LMFDB - Embedded newform 441.7.d.c.244.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,7,Mod(244,441)]

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([0, 1]))

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("441.244");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 441 = 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	441.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:	$101.453850876$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} + 212x^{6} + 473x^{5} + 39800x^{4} + 36821x^{3} + 985651x^{2} - 601290x + 21068100 $ x^8 - x^7 + 212x^6 + 473x^5 + 39800x^4 + 36821x^3 + 985651x^2 - 601290x + 21068100
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6}\cdot 3\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 21)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			244.3
Root			$2.26350 - 3.92050i$ of defining polynomial
Character	$\chi$	$=$	441.244
Dual form			441.7.d.c.244.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.52700 q^{2} -33.4522 q^{4} -66.9661i q^{5} +538.619 q^{8} +O(q^{10})$	$q-5.52700 q^{2} -33.4522 q^{4} -66.9661i q^{5} +538.619 q^{8} +370.122i q^{10} -1725.74 q^{11} +2807.43i q^{13} -836.004 q^{16} +6147.14i q^{17} -8934.27i q^{19} +2240.17i q^{20} +9538.15 q^{22} -9901.28 q^{23} +11140.5 q^{25} -15516.6i q^{26} +13610.4 q^{29} -25210.5i q^{31} -29851.0 q^{32} -33975.2i q^{34} +22732.7 q^{37} +49379.7i q^{38} -36069.2i q^{40} +37897.2i q^{41} +73646.2 q^{43} +57729.7 q^{44} +54724.4 q^{46} +139168. i q^{47} -61573.8 q^{50} -93914.7i q^{52} +16517.6 q^{53} +115566. i q^{55} -75224.5 q^{58} +71579.5i q^{59} +263915. i q^{61} +139339. i q^{62} +218491. q^{64} +188002. q^{65} +548667. q^{67} -205635. i q^{68} -465655. q^{71} -333732. i q^{73} -125644. q^{74} +298871. i q^{76} -410024. q^{79} +55983.9i q^{80} -209458. i q^{82} -71948.9i q^{83} +411649. q^{85} -407043. q^{86} -929514. q^{88} -163396. i q^{89} +331220. q^{92} -769180. i q^{94} -598293. q^{95} +651914. i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 10 q^{2} + 346 q^{4} - 3326 q^{8}+O(q^{10}) $ 8 * q - 10 * q^2 + 346 * q^4 - 3326 * q^8	$ 8 q - 10 q^{2} + 346 q^{4} - 3326 q^{8} - 628 q^{11} + 25442 q^{16} + 86106 q^{22} + 7856 q^{23} + 34076 q^{25} + 8300 q^{29} - 372414 q^{32} - 129412 q^{37} + 45740 q^{43} + 185058 q^{44} + 223008 q^{46} - 967216 q^{50} - 1081948 q^{53} - 1079598 q^{58} + 2378626 q^{64} - 828408 q^{65} + 2317804 q^{67} - 1442344 q^{71} - 865880 q^{74} - 1222904 q^{79} - 275112 q^{85} + 1632448 q^{86} + 732882 q^{88} - 678720 q^{92} - 1183584 q^{95}+O(q^{100}) $ 8 * q - 10 * q^2 + 346 * q^4 - 3326 * q^8 - 628 * q^11 + 25442 * q^16 + 86106 * q^22 + 7856 * q^23 + 34076 * q^25 + 8300 * q^29 - 372414 * q^32 - 129412 * q^37 + 45740 * q^43 + 185058 * q^44 + 223008 * q^46 - 967216 * q^50 - 1081948 * q^53 - 1079598 * q^58 + 2378626 * q^64 - 828408 * q^65 + 2317804 * q^67 - 1442344 * q^71 - 865880 * q^74 - 1222904 * q^79 - 275112 * q^85 + 1632448 * q^86 + 732882 * q^88 - 678720 * q^92 - 1183584 * q^95

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 $

$2$	−5.52700	−0.690875	−0.345438	−	0.938442i	$-0.612270\pi$
$2$	−5.52700	−0.690875	−0.345438	+	0.938442i	$0.612270\pi$
$3$	0	0
$3$	0	0
$4$	−33.4522	−0.522691
$5$	− 66.9661i	− 0.535729i	−0.963457	−	0.267864i	$-0.913682\pi$
$5$	− 66.9661i	− 0.535729i	0.963457	−	0.267864i	$-0.0863179\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	538.619	1.05199
$9$	0	0
$10$	370.122i	0.370122i
$11$	−1725.74	−1.29657	−0.648285	−	0.761397i	$-0.724514\pi$
$11$	−1725.74	−1.29657	−0.648285	+	0.761397i	$0.724514\pi$
$12$	0	0
$13$	2807.43i	1.27785i	0.769271	+	0.638923i	$0.220620\pi$
$13$	2807.43i	1.27785i	−0.769271	+	0.638923i	$0.779380\pi$
$14$	0	0
$15$	0	0
$16$	−836.004	−0.204102
$17$	6147.14i	1.25120i	0.780145	+	0.625599i	$0.215145\pi$
$17$	6147.14i	1.25120i	−0.780145	+	0.625599i	$0.784855\pi$
$18$	0	0
$19$	− 8934.27i	− 1.30256i	−0.758837	−	0.651281i	$-0.774232\pi$
$19$	− 8934.27i	− 1.30256i	0.758837	−	0.651281i	$-0.225768\pi$
$20$	2240.17i	0.280021i
$21$	0	0
$22$	9538.15	0.895769
$23$	−9901.28	−0.813781	−0.406891	−	0.913477i	$-0.633387\pi$
$23$	−9901.28	−0.813781	−0.406891	+	0.913477i	$0.633387\pi$
$24$	0	0
$25$	11140.5	0.712995
$26$	− 15516.6i	− 0.882832i
$27$	0	0
$28$	0	0
$29$	13610.4	0.558054	0.279027	−	0.960283i	$-0.409988\pi$
$29$	13610.4	0.558054	0.279027	+	0.960283i	$0.409988\pi$
$30$	0	0
$31$	− 25210.5i	− 0.846246i	−0.906072	−	0.423123i	$-0.860934\pi$
$31$	− 25210.5i	− 0.846246i	0.906072	−	0.423123i	$-0.139066\pi$
$32$	−29851.0	−0.910980
$33$	0	0
$34$	− 33975.2i	− 0.864422i
$35$	0	0
$36$	0	0
$37$	22732.7	0.448793	0.224397	−	0.974498i	$-0.427959\pi$
$37$	22732.7	0.448793	0.224397	+	0.974498i	$0.427959\pi$
$38$	49379.7i	0.899907i
$39$	0	0
$40$	− 36069.2i	− 0.563581i
$41$	37897.2i	0.549865i	0.961464	+	0.274932i	$0.0886554\pi$
$41$	37897.2i	0.549865i	−0.961464	+	0.274932i	$0.911345\pi$
$42$	0	0
$43$	73646.2	0.926286	0.463143	−	0.886284i	$-0.346722\pi$
$43$	73646.2	0.926286	0.463143	+	0.886284i	$0.346722\pi$
$44$	57729.7	0.677706
$45$	0	0
$46$	54724.4	0.562221
$47$	139168.i	1.34043i	0.742166	+	0.670216i	$0.233799\pi$
$47$	139168.i	1.34043i	−0.742166	+	0.670216i	$0.766201\pi$
$48$	0	0
$49$	0	0
$50$	−61573.8	−0.492591
$51$	0	0
$52$	− 93914.7i	− 0.667919i
$53$	16517.6	0.110948	0.0554739	−	0.998460i	$-0.482333\pi$
