LMFDB - Embedded newform 384.6.d.g.193.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,6,Mod(193,384)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(384, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("384.193");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	384.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$61.5873868082$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{61})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 31x^{2} + 225 $ x^4 + 31*x^2 + 225
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			193.4
Root			$3.40512i$ of defining polynomial
Character	$\chi$	$=$	384.193
Dual form			384.6.d.g.193.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+9.00000i q^{3} +11.2410i q^{5} +148.205 q^{7} -81.0000 q^{9} +O(q^{10})$	$q+9.00000i q^{3} +11.2410i q^{5} +148.205 q^{7} -81.0000 q^{9} -47.0360i q^{11} +545.374i q^{13} -101.169 q^{15} -1162.53 q^{17} +606.748i q^{19} +1333.84i q^{21} -1104.70 q^{23} +2998.64 q^{25} -729.000i q^{27} +2355.82i q^{29} +2949.39 q^{31} +423.324 q^{33} +1665.97i q^{35} -3567.52i q^{37} -4908.37 q^{39} +14129.3 q^{41} +17714.4i q^{43} -910.521i q^{45} -11597.8 q^{47} +5157.72 q^{49} -10462.8i q^{51} -1514.95i q^{53} +528.732 q^{55} -5460.73 q^{57} -10085.1i q^{59} +35214.0i q^{61} -12004.6 q^{63} -6130.55 q^{65} +28755.0i q^{67} -9942.28i q^{69} -53443.0 q^{71} -43934.9 q^{73} +26987.8i q^{75} -6970.97i q^{77} -26786.0 q^{79} +6561.00 q^{81} +105107. i q^{83} -13068.0i q^{85} -21202.4 q^{87} -47927.2 q^{89} +80827.1i q^{91} +26544.5i q^{93} -6820.45 q^{95} +38270.6 q^{97} +3809.92i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 32 q^{7} - 324 q^{9}+O(q^{10}) $ 4 * q - 32 * q^7 - 324 * q^9	$ 4 q - 32 q^{7} - 324 q^{9} + 720 q^{15} + 1848 q^{17} - 7168 q^{23} + 6996 q^{25} - 7072 q^{31} + 6192 q^{33} - 3888 q^{39} + 5032 q^{41} - 9152 q^{47} + 30628 q^{49} - 29376 q^{55} + 9648 q^{57} + 2592 q^{63} - 46016 q^{65} - 151040 q^{71} + 34200 q^{73} - 200992 q^{79} + 26244 q^{81} - 155664 q^{87} - 180712 q^{89} - 130752 q^{95} - 199816 q^{97}+O(q^{100}) $ 4 * q - 32 * q^7 - 324 * q^9 + 720 * q^15 + 1848 * q^17 - 7168 * q^23 + 6996 * q^25 - 7072 * q^31 + 6192 * q^33 - 3888 * q^39 + 5032 * q^41 - 9152 * q^47 + 30628 * q^49 - 29376 * q^55 + 9648 * q^57 + 2592 * q^63 - 46016 * q^65 - 151040 * q^71 + 34200 * q^73 - 200992 * q^79 + 26244 * q^81 - 155664 * q^87 - 180712 * q^89 - 130752 * q^95 - 199816 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/384\mathbb{Z}\right)^\times$.

$n$	$127$	$133$	$257$
$\chi(n)$	$1$	$-1$	$1$

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
$5$	11.2410i	0.201085i	0.994933	+	0.100543i	$0.0320579\pi$
$5$	11.2410i	0.201085i	−0.994933	+	0.100543i	$0.967942\pi$
$6$	0	0
$7$	148.205	1.14319	0.571594	−	0.820537i	$-0.306325\pi$
$7$	148.205	1.14319	0.571594	+	0.820537i	$0.306325\pi$
$8$	0	0
$9$	−81.0000	−0.333333
$10$	0	0
$11$	− 47.0360i	− 0.117206i	−0.998281	−	0.0586028i	$-0.981335\pi$
$11$	− 47.0360i	− 0.117206i	0.998281	−	0.0586028i	$-0.0186646\pi$
$12$	0	0
$13$	545.374i	0.895027i	0.894277	+	0.447513i	$0.147690\pi$
$13$	545.374i	0.895027i	−0.894277	+	0.447513i	$0.852310\pi$
$14$	0	0
$15$	−101.169	−0.116097
$16$	0	0
$17$	−1162.53	−0.975624	−0.487812	−	0.872949i	$-0.662205\pi$
$17$	−1162.53	−0.975624	−0.487812	+	0.872949i	$0.662205\pi$
$18$	0	0
$19$	606.748i	0.385589i	0.981239	+	0.192794i	$0.0617550\pi$
$19$	606.748i	0.385589i	−0.981239	+	0.192794i	$0.938245\pi$
$20$	0	0
$21$	1333.84i	0.660020i
$22$	0	0
$23$	−1104.70	−0.435436	−0.217718	−	0.976012i	$-0.569861\pi$
$23$	−1104.70	−0.435436	−0.217718	+	0.976012i	$0.569861\pi$
$24$	0	0
$25$	2998.64	0.959565
$26$	0	0
$27$	− 729.000i	− 0.192450i
$28$	0	0
$29$	2355.82i	0.520172i	0.965586	+	0.260086i	$0.0837508\pi$
$29$	2355.82i	0.520172i	−0.965586	+	0.260086i	$0.916249\pi$
$30$	0	0
$31$	2949.39	0.551224	0.275612	−	0.961269i	$-0.411120\pi$
$31$	2949.39	0.551224	0.275612	+	0.961269i	$0.411120\pi$
$32$	0	0
$33$	423.324	0.0676687
$34$	0	0
$35$	1665.97i	0.229878i
$36$	0	0
$37$	− 3567.52i	− 0.428413i	−0.976788	−	0.214206i	$-0.931283\pi$
$37$	− 3567.52i	− 0.428413i	0.976788	−	0.214206i	$-0.0687165\pi$
$38$	0	0
$39$	−4908.37	−0.516744
$40$	0	0
$41$	14129.3	1.31269	0.656343	−	0.754463i	$-0.272103\pi$
$41$	14129.3	1.31269	0.656343	+	0.754463i	$0.272103\pi$
$42$	0	0
$43$	17714.4i	1.46102i	0.682902	+	0.730510i	$0.260717\pi$
$43$	17714.4i	1.46102i	−0.682902	+	0.730510i	$0.739283\pi$
$44$	0	0
$45$	− 910.521i	− 0.0670284i
$46$	0	0
$47$	−11597.8	−0.765829	−0.382914	−	0.923784i	$-0.625080\pi$
$47$	−11597.8	−0.765829	−0.382914	+	0.923784i	$0.625080\pi$
$48$	0	0
$49$	5157.72	0.306879
$50$	0	0
$51$	− 10462.8i	− 0.563277i
$52$	0	0
$53$	− 1514.95i	− 0.0740814i	−0.999314	−	0.0370407i	$-0.988207\pi$
$53$	− 1514.95i	− 0.0740814i	0.999314	−	0.0370407i	$-0.0117931\pi$
$54$	0	0
$55$	528.732	0.0235683
$56$	0	0
$57$	−5460.73	−0.222620
$58$	0	0
$59$	− 10085.1i	− 0.377181i	−0.982056	−	0.188590i	$-0.939608\pi$
$59$	− 10085.1i	− 0.377181i	0.982056	−	0.188590i	$-0.0603919\pi$
