LMFDB - Embedded newform 96.6.d.a.49.10

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [96,6,Mod(49,96)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(96, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 6, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("96.49"); S:= CuspForms(chi, 6); N := Newforms(S);

Level:	$ N $	$=$	$ 96 = 2^{5} \cdot 3 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	96.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

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

Self dual:	no
Analytic conductor:	$15.3968467020$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 4 x^{9} + 20 x^{8} - 16 x^{7} + 290 x^{6} + 896 x^{5} + 416 x^{4} + 13952 x^{3} + 60997 x^{2} + \cdots + 750564 $ x^10 - 4x^9 + 20x^8 - 16x^7 + 290x^6 + 896x^5 + 416x^4 + 13952x^3 + 60997x^2 + 221460*x + 750564
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{40}\cdot 3^{8} $
Twist minimal:	no (minimal twist has level 24)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.10
Root			$-2.89538 + 0.908872i$ of defining polynomial
Character	$\chi$	$=$	96.49
Dual form			96.6.d.a.49.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+9.00000i q^{3} +100.204i q^{5} -154.395 q^{7} -81.0000 q^{9} +85.0809i q^{11} -812.927i q^{13} -901.832 q^{15} +1275.57 q^{17} -393.875i q^{19} -1389.55i q^{21} -2533.66 q^{23} -6915.74 q^{25} -729.000i q^{27} +254.097i q^{29} -1801.33 q^{31} -765.728 q^{33} -15470.9i q^{35} +686.112i q^{37} +7316.35 q^{39} -13255.9 q^{41} +13936.2i q^{43} -8116.48i q^{45} -11557.4 q^{47} +7030.74 q^{49} +11480.1i q^{51} +8023.45i q^{53} -8525.40 q^{55} +3544.88 q^{57} -7967.95i q^{59} +15828.6i q^{61} +12506.0 q^{63} +81458.2 q^{65} +33836.9i q^{67} -22803.0i q^{69} +72786.0 q^{71} -717.727 q^{73} -62241.7i q^{75} -13136.0i q^{77} -32844.2 q^{79} +6561.00 q^{81} +46129.8i q^{83} +127816. i q^{85} -2286.88 q^{87} -15255.6 q^{89} +125512. i q^{91} -16211.9i q^{93} +39467.7 q^{95} -173584. q^{97} -6891.55i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 196 q^{7} - 810 q^{9} - 900 q^{15} - 404 q^{17} - 1672 q^{23} - 9366 q^{25} - 18692 q^{31} + 12168 q^{39} - 2476 q^{41} - 65256 q^{47} + 9378 q^{49} + 103424 q^{55} - 12888 q^{57} - 15876 q^{63}+ \cdots + 59372 q^{97}+O(q^{100}) $ 10 * q + 196 * q^7 - 810 * q^9 - 900 * q^15 - 404 * q^17 - 1672 * q^23 - 9366 * q^25 - 18692 * q^31 + 12168 * q^39 - 2476 * q^41 - 65256 * q^47 + 9378 * q^49 + 103424 * q^55 - 12888 * q^57 - 15876 * q^63 + 51536 * q^65 + 272232 * q^71 - 27388 * q^73 - 252308 * q^79 + 65610 * q^81 + 90828 * q^87 + 175580 * q^89 - 243184 * q^95 + 59372 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/96\mathbb{Z}\right)^\times$.

$n$	$31$	$37$	$65$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	9.00000i	0.577350i
$3$	9.00000i	0.577350i
$4$	0	0
$5$	100.204i	1.79249i	0.443554	+	0.896247i	$0.353717\pi$
$5$	100.204i	1.79249i	−0.443554	+	0.896247i	$0.646283\pi$
$6$	0	0
$7$	−154.395	−1.19093	−0.595467	−	0.803380i	$-0.703033\pi$
$7$	−154.395	−1.19093	−0.595467	+	0.803380i	$0.703033\pi$
$8$	0	0
$9$	−81.0000	−0.333333
$10$	0	0
$11$	85.0809i	0.212007i	0.994366	+	0.106003i	$0.0338055\pi$
$11$	85.0809i	0.212007i	−0.994366	+	0.106003i	$0.966195\pi$
$12$	0	0
$13$	− 812.927i	− 1.33412i	−0.745006	−	0.667058i	$-0.767553\pi$
$13$	− 812.927i	− 1.33412i	0.745006	−	0.667058i	$-0.232447\pi$
$14$	0	0
$15$	−901.832	−1.03490
$16$	0	0
$17$	1275.57	1.07049	0.535244	−	0.844698i	$-0.320220\pi$
$17$	1275.57	1.07049	0.535244	+	0.844698i	$0.320220\pi$
$18$	0	0
$19$	− 393.875i	− 0.250308i	−0.992137	−	0.125154i	$-0.960057\pi$
$19$	− 393.875i	− 0.250308i	0.992137	−	0.125154i	$-0.0399425\pi$
$20$	0	0
$21$	− 1389.55i	− 0.687586i
$22$	0	0
$23$	−2533.66	−0.998686	−0.499343	−	0.866404i	$-0.666425\pi$
$23$	−2533.66	−0.998686	−0.499343	+	0.866404i	$0.666425\pi$
$24$	0	0
$25$	−6915.74	−2.21304
$26$	0	0
$27$	− 729.000i	− 0.192450i
$28$	0	0
$29$	254.097i	0.0561055i	0.999606	+	0.0280527i	$0.00893064\pi$
$29$	254.097i	0.0561055i	−0.999606	+	0.0280527i	$0.991069\pi$
$30$	0	0
$31$	−1801.33	−0.336658	−0.168329	−	0.985731i	$-0.553837\pi$
$31$	−1801.33	−0.336658	−0.168329	+	0.985731i	$0.553837\pi$
$32$	0	0
$33$	−765.728	−0.122402
$34$	0	0
$35$	− 15470.9i	− 2.13474i
$36$	0	0
$37$	686.112i	0.0823930i	0.999151	+	0.0411965i	$0.0131170\pi$
$37$	686.112i	0.0823930i	−0.999151	+	0.0411965i	$0.986883\pi$
$38$	0	0
$39$	7316.35	0.770252
$40$	0	0
$41$	−13255.9	−1.23154	−0.615772	−	0.787924i	$-0.711156\pi$
$41$	−13255.9	−1.23154	−0.615772	+	0.787924i	$0.711156\pi$
$42$	0	0
$43$	13936.2i	1.14940i	0.818363	+	0.574701i	$0.194882\pi$
$43$	13936.2i	1.14940i	−0.818363	+	0.574701i	$0.805118\pi$
$44$	0	0
$45$	− 8116.48i	− 0.597498i
$46$	0	0
$47$	−11557.4	−0.763158	−0.381579	−	0.924336i	$-0.624620\pi$
$47$	−11557.4	−0.763158	−0.381579	+	0.924336i	$0.624620\pi$
$48$	0	0
$49$	7030.74	0.418322
$50$	0	0
$51$	11480.1i	0.618046i
$52$	0	0
$53$	8023.45i	0.392348i	0.980569	+	0.196174i	$0.0628518\pi$
$53$	8023.45i	0.392348i	−0.980569	+	0.196174i	$0.937148\pi$
