LMFDB - Embedded newform 441.6.c.c.440.15

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:	$40$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			440.15
Character	$\chi$	$=$	441.440
Dual form			441.6.c.c.440.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.51435i q^{2} +19.6494 q^{4} +28.2666 q^{5} -181.514i q^{8} +O(q^{10})$	$q-3.51435i q^{2} +19.6494 q^{4} +28.2666 q^{5} -181.514i q^{8} -99.3387i q^{10} -265.746i q^{11} -882.168i q^{13} -9.12380 q^{16} +337.576 q^{17} -694.092i q^{19} +555.420 q^{20} -933.925 q^{22} +713.272i q^{23} -2326.00 q^{25} -3100.25 q^{26} +4112.32i q^{29} -646.714i q^{31} -5776.38i q^{32} -1186.36i q^{34} -7742.41 q^{37} -2439.28 q^{38} -5130.78i q^{40} -15271.7 q^{41} +11501.8 q^{43} -5221.74i q^{44} +2506.69 q^{46} +9282.44 q^{47} +8174.37i q^{50} -17334.0i q^{52} +838.025i q^{53} -7511.74i q^{55} +14452.1 q^{58} +14092.8 q^{59} -27138.3i q^{61} -2272.78 q^{62} -20592.2 q^{64} -24935.9i q^{65} -35950.6 q^{67} +6633.15 q^{68} +49835.3i q^{71} -86542.1i q^{73} +27209.5i q^{74} -13638.5i q^{76} +73536.7 q^{79} -257.899 q^{80} +53670.2i q^{82} -57161.3 q^{83} +9542.12 q^{85} -40421.4i q^{86} -48236.6 q^{88} +99813.8 q^{89} +14015.3i q^{92} -32621.7i q^{94} -19619.6i q^{95} -93281.8i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 640 q^{4}+O(q^{10}) $ 40 * q - 640 * q^4	$ 40 q - 640 q^{4} + 13584 q^{16} - 16208 q^{22} + 14792 q^{25} + 52448 q^{37} - 43168 q^{43} + 58192 q^{46} + 383664 q^{58} - 278432 q^{64} + 228480 q^{67} + 154592 q^{79} - 357824 q^{85} + 1796656 q^{88}+O(q^{100}) $ 40 * q - 640 * q^4 + 13584 * q^16 - 16208 * q^22 + 14792 * q^25 + 52448 * q^37 - 43168 * q^43 + 58192 * q^46 + 383664 * q^58 - 278432 * q^64 + 228480 * q^67 + 154592 * q^79 - 357824 * q^85 + 1796656 * 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 $

$2$	− 3.51435i	− 0.621255i	−0.950532	−	0.310628i	$-0.899461\pi$
$2$	− 3.51435i	− 0.621255i	0.950532	−	0.310628i	$-0.100539\pi$
$3$	0	0
$3$	0	0
$4$	19.6494	0.614042
$5$	28.2666	0.505648	0.252824	−	0.967512i	$-0.418641\pi$
$5$	28.2666	0.505648	0.252824	+	0.967512i	$0.418641\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	− 181.514i	− 1.00273i
$9$	0	0
$10$	− 99.3387i	− 0.314136i
$11$	− 265.746i	− 0.662194i	−0.943597	−	0.331097i	$-0.892581\pi$
$11$	− 265.746i	− 0.662194i	0.943597	−	0.331097i	$-0.107419\pi$
$12$	0	0
$13$	− 882.168i	− 1.44775i	−0.689932	−	0.723874i	$-0.742360\pi$
$13$	− 882.168i	− 1.44775i	0.689932	−	0.723874i	$-0.257640\pi$
$14$	0	0
$15$	0	0
$16$	−9.12380	−0.00890996
$17$	337.576	0.283302	0.141651	−	0.989917i	$-0.454759\pi$
$17$	337.576	0.283302	0.141651	+	0.989917i	$0.454759\pi$
$18$	0	0
$19$	− 694.092i	− 0.441096i	−0.975376	−	0.220548i	$-0.929215\pi$
$19$	− 694.092i	− 0.441096i	0.975376	−	0.220548i	$-0.0707845\pi$
$20$	555.420	0.310489
$21$	0	0
$22$	−933.925	−0.411391
$23$	713.272i	0.281148i	0.990070	+	0.140574i	$0.0448949\pi$
$23$	713.272i	0.281148i	−0.990070	+	0.140574i	$0.955105\pi$
$24$	0	0
$25$	−2326.00	−0.744320
$26$	−3100.25	−0.899420
$27$	0	0
$28$	0	0
$29$	4112.32i	0.908012i	0.890999	+	0.454006i	$0.150006\pi$
$29$	4112.32i	0.908012i	−0.890999	+	0.454006i	$0.849994\pi$
$30$	0	0
$31$	− 646.714i	− 0.120867i	−0.998172	−	0.0604335i	$-0.980752\pi$
$31$	− 646.714i	− 0.120867i	0.998172	−	0.0604335i	$-0.0192483\pi$
$32$	− 5776.38i	− 0.997196i
$33$	0	0
$34$	− 1186.36i	− 0.176003i
$35$	0	0
$36$	0	0
$37$	−7742.41	−0.929763	−0.464881	−	0.885373i	$-0.653903\pi$
$37$	−7742.41	−0.929763	−0.464881	+	0.885373i	$0.653903\pi$
$38$	−2439.28	−0.274033
$39$	0	0
$40$	− 5130.78i	− 0.507030i
$41$	−15271.7	−1.41883	−0.709413	−	0.704793i	$-0.751040\pi$
$41$	−15271.7	−1.41883	−0.709413	+	0.704793i	$0.751040\pi$
$42$	0	0