$60$	0	0
$61$	35214.0i	1.21169i	0.795584	+	0.605843i	$0.207164\pi$
$61$	35214.0i	1.21169i	−0.795584	+	0.605843i	$0.792836\pi$
$62$	0	0
$63$	−12004.6	−0.381063
$64$	0	0
$65$	−6130.55	−0.179977
$66$	0	0
$67$	28755.0i	0.782576i	0.920268	+	0.391288i	$0.127970\pi$
$67$	28755.0i	0.782576i	−0.920268	+	0.391288i	$0.872030\pi$
$68$	0	0
$69$	− 9942.28i	− 0.251399i
$70$	0	0
$71$	−53443.0	−1.25819	−0.629093	−	0.777330i	$-0.716574\pi$
$71$	−53443.0	−1.25819	−0.629093	+	0.777330i	$0.716574\pi$
$72$	0	0
$73$	−43934.9	−0.964944	−0.482472	−	0.875911i	$-0.660261\pi$
$73$	−43934.9	−0.964944	−0.482472	+	0.875911i	$0.660261\pi$
$74$	0	0
$75$	26987.8i	0.554005i
$76$	0	0
$77$	− 6970.97i	− 0.133988i
$78$	0	0
$79$	−26786.0	−0.482881	−0.241441	−	0.970416i	$-0.577620\pi$
$79$	−26786.0	−0.482881	−0.241441	+	0.970416i	$0.577620\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	105107.i	1.67470i	0.546671	+	0.837348i	$0.315895\pi$
$83$	105107.i	1.67470i	−0.546671	+	0.837348i	$0.684105\pi$
$84$	0	0
$85$	− 13068.0i	− 0.196183i
$86$	0	0
$87$	−21202.4	−0.300321
$88$	0	0
$89$	−47927.2	−0.641368	−0.320684	−	0.947186i	$-0.603913\pi$
$89$	−47927.2	−0.641368	−0.320684	+	0.947186i	$0.603913\pi$
$90$	0	0
$91$	80827.1i	1.02318i
$92$	0	0
$93$	26544.5i	0.318249i
$94$	0	0
$95$	−6820.45	−0.0775361
$96$	0	0
$97$	38270.6	0.412986	0.206493	−	0.978448i	$-0.433795\pi$
$97$	38270.6	0.412986	0.206493	+	0.978448i	$0.433795\pi$
$98$	0	0
$99$	3809.92i	0.0390686i
$100$	0	0
$101$	− 128974.i	− 1.25805i	−0.777385	−	0.629025i	$-0.783454\pi$
$101$	− 128974.i	− 1.25805i	0.777385	−	0.629025i	$-0.216546\pi$
$102$	0	0
$103$	−202213.	−1.87809	−0.939044	−	0.343798i	$-0.888287\pi$
$103$	−202213.	−1.87809	−0.939044	+	0.343798i	$0.888287\pi$
$104$	0	0
$105$	−14993.7	−0.132720
$106$	0	0
$107$	50910.8i	0.429883i	0.976627	+	0.214942i	$0.0689561\pi$
$107$	50910.8i	0.429883i	−0.976627	+	0.214942i	$0.931044\pi$
$108$	0	0
$109$	− 86786.3i	− 0.699656i	−0.936814	−	0.349828i	$-0.886240\pi$
$109$	− 86786.3i	− 0.699656i	0.936814	−	0.349828i	$-0.113760\pi$
$110$	0	0
$111$	32107.7	0.247344
$112$	0	0
$113$	10281.4	0.0757457	0.0378729	−	0.999283i	$-0.487942\pi$
$113$	10281.4	0.0757457	0.0378729	+	0.999283i	$0.487942\pi$
$114$	0	0
$115$	− 12417.9i	− 0.0875596i
$116$	0	0
$117$	− 44175.3i	− 0.298342i
$118$	0	0
$119$	−172293.	−1.11532
$120$	0	0
$121$	158839.	0.986263
$122$	0	0
$123$	127164.i	0.757879i
$124$	0	0
$125$	68835.8i	0.394039i
$126$	0	0
$127$	−118323.	−0.650966	−0.325483	−	0.945548i	$-0.605527\pi$
$127$	−118323.	−0.650966	−0.325483	+	0.945548i	$0.605527\pi$
$128$	0	0
$129$	−159430.	−0.843520
$130$	0	0
$131$	328293.i	1.67141i	0.549179	+	0.835705i	$0.314940\pi$
$131$	328293.i	1.67141i	−0.549179	+	0.835705i	$0.685060\pi$
$132$	0	0
$133$	89923.1i	0.440800i
$134$	0	0
$135$	8194.69	0.0386988
$136$	0	0
$137$	−11172.7	−0.0508579	−0.0254289	−	0.999677i	$-0.508095\pi$
$137$	−11172.7	−0.0508579	−0.0254289	+	0.999677i	$0.508095\pi$
$138$	0	0
$139$	109510.i	0.480748i	0.970680	+	0.240374i	$0.0772701\pi$
$139$	109510.i	0.480748i	−0.970680	+	0.240374i	$0.922730\pi$
$140$	0	0
$141$	− 104380.i	− 0.442151i
$142$	0	0
$143$	25652.2	0.104902
$144$	0	0
$145$	−26481.7	−0.104599
$146$	0	0
$147$	46419.5i	0.177177i
$148$	0	0
$149$	309729.i	1.14292i	0.820629	+	0.571461i	$0.193623\pi$
$149$	309729.i	1.14292i	−0.820629	+	0.571461i	$0.806377\pi$
$150$	0	0
$151$	−461953.	−1.64875	−0.824377	−	0.566041i	$-0.808474\pi$
$151$	−461953.	−1.64875	−0.824377	+	0.566041i	$0.808474\pi$
$152$	0	0
$153$	94165.1	0.325208
$154$	0	0
$155$	33154.1i	0.110843i
$156$	0	0
$157$	434846.i	1.40795i	0.710227	+	0.703973i	$0.248592\pi$
$157$	434846.i	1.40795i	−0.710227	+	0.703973i	$0.751408\pi$
$158$	0	0
$159$	13634.6	0.0427709
$160$	0	0
$161$	−163722.	−0.497785
$162$	0	0
$163$	119362.i	0.351883i	0.984401	+	0.175941i	$0.0562969\pi$
$163$	119362.i	0.351883i	−0.984401	+	0.175941i	$0.943703\pi$
$164$	0	0
$165$	4758.59i	0.0136072i
$166$	0	0
$167$	−460259.	−1.27706	−0.638530	−	0.769597i	$-0.720457\pi$
$167$	−460259.	−1.27706	−0.638530	+	0.769597i	$0.720457\pi$
$168$	0	0
$169$	73860.2	0.198927
$170$	0	0
$171$	− 49146.6i	− 0.128530i
$172$	0	0
$173$	− 465904.i	− 1.18353i	−0.806109	−	0.591767i	$-0.798431\pi$
$173$	− 465904.i	− 1.18353i	0.806109	−	0.591767i	$-0.201569\pi$
$174$	0	0
$175$	444413.	1.09696
$176$	0	0
$177$	90765.8	0.217765
$178$	0	0
$179$	− 143810.i	− 0.335471i	−0.985832	−	0.167736i	$-0.946355\pi$
$179$	− 143810.i	− 0.335471i	0.985832	−	0.167736i	$-0.0536455\pi$
$180$	0	0
$181$	281826.i	0.639418i	0.947516	+	0.319709i	$0.103585\pi$
$181$	281826.i	0.639418i	−0.947516	+	0.319709i	$0.896415\pi$
$182$	0	0
$183$	−316926.	−0.699567
$184$	0	0
$185$	40102.5	0.0861475
$186$	0	0
$187$	54680.9i	0.114349i
$188$	0	0
$189$	− 108041.i	− 0.220007i
$190$	0	0
$191$	319313.	0.633335	0.316667	−	0.948537i	$-0.397436\pi$
$191$	319313.	0.633335	0.316667	+	0.948537i	$0.397436\pi$
$192$	0	0
$193$	755250.	1.45948	0.729739	−	0.683726i	$-0.239642\pi$
$193$	755250.	1.45948	0.729739	+	0.683726i	$0.239642\pi$
$194$	0	0