$53$	16517.6	0.110948	0.0554739	+	0.998460i	$0.482333\pi$
$54$	0	0
$55$	115566.i	0.694610i
$56$	0	0
$57$	0	0
$58$	−75224.5	−0.385545
$59$	71579.5i	0.348524i	0.984699	+	0.174262i	$0.0557540\pi$
$59$	71579.5i	0.348524i	−0.984699	+	0.174262i	$0.944246\pi$
$60$	0	0
$61$	263915.i	1.16272i	0.813646	+	0.581360i	$0.197479\pi$
$61$	263915.i	1.16272i	−0.813646	+	0.581360i	$0.802521\pi$
$62$	139339.i	0.584651i
$63$	0	0
$64$	218491.	0.833476
$65$	188002.	0.684578
$66$	0	0
$67$	548667.	1.82425	0.912125	−	0.409912i	$-0.134441\pi$
$67$	548667.	1.82425	0.912125	+	0.409912i	$0.134441\pi$
$68$	− 205635.i	− 0.653990i
$69$	0	0
$70$	0	0
$71$	−465655.	−1.30103	−0.650517	−	0.759491i	$-0.725448\pi$
$71$	−465655.	−1.30103	−0.650517	+	0.759491i	$0.725448\pi$
$72$	0	0
$73$	− 333732.i	− 0.857885i	−0.903332	−	0.428942i	$-0.858886\pi$
$73$	− 333732.i	− 0.857885i	0.903332	−	0.428942i	$-0.141114\pi$
$74$	−125644.	−0.310060
$75$	0	0
$76$	298871.i	0.680837i
$77$	0	0
$78$	0	0
$79$	−410024.	−0.831626	−0.415813	−	0.909450i	$-0.636503\pi$
$79$	−410024.	−0.831626	−0.415813	+	0.909450i	$0.636503\pi$
$80$	55983.9i	0.109343i
$81$	0	0
$82$	− 209458.i	− 0.379888i
$83$	− 71948.9i	− 0.125832i	−0.998019	−	0.0629158i	$-0.979960\pi$
$83$	− 71948.9i	− 0.125832i	0.998019	−	0.0629158i	$-0.0200400\pi$
$84$	0	0
$85$	411649.	0.670302
$86$	−407043.	−0.639948
$87$	0	0
$88$	−929514.	−1.36398
$89$	− 163396.i	− 0.231777i	−0.993262	−	0.115889i	$-0.963028\pi$
$89$	− 163396.i	− 0.231777i	0.993262	−	0.115889i	$-0.0369716\pi$
$90$	0	0
$91$	0	0
$92$	331220.	0.425357
$93$	0	0
$94$	− 769180.i	− 0.926072i
$95$	−598293.	−0.697819
$96$	0	0
$97$	651914.i	0.714291i	0.934049	+	0.357146i	$0.116250\pi$
$97$	651914.i	0.714291i	−0.934049	+	0.357146i	$0.883750\pi$
$98$	0	0
$99$	0	0
$100$	−372676.	−0.372676
$101$	− 1.02058e6i	− 0.990567i	−0.868731	−	0.495283i	$-0.835064\pi$
$101$	− 1.02058e6i	− 0.990567i	0.868731	−	0.495283i	$-0.164936\pi$
$102$	0	0
$103$	− 419326.i	− 0.383742i	−0.981420	−	0.191871i	$-0.938544\pi$
$103$	− 419326.i	− 0.383742i	0.981420	−	0.191871i	$-0.0614556\pi$
$104$	1.51213e6i	1.34428i
$105$	0	0
$106$	−91292.8	−0.0766512
$107$	−2.21241e6	−1.80599	−0.902993	−	0.429655i	$-0.858635\pi$
$107$	−2.21241e6	−1.80599	−0.902993	+	0.429655i	$0.858635\pi$
$108$	0	0
$109$	765910.	0.591423	0.295711	−	0.955277i	$-0.404443\pi$
$109$	765910.	0.591423	0.295711	+	0.955277i	$0.404443\pi$
$110$	− 638732.i	− 0.479889i
$111$	0	0
$112$	0	0
$113$	2.66096e6	1.84418	0.922090	−	0.386976i	$-0.126480\pi$
$113$	2.66096e6	1.84418	0.922090	+	0.386976i	$0.126480\pi$
$114$	0	0
$115$	663050.i	0.435966i
$116$	−455297.	−0.291690
$117$	0	0
$118$	− 395620.i	− 0.240787i
$119$	0	0
$120$	0	0
$121$	1.20660e6	0.681096
$122$	− 1.45866e6i	− 0.803295i
$123$	0	0
$124$	843349.i	0.442326i
$125$	− 1.79238e6i	− 0.917700i
$126$	0	0
$127$	−1.11648e6	−0.545055	−0.272527	−	0.962148i	$-0.587859\pi$
$127$	−1.11648e6	−0.545055	−0.272527	+	0.962148i	$0.587859\pi$
$128$	702865.	0.335152
$129$	0	0
$130$	−1.03909e6	−0.472958
$131$	286165.i	0.127293i	0.997973	+	0.0636463i	$0.0202730\pi$
$131$	286165.i	0.127293i	−0.997973	+	0.0636463i	$0.979727\pi$
$132$	0	0
$133$	0	0
$134$	−3.03248e6	−1.26033
$135$	0	0
$136$	3.31096e6i	1.31625i
$137$	3.73393e6	1.45213	0.726063	−	0.687628i	$-0.241348\pi$
$137$	3.73393e6	1.45213	0.726063	+	0.687628i	$0.241348\pi$
$138$	0	0
$139$	618216.i	0.230195i	0.993354	+	0.115097i	$0.0367180\pi$
$139$	618216.i	0.230195i	−0.993354	+	0.115097i	$0.963282\pi$
$140$	0	0
$141$	0	0
$142$	2.57367e6	0.898853
$143$	− 4.84488e6i	− 1.65682i
$144$	0	0
$145$	− 911433.i	− 0.298965i
$146$	1.84454e6i	0.592691i
$147$	0	0
$148$	−760461.	−0.234580
$149$	−3.72319e6	−1.12553	−0.562765	−	0.826617i	$-0.690262\pi$
$149$	−3.72319e6	−1.12553	−0.562765	+	0.826617i	$0.690262\pi$
$150$	0	0
$151$	−2.55199e6	−0.741221	−0.370611	−	0.928788i	$-0.620852\pi$
$151$	−2.55199e6	−0.741221	−0.370611	+	0.928788i	$0.620852\pi$
$152$	− 4.81216e6i	− 1.37028i
$153$	0	0
$154$	0	0
$155$	−1.68825e6	−0.453358
$156$	0	0
$157$	− 6.77876e6i	− 1.75166i	−0.482615	−	0.875832i	$-0.660313\pi$
$157$	− 6.77876e6i	− 1.75166i	0.482615	−	0.875832i	$-0.339687\pi$
$158$	2.26620e6	0.574550
$159$	0	0
$160$	1.99900e6i	0.488038i
$161$	0	0
$162$	0	0
$163$	−8.27141e6	−1.90993	−0.954964	−	0.296722i	$-0.904106\pi$
$163$	−8.27141e6	−1.90993	−0.954964	+	0.296722i	$0.904106\pi$
$164$	− 1.26775e6i	− 0.287409i
$165$	0	0
$166$	397662.i	0.0869340i
$167$	2.59960e6i	0.558158i	0.960268	+	0.279079i	$0.0900291\pi$
$167$	2.59960e6i	0.558158i	−0.960268	+	0.279079i	$0.909971\pi$
$168$	0	0
$169$	−3.05483e6	−0.632888
$170$	−2.27519e6	−0.463095
$171$	0	0
$172$	−2.46363e6	−0.484162
$173$	− 8.20544e6i	− 1.58476i	−0.610027	−	0.792380i	$-0.708842\pi$
$173$	− 8.20544e6i	− 1.58476i	0.610027	−	0.792380i	$-0.291158\pi$
$174$	0	0
$175$	0	0
$176$	1.44272e6	0.264633
$177$	0	0
$178$	903090.i	0.160129i
$179$	−8.18803e6	−1.42764	−0.713822	−	0.700327i	$-0.753038\pi$
$179$	−8.18803e6	−1.42764	−0.713822	+	0.700327i	$0.753038\pi$
$180$	0	0
$181$	− 3.25606e6i	− 0.549106i	−0.961572	−	0.274553i	$-0.911470\pi$
$181$	− 3.25606e6i	− 0.549106i	0.961572	−	0.274553i	$-0.0885298\pi$
$182$	0	0
$183$	0	0
$184$	−5.33301e6	−0.856090
$185$	− 1.52232e6i	− 0.240431i
$186$	0	0
$187$	− 1.06083e7i	− 1.62227i
$188$	− 4.65547e6i	− 0.700633i
$189$	0	0
$190$	3.30677e6	0.482106
$191$	−8.60850e6	−1.23546	−0.617728	−	0.786392i	$-0.711947\pi$