$43$	11501.8	0.948628	0.474314	−	0.880356i	$-0.342696\pi$
$43$	11501.8	0.948628	0.474314	+	0.880356i	$0.342696\pi$
$44$	− 5221.74i	− 0.406615i
$45$	0	0
$46$	2506.69	0.174665
$47$	9282.44	0.612940	0.306470	−	0.951880i	$-0.400852\pi$
$47$	9282.44	0.612940	0.306470	+	0.951880i	$0.400852\pi$
$48$	0	0
$49$	0	0
$50$	8174.37i	0.462412i
$51$	0	0
$52$	− 17334.0i	− 0.888978i
$53$	838.025i	0.0409796i	0.999790	+	0.0204898i	$0.00652256\pi$
$53$	838.025i	0.0409796i	−0.999790	+	0.0204898i	$0.993477\pi$
$54$	0	0
$55$	− 7511.74i	− 0.334837i
$56$	0	0
$57$	0	0
$58$	14452.1	0.564107
$59$	14092.8	0.527067	0.263534	−	0.964650i	$-0.415112\pi$
$59$	14092.8	0.527067	0.263534	+	0.964650i	$0.415112\pi$
$60$	0	0
$61$	− 27138.3i	− 0.933810i	−0.884308	−	0.466905i	$-0.845369\pi$
$61$	− 27138.3i	− 0.933810i	0.884308	−	0.466905i	$-0.154631\pi$
$62$	−2272.78	−0.0750892
$63$	0	0
$64$	−20592.2	−0.628423
$65$	− 24935.9i	− 0.732051i
$66$	0	0
$67$	−35950.6	−0.978407	−0.489204	−	0.872170i	$-0.662712\pi$
$67$	−35950.6	−0.978407	−0.489204	+	0.872170i	$0.662712\pi$
$68$	6633.15	0.173959
$69$	0	0
$70$	0	0
$71$	49835.3i	1.17325i	0.809858	+	0.586625i	$0.199544\pi$
$71$	49835.3i	1.17325i	−0.809858	+	0.586625i	$0.800456\pi$
$72$	0	0
$73$	− 86542.1i	− 1.90073i	−0.311137	−	0.950365i	$-0.600710\pi$
$73$	− 86542.1i	− 1.90073i	0.311137	−	0.950365i	$-0.399290\pi$
$74$	27209.5i	0.577620i
$75$	0	0
$76$	− 13638.5i	− 0.270851i
$77$	0	0
$78$	0	0
$79$	73536.7	1.32567	0.662836	−	0.748764i	$-0.269353\pi$
$79$	73536.7	1.32567	0.662836	+	0.748764i	$0.269353\pi$
$80$	−257.899	−0.00450530
$81$	0	0
$82$	53670.2i	0.881452i
$83$	−57161.3	−0.910766	−0.455383	−	0.890296i	$-0.650498\pi$
$83$	−57161.3	−0.910766	−0.455383	+	0.890296i	$0.650498\pi$
$84$	0	0
$85$	9542.12	0.143251
$86$	− 40421.4i	− 0.589340i
$87$	0	0
$88$	−48236.6	−0.664003
$89$	99813.8	1.33572	0.667860	−	0.744287i	$-0.267210\pi$
$89$	99813.8	1.33572	0.667860	+	0.744287i	$0.267210\pi$
$90$	0	0
$91$	0	0
$92$	14015.3i	0.172637i
$93$	0	0
$94$	− 32621.7i	− 0.380792i
$95$	− 19619.6i	− 0.223039i
$96$	0	0
$97$	− 93281.8i	− 1.00662i	−0.864105	−	0.503312i	$-0.832115\pi$
$97$	− 93281.8i	− 1.00662i	0.864105	−	0.503312i	$-0.167885\pi$
$98$	0	0
$99$	0	0
$100$	−45704.4	−0.457044
$101$	−159364.	−1.55449	−0.777243	−	0.629201i	$-0.783382\pi$
$101$	−159364.	−1.55449	−0.777243	+	0.629201i	$0.783382\pi$
$102$	0	0
$103$	− 44279.9i	− 0.411257i	−0.978630	−	0.205629i	$-0.934076\pi$
$103$	− 44279.9i	− 0.411257i	0.978630	−	0.205629i	$-0.0659239\pi$
$104$	−160126.	−1.45170
$105$	0	0
$106$	2945.11	0.0254588
$107$	− 214350.i	− 1.80994i	−0.425473	−	0.904971i	$-0.639892\pi$
$107$	− 214350.i	− 1.80994i	0.425473	−	0.904971i	$-0.360108\pi$
$108$	0	0
$109$	−4074.08	−0.0328445	−0.0164223	−	0.999865i	$-0.505228\pi$
$109$	−4074.08	−0.0328445	−0.0164223	+	0.999865i	$0.505228\pi$
$110$	−26398.9	−0.208019
$111$	0	0
$112$	0	0
$113$	221000.i	1.62815i	0.580758	+	0.814076i	$0.302756\pi$
$113$	221000.i	1.62815i	−0.580758	+	0.814076i	$0.697244\pi$
$114$	0	0
$115$	20161.8i	0.142162i
$116$	80804.3i	0.557558i
$117$	0	0
$118$	− 49526.9i	− 0.327443i
$119$	0	0
$120$	0	0
$121$	90430.0	0.561499
$122$	−95373.5	−0.580134
$123$	0	0
$124$	− 12707.5i	− 0.0742174i
$125$	−154081.	−0.882012
$126$	0	0
$127$	−290068.	−1.59585	−0.797923	−	0.602760i	$-0.794068\pi$
$127$	−290068.	−1.59585	−0.797923	+	0.602760i	$0.794068\pi$
$128$	− 112476.i	− 0.606785i
$129$	0	0
$130$	−87633.4	−0.454790
$131$	−168549.	−0.858119	−0.429060	−	0.903276i	$-0.641155\pi$
$131$	−168549.	−0.858119	−0.429060	+	0.903276i	$0.641155\pi$
$132$	0	0
$133$	0	0
$134$	126343.i	0.607840i
$135$	0	0
$136$	− 61274.7i	− 0.284076i
$137$	108116.i	0.492141i	0.969252	+	0.246070i	$0.0791394\pi$
$137$	108116.i	0.492141i	−0.969252	+	0.246070i	$0.920861\pi$