$195$	− 55174.9i	− 0.103910i
$196$	0	0
$197$	− 69303.0i	− 0.127229i	−0.997975	−	0.0636146i	$-0.979737\pi$
$197$	− 69303.0i	− 0.127229i	0.997975	−	0.0636146i	$-0.0202628\pi$
$198$	0	0
$199$	−16621.2	−0.0297530	−0.0148765	−	0.999889i	$-0.504736\pi$
$199$	−16621.2	−0.0297530	−0.0148765	+	0.999889i	$0.504736\pi$
$200$	0	0
$201$	−258795.	−0.451821
$202$	0	0
$203$	349144.i	0.594654i
$204$	0	0
$205$	158827.i	0.263962i
$206$	0	0
$207$	89480.5	0.145145
$208$	0	0
$209$	28539.0	0.0451932
$210$	0	0
$211$	412143.i	0.637297i	0.947873	+	0.318648i	$0.103229\pi$
$211$	412143.i	0.637297i	−0.947873	+	0.318648i	$0.896771\pi$
$212$	0	0
$213$	− 480987.i	− 0.726414i
$214$	0	0
$215$	−199128.	−0.293789
$216$	0	0
$217$	437114.	0.630153
$218$	0	0
$219$	− 395414.i	− 0.557111i
$220$	0	0
$221$	− 634015.i	− 0.873210i
$222$	0	0
$223$	1.19123e6	1.60411	0.802053	−	0.597252i	$-0.203741\pi$
$223$	1.19123e6	1.60411	0.802053	+	0.597252i	$0.203741\pi$
$224$	0	0
$225$	−242890.	−0.319855
$226$	0	0
$227$	− 1.41130e6i	− 1.81783i	−0.416980	−	0.908916i	$-0.636911\pi$
$227$	− 1.41130e6i	− 1.81783i	0.416980	−	0.908916i	$-0.363089\pi$
$228$	0	0
$229$	546374.i	0.688496i	0.938879	+	0.344248i	$0.111866\pi$
$229$	546374.i	0.688496i	−0.938879	+	0.344248i	$0.888134\pi$
$230$	0	0
$231$	62738.7	0.0773581
$232$	0	0
$233$	−919704.	−1.10984	−0.554918	−	0.831905i	$-0.687250\pi$
$233$	−919704.	−1.10984	−0.554918	+	0.831905i	$0.687250\pi$
$234$	0	0
$235$	− 130371.i	− 0.153997i
$236$	0	0
$237$	− 241074.i	− 0.278792i
$238$	0	0
$239$	1.38063e6	1.56345	0.781724	−	0.623624i	$-0.214340\pi$
$239$	1.38063e6	1.56345	0.781724	+	0.623624i	$0.214340\pi$
$240$	0	0
$241$	650375.	0.721308	0.360654	−	0.932700i	$-0.382553\pi$
$241$	650375.	0.721308	0.360654	+	0.932700i	$0.382553\pi$
$242$	0	0
$243$	59049.0i	0.0641500i
$244$	0	0
$245$	57977.9i	0.0617089i
$246$	0	0
$247$	−330905.	−0.345112
$248$	0	0
$249$	−945962.	−0.966886
$250$	0	0
$251$	− 393270.i	− 0.394009i	−0.980403	−	0.197005i	$-0.936879\pi$
$251$	− 393270.i	− 0.394009i	0.980403	−	0.197005i	$-0.0631214\pi$
$252$	0	0
$253$	51960.6i	0.0510355i
$254$	0	0
$255$	117612.	0.113267
$256$	0	0
$257$	−1.23955e6	−1.17066	−0.585331	−	0.810794i	$-0.699036\pi$
$257$	−1.23955e6	−1.17066	−0.585331	+	0.810794i	$0.699036\pi$
$258$	0	0
$259$	− 528725.i	− 0.489757i
$260$	0	0
$261$	− 190821.i	− 0.173391i
$262$	0	0
$263$	−1.47951e6	−1.31895	−0.659476	−	0.751726i	$-0.729222\pi$
$263$	−1.47951e6	−1.31895	−0.659476	+	0.751726i	$0.729222\pi$
$264$	0	0
$265$	17029.6	0.0148967
$266$	0	0
$267$	− 431345.i	− 0.370294i
$268$	0	0
$269$	345680.i	0.291269i	0.989338	+	0.145634i	$0.0465223\pi$
$269$	345680.i	0.291269i	−0.989338	+	0.145634i	$0.953478\pi$
$270$	0	0
$271$	1.53008e6	1.26558	0.632791	−	0.774322i	$-0.281909\pi$
$271$	1.53008e6	1.26558	0.632791	+	0.774322i	$0.281909\pi$
$272$	0	0
$273$	−727444.	−0.590736
$274$	0	0
$275$	− 141044.i	− 0.112466i
$276$	0	0
$277$	464109.i	0.363429i	0.983351	+	0.181715i	$0.0581648\pi$
$277$	464109.i	0.363429i	−0.983351	+	0.181715i	$0.941835\pi$
$278$	0	0
$279$	−238901.	−0.183741
$280$	0	0
$281$	847312.	0.640144	0.320072	−	0.947393i	$-0.396293\pi$
$281$	847312.	0.640144	0.320072	+	0.947393i	$0.396293\pi$
$282$	0	0
$283$	− 2.08212e6i	− 1.54540i	−0.634773	−	0.772698i	$-0.718907\pi$
$283$	− 2.08212e6i	− 1.54540i	0.634773	−	0.772698i	$-0.281093\pi$
$284$	0	0
$285$	− 61384.1i	− 0.0447655i
$286$	0	0
$287$	2.09403e6	1.50065
$288$	0	0
$289$	−68376.5	−0.0481573
$290$	0	0
$291$	344435.i	0.238438i
$292$	0	0
$293$	− 1.23146e6i	− 0.838017i	−0.907982	−	0.419008i	$-0.862378\pi$
$293$	− 1.23146e6i	− 0.838017i	0.907982	−	0.419008i	$-0.137622\pi$
$294$	0	0
$295$	113366.	0.0758454
$296$	0	0
$297$	−34289.2	−0.0225562
$298$	0	0
$299$	− 602474.i	− 0.389727i
$300$	0	0
$301$	2.62537e6i	1.67022i
$302$	0	0
$303$	1.16076e6	0.726335
$304$	0	0
$305$	−395840.	−0.243652
$306$	0	0
$307$	− 1.70679e6i	− 1.03356i	−0.856120	−	0.516778i	$-0.827131\pi$
$307$	− 1.70679e6i	− 1.03356i	0.856120	−	0.516778i	$-0.172869\pi$
$308$	0	0
$309$	− 1.81992e6i	− 1.08431i
$310$	0	0
$311$	2.97615e6	1.74484	0.872418	−	0.488761i	$-0.162551\pi$
$311$	2.97615e6	1.74484	0.872418	+	0.488761i	$0.162551\pi$
$312$	0	0
$313$	3.41990e6	1.97312	0.986558	−	0.163411i	$-0.0522497\pi$
$313$	3.41990e6	1.97312	0.986558	+	0.163411i	$0.0522497\pi$
$314$	0	0
$315$	− 134944.i	− 0.0766260i
$316$	0	0
$317$	− 3.30728e6i	− 1.84851i	−0.381770	−	0.924257i	$-0.624685\pi$
$317$	− 3.30728e6i	− 1.84851i	0.381770	−	0.924257i	$-0.375315\pi$
$318$	0	0
$319$	110808.	0.0609671
$320$	0	0
$321$	−458197.	−0.248193
$322$	0	0
$323$	− 705364.i	− 0.376190i
$324$	0	0
$325$	1.63538e6i	0.858836i
$326$	0	0
$327$	781077.	0.403947
$328$	0	0
$329$	−1.71885e6	−0.875486
$330$	0	0
$331$	− 858062.i	− 0.430476i	−0.976562	−	0.215238i	$-0.930947\pi$
$331$	− 858062.i	− 0.430476i	0.976562	−	0.215238i	$-0.0690527\pi$
$332$	0	0
$333$	288969.i	0.142804i
$334$	0	0
$335$	−323235.	−0.157364
$336$	0	0
$337$	2.31479e6	1.11029	0.555146	−	0.831753i	$-0.312662\pi$
$337$	2.31479e6	1.11029	0.555146	+	0.831753i	$0.312662\pi$
