LMFDB - Embedded newform 441.6.c.a.440.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,6,Mod(440,441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("441.440");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 441 = 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	441.c (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:	$70.7292645375$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{2} + 4 $ x^4 - 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2}\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 63)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			440.2
Root			$-1.22474 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	441.440
Dual form			441.6.c.a.440.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.82843i q^{2} +24.0000 q^{4} +51.4393 q^{5} -158.392i q^{8} +O(q^{10})$	$q-2.82843i q^{2} +24.0000 q^{4} +51.4393 q^{5} -158.392i q^{8} -145.492i q^{10} +578.413i q^{11} -1151.81i q^{13} +320.000 q^{16} +377.221 q^{17} +1733.78i q^{19} +1234.54 q^{20} +1636.00 q^{22} -4610.34i q^{23} -479.000 q^{25} -3257.82 q^{26} -4601.85i q^{29} -2267.25i q^{31} -5973.64i q^{32} -1066.94i q^{34} +203.000 q^{37} +4903.88 q^{38} -8147.57i q^{40} -85.7321 q^{41} +4697.00 q^{43} +13881.9i q^{44} -13040.0 q^{46} +23987.9 q^{47} +1354.82i q^{50} -27643.5i q^{52} -7418.96i q^{53} +29753.2i q^{55} -13016.0 q^{58} +4972.46 q^{59} +17434.8i q^{61} -6412.76 q^{62} -6656.00 q^{64} -59248.5i q^{65} +30197.0 q^{67} +9053.31 q^{68} -3961.21i q^{71} +53335.0i q^{73} -574.171i q^{74} +41610.8i q^{76} -39385.0 q^{79} +16460.6 q^{80} +242.487i q^{82} +63561.8 q^{83} +19404.0 q^{85} -13285.1i q^{86} +91616.0 q^{88} +105382. q^{89} -110648. i q^{92} -67847.9i q^{94} +89184.5i q^{95} +2061.14i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 96 q^{4}+O(q^{10}) $ 4 * q + 96 * q^4	$ 4 q + 96 q^{4} + 1280 q^{16} + 6544 q^{22} - 1916 q^{25} + 812 q^{37} + 18788 q^{43} - 52160 q^{46} - 52064 q^{58} - 26624 q^{64} + 120788 q^{67} - 157540 q^{79} + 77616 q^{85} + 366464 q^{88}+O(q^{100}) $ 4 * q + 96 * q^4 + 1280 * q^16 + 6544 * q^22 - 1916 * q^25 + 812 * q^37 + 18788 * q^43 - 52160 * q^46 - 52064 * q^58 - 26624 * q^64 + 120788 * q^67 - 157540 * q^79 + 77616 * q^85 + 366464 * q^88

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/441\mathbb{Z}\right)^\times$.

$n$	$199$	$344$
$\chi(n)$	$-1$	$-1$

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 $

$2$	− 2.82843i	− 0.500000i	−0.968246	−	0.250000i	$-0.919569\pi$
$2$	− 2.82843i	− 0.500000i	0.968246	−	0.250000i	$-0.0804306\pi$
$3$	0	0
$3$	0	0
$4$	24.0000	0.750000
$5$	51.4393	0.920174	0.460087	−	0.887874i	$-0.347818\pi$
$5$	51.4393	0.920174	0.460087	+	0.887874i	$0.347818\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	− 158.392i	− 0.875000i
$9$	0	0
$10$	− 145.492i	− 0.460087i
$11$	578.413i	1.44131i	0.693295	+	0.720654i	$0.256158\pi$
$11$	578.413i	1.44131i	−0.693295	+	0.720654i	$0.743842\pi$
$12$	0	0
$13$	− 1151.81i	− 1.89027i	−0.326680	−	0.945135i	$-0.605930\pi$
$13$	− 1151.81i	− 1.89027i	0.326680	−	0.945135i	$-0.394070\pi$
$14$	0	0
$15$	0	0
$16$	320.000	0.312500
$17$	377.221	0.316573	0.158287	−	0.987393i	$-0.449403\pi$
$17$	377.221	0.316573	0.158287	+	0.987393i	$0.449403\pi$
$18$	0	0
$19$	1733.78i	1.10182i	0.834565	+	0.550910i	$0.185719\pi$
$19$	1733.78i	1.10182i	−0.834565	+	0.550910i	$0.814281\pi$
$20$	1234.54	0.690130
$21$	0	0
$22$	1636.00	0.720654
$23$	− 4610.34i	− 1.81724i	−0.417621	−	0.908622i	$-0.637136\pi$
$23$	− 4610.34i	− 1.81724i	0.417621	−	0.908622i	$-0.362864\pi$
$24$	0	0
$25$	−479.000	−0.153280
$26$	−3257.82	−0.945135
$27$	0	0
$28$	0	0
$29$	− 4601.85i	− 1.01610i	−0.861327	−	0.508051i	$-0.830366\pi$
$29$	− 4601.85i	− 1.01610i	0.861327	−	0.508051i	$-0.169634\pi$
$30$	0	0
$31$	− 2267.25i	− 0.423737i	−0.977298	−	0.211868i	$-0.932045\pi$
$31$	− 2267.25i	− 0.423737i	0.977298	−	0.211868i	$-0.0679548\pi$
$32$	− 5973.64i	− 1.03125i
$33$	0	0
$34$	− 1066.94i	− 0.158287i
$35$	0	0
$36$	0	0
$37$	203.000	0.0243776	0.0121888	−	0.999926i	$-0.496120\pi$
$37$	203.000	0.0243776	0.0121888	+	0.999926i	$0.496120\pi$
$38$	4903.88	0.550910
$39$	0	0
$40$	− 8147.57i	− 0.805152i
$41$	−85.7321	−0.00796497	−0.00398248	−	0.999992i	$-0.501268\pi$
$41$	−85.7321	−0.00796497	−0.00398248	+	0.999992i	$0.501268\pi$
$42$	0	0
$43$	4697.00	0.387391	0.193695	−	0.981062i	$-0.437953\pi$
$43$	4697.00	0.387391	0.193695	+	0.981062i	$0.437953\pi$
$44$	13881.9i	1.08098i
$45$	0	0
$46$	−13040.0	−0.908622
$47$	23987.9	1.58397	0.791985	−	0.610541i	$-0.209048\pi$
$47$	23987.9	1.58397	0.791985	+	0.610541i	$0.209048\pi$
$48$	0	0
$49$	0	0
$50$	1354.82i	0.0766400i
$51$	0	0
$52$	− 27643.5i	− 1.41770i
$53$	− 7418.96i	− 0.362789i	−0.983410	−	0.181394i	$-0.941939\pi$
$53$	− 7418.96i	− 0.362789i	0.983410	−	0.181394i	$-0.0580610\pi$
$54$	0	0
$55$	29753.2i	1.32625i
$56$	0	0
$57$	0	0
$58$	−13016.0	−0.508051
$59$	4972.46	0.185969	0.0929847	−	0.995668i	$-0.470359\pi$
$59$	4972.46	0.185969	0.0929847	+	0.995668i	$0.470359\pi$
$60$	0	0
$61$	17434.8i	0.599919i	0.953952	+	0.299960i	$0.0969733\pi$
$61$	17434.8i	0.599919i	−0.953952	+	0.299960i	$0.903027\pi$