$138$	0	0
$139$	− 420589.i	− 1.84638i	−0.384346	−	0.923189i	$-0.625573\pi$
$139$	− 420589.i	− 1.84638i	0.384346	−	0.923189i	$-0.374427\pi$
$140$	0	0
$141$	0	0
$142$	175138.	0.728888
$143$	−234433.	−0.958690
$144$	0	0
$145$	116241.i	0.459135i
$146$	−304139.	−1.18084
$147$	0	0
$148$	−152133.	−0.570913
$149$	99847.4i	0.368443i	0.982885	+	0.184222i	$0.0589765\pi$
$149$	99847.4i	0.368443i	−0.982885	+	0.184222i	$0.941024\pi$
$150$	0	0
$151$	172525.	0.615758	0.307879	−	0.951426i	$-0.400381\pi$
$151$	172525.	0.615758	0.307879	+	0.951426i	$0.400381\pi$
$152$	−125987.	−0.442301
$153$	0	0
$154$	0	0
$155$	− 18280.4i	− 0.0611162i
$156$	0	0
$157$	− 62411.0i	− 0.202075i	−0.994883	−	0.101037i	$-0.967784\pi$
$157$	− 62411.0i	− 0.202075i	0.994883	−	0.101037i	$-0.0322161\pi$
$158$	− 258434.i	− 0.823581i
$159$	0	0
$160$	− 163279.i	− 0.504231i
$161$	0	0
$162$	0	0
$163$	368220.	1.08552	0.542761	−	0.839887i	$-0.317379\pi$
$163$	368220.	1.08552	0.542761	+	0.839887i	$0.317379\pi$
$164$	−300080.	−0.871219
$165$	0	0
$166$	200885.i	0.565818i
$167$	−532956.	−1.47877	−0.739384	−	0.673284i	$-0.764883\pi$
$167$	−532956.	−1.47877	−0.739384	+	0.673284i	$0.764883\pi$
$168$	0	0
$169$	−406927.	−1.09597
$170$	− 33534.4i	− 0.0889954i
$171$	0	0
$172$	226003.	0.582497
$173$	293518.	0.745623	0.372812	−	0.927907i	$-0.378394\pi$
$173$	293518.	0.745623	0.372812	+	0.927907i	$0.378394\pi$
$174$	0	0
$175$	0	0
$176$	2424.61i	0.00590012i
$177$	0	0
$178$	− 350781.i	− 0.829823i
$179$	688630.i	1.60640i	0.595710	+	0.803200i	$0.296871\pi$
$179$	688630.i	1.60640i	−0.595710	+	0.803200i	$0.703129\pi$
$180$	0	0
$181$	− 241111.i	− 0.547041i	−0.961866	−	0.273520i	$-0.911812\pi$
$181$	− 241111.i	− 0.547041i	0.961866	−	0.273520i	$-0.0881881\pi$
$182$	0	0
$183$	0	0
$184$	129469.	0.281916
$185$	−218852.	−0.470133
$186$	0	0
$187$	− 89709.5i	− 0.187601i
$188$	182394.	0.376371
$189$	0	0
$190$	−68950.2	−0.138564
$191$	− 662372.i	− 1.31377i	−0.753992	−	0.656884i	$-0.771874\pi$
$191$	− 662372.i	− 1.31377i	0.753992	−	0.656884i	$-0.228126\pi$
$192$	0	0
$193$	230763.	0.445936	0.222968	−	0.974826i	$-0.428425\pi$
$193$	230763.	0.445936	0.222968	+	0.974826i	$0.428425\pi$
$194$	−327825.	−0.625370
$195$	0	0
$196$	0	0
$197$	488751.i	0.897267i	0.893716	+	0.448634i	$0.148089\pi$
$197$	488751.i	0.897267i	−0.893716	+	0.448634i	$0.851911\pi$
$198$	0	0
$199$	865680.i	1.54962i	0.632196	+	0.774809i	$0.282154\pi$
$199$	865680.i	1.54962i	−0.632196	+	0.774809i	$0.717846\pi$
$200$	422201.i	0.746353i
$201$	0	0
$202$	560061.i	0.965732i
$203$	0	0
$204$	0	0
$205$	−431680.	−0.717427
$206$	−155615.	−0.255496
$207$	0	0
$208$	8048.72i	0.0128994i
$209$	−184452.	−0.292091
$210$	0	0
$211$	−39578.0	−0.0611995	−0.0305998	−	0.999532i	$-0.509742\pi$
$211$	−39578.0	−0.0611995	−0.0305998	+	0.999532i	$0.509742\pi$
$212$	16466.7i	0.0251632i
$213$	0	0
$214$	−753302.	−1.12444
$215$	325118.	0.479672
$216$	0	0
$217$	0	0
$218$	14317.7i	0.0204048i
$219$	0	0
$220$	− 147601.i	− 0.205604i
$221$	− 297799.i	− 0.410149i
$222$	0	0
$223$	194472.i	0.261875i	0.991391	+	0.130937i	$0.0417987\pi$
$223$	194472.i	0.261875i	−0.991391	+	0.130937i	$0.958201\pi$
$224$	0	0
$225$	0	0
$226$	776669.	1.01150
$227$	161317.	0.207785	0.103893	−	0.994589i	$-0.466870\pi$
$227$	161317.	0.207785	0.103893	+	0.994589i	$0.466870\pi$
$228$	0	0
$229$	334923.i	0.422043i	0.977481	+	0.211021i	$0.0676790\pi$
$229$	334923.i	0.422043i	−0.977481	+	0.211021i	$0.932321\pi$
$230$	70855.5	0.0883190
$231$	0	0
$232$	746442.	0.910492
$233$	− 373555.i	− 0.450780i	−0.974269	−	0.225390i	$-0.927634\pi$
$233$	− 373555.i	− 0.450780i	0.974269	−	0.225390i	$-0.0723657\pi$
$234$	0	0
$235$	262383.	0.309932
$236$	276914.	0.323642
$237$	0	0
$238$	0	0