$338$	0	0
$339$	92533.0i	0.0437318i
$340$	0	0
$341$	− 138728.i	− 0.0646066i
$342$	0	0
$343$	−1.72648e6	−0.792367
$344$	0	0
$345$	111761.	0.0505526
$346$	0	0
$347$	42663.1i	0.0190208i	0.999955	+	0.00951039i	$0.00302730\pi$
$347$	42663.1i	0.0190208i	−0.999955	+	0.00951039i	$0.996973\pi$
$348$	0	0
$349$	4.06310e6i	1.78564i	0.450413	+	0.892820i	$0.351277\pi$
$349$	4.06310e6i	1.78564i	−0.450413	+	0.892820i	$0.648723\pi$
$350$	0	0
$351$	397578.	0.172248
$352$	0	0
$353$	1.25470e6	0.535922	0.267961	−	0.963430i	$-0.413650\pi$
$353$	1.25470e6	0.535922	0.267961	+	0.963430i	$0.413650\pi$
$354$	0	0
$355$	− 600752.i	− 0.253002i
$356$	0	0
$357$	− 1.55064e6i	− 0.643932i
$358$	0	0
$359$	−1.93310e6	−0.791621	−0.395811	−	0.918332i	$-0.629536\pi$
$359$	−1.93310e6	−0.791621	−0.395811	+	0.918332i	$0.629536\pi$
$360$	0	0
$361$	2.10796e6	0.851321
$362$	0	0
$363$	1.42955e6i	0.569419i
$364$	0	0
$365$	− 493872.i	− 0.194036i
$366$	0	0
$367$	932855.	0.361534	0.180767	−	0.983526i	$-0.442142\pi$
$367$	932855.	0.361534	0.180767	+	0.983526i	$0.442142\pi$
$368$	0	0
$369$	−1.14447e6	−0.437562
$370$	0	0
$371$	− 224524.i	− 0.0846890i
$372$	0	0
$373$	− 636513.i	− 0.236884i	−0.992961	−	0.118442i	$-0.962210\pi$
$373$	− 636513.i	− 0.236884i	0.992961	−	0.118442i	$-0.0377899\pi$
$374$	0	0
$375$	−619522.	−0.227499
$376$	0	0
$377$	−1.28480e6	−0.465567
$378$	0	0
$379$	436637.i	0.156143i	0.996948	+	0.0780715i	$0.0248762\pi$
$379$	436637.i	0.156143i	−0.996948	+	0.0780715i	$0.975124\pi$
$380$	0	0
$381$	− 1.06490e6i	− 0.375835i
$382$	0	0
$383$	2.77960e6	0.968244	0.484122	−	0.875001i	$-0.339139\pi$
$383$	2.77960e6	0.968244	0.484122	+	0.875001i	$0.339139\pi$
$384$	0	0
$385$	78360.7	0.0269430
$386$	0	0
$387$	− 1.43487e6i	− 0.487006i
$388$	0	0
$389$	4.22186e6i	1.41459i	0.706919	+	0.707294i	$0.250084\pi$
$389$	4.22186e6i	1.41459i	−0.706919	+	0.707294i	$0.749916\pi$
$390$	0	0
$391$	1.28425e6	0.424822
$392$	0	0
$393$	−2.95463e6	−0.964989
$394$	0	0
$395$	− 301102.i	− 0.0971002i
$396$	0	0
$397$	3.89554e6i	1.24048i	0.784411	+	0.620242i	$0.212966\pi$
$397$	3.89554e6i	1.24048i	−0.784411	+	0.620242i	$0.787034\pi$
$398$	0	0
$399$	−809308.	−0.254496
$400$	0	0
$401$	−82294.6	−0.0255570	−0.0127785	−	0.999918i	$-0.504068\pi$
$401$	−82294.6	−0.0255570	−0.0127785	+	0.999918i	$0.504068\pi$
$402$	0	0
$403$	1.60852e6i	0.493360i
$404$	0	0
$405$	73752.2i	0.0223428i
$406$	0	0
$407$	−167802.	−0.0502124
$408$	0	0
$409$	−133394.	−0.0394300	−0.0197150	−	0.999806i	$-0.506276\pi$
$409$	−133394.	−0.0394300	−0.0197150	+	0.999806i	$0.506276\pi$
$410$	0	0
$411$	− 100555.i	− 0.0293628i
$412$	0	0
$413$	− 1.49466e6i	− 0.431189i
$414$	0	0
$415$	−1.18151e6	−0.336756
$416$	0	0
$417$	−985591.	−0.277560
$418$	0	0
$419$	4.85459e6i	1.35088i	0.737414	+	0.675441i	$0.236047\pi$
$419$	4.85459e6i	1.35088i	−0.737414	+	0.675441i	$0.763953\pi$
$420$	0	0
$421$	− 674802.i	− 0.185554i	−0.995687	−	0.0927772i	$-0.970426\pi$
$421$	− 674802.i	− 0.185554i	0.995687	−	0.0927772i	$-0.0295744\pi$
$422$	0	0
$423$	939423.	0.255276
$424$	0	0
$425$	−3.48601e6	−0.936175
$426$	0	0
$427$	5.21888e6i	1.38519i
$428$	0	0
$429$	230870.i	0.0605653i
$430$	0	0
$431$	−5.62432e6	−1.45840	−0.729200	−	0.684301i	$-0.760108\pi$
$431$	−5.62432e6	−1.45840	−0.729200	+	0.684301i	$0.760108\pi$
$432$	0	0
$433$	5.89978e6	1.51222	0.756112	−	0.654442i	$-0.227097\pi$
$433$	5.89978e6	1.51222	0.756112	+	0.654442i	$0.227097\pi$
$434$	0	0
$435$	− 238336.i	− 0.0603901i
$436$	0	0
$437$	− 670273.i	− 0.167899i
$438$	0	0
$439$	−417376.	−0.103363	−0.0516817	−	0.998664i	$-0.516458\pi$
$439$	−417376.	−0.103363	−0.0516817	+	0.998664i	$0.516458\pi$
$440$	0	0
$441$	−417775.	−0.102293
$442$	0	0
$443$	− 5.30672e6i	− 1.28474i	−0.766393	−	0.642372i	$-0.777950\pi$
$443$	− 5.30672e6i	− 1.28474i	0.766393	−	0.642372i	$-0.222050\pi$
$444$	0	0
$445$	− 538750.i	− 0.128969i
$446$	0	0
$447$	−2.78756e6	−0.659866
$448$	0	0
$449$	−2.85104e6	−0.667402	−0.333701	−	0.942679i	$-0.608298\pi$
$449$	−2.85104e6	−0.667402	−0.333701	+	0.942679i	$0.608298\pi$
$450$	0	0
$451$	− 664585.i	− 0.153854i
$452$	0	0
$453$	− 4.15758e6i	− 0.951909i
$454$	0	0
$455$	−908578.	−0.205747
$456$	0	0
$457$	2.94965e6	0.660662	0.330331	−	0.943865i	$-0.392840\pi$
$457$	2.94965e6	0.660662	0.330331	+	0.943865i	$0.392840\pi$
$458$	0	0
$459$	847486.i	0.187759i
$460$	0	0
$461$	− 7.87075e6i	− 1.72490i	−0.506143	−	0.862449i	$-0.668929\pi$
$461$	− 7.87075e6i	− 1.72490i	0.506143	−	0.862449i	$-0.331071\pi$
$462$	0	0
$463$	−3.22083e6	−0.698257	−0.349128	−	0.937075i	$-0.613522\pi$
$463$	−3.22083e6	−0.698257	−0.349128	+	0.937075i	$0.613522\pi$
$464$	0	0
$465$	−298387.	−0.0639952
$466$	0	0
$467$	3.23362e6i	0.686115i	0.939314	+	0.343057i	$0.111463\pi$
$467$	3.23362e6i	0.686115i	−0.939314	+	0.343057i	$0.888537\pi$
$468$	0	0
$469$	4.26164e6i	0.894632i
$470$	0	0
$471$	−3.91361e6	−0.812878
$472$	0	0
$473$	833216.	0.171240
$474$	0	0
$475$	1.81942e6i	0.369997i
$476$	0	0
$477$	122711.i	0.0246938i
$478$	0	0
$479$	−7.98448e6	−1.59004	−0.795020	−	0.606583i	$-0.792540\pi$