$54$	0	0
$55$	−8525.40	−0.380021
$56$	0	0
$57$	3544.88	0.144515
$58$	0	0
$59$	− 7967.95i	− 0.298000i	−0.988837	−	0.149000i	$-0.952395\pi$
$59$	− 7967.95i	− 0.298000i	0.988837	−	0.149000i	$-0.0476055\pi$
$60$	0	0
$61$	15828.6i	0.544651i	0.962205	+	0.272326i	$0.0877928\pi$
$61$	15828.6i	0.544651i	−0.962205	+	0.272326i	$0.912207\pi$
$62$	0	0
$63$	12506.0	0.396978
$64$	0	0
$65$	81458.2	2.39140
$66$	0	0
$67$	33836.9i	0.920881i	0.887691	+	0.460440i	$0.152309\pi$
$67$	33836.9i	0.920881i	−0.887691	+	0.460440i	$0.847691\pi$
$68$	0	0
$69$	− 22803.0i	− 0.576592i
$70$	0	0
$71$	72786.0	1.71357	0.856786	−	0.515672i	$-0.172458\pi$
$71$	72786.0	1.71357	0.856786	+	0.515672i	$0.172458\pi$
$72$	0	0
$73$	−717.727	−0.0157635	−0.00788174	−	0.999969i	$-0.502509\pi$
$73$	−717.727	−0.0157635	−0.00788174	+	0.999969i	$0.502509\pi$
$74$	0	0
$75$	− 62241.7i	− 1.27770i
$76$	0	0
$77$	− 13136.0i	− 0.252486i
$78$	0	0
$79$	−32844.2	−0.592094	−0.296047	−	0.955173i	$-0.595669\pi$
$79$	−32844.2	−0.592094	−0.296047	+	0.955173i	$0.595669\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	46129.8i	0.734998i	0.930024	+	0.367499i	$0.119786\pi$
$83$	46129.8i	0.734998i	−0.930024	+	0.367499i	$0.880214\pi$
$84$	0	0
$85$	127816.i	1.91884i
$86$	0	0
$87$	−2286.88	−0.0323925
$88$	0	0
$89$	−15255.6	−0.204152	−0.102076	−	0.994777i	$-0.532549\pi$
$89$	−15255.6	−0.204152	−0.102076	+	0.994777i	$0.532549\pi$
$90$	0	0
$91$	125512.i	1.58884i
$92$	0	0
$93$	− 16211.9i	− 0.194369i
$94$	0	0
$95$	39467.7	0.448676
$96$	0	0
$97$	−173584.	−1.87318	−0.936591	−	0.350426i	$-0.886037\pi$
$97$	−173584.	−1.87318	−0.936591	+	0.350426i	$0.886037\pi$
$98$	0	0
$99$	− 6891.55i	− 0.0706690i
$100$	0	0
$101$	− 76270.0i	− 0.743961i	−0.928240	−	0.371981i	$-0.878679\pi$
$101$	− 76270.0i	− 0.743961i	0.928240	−	0.371981i	$-0.121321\pi$
$102$	0	0
$103$	71216.8	0.661439	0.330719	−	0.943729i	$-0.392709\pi$
$103$	71216.8	0.661439	0.330719	+	0.943729i	$0.392709\pi$
$104$	0	0
$105$	139238.	1.23249
$106$	0	0
$107$	− 144823.i	− 1.22286i	−0.791297	−	0.611432i	$-0.790594\pi$
$107$	− 144823.i	− 1.22286i	0.791297	−	0.611432i	$-0.209406\pi$
$108$	0	0
$109$	16321.4i	0.131580i	0.997833	+	0.0657900i	$0.0209568\pi$
$109$	16321.4i	0.131580i	−0.997833	+	0.0657900i	$0.979043\pi$
$110$	0	0
$111$	−6175.01	−0.0475696
$112$	0	0
$113$	−167326.	−1.23273	−0.616366	−	0.787460i	$-0.711396\pi$
$113$	−167326.	−1.23273	−0.616366	+	0.787460i	$0.711396\pi$
$114$	0	0
$115$	− 253882.i	− 1.79014i
$116$	0	0
$117$	65847.1i	0.444705i
$118$	0	0
$119$	−196941.	−1.27488
$120$	0	0
$121$	153812.	0.955053
$122$	0	0
$123$	− 119303.i	− 0.711032i
$124$	0	0
$125$	− 379846.i	− 2.17436i
$126$	0	0
$127$	−307665.	−1.69265	−0.846327	−	0.532663i	$-0.821191\pi$
$127$	−307665.	−1.69265	−0.846327	+	0.532663i	$0.821191\pi$
$128$	0	0
$129$	−125425.	−0.663608
$130$	0	0
$131$	197124.i	1.00360i	0.864983	+	0.501802i	$0.167329\pi$
$131$	197124.i	1.00360i	−0.864983	+	0.501802i	$0.832671\pi$
$132$	0	0
$133$	60812.3i	0.298100i
$134$	0	0
$135$	73048.4	0.344966
$136$	0	0
$137$	−41854.6	−0.190521	−0.0952603	−	0.995452i	$-0.530368\pi$
$137$	−41854.6	−0.190521	−0.0952603	+	0.995452i	$0.530368\pi$
$138$	0	0
$139$	447375.i	1.96397i	0.188965	+	0.981984i	$0.439487\pi$
$139$	447375.i	1.96397i	−0.188965	+	0.981984i	$0.560513\pi$
$140$	0	0
$141$	− 104016.i	− 0.440610i
$142$	0	0
$143$	69164.6	0.282842
$144$	0	0
$145$	−25461.5	−0.100569
$146$	0	0
$147$	63276.6i	0.241518i
$148$	0	0
$149$	367279.i	1.35529i	0.735391	+	0.677643i	$0.236999\pi$
$149$	367279.i	1.35529i	−0.735391	+	0.677643i	$0.763001\pi$
$150$	0	0
$151$	−357113.	−1.27457	−0.637285	−	0.770628i	$-0.719942\pi$
$151$	−357113.	−1.27457	−0.637285	+	0.770628i	$0.719942\pi$
$152$	0	0
$153$	−103321.	−0.356829
$154$	0	0
$155$	− 180499.i	− 0.603457i
$156$	0	0
$157$	− 256597.i	− 0.830810i	−0.909637	−	0.415405i	$-0.863640\pi$
$157$	− 256597.i	− 0.830810i	0.909637	−	0.415405i	$-0.136360\pi$
$158$	0	0
$159$	−72211.1	−0.226522
$160$	0	0
$161$	391184.	1.18937
$162$	0	0
$163$	− 560618.i	− 1.65272i	−0.563145	−	0.826358i	$-0.690409\pi$
$163$	− 560618.i	− 1.65272i	0.563145	−	0.826358i	$-0.309591\pi$
$164$	0	0
$165$	− 76728.6i	− 0.219405i
$166$	0	0
$167$	410936.	1.14021	0.570103	−	0.821573i	$-0.306903\pi$
$167$	410936.	1.14021	0.570103	+	0.821573i	$0.306903\pi$
$168$	0	0
$169$	−289558.	−0.779864
$170$	0	0
$171$	31903.9i	0.0834360i
$172$	0	0
$173$	407557.i	1.03532i	0.855587	+	0.517658i	$0.173196\pi$
$173$	407557.i	1.03532i	−0.855587	+	0.517658i	$0.826804\pi$
$174$	0	0
$175$	1.06775e6	2.63558
$176$	0	0
$177$	71711.5	0.172050
$178$	0	0
$179$	73705.9i	0.171937i	0.996298	+	0.0859686i	$0.0273985\pi$
$179$	73705.9i	0.171937i	−0.996298	+	0.0859686i	$0.972602\pi$
$180$	0	0
$181$	194327.i	0.440896i	0.975399	+	0.220448i	$0.0707520\pi$
$181$	194327.i	0.440896i	−0.975399	+	0.220448i	$0.929248\pi$
$182$	0	0
$183$	−142458.	−0.314455
$184$	0	0
$185$	−68750.8	−0.147689
$186$	0	0
$187$	108526.i	0.226951i
$188$	0	0
$189$	112554.i	0.229195i
$190$	0	0
$191$	−171483.	−0.340125	−0.170063	−	0.985433i	$-0.554397\pi$
