i	$0.712916\pi$
$908$	0	0
$909$	961532.i	0.0385970i
$910$	0	0
$911$	2.73840e7	1.09320	0.546602	−	0.837393i	$-0.315921\pi$
$911$	2.73840e7	1.09320	0.546602	+	0.837393i	$0.315921\pi$
$912$	0	0
$913$	1.82962e7	0.726414
$914$	0	0
$915$	1.79513e7i	0.708831i
$916$	0	0
$917$	− 4.22576e6i	− 0.165952i
$918$	0	0
$919$	3.14731e7	1.22928	0.614640	−	0.788808i	$-0.289301\pi$
$919$	3.14731e7	1.22928	0.614640	+	0.788808i	$0.289301\pi$
$920$	0	0
$921$	−4.11728e7	−1.59942
$922$	0	0
$923$	− 2.14119e7i	− 0.827277i
$924$	0	0
$925$	− 4.10026e6i	− 0.157564i
$926$	0	0
$927$	−1.45931e6	−0.0557761
$928$	0	0
$929$	−2.58150e7	−0.981371	−0.490685	−	0.871337i	$-0.663254\pi$
$929$	−2.58150e7	−0.981371	−0.490685	+	0.871337i	$0.663254\pi$
$930$	0	0
$931$	− 3.25139e7i	− 1.22941i
$932$	0	0
$933$	− 3.80243e7i	− 1.43007i
$934$	0	0
$935$	−2.43186e6	−0.0909723
$936$	0	0
$937$	−1.77379e7	−0.660014	−0.330007	−	0.943979i	$-0.607051\pi$
$937$	−1.77379e7	−0.660014	−0.330007	+	0.943979i	$0.607051\pi$
$938$	0	0
$939$	− 4.45265e7i	− 1.64799i
$940$	0	0
$941$	3.65458e7i	1.34544i	0.739898	+	0.672719i	$0.234874\pi$
$941$	3.65458e7i	1.34544i	−0.739898	+	0.672719i	$0.765126\pi$
$942$	0	0
$943$	2.87348e7	1.05227
$944$	0	0
$945$	3.77139e6	0.137380
$946$	0	0
$947$	4.00234e7i	1.45024i	0.688623	+	0.725119i	$0.258215\pi$
$947$	4.00234e7i	1.45024i	−0.688623	+	0.725119i	$0.741785\pi$
$948$	0	0
$949$	4.26134e7i	1.53596i
$950$	0	0
$951$	2.46250e7	0.882928
$952$	0	0
$953$	1.61525e7	0.576114	0.288057	−	0.957613i	$-0.406991\pi$
$953$	1.61525e7	0.576114	0.288057	+	0.957613i	$0.406991\pi$
$954$	0	0
$955$	1.87204e7i	0.664213i
$956$	0	0
$957$	1.92001e7i	0.677680i
$958$	0	0
$959$	−4.11941e6	−0.144640
$960$	0	0
$961$	4.68683e7	1.63708
$962$	0	0
$963$	− 966579.i	− 0.0335870i
$964$	0	0
$965$	1.19714e7i	0.413833i
$966$	0	0
$967$	−1.46946e7	−0.505349	−0.252674	−	0.967551i	$-0.581310\pi$
$967$	−1.46946e7	−0.505349	−0.252674	+	0.967551i	$0.581310\pi$
$968$	0	0
$969$	−6.62546e6	−0.226676
$970$	0	0
$971$	2.33873e7i	0.796034i	0.917378	+	0.398017i	$0.130302\pi$
$971$	2.33873e7i	0.796034i	−0.917378	+	0.398017i	$0.869698\pi$
$972$	0	0
$973$	1.02120e6i	0.0345803i
$974$	0	0
$975$	1.91524e7	0.645225
$976$	0	0
$977$	3.68638e7	1.23556	0.617779	−	0.786351i	$-0.288032\pi$
$977$	3.68638e7	1.23556	0.617779	+	0.786351i	$0.288032\pi$
$978$	0	0
$979$	− 2.43063e7i	− 0.810519i
$980$	0	0
$981$	1.33970e6i	0.0444462i
$982$	0	0
$983$	1.35629e7	0.447682	0.223841	−	0.974626i	$-0.428140\pi$
$983$	1.35629e7	0.447682	0.223841	+	0.974626i	$0.428140\pi$
$984$	0	0
$985$	−4.33135e7	−1.42243
$986$	0	0
$987$	− 6.25444e6i	− 0.204360i
$988$	0	0
$989$	− 2.34190e6i	− 0.0761338i
$990$	0	0