$479$	−7.98448e6	−1.59004	−0.795020	+	0.606583i	$0.792540\pi$
$480$	0	0
$481$	1.94563e6	0.383441
$482$	0	0
$483$	− 1.47350e6i	− 0.287396i
$484$	0	0
$485$	430200.i	0.0830454i
$486$	0	0
$487$	4.14364e6	0.791698	0.395849	−	0.918316i	$-0.370450\pi$
$487$	4.14364e6	0.791698	0.395849	+	0.918316i	$0.370450\pi$
$488$	0	0
$489$	−1.07426e6	−0.203160
$490$	0	0
$491$	8.09427e6i	1.51521i	0.652711	+	0.757607i	$0.273632\pi$
$491$	8.09427e6i	1.51521i	−0.652711	+	0.757607i	$0.726368\pi$
$492$	0	0
$493$	− 2.73871e6i	− 0.507492i
$494$	0	0
$495$	−42827.3	−0.00785611
$496$	0	0
$497$	−7.92052e6	−1.43834
$498$	0	0
$499$	6.18633e6i	1.11220i	0.831117	+	0.556098i	$0.187702\pi$
$499$	6.18633e6i	1.11220i	−0.831117	+	0.556098i	$0.812298\pi$
$500$	0	0
$501$	− 4.14233e6i	− 0.737311i
$502$	0	0
$503$	1.12950e6	0.199052	0.0995260	−	0.995035i	$-0.468267\pi$
$503$	1.12950e6	0.199052	0.0995260	+	0.995035i	$0.468267\pi$
$504$	0	0
$505$	1.44979e6	0.252975
$506$	0	0
$507$	664742.i	0.114851i
$508$	0	0
$509$	− 5.44872e6i	− 0.932181i	−0.884737	−	0.466091i	$-0.845662\pi$
$509$	− 5.44872e6i	− 0.932181i	0.884737	−	0.466091i	$-0.154338\pi$
$510$	0	0
$511$	−6.51137e6	−1.10311
$512$	0	0
$513$	442319.	0.0742066
$514$	0	0
$515$	− 2.27308e6i	− 0.377655i
$516$	0	0
$517$	545515.i	0.0897595i
$518$	0	0
$519$	4.19313e6	0.683314
$520$	0	0
$521$	5.87834e6	0.948769	0.474385	−	0.880318i	$-0.342671\pi$
$521$	5.87834e6	0.948769	0.474385	+	0.880318i	$0.342671\pi$
$522$	0	0
$523$	1.00787e7i	1.61120i	0.592458	+	0.805601i	$0.298158\pi$
$523$	1.00787e7i	1.61120i	−0.592458	+	0.805601i	$0.701842\pi$
$524$	0	0
$525$	3.99972e6i	0.633332i
$526$	0	0
$527$	−3.42876e6	−0.537787
$528$	0	0
$529$	−5.21599e6	−0.810396
$530$	0	0
$531$	816892.i	0.125727i
$532$	0	0
$533$	7.70575e6i	1.17489i
$534$	0	0
$535$	−572288.	−0.0864431
$536$	0	0
$537$	1.29429e6	0.193684
$538$	0	0
$539$	− 242599.i	− 0.0359680i
$540$	0	0
$541$	1.07470e7i	1.57869i	0.613953	+	0.789343i	$0.289579\pi$
$541$	1.07470e7i	1.57869i	−0.613953	+	0.789343i	$0.710421\pi$
$542$	0	0
$543$	−2.53643e6	−0.369168
$544$	0	0
$545$	975565.	0.140690
$546$	0	0
$547$	1.90847e6i	0.272719i	0.990659	+	0.136360i	$0.0435403\pi$
$547$	1.90847e6i	0.272719i	−0.990659	+	0.136360i	$0.956460\pi$
$548$	0	0
$549$	− 2.85233e6i	− 0.403895i
$550$	0	0
$551$	−1.42939e6	−0.200572
$552$	0	0
$553$	−3.96982e6	−0.552024
$554$	0	0
$555$	360923.i	0.0497373i
$556$	0	0
$557$	− 9.16492e6i	− 1.25167i	−0.779954	−	0.625837i	$-0.784758\pi$
$557$	− 9.16492e6i	− 1.25167i	0.779954	−	0.625837i	$-0.215242\pi$
$558$	0	0
$559$	−9.66099e6	−1.30765
$560$	0	0
$561$	−492128.	−0.0660193
$562$	0	0
$563$	7.86037e6i	1.04513i	0.852598	+	0.522567i	$0.175026\pi$
$563$	7.86037e6i	1.04513i	−0.852598	+	0.522567i	$0.824974\pi$
$564$	0	0
$565$	115574.i	0.0152313i
$566$	0	0
$567$	972373.	0.127021
$568$	0	0
$569$	−3.16208e6	−0.409442	−0.204721	−	0.978820i	$-0.565629\pi$
$569$	−3.16208e6	−0.409442	−0.204721	+	0.978820i	$0.565629\pi$
$570$	0	0
$571$	− 1.36016e7i	− 1.74582i	−0.487881	−	0.872910i	$-0.662230\pi$
$571$	− 1.36016e7i	− 1.74582i	0.487881	−	0.872910i	$-0.337770\pi$
$572$	0	0
$573$	2.87382e6i	0.365656i
$574$	0	0
$575$	−3.31259e6	−0.417829
$576$	0	0
$577$	−1.53482e6	−0.191919	−0.0959594	−	0.995385i	$-0.530592\pi$
$577$	−1.53482e6	−0.191919	−0.0959594	+	0.995385i	$0.530592\pi$
$578$	0	0
$579$	6.79725e6i	0.842630i
$580$	0	0
$581$	1.55774e7i	1.91449i
$582$	0	0
$583$	−71257.3	−0.00868277
$584$	0	0
$585$	496574.	0.0599922
$586$	0	0
$587$	7.35171e6i	0.880629i	0.897844	+	0.440315i	$0.145133\pi$
$587$	7.35171e6i	0.880629i	−0.897844	+	0.440315i	$0.854867\pi$
$588$	0	0
$589$	1.78954e6i	0.212546i
$590$	0	0
$591$	623727.	0.0734558
$592$	0	0
$593$	7.72521e6	0.902139	0.451069	−	0.892489i	$-0.351043\pi$
$593$	7.72521e6	0.902139	0.451069	+	0.892489i	$0.351043\pi$
$594$	0	0
$595$	− 1.93675e6i	− 0.224275i
$596$	0	0
$597$	− 149591.i	− 0.0171779i
$598$	0	0
$599$	1.53129e7	1.74378	0.871890	−	0.489702i	$-0.162894\pi$
$599$	1.53129e7	1.74378	0.871890	+	0.489702i	$0.162894\pi$
$600$	0	0
$601$	3.69885e6	0.417715	0.208857	−	0.977946i	$-0.433026\pi$
$601$	3.69885e6	0.417715	0.208857	+	0.977946i	$0.433026\pi$
$602$	0	0
$603$	− 2.32916e6i	− 0.260859i
$604$	0	0
$605$	1.78550e6i	0.198323i
$606$	0	0
$607$	−4.37471e6	−0.481923	−0.240962	−	0.970535i	$-0.577463\pi$
$607$	−4.37471e6	−0.481923	−0.240962	+	0.970535i	$0.577463\pi$
$608$	0	0
$609$	−3.14229e6	−0.343324
$610$	0	0
$611$	− 6.32515e6i	− 0.685437i
$612$	0	0
$613$	− 1.46082e7i	− 1.57016i	−0.619392	−	0.785082i	$-0.712621\pi$
$613$	− 1.46082e7i	− 1.57016i	0.619392	−	0.785082i	$-0.287379\pi$
$614$	0	0
$615$	−1.42945e6	−0.152398
$616$	0	0
$617$	576151.	0.0609289	0.0304644	−	0.999536i	$-0.490301\pi$
$617$	576151.	0.0609289	0.0304644	+	0.999536i	$0.490301\pi$
$618$	0	0
$619$	− 4.66461e6i	− 0.489315i	−0.969610	−	0.244657i	$-0.921325\pi$
$619$	− 4.66461e6i	− 0.489315i	0.969610	−	0.244657i	$-0.0786755\pi$
$620$	0	0
$621$	805325.i	0.0837996i
$622$	0	0
$623$	−7.10305e6	−0.733204
$624$	0	0
$625$	8.59697e6	0.880329
$626$	0	0
$627$	256851.i	0.0260923i
$628$	0	0
$629$	4.14736e6i	0.417970i