$191$	−8.60850e6	−1.23546	−0.617728	+	0.786392i	$0.711947\pi$
$192$	0	0
$193$	−179886.	−0.0250222	−0.0125111	−	0.999922i	$-0.503983\pi$
$193$	−179886.	−0.0250222	−0.0125111	+	0.999922i	$0.503983\pi$
$194$	− 3.60313e6i	− 0.493486i
$195$	0	0
$196$	0	0
$197$	−2.69333e6	−0.352283	−0.176141	−	0.984365i	$-0.556362\pi$
$197$	−2.69333e6	−0.352283	−0.176141	+	0.984365i	$0.556362\pi$
$198$	0	0
$199$	6.04221e6i	0.766720i	0.923599	+	0.383360i	$0.125233\pi$
$199$	6.04221e6i	0.766720i	−0.923599	+	0.383360i	$0.874767\pi$
$200$	6.00051e6	0.750063
$201$	0	0
$202$	5.64076e6i	0.684358i
$203$	0	0
$204$	0	0
$205$	2.53783e6	0.294578
$206$	2.31761e6i	0.265118i
$207$	0	0
$208$	− 2.34702e6i	− 0.260811i
$209$	1.54182e7i	1.68886i
$210$	0	0
$211$	−1.38113e7	−1.47024	−0.735118	−	0.677939i	$-0.762873\pi$
$211$	−1.38113e7	−1.47024	−0.735118	+	0.677939i	$0.762873\pi$
$212$	−552550.	−0.0579915
$213$	0	0
$214$	1.22280e7	1.24771
$215$	− 4.93180e6i	− 0.496238i
$216$	0	0
$217$	0	0
$218$	−4.23318e6	−0.408599
$219$	0	0
$220$	− 3.86593e6i	− 0.363067i
$221$	−1.72576e7	−1.59884
$222$	0	0
$223$	− 9.81400e6i	− 0.884976i	−0.896774	−	0.442488i	$-0.854096\pi$
$223$	− 9.81400e6i	− 0.884976i	0.896774	−	0.442488i	$-0.145904\pi$
$224$	0	0
$225$	0	0
$226$	−1.47071e7	−1.27410
$227$	1.45743e7i	1.24598i	0.782231	+	0.622989i	$0.214082\pi$
$227$	1.45743e7i	1.24598i	−0.782231	+	0.622989i	$0.785918\pi$
$228$	0	0
$229$	− 1.46985e7i	− 1.22396i	−0.790875	−	0.611978i	$-0.790374\pi$
$229$	− 1.46985e7i	− 1.22396i	0.790875	−	0.611978i	$-0.209626\pi$
$230$	− 3.66468e6i	− 0.301198i
$231$	0	0
$232$	7.33080e6	0.587067
$233$	−5.75195e6	−0.454723	−0.227362	−	0.973810i	$-0.573010\pi$
$233$	−5.75195e6	−0.454723	−0.227362	+	0.973810i	$0.573010\pi$
$234$	0	0
$235$	9.31952e6	0.718108
$236$	− 2.39450e6i	− 0.182171i
$237$	0	0
$238$	0	0
$239$	−1.60167e7	−1.17322	−0.586610	−	0.809869i	$-0.699538\pi$
$239$	−1.60167e7	−1.17322	−0.586610	+	0.809869i	$0.699538\pi$
$240$	0	0
$241$	− 2.58206e7i	− 1.84465i	−0.386410	−	0.922327i	$-0.626285\pi$
$241$	− 2.58206e7i	− 1.84465i	0.386410	−	0.922327i	$-0.373715\pi$
$242$	−6.66890e6	−0.470553
$243$	0	0
$244$	− 8.82856e6i	− 0.607744i
$245$	0	0
$246$	0	0
$247$	2.50823e7	1.66447
$248$	− 1.35789e7i	− 0.890243i
$249$	0	0
$250$	9.90651e6i	0.634016i
$251$	− 9.51912e6i	− 0.601971i	−0.953629	−	0.300985i	$-0.902684\pi$
$251$	− 9.51912e6i	− 0.601971i	0.953629	−	0.300985i	$-0.0973156\pi$
$252$	0	0
$253$	1.70870e7	1.05513
$254$	6.17079e6	0.376565
$255$	0	0
$256$	−1.78682e7	−1.06502
$257$	2.23864e7i	1.31882i	0.751785	+	0.659408i	$0.229193\pi$
$257$	2.23864e7i	1.31882i	−0.751785	+	0.659408i	$0.770807\pi$
$258$	0	0
$259$	0	0
$260$	−6.28910e6	−0.357823
$261$	0	0
$262$	− 1.58164e6i	− 0.0879433i
$263$	−1.04932e7	−0.576819	−0.288410	−	0.957507i	$-0.593126\pi$
$263$	−1.04932e7	−0.576819	−0.288410	+	0.957507i	$0.593126\pi$
$264$	0	0
$265$	− 1.10612e6i	− 0.0594380i
$266$	0	0
$267$	0	0
$268$	−1.83541e7	−0.953520
$269$	− 2.69811e7i	− 1.38613i	−0.720877	−	0.693063i	$-0.756261\pi$
$269$	− 2.69811e7i	− 1.38613i	0.720877	−	0.693063i	$-0.243739\pi$
$270$	0	0
$271$	1.26965e6i	0.0637935i	0.999491	+	0.0318968i	$0.0101548\pi$
$271$	1.26965e6i	0.0637935i	−0.999491	+	0.0318968i	$0.989845\pi$
$272$	− 5.13903e6i	− 0.255373i
$273$	0	0
$274$	−2.06374e7	−1.00324
$275$	−1.92256e7	−0.924449
$276$	0	0
$277$	−6.70033e6	−0.315251	−0.157626	−	0.987499i	$-0.550384\pi$
$277$	−6.70033e6	−0.315251	−0.157626	+	0.987499i	$0.550384\pi$
$278$	− 3.41688e6i	− 0.159036i
$279$	0	0
$280$	0	0
$281$	321000.	0.0144673	0.00723363	−	0.999974i	$-0.497697\pi$
$281$	321000.	0.0144673	0.00723363	+	0.999974i	$0.497697\pi$
$282$	0	0
$283$	3.20211e6i	0.141279i	0.997502	+	0.0706394i	$0.0225040\pi$
$283$	3.20211e6i	0.141279i	−0.997502	+	0.0706394i	$0.977496\pi$
$284$	1.55772e7	0.680039
$285$	0	0
$286$	2.67776e7i	1.14465i
$287$	0	0
$288$	0	0
$289$	−1.36497e7	−0.565496
$290$	5.03749e6i	0.206548i
$291$	0	0
$292$	1.11641e7i	0.448409i
$293$	− 2.67440e6i	− 0.106322i	−0.998586	−	0.0531610i	$-0.983070\pi$
$293$	− 2.67440e6i	− 0.106322i	0.998586	−	0.0531610i	$-0.0169297\pi$
$294$	0	0
$295$	4.79340e6	0.186714
$296$	1.22443e7	0.472126
$297$	0	0
$298$	2.05781e7	0.777601
$299$	− 2.77971e7i	− 1.03989i
$300$	0	0
$301$	0	0
$302$	1.41048e7	0.512092
$303$	0	0
$304$	7.46908e6i	0.265856i
$305$	1.76734e7	0.622903
$306$	0	0
$307$	− 1.98765e7i	− 0.686950i	−0.939162	−	0.343475i	$-0.888396\pi$
$307$	− 1.98765e7i	− 0.686950i	0.939162	−	0.343475i	$-0.111604\pi$
$308$	0	0
$309$	0	0
$310$	9.33096e6	0.313214
$311$	1.57710e7i	0.524296i	0.965028	+	0.262148i	$0.0844309\pi$
$311$	1.57710e7i	0.524296i	−0.965028	+	0.262148i	$0.915569\pi$
$312$	0	0
$313$	1.05533e7i	0.344154i	0.985083	+	0.172077i	$0.0550479\pi$
$313$	1.05533e7i	0.344154i	−0.985083	+	0.172077i	$0.944952\pi$
$314$	3.74662e7i	1.21018i
$315$	0	0
$316$	1.37162e7	0.434684
$317$	3.23227e7	1.01468	0.507341	−	0.861745i	$-0.330629\pi$
$317$	3.23227e7	1.01468	0.507341	+	0.861745i	$0.330629\pi$
$318$	0	0
$319$	−2.34879e7	−0.723556
$320$	− 1.46315e7i	− 0.446517i
$321$	0	0
$322$	0	0
$323$	5.49201e7	1.62976
$324$	0	0
$325$	3.12763e7i	0.911097i
$326$	4.57161e7	1.31952
$327$	0	0
$328$	2.04122e7i	0.578452i
$329$	0	0
$330$	0	0
$331$	2.00329e7	0.552409	0.276205	−	0.961099i	$-0.410923\pi$
$331$	2.00329e7	0.552409	0.276205	+	0.961099i	$0.410923\pi$
$332$	2.40685e6i	0.0657711i