$991$	3.33321e7	1.07815	0.539074	−	0.842258i	$-0.318774\pi$
$991$	3.33321e7	1.07815	0.539074	+	0.842258i	$0.318774\pi$
$992$	0	0
$993$	−1.74819e7	−0.562620
$994$	0	0
$995$	− 2.05716e7i	− 0.658735i
$996$	0	0
$997$	3.52048e6i	0.112167i	0.998426	+	0.0560833i	$0.0178612\pi$
$997$	3.52048e6i	0.112167i	−0.998426	+	0.0560833i	$0.982139\pi$
$998$	0	0
$999$	−1.25005e7	−0.396291

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	128.6.b.d.65.2	yes	4
4.3	odd	2	inner	128.6.b.d.65.4	yes	4
8.3	odd	2	inner	128.6.b.d.65.1	✓	4
8.5	even	2	inner	128.6.b.d.65.3	yes	4
16.3	odd	4		256.6.a.m.1.2		4
16.5	even	4		256.6.a.m.1.1		4
16.11	odd	4		256.6.a.m.1.3		4
16.13	even	4		256.6.a.m.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.6.b.d.65.1	✓	4	8.3	odd	2	inner
128.6.b.d.65.2	yes	4	1.1	even	1	trivial
128.6.b.d.65.3	yes	4	8.5	even	2	inner
128.6.b.d.65.4	yes	4	4.3	odd	2	inner
256.6.a.m.1.1		4	16.5	even	4
256.6.a.m.1.2		4	16.3	odd	4
256.6.a.m.1.3		4	16.11	odd	4
256.6.a.m.1.4		4	16.13	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:	\( N \)	\(=\)	\( 128 = 2^{7} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	128.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-15.2315i q^{3} +43.0813i q^{5} +22.6274 q^{7} +11.0000 q^{9} +O(q^{10})\)	\(q-15.2315i q^{3} +43.0813i q^{5} +22.6274 q^{7} +11.0000 q^{9} -258.936i q^{11} +990.870i q^{13} +656.195 q^{15} -218.000 q^{17} +1995.33i q^{19} -344.651i q^{21} +3416.74 q^{23} +1269.00 q^{25} -3868.81i q^{27} -4868.19i q^{29} +8688.93 q^{31} -3944.00 q^{33} +974.819i q^{35} -3231.10i q^{37} +15092.5 q^{39} +8410.00 q^{41} -685.420i q^{43} +473.895i q^{45} +18147.2 q^{47} -16295.0 q^{49} +3320.48i q^{51} +26581.2i q^{53} +11155.3 q^{55} +30392.0 q^{57} +33798.8i q^{59} +27356.6i q^{61} +248.902 q^{63} -42688.0 q^{65} -66729.4i q^{67} -52042.2i q^{69} -21609.2 q^{71} +43006.0 q^{73} -19328.8i q^{75} -5859.06i q^{77} -64759.7 q^{79} -56255.0 q^{81} +70659.1i q^{83} -9391.73i q^{85} -74150.0 q^{87} +93870.0 q^{89} +22420.8i q^{91} -132346. i q^{93} -85961.6 q^{95} -81706.0 q^{97} -2848.30i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 44 q^{9}+O(q^{10}) \) 4 * q + 44 * q^9	\( 4 q + 44 q^{9} - 872 q^{17} + 5076 q^{25} - 15776 q^{33} + 33640 q^{41} - 65180 q^{49} + 121568 q^{57} - 170752 q^{65} + 172024 q^{73} - 225020 q^{81} + 375480 q^{89} - 326824 q^{97}+O(q^{100}) \) 4 * q + 44 * q^9 - 872 * q^17 + 5076 * q^25 - 15776 * q^33 + 33640 * q^41 - 65180 * q^49 + 121568 * q^57 - 170752 * q^65 + 172024 * q^73 - 225020 * q^81 + 375480 * q^89 - 326824 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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