$239$	− 811814.i	− 0.919310i	−0.888098	−	0.459655i	$-0.847973\pi$
$239$	− 811814.i	− 0.919310i	0.888098	−	0.459655i	$-0.152027\pi$
$240$	0	0
$241$	− 785799.i	− 0.871503i	−0.900067	−	0.435752i	$-0.856483\pi$
$241$	− 785799.i	− 0.871503i	0.900067	−	0.435752i	$-0.143517\pi$
$242$	− 317802.i	− 0.348834i
$243$	0	0
$244$	− 533250.i	− 0.573399i
$245$	0	0
$246$	0	0
$247$	−612305.	−0.638595
$248$	−117387.	−0.121197
$249$	0	0
$250$	541495.i	0.547955i
$251$	1.35263e6	1.35518	0.677588	−	0.735442i	$-0.263025\pi$
$251$	1.35263e6	1.35518	0.677588	+	0.735442i	$0.263025\pi$
$252$	0	0
$253$	189549.	0.186175
$254$	1.01940e6i	0.991427i
$255$	0	0
$256$	−1.05423e6	−1.00539
$257$	680939.	0.643095	0.321548	−	0.946893i	$-0.395797\pi$
$257$	680939.	0.643095	0.321548	+	0.946893i	$0.395797\pi$
$258$	0	0
$259$	0	0
$260$	− 489974.i	− 0.449510i
$261$	0	0
$262$	592340.i	0.533111i
$263$	− 1.68409e6i	− 1.50133i	−0.660685	−	0.750663i	$-0.729734\pi$
$263$	− 1.68409e6i	− 1.50133i	0.660685	−	0.750663i	$-0.270266\pi$
$264$	0	0
$265$	23688.1i	0.0207212i
$266$	0	0
$267$	0	0
$268$	−706407.	−0.600783
$269$	2.04797e6	1.72561	0.862807	−	0.505533i	$-0.168704\pi$
$269$	2.04797e6	1.72561	0.862807	+	0.505533i	$0.168704\pi$
$270$	0	0
$271$	− 949006.i	− 0.784957i	−0.919761	−	0.392479i	$-0.871618\pi$
$271$	− 949006.i	− 0.784957i	0.919761	−	0.392479i	$-0.128382\pi$
$272$	−3079.98	−0.00252421
$273$	0	0
$274$	379958.	0.305745
$275$	618126.i	0.492884i
$276$	0	0
$277$	256024.	0.200485	0.100242	−	0.994963i	$-0.468038\pi$
$277$	256024.	0.200485	0.100242	+	0.994963i	$0.468038\pi$
$278$	−1.47810e6	−1.14707
$279$	0	0
$280$	0	0
$281$	598494.i	0.452162i	0.974108	+	0.226081i	$0.0725914\pi$
$281$	598494.i	0.452162i	−0.974108	+	0.226081i	$0.927409\pi$
$282$	0	0
$283$	− 718851.i	− 0.533547i	−0.963759	−	0.266773i	$-0.914042\pi$
$283$	− 718851.i	− 0.533547i	0.963759	−	0.266773i	$-0.0859576\pi$
$284$	979230.i	0.720425i
$285$	0	0
$286$	823878.i	0.595591i
$287$	0	0
$288$	0	0
$289$	−1.30590e6	−0.919740
$290$	408512.	0.285240
$291$	0	0
$292$	− 1.70050e6i	− 1.16713i
$293$	2.53255e6	1.72341	0.861704	−	0.507411i	$-0.169397\pi$
$293$	2.53255e6	1.72341	0.861704	+	0.507411i	$0.169397\pi$
$294$	0	0
$295$	398355.	0.266511
$296$	1.40536e6i	0.932303i
$297$	0	0
$298$	350898.	0.228897
$299$	629226.	0.407032
$300$	0	0
$301$	0	0
$302$	− 606313.i	− 0.382543i
$303$	0	0
$304$	6332.75i	0.00393015i
$305$	− 767108.i	− 0.472179i
$306$	0	0
$307$	1.51481e6i	0.917303i	0.888616	+	0.458652i	$0.151667\pi$
$307$	1.51481e6i	0.917303i	−0.888616	+	0.458652i	$0.848333\pi$
$308$	0	0
$309$	0	0
$310$	−64243.7	−0.0379687
$311$	2.78613e6	1.63343	0.816714	−	0.577042i	$-0.195793\pi$
$311$	2.78613e6	1.63343	0.816714	+	0.577042i	$0.195793\pi$
$312$	0	0
$313$	− 1.99580e6i	− 1.15148i	−0.817633	−	0.575740i	$-0.804714\pi$
$313$	− 1.99580e6i	− 1.15148i	0.817633	−	0.575740i	$-0.195286\pi$
$314$	−219334.	−0.125540
$315$	0	0
$316$	1.44495e6	0.814019
$317$	18723.6i	0.0104651i	0.999986	+	0.00523253i	$0.00166557\pi$
$317$	18723.6i	0.0104651i	−0.999986	+	0.00523253i	$0.998334\pi$
$318$	0	0
$319$	1.09283e6	0.601280
$320$	−582071.	−0.317761
$321$	0	0
$322$	0	0
$323$	− 234309.i	− 0.124963i
$324$	0	0
$325$	2.05192e6i	1.07759i
$326$	− 1.29405e6i	− 0.674386i
$327$	0	0
$328$	2.77203e6i	1.42270i
$329$	0	0
$330$	0	0
$331$	−1.54569e6	−0.775446	−0.387723	−	0.921776i	$-0.626738\pi$
$331$	−1.54569e6	−0.775446	−0.387723	+	0.921776i	$0.626738\pi$
$332$	−1.12318e6	−0.559249
$333$	0	0
$334$	1.87299e6i	0.918692i
$335$	−1.01620e6	−0.494730
$336$	0	0
$337$	2.51878e6	1.20814	0.604068	−	0.796933i	$-0.293546\pi$
$337$	2.51878e6	1.20814	0.604068	+	0.796933i	$0.293546\pi$
$338$	1.43008e6i	0.680879i
$339$	0	0
$340$	187497.	0.0879622
$341$	−171862.	−0.0800374
$342$	0	0