$62$	−6412.76	−0.211868
$63$	0	0
$64$	−6656.00	−0.203125
$65$	− 59248.5i	− 1.73938i
$66$	0	0
$67$	30197.0	0.821820	0.410910	−	0.911676i	$-0.365211\pi$
$67$	30197.0	0.821820	0.410910	+	0.911676i	$0.365211\pi$
$68$	9053.31	0.237430
$69$	0	0
$70$	0	0
$71$	− 3961.21i	− 0.0932572i	−0.998912	−	0.0466286i	$-0.985152\pi$
$71$	− 3961.21i	− 0.0932572i	0.998912	−	0.0466286i	$-0.0148477\pi$
$72$	0	0
$73$	53335.0i	1.17140i	0.810528	+	0.585700i	$0.199180\pi$
$73$	53335.0i	1.17140i	−0.810528	+	0.585700i	$0.800820\pi$
$74$	− 574.171i	− 0.0121888i
$75$	0	0
$76$	41610.8i	0.826365i
$77$	0	0
$78$	0	0
$79$	−39385.0	−0.710008	−0.355004	−	0.934865i	$-0.615520\pi$
$79$	−39385.0	−0.710008	−0.355004	+	0.934865i	$0.615520\pi$
$80$	16460.6	0.287554
$81$	0	0
$82$	242.487i	0.00398248i
$83$	63561.8	1.01275	0.506374	−	0.862314i	$-0.330986\pi$
$83$	63561.8	1.01275	0.506374	+	0.862314i	$0.330986\pi$
$84$	0	0
$85$	19404.0	0.291302
$86$	− 13285.1i	− 0.193695i
$87$	0	0
$88$	91616.0	1.26114
$89$	105382.	1.41023	0.705117	−	0.709091i	$-0.250894\pi$
$89$	105382.	1.41023	0.705117	+	0.709091i	$0.250894\pi$
$90$	0	0
$91$	0	0
$92$	− 110648.i	− 1.36293i
$93$	0	0
$94$	− 67847.9i	− 0.791985i
$95$	89184.5i	1.01387i
$96$	0	0
$97$	2061.14i	0.0222422i	0.999938	+	0.0111211i	$0.00354003\pi$
$97$	2061.14i	0.0222422i	−0.999938	+	0.0111211i	$0.996460\pi$
$98$	0	0
$99$	0	0
$100$	−11496.0	−0.114960
$101$	31738.0	0.309583	0.154791	−	0.987947i	$-0.450529\pi$
$101$	31738.0	0.309583	0.154791	+	0.987947i	$0.450529\pi$
$102$	0	0
$103$	− 168953.i	− 1.56918i	−0.620015	−	0.784590i	$-0.712874\pi$
$103$	− 168953.i	− 1.56918i	0.620015	−	0.784590i	$-0.287126\pi$
$104$	−182438.	−1.65399
$105$	0	0
$106$	−20984.0	−0.181394
$107$	− 137091.i	− 1.15758i	−0.815478	−	0.578788i	$-0.803526\pi$
$107$	− 137091.i	− 1.15758i	0.815478	−	0.578788i	$-0.196474\pi$
$108$	0	0
$109$	−206605.	−1.66561	−0.832807	−	0.553563i	$-0.813268\pi$
$109$	−206605.	−1.66561	−0.832807	+	0.553563i	$0.813268\pi$
$110$	84154.7	0.663127
$111$	0	0
$112$	0	0
$113$	70505.6i	0.519431i	0.965685	+	0.259715i	$0.0836287\pi$
$113$	70505.6i	0.519431i	−0.965685	+	0.259715i	$0.916371\pi$
$114$	0	0
$115$	− 237152.i	− 1.67218i
$116$	− 110444.i	− 0.762077i
$117$	0	0
$118$	− 14064.3i	− 0.0929847i
$119$	0	0
$120$	0	0
$121$	−173511.	−1.07737
$122$	49313.1	0.299960
$123$	0	0
$124$	− 54414.1i	− 0.317802i
$125$	−185387.	−1.06122
$126$	0	0
$127$	25289.0	0.139131	0.0695653	−	0.997577i	$-0.477839\pi$
$127$	25289.0	0.139131	0.0695653	+	0.997577i	$0.477839\pi$
$128$	− 172330.i	− 0.929688i
$129$	0	0
$130$	−167580.	−0.869689
$131$	145419.	0.740359	0.370180	−	0.928960i	$-0.379296\pi$
$131$	145419.	0.740359	0.370180	+	0.928960i	$0.379296\pi$
$132$	0	0
$133$	0	0
$134$	− 85410.0i	− 0.410910i
$135$	0	0
$136$	− 59748.8i	− 0.277001i
$137$	− 336062.i	− 1.52974i	−0.644183	−	0.764872i	$-0.722802\pi$
$137$	− 336062.i	− 1.52974i	0.644183	−	0.764872i	$-0.277198\pi$
$138$	0	0
$139$	108768.i	0.477488i	0.971083	+	0.238744i	$0.0767357\pi$
$139$	108768.i	0.477488i	−0.971083	+	0.238744i	$0.923264\pi$
$140$	0	0
$141$	0	0
$142$	−11204.0	−0.0466286
$143$	666224.	2.72446
$144$	0	0
$145$	− 236716.i	− 0.934991i
$146$	150854.	0.585700
$147$	0	0
$148$	4872.00	0.0182832
$149$	− 293721.i	− 1.08385i	−0.840427	−	0.541925i	$-0.817696\pi$
$149$	− 293721.i	− 1.08385i	0.840427	−	0.541925i	$-0.182304\pi$
$150$	0	0
$151$	190694.	0.680604	0.340302	−	0.940316i	$-0.389471\pi$
$151$	190694.	0.680604	0.340302	+	0.940316i	$0.389471\pi$
$152$	274617.	0.964093
$153$	0	0
$154$	0	0
$155$	− 116626.i	− 0.389911i
$156$	0	0
$157$	272822.i	0.883346i	0.897176	+	0.441673i	$0.145615\pi$
$157$	272822.i	0.883346i	−0.897176	+	0.441673i	$0.854385\pi$
$158$	111398.i	0.355004i
$159$	0	0
$160$	− 307280.i	− 0.948929i
$161$	0	0
$162$	0	0
$163$	−187366.	−0.552359	−0.276180	−	0.961106i	$-0.589068\pi$
$163$	−187366.	−0.552359	−0.276180	+	0.961106i	$0.589068\pi$
$164$	−2057.57	−0.00597373
$165$	0	0
$166$	− 179780.i	− 0.506374i
$167$	−218908.	−0.607395	−0.303698	−	0.952768i	$-0.598221\pi$
$167$	−218908.	−0.607395	−0.303698	+	0.952768i	$0.598221\pi$
$168$	0	0
$169$	−955382.	−2.57312
$170$	− 54882.8i	− 0.145651i
$171$	0	0
$172$	112728.	0.290543
$173$	−98283.3	−0.249669	−0.124834	−	0.992178i	$-0.539840\pi$
$173$	−98283.3	−0.249669	−0.124834	+	0.992178i	$0.539840\pi$
$174$	0	0
$175$	0	0
$176$	185092.i	0.450409i
$177$	0	0
$178$	− 298065.i	− 0.705117i
$179$	347635.i	0.810944i	0.914107	+	0.405472i	$0.132893\pi$
$179$	347635.i	0.810944i	−0.914107	+	0.405472i	$0.867107\pi$
$180$	0	0
$181$	275526.i	0.625124i	0.949897	+	0.312562i	$0.101187\pi$
$181$	275526.i	0.625124i	−0.949897	+	0.312562i	$0.898813\pi$
$182$	0	0
$183$	0	0
$184$	−730240.	−1.59009
$185$	10442.2	0.0224317
$186$	0	0
$187$	218190.i	0.456279i
$188$	575708.	1.18798
$189$	0	0
$190$	252252.	0.506933
$191$	− 277688.i	− 0.550774i	−0.961333	−	0.275387i	$-0.911194\pi$
$191$	− 277688.i	− 0.550774i	0.961333	−	0.275387i	$-0.0888060\pi$
$192$	0	0
$193$	−229621.	−0.443730	−0.221865	−	0.975077i	$-0.571214\pi$
$193$	−229621.	−0.443730	−0.221865	+	0.975077i	$0.571214\pi$
$194$	5829.79	0.0111211
$195$	0	0
$196$	0	0