$333$	0	0
$334$	− 1.43680e7i	− 0.385618i
$335$	− 3.67421e7i	− 0.977303i
$336$	0	0
$337$	−3.66702e6	−0.0958128	−0.0479064	−	0.998852i	$-0.515255\pi$
$337$	−3.66702e6	−0.0958128	−0.0479064	+	0.998852i	$0.515255\pi$
$338$	1.68841e7	0.437247
$339$	0	0
$340$	−1.37706e7	−0.350361
$341$	4.35067e7i	1.09722i
$342$	0	0
$343$	0	0
$344$	3.96672e7	0.974443
$345$	0	0
$346$	4.53515e7i	1.09487i
$347$	−6.45603e6	−0.154517	−0.0772587	−	0.997011i	$-0.524617\pi$
$347$	−6.45603e6	−0.154517	−0.0772587	+	0.997011i	$0.524617\pi$
$348$	0	0
$349$	4.01536e7i	0.944600i	0.881438	+	0.472300i	$0.156576\pi$
$349$	4.01536e7i	0.944600i	−0.881438	+	0.472300i	$0.843424\pi$
$350$	0	0
$351$	0	0
$352$	5.15150e7	1.18115
$353$	− 8.11656e6i	− 0.184522i	−0.995735	−	0.0922610i	$-0.970591\pi$
$353$	− 8.11656e6i	− 0.184522i	0.995735	−	0.0922610i	$-0.0294094\pi$
$354$	0	0
$355$	3.11831e7i	0.697001i
$356$	5.46596e6i	0.121148i
$357$	0	0
$358$	4.52552e7	0.986324
$359$	−3.69171e7	−0.797893	−0.398946	−	0.916974i	$-0.630624\pi$
$359$	−3.69171e7	−0.797893	−0.398946	+	0.916974i	$0.630624\pi$
$360$	0	0
$361$	−3.27752e7	−0.696665
$362$	1.79962e7i	0.379364i
$363$	0	0
$364$	0	0
$365$	−2.23487e7	−0.459593
$366$	0	0
$367$	1.97155e7i	0.398851i	0.979913	+	0.199425i	$0.0639076\pi$
$367$	1.97155e7i	0.398851i	−0.979913	+	0.199425i	$0.936092\pi$
$368$	8.27750e6	0.166095
$369$	0	0
$370$	8.41387e6i	0.166108i
$371$	0	0
$372$	0	0
$373$	1.04331e7	0.201041	0.100521	−	0.994935i	$-0.467949\pi$
$373$	1.04331e7	0.201041	0.100521	+	0.994935i	$0.467949\pi$
$374$	5.86323e7i	1.12078i
$375$	0	0
$376$	7.49584e7i	1.41012i
$377$	3.82101e7i	0.713106i
$378$	0	0
$379$	−5.26018e7	−0.966235	−0.483118	−	0.875556i	$-0.660496\pi$
$379$	−5.26018e7	−0.966235	−0.483118	+	0.875556i	$0.660496\pi$
$380$	2.00142e7	0.364744
$381$	0	0
$382$	4.75792e7	0.853547
$383$	7.35366e7i	1.30890i	0.756104	+	0.654451i	$0.227100\pi$
$383$	7.35366e7i	1.30890i	−0.756104	+	0.654451i	$0.772900\pi$
$384$	0	0
$385$	0	0
$386$	994230.	0.0172872
$387$	0	0
$388$	− 2.18080e7i	− 0.373354i
$389$	−1.72139e7	−0.292436	−0.146218	−	0.989252i	$-0.546710\pi$
$389$	−1.72139e7	−0.292436	−0.146218	+	0.989252i	$0.546710\pi$
$390$	0	0
$391$	− 6.08645e7i	− 1.01820i
$392$	0	0
$393$	0	0
$394$	1.48861e7	0.243384
$395$	2.74577e7i	0.445526i
$396$	0	0
$397$	6.13174e7i	0.979969i	0.871731	+	0.489984i	$0.162998\pi$
$397$	6.13174e7i	0.979969i	−0.871731	+	0.489984i	$0.837002\pi$
$398$	− 3.33953e7i	− 0.529708i
$399$	0	0
$400$	−9.31354e6	−0.145524
$401$	4.61470e7	0.715667	0.357833	−	0.933785i	$-0.383516\pi$
$401$	4.61470e7	0.715667	0.357833	+	0.933785i	$0.383516\pi$
$402$	0	0
$403$	7.07767e7	1.08137
$404$	3.41408e7i	0.517761i
$405$	0	0
$406$	0	0
$407$	−3.92307e7	−0.581892
$408$	0	0
$409$	− 6.06926e7i	− 0.887086i	−0.896253	−	0.443543i	$-0.853721\pi$
$409$	− 6.06926e7i	− 0.887086i	0.896253	−	0.443543i	$-0.146279\pi$
$410$	−1.40266e7	−0.203517
$411$	0	0
$412$	1.40274e7i	0.200579i
$413$	0	0
$414$	0	0
$415$	−4.81814e6	−0.0674116
$416$	− 8.38045e7i	− 1.16409i
$417$	0	0
$418$	− 8.52163e7i	− 1.16679i
$419$	− 1.17371e8i	− 1.59558i	−0.602939	−	0.797788i	$-0.706004\pi$
$419$	− 1.17371e8i	− 1.59558i	0.602939	−	0.797788i	$-0.293996\pi$
$420$	0	0
$421$	−1.28510e7	−0.172222	−0.0861111	−	0.996286i	$-0.527444\pi$
$421$	−1.28510e7	−0.172222	−0.0861111	+	0.996286i	$0.527444\pi$
$422$	7.63351e7	1.01575
$423$	0	0
$424$	8.89668e6	0.116716
$425$	6.84824e7i	0.892098i
$426$	0	0
$427$	0	0
$428$	7.40101e7	0.943973
$429$	0	0
$430$	2.72581e7i	0.342838i
$431$	1.13856e8	1.42209	0.711043	−	0.703149i	$-0.248223\pi$
$431$	1.13856e8	1.42209	0.711043	+	0.703149i	$0.248223\pi$
$432$	0	0
$433$	2.76112e7i	0.340111i	0.985434	+	0.170056i	$0.0543947\pi$
$433$	2.76112e7i	0.340111i	−0.985434	+	0.170056i	$0.945605\pi$
$434$	0	0
$435$	0	0
$436$	−2.56214e7	−0.309132
$437$	8.84607e7i	1.06000i
$438$	0	0
$439$	− 7.18746e7i	− 0.849536i	−0.905302	−	0.424768i	$-0.860356\pi$
$439$	− 7.18746e7i	− 0.849536i	0.905302	−	0.424768i	$-0.139644\pi$
$440$	6.22459e7i	0.730723i
$441$	0	0
$442$	9.53829e7	1.10460
$443$	9.50290e7	1.09306	0.546531	−	0.837439i	$-0.315948\pi$
$443$	9.50290e7	1.09306	0.546531	+	0.837439i	$0.315948\pi$
$444$	0	0
$445$	−1.09420e7	−0.124170
$446$	5.42420e7i	0.611408i
$447$	0	0
$448$	0	0
$449$	−1.15401e8	−1.27488	−0.637440	−	0.770500i	$-0.720007\pi$
$449$	−1.15401e8	−1.27488	−0.637440	+	0.770500i	$0.720007\pi$
$450$	0	0
$451$	− 6.54006e7i	− 0.712938i
$452$	−8.90151e7	−0.963937
$453$	0	0
$454$	− 8.05522e7i	− 0.860815i
$455$	0	0
$456$	0	0
$457$	9.17526e6	0.0961324	0.0480662	−	0.998844i	$-0.484694\pi$
$457$	9.17526e6	0.0961324	0.0480662	+	0.998844i	$0.484694\pi$
$458$	8.12384e7i	0.845600i
$459$	0	0
$460$	− 2.21805e7i	− 0.227876i
$461$	− 5.79354e7i	− 0.591346i	−0.955289	−	0.295673i	$-0.904456\pi$
$461$	− 5.79354e7i	− 0.591346i	0.955289	−	0.295673i	$-0.0955438\pi$
$462$	0	0
$463$	6.51781e7	0.656687	0.328344	−	0.944558i	$-0.393510\pi$
$463$	6.51781e7	0.656687	0.328344	+	0.944558i	$0.393510\pi$
$464$	−1.13783e7	−0.113900
$465$	0	0
$466$	3.17910e7	0.314157
$467$	− 1.69452e8i	− 1.66378i	−0.554943	−	0.831888i	$-0.687260\pi$
$467$	− 1.69452e8i	− 1.66378i	0.554943	−	0.831888i	$-0.312740\pi$
$468$	0	0
$469$	0	0
$470$	−5.15090e7	−0.496123
$471$	0	0
$472$	3.85541e7i	0.366644i
$473$	−1.27094e8	−1.20100
$474$	0	0
$475$	− 9.95326e7i	− 0.928719i
$476$	0	0
$477$	0	0
$478$	8.85244e7	0.810549