$343$	0	0
$344$	− 2.08774e6i	− 0.951219i
$345$	0	0
$346$	− 1.03152e6i	− 0.463222i
$347$	− 898571.i	− 0.400616i	−0.979733	−	0.200308i	$-0.935806\pi$
$347$	− 898571.i	− 0.400616i	0.979733	−	0.200308i	$-0.0641943\pi$
$348$	0	0
$349$	379764.i	0.166898i	0.996512	+	0.0834488i	$0.0265935\pi$
$349$	379764.i	0.166898i	−0.996512	+	0.0834488i	$0.973406\pi$
$350$	0	0
$351$	0	0
$352$	−1.53505e6	−0.660338
$353$	2.68671e6	1.14758	0.573791	−	0.819002i	$-0.305472\pi$
$353$	2.68671e6	1.14758	0.573791	+	0.819002i	$0.305472\pi$
$354$	0	0
$355$	1.40867e6i	0.593252i
$356$	1.96128e6	0.820189
$357$	0	0
$358$	2.42009e6	0.997984
$359$	3.20634e6i	1.31303i	0.754314	+	0.656514i	$0.227970\pi$
$359$	3.20634e6i	1.31303i	−0.754314	+	0.656514i	$0.772030\pi$
$360$	0	0
$361$	1.99434e6	0.805434
$362$	−847347.	−0.339852
$363$	0	0
$364$	0	0
$365$	− 2.44625e6i	− 0.961101i
$366$	0	0
$367$	2.24939e6i	0.871763i	0.900004	+	0.435882i	$0.143563\pi$
$367$	2.24939e6i	0.871763i	−0.900004	+	0.435882i	$0.856437\pi$
$368$	− 6507.75i	− 0.00250502i
$369$	0	0
$370$	769121.i	0.292072i
$371$	0	0
$372$	0	0
$373$	887206.	0.330181	0.165091	−	0.986278i	$-0.447208\pi$
$373$	887206.	0.330181	0.165091	+	0.986278i	$0.447208\pi$
$374$	−315271.	−0.116548
$375$	0	0
$376$	− 1.68489e6i	− 0.614614i
$377$	3.62775e6	1.31457
$378$	0	0
$379$	−4.90816e6	−1.75518	−0.877588	−	0.479415i	$-0.840849\pi$
$379$	−4.90816e6	−1.75518	−0.877588	+	0.479415i	$0.840849\pi$
$380$	− 385513.i	− 0.136956i
$381$	0	0
$382$	−2.32781e6	−0.816185
$383$	3.68615e6	1.28403	0.642017	−	0.766691i	$-0.278098\pi$
$383$	3.68615e6	1.28403	0.642017	+	0.766691i	$0.278098\pi$
$384$	0	0
$385$	0	0
$386$	− 810981.i	− 0.277040i
$387$	0	0
$388$	− 1.83293e6i	− 0.618110i
$389$	2.09713e6i	0.702671i	0.936250	+	0.351336i	$0.114272\pi$
$389$	2.09713e6i	0.702671i	−0.936250	+	0.351336i	$0.885728\pi$
$390$	0	0
$391$	240784.i	0.0796498i
$392$	0	0
$393$	0	0
$394$	1.71764e6	0.557432
$395$	2.07863e6	0.670324
$396$	0	0
$397$	2.27896e6i	0.725706i	0.931846	+	0.362853i	$0.118197\pi$
$397$	2.27896e6i	0.725706i	−0.931846	+	0.362853i	$0.881803\pi$
$398$	3.04230e6	0.962708
$399$	0	0
$400$	21221.9	0.00663186
$401$	− 758406.i	− 0.235527i	−0.993042	−	0.117764i	$-0.962428\pi$
$401$	− 758406.i	− 0.235527i	0.993042	−	0.117764i	$-0.0375725\pi$
$402$	0	0
$403$	−570510.	−0.174985
$404$	−3.13140e6	−0.954520
$405$	0	0
$406$	0	0
$407$	2.05752e6i	0.615683i
$408$	0	0
$409$	5.93180e6i	1.75339i	0.481047	+	0.876695i	$0.340257\pi$
$409$	5.93180e6i	1.75339i	−0.481047	+	0.876695i	$0.659743\pi$
$410$	1.51707e6i	0.445705i
$411$	0	0
$412$	− 870071.i	− 0.252529i
$413$	0	0
$414$	0	0
$415$	−1.61575e6	−0.460527
$416$	−5.09574e6	−1.44369
$417$	0	0
$418$	648230.i	0.181463i
$419$	4.42360e6	1.23095	0.615475	−	0.788156i	$-0.288964\pi$
$419$	4.42360e6	1.23095	0.615475	+	0.788156i	$0.288964\pi$
$420$	0	0
$421$	−582124.	−0.160070	−0.0800350	−	0.996792i	$-0.525503\pi$
$421$	−582124.	−0.160070	−0.0800350	+	0.996792i	$0.525503\pi$
$422$	139091.i	0.0380205i
$423$	0	0
$424$	152113.	0.0410915
$425$	−785202.	−0.210867
$426$	0	0
$427$	0	0
$428$	− 4.21184e6i	− 1.11138i
$429$	0	0
$430$	− 1.14258e6i	− 0.297999i
$431$	− 2.43223e6i	− 0.630683i	−0.948978	−	0.315341i	$-0.897881\pi$
$431$	− 2.43223e6i	− 0.630683i	0.948978	−	0.315341i	$-0.102119\pi$
$432$	0	0
$433$	− 1.26435e6i	− 0.324076i	−0.986785	−	0.162038i	$-0.948193\pi$
$433$	− 1.26435e6i	− 0.324076i	0.986785	−	0.162038i	$-0.0518066\pi$
$434$	0	0
$435$	0	0
$436$	−80053.0	−0.0201679
$437$	495076.	0.124013
$438$	0	0
$439$	1.45182e6i	0.359544i	0.983708	+	0.179772i	$0.0575360\pi$
$439$	1.45182e6i	0.359544i	−0.983708	+	0.179772i	$0.942464\pi$
$440$	−1.36348e6	−0.335752
$441$	0	0
$442$	−1.04657e6	−0.254807