$197$	21074.6i	0.0386896i	0.999813	+	0.0193448i	$0.00615802\pi$
$197$	21074.6i	0.0386896i	−0.999813	+	0.0193448i	$0.993842\pi$
$198$	0	0
$199$	− 513103.i	− 0.918484i	−0.888311	−	0.459242i	$-0.848121\pi$
$199$	− 513103.i	− 0.918484i	0.888311	−	0.459242i	$-0.151879\pi$
$200$	75869.7i	0.134120i
$201$	0	0
$202$	− 89768.7i	− 0.154791i
$203$	0	0
$204$	0	0
$205$	−4410.00	−0.00732916
$206$	−477871.	−0.784590
$207$	0	0
$208$	− 368580.i	− 0.590709i
$209$	−1.00284e6	−1.58806
$210$	0	0
$211$	444710.	0.687655	0.343828	−	0.939033i	$-0.388276\pi$
$211$	444710.	0.687655	0.343828	+	0.939033i	$0.388276\pi$
$212$	− 178055.i	− 0.272091i
$213$	0	0
$214$	−387752.	−0.578788
$215$	241610.	0.356467
$216$	0	0
$217$	0	0
$218$	584367.i	0.832807i
$219$	0	0
$220$	714076.i	0.994690i
$221$	− 434489.i	− 0.598409i
$222$	0	0
$223$	286280.i	0.385504i	0.981247	+	0.192752i	$0.0617413\pi$
$223$	286280.i	0.385504i	−0.981247	+	0.192752i	$0.938259\pi$
$224$	0	0
$225$	0	0
$226$	199420.	0.259715
$227$	490576.	0.631891	0.315945	−	0.948777i	$-0.397678\pi$
$227$	490576.	0.631891	0.315945	+	0.948777i	$0.397678\pi$
$228$	0	0
$229$	− 920566.i	− 1.16002i	−0.814609	−	0.580011i	$-0.803048\pi$
$229$	− 920566.i	− 1.16002i	0.814609	−	0.580011i	$-0.196952\pi$
$230$	−670768.	−0.836090
$231$	0	0
$232$	−728896.	−0.889090
$233$	1.26275e6i	1.52380i	0.647698	+	0.761898i	$0.275732\pi$
$233$	1.26275e6i	1.52380i	−0.647698	+	0.761898i	$0.724268\pi$
$234$	0	0
$235$	1.23392e6	1.45753
$236$	119339.	0.139477
$237$	0	0
$238$	0	0
$239$	− 723615.i	− 0.819432i	−0.912213	−	0.409716i	$-0.865628\pi$
$239$	− 723615.i	− 0.819432i	0.912213	−	0.409716i	$-0.134372\pi$
$240$	0	0
$241$	567711.i	0.629629i	0.949153	+	0.314814i	$0.101942\pi$
$241$	567711.i	0.629629i	−0.949153	+	0.314814i	$0.898058\pi$
$242$	490763.i	0.538683i
$243$	0	0
$244$	418436.i	0.449940i
$245$	0	0
$246$	0	0
$247$	1.99700e6	2.08274
$248$	−359115.	−0.370770
$249$	0	0
$250$	524354.i	0.530609i
$251$	621798.	0.622967	0.311484	−	0.950252i	$-0.399174\pi$
$251$	621798.	0.622967	0.311484	+	0.950252i	$0.399174\pi$
$252$	0	0
$253$	2.66668e6	2.61921
$254$	− 71528.1i	− 0.0695653i
$255$	0	0
$256$	−700416.	−0.667969
$257$	240993.	0.227600	0.113800	−	0.993504i	$-0.463698\pi$
$257$	240993.	0.227600	0.113800	+	0.993504i	$0.463698\pi$
$258$	0	0
$259$	0	0
$260$	− 1.42196e6i	− 1.30453i
$261$	0	0
$262$	− 411307.i	− 0.370180i
$263$	− 392549.i	− 0.349949i	−0.984573	−	0.174974i	$-0.944016\pi$
$263$	− 392549.i	− 0.349949i	0.984573	−	0.174974i	$-0.0559843\pi$
$264$	0	0
$265$	− 381626.i	− 0.333829i
$266$	0	0
$267$	0	0
$268$	724728.	0.616365
$269$	−1.04039e6	−0.876631	−0.438316	−	0.898821i	$-0.644425\pi$
$269$	−1.04039e6	−0.876631	−0.438316	+	0.898821i	$0.644425\pi$
$270$	0	0
$271$	2.21245e6i	1.83000i	0.403456	+	0.914999i	$0.367809\pi$
$271$	2.21245e6i	1.83000i	−0.403456	+	0.914999i	$0.632191\pi$
$272$	120711.	0.0989291
$273$	0	0
$274$	−950528.	−0.764872
$275$	− 277060.i	− 0.220924i
$276$	0	0
$277$	−172717.	−0.135249	−0.0676247	−	0.997711i	$-0.521542\pi$
$277$	−172717.	−0.135249	−0.0676247	+	0.997711i	$0.521542\pi$
$278$	307641.	0.238744
$279$	0	0
$280$	0	0
$281$	1.93894e6i	1.46487i	0.680837	+	0.732435i	$0.261616\pi$
$281$	1.93894e6i	1.46487i	−0.680837	+	0.732435i	$0.738384\pi$
$282$	0	0
$283$	− 374873.i	− 0.278239i	−0.990276	−	0.139120i	$-0.955573\pi$
$283$	− 374873.i	− 0.278239i	0.990276	−	0.139120i	$-0.0444272\pi$
$284$	− 95069.1i	− 0.0699429i
$285$	0	0
$286$	− 1.88437e6i	− 1.36223i
$287$	0	0
$288$	0	0
$289$	−1.27756e6	−0.899781
$290$	−669534.	−0.467496
$291$	0	0
$292$	1.28004e6i	0.878551i
$293$	2.75403e6	1.87413	0.937063	−	0.349160i	$-0.113533\pi$
$293$	2.75403e6	1.87413	0.937063	+	0.349160i	$0.113533\pi$
$294$	0	0
$295$	255780.	0.171124
$296$	− 32153.6i	− 0.0213304i
$297$	0	0
$298$	−830768.	−0.541925
$299$	−5.31025e6	−3.43508
$300$	0	0
$301$	0	0
$302$	− 539364.i	− 0.340302i
$303$	0	0
$304$	554811.i	0.344319i
$305$	896835.i	0.552030i
$306$	0	0
$307$	1.10731e6i	0.670535i	0.942123	+	0.335267i	$0.108827\pi$
$307$	1.10731e6i	0.670535i	−0.942123	+	0.335267i	$0.891173\pi$
$308$	0	0
$309$	0	0
$310$	−329868.	−0.194956
$311$	−903600.	−0.529755	−0.264878	−	0.964282i	$-0.585332\pi$
$311$	−903600.	−0.529755	−0.264878	+	0.964282i	$0.585332\pi$
$312$	0	0
$313$	953641.i	0.550205i	0.961415	+	0.275102i	$0.0887117\pi$
$313$	953641.i	0.550205i	−0.961415	+	0.275102i	$0.911288\pi$
$314$	771658.	0.441673
$315$	0	0
$316$	−945240.	−0.532506
$317$	1.88142e6i	1.05157i	0.850618	+	0.525785i	$0.176228\pi$
$317$	1.88142e6i	1.05157i	−0.850618	+	0.525785i	$0.823772\pi$
$318$	0	0
$319$	2.66177e6	1.46452
$320$	−342380.	−0.186910
$321$	0	0
$322$	0	0
$323$	654020.i	0.348807i
$324$	0	0
$325$	551719.i	0.289741i
$326$	529951.i	0.276180i
$327$	0	0
$328$	13579.3i	0.00696935i
$329$	0	0
$330$	0	0
$331$	−433825.	−0.217643	−0.108821	−	0.994061i	$-0.534708\pi$
$331$	−433825.	−0.217643	−0.108821	+	0.994061i	$0.534708\pi$
$332$	1.52548e6	0.759560
$333$	0	0
$334$	619167.i	0.303698i
$335$	1.55331e6	0.756217
$336$	0	0
$337$	1.18200e6	0.566947	0.283473	−	0.958980i	$-0.408513\pi$
$337$	1.18200e6	0.566947	0.283473	+	0.958980i	$0.408513\pi$
$338$	2.70223e6i	1.28656i