$630$	0	0
$631$	4.93025e6	0.492942	0.246471	−	0.969150i	$-0.420729\pi$
$631$	4.93025e6	0.492942	0.246471	+	0.969150i	$0.420729\pi$
$632$	0	0
$633$	−3.70929e6	−0.367944
$634$	0	0
$635$	− 1.33006e6i	− 0.130900i
$636$	0	0
$637$	2.81289e6i	0.274665i
$638$	0	0
$639$	4.32888e6	0.419395
$640$	0	0
$641$	7.42308e6	0.713574	0.356787	−	0.934186i	$-0.383872\pi$
$641$	7.42308e6	0.713574	0.356787	+	0.934186i	$0.383872\pi$
$642$	0	0
$643$	− 1.28737e6i	− 0.122794i	−0.998113	−	0.0613969i	$-0.980444\pi$
$643$	− 1.28737e6i	− 0.122794i	0.998113	−	0.0613969i	$-0.0195555\pi$
$644$	0	0
$645$	− 1.79215e6i	− 0.169619i
$646$	0	0
$647$	5.25779e6	0.493790	0.246895	−	0.969042i	$-0.420590\pi$
$647$	5.25779e6	0.493790	0.246895	+	0.969042i	$0.420590\pi$
$648$	0	0
$649$	−474362.	−0.0442077
$650$	0	0
$651$	3.93403e6i	0.363819i
$652$	0	0
$653$	− 1.03822e7i	− 0.952811i	−0.879226	−	0.476405i	$-0.841939\pi$
$653$	− 1.03822e7i	− 0.952811i	0.879226	−	0.476405i	$-0.158061\pi$
$654$	0	0
$655$	−3.69034e6	−0.336096
$656$	0	0
$657$	3.55873e6	0.321648
$658$	0	0
$659$	− 5.31093e6i	− 0.476384i	−0.971218	−	0.238192i	$-0.923445\pi$
$659$	− 5.31093e6i	− 0.476384i	0.971218	−	0.238192i	$-0.0765548\pi$
$660$	0	0
$661$	− 5.12917e6i	− 0.456608i	−0.973590	−	0.228304i	$-0.926682\pi$
$661$	− 5.12917e6i	− 0.456608i	0.973590	−	0.228304i	$-0.0733180\pi$
$662$	0	0
$663$	5.70613e6	0.504148
$664$	0	0
$665$	−1.01083e6	−0.0886384
$666$	0	0
$667$	− 2.60247e6i	− 0.226501i
$668$	0	0
$669$	1.07211e7i	0.926132i
$670$	0	0
$671$	1.65632e6	0.142017
$672$	0	0
$673$	7.93890e6	0.675652	0.337826	−	0.941209i	$-0.390309\pi$
$673$	7.93890e6	0.675652	0.337826	+	0.941209i	$0.390309\pi$
$674$	0	0
$675$	− 2.18601e6i	− 0.184668i
$676$	0	0
$677$	381860.i	0.0320208i	0.999872	+	0.0160104i	$0.00509649\pi$
$677$	381860.i	0.0320208i	−0.999872	+	0.0160104i	$0.994904\pi$
$678$	0	0
$679$	5.67189e6	0.472121
$680$	0	0
$681$	1.27017e7	1.04953
$682$	0	0
$683$	8.87069e6i	0.727621i	0.931473	+	0.363811i	$0.118524\pi$
$683$	8.87069e6i	0.727621i	−0.931473	+	0.363811i	$0.881476\pi$
$684$	0	0
$685$	− 125593.i	− 0.0102268i
$686$	0	0
$687$	−4.91736e6	−0.397503
$688$	0	0
$689$	826216.	0.0663049
$690$	0	0
$691$	1.54329e6i	0.122957i	0.998108	+	0.0614783i	$0.0195815\pi$
$691$	1.54329e6i	0.122957i	−0.998108	+	0.0614783i	$0.980419\pi$
$692$	0	0
$693$	564649.i	0.0446627i
$694$	0	0
$695$	−1.23100e6	−0.0966712
$696$	0	0
$697$	−1.64258e7	−1.28069
$698$	0	0
$699$	− 8.27734e6i	− 0.640764i
$700$	0	0
$701$	− 7.51477e6i	− 0.577592i	−0.957391	−	0.288796i	$-0.906745\pi$
$701$	− 7.51477e6i	− 0.577592i	0.957391	−	0.288796i	$-0.0932549\pi$
$702$	0	0
$703$	2.16459e6	0.165191
$704$	0	0
$705$	1.17334e6	0.0889101
$706$	0	0
$707$	− 1.91145e7i	− 1.43819i
$708$	0	0
$709$	− 1.54954e7i	− 1.15768i	−0.815442	−	0.578839i	$-0.803506\pi$
$709$	− 1.54954e7i	− 1.15768i	0.815442	−	0.578839i	$-0.196494\pi$
$710$	0	0
$711$	2.16967e6	0.160960
$712$	0	0
$713$	−3.25819e6	−0.240023
$714$	0	0
$715$	288356.i	0.0210943i
$716$	0	0
$717$	1.24257e7i	0.902657i
$718$	0	0
$719$	1.10624e7	0.798045	0.399022	−	0.916941i	$-0.369350\pi$
$719$	1.10624e7	0.798045	0.399022	+	0.916941i	$0.369350\pi$
$720$	0	0
$721$	−2.99690e7	−2.14701
$722$	0	0
$723$	5.85337e6i	0.416448i
$724$	0	0
$725$	7.06425e6i	0.499138i
$726$	0	0
$727$	1.63858e7	1.14982	0.574912	−	0.818216i	$-0.305036\pi$
$727$	1.63858e7	1.14982	0.574912	+	0.818216i	$0.305036\pi$
$728$	0	0
$729$	−531441.	−0.0370370
$730$	0	0
$731$	− 2.05936e7i	− 1.42541i
$732$	0	0
$733$	4.09602e6i	0.281580i	0.990039	+	0.140790i	$0.0449642\pi$
$733$	4.09602e6i	0.281580i	−0.990039	+	0.140790i	$0.955036\pi$
$734$	0	0
$735$	−521801.	−0.0356276
$736$	0	0
$737$	1.35252e6	0.0917224
$738$	0	0
$739$	− 2.35232e7i	− 1.58447i	−0.610213	−	0.792237i	$-0.708916\pi$
$739$	− 2.35232e7i	− 1.58447i	0.610213	−	0.792237i	$-0.291084\pi$
$740$	0	0
$741$	− 2.97814e6i	− 0.199251i
$742$	0	0
$743$	8.62364e6	0.573084	0.286542	−	0.958068i	$-0.407494\pi$
$743$	8.62364e6	0.573084	0.286542	+	0.958068i	$0.407494\pi$
$744$	0	0
$745$	−3.48167e6	−0.229825
$746$	0	0
$747$	− 8.51366e6i	− 0.558232i
$748$	0	0
$749$	7.54524e6i	0.491438i
$750$	0	0
$751$	1.62269e7	1.04987	0.524937	−	0.851141i	$-0.324089\pi$
$751$	1.62269e7	1.04987	0.524937	+	0.851141i	$0.324089\pi$
$752$	0	0
$753$	3.53943e6	0.227481
$754$	0	0
$755$	− 5.19282e6i	− 0.331540i
$756$	0	0
$757$	− 2.05486e7i	− 1.30330i	−0.758522	−	0.651648i	$-0.774078\pi$
$757$	− 2.05486e7i	− 1.30330i	0.758522	−	0.651648i	$-0.225922\pi$
$758$	0	0
$759$	−467645.	−0.0294654
$760$	0	0
$761$	4.33531e6	0.271368	0.135684	−	0.990752i	$-0.456677\pi$
$761$	4.33531e6	0.271368	0.135684	+	0.990752i	$0.456677\pi$
$762$	0	0
$763$	− 1.28622e7i	− 0.799839i
$764$	0	0
$765$	1.05851e6i	0.0653945i
$766$	0	0
$767$	5.50014e6	0.337587
$768$	0	0
$769$	−1.72697e7	−1.05310	−0.526548	−	0.850145i	$-0.676514\pi$
$769$	−1.72697e7	−1.05310	−0.526548	+	0.850145i	$0.676514\pi$
$770$	0	0
$771$	− 1.11560e7i	− 0.675882i
$772$	0	0
$773$	2.64326e7i	1.59108i	0.605903	+	0.795539i	$0.292812\pi$
$773$	2.64326e7i	1.59108i	−0.605903	+	0.795539i	$0.707188\pi$