$479$	9.15319e7i	0.832848i	0.909171	+	0.416424i	$0.136717\pi$
$479$	9.15319e7i	0.832848i	−0.909171	+	0.416424i	$0.863283\pi$
$480$	0	0
$481$	6.38205e7i	0.573488i
$482$	1.42710e8i	1.27443i
$483$	0	0
$484$	−4.03636e7	−0.356003
$485$	4.36561e7	0.382666
$486$	0	0
$487$	−1.31240e8	−1.13626	−0.568131	−	0.822938i	$-0.692333\pi$
$487$	−1.31240e8	−1.13626	−0.568131	+	0.822938i	$0.692333\pi$
$488$	1.42150e8i	1.22317i
$489$	0	0
$490$	0	0
$491$	8.81776e7	0.744927	0.372463	−	0.928047i	$-0.378513\pi$
$491$	8.81776e7	0.744927	0.372463	+	0.928047i	$0.378513\pi$
$492$	0	0
$493$	8.36648e7i	0.698235i
$494$	−1.38630e8	−1.14994
$495$	0	0
$496$	2.10761e7i	0.172721i
$497$	0	0
$498$	0	0
$499$	−2.28471e8	−1.83878	−0.919389	−	0.393350i	$-0.871316\pi$
$499$	−2.28471e8	−1.83878	−0.919389	+	0.393350i	$0.871316\pi$
$500$	5.99592e7i	0.479674i
$501$	0	0
$502$	5.26122e7i	0.415887i
$503$	− 2.07735e8i	− 1.63232i	−0.577824	−	0.816161i	$-0.696098\pi$
$503$	− 2.07735e8i	− 1.63232i	0.577824	−	0.816161i	$-0.303902\pi$
$504$	0	0
$505$	−6.83444e7	−0.530675
$506$	−9.44398e7	−0.728960
$507$	0	0
$508$	3.73488e7	0.284895
$509$	− 5.48436e7i	− 0.415885i	−0.978141	−	0.207942i	$-0.933323\pi$
$509$	− 5.48436e7i	− 0.415885i	0.978141	−	0.207942i	$-0.0666767\pi$
$510$	0	0
$511$	0	0
$512$	5.37739e7	0.400647
$513$	0	0
$514$	− 1.23729e8i	− 0.911137i
$515$	−2.80806e7	−0.205582
$516$	0	0
$517$	− 2.40167e8i	− 1.73797i
$518$	0	0
$519$	0	0
$520$	1.01262e8	0.720169
$521$	2.85252e7i	0.201704i	0.994901	+	0.100852i	$0.0321569\pi$
$521$	2.85252e7i	0.201704i	−0.994901	+	0.100852i	$0.967843\pi$
$522$	0	0
$523$	1.07664e8i	0.752599i	0.926498	+	0.376300i	$0.122804\pi$
$523$	1.07664e8i	0.752599i	−0.926498	+	0.376300i	$0.877196\pi$
$524$	− 9.57287e6i	− 0.0665347i
$525$	0	0
$526$	5.79958e7	0.398510
$527$	1.54973e8	1.05882
$528$	0	0
$529$	−5.00006e7	−0.337760
$530$	6.11352e6i	0.0410642i
$531$	0	0
$532$	0	0
$533$	−1.06394e8	−0.702642
$534$	0	0
$535$	1.48156e8i	0.967518i
$536$	2.95522e8	1.91909
$537$	0	0
$538$	1.49125e8i	0.957640i
$539$	0	0
$540$	0	0
$541$	5.55754e7	0.350987	0.175493	−	0.984481i	$-0.443848\pi$
$541$	5.55754e7	0.350987	0.175493	+	0.984481i	$0.443848\pi$
$542$	− 7.01736e6i	− 0.0440734i
$543$	0	0
$544$	− 1.83498e8i	− 1.13982i
$545$	− 5.12900e7i	− 0.316842i
$546$	0	0
$547$	5.44170e7	0.332485	0.166243	−	0.986085i	$-0.446837\pi$
$547$	5.44170e7	0.332485	0.166243	+	0.986085i	$0.446837\pi$
$548$	−1.24908e8	−0.759014
$549$	0	0
$550$	1.06260e8	0.638679
$551$	− 1.21599e8i	− 0.726899i
$552$	0	0
$553$	0	0
$554$	3.70327e7	0.217799
$555$	0	0
$556$	− 2.06807e7i	− 0.120321i
$557$	−1.25227e7	−0.0724657	−0.0362329	−	0.999343i	$-0.511536\pi$
$557$	−1.25227e7	−0.0724657	−0.0362329	+	0.999343i	$0.511536\pi$
$558$	0	0
$559$	2.06756e8i	1.18365i
$560$	0	0
$561$	0	0
$562$	−1.77417e6	−0.00999508
$563$	− 1.74062e8i	− 0.975391i	−0.873014	−	0.487695i	$-0.837838\pi$
$563$	− 1.74062e8i	− 0.975391i	0.873014	−	0.487695i	$-0.162162\pi$
$564$	0	0
$565$	− 1.78194e8i	− 0.987980i
$566$	− 1.76981e7i	− 0.0976060i
$567$	0	0
$568$	−2.50810e8	−1.36868
$569$	−4.75605e7	−0.258172	−0.129086	−	0.991633i	$-0.541204\pi$
$569$	−4.75605e7	−0.258172	−0.129086	+	0.991633i	$0.541204\pi$
$570$	0	0
$571$	1.65541e8	0.889198	0.444599	−	0.895730i	$-0.353346\pi$
$571$	1.65541e8	0.889198	0.444599	+	0.895730i	$0.353346\pi$
$572$	1.62072e8i	0.866004i
$573$	0	0
$574$	0	0
$575$	−1.10306e8	−0.580222
$576$	0	0
$577$	− 5.13486e7i	− 0.267301i	−0.991029	−	0.133651i	$-0.957330\pi$
$577$	− 5.13486e7i	− 0.267301i	0.991029	−	0.133651i	$-0.0426700\pi$
$578$	7.54419e7	0.390687
$579$	0	0
$580$	3.04895e7i	0.156267i
$581$	0	0
$582$	0	0
$583$	−2.85050e7	−0.143852
$584$	− 1.79754e8i	− 0.902486i
$585$	0	0
$586$	1.47814e7i	0.0734552i
$587$	2.37624e8i	1.17483i	0.809285	+	0.587416i	$0.199855\pi$
$587$	2.37624e8i	1.17483i	−0.809285	+	0.587416i	$0.800145\pi$
$588$	0	0
$589$	−2.25238e8	−1.10229
$590$	−2.64931e7	−0.128996
$591$	0	0
$592$	−1.90046e7	−0.0915998
$593$	− 3.70225e8i	− 1.77542i	−0.460401	−	0.887711i	$-0.652295\pi$
$593$	− 3.70225e8i	− 1.77542i	0.460401	−	0.887711i	$-0.347705\pi$
$594$	0	0
$595$	0	0
$596$	1.24549e8	0.588305
$597$	0	0
$598$	1.53635e8i	0.718432i
$599$	3.75877e7	0.174890	0.0874451	−	0.996169i	$-0.472130\pi$
$599$	3.75877e7	0.174890	0.0874451	+	0.996169i	$0.472130\pi$
$600$	0	0
$601$	− 2.67605e7i	− 0.123274i	−0.998099	−	0.0616369i	$-0.980368\pi$
$601$	− 2.67605e7i	− 0.123274i	0.998099	−	0.0616369i	$-0.0196321\pi$
$602$	0	0
$603$	0	0
$604$	8.53698e7	0.387430
$605$	− 8.08015e7i	− 0.364883i
$606$	0	0
$607$	− 7.72576e7i	− 0.345442i	−0.984971	−	0.172721i	$-0.944744\pi$
$607$	− 7.72576e7i	− 0.345442i	0.984971	−	0.172721i	$-0.0552559\pi$
$608$	2.66697e8i	1.18661i
$609$	0	0
$610$	−9.76808e7	−0.430348
$611$	−3.90703e8	−1.71287
$612$	0	0
$613$	−3.52971e8	−1.53235	−0.766174	−	0.642633i	$-0.777842\pi$
$613$	−3.52971e8	−1.53235	−0.766174	+	0.642633i	$0.777842\pi$
$614$	1.09858e8i	0.474597i
$615$	0	0
$616$	0	0
$617$	−1.82109e8	−0.775310	−0.387655	−	0.921805i	$-0.626715\pi$
$617$	−1.82109e8	−0.775310	−0.387655	+	0.921805i	$0.626715\pi$
$618$	0	0
$619$	9.73800e7i	0.410580i	0.978701	+	0.205290i	$0.0658137\pi$
$619$	9.73800e7i	0.410580i	−0.978701	+	0.205290i	$0.934186\pi$
$620$	5.64757e7	0.236966
$621$	0	0
$622$	− 8.71661e7i	− 0.362223i
$623$	0	0
$624$	0	0
$625$	5.40422e7	0.221357
$626$	− 5.83279e7i	− 0.237768i
$627$	0	0