$191$	−171483.	−0.340125	−0.170063	+	0.985433i	$0.554397\pi$
$192$	0	0
$193$	774459.	1.49660	0.748299	−	0.663362i	$-0.230871\pi$
$193$	774459.	1.49660	0.748299	+	0.663362i	$0.230871\pi$
$194$	0	0
$195$	733124.i	1.38067i
$196$	0	0
$197$	864723.i	1.58749i	0.608250	+	0.793746i	$0.291872\pi$
$197$	864723.i	1.58749i	−0.608250	+	0.793746i	$0.708128\pi$
$198$	0	0
$199$	752715.	1.34740	0.673702	−	0.739003i	$-0.264703\pi$
$199$	752715.	1.34740	0.673702	+	0.739003i	$0.264703\pi$
$200$	0	0
$201$	−304532.	−0.531671
$202$	0	0
$203$	− 39231.3i	− 0.0668179i
$204$	0	0
$205$	− 1.32829e6i	− 2.20754i
$206$	0	0
$207$	205227.	0.332895
$208$	0	0
$209$	33511.3	0.0530671
$210$	0	0
$211$	265517.i	0.410569i	0.978702	+	0.205285i	$0.0658120\pi$
$211$	265517.i	0.410569i	−0.978702	+	0.205285i	$0.934188\pi$
$212$	0	0
$213$	655074.i	0.989331i
$214$	0	0
$215$	−1.39645e6	−2.06030
$216$	0	0
$217$	278115.	0.400937
$218$	0	0
$219$	− 6459.54i	− 0.00910105i
$220$	0	0
$221$	− 1.03694e6i	− 1.42815i
$222$	0	0
$223$	−209136.	−0.281623	−0.140811	−	0.990036i	$-0.544971\pi$
$223$	−209136.	−0.281623	−0.140811	+	0.990036i	$0.544971\pi$
$224$	0	0
$225$	560175.	0.737679
$226$	0	0
$227$	120120.i	0.154722i	0.997003	+	0.0773610i	$0.0246494\pi$
$227$	120120.i	0.154722i	−0.997003	+	0.0773610i	$0.975351\pi$
$228$	0	0
$229$	440615.i	0.555227i	0.960693	+	0.277613i	$0.0895434\pi$
$229$	440615.i	0.555227i	−0.960693	+	0.277613i	$0.910457\pi$
$230$	0	0
$231$	118224.	0.145773
$232$	0	0
$233$	167121.	0.201670	0.100835	−	0.994903i	$-0.467849\pi$
$233$	167121.	0.201670	0.100835	+	0.994903i	$0.467849\pi$
$234$	0	0
$235$	− 1.15809e6i	− 1.36796i
$236$	0	0
$237$	− 295598.i	− 0.341846i
$238$	0	0
$239$	445757.	0.504781	0.252391	−	0.967625i	$-0.418783\pi$
$239$	445757.	0.504781	0.252391	+	0.967625i	$0.418783\pi$
$240$	0	0
$241$	153018.	0.169708	0.0848538	−	0.996393i	$-0.472958\pi$
$241$	153018.	0.169708	0.0848538	+	0.996393i	$0.472958\pi$
$242$	0	0
$243$	59049.0i	0.0641500i
$244$	0	0
$245$	704505.i	0.749840i
$246$	0	0
$247$	−320192.	−0.333940
$248$	0	0
$249$	−415168.	−0.424351
$250$	0	0
$251$	693358.i	0.694662i	0.937743	+	0.347331i	$0.112912\pi$
$251$	693358.i	0.694662i	−0.937743	+	0.347331i	$0.887088\pi$
$252$	0	0
$253$	− 215566.i	− 0.211728i
$254$	0	0
$255$	−1.15035e6	−1.10784
$256$	0	0
$257$	−1.71886e6	−1.62334	−0.811668	−	0.584119i	$-0.801440\pi$
$257$	−1.71886e6	−1.62334	−0.811668	+	0.584119i	$0.801440\pi$
$258$	0	0
$259$	− 105932.i	− 0.0981246i
$260$	0	0
$261$	− 20581.9i	− 0.0187018i
$262$	0	0
$263$	156631.	0.139633	0.0698167	−	0.997560i	$-0.477759\pi$
$263$	156631.	0.139633	0.0698167	+	0.997560i	$0.477759\pi$
$264$	0	0
$265$	−803978.	−0.703282
$266$	0	0
$267$	− 137300.i	− 0.117867i
$268$	0	0
$269$	− 800001.i	− 0.674078i	−0.941491	−	0.337039i	$-0.890575\pi$
$269$	− 800001.i	− 0.674078i	0.941491	−	0.337039i	$-0.109425\pi$
$270$	0	0
$271$	−1.89597e6	−1.56823	−0.784113	−	0.620618i	$-0.786882\pi$
$271$	−1.89597e6	−1.56823	−0.784113	+	0.620618i	$0.786882\pi$
$272$	0	0
$273$	−1.12961e6	−0.917319
$274$	0	0
$275$	− 588397.i	− 0.469180i
$276$	0	0
$277$	− 279803.i	− 0.219105i	−0.993981	−	0.109553i	$-0.965058\pi$
$277$	− 279803.i	− 0.219105i	0.993981	−	0.109553i	$-0.0349418\pi$
$278$	0	0
$279$	145907.	0.112219
$280$	0	0
$281$	554219.	0.418712	0.209356	−	0.977840i	$-0.432863\pi$
$281$	554219.	0.418712	0.209356	+	0.977840i	$0.432863\pi$
$282$	0	0
$283$	− 1.57678e6i	− 1.17032i	−0.810917	−	0.585161i	$-0.801031\pi$
$283$	− 1.57678e6i	− 1.17032i	0.810917	−	0.585161i	$-0.198969\pi$
$284$	0	0
$285$	355209.i	0.259043i
$286$	0	0
$287$	2.04664e6	1.46669
$288$	0	0
$289$	207218.	0.145943
$290$	0	0
$291$	− 1.56225e6i	− 1.08148i
$292$	0	0
$293$	− 184563.i	− 0.125596i	−0.998026	−	0.0627979i	$-0.979998\pi$
$293$	− 184563.i	− 0.125596i	0.998026	−	0.0627979i	$-0.0200024\pi$
$294$	0	0
$295$	798416.	0.534164
$296$	0	0
$297$	62023.9	0.0408008
$298$	0	0
$299$	2.05968e6i	1.33236i
$300$	0	0
$301$	− 2.15167e6i	− 1.36886i
$302$	0	0
$303$	686430.	0.429526
$304$	0	0
$305$	−1.58608e6	−0.976285
$306$	0	0
$307$	− 405878.i	− 0.245782i	−0.992420	−	0.122891i	$-0.960783\pi$
$307$	− 405878.i	− 0.245782i	0.992420	−	0.122891i	$-0.0392165\pi$
$308$	0	0
$309$	640952.i	0.381882i
$310$	0	0
$311$	−254438.	−0.149170	−0.0745849	−	0.997215i	$-0.523763\pi$
$311$	−254438.	−0.149170	−0.0745849	+	0.997215i	$0.523763\pi$
$312$	0	0
$313$	1.03862e6	0.599234	0.299617	−	0.954060i	$-0.403141\pi$
$313$	1.03862e6	0.599234	0.299617	+	0.954060i	$0.403141\pi$
$314$	0	0
$315$	1.25314e6i	0.711581i
$316$	0	0
$317$	− 1.39910e6i	− 0.781990i	−0.920393	−	0.390995i	$-0.872131\pi$
$317$	− 1.39910e6i	− 0.781990i	0.920393	−	0.390995i	$-0.127869\pi$
$318$	0	0
$319$	−21618.8	−0.0118948
$320$	0	0
$321$	1.30341e6	0.706021
$322$	0	0
$323$	− 502415.i	− 0.267952i
$324$	0	0
$325$	5.62200e6i	2.95245i
$326$	0	0
$327$	−146892.	−0.0759678
$328$	0	0
$329$	1.78440e6	0.908870
$330$	0	0
$331$	2.04667e6i	1.02678i	0.858156	+	0.513390i	$0.171610\pi$
$331$	2.04667e6i	1.02678i	−0.858156	+	0.513390i	$0.828390\pi$
$332$	0	0
$333$	− 55575.0i	− 0.0274643i