$339$	0	0
$340$	465696.	0.218477
$341$	1.31141e6	0.610735
$342$	0	0
$343$	0	0
$344$	− 743967.i	− 0.338967i
$345$	0	0
$346$	277987.i	0.124834i
$347$	− 1.65776e6i	− 0.739089i	−0.929213	−	0.369544i	$-0.879514\pi$
$347$	− 1.65776e6i	− 0.739089i	0.929213	−	0.369544i	$-0.120486\pi$
$348$	0	0
$349$	1.90976e6i	0.839294i	0.907687	+	0.419647i	$0.137846\pi$
$349$	1.90976e6i	0.839294i	−0.907687	+	0.419647i	$0.862154\pi$
$350$	0	0
$351$	0	0
$352$	3.45523e6	1.48635
$353$	−2.83919e6	−1.21271	−0.606356	−	0.795193i	$-0.707369\pi$
$353$	−2.83919e6	−1.21271	−0.606356	+	0.795193i	$0.707369\pi$
$354$	0	0
$355$	− 203762.i	− 0.0858128i
$356$	2.52917e6	1.05768
$357$	0	0
$358$	983260.	0.405472
$359$	2.73644e6i	1.12060i	0.828290	+	0.560299i	$0.189314\pi$
$359$	2.73644e6i	1.12060i	−0.828290	+	0.560299i	$0.810686\pi$
$360$	0	0
$361$	−529904.	−0.214008
$362$	779305.	0.312562
$363$	0	0
$364$	0	0
$365$	2.74352e6i	1.07789i
$366$	0	0
$367$	2.33832e6i	0.906229i	0.891452	+	0.453114i	$0.149687\pi$
$367$	2.33832e6i	0.906229i	−0.891452	+	0.453114i	$0.850313\pi$
$368$	− 1.47531e6i	− 0.567888i
$369$	0	0
$370$	− 29534.9i	− 0.0112158i
$371$	0	0
$372$	0	0
$373$	2.30750e6	0.858755	0.429377	−	0.903125i	$-0.358733\pi$
$373$	2.30750e6	0.858755	0.429377	+	0.903125i	$0.358733\pi$
$374$	617134.	0.228140
$375$	0	0
$376$	− 3.79948e6i	− 1.38597i
$377$	−5.30048e6	−1.92071
$378$	0	0
$379$	−2.36146e6	−0.844466	−0.422233	−	0.906487i	$-0.638754\pi$
$379$	−2.36146e6	−0.844466	−0.422233	+	0.906487i	$0.638754\pi$
$380$	2.14043e6i	0.760400i
$381$	0	0
$382$	−785420.	−0.275387
$383$	−4.18592e6	−1.45812	−0.729062	−	0.684448i	$-0.760043\pi$
$383$	−4.18592e6	−1.45812	−0.729062	+	0.684448i	$0.760043\pi$
$384$	0	0
$385$	0	0
$386$	649466.i	0.221865i
$387$	0	0
$388$	49467.4i	0.0166817i
$389$	2.00582e6i	0.672075i	0.941849	+	0.336038i	$0.109087\pi$
$389$	2.00582e6i	0.672075i	−0.941849	+	0.336038i	$0.890913\pi$
$390$	0	0
$391$	− 1.73912e6i	− 0.575290i
$392$	0	0
$393$	0	0
$394$	59608.0	0.0193448
$395$	−2.02594e6	−0.653331
$396$	0	0
$397$	5.84274e6i	1.86055i	0.366869	+	0.930273i	$0.380430\pi$
$397$	5.84274e6i	1.86055i	−0.366869	+	0.930273i	$0.619570\pi$
$398$	−1.45127e6	−0.459242
$399$	0	0
$400$	−153280.	−0.0479000
$401$	− 4.17754e6i	− 1.29736i	−0.761062	−	0.648679i	$-0.775322\pi$
$401$	− 4.17754e6i	− 1.29736i	0.761062	−	0.648679i	$-0.224678\pi$
$402$	0	0
$403$	−2.61146e6	−0.800977
$404$	761713.	0.232187
$405$	0	0
$406$	0	0
$407$	117418.i	0.0351357i
$408$	0	0
$409$	2.09137e6i	0.618190i	0.951031	+	0.309095i	$0.100026\pi$
$409$	2.09137e6i	0.618190i	−0.951031	+	0.309095i	$0.899974\pi$
$410$	12473.4i	0.00366458i
$411$	0	0
$412$	− 4.05487e6i	− 1.17688i
$413$	0	0
$414$	0	0
$415$	3.26957e6	0.931903
$416$	−6.88052e6	−1.94934
$417$	0	0
$418$	2.83647e6i	0.794031i
$419$	−6.55448e6	−1.82391	−0.911954	−	0.410291i	$-0.865427\pi$
$419$	−6.55448e6	−1.82391	−0.911954	+	0.410291i	$0.865427\pi$
$420$	0	0
$421$	5.99908e6	1.64960	0.824801	−	0.565423i	$-0.191287\pi$
$421$	5.99908e6	1.64960	0.824801	+	0.565423i	$0.191287\pi$
$422$	− 1.25783e6i	− 0.343828i
$423$	0	0
$424$	−1.17510e6	−0.317440
$425$	−180689.	−0.0485243
$426$	0	0
$427$	0	0
$428$	− 3.29018e6i	− 0.868182i
$429$	0	0
$430$	− 683377.i	− 0.178233i
$431$	5.26665e6i	1.36565i	0.730580	+	0.682827i	$0.239250\pi$
$431$	5.26665e6i	1.36565i	−0.730580	+	0.682827i	$0.760750\pi$
$432$	0	0
$433$	− 809992.i	− 0.207616i	−0.994597	−	0.103808i	$-0.966897\pi$
$433$	− 809992.i	− 0.207616i	0.994597	−	0.103808i	$-0.0331028\pi$
$434$	0	0
$435$	0	0
$436$	−4.95852e6	−1.24921
$437$	7.99332e6	2.00227
$438$	0	0
$439$	− 5.41939e6i	− 1.34211i	−0.741406	−	0.671057i	$-0.765841\pi$
$439$	− 5.41939e6i	− 1.34211i	0.741406	−	0.671057i	$-0.234159\pi$
$440$	4.71266e6	1.16047
$441$	0	0
$442$	−1.22892e6	−0.299204
$443$	2.82652e6i	0.684294i	0.939646	+	0.342147i	$0.111154\pi$
$443$	2.82652e6i	0.684294i	−0.939646	+	0.342147i	$0.888846\pi$
$444$	0	0
$445$	5.42077e6	1.29766
$446$	809723.	0.192752
$447$	0	0
$448$	0	0
$449$	3.89941e6i	0.912816i	0.889771	+	0.456408i	$0.150864\pi$
$449$	3.89941e6i	0.912816i	−0.889771	+	0.456408i	$0.849136\pi$
$450$	0	0
$451$	− 49588.6i	− 0.0114800i
$452$	1.69213e6i	0.389573i
$453$	0	0
$454$	− 1.38756e6i	− 0.315945i
$455$	0	0
$456$	0	0
$457$	−646723.	−0.144853	−0.0724266	−	0.997374i	$-0.523074\pi$
$457$	−646723.	−0.144853	−0.0724266	+	0.997374i	$0.523074\pi$
$458$	−2.60375e6	−0.580011
$459$	0	0
$460$	− 5.69166e6i	− 1.25413i
$461$	4.99109e6	1.09381	0.546906	−	0.837194i	$-0.315806\pi$
$461$	4.99109e6	1.09381	0.546906	+	0.837194i	$0.315806\pi$
$462$	0	0
$463$	−6.41938e6	−1.39168	−0.695842	−	0.718195i	$-0.744969\pi$
$463$	−6.41938e6	−1.39168	−0.695842	+	0.718195i	$0.744969\pi$
$464$	− 1.47259e6i	− 0.317532i
$465$	0	0
$466$	3.57159e6	0.761898
$467$	−4.66484e6	−0.989793	−0.494897	−	0.868952i	$-0.664794\pi$
$467$	−4.66484e6	−0.989793	−0.494897	+	0.868952i	$0.664794\pi$
$468$	0	0
$469$	0	0
$470$	− 3.49005e6i	− 0.728764i
$471$	0	0
$472$	− 787598.i	− 0.162723i
$473$	2.71681e6i	0.558349i
$474$	0	0
$475$	− 830482.i	− 0.168887i
$476$	0	0
$477$	0	0
$478$	−2.04669e6	−0.409716
$479$	7.81785e6	1.55686	0.778428	−	0.627734i	$-0.216018\pi$