$443$	− 1.61997e6i	− 0.392191i	−0.980585	−	0.196096i	$-0.937174\pi$
$443$	− 1.61997e6i	− 0.392191i	0.980585	−	0.196096i	$-0.0628263\pi$
$444$	0	0
$445$	2.82140e6	0.675405
$446$	683441.	0.162691
$447$	0	0
$448$	0	0
$449$	− 694748.i	− 0.162634i	−0.996688	−	0.0813171i	$-0.974087\pi$
$449$	− 694748.i	− 0.162634i	0.996688	−	0.0813171i	$-0.0259126\pi$
$450$	0	0
$451$	4.05841e6i	0.939538i
$452$	4.34250e6i	0.999755i
$453$	0	0
$454$	− 566923.i	− 0.129088i
$455$	0	0
$456$	0	0
$457$	2.89814e6	0.649125	0.324562	−	0.945864i	$-0.394783\pi$
$457$	2.89814e6	0.649125	0.324562	+	0.945864i	$0.394783\pi$
$458$	1.17704e6	0.262196
$459$	0	0
$460$	396166.i	0.0872936i
$461$	−8.94894e6	−1.96119	−0.980594	−	0.196048i	$-0.937189\pi$
$461$	−8.94894e6	−1.96119	−0.980594	+	0.196048i	$0.937189\pi$
$462$	0	0
$463$	1.35866e6	0.294549	0.147275	−	0.989096i	$-0.452950\pi$
$463$	1.35866e6	0.294549	0.147275	+	0.989096i	$0.452950\pi$
$464$	− 37519.9i	− 0.00809035i
$465$	0	0
$466$	−1.31280e6	−0.280049
$467$	3.01191e6	0.639072	0.319536	−	0.947574i	$-0.396473\pi$
$467$	3.01191e6	0.639072	0.319536	+	0.947574i	$0.396473\pi$
$468$	0	0
$469$	0	0
$470$	− 922106.i	− 0.192547i
$471$	0	0
$472$	− 2.55803e6i	− 0.528507i
$473$	− 3.05657e6i	− 0.628176i
$474$	0	0
$475$	1.61446e6i	0.328316i
$476$	0	0
$477$	0	0
$478$	−2.85300e6	−0.571126
$479$	−7.28818e6	−1.45138	−0.725689	−	0.688023i	$-0.758479\pi$
$479$	−7.28818e6	−1.45138	−0.725689	+	0.688023i	$0.758479\pi$
$480$	0	0
$481$	6.83011e6i	1.34606i
$482$	−2.76157e6	−0.541426
$483$	0	0
$484$	1.77689e6	0.344784
$485$	− 2.63676e6i	− 0.508998i
$486$	0	0
$487$	4.82888e6	0.922622	0.461311	−	0.887238i	$-0.347379\pi$
$487$	4.82888e6	0.922622	0.461311	+	0.887238i	$0.347379\pi$
$488$	−4.92598e6	−0.936361
$489$	0	0
$490$	0	0
$491$	7.22790e6i	1.35303i	0.736428	+	0.676516i	$0.236511\pi$
$491$	7.22790e6i	1.35303i	−0.736428	+	0.676516i	$0.763489\pi$
$492$	0	0
$493$	1.38822e6i	0.257241i
$494$	2.15186e6i	0.396731i
$495$	0	0
$496$	5900.48i	0.00107692i
$497$	0	0
$498$	0	0
$499$	6.14964e6	1.10560	0.552800	−	0.833314i	$-0.313559\pi$
$499$	6.14964e6	1.10560	0.552800	+	0.833314i	$0.313559\pi$
$500$	−3.02760e6	−0.541593
$501$	0	0
$502$	− 4.75362e6i	− 0.841909i
$503$	−5.02424e6	−0.885422	−0.442711	−	0.896664i	$-0.645983\pi$
$503$	−5.02424e6	−0.885422	−0.442711	+	0.896664i	$0.645983\pi$
$504$	0	0
$505$	−4.50468e6	−0.786023
$506$	− 666142.i	− 0.115662i
$507$	0	0
$508$	−5.69965e6	−0.979916
$509$	−2.18309e6	−0.373489	−0.186745	−	0.982409i	$-0.559794\pi$
$509$	−2.18309e6	−0.373489	−0.186745	+	0.982409i	$0.559794\pi$
$510$	0	0
$511$	0	0
$512$	105698.i	0.0178193i
$513$	0	0
$514$	− 2.39306e6i	− 0.399526i
$515$	− 1.25164e6i	− 0.207952i
$516$	0	0
$517$	− 2.46677e6i	− 0.405885i
$518$	0	0
$519$	0	0
$520$	−4.52621e6	−0.734051
$521$	1.42952e6	0.230725	0.115363	−	0.993323i	$-0.463197\pi$
$521$	1.42952e6	0.230725	0.115363	+	0.993323i	$0.463197\pi$
$522$	0	0
$523$	− 8.53728e6i	− 1.36479i	−0.730984	−	0.682394i	$-0.760939\pi$
$523$	− 8.53728e6i	− 1.36479i	0.730984	−	0.682394i	$-0.239061\pi$
$524$	−3.31188e6	−0.526921
$525$	0	0
$526$	−5.91847e6	−0.932706
$527$	− 218315.i	− 0.0342418i
$528$	0	0
$529$	5.92759e6	0.920956
$530$	83248.3	0.0128732
$531$	0	0
$532$	0	0
$533$	1.34722e7i	2.05410i
$534$	0	0
$535$	− 6.05895e6i	− 0.915194i
$536$	6.52554e6i	0.981080i
$537$	0	0
$538$	− 7.19730e6i	− 1.07205i
$539$	0	0
$540$	0	0
$541$	8.08384e6	1.18747	0.593737	−	0.804659i	$-0.297652\pi$
$541$	8.08384e6	1.18747	0.593737	+	0.804659i	$0.297652\pi$
$542$	−3.33514e6	−0.487659
$543$	0	0
$544$	− 1.94997e6i	− 0.282508i
$545$	−115160.	−0.0166078
$546$	0	0
$547$	−1.00382e7	−1.43446	−0.717230	−	0.696836i	$-0.754590\pi$
$547$	−1.00382e7	−1.43446	−0.717230	+	0.696836i	$0.754590\pi$
$548$	2.12441e6i	0.302195i
$549$	0	0