$334$	0	0
$335$	−3.39058e6	−1.65067
$336$	0	0
$337$	−3.22090e6	−1.54491	−0.772453	−	0.635072i	$-0.780970\pi$
$337$	−3.22090e6	−1.54491	−0.772453	+	0.635072i	$0.780970\pi$
$338$	0	0
$339$	− 1.50594e6i	− 0.711718i
$340$	0	0
$341$	− 153258.i	− 0.0713737i
$342$	0	0
$343$	1.50940e6	0.692740
$344$	0	0
$345$	2.28494e6	1.03354
$346$	0	0
$347$	− 1.80278e6i	− 0.803748i	−0.915695	−	0.401874i	$-0.868359\pi$
$347$	− 1.80278e6i	− 0.803748i	0.915695	−	0.401874i	$-0.131641\pi$
$348$	0	0
$349$	3.09976e6i	1.36228i	0.732155	+	0.681138i	$0.238515\pi$
$349$	3.09976e6i	1.36228i	−0.732155	+	0.681138i	$0.761485\pi$
$350$	0	0
$351$	−592624.	−0.256751
$352$	0	0
$353$	2.54655e6	1.08772	0.543859	−	0.839177i	$-0.316963\pi$
$353$	2.54655e6	1.08772	0.543859	+	0.839177i	$0.316963\pi$
$354$	0	0
$355$	7.29342e6i	3.07157i
$356$	0	0
$357$	− 1.77247e6i	− 0.736052i
$358$	0	0
$359$	790598.	0.323757	0.161879	−	0.986811i	$-0.448245\pi$
$359$	790598.	0.323757	0.161879	+	0.986811i	$0.448245\pi$
$360$	0	0
$361$	2.32096e6	0.937346
$362$	0	0
$363$	1.38431e6i	0.551400i
$364$	0	0
$365$	− 71918.7i	− 0.0282560i
$366$	0	0
$367$	1.30401e6	0.505376	0.252688	−	0.967548i	$-0.418685\pi$
$367$	1.30401e6	0.505376	0.252688	+	0.967548i	$0.418685\pi$
$368$	0	0
$369$	1.07373e6	0.410515
$370$	0	0
$371$	− 1.23878e6i	− 0.467260i
$372$	0	0
$373$	− 4.94317e6i	− 1.83964i	−0.392337	−	0.919822i	$-0.628333\pi$
$373$	− 4.94317e6i	− 1.83964i	0.392337	−	0.919822i	$-0.371667\pi$
$374$	0	0
$375$	3.41861e6	1.25537
$376$	0	0
$377$	206563.	0.0748512
$378$	0	0
$379$	− 1.92614e6i	− 0.688795i	−0.938824	−	0.344397i	$-0.888083\pi$
$379$	− 1.92614e6i	− 0.688795i	0.938824	−	0.344397i	$-0.111917\pi$
$380$	0	0
$381$	− 2.76898e6i	− 0.977255i
$382$	0	0
$383$	4.69380e6	1.63504	0.817519	−	0.575902i	$-0.195349\pi$
$383$	4.69380e6	1.63504	0.817519	+	0.575902i	$0.195349\pi$
$384$	0	0
$385$	1.31628e6	0.452580
$386$	0	0
$387$	− 1.12883e6i	− 0.383134i
$388$	0	0
$389$	2.35028e6i	0.787492i	0.919219	+	0.393746i	$0.128821\pi$
$389$	2.35028e6i	0.787492i	−0.919219	+	0.393746i	$0.871179\pi$
$390$	0	0
$391$	−3.23186e6	−1.06908
$392$	0	0
$393$	−1.77412e6	−0.579430
$394$	0	0
$395$	− 3.29110e6i	− 1.06133i
$396$	0	0
$397$	1.84236e6i	0.586675i	0.956009	+	0.293338i	$0.0947660\pi$
$397$	1.84236e6i	0.586675i	−0.956009	+	0.293338i	$0.905234\pi$
$398$	0	0
$399$	−547311.	−0.172108
$400$	0	0
$401$	4.73319e6	1.46992	0.734959	−	0.678112i	$-0.237201\pi$
$401$	4.73319e6	1.46992	0.734959	+	0.678112i	$0.237201\pi$
$402$	0	0
$403$	1.46435e6i	0.449140i
$404$	0	0
$405$	657435.i	0.199166i
$406$	0	0
$407$	−58375.0	−0.0174679
$408$	0	0
$409$	−6.34620e6	−1.87588	−0.937941	−	0.346796i	$-0.887270\pi$
$409$	−6.34620e6	−1.87588	−0.937941	+	0.346796i	$0.887270\pi$
$410$	0	0
$411$	− 376692.i	− 0.109997i
$412$	0	0
$413$	1.23021e6i	0.354898i
$414$	0	0
$415$	−4.62237e6	−1.31748
$416$	0	0
$417$	−4.02637e6	−1.13390
$418$	0	0
$419$	1.21837e6i	0.339034i	0.985527	+	0.169517i	$0.0542208\pi$
$419$	1.21837e6i	0.339034i	−0.985527	+	0.169517i	$0.945779\pi$
$420$	0	0
$421$	3.51877e6i	0.967579i	0.875185	+	0.483789i	$0.160740\pi$
$421$	3.51877e6i	0.967579i	−0.875185	+	0.483789i	$0.839260\pi$
$422$	0	0
$423$	936147.	0.254386
$424$	0	0
$425$	−8.82151e6	−2.36903
$426$	0	0
$427$	− 2.44386e6i	− 0.648643i
$428$	0	0
$429$	622481.i	0.163299i
$430$	0	0
$431$	3.01231e6	0.781099	0.390549	−	0.920582i	$-0.372285\pi$
$431$	3.01231e6	0.781099	0.390549	+	0.920582i	$0.372285\pi$
$432$	0	0
$433$	301545.	0.0772917	0.0386459	−	0.999253i	$-0.487696\pi$
$433$	301545.	0.0772917	0.0386459	+	0.999253i	$0.487696\pi$
$434$	0	0
$435$	− 229153.i	− 0.0580634i
$436$	0	0
$437$	997947.i	0.249979i
$438$	0	0
$439$	4.77660e6	1.18293	0.591464	−	0.806332i	$-0.298550\pi$
$439$	4.77660e6	1.18293	0.591464	+	0.806332i	$0.298550\pi$
$440$	0	0
$441$	−569490.	−0.139441
$442$	0	0
$443$	2.87476e6i	0.695973i	0.937500	+	0.347986i	$0.113134\pi$
$443$	2.87476e6i	0.695973i	−0.937500	+	0.347986i	$0.886866\pi$
$444$	0	0
$445$	− 1.52867e6i	− 0.365942i
$446$	0	0
$447$	−3.30552e6	−0.782475
$448$	0	0
$449$	5.95349e6	1.39366	0.696828	−	0.717238i	$-0.254594\pi$
$449$	5.95349e6	1.39366	0.696828	+	0.717238i	$0.254594\pi$
$450$	0	0
$451$	− 1.12782e6i	− 0.261096i
$452$	0	0
$453$	− 3.21402e6i	− 0.735873i
$454$	0	0
$455$	−1.25767e7	−2.84799
$456$	0	0
$457$	−2.01722e6	−0.451818	−0.225909	−	0.974148i	$-0.572535\pi$
$457$	−2.01722e6	−0.451818	−0.225909	+	0.974148i	$0.572535\pi$
$458$	0	0
$459$	− 929889.i	− 0.206015i
$460$	0	0
$461$	2.25196e6i	0.493525i	0.969076	+	0.246763i	$0.0793668\pi$
$461$	2.25196e6i	0.493525i	−0.969076	+	0.246763i	$0.920633\pi$
$462$	0	0
$463$	−3.74465e6	−0.811819	−0.405909	−	0.913913i	$-0.633045\pi$
$463$	−3.74465e6	−0.811819	−0.405909	+	0.913913i	$0.633045\pi$
$464$	0	0
$465$	1.62449e6	0.348406
$466$	0	0
$467$	− 8.62528e6i	− 1.83013i	−0.403311	−	0.915063i	$-0.632141\pi$
$467$	− 8.62528e6i	− 1.83013i	0.403311	−	0.915063i	$-0.367859\pi$
$468$	0	0
$469$	− 5.22424e6i	− 1.09671i
$470$	0	0
$471$	2.30937e6	0.479668
$472$	0	0
$473$	−1.18570e6	−0.243681
$474$	0	0