$774$	0	0
$775$	8.84416e6	0.528935
$776$	0	0
$777$	4.75852e6	0.282761
$778$	0	0
$779$	8.57292e6i	0.506157i
$780$	0	0
$781$	2.51374e6i	0.147467i
$782$	0	0
$783$	1.71739e6	0.100107
$784$	0	0
$785$	−4.88810e6	−0.283117
$786$	0	0
$787$	2.17967e7i	1.25445i	0.778837	+	0.627226i	$0.215810\pi$
$787$	2.17967e7i	1.25445i	−0.778837	+	0.627226i	$0.784190\pi$
$788$	0	0
$789$	− 1.33156e7i	− 0.761497i
$790$	0	0
$791$	1.52376e6	0.0865916
$792$	0	0
$793$	−1.92048e7	−1.08449
$794$	0	0
$795$	153266.i	0.00860060i
$796$	0	0
$797$	− 2.45761e7i	− 1.37046i	−0.728326	−	0.685231i	$-0.759701\pi$
$797$	− 2.45761e7i	− 1.37046i	0.728326	−	0.685231i	$-0.240299\pi$
$798$	0	0
$799$	1.34828e7	0.747161
$800$	0	0
$801$	3.88210e6	0.213789
$802$	0	0
$803$	2.06652e6i	0.113097i
$804$	0	0
$805$	− 1.84040e6i	− 0.100097i
$806$	0	0
$807$	−3.11112e6	−0.168164
$808$	0	0
$809$	2.10063e7	1.12844	0.564221	−	0.825624i	$-0.309177\pi$
$809$	2.10063e7	1.12844	0.564221	+	0.825624i	$0.309177\pi$
$810$	0	0
$811$	2.68688e7i	1.43448i	0.696824	+	0.717242i	$0.254596\pi$
$811$	2.68688e7i	1.43448i	−0.696824	+	0.717242i	$0.745404\pi$
$812$	0	0
$813$	1.37707e7i	0.730685i
$814$	0	0
$815$	−1.34175e6	−0.0707583
$816$	0	0
$817$	−1.07482e7	−0.563353
$818$	0	0
$819$	− 6.54700e6i	− 0.341061i
$820$	0	0
$821$	1.90654e7i	0.987161i	0.869700	+	0.493581i	$0.164312\pi$
$821$	1.90654e7i	0.987161i	−0.869700	+	0.493581i	$0.835688\pi$
$822$	0	0
$823$	3.64112e6	0.187385	0.0936926	−	0.995601i	$-0.470133\pi$
$823$	3.64112e6	0.187385	0.0936926	+	0.995601i	$0.470133\pi$
$824$	0	0
$825$	1.26940e6	0.0649325
$826$	0	0
$827$	− 3.44261e7i	− 1.75034i	−0.483812	−	0.875172i	$-0.660748\pi$
$827$	− 3.44261e7i	− 1.75034i	0.483812	−	0.875172i	$-0.339252\pi$
$828$	0	0
$829$	− 2.51490e7i	− 1.27097i	−0.772114	−	0.635485i	$-0.780800\pi$
$829$	− 2.51490e7i	− 1.27097i	0.772114	−	0.635485i	$-0.219200\pi$
$830$	0	0
$831$	−4.17698e6	−0.209826
$832$	0	0
$833$	−5.99601e6	−0.299399
$834$	0	0
$835$	− 5.17377e6i	− 0.256798i
$836$	0	0
$837$	− 2.15011e6i	− 0.106083i
$838$	0	0
$839$	1.37585e7	0.674785	0.337392	−	0.941364i	$-0.390455\pi$
$839$	1.37585e7	0.674785	0.337392	+	0.941364i	$0.390455\pi$
$840$	0	0
$841$	1.49613e7	0.729422
$842$	0	0
$843$	7.62581e6i	0.369587i
$844$	0	0
$845$	830263.i	0.0400013i
$846$	0	0
$847$	2.35407e7	1.12748
$848$	0	0
$849$	1.87391e7	0.892235
$850$	0	0
$851$	3.94104e6i	0.186546i
$852$	0	0
$853$	− 2.71986e7i	− 1.27989i	−0.768419	−	0.639946i	$-0.778957\pi$
$853$	− 2.71986e7i	− 1.27989i	0.768419	−	0.639946i	$-0.221043\pi$
$854$	0	0
$855$	552457.	0.0258454
$856$	0	0
$857$	1.90034e7	0.883851	0.441925	−	0.897052i	$-0.354296\pi$
$857$	1.90034e7	0.883851	0.441925	+	0.897052i	$0.354296\pi$
$858$	0	0
$859$	− 4.23596e7i	− 1.95871i	−0.202157	−	0.979353i	$-0.564795\pi$
$859$	− 4.23596e7i	− 1.95871i	0.202157	−	0.979353i	$-0.435205\pi$
$860$	0	0
$861$	1.88463e7i	0.866399i
$862$	0	0
$863$	1.43661e6	0.0656616	0.0328308	−	0.999461i	$-0.489548\pi$
$863$	1.43661e6	0.0656616	0.0328308	+	0.999461i	$0.489548\pi$
$864$	0	0
$865$	5.23722e6	0.237991
$866$	0	0
$867$	− 615389.i	− 0.0278036i
$868$	0	0
$869$	1.25991e6i	0.0565964i
$870$	0	0
$871$	−1.56822e7	−0.700427
$872$	0	0
$873$	−3.09992e6	−0.137662
$874$	0	0
$875$	1.02018e7i	0.450461i
$876$	0	0
$877$	− 1.32043e7i	− 0.579718i	−0.957069	−	0.289859i	$-0.906392\pi$
$877$	− 1.32043e7i	− 0.579718i	0.957069	−	0.289859i	$-0.0936085\pi$
$878$	0	0
$879$	1.10832e7	0.483829
$880$	0	0
$881$	9.00664e6	0.390952	0.195476	−	0.980709i	$-0.437375\pi$
$881$	9.00664e6	0.390952	0.195476	+	0.980709i	$0.437375\pi$
$882$	0	0
$883$	− 2.18359e7i	− 0.942473i	−0.882007	−	0.471237i	$-0.843808\pi$
$883$	− 2.18359e7i	− 0.942473i	0.882007	−	0.471237i	$-0.156192\pi$
$884$	0	0
$885$	1.02030e6i	0.0437894i
$886$	0	0
$887$	−4.58096e7	−1.95500	−0.977502	−	0.210926i	$-0.932352\pi$
$887$	−4.58096e7	−1.95500	−0.977502	+	0.210926i	$0.932352\pi$
$888$	0	0
$889$	−1.75360e7	−0.744176
$890$	0	0
$891$	− 308603.i	− 0.0130229i
$892$	0	0
$893$	− 7.03695e6i	− 0.295295i
$894$	0	0
$895$	1.61656e6	0.0674582
$896$	0	0
$897$	5.42226e6	0.225009
$898$	0	0
$899$	6.94823e6i	0.286731i
$900$	0	0
$901$	1.76118e6i	0.0722757i
$902$	0	0
$903$	−2.36283e7	−0.964302
$904$	0	0
$905$	−3.16801e6	−0.128577
$906$	0	0
$907$	3.25677e7i	1.31452i	0.753662	+	0.657262i	$0.228286\pi$
$907$	3.25677e7i	1.31452i	−0.753662	+	0.657262i	$0.771714\pi$
$908$	0	0
$909$	1.04469e7i	0.419350i
$910$	0	0
$911$	4.24433e7	1.69439	0.847194	−	0.531283i	$-0.178290\pi$
$911$	4.24433e7	1.69439	0.847194	+	0.531283i	$0.178290\pi$
$912$	0	0
$913$	4.94381e6	0.196284
$914$	0	0
$915$	− 3.56256e6i	− 0.140673i
$916$	0	0
$917$	4.86546e7i	1.91074i
$918$	0	0
$919$	−3.77474e7	−1.47434	−0.737172	−	0.675705i	$-0.763839\pi$
$919$	−3.77474e7	−1.47434	−0.737172	+	0.675705i	$0.763839\pi$
$920$	0	0
$921$	1.53611e7	0.596724
$922$	0	0
$923$	− 2.91464e7i	− 1.12611i
$924$	0	0
$925$	− 1.06977e7i	− 0.411090i
$926$	0	0
$927$	1.63792e7	0.626029
$928$	0	0
$929$	3.41047e7	1.29651	0.648254	−	0.761424i	$-0.275499\pi$