$628$	2.26765e8i	0.915580i
$629$	1.39741e8i	0.561529i
$630$	0	0
$631$	−1.37285e8	−0.546430	−0.273215	−	0.961953i	$-0.588087\pi$
$631$	−1.37285e8	−0.546430	−0.273215	+	0.961953i	$0.588087\pi$
$632$	−2.20847e8	−0.874862
$633$	0	0
$634$	−1.78648e8	−0.701019
$635$	7.47663e7i	0.292001i
$636$	0	0
$637$	0	0
$638$	1.29818e8	0.499887
$639$	0	0
$640$	− 4.70681e7i	− 0.179551i
$641$	2.09345e8	0.794857	0.397428	−	0.917633i	$-0.369903\pi$
$641$	2.09345e8	0.794857	0.397428	+	0.917633i	$0.369903\pi$
$642$	0	0
$643$	3.49818e8i	1.31586i	0.753079	+	0.657930i	$0.228568\pi$
$643$	3.49818e8i	1.31586i	−0.753079	+	0.657930i	$0.771432\pi$
$644$	0	0
$645$	0	0
$646$	−3.03544e8	−1.12596
$647$	− 3.69732e8i	− 1.36513i	−0.730824	−	0.682566i	$-0.760864\pi$
$647$	− 3.69732e8i	− 1.36513i	0.730824	−	0.682566i	$-0.239136\pi$
$648$	0	0
$649$	− 1.23527e8i	− 0.451886i
$650$	− 1.72864e8i	− 0.629454i
$651$	0	0
$652$	2.76697e8	0.998303
$653$	3.95246e7	0.141947	0.0709737	−	0.997478i	$-0.477389\pi$
$653$	3.95246e7	0.141947	0.0709737	+	0.997478i	$0.477389\pi$
$654$	0	0
$655$	1.91634e7	0.0681943
$656$	− 3.16822e7i	− 0.112229i
$657$	0	0
$658$	0	0
$659$	3.29124e8	1.15002	0.575008	−	0.818148i	$-0.304999\pi$
$659$	3.29124e8	1.15002	0.575008	+	0.818148i	$0.304999\pi$
$660$	0	0
$661$	− 1.60841e8i	− 0.556919i	−0.960448	−	0.278460i	$-0.910176\pi$
$661$	− 1.60841e8i	− 0.556919i	0.960448	−	0.278460i	$-0.0898238\pi$
$662$	−1.10722e8	−0.381646
$663$	0	0
$664$	− 3.87530e7i	− 0.132374i
$665$	0	0
$666$	0	0
$667$	−1.34760e8	−0.454134
$668$	− 8.69625e7i	− 0.291744i
$669$	0	0
$670$	2.03073e8i	0.675194i
$671$	− 4.55448e8i	− 1.50755i
$672$	0	0
$673$	5.32162e8	1.74582	0.872908	−	0.487886i	$-0.162232\pi$
$673$	5.32162e8	1.74582	0.872908	+	0.487886i	$0.162232\pi$
$674$	2.02676e7	0.0661947
$675$	0	0
$676$	1.02191e8	0.330805
$677$	− 5.41413e8i	− 1.74487i	−0.488730	−	0.872435i	$-0.662540\pi$
$677$	− 5.41413e8i	− 1.74487i	0.488730	−	0.872435i	$-0.337460\pi$
$678$	0	0
$679$	0	0
$680$	2.21722e8	0.705151
$681$	0	0
$682$	− 2.40462e8i	− 0.758041i
$683$	−2.48939e8	−0.781323	−0.390662	−	0.920534i	$-0.627754\pi$
$683$	−2.48939e8	−0.781323	−0.390662	+	0.920534i	$0.627754\pi$
$684$	0	0
$685$	− 2.50047e8i	− 0.777946i
$686$	0	0
$687$	0	0
$688$	−6.15685e7	−0.189057
$689$	4.63719e7i	0.141774i
$690$	0	0
$691$	− 9.24413e7i	− 0.280177i	−0.990139	−	0.140088i	$-0.955261\pi$
$691$	− 9.24413e7i	− 0.280177i	0.990139	−	0.140088i	$-0.0447386\pi$
$692$	2.74491e8i	0.828341i
$693$	0	0
$694$	3.56825e7	0.106752
$695$	4.13995e7	0.123322
$696$	0	0
$697$	−2.32959e8	−0.687989
$698$	− 2.21929e8i	− 0.652601i
$699$	0	0
$700$	0	0
$701$	−5.38435e8	−1.56307	−0.781537	−	0.623859i	$-0.785564\pi$
$701$	−5.38435e8	−1.56307	−0.781537	+	0.623859i	$0.785564\pi$
$702$	0	0
$703$	− 2.03100e8i	− 0.584581i
$704$	−3.77057e8	−1.08066
$705$	0	0
$706$	4.48603e7i	0.127482i
$707$	0	0
$708$	0	0
$709$	7.06910e7	0.198347	0.0991735	−	0.995070i	$-0.468380\pi$
$709$	7.06910e7	0.198347	0.0991735	+	0.995070i	$0.468380\pi$
$710$	− 1.72349e8i	− 0.481541i
$711$	0	0
$712$	− 8.80081e7i	− 0.243828i
$713$	2.49616e8i	0.688660i
$714$	0	0
$715$	−3.24442e8	−0.887604
$716$	2.73908e8	0.746217
$717$	0	0
$718$	2.04041e8	0.551245
$719$	− 3.91374e8i	− 1.05294i	−0.850193	−	0.526472i	$-0.823515\pi$
$719$	− 3.91374e8i	− 1.05294i	0.850193	−	0.526472i	$-0.176485\pi$
$720$	0	0
$721$	0	0
$722$	1.81149e8	0.481309
$723$	0	0
$724$	1.08922e8i	0.287013i
$725$	1.51627e8	0.397889
$726$	0	0
$727$	4.83550e8i	1.25846i	0.777221	+	0.629228i	$0.216629\pi$
$727$	4.83550e8i	1.25846i	−0.777221	+	0.629228i	$0.783371\pi$
$728$	0	0
$729$	0	0
$730$	1.23521e8	0.317522
$731$	4.52713e8i	1.15897i
$732$	0	0
$733$	3.19519e8i	0.811305i	0.914027	+	0.405653i	$0.132956\pi$
$733$	3.19519e8i	0.811305i	−0.914027	+	0.405653i	$0.867044\pi$
$734$	− 1.08968e8i	− 0.275556i
$735$	0	0
$736$	2.95563e8	0.741339
$737$	−9.46854e8	−2.36527
$738$	0	0
$739$	3.53513e8	0.875935	0.437968	−	0.898991i	$-0.355698\pi$
$739$	3.53513e8	0.875935	0.437968	+	0.898991i	$0.355698\pi$
$740$	5.09251e7i	0.125671i
$741$	0	0
$742$	0	0
$743$	−4.63396e8	−1.12976	−0.564880	−	0.825173i	$-0.691077\pi$
$743$	−4.63396e8	−1.12976	−0.564880	+	0.825173i	$0.691077\pi$
$744$	0	0
$745$	2.49328e8i	0.602978i
$746$	−5.76636e7	−0.138894
$747$	0	0
$748$	3.54873e8i	0.847945i
$749$	0	0
$750$	0	0
$751$	−2.97019e7	−0.0701237	−0.0350618	−	0.999385i	$-0.511163\pi$
$751$	−2.97019e7	−0.0701237	−0.0350618	+	0.999385i	$0.511163\pi$
$752$	− 1.16345e8i	− 0.273586i
$753$	0	0
$754$	− 2.11187e8i	− 0.492667i
$755$	1.70897e8i	0.397093i
$756$	0	0
$757$	7.11235e8	1.63955	0.819776	−	0.572684i	$-0.194098\pi$
$757$	7.11235e8	1.63955	0.819776	+	0.572684i	$0.194098\pi$
$758$	2.90730e8	0.667548
$759$	0	0
$760$	−3.22252e8	−0.734099
$761$	3.67489e8i	0.833856i	0.908940	+	0.416928i	$0.136893\pi$
$761$	3.67489e8i	0.833856i	−0.908940	+	0.416928i	$0.863107\pi$
$762$	0	0
$763$	0	0
$764$	2.87974e8	0.645763
$765$	0	0
$766$	− 4.06437e8i	− 0.904288i
$767$	−2.00954e8	−0.445360
$768$	0	0
$769$	− 3.18891e8i	− 0.701235i	−0.936519	−	0.350617i	$-0.885972\pi$
$769$	− 3.18891e8i	− 0.701235i	0.936519	−	0.350617i	$-0.114028\pi$
$770$	0	0
$771$	0	0
$772$	6.01759e6	0.0130789
$773$	− 1.56061e8i	− 0.337876i	−0.985627	−	0.168938i	$-0.945966\pi$
$773$	− 1.56061e8i	− 0.337876i	0.985627	−	0.168938i	$-0.0540337\pi$
$774$	0	0
$775$	− 2.80859e8i	− 0.603369i
$776$	3.51133e8i	0.751427i
$777$	0	0