$550$	2.17231e6	0.306207
$551$	2.85433e6	0.400520
$552$	0	0
$553$	0	0
$554$	− 899757.i	− 0.124552i
$555$	0	0
$556$	− 8.26430e6i	− 1.13375i
$557$	− 2.76036e6i	− 0.376989i	−0.982074	−	0.188494i	$-0.939639\pi$
$557$	− 2.76036e6i	− 0.376989i	0.982074	−	0.188494i	$-0.0603607\pi$
$558$	0	0
$559$	− 1.01465e7i	− 1.37337i
$560$	0	0
$561$	0	0
$562$	2.10332e6	0.280908
$563$	1.13062e7	1.50330	0.751649	−	0.659563i	$-0.229259\pi$
$563$	1.13062e7	1.50330	0.751649	+	0.659563i	$0.229259\pi$
$564$	0	0
$565$	6.24690e6i	0.823273i
$566$	−2.52629e6	−0.331469
$567$	0	0
$568$	9.04579e6	1.17646
$569$	− 1.04122e7i	− 1.34823i	−0.738627	−	0.674114i	$-0.764525\pi$
$569$	− 1.04122e7i	− 1.34823i	0.738627	−	0.674114i	$-0.235475\pi$
$570$	0	0
$571$	−1.17519e7	−1.50841	−0.754203	−	0.656641i	$-0.771977\pi$
$571$	−1.17519e7	−1.50841	−0.754203	+	0.656641i	$0.771977\pi$
$572$	−4.60645e6	−0.588676
$573$	0	0
$574$	0	0
$575$	− 1.65907e6i	− 0.209264i
$576$	0	0
$577$	− 7.27148e6i	− 0.909250i	−0.890683	−	0.454625i	$-0.849773\pi$
$577$	− 7.27148e6i	− 0.909250i	0.890683	−	0.454625i	$-0.150227\pi$
$578$	4.58939e6i	0.571393i
$579$	0	0
$580$	2.28406e6i	0.281928i
$581$	0	0
$582$	0	0
$583$	222702.	0.0271364
$584$	−1.57086e7	−1.90592
$585$	0	0
$586$	− 8.90025e6i	− 1.07068i
$587$	−1.53792e7	−1.84220	−0.921102	−	0.389321i	$-0.872710\pi$
$587$	−1.53792e7	−1.84220	−0.921102	+	0.389321i	$0.872710\pi$
$588$	0	0
$589$	−448879.	−0.0533139
$590$	− 1.39996e6i	− 0.165571i
$591$	0	0
$592$	70640.2	0.00828415
$593$	1.31776e7	1.53886	0.769431	−	0.638730i	$-0.220540\pi$
$593$	1.31776e7	1.53886	0.769431	+	0.638730i	$0.220540\pi$
$594$	0	0
$595$	0	0
$596$	1.96194e6i	0.226240i
$597$	0	0
$598$	− 2.21132e6i	− 0.252871i
$599$	− 1.26631e7i	− 1.44203i	−0.692922	−	0.721013i	$-0.743677\pi$
$599$	− 1.26631e7i	− 1.44203i	0.692922	−	0.721013i	$-0.256323\pi$
$600$	0	0
$601$	− 7.41777e6i	− 0.837697i	−0.908056	−	0.418849i	$-0.862434\pi$
$601$	− 7.41777e6i	− 0.837697i	0.908056	−	0.418849i	$-0.137566\pi$
$602$	0	0
$603$	0	0
$604$	3.39001e6	0.378101
$605$	2.55615e6	0.283921
$606$	0	0
$607$	1.03313e7i	1.13811i	0.822299	+	0.569056i	$0.192691\pi$
$607$	1.03313e7i	1.13811i	−0.822299	+	0.569056i	$0.807309\pi$
$608$	−4.00934e6	−0.439859
$609$	0	0
$610$	−2.69588e6	−0.293344
$611$	− 8.18867e6i	− 0.887382i
$612$	0	0
$613$	−9.88461e6	−1.06245	−0.531225	−	0.847231i	$-0.678268\pi$
$613$	−9.88461e6	−1.06245	−0.531225	+	0.847231i	$0.678268\pi$
$614$	5.32358e6	0.569879
$615$	0	0
$616$	0	0
$617$	1.00797e7i	1.06594i	0.846133	+	0.532971i	$0.178925\pi$
$617$	1.00797e7i	1.06594i	−0.846133	+	0.532971i	$0.821075\pi$
$618$	0	0
$619$	− 126877.i	− 0.0133093i	−0.999978	−	0.00665465i	$-0.997882\pi$
$619$	− 126877.i	− 0.0133093i	0.999978	−	0.00665465i	$-0.00211826\pi$
$620$	− 359198.i	− 0.0375279i
$621$	0	0
$622$	− 9.79143e6i	− 1.01478i
$623$	0	0
$624$	0	0
$625$	2.91340e6	0.298332
$626$	−7.01395e6	−0.715363
$627$	0	0
$628$	− 1.22633e6i	− 0.124082i
$629$	−2.61365e6	−0.263403
$630$	0	0
$631$	6.20470e6	0.620366	0.310183	−	0.950677i	$-0.399610\pi$
$631$	6.20470e6	0.620366	0.310183	+	0.950677i	$0.399610\pi$
$632$	− 1.33479e7i	− 1.32929i
$633$	0	0
$634$	65801.3	0.00650147
$635$	−8.19924e6	−0.806936
$636$	0	0
$637$	0	0
$638$	− 3.84059e6i	− 0.373548i
$639$	0	0
$640$	− 3.17931e6i	− 0.306820i
$641$	4.96269e6i	0.477059i	0.971135	+	0.238529i	$0.0766653\pi$
$641$	4.96269e6i	0.477059i	−0.971135	+	0.238529i	$0.923335\pi$
$642$	0	0
$643$	1.13203e7i	1.07977i	0.841738	+	0.539886i	$0.181533\pi$
$643$	1.13203e7i	1.07977i	−0.841738	+	0.539886i	$0.818467\pi$
$644$	0	0
$645$	0	0
$646$	−823443.	−0.0776340
$647$	6.60661e6	0.620466	0.310233	−	0.950661i	$-0.399593\pi$
$647$	6.60661e6	0.620466	0.310233	+	0.950661i	$0.399593\pi$
$648$	0	0