$475$	2.72394e6i	0.553941i
$476$	0	0
$477$	− 649900.i	− 0.130783i
$478$	0	0
$479$	6.32087e6	1.25875	0.629373	−	0.777103i	$-0.283312\pi$
$479$	6.32087e6	1.25875	0.629373	+	0.777103i	$0.283312\pi$
$480$	0	0
$481$	557759.	0.109922
$482$	0	0
$483$	3.52066e6i	0.686682i
$484$	0	0
$485$	− 1.73937e7i	− 3.35767i
$486$	0	0
$487$	−4.53590e6	−0.866644	−0.433322	−	0.901239i	$-0.642659\pi$
$487$	−4.53590e6	−0.866644	−0.433322	+	0.901239i	$0.642659\pi$
$488$	0	0
$489$	5.04556e6	0.954196
$490$	0	0
$491$	2.68402e6i	0.502438i	0.967930	+	0.251219i	$0.0808314\pi$
$491$	2.68402e6i	0.502438i	−0.967930	+	0.251219i	$0.919169\pi$
$492$	0	0
$493$	324119.i	0.0600602i
$494$	0	0
$495$	690557.	0.126674
$496$	0	0
$497$	−1.12378e7	−2.04075
$498$	0	0
$499$	− 2.57253e6i	− 0.462496i	−0.972895	−	0.231248i	$-0.925719\pi$
$499$	− 2.57253e6i	− 0.462496i	0.972895	−	0.231248i	$-0.0742810\pi$
$500$	0	0
$501$	3.69843e6i	0.658298i
$502$	0	0
$503$	−4.22685e6	−0.744898	−0.372449	−	0.928053i	$-0.621482\pi$
$503$	−4.22685e6	−0.744898	−0.372449	+	0.928053i	$0.621482\pi$
$504$	0	0
$505$	7.64252e6	1.33355
$506$	0	0
$507$	− 2.60602e6i	− 0.450255i
$508$	0	0
$509$	2.93969e6i	0.502929i	0.967866	+	0.251465i	$0.0809122\pi$
$509$	2.93969e6i	0.502929i	−0.967866	+	0.251465i	$0.919088\pi$
$510$	0	0
$511$	110813.	0.0187732
$512$	0	0
$513$	−287135.	−0.0481718
$514$	0	0
$515$	7.13618e6i	1.18563i
$516$	0	0
$517$	− 983311.i	− 0.161795i
$518$	0	0
$519$	−3.66801e6	−0.597740
$520$	0	0
$521$	5.13499e6	0.828792	0.414396	−	0.910097i	$-0.363993\pi$
$521$	5.13499e6	0.828792	0.414396	+	0.910097i	$0.363993\pi$
$522$	0	0
$523$	479078.i	0.0765865i	0.999267	+	0.0382932i	$0.0121921\pi$
$523$	479078.i	0.0765865i	−0.999267	+	0.0382932i	$0.987808\pi$
$524$	0	0
$525$	9.60979e6i	1.52165i
$526$	0	0
$527$	−2.29772e6	−0.360388
$528$	0	0
$529$	−16898.0	−0.00262540
$530$	0	0
$531$	645404.i	0.0993334i
$532$	0	0
$533$	1.07761e7i	1.64302i
$534$	0	0
$535$	1.45118e7	2.19198
$536$	0	0
$537$	−663353.	−0.0992680
$538$	0	0
$539$	598181.i	0.0886872i
$540$	0	0
$541$	1.03133e7i	1.51497i	0.652852	+	0.757486i	$0.273572\pi$
$541$	1.03133e7i	1.51497i	−0.652852	+	0.757486i	$0.726428\pi$
$542$	0	0
$543$	−1.74894e6	−0.254552
$544$	0	0
$545$	−1.63546e6	−0.235857
$546$	0	0
$547$	− 1.10504e7i	− 1.57910i	−0.613684	−	0.789552i	$-0.710313\pi$
$547$	− 1.10504e7i	− 1.57910i	0.613684	−	0.789552i	$-0.289687\pi$
$548$	0	0
$549$	− 1.28212e6i	− 0.181550i
$550$	0	0
$551$	100083.	0.0140437
$552$	0	0
$553$	5.07097e6	0.705145
$554$	0	0
$555$	− 618757.i	− 0.0852683i
$556$	0	0
$557$	1.09232e6i	0.149181i	0.997214	+	0.0745904i	$0.0237649\pi$
$557$	1.09232e6i	0.149181i	−0.997214	+	0.0745904i	$0.976235\pi$
$558$	0	0
$559$	1.13291e7	1.53344
$560$	0	0
$561$	−976738.	−0.131030
$562$	0	0
$563$	8.03622e6i	1.06852i	0.845322	+	0.534258i	$0.179409\pi$
$563$	8.03622e6i	1.06852i	−0.845322	+	0.534258i	$0.820591\pi$
$564$	0	0
$565$	− 1.67667e7i	− 2.20967i
$566$	0	0
$567$	−1.01298e6	−0.132326
$568$	0	0
$569$	4.89482e6	0.633806	0.316903	−	0.948458i	$-0.397357\pi$
$569$	4.89482e6	0.633806	0.316903	+	0.948458i	$0.397357\pi$
$570$	0	0
$571$	629963.i	0.0808583i	0.999182	+	0.0404291i	$0.0128725\pi$
$571$	629963.i	0.0808583i	−0.999182	+	0.0404291i	$0.987127\pi$
$572$	0	0
$573$	− 1.54335e6i	− 0.196371i
$574$	0	0
$575$	1.75222e7	2.21013
$576$	0	0
$577$	−6.90487e6	−0.863408	−0.431704	−	0.902015i	$-0.642087\pi$
$577$	−6.90487e6	−0.863408	−0.431704	+	0.902015i	$0.642087\pi$
$578$	0	0
$579$	6.97013e6i	0.864061i
$580$	0	0
$581$	− 7.12220e6i	− 0.875333i
$582$	0	0
$583$	−682642.	−0.0831805
$584$	0	0
$585$	−6.59811e6	−0.797132
$586$	0	0
$587$	1.36076e7i	1.63000i	0.579463	+	0.814998i	$0.303262\pi$
$587$	1.36076e7i	1.63000i	−0.579463	+	0.814998i	$0.696738\pi$
$588$	0	0
$589$	709498.i	0.0842681i
$590$	0	0
$591$	−7.78250e6	−0.916538
$592$	0	0
$593$	2.10052e6	0.245296	0.122648	−	0.992450i	$-0.460861\pi$
$593$	2.10052e6	0.245296	0.122648	+	0.992450i	$0.460861\pi$
$594$	0	0
$595$	− 1.97342e7i	− 2.28521i
$596$	0	0
$597$	6.77444e6i	0.777924i
$598$	0	0
$599$	1.62682e6	0.185256	0.0926278	−	0.995701i	$-0.470473\pi$
$599$	1.62682e6	0.185256	0.0926278	+	0.995701i	$0.470473\pi$
$600$	0	0
$601$	4.49767e6	0.507928	0.253964	−	0.967214i	$-0.418266\pi$
$601$	4.49767e6	0.507928	0.253964	+	0.967214i	$0.418266\pi$
$602$	0	0
$603$	− 2.74079e6i	− 0.306960i
$604$	0	0
$605$	1.54125e7i	1.71193i
$606$	0	0
$607$	−5.96865e6	−0.657513	−0.328756	−	0.944415i	$-0.606630\pi$
$607$	−5.96865e6	−0.657513	−0.328756	+	0.944415i	$0.606630\pi$
$608$	0	0
$609$	353082.	0.0385773
$610$	0	0
$611$	9.39531e6i	1.01814i
$612$	0	0
$613$	3.34021e6i	0.359023i	0.983756	+	0.179511i	$0.0574517\pi$
$613$	3.34021e6i	0.359023i	−0.983756	+	0.179511i	$0.942548\pi$
$614$	0	0
$615$	1.19546e7	1.27452
$616$	0	0
$617$	7.12648e6	0.753637	0.376818	−	0.926287i	$-0.377018\pi$
$617$	7.12648e6	0.753637	0.376818	+	0.926287i	$0.377018\pi$
$618$	0	0
$619$	1.20586e7i	1.26494i	0.774586	+	0.632469i	$0.217958\pi$
$619$	1.20586e7i	1.26494i	−0.774586	+	0.632469i	$0.782042\pi$
$620$	0	0
$621$	1.84704e6i	0.192197i
$622$	0	0