$479$	7.81785e6	1.55686	0.778428	+	0.627734i	$0.216018\pi$
$480$	0	0
$481$	− 233818.i	− 0.0460803i
$482$	1.60573e6	0.314814
$483$	0	0
$484$	−4.16426e6	−0.808025
$485$	106024.i	0.0204667i
$486$	0	0
$487$	−2.06216e6	−0.394004	−0.197002	−	0.980403i	$-0.563121\pi$
$487$	−2.06216e6	−0.394004	−0.197002	+	0.980403i	$0.563121\pi$
$488$	2.76154e6	0.524930
$489$	0	0
$490$	0	0
$491$	3.97446e6i	0.744003i	0.928232	+	0.372001i	$0.121328\pi$
$491$	3.97446e6i	0.744003i	−0.928232	+	0.372001i	$0.878672\pi$
$492$	0	0
$493$	− 1.73592e6i	− 0.321671i
$494$	− 5.64835e6i	− 1.04137i
$495$	0	0
$496$	− 725521.i	− 0.132418i
$497$	0	0
$498$	0	0
$499$	2.75654e6	0.495579	0.247789	−	0.968814i	$-0.420296\pi$
$499$	2.75654e6	0.495579	0.247789	+	0.968814i	$0.420296\pi$
$500$	−4.44929e6	−0.795914
$501$	0	0
$502$	− 1.75871e6i	− 0.311484i
$503$	−4.61230e6	−0.812827	−0.406413	−	0.913689i	$-0.633221\pi$
$503$	−4.61230e6	−0.812827	−0.406413	+	0.913689i	$0.633221\pi$
$504$	0	0
$505$	1.63258e6	0.284870
$506$	− 7.54251e6i	− 1.30960i
$507$	0	0
$508$	606936.	0.104348
$509$	5.49047e6	0.939324	0.469662	−	0.882846i	$-0.344376\pi$
$509$	5.49047e6	0.939324	0.469662	+	0.882846i	$0.344376\pi$
$510$	0	0
$511$	0	0
$512$	− 3.53350e6i	− 0.595703i
$513$	0	0
$514$	− 681631.i	− 0.113800i
$515$	− 8.69082e6i	− 1.44392i
$516$	0	0
$517$	1.38749e7i	2.28299i
$518$	0	0
$519$	0	0
$520$	−9.38448e6	−1.52196
$521$	5.95211e6	0.960675	0.480338	−	0.877084i	$-0.340514\pi$
$521$	5.95211e6	0.960675	0.480338	+	0.877084i	$0.340514\pi$
$522$	0	0
$523$	3.62100e6i	0.578861i	0.957199	+	0.289430i	$0.0934659\pi$
$523$	3.62100e6i	0.578861i	−0.957199	+	0.289430i	$0.906534\pi$
$524$	3.49005e6	0.555269
$525$	0	0
$526$	−1.11030e6	−0.174974
$527$	− 855257.i	− 0.134144i
$528$	0	0
$529$	−1.48189e7	−2.30237
$530$	−1.07940e6	−0.166914
$531$	0	0
$532$	0	0
$533$	98747.5i	0.0150559i
$534$	0	0
$535$	− 7.05186e6i	− 1.06517i
$536$	− 4.78296e6i	− 0.719093i
$537$	0	0
$538$	2.94268e6i	0.438316i
$539$	0	0
$540$	0	0
$541$	−1.08081e7	−1.58765	−0.793825	−	0.608147i	$-0.791913\pi$
$541$	−1.08081e7	−1.58765	−0.793825	+	0.608147i	$0.791913\pi$
$542$	6.25776e6	0.914999
$543$	0	0
$544$	− 2.25338e6i	− 0.326466i
$545$	−1.06276e7	−1.53266
$546$	0	0
$547$	−87526.0	−0.0125074	−0.00625372	−	0.999980i	$-0.501991\pi$
$547$	−87526.0	−0.0125074	−0.00625372	+	0.999980i	$0.501991\pi$
$548$	− 8.06550e6i	− 1.14731i
$549$	0	0
$550$	−783644.	−0.110462
$551$	7.97861e6	1.11956
$552$	0	0
$553$	0	0
$554$	488517.i	0.0676247i
$555$	0	0
$556$	2.61042e6i	0.358116i
$557$	− 5.45592e6i	− 0.745127i	−0.928007	−	0.372564i	$-0.878479\pi$
$557$	− 5.45592e6i	− 0.745127i	0.928007	−	0.372564i	$-0.121521\pi$
$558$	0	0
$559$	− 5.41007e6i	− 0.732273i
$560$	0	0
$561$	0	0
$562$	5.48416e6	0.732435
$563$	4.45812e6	0.592763	0.296381	−	0.955070i	$-0.404220\pi$
$563$	4.45812e6	0.592763	0.296381	+	0.955070i	$0.404220\pi$
$564$	0	0
$565$	3.62676e6i	0.477967i
$566$	−1.06030e6	−0.139120
$567$	0	0
$568$	−627424.	−0.0816000
$569$	8.54666e6i	1.10666i	0.832961	+	0.553332i	$0.186644\pi$
$569$	8.54666e6i	1.10666i	−0.832961	+	0.553332i	$0.813356\pi$
$570$	0	0
$571$	3.30048e6	0.423631	0.211815	−	0.977310i	$-0.432062\pi$
$571$	3.30048e6	0.423631	0.211815	+	0.977310i	$0.432062\pi$
$572$	1.59894e7	2.04335
$573$	0	0
$574$	0	0
$575$	2.20835e6i	0.278547i
$576$	0	0
$577$	− 1.57698e7i	− 1.97191i	−0.167009	−	0.985955i	$-0.553411\pi$
$577$	− 1.57698e7i	− 1.97191i	0.167009	−	0.985955i	$-0.446589\pi$
$578$	3.61349e6i	0.449891i
$579$	0	0
$580$	− 5.68118e6i	− 0.701243i
$581$	0	0
$582$	0	0
$583$	4.29123e6	0.522890
$584$	8.44784e6	1.02498
$585$	0	0
$586$	− 7.78956e6i	− 0.937063i
$587$	−3.36204e6	−0.402724	−0.201362	−	0.979517i	$-0.564537\pi$
$587$	−3.36204e6	−0.402724	−0.201362	+	0.979517i	$0.564537\pi$
$588$	0	0
$589$	3.93093e6	0.466882
$590$	− 723455.i	− 0.0855621i
$591$	0	0
$592$	64960.0	0.00761801
$593$	962823.	0.112437	0.0562186	−	0.998418i	$-0.482096\pi$
$593$	962823.	0.112437	0.0562186	+	0.998418i	$0.482096\pi$
$594$	0	0
$595$	0	0
$596$	− 7.04930e6i	− 0.812887i
$597$	0	0
$598$	1.50197e7i	1.71754i
$599$	7.02857e6i	0.800387i	0.916431	+	0.400193i	$0.131057\pi$
$599$	7.02857e6i	0.800387i	−0.916431	+	0.400193i	$0.868943\pi$
$600$	0	0
$601$	9.08125e6i	1.02556i	0.858521	+	0.512778i	$0.171384\pi$
$601$	9.08125e6i	1.02556i	−0.858521	+	0.512778i	$0.828616\pi$
$602$	0	0
$603$	0	0
$604$	4.57666e6	0.510453
$605$	−8.92528e6	−0.991365
$606$	0	0
$607$	− 1.23767e7i	− 1.36343i	−0.731617	−	0.681715i	$-0.761234\pi$
$607$	− 1.23767e7i	− 1.36343i	0.731617	−	0.681715i	$-0.238766\pi$
$608$	1.03570e7	1.13625
$609$	0	0
$610$	2.53663e6	0.276015
$611$	− 2.76295e7i	− 2.99413i
$612$	0	0
$613$	1.14616e7	1.23195	0.615975	−	0.787766i	$-0.288762\pi$
$613$	1.14616e7	1.23195	0.615975	+	0.787766i	$0.288762\pi$
$614$	3.13193e6	0.335267
$615$	0	0
$616$	0	0
$617$	− 1.38242e7i	− 1.46193i	−0.682417	−	0.730963i	$-0.739071\pi$
$617$	− 1.38242e7i	− 1.46193i	0.682417	−	0.730963i	$-0.260929\pi$
$618$	0	0
$619$	− 8.52363e6i	− 0.894124i	−0.894503	−	0.447062i	$-0.852470\pi$
$619$	− 8.52363e6i	− 0.894124i	0.894503	−	0.447062i	$-0.147530\pi$
$620$	− 2.79902e6i	− 0.292434i