$778$	9.51413e7	0.202037
$779$	3.38584e8	0.716232
$780$	0	0
$781$	8.03597e8	1.68688
$782$	3.36398e8i	0.703450i
$783$	0	0
$784$	0	0
$785$	−4.53947e8	−0.938417
$786$	0	0
$787$	− 7.50607e8i	− 1.53988i	−0.638114	−	0.769942i	$-0.720285\pi$
$787$	− 7.50607e8i	− 1.53988i	0.638114	−	0.769942i	$-0.279715\pi$
$788$	9.00981e7	0.184135
$789$	0	0
$790$	− 1.51759e8i	− 0.307803i
$791$	0	0
$792$	0	0
$793$	−7.40923e8	−1.48578
$794$	− 3.38901e8i	− 0.677036i
$795$	0	0
$796$	− 2.02126e8i	− 0.400758i
$797$	− 1.08408e6i	− 0.00214134i	−0.999999	−	0.00107067i	$-0.999659\pi$
$797$	− 1.08408e6i	− 0.00214134i	0.999999	−	0.00107067i	$-0.000340806\pi$
$798$	0	0
$799$	−8.55483e8	−1.67715
$800$	−3.32557e8	−0.649524
$801$	0	0
$802$	−2.55055e8	−0.494436
$803$	5.75933e8i	1.11231i
$804$	0	0
$805$	0	0
$806$	−3.91183e8	−0.747093
$807$	0	0
$808$	− 5.49705e8i	− 1.04207i
$809$	−4.47774e8	−0.845694	−0.422847	−	0.906201i	$-0.638969\pi$
$809$	−4.47774e8	−0.845694	−0.422847	+	0.906201i	$0.638969\pi$
$810$	0	0
$811$	− 7.02416e8i	− 1.31684i	−0.752652	−	0.658418i	$-0.771226\pi$
$811$	− 7.02416e8i	− 1.31684i	0.752652	−	0.658418i	$-0.228774\pi$
$812$	0	0
$813$	0	0
$814$	2.16828e8	0.402015
$815$	5.53904e8i	1.02320i
$816$	0	0
$817$	− 6.57975e8i	− 1.20654i
$818$	3.35448e8i	0.612866i
$819$	0	0
$820$	−8.48960e7	−0.153973
$821$	−6.63875e8	−1.19966	−0.599828	−	0.800129i	$-0.704764\pi$
$821$	−6.63875e8	−1.19966	−0.599828	+	0.800129i	$0.704764\pi$
$822$	0	0
$823$	4.16855e8	0.747800	0.373900	−	0.927469i	$-0.378020\pi$
$823$	4.16855e8	0.747800	0.373900	+	0.927469i	$0.378020\pi$
$824$	− 2.25857e8i	− 0.403693i
$825$	0	0
$826$	0	0
$827$	−1.14598e8	−0.202609	−0.101305	−	0.994855i	$-0.532302\pi$
$827$	−1.14598e8	−0.202609	−0.101305	+	0.994855i	$0.532302\pi$
$828$	0	0
$829$	4.68094e8i	0.821617i	0.911722	+	0.410808i	$0.134754\pi$
$829$	4.68094e8i	0.821617i	−0.911722	+	0.410808i	$0.865246\pi$
$830$	2.66299e7	0.0465730
$831$	0	0
$832$	6.13397e8i	1.06505i
$833$	0	0
$834$	0	0
$835$	1.74085e8	0.299021
$836$	− 5.15773e8i	− 0.882754i
$837$	0	0
$838$	6.48708e8i	1.10234i
$839$	− 6.49855e8i	− 1.10035i	−0.835050	−	0.550174i	$-0.814561\pi$
$839$	− 6.49855e8i	− 1.10035i	0.835050	−	0.550174i	$-0.185439\pi$
$840$	0	0
$841$	−4.09581e8	−0.688576
$842$	7.10272e7	0.118984
$843$	0	0
$844$	4.62019e8	0.768480
$845$	2.04570e8i	0.339056i
$846$	0	0
$847$	0	0
$848$	−1.38088e7	−0.0226447
$849$	0	0
$850$	− 3.78503e8i	− 0.616328i
$851$	−2.25083e8	−0.365220
$852$	0	0
$853$	− 7.47082e7i	− 0.120371i	−0.998187	−	0.0601854i	$-0.980831\pi$
$853$	− 7.47082e7i	− 0.120371i	0.998187	−	0.0601854i	$-0.0191692\pi$
$854$	0	0
$855$	0	0
$856$	−1.19165e9	−1.89988
$857$	8.66704e8i	1.37698i	0.725245	+	0.688491i	$0.241727\pi$
$857$	8.66704e8i	1.37698i	−0.725245	+	0.688491i	$0.758273\pi$
$858$	0	0
$859$	1.74675e7i	0.0275583i	0.999905	+	0.0137791i	$0.00438617\pi$
$859$	1.74675e7i	0.0275583i	−0.999905	+	0.0137791i	$0.995614\pi$
$860$	1.64980e8i	0.259379i
$861$	0	0
$862$	−6.29285e8	−0.982484
$863$	1.11908e9	1.74113	0.870563	−	0.492056i	$-0.163755\pi$
$863$	1.11908e9	1.74113	0.870563	+	0.492056i	$0.163755\pi$
$864$	0	0
$865$	−5.49486e8	−0.849002
$866$	− 1.52607e8i	− 0.234974i
$867$	0	0
$868$	0	0
$869$	7.07593e8	1.07826
$870$	0	0
$871$	1.54034e9i	2.33111i
$872$	4.12533e8	0.622171
$873$	0	0
$874$	− 4.88922e8i	− 0.732328i
$875$	0	0
$876$	0	0
$877$	1.16546e9	1.72782	0.863910	−	0.503646i	$-0.168008\pi$
$877$	1.16546e9	1.72782	0.863910	+	0.503646i	$0.168008\pi$
$878$	3.97251e8i	0.586923i
$879$	0	0
$880$	− 9.66134e7i	− 0.141772i
$881$	− 2.07806e8i	− 0.303900i	−0.988388	−	0.151950i	$-0.951445\pi$
$881$	− 2.07806e8i	− 0.303900i	0.988388	−	0.151950i	$-0.0485553\pi$
$882$	0	0
$883$	−3.48816e8	−0.506657	−0.253329	−	0.967380i	$-0.581525\pi$
$883$	−3.48816e8	−0.506657	−0.253329	+	0.967380i	$0.581525\pi$
$884$	5.77306e8	0.835698
$885$	0	0
$886$	−5.25225e8	−0.755170
$887$	− 8.44527e8i	− 1.21016i	−0.796165	−	0.605080i	$-0.793141\pi$
$887$	− 8.44527e8i	− 1.21016i	0.796165	−	0.605080i	$-0.206859\pi$
$888$	0	0
$889$	0	0
$890$	6.04764e7	0.0857858
$891$	0	0
$892$	3.28300e8i	0.462569i
$893$	1.24336e9	1.74600
$894$	0	0
$895$	5.48320e8i	0.764830i
$896$	0	0
$897$	0	0
$898$	6.37820e8	0.880783
$899$	− 3.43125e8i	− 0.472251i
$900$	0	0
$901$	1.01536e8i	0.138818i
$902$	3.61469e8i	0.492552i
$903$	0	0
$904$	1.43324e9	1.94006
$905$	−2.18045e8	−0.294172
$906$	0	0
$907$	−7.99849e8	−1.07198	−0.535989	−	0.844225i	$-0.680061\pi$
$907$	−7.99849e8	−1.07198	−0.535989	+	0.844225i	$0.680061\pi$
$908$	− 4.87543e8i	− 0.651261i
$909$	0	0
$910$	0	0
$911$	−1.54047e8	−0.203750	−0.101875	−	0.994797i	$-0.532484\pi$
$911$	−1.54047e8	−0.203750	−0.101875	+	0.994797i	$0.532484\pi$
$912$	0	0
$913$	1.24165e8i	0.163150i
$914$	−5.07117e7	−0.0664155
$915$	0	0
$916$	4.91697e8i	0.639751i
$917$	0	0
$918$	0	0
$919$	6.87088e8	0.885250	0.442625	−	0.896707i	$-0.354047\pi$
$919$	6.87088e8	0.885250	0.442625	+	0.896707i	$0.354047\pi$
$920$	3.57131e8i	0.458632i
$921$	0	0
$922$	3.20209e8i	0.408546i
$923$	− 1.30729e9i	− 1.66252i
$924$	0	0
$925$	2.53255e8	0.319987
$926$	−3.60239e8	−0.453689
$927$	0	0
$928$	−4.06283e8	−0.508376
$929$	3.78043e8i	0.471513i	0.971812	+	0.235757i	$0.0757568\pi$
$929$	3.78043e8i	0.471513i	−0.971812	+	0.235757i	$0.924243\pi$
$930$	0	0
$931$	0	0
$932$	1.92416e8	0.237680
$933$	0	0
$934$	9.36559e8i	1.14946i
$935$	−7.10398e8	−0.869095
$936$	0	0