$623$	2.35539e6	0.243132
$624$	0	0
$625$	1.64502e7	1.68450
$626$	0	0
$627$	301601.i	0.0306383i
$628$	0	0
$629$	875182.i	0.0882007i
$630$	0	0
$631$	−7.68926e6	−0.768796	−0.384398	−	0.923168i	$-0.625591\pi$
$631$	−7.68926e6	−0.768796	−0.384398	+	0.923168i	$0.625591\pi$
$632$	0	0
$633$	−2.38965e6	−0.237042
$634$	0	0
$635$	− 3.08291e7i	− 3.03408i
$636$	0	0
$637$	− 5.71548e6i	− 0.558090i
$638$	0	0
$639$	−5.89567e6	−0.571191
$640$	0	0
$641$	−1.55180e7	−1.49174	−0.745868	−	0.666094i	$-0.767965\pi$
$641$	−1.55180e7	−1.49174	−0.745868	+	0.666094i	$0.767965\pi$
$642$	0	0
$643$	1.57536e7i	1.50263i	0.659945	+	0.751314i	$0.270580\pi$
$643$	1.57536e7i	1.50263i	−0.659945	+	0.751314i	$0.729420\pi$
$644$	0	0
$645$	− 1.25681e7i	− 1.18951i
$646$	0	0
$647$	−1.36896e7	−1.28567	−0.642834	−	0.766006i	$-0.722241\pi$
$647$	−1.36896e7	−1.28567	−0.642834	+	0.766006i	$0.722241\pi$
$648$	0	0
$649$	677920.	0.0631781
$650$	0	0
$651$	2.50304e6i	0.231481i
$652$	0	0
$653$	4.41496e6i	0.405176i	0.979264	+	0.202588i	$0.0649352\pi$
$653$	4.41496e6i	0.405176i	−0.979264	+	0.202588i	$0.935065\pi$
$654$	0	0
$655$	−1.97526e7	−1.79895
$656$	0	0
$657$	58135.9	0.00525449
$658$	0	0
$659$	− 1.73108e6i	− 0.155276i	−0.996982	−	0.0776381i	$-0.975262\pi$
$659$	− 1.73108e6i	− 0.155276i	0.996982	−	0.0776381i	$-0.0247379\pi$
$660$	0	0
$661$	− 8.30967e6i	− 0.739742i	−0.929083	−	0.369871i	$-0.879402\pi$
$661$	− 8.30967e6i	− 0.739742i	0.929083	−	0.369871i	$-0.120598\pi$
$662$	0	0
$663$	9.33250e6	0.824545
$664$	0	0
$665$	−6.09361e6	−0.534343
$666$	0	0
$667$	− 643797.i	− 0.0560318i
$668$	0	0
$669$	− 1.88223e6i	− 0.162595i
$670$	0	0
$671$	−1.34671e6	−0.115470
$672$	0	0
$673$	−1.56921e6	−0.133550	−0.0667751	−	0.997768i	$-0.521271\pi$
$673$	−1.56921e6	−0.133550	−0.0667751	+	0.997768i	$0.521271\pi$
$674$	0	0
$675$	5.04158e6i	0.425899i
$676$	0	0
$677$	− 1.17260e7i	− 0.983282i	−0.870798	−	0.491641i	$-0.836397\pi$
$677$	− 1.17260e7i	− 0.983282i	0.870798	−	0.491641i	$-0.163603\pi$
$678$	0	0
$679$	2.68004e7	2.23083
$680$	0	0
$681$	−1.08108e6	−0.0893288
$682$	0	0
$683$	− 2.11345e7i	− 1.73356i	−0.498687	−	0.866782i	$-0.666184\pi$
$683$	− 2.11345e7i	− 1.73356i	0.498687	−	0.866782i	$-0.333816\pi$
$684$	0	0
$685$	− 4.19398e6i	− 0.341507i
$686$	0	0
$687$	−3.96553e6	−0.320560
$688$	0	0
$689$	6.52248e6	0.523438
$690$	0	0
$691$	− 1.84062e7i	− 1.46646i	−0.679981	−	0.733230i	$-0.738012\pi$
$691$	− 1.84062e7i	− 1.46646i	0.679981	−	0.733230i	$-0.261988\pi$
$692$	0	0
$693$	1.06402e6i	0.0841621i
$694$	0	0
$695$	−4.48285e7	−3.52040
$696$	0	0
$697$	−1.69088e7	−1.31835
$698$	0	0
$699$	1.50409e6i	0.116434i
$700$	0	0
$701$	− 9.57182e6i	− 0.735698i	−0.929886	−	0.367849i	$-0.880094\pi$
$701$	− 9.57182e6i	− 0.735698i	0.929886	−	0.367849i	$-0.119906\pi$
$702$	0	0
$703$	270243.	0.0206236
$704$	0	0
$705$	1.04228e7	0.789790
$706$	0	0
$707$	1.17757e7i	0.886008i
$708$	0	0
$709$	5.35207e6i	0.399859i	0.979810	+	0.199929i	$0.0640713\pi$
$709$	5.35207e6i	0.399859i	−0.979810	+	0.199929i	$0.935929\pi$
$710$	0	0
$711$	2.66038e6	0.197365
$712$	0	0
$713$	4.56395e6	0.336215
$714$	0	0
$715$	6.93053e6i	0.506992i
$716$	0	0
$717$	4.01181e6i	0.291436i
$718$	0	0
$719$	1.13450e7	0.818435	0.409217	−	0.912437i	$-0.365802\pi$
$719$	1.13450e7	0.818435	0.409217	+	0.912437i	$0.365802\pi$
$720$	0	0
$721$	−1.09955e7	−0.787730
$722$	0	0
$723$	1.37717e6i	0.0979807i
$724$	0	0
$725$	− 1.75727e6i	− 0.124164i
$726$	0	0
$727$	−5.36024e6	−0.376139	−0.188069	−	0.982156i	$-0.560223\pi$
$727$	−5.36024e6	−0.376139	−0.188069	+	0.982156i	$0.560223\pi$
$728$	0	0
$729$	−531441.	−0.0370370
$730$	0	0
$731$	1.77765e7i	1.23042i
$732$	0	0
$733$	2.07596e7i	1.42712i	0.700596	+	0.713558i	$0.252918\pi$
$733$	2.07596e7i	1.42712i	−0.700596	+	0.713558i	$0.747082\pi$
$734$	0	0
$735$	−6.34054e6	−0.432920
$736$	0	0
$737$	−2.87887e6	−0.195233
$738$	0	0
$739$	2.02904e7i	1.36672i	0.730082	+	0.683359i	$0.239482\pi$
$739$	2.02904e7i	1.36672i	−0.730082	+	0.683359i	$0.760518\pi$
$740$	0	0
$741$	− 2.88173e6i	− 0.192800i
$742$	0	0
$743$	2.10215e7	1.39699	0.698493	−	0.715617i	$-0.253855\pi$
$743$	2.10215e7	1.39699	0.698493	+	0.715617i	$0.253855\pi$
$744$	0	0
$745$	−3.68027e7	−2.42934
$746$	0	0
$747$	− 3.73651e6i	− 0.244999i
$748$	0	0
$749$	2.23599e7i	1.45635i
$750$	0	0
$751$	−3.49173e6	−0.225913	−0.112956	−	0.993600i	$-0.536032\pi$
$751$	−3.49173e6	−0.225913	−0.112956	+	0.993600i	$0.536032\pi$
$752$	0	0
$753$	−6.24023e6	−0.401063
$754$	0	0
$755$	− 3.57840e7i	− 2.28466i
$756$	0	0
$757$	− 2.15165e7i	− 1.36468i	−0.731033	−	0.682342i	$-0.760961\pi$
$757$	− 2.15165e7i	− 1.36468i	0.731033	−	0.682342i	$-0.239039\pi$
$758$	0	0
$759$	1.94010e6	0.122242
$760$	0	0
$761$	−1.62464e7	−1.01694	−0.508469	−	0.861080i	$-0.669788\pi$
$761$	−1.62464e7	−1.01694	−0.508469	+	0.861080i	$0.669788\pi$
$762$	0	0
$763$	− 2.51993e6i	− 0.156703i
$764$	0	0
$765$	− 1.03531e7i	− 0.639614i
$766$	0	0
$767$	−6.47736e6	−0.397567
$768$	0	0
$769$	1.89849e7	1.15769	0.578846	−	0.815437i	$-0.303503\pi$
$769$	1.89849e7	1.15769	0.578846	+	0.815437i	$0.303503\pi$
$770$	0	0