$621$	0	0
$622$	2.55577e6i	0.264878i
$623$	0	0
$624$	0	0
$625$	−8.03931e6	−0.823225
$626$	2.69730e6	0.275102
$627$	0	0
$628$	6.54773e6i	0.662509i
$629$	76575.9	0.00771731
$630$	0	0
$631$	8.57038e6	0.856893	0.428446	−	0.903567i	$-0.359061\pi$
$631$	8.57038e6	0.856893	0.428446	+	0.903567i	$0.359061\pi$
$632$	6.23827e6i	0.621257i
$633$	0	0
$634$	5.32146e6	0.525785
$635$	1.30085e6	0.128024
$636$	0	0
$637$	0	0
$638$	− 7.52863e6i	− 0.732258i
$639$	0	0
$640$	− 8.86455e6i	− 0.855474i
$641$	1.53744e7i	1.47793i	0.673743	+	0.738966i	$0.264686\pi$
$641$	1.53744e7i	1.47793i	−0.673743	+	0.738966i	$0.735314\pi$
$642$	0	0
$643$	1.52086e7i	1.45065i	0.688408	+	0.725324i	$0.258310\pi$
$643$	1.52086e7i	1.45065i	−0.688408	+	0.725324i	$0.741690\pi$
$644$	0	0
$645$	0	0
$646$	1.84985e6	0.174403
$647$	−619106.	−0.0581439	−0.0290720	−	0.999577i	$-0.509255\pi$
$647$	−619106.	−0.0581439	−0.0290720	+	0.999577i	$0.509255\pi$
$648$	0	0
$649$	2.87614e6i	0.268039i
$650$	1.56050e6	0.144870
$651$	0	0
$652$	−4.49678e6	−0.414270
$653$	− 9.23344e6i	− 0.847385i	−0.905806	−	0.423693i	$-0.860734\pi$
$653$	− 9.23344e6i	− 0.847385i	0.905806	−	0.423693i	$-0.139266\pi$
$654$	0	0
$655$	7.48024e6	0.681259
$656$	−27434.3	−0.00248905
$657$	0	0
$658$	0	0
$659$	1.34210e7i	1.20384i	0.798555	+	0.601922i	$0.205598\pi$
$659$	1.34210e7i	1.20384i	−0.798555	+	0.601922i	$0.794402\pi$
$660$	0	0
$661$	1.00329e7i	0.893148i	0.894747	+	0.446574i	$0.147356\pi$
$661$	1.00329e7i	0.893148i	−0.894747	+	0.446574i	$0.852644\pi$
$662$	1.22704e6i	0.108821i
$663$	0	0
$664$	− 1.00677e7i	− 0.886154i
$665$	0	0
$666$	0	0
$667$	−2.12161e7	−1.84651
$668$	−5.25380e6	−0.455546
$669$	0	0
$670$	− 4.39343e6i	− 0.378109i
$671$	−1.00845e7	−0.864668
$672$	0	0
$673$	−2.68578e6	−0.228577	−0.114289	−	0.993448i	$-0.536459\pi$
$673$	−2.68578e6	−0.228577	−0.114289	+	0.993448i	$0.536459\pi$
$674$	− 3.34320e6i	− 0.283473i
$675$	0	0
$676$	−2.29292e7	−1.92984
$677$	−1.99641e7	−1.67409	−0.837045	−	0.547134i	$-0.815719\pi$
$677$	−1.99641e7	−1.67409	−0.837045	+	0.547134i	$0.815719\pi$
$678$	0	0
$679$	0	0
$680$	− 3.07344e6i	− 0.254890i
$681$	0	0
$682$	− 3.70923e6i	− 0.305367i
$683$	− 6.40946e6i	− 0.525739i	−0.964831	−	0.262869i	$-0.915331\pi$
$683$	− 6.40946e6i	− 0.525739i	0.964831	−	0.262869i	$-0.0846688\pi$
$684$	0	0
$685$	− 1.72868e7i	− 1.40763i
$686$	0	0
$687$	0	0
$688$	1.50304e6	0.121060
$689$	−8.54527e6	−0.685768
$690$	0	0
$691$	1.20899e7i	0.963225i	0.876384	+	0.481613i	$0.159949\pi$
$691$	1.20899e7i	0.963225i	−0.876384	+	0.481613i	$0.840051\pi$
$692$	−2.35880e6	−0.187252
$693$	0	0
$694$	−4.68884e6	−0.369544
$695$	5.59493e6i	0.439372i
$696$	0	0
$697$	−32340.0	−0.00252149
$698$	5.40160e6	0.419647
$699$	0	0
$700$	0	0
$701$	1.17104e6i	0.0900073i	0.998987	+	0.0450036i	$0.0143299\pi$
$701$	1.17104e6i	0.0900073i	−0.998987	+	0.0450036i	$0.985670\pi$
$702$	0	0
$703$	351958.i	0.0268598i
$704$	− 3.84992e6i	− 0.292766i
$705$	0	0
$706$	8.03045e6i	0.606356i
$707$	0	0
$708$	0	0
$709$	1.34096e7	1.00185	0.500924	−	0.865491i	$-0.332994\pi$
$709$	1.34096e7	1.00185	0.500924	+	0.865491i	$0.332994\pi$
$710$	−576326.	−0.0429064
$711$	0	0
$712$	− 1.66916e7i	− 1.23395i
$713$	−1.04528e7	−0.770032
$714$	0	0
$715$	3.42701e7	2.50698
$716$	8.34324e6i	0.608208i
$717$	0	0
$718$	7.73982e6	0.560299
$719$	1.77543e7	1.28080	0.640398	−	0.768043i	$-0.278769\pi$
$719$	1.77543e7	1.28080	0.640398	+	0.768043i	$0.278769\pi$
$720$	0	0
$721$	0	0
$722$	1.49879e6i	0.107004i
$723$	0	0
$724$	6.61262e6i	0.468843i
$725$	2.20429e6i	0.155748i
$726$	0	0
$727$	5.19333e6i	0.364427i	0.983259	+	0.182213i	$0.0583262\pi$
$727$	5.19333e6i	0.364427i	−0.983259	+	0.182213i	$0.941674\pi$
$728$	0	0
$729$	0	0
$730$	7.75984e6	0.538946
$731$	1.77181e6	0.122638
$732$	0	0
$733$	1.28903e6i	0.0886138i	0.999018	+	0.0443069i	$0.0141079\pi$
$733$	1.28903e6i	0.0886138i	−0.999018	+	0.0443069i	$0.985892\pi$
$734$	6.61375e6	0.453114
$735$	0	0
$736$	−2.75405e7	−1.87403
$737$	1.74663e7i	1.18450i
$738$	0	0
$739$	−1.29124e7	−0.869751	−0.434875	−	0.900491i	$-0.643208\pi$
$739$	−1.29124e7	−0.869751	−0.434875	+	0.900491i	$0.643208\pi$
$740$	250612.	0.0168238
$741$	0	0
$742$	0	0
$743$	8.96758e6i	0.595941i	0.954575	+	0.297970i	$0.0963097\pi$
$743$	8.96758e6i	0.595941i	−0.954575	+	0.297970i	$0.903690\pi$
$744$	0	0
$745$	− 1.51088e7i	− 0.997330i
$746$	− 6.52659e6i	− 0.429377i
$747$	0	0
$748$	5.23656e6i	0.342209i
$749$	0	0
$750$	0	0
$751$	3.70586e6	0.239767	0.119884	−	0.992788i	$-0.461748\pi$
$751$	3.70586e6	0.239767	0.119884	+	0.992788i	$0.461748\pi$
$752$	7.67611e6	0.494990
$753$	0	0
$754$	1.49920e7i	0.960354i
$755$	9.80916e6	0.626274
$756$	0	0
$757$	−2.81893e7	−1.78791	−0.893953	−	0.448161i	$-0.852079\pi$
$757$	−2.81893e7	−1.78791	−0.893953	+	0.448161i	$0.852079\pi$
$758$	6.67921e6i	0.422233i
$759$	0	0
$760$	1.41261e7	0.887133
$761$	9.87315e6	0.618008	0.309004	−	0.951061i	$-0.400004\pi$
$761$	9.87315e6	0.618008	0.309004	+	0.951061i	$0.400004\pi$
$762$	0	0
$763$	0	0
$764$	− 6.66451e6i	− 0.413081i
$765$	0	0
$766$	1.18396e7i	0.729062i
$767$	− 5.72735e6i	− 0.351533i
$768$	0	0
$769$	2.61429e6i	0.159418i	0.996818	+	0.0797091i	$0.0253991\pi$