$929$	3.41047e7	1.29651	0.648254	+	0.761424i	$0.275499\pi$
$930$	0	0
$931$	3.12944e6i	0.118329i
$932$	0	0
$933$	2.67854e7i	1.00738i
$934$	0	0
$935$	−614667.	−0.0229938
$936$	0	0
$937$	−2.26592e7	−0.843130	−0.421565	−	0.906798i	$-0.638519\pi$
$937$	−2.26592e7	−0.843130	−0.421565	+	0.906798i	$0.638519\pi$
$938$	0	0
$939$	3.07791e7i	1.13918i
$940$	0	0
$941$	1.44909e7i	0.533484i	0.963768	+	0.266742i	$0.0859471\pi$
$941$	1.44909e7i	0.533484i	−0.963768	+	0.266742i	$0.914053\pi$
$942$	0	0
$943$	−1.56086e7	−0.571590
$944$	0	0
$945$	1.21449e6	0.0442401
$946$	0	0
$947$	3.43018e7i	1.24292i	0.783447	+	0.621459i	$0.213460\pi$
$947$	3.43018e7i	1.24292i	−0.783447	+	0.621459i	$0.786540\pi$
$948$	0	0
$949$	− 2.39609e7i	− 0.863651i
$950$	0	0
$951$	2.97655e7	1.06724
$952$	0	0
$953$	−3.23134e7	−1.15252	−0.576262	−	0.817265i	$-0.695489\pi$
$953$	−3.23134e7	−1.15252	−0.576262	+	0.817265i	$0.695489\pi$
$954$	0	0
$955$	3.58940e6i	0.127354i
$956$	0	0
$957$	997274.i	0.0351994i
$958$	0	0
$959$	−1.65586e6	−0.0581401
$960$	0	0
$961$	−1.99302e7	−0.696152
$962$	0	0
$963$	− 4.12378e6i	− 0.143294i
$964$	0	0
$965$	8.48976e6i	0.293479i
$966$	0	0
$967$	−2.73182e7	−0.939477	−0.469738	−	0.882806i	$-0.655652\pi$
$967$	−2.73182e7	−0.939477	−0.469738	+	0.882806i	$0.655652\pi$
$968$	0	0
$969$	6.34827e6	0.217193
$970$	0	0
$971$	2.99922e7i	1.02085i	0.859923	+	0.510423i	$0.170511\pi$
$971$	2.99922e7i	1.02085i	−0.859923	+	0.510423i	$0.829489\pi$
$972$	0	0
$973$	1.62300e7i	0.549585i
$974$	0	0
$975$	−1.47184e7	−0.495849
$976$	0	0
$977$	−2.33987e7	−0.784251	−0.392126	−	0.919912i	$-0.628260\pi$
$977$	−2.33987e7	−0.784251	−0.392126	+	0.919912i	$0.628260\pi$
$978$	0	0
$979$	2.25430e6i	0.0751720i
$980$	0	0
$981$	7.02969e6i	0.233219i
$982$	0	0
$983$	−1.38086e7	−0.455792	−0.227896	−	0.973686i	$-0.573185\pi$
$983$	−1.38086e7	−0.455792	−0.227896	+	0.973686i	$0.573185\pi$
$984$	0	0
$985$	779035.	0.0255839
$986$	0	0
$987$	− 1.54697e7i	− 0.505462i
$988$	0	0
$989$	− 1.95691e7i	− 0.636180i
$990$	0	0
$991$	3.17280e6	0.102626	0.0513131	−	0.998683i	$-0.483659\pi$
$991$	3.17280e6	0.102626	0.0513131	+	0.998683i	$0.483659\pi$
$992$	0	0
$993$	7.72256e6	0.248535
$994$	0	0
$995$	− 186839.i	− 0.00598288i
$996$	0	0
$997$	− 4.02543e6i	− 0.128255i	−0.997942	−	0.0641276i	$-0.979574\pi$
$997$	− 4.02543e6i	− 0.128255i	0.997942	−	0.0641276i	$-0.0204265\pi$
$998$	0	0
$999$	−2.60072e6	−0.0824481

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.6.d.g.193.4	yes	4
4.3	odd	2		384.6.d.h.193.2	yes	4
8.3	odd	2		384.6.d.h.193.3	yes	4
8.5	even	2	inner	384.6.d.g.193.1	✓	4
16.3	odd	4		768.6.a.r.1.2		2
16.5	even	4		768.6.a.v.1.1		2
16.11	odd	4		768.6.a.q.1.1		2
16.13	even	4		768.6.a.m.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.6.d.g.193.1	✓	4	8.5	even	2	inner
384.6.d.g.193.4	yes	4	1.1	even	1	trivial
384.6.d.h.193.2	yes	4	4.3	odd	2
384.6.d.h.193.3	yes	4	8.3	odd	2
768.6.a.m.1.2		2	16.13	even	4
768.6.a.q.1.1		2	16.11	odd	4
768.6.a.r.1.2		2	16.3	odd	4
768.6.a.v.1.1		2	16.5	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:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	384.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+9.00000i q^{3} +11.2410i q^{5} +148.205 q^{7} -81.0000 q^{9} +O(q^{10})\)	\(q+9.00000i q^{3} +11.2410i q^{5} +148.205 q^{7} -81.0000 q^{9} -47.0360i q^{11} +545.374i q^{13} -101.169 q^{15} -1162.53 q^{17} +606.748i q^{19} +1333.84i q^{21} -1104.70 q^{23} +2998.64 q^{25} -729.000i q^{27} +2355.82i q^{29} +2949.39 q^{31} +423.324 q^{33} +1665.97i q^{35} -3567.52i q^{37} -4908.37 q^{39} +14129.3 q^{41} +17714.4i q^{43} -910.521i q^{45} -11597.8 q^{47} +5157.72 q^{49} -10462.8i q^{51} -1514.95i q^{53} +528.732 q^{55} -5460.73 q^{57} -10085.1i q^{59} +35214.0i q^{61} -12004.6 q^{63} -6130.55 q^{65} +28755.0i q^{67} -9942.28i q^{69} -53443.0 q^{71} -43934.9 q^{73} +26987.8i q^{75} -6970.97i q^{77} -26786.0 q^{79} +6561.00 q^{81} +105107. i q^{83} -13068.0i q^{85} -21202.4 q^{87} -47927.2 q^{89} +80827.1i q^{91} +26544.5i q^{93} -6820.45 q^{95} +38270.6 q^{97} +3809.92i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 32 q^{7} - 324 q^{9}+O(q^{10}) \) 4 * q - 32 * q^7 - 324 * q^9	\( 4 q - 32 q^{7} - 324 q^{9} + 720 q^{15} + 1848 q^{17} - 7168 q^{23} + 6996 q^{25} - 7072 q^{31} + 6192 q^{33} - 3888 q^{39} + 5032 q^{41} - 9152 q^{47} + 30628 q^{49} - 29376 q^{55} + 9648 q^{57} + 2592 q^{63} - 46016 q^{65} - 151040 q^{71} + 34200 q^{73} - 200992 q^{79} + 26244 q^{81} - 155664 q^{87} - 180712 q^{89} - 130752 q^{95} - 199816 q^{97}+O(q^{100}) \) 4 * q - 32 * q^7 - 324 * q^9 + 720 * q^15 + 1848 * q^17 - 7168 * q^23 + 6996 * q^25 - 7072 * q^31 + 6192 * q^33 - 3888 * q^39 + 5032 * q^41 - 9152 * q^47 + 30628 * q^49 - 29376 * q^55 + 9648 * q^57 + 2592 * q^63 - 46016 * q^65 - 151040 * q^71 + 34200 * q^73 - 200992 * q^79 + 26244 * q^81 - 155664 * q^87 - 180712 * q^89 - 130752 * q^95 - 199816 * q^97

Introduction

L-functions

Modular forms

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Database

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