$771$	− 1.54698e7i	− 0.937233i
$772$	0	0
$773$	9.61884e6i	0.578994i	0.957179	+	0.289497i	$0.0934880\pi$
$773$	9.61884e6i	0.578994i	−0.957179	+	0.289497i	$0.906512\pi$
$774$	0	0
$775$	1.24575e7	0.745036
$776$	0	0
$777$	953388.	0.0566523
$778$	0	0
$779$	5.22118e6i	0.308265i
$780$	0	0
$781$	6.19270e6i	0.363289i
$782$	0	0
$783$	185237.	0.0107975
$784$	0	0
$785$	2.57119e7	1.48922
$786$	0	0
$787$	− 9.93586e6i	− 0.571832i	−0.958255	−	0.285916i	$-0.907702\pi$
$787$	− 9.93586e6i	− 0.571832i	0.958255	−	0.285916i	$-0.0922979\pi$
$788$	0	0
$789$	1.40968e6i	0.0806174i
$790$	0	0
$791$	2.58343e7	1.46810
$792$	0	0
$793$	1.28675e7	0.726628
$794$	0	0
$795$	− 7.23580e6i	− 0.406040i
$796$	0	0
$797$	− 2.77653e7i	− 1.54831i	−0.632998	−	0.774154i	$-0.718176\pi$
$797$	− 2.77653e7i	− 1.54831i	0.632998	−	0.774154i	$-0.281824\pi$
$798$	0	0
$799$	−1.47422e7	−0.816951
$800$	0	0
$801$	1.23570e6	0.0680508
$802$	0	0
$803$	− 61064.8i	− 0.00334197i
$804$	0	0
$805$	3.91980e7i	2.13194i
$806$	0	0
$807$	7.20001e6	0.389179
$808$	0	0
$809$	2.70847e7	1.45497	0.727484	−	0.686125i	$-0.240689\pi$
$809$	2.70847e7	1.45497	0.727484	+	0.686125i	$0.240689\pi$
$810$	0	0
$811$	2.00386e7i	1.06983i	0.844905	+	0.534917i	$0.179657\pi$
$811$	2.00386e7i	1.06983i	−0.844905	+	0.534917i	$0.820343\pi$
$812$	0	0
$813$	− 1.70637e7i	− 0.905416i
$814$	0	0
$815$	5.61759e7	2.96249
$816$	0	0
$817$	5.48911e6	0.287705
$818$	0	0
$819$	− 1.01665e7i	− 0.529614i
$820$	0	0
$821$	3.68572e6i	0.190838i	0.995437	+	0.0954188i	$0.0304190\pi$
$821$	3.68572e6i	0.190838i	−0.995437	+	0.0954188i	$0.969581\pi$
$822$	0	0
$823$	−2.22928e7	−1.14727	−0.573633	−	0.819112i	$-0.694467\pi$
$823$	−2.22928e7	−1.14727	−0.573633	+	0.819112i	$0.694467\pi$
$824$	0	0
$825$	5.29558e6	0.270881
$826$	0	0
$827$	1.98390e7i	1.00869i	0.863503	+	0.504343i	$0.168265\pi$
$827$	1.98390e7i	1.00869i	−0.863503	+	0.504343i	$0.831735\pi$
$828$	0	0
$829$	4.83084e6i	0.244139i	0.992522	+	0.122069i	$0.0389530\pi$
$829$	4.83084e6i	0.244139i	−0.992522	+	0.122069i	$0.961047\pi$
$830$	0	0
$831$	2.51823e6	0.126500
$832$	0	0
$833$	8.96819e6	0.447808
$834$	0	0
$835$	4.11773e7i	2.04381i
$836$	0	0
$837$	1.31317e6i	0.0647898i
$838$	0	0
$839$	−2.24080e7	−1.09900	−0.549501	−	0.835493i	$-0.685182\pi$
$839$	−2.24080e7	−1.09900	−0.549501	+	0.835493i	$0.685182\pi$
$840$	0	0
$841$	2.04466e7	0.996852
$842$	0	0
$843$	4.98797e6i	0.241743i
$844$	0	0
$845$	− 2.90147e7i	− 1.39790i
$846$	0	0
$847$	−2.37478e7	−1.13740
$848$	0	0
$849$	1.41910e7	0.675686
$850$	0	0
$851$	− 1.73838e6i	− 0.0822848i
$852$	0	0
$853$	2.55770e7i	1.20358i	0.798653	+	0.601792i	$0.205547\pi$
$853$	2.55770e7i	1.20358i	−0.798653	+	0.601792i	$0.794453\pi$
$854$	0	0
$855$	−3.19688e6	−0.149559
$856$	0	0
$857$	−1.94781e7	−0.905929	−0.452964	−	0.891529i	$-0.649634\pi$
$857$	−1.94781e7	−0.905929	−0.452964	+	0.891529i	$0.649634\pi$
$858$	0	0
$859$	− 2.93437e7i	− 1.35685i	−0.734670	−	0.678424i	$-0.762663\pi$
$859$	− 2.93437e7i	− 1.35685i	0.734670	−	0.678424i	$-0.237337\pi$
$860$	0	0
$861$	1.84198e7i	0.846792i
$862$	0	0
$863$	−9.84655e6	−0.450046	−0.225023	−	0.974353i	$-0.572246\pi$
$863$	−9.84655e6	−0.450046	−0.225023	+	0.974353i	$0.572246\pi$
$864$	0	0
$865$	−4.08387e7	−1.85580
$866$	0	0
$867$	1.86496e6i	0.0842601i
$868$	0	0
$869$	− 2.79441e6i	− 0.125528i
$870$	0	0
$871$	2.75069e7	1.22856
$872$	0	0
$873$	1.40603e7	0.624394
$874$	0	0
$875$	5.86462e7i	2.58952i
$876$	0	0
$877$	− 1.09591e7i	− 0.481147i	−0.970631	−	0.240573i	$-0.922665\pi$
$877$	− 1.09591e7i	− 0.481147i	0.970631	−	0.240573i	$-0.0773355\pi$
$878$	0	0
$879$	1.66107e6	0.0725128
$880$	0	0
$881$	−4.72688e6	−0.205180	−0.102590	−	0.994724i	$-0.532713\pi$
$881$	−4.72688e6	−0.205180	−0.102590	+	0.994724i	$0.532713\pi$
$882$	0	0
$883$	3.23858e7i	1.39782i	0.715207	+	0.698912i	$0.246332\pi$
$883$	3.23858e7i	1.39782i	−0.715207	+	0.698912i	$0.753668\pi$
$884$	0	0
$885$	7.18575e6i	0.308400i
$886$	0	0
$887$	−4.35277e7	−1.85762	−0.928809	−	0.370559i	$-0.879166\pi$
$887$	−4.35277e7	−1.85762	−0.928809	+	0.370559i	$0.879166\pi$
$888$	0	0
$889$	4.75018e7	2.01584
$890$	0	0
$891$	558216.i	0.0235563i
$892$	0	0
$893$	4.55217e6i	0.191025i
$894$	0	0
$895$	−7.38559e6	−0.308197
$896$	0	0
$897$	−1.85372e7	−0.769240
$898$	0	0
$899$	− 457713.i	− 0.0188883i
$900$	0	0
$901$	1.02345e7i	0.420004i
$902$	0	0
$903$	1.93650e7	0.790312
$904$	0	0
$905$	−1.94722e7	−0.790304
$906$	0	0
$907$	− 4.49574e7i	− 1.81461i	−0.420477	−	0.907303i	$-0.638137\pi$
$907$	− 4.49574e7i	− 1.81461i	0.420477	−	0.907303i	$-0.361863\pi$
$908$	0	0
$909$	6.17787e6i	0.247987i
$910$	0	0
$911$	3.43483e7	1.37123	0.685614	−	0.727965i	$-0.259534\pi$
$911$	3.43483e7	1.37123	0.685614	+	0.727965i	$0.259534\pi$
$912$	0	0
$913$	−3.92476e6	−0.155825
$914$	0	0
$915$	− 1.42748e7i	− 0.563658i
$916$	0	0
$917$	− 3.04350e7i	− 1.19522i
$918$	0	0
$919$	1.51792e7	0.592871	0.296435	−	0.955053i	$-0.404202\pi$
$919$	1.51792e7	0.592871	0.296435	+	0.955053i	$0.404202\pi$
$920$	0	0
$921$	3.65290e6	0.141902
$922$	0	0
$923$	− 5.91698e7i	− 2.28610i
$924$	0	0
$925$	− 4.74497e6i	− 0.182339i
$926$	0	0