$769$	2.61429e6i	0.159418i	−0.996818	+	0.0797091i	$0.974601\pi$
$770$	0	0
$771$	0	0
$772$	−5.51090e6	−0.332797
$773$	8.54100e6	0.514114	0.257057	−	0.966396i	$-0.417247\pi$
$773$	8.54100e6	0.514114	0.257057	+	0.966396i	$0.417247\pi$
$774$	0	0
$775$	1.08601e6i	0.0649504i
$776$	326468.	0.0194619
$777$	0	0
$778$	5.67332e6	0.336038
$779$	− 148641.i	− 0.00877596i
$780$	0	0
$781$	2.29122e6	0.134412
$782$	−4.91897e6	−0.287645
$783$	0	0
$784$	0	0
$785$	1.40338e7i	0.812832i
$786$	0	0
$787$	2.10654e7i	1.21236i	0.795326	+	0.606182i	$0.207300\pi$
$787$	2.10654e7i	1.21236i	−0.795326	+	0.606182i	$0.792700\pi$
$788$	505791.i	0.0290172i
$789$	0	0
$790$	5.73021e6i	0.326665i
$791$	0	0
$792$	0	0
$793$	2.00817e7	1.13401
$794$	1.65258e7	0.930273
$795$	0	0
$796$	− 1.23145e7i	− 0.688863i
$797$	3.23910e6	0.180625	0.0903126	−	0.995913i	$-0.471213\pi$
$797$	3.23910e6	0.180625	0.0903126	+	0.995913i	$0.471213\pi$
$798$	0	0
$799$	9.04873e6	0.501442
$800$	2.86137e6i	0.158070i
$801$	0	0
$802$	−1.18159e7	−0.648679
$803$	−3.08497e7	−1.68835
$804$	0	0
$805$	0	0
$806$	7.38631e6i	0.400488i
$807$	0	0
$808$	− 5.02705e6i	− 0.270885i
$809$	− 6.31534e6i	− 0.339254i	−0.985508	−	0.169627i	$-0.945744\pi$
$809$	− 6.31534e6i	− 0.339254i	0.985508	−	0.169627i	$-0.0542563\pi$
$810$	0	0
$811$	− 2.96115e7i	− 1.58092i	−0.612516	−	0.790458i	$-0.709843\pi$
$811$	− 2.96115e7i	− 1.58092i	0.612516	−	0.790458i	$-0.290157\pi$
$812$	0	0
$813$	0	0
$814$	332108.	0.0175678
$815$	−9.63797e6	−0.508267
$816$	0	0
$817$	8.14358e6i	0.426835i
$818$	5.91528e6	0.309095
$819$	0	0
$820$	−105840.	−0.00549687
$821$	1.33551e7i	0.691497i	0.938327	+	0.345749i	$0.112375\pi$
$821$	1.33551e7i	0.691497i	−0.938327	+	0.345749i	$0.887625\pi$
$822$	0	0
$823$	−1.43433e6	−0.0738160	−0.0369080	−	0.999319i	$-0.511751\pi$
$823$	−1.43433e6	−0.0738160	−0.0369080	+	0.999319i	$0.511751\pi$
$824$	−2.67608e7	−1.37303
$825$	0	0
$826$	0	0
$827$	2.58805e7i	1.31586i	0.753080	+	0.657929i	$0.228567\pi$
$827$	2.58805e7i	1.31586i	−0.753080	+	0.657929i	$0.771433\pi$
$828$	0	0
$829$	2.21974e7i	1.12180i	0.827883	+	0.560901i	$0.189545\pi$
$829$	2.21974e7i	1.12180i	−0.827883	+	0.560901i	$0.810455\pi$
$830$	− 9.24775e6i	− 0.465952i
$831$	0	0
$832$	7.66647e6i	0.383961i
$833$	0	0
$834$	0	0
$835$	−1.12605e7	−0.558909
$836$	−2.40682e7	−1.19105
$837$	0	0
$838$	1.85389e7i	0.911954i
$839$	2.56110e7	1.25609	0.628047	−	0.778175i	$-0.283854\pi$
$839$	2.56110e7	1.25609	0.628047	+	0.778175i	$0.283854\pi$
$840$	0	0
$841$	−665883.	−0.0324644
$842$	− 1.69679e7i	− 0.824801i
$843$	0	0
$844$	1.06730e7	0.515741
$845$	−4.91442e7	−2.36772
$846$	0	0
$847$	0	0
$848$	− 2.37407e6i	− 0.113371i
$849$	0	0
$850$	511066.i	0.0242622i
$851$	− 935898.i	− 0.0443001i
$852$	0	0
$853$	− 3.68131e7i	− 1.73233i	−0.499760	−	0.866164i	$-0.666579\pi$
$853$	− 3.68131e7i	− 1.73233i	0.499760	−	0.866164i	$-0.333421\pi$
$854$	0	0
$855$	0	0
$856$	−2.17141e7	−1.01288
$857$	−1.82523e7	−0.848918	−0.424459	−	0.905447i	$-0.639536\pi$
$857$	−1.82523e7	−0.848918	−0.424459	+	0.905447i	$0.639536\pi$
$858$	0	0
$859$	1.11138e7i	0.513901i	0.966425	+	0.256950i	$0.0827177\pi$
$859$	1.11138e7i	0.513901i	−0.966425	+	0.256950i	$0.917282\pi$
$860$	5.79865e6	0.267350
$861$	0	0
$862$	1.48963e7	0.682827
$863$	3.85839e7i	1.76352i	0.471703	+	0.881758i	$0.343640\pi$
$863$	3.85839e7i	1.76352i	−0.471703	+	0.881758i	$0.656360\pi$
$864$	0	0
$865$	−5.05562e6	−0.229739
$866$	−2.29100e6	−0.103808
$867$	0	0
$868$	0	0
$869$	− 2.27808e7i	− 1.02334i
$870$	0	0
$871$	− 3.47813e7i	− 1.55346i
$872$	3.27246e7i	1.45741i
$873$	0	0
$874$	− 2.26085e7i	− 1.00114i
$875$	0	0
$876$	0	0
$877$	2.28950e7	1.00517	0.502587	−	0.864527i	$-0.332382\pi$
$877$	2.28950e7	1.00517	0.502587	+	0.864527i	$0.332382\pi$
$878$	−1.53284e7	−0.671057
$879$	0	0
$880$	9.52101e6i	0.414454i
$881$	−2.09177e7	−0.907975	−0.453987	−	0.891008i	$-0.649999\pi$
$881$	−2.09177e7	−0.907975	−0.453987	+	0.891008i	$0.649999\pi$
$882$	0	0
$883$	−2.26473e7	−0.977497	−0.488748	−	0.872425i	$-0.662546\pi$
$883$	−2.26473e7	−0.977497	−0.488748	+	0.872425i	$0.662546\pi$
$884$	− 1.04277e7i	− 0.448807i
$885$	0	0
$886$	7.99461e6	0.342147
$887$	−4.82907e6	−0.206089	−0.103044	−	0.994677i	$-0.532858\pi$
$887$	−4.82907e6	−0.206089	−0.103044	+	0.994677i	$0.532858\pi$
$888$	0	0
$889$	0	0
$890$	− 1.53323e7i	− 0.648830i
$891$	0	0
$892$	6.87073e6i	0.289128i
$893$	4.15897e7i	1.74525i
$894$	0	0
$895$	1.78821e7i	0.746209i
$896$	0	0
$897$	0	0
$898$	1.10292e7	0.456408
$899$	−1.04336e7	−0.430560
$900$	0	0
$901$	− 2.79859e6i	− 0.114849i
$902$	−140258.	−0.00573998
$903$	0	0
$904$	1.11675e7	0.454502
$905$	1.41729e7i	0.575223i
$906$	0	0
$907$	−1.19152e7	−0.480931	−0.240466	−	0.970658i	$-0.577300\pi$
$907$	−1.19152e7	−0.480931	−0.240466	+	0.970658i	$0.577300\pi$
$908$	1.17738e7	0.473918
$909$	0	0
$910$	0	0
$911$	− 1.33120e7i	− 0.531432i	−0.964051	−	0.265716i	$-0.914392\pi$
$911$	− 1.33120e7i	− 0.531432i	0.964051	−	0.265716i	$-0.0856083\pi$
$912$	0	0
$913$	3.67650e7i	1.45968i
$914$	1.82921e6i	0.0724266i
$915$	0	0
$916$	− 2.20936e7i	− 0.870016i
$917$	0	0
$918$	0	0
$919$	2.23234e7	0.871909	0.435954	−	0.899969i	$-0.356411\pi$