$937$	− 1.27190e9i	− 1.54609i	−0.634351	−	0.773045i	$-0.718732\pi$
$937$	− 1.27190e9i	− 1.54609i	0.634351	−	0.773045i	$-0.281268\pi$
$938$	0	0
$939$	0	0
$940$	−3.11759e8	−0.375349
$941$	6.98803e8i	0.838660i	0.907834	+	0.419330i	$0.137735\pi$
$941$	6.98803e8i	0.838660i	−0.907834	+	0.419330i	$0.862265\pi$
$942$	0	0
$943$	− 3.75231e8i	− 0.447470i
$944$	− 5.98408e7i	− 0.0711346i
$945$	0	0
$946$	7.02448e8	0.829738
$947$	3.91438e8	0.460907	0.230453	−	0.973083i	$-0.425979\pi$
$947$	3.91438e8	0.460907	0.230453	+	0.973083i	$0.425979\pi$
$948$	0	0
$949$	9.36927e8	1.09624
$950$	5.50117e8i	0.641629i
$951$	0	0
$952$	0	0
$953$	9.05418e8	1.04609	0.523047	−	0.852304i	$-0.324795\pi$
$953$	9.05418e8	1.04609	0.523047	+	0.852304i	$0.324795\pi$
$954$	0	0
$955$	5.76478e8i	0.661869i
$956$	5.35795e8	0.613232
$957$	0	0
$958$	− 5.05897e8i	− 0.575394i
$959$	0	0
$960$	0	0
$961$	2.51933e8	0.283867
$962$	− 3.52736e8i	− 0.396209i
$963$	0	0
$964$	8.63756e8i	0.964185i
$965$	1.20463e7i	0.0134051i
$966$	0	0
$967$	−1.33523e9	−1.47665	−0.738323	−	0.674448i	$-0.764382\pi$
$967$	−1.33523e9	−1.47665	−0.738323	+	0.674448i	$0.764382\pi$
$968$	6.49899e8	0.716506
$969$	0	0
$970$	−2.41288e8	−0.264375
$971$	− 6.30242e8i	− 0.688414i	−0.938894	−	0.344207i	$-0.888148\pi$
$971$	− 6.30242e8i	− 0.688414i	0.938894	−	0.344207i	$-0.111852\pi$
$972$	0	0
$973$	0	0
$974$	7.25362e8	0.785015
$975$	0	0
$976$	− 2.20634e8i	− 0.237314i
$977$	4.09104e8	0.438682	0.219341	−	0.975648i	$-0.429609\pi$
$977$	4.09104e8	0.438682	0.219341	+	0.975648i	$0.429609\pi$
$978$	0	0
$979$	2.81978e8i	0.300516i
$980$	0	0
$981$	0	0
$982$	−4.87358e8	−0.514652
$983$	− 9.18323e8i	− 0.966796i	−0.875401	−	0.483398i	$-0.839402\pi$
$983$	− 9.18323e8i	− 0.966796i	0.875401	−	0.483398i	$-0.160598\pi$
$984$	0	0
$985$	1.80362e8i	0.188728i
$986$	− 4.62415e8i	− 0.482394i
$987$	0	0
$988$	−8.39059e8	−0.870005
$989$	−7.29192e8	−0.753794
$990$	0	0
$991$	−5.18738e8	−0.533000	−0.266500	−	0.963835i	$-0.585867\pi$
$991$	−5.18738e8	−0.533000	−0.266500	+	0.963835i	$0.585867\pi$
$992$	7.52560e8i	0.770914i
$993$	0	0
$994$	0	0
$995$	4.04623e8	0.410754
$996$	0	0
$997$	3.60955e8i	0.364223i	0.983278	+	0.182111i	$0.0582931\pi$
$997$	3.60955e8i	0.364223i	−0.983278	+	0.182111i	$0.941707\pi$
$998$	1.26276e9	1.27037
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.7.d.c.244.3		8
3.2	odd	2		147.7.d.b.97.5		8
7.4	even	3		63.7.m.d.19.3		8
7.5	odd	6		63.7.m.d.10.3		8
7.6	odd	2	inner	441.7.d.c.244.4		8
21.2	odd	6		147.7.f.d.31.2		8
21.5	even	6		21.7.f.a.10.2	✓	8
21.11	odd	6		21.7.f.a.19.2	yes	8
21.17	even	6		147.7.f.d.19.2		8
21.20	even	2		147.7.d.b.97.6		8
84.11	even	6		336.7.bh.d.145.2		8
84.47	odd	6		336.7.bh.d.241.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.7.f.a.10.2	✓	8	21.5	even	6
21.7.f.a.19.2	yes	8	21.11	odd	6
63.7.m.d.10.3		8	7.5	odd	6
63.7.m.d.19.3		8	7.4	even	3
147.7.d.b.97.5		8	3.2	odd	2
147.7.d.b.97.6		8	21.20	even	2
147.7.f.d.19.2		8	21.17	even	6
147.7.f.d.31.2		8	21.2	odd	6
336.7.bh.d.145.2		8	84.11	even	6
336.7.bh.d.241.2		8	84.47	odd	6
441.7.d.c.244.3		8	1.1	even	1	trivial
441.7.d.c.244.4		8	7.6	odd	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:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	441.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-5.52700 q^{2} -33.4522 q^{4} -66.9661i q^{5} +538.619 q^{8} +O(q^{10})\)	\(q-5.52700 q^{2} -33.4522 q^{4} -66.9661i q^{5} +538.619 q^{8} +370.122i q^{10} -1725.74 q^{11} +2807.43i q^{13} -836.004 q^{16} +6147.14i q^{17} -8934.27i q^{19} +2240.17i q^{20} +9538.15 q^{22} -9901.28 q^{23} +11140.5 q^{25} -15516.6i q^{26} +13610.4 q^{29} -25210.5i q^{31} -29851.0 q^{32} -33975.2i q^{34} +22732.7 q^{37} +49379.7i q^{38} -36069.2i q^{40} +37897.2i q^{41} +73646.2 q^{43} +57729.7 q^{44} +54724.4 q^{46} +139168. i q^{47} -61573.8 q^{50} -93914.7i q^{52} +16517.6 q^{53} +115566. i q^{55} -75224.5 q^{58} +71579.5i q^{59} +263915. i q^{61} +139339. i q^{62} +218491. q^{64} +188002. q^{65} +548667. q^{67} -205635. i q^{68} -465655. q^{71} -333732. i q^{73} -125644. q^{74} +298871. i q^{76} -410024. q^{79} +55983.9i q^{80} -209458. i q^{82} -71948.9i q^{83} +411649. q^{85} -407043. q^{86} -929514. q^{88} -163396. i q^{89} +331220. q^{92} -769180. i q^{94} -598293. q^{95} +651914. i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 10 q^{2} + 346 q^{4} - 3326 q^{8}+O(q^{10}) \) 8 * q - 10 * q^2 + 346 * q^4 - 3326 * q^8	\( 8 q - 10 q^{2} + 346 q^{4} - 3326 q^{8} - 628 q^{11} + 25442 q^{16} + 86106 q^{22} + 7856 q^{23} + 34076 q^{25} + 8300 q^{29} - 372414 q^{32} - 129412 q^{37} + 45740 q^{43} + 185058 q^{44} + 223008 q^{46} - 967216 q^{50} - 1081948 q^{53} - 1079598 q^{58} + 2378626 q^{64} - 828408 q^{65} + 2317804 q^{67} - 1442344 q^{71} - 865880 q^{74} - 1222904 q^{79} - 275112 q^{85} + 1632448 q^{86} + 732882 q^{88} - 678720 q^{92} - 1183584 q^{95}+O(q^{100}) \) 8 * q - 10 * q^2 + 346 * q^4 - 3326 * q^8 - 628 * q^11 + 25442 * q^16 + 86106 * q^22 + 7856 * q^23 + 34076 * q^25 + 8300 * q^29 - 372414 * q^32 - 129412 * q^37 + 45740 * q^43 + 185058 * q^44 + 223008 * q^46 - 967216 * q^50 - 1081948 * q^53 - 1079598 * q^58 + 2378626 * q^64 - 828408 * q^65 + 2317804 * q^67 - 1442344 * q^71 - 865880 * q^74 - 1222904 * q^79 - 275112 * q^85 + 1632448 * q^86 + 732882 * q^88 - 678720 * q^92 - 1183584 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more