$927$	−5.76856e6	−0.220480
$928$	0	0
$929$	−3.12691e7	−1.18871	−0.594356	−	0.804202i	$-0.702593\pi$
$929$	−3.12691e7	−1.18871	−0.594356	+	0.804202i	$0.702593\pi$
$930$	0	0
$931$	− 2.76923e6i	− 0.104709i
$932$	0	0
$933$	− 2.28994e6i	− 0.0861232i
$934$	0	0
$935$	−1.08747e7	−0.406808
$936$	0	0
$937$	2.67948e7	0.997015	0.498507	−	0.866885i	$-0.333882\pi$
$937$	2.67948e7	0.997015	0.498507	+	0.866885i	$0.333882\pi$
$938$	0	0
$939$	9.34759e6i	0.345968i
$940$	0	0
$941$	− 8.36072e6i	− 0.307801i	−0.988086	−	0.153900i	$-0.950817\pi$
$941$	− 8.36072e6i	− 0.307801i	0.988086	−	0.153900i	$-0.0491835\pi$
$942$	0	0
$943$	3.35860e7	1.22993
$944$	0	0
$945$	−1.12783e7	−0.410831
$946$	0	0
$947$	− 4.40994e7i	− 1.59793i	−0.601377	−	0.798965i	$-0.705381\pi$
$947$	− 4.40994e7i	− 1.59793i	0.601377	−	0.798965i	$-0.294619\pi$
$948$	0	0
$949$	583460.i	0.0210303i
$950$	0	0
$951$	1.25919e7	0.451482
$952$	0	0
$953$	1.99247e7	0.710658	0.355329	−	0.934741i	$-0.384369\pi$
$953$	1.99247e7	0.710658	0.355329	+	0.934741i	$0.384369\pi$
$954$	0	0
$955$	− 1.71832e7i	− 0.609673i
$956$	0	0
$957$	− 194569.i	− 0.00686744i
$958$	0	0
$959$	6.46213e6	0.226897
$960$	0	0
$961$	−2.53844e7	−0.886662
$962$	0	0
$963$	1.17307e7i	0.407621i
$964$	0	0
$965$	7.76035e7i	2.68264i
$966$	0	0
$967$	−4.56359e7	−1.56942	−0.784712	−	0.619860i	$-0.787189\pi$
$967$	−4.56359e7	−1.56942	−0.784712	+	0.619860i	$0.787189\pi$
$968$	0	0
$969$	4.52173e6	0.154702
$970$	0	0
$971$	− 2.09777e7i	− 0.714020i	−0.934101	−	0.357010i	$-0.883796\pi$
$971$	− 2.09777e7i	− 0.714020i	0.934101	−	0.357010i	$-0.116204\pi$
$972$	0	0
$973$	− 6.90723e7i	− 2.33895i
$974$	0	0
$975$	−5.05980e7	−1.70460
$976$	0	0
$977$	−3.03128e7	−1.01599	−0.507995	−	0.861360i	$-0.669613\pi$
$977$	−3.03128e7	−1.01599	−0.507995	+	0.861360i	$0.669613\pi$
$978$	0	0
$979$	− 1.29796e6i	− 0.0432817i
$980$	0	0
$981$	− 1.32203e6i	− 0.0438600i
$982$	0	0
$983$	−2.04558e7	−0.675200	−0.337600	−	0.941290i	$-0.609615\pi$
$983$	−2.04558e7	−0.675200	−0.337600	+	0.941290i	$0.609615\pi$
$984$	0	0
$985$	−8.66483e7	−2.84557
$986$	0	0
$987$	1.60596e7i	0.524737i
$988$	0	0
$989$	− 3.53095e7i	− 1.14789i
$990$	0	0
$991$	1.72449e7	0.557797	0.278898	−	0.960321i	$-0.410031\pi$
$991$	1.72449e7	0.557797	0.278898	+	0.960321i	$0.410031\pi$
$992$	0	0
$993$	−1.84200e7	−0.592811
$994$	0	0
$995$	7.54247e7i	2.41522i
$996$	0	0
$997$	− 7.27445e6i	− 0.231773i	−0.993262	−	0.115886i	$-0.963029\pi$
$997$	− 7.27445e6i	− 0.231773i	0.993262	−	0.115886i	$-0.0369708\pi$
$998$	0	0
$999$	500175.	0.0158565

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	96.6.d.a.49.10		10
3.2	odd	2		288.6.d.d.145.1		10
4.3	odd	2		24.6.d.a.13.8	yes	10
8.3	odd	2		24.6.d.a.13.7	✓	10
8.5	even	2	inner	96.6.d.a.49.1		10
12.11	even	2		72.6.d.d.37.3		10
16.3	odd	4		768.6.a.bd.1.5		5
16.5	even	4		768.6.a.bc.1.1		5
16.11	odd	4		768.6.a.ba.1.1		5
16.13	even	4		768.6.a.bb.1.5		5
24.5	odd	2		288.6.d.d.145.10		10
24.11	even	2		72.6.d.d.37.4		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.6.d.a.13.7	✓	10	8.3	odd	2
24.6.d.a.13.8	yes	10	4.3	odd	2
72.6.d.d.37.3		10	12.11	even	2
72.6.d.d.37.4		10	24.11	even	2
96.6.d.a.49.1		10	8.5	even	2	inner
96.6.d.a.49.10		10	1.1	even	1	trivial
288.6.d.d.145.1		10	3.2	odd	2
288.6.d.d.145.10		10	24.5	odd	2
768.6.a.ba.1.1		5	16.11	odd	4
768.6.a.bb.1.5		5	16.13	even	4
768.6.a.bc.1.1		5	16.5	even	4
768.6.a.bd.1.5		5	16.3	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 96 = 2^{5} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	96.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+9.00000i q^{3} +100.204i q^{5} -154.395 q^{7} -81.0000 q^{9} +85.0809i q^{11} -812.927i q^{13} -901.832 q^{15} +1275.57 q^{17} -393.875i q^{19} -1389.55i q^{21} -2533.66 q^{23} -6915.74 q^{25} -729.000i q^{27} +254.097i q^{29} -1801.33 q^{31} -765.728 q^{33} -15470.9i q^{35} +686.112i q^{37} +7316.35 q^{39} -13255.9 q^{41} +13936.2i q^{43} -8116.48i q^{45} -11557.4 q^{47} +7030.74 q^{49} +11480.1i q^{51} +8023.45i q^{53} -8525.40 q^{55} +3544.88 q^{57} -7967.95i q^{59} +15828.6i q^{61} +12506.0 q^{63} +81458.2 q^{65} +33836.9i q^{67} -22803.0i q^{69} +72786.0 q^{71} -717.727 q^{73} -62241.7i q^{75} -13136.0i q^{77} -32844.2 q^{79} +6561.00 q^{81} +46129.8i q^{83} +127816. i q^{85} -2286.88 q^{87} -15255.6 q^{89} +125512. i q^{91} -16211.9i q^{93} +39467.7 q^{95} -173584. q^{97} -6891.55i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 196 q^{7} - 810 q^{9} - 900 q^{15} - 404 q^{17} - 1672 q^{23} - 9366 q^{25} - 18692 q^{31} + 12168 q^{39} - 2476 q^{41} - 65256 q^{47} + 9378 q^{49} + 103424 q^{55} - 12888 q^{57} - 15876 q^{63}+ \cdots + 59372 q^{97}+O(q^{100}) \) 10 * q + 196 * q^7 - 810 * q^9 - 900 * q^15 - 404 * q^17 - 1672 * q^23 - 9366 * q^25 - 18692 * q^31 + 12168 * q^39 - 2476 * q^41 - 65256 * q^47 + 9378 * q^49 + 103424 * q^55 - 12888 * q^57 - 15876 * q^63 + 51536 * q^65 + 272232 * q^71 - 27388 * q^73 - 252308 * q^79 + 65610 * q^81 + 90828 * q^87 + 175580 * q^89 - 243184 * q^95 + 59372 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more