$919$	2.23234e7	0.871909	0.435954	+	0.899969i	$0.356411\pi$
$920$	−3.75630e7	−1.46316
$921$	0	0
$922$	− 1.41169e7i	− 0.546906i
$923$	−4.56258e6	−0.176281
$924$	0	0
$925$	−97237.0	−0.00373661
$926$	1.81568e7i	0.695842i
$927$	0	0
$928$	−2.74898e7	−1.04786
$929$	9.56783e6	0.363726	0.181863	−	0.983324i	$-0.441787\pi$
$929$	9.56783e6	0.363726	0.181863	+	0.983324i	$0.441787\pi$
$930$	0	0
$931$	0	0
$932$	3.03059e7i	1.14285i
$933$	0	0
$934$	1.31942e7i	0.494897i
$935$	1.12235e7i	0.419856i
$936$	0	0
$937$	6.81897e6i	0.253729i	0.991920	+	0.126864i	$0.0404913\pi$
$937$	6.81897e6i	0.253729i	−0.991920	+	0.126864i	$0.959509\pi$
$938$	0	0
$939$	0	0
$940$	2.96140e7	1.09315
$941$	2.57175e7	0.946793	0.473396	−	0.880849i	$-0.343028\pi$
$941$	2.57175e7	0.946793	0.473396	+	0.880849i	$0.343028\pi$
$942$	0	0
$943$	395254.i	0.0144743i
$944$	1.59119e6	0.0581155
$945$	0	0
$946$	7.68429e6	0.279175
$947$	7.84001e6i	0.284081i	0.989861	+	0.142040i	$0.0453663\pi$
$947$	7.84001e6i	0.284081i	−0.989861	+	0.142040i	$0.954634\pi$
$948$	0	0
$949$	6.14320e7	2.21426
$950$	−2.34896e6	−0.0844435
$951$	0	0
$952$	0	0
$953$	2.65199e7i	0.945888i	0.881092	+	0.472944i	$0.156809\pi$
$953$	2.65199e7i	0.945888i	−0.881092	+	0.472944i	$0.843191\pi$
$954$	0	0
$955$	− 1.42841e7i	− 0.506808i
$956$	− 1.73668e7i	− 0.614574i
$957$	0	0
$958$	− 2.21122e7i	− 0.778428i
$959$	0	0
$960$	0	0
$961$	2.34887e7	0.820447
$962$	−661338.	−0.0230402
$963$	0	0
$964$	1.36251e7i	0.472222i
$965$	−1.18115e7	−0.408308
$966$	0	0
$967$	1.56461e7	0.538070	0.269035	−	0.963130i	$-0.413295\pi$
$967$	1.56461e7	0.538070	0.269035	+	0.963130i	$0.413295\pi$
$968$	2.74827e7i	0.942696i
$969$	0	0
$970$	299880.	0.0102334
$971$	2.71185e7	0.923034	0.461517	−	0.887131i	$-0.347305\pi$
$971$	2.71185e7	0.923034	0.461517	+	0.887131i	$0.347305\pi$
$972$	0	0
$973$	0	0
$974$	5.83268e6i	0.197002i
$975$	0	0
$976$	5.57914e6i	0.187475i
$977$	− 2.43034e7i	− 0.814573i	−0.913301	−	0.407286i	$-0.866475\pi$
$977$	− 2.43034e7i	− 0.814573i	0.913301	−	0.407286i	$-0.133525\pi$
$978$	0	0
$979$	6.09543e7i	2.03258i
$980$	0	0
$981$	0	0
$982$	1.12415e7	0.372001
$983$	−1.30282e7	−0.430033	−0.215016	−	0.976610i	$-0.568981\pi$
$983$	−1.30282e7	−0.430033	−0.215016	+	0.976610i	$0.568981\pi$
$984$	0	0
$985$	1.08406e6i	0.0356011i
$986$	−4.90991e6	−0.160835
$987$	0	0
$988$	4.79279e7	1.56205
$989$	− 2.16547e7i	− 0.703983i
$990$	0	0
$991$	−3.27476e7	−1.05924	−0.529621	−	0.848235i	$-0.677666\pi$
$991$	−3.27476e7	−1.05924	−0.529621	+	0.848235i	$0.677666\pi$
$992$	−1.35438e7	−0.436978
$993$	0	0
$994$	0	0
$995$	− 2.63936e7i	− 0.845165i
$996$	0	0
$997$	2.16492e7i	0.689770i	0.938645	+	0.344885i	$0.112082\pi$
$997$	2.16492e7i	0.689770i	−0.938645	+	0.344885i	$0.887918\pi$
$998$	− 7.79666e6i	− 0.247789i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.6.c.a.440.2		4
3.2	odd	2	inner	441.6.c.a.440.3		4
7.4	even	3		63.6.p.a.26.2	yes	4
7.5	odd	6		63.6.p.a.17.1	✓	4
7.6	odd	2	inner	441.6.c.a.440.1		4
21.5	even	6		63.6.p.a.17.2	yes	4
21.11	odd	6		63.6.p.a.26.1	yes	4
21.20	even	2	inner	441.6.c.a.440.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.6.p.a.17.1	✓	4	7.5	odd	6
63.6.p.a.17.2	yes	4	21.5	even	6
63.6.p.a.26.1	yes	4	21.11	odd	6
63.6.p.a.26.2	yes	4	7.4	even	3
441.6.c.a.440.1		4	7.6	odd	2	inner
441.6.c.a.440.2		4	1.1	even	1	trivial
441.6.c.a.440.3		4	3.2	odd	2	inner
441.6.c.a.440.4		4	21.20	even	2	inner

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:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	441.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.82843i q^{2} +24.0000 q^{4} +51.4393 q^{5} -158.392i q^{8} +O(q^{10})\)	\(q-2.82843i q^{2} +24.0000 q^{4} +51.4393 q^{5} -158.392i q^{8} -145.492i q^{10} +578.413i q^{11} -1151.81i q^{13} +320.000 q^{16} +377.221 q^{17} +1733.78i q^{19} +1234.54 q^{20} +1636.00 q^{22} -4610.34i q^{23} -479.000 q^{25} -3257.82 q^{26} -4601.85i q^{29} -2267.25i q^{31} -5973.64i q^{32} -1066.94i q^{34} +203.000 q^{37} +4903.88 q^{38} -8147.57i q^{40} -85.7321 q^{41} +4697.00 q^{43} +13881.9i q^{44} -13040.0 q^{46} +23987.9 q^{47} +1354.82i q^{50} -27643.5i q^{52} -7418.96i q^{53} +29753.2i q^{55} -13016.0 q^{58} +4972.46 q^{59} +17434.8i q^{61} -6412.76 q^{62} -6656.00 q^{64} -59248.5i q^{65} +30197.0 q^{67} +9053.31 q^{68} -3961.21i q^{71} +53335.0i q^{73} -574.171i q^{74} +41610.8i q^{76} -39385.0 q^{79} +16460.6 q^{80} +242.487i q^{82} +63561.8 q^{83} +19404.0 q^{85} -13285.1i q^{86} +91616.0 q^{88} +105382. q^{89} -110648. i q^{92} -67847.9i q^{94} +89184.5i q^{95} +2061.14i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 96 q^{4}+O(q^{10}) \) 4 * q + 96 * q^4	\( 4 q + 96 q^{4} + 1280 q^{16} + 6544 q^{22} - 1916 q^{25} + 812 q^{37} + 18788 q^{43} - 52160 q^{46} - 52064 q^{58} - 26624 q^{64} + 120788 q^{67} - 157540 q^{79} + 77616 q^{85} + 366464 q^{88}+O(q^{100}) \) 4 * q + 96 * q^4 + 1280 * q^16 + 6544 * q^22 - 1916 * q^25 + 812 * q^37 + 18788 * q^43 - 52160 * q^46 - 52064 * q^58 - 26624 * q^64 + 120788 * q^67 - 157540 * q^79 + 77616 * q^85 + 366464 * q^88

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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