$649$	− 3.74510e6i	− 0.349021i
$650$	7.21117e6	0.669456
$651$	0	0
$652$	7.23529e6	0.666556
$653$	7.23156e6i	0.663665i	0.943338	+	0.331832i	$0.107667\pi$
$653$	7.23156e6i	0.663665i	−0.943338	+	0.331832i	$0.892333\pi$
$654$	0	0
$655$	−4.76430e6	−0.433906
$656$	139336.	0.0126417
$657$	0	0
$658$	0	0
$659$	2.08607e7i	1.87118i	0.353089	+	0.935590i	$0.385131\pi$
$659$	2.08607e7i	1.87118i	−0.353089	+	0.935590i	$0.614869\pi$
$660$	0	0
$661$	− 981732.i	− 0.0873956i	−0.999045	−	0.0436978i	$-0.986086\pi$
$661$	− 981732.i	− 0.0873956i	0.999045	−	0.0436978i	$-0.0139139\pi$
$662$	5.43208e6i	0.481750i
$663$	0	0
$664$	1.03756e7i	0.913254i
$665$	0	0
$666$	0	0
$667$	−2.93320e6	−0.255286
$668$	−1.04722e7	−0.908026
$669$	0	0
$670$	3.57129e6i	0.307353i
$671$	−7.21191e6	−0.618363
$672$	0	0
$673$	−5.57007e6	−0.474048	−0.237024	−	0.971504i	$-0.576172\pi$
$673$	−5.57007e6	−0.474048	−0.237024	+	0.971504i	$0.576172\pi$
$674$	− 8.85188e6i	− 0.750560i
$675$	0	0
$676$	−7.99585e6	−0.672974
$677$	1.25749e7	1.05447	0.527234	−	0.849720i	$-0.323229\pi$
$677$	1.25749e7	1.05447	0.527234	+	0.849720i	$0.323229\pi$
$678$	0	0
$679$	0	0
$680$	− 1.73203e6i	− 0.143642i
$681$	0	0
$682$	603982.i	0.0497237i
$683$	5.55008e6i	0.455247i	0.973749	+	0.227624i	$0.0730956\pi$
$683$	5.55008e6i	0.455247i	−0.973749	+	0.227624i	$0.926904\pi$
$684$	0	0
$685$	3.05608e6i	0.248850i
$686$	0	0
$687$	0	0
$688$	−104940.	−0.00845223
$689$	739279.	0.0593281
$690$	0	0
$691$	2.33744e6i	0.186228i	0.995655	+	0.0931142i	$0.0296822\pi$
$691$	2.33744e6i	0.186228i	−0.995655	+	0.0931142i	$0.970318\pi$
$692$	5.76744e6	0.457844
$693$	0	0
$694$	−3.15789e6	−0.248885
$695$	− 1.18886e7i	− 0.933618i
$696$	0	0
$697$	−5.15537e6	−0.401956
$698$	1.33462e6	0.103686
$699$	0	0
$700$	0	0
$701$	1.90286e6i	0.146255i	0.997323	+	0.0731276i	$0.0232980\pi$
$701$	1.90286e6i	0.146255i	−0.997323	+	0.0731276i	$0.976702\pi$
$702$	0	0
$703$	5.37395e6i	0.410114i
$704$	5.47229e6i	0.416138i
$705$	0	0
$706$	− 9.44203e6i	− 0.712941i
$707$	0	0
$708$	0	0
$709$	−5.40287e6	−0.403654	−0.201827	−	0.979421i	$-0.564688\pi$
$709$	−5.40287e6	−0.403654	−0.201827	+	0.979421i	$0.564688\pi$
$710$	4.95057e6	0.368561
$711$	0	0
$712$	− 1.81176e7i	− 1.33937i
$713$	461283.	0.0339816
$714$	0	0
$715$	−6.62661e6	−0.484760
$716$	1.35311e7i	0.986397i
$717$	0	0
$718$	1.12682e7	0.815725
$719$	−1.67267e6	−0.120667	−0.0603335	−	0.998178i	$-0.519216\pi$
$719$	−1.67267e6	−0.120667	−0.0603335	+	0.998178i	$0.519216\pi$
$720$	0	0
$721$	0	0
$722$	− 7.00879e6i	− 0.500380i
$723$	0	0
$724$	− 4.73767e6i	− 0.335906i
$725$	− 9.56525e6i	− 0.675851i
$726$	0	0
$727$	− 1.30343e7i	− 0.914640i	−0.889302	−	0.457320i	$-0.848809\pi$
$727$	− 1.30343e7i	− 0.914640i	0.889302	−	0.457320i	$-0.151191\pi$
$728$	0	0
$729$	0	0
$730$	−8.59698e6	−0.597089
$731$	3.88274e6	0.268748
$732$	0	0
$733$	2.33641e7i	1.60616i	0.595871	+	0.803080i	$0.296807\pi$
$733$	2.33641e7i	1.60616i	−0.595871	+	0.803080i	$0.703193\pi$
$734$	7.90512e6	0.541587
$735$	0	0
$736$	4.12013e6	0.280360
$737$	9.55375e6i	0.647896i
$738$	0	0
$739$	2.36683e7	1.59425	0.797124	−	0.603815i	$-0.206354\pi$
$739$	2.36683e7	1.59425	0.797124	+	0.603815i	$0.206354\pi$
$740$	−4.30029e6	−0.288681
$741$	0	0
$742$	0	0
$743$	− 2.56105e7i	− 1.70195i	−0.525209	−	0.850973i	$-0.676013\pi$
$743$	− 2.56105e7i	− 1.70195i	0.525209	−	0.850973i	$-0.323987\pi$
$744$	0	0
$745$	2.82234e6i	0.186303i
$746$	− 3.11795e6i	− 0.205127i
$747$	0	0
$748$	− 1.76273e6i	− 0.115195i
$749$	0	0
$750$	0	0
$751$	2.79835e7	1.81052	0.905258	−	0.424861i	$-0.139677\pi$
$751$	2.79835e7	1.81052	0.905258	+	0.424861i	$0.139677\pi$
$752$	−84691.1	−0.00546127
$753$	0	0
$754$	− 1.27492e7i	− 0.816684i
$755$	4.87670e6	0.311357
$756$	0	0
$757$	−3.34651e6	−0.212253	−0.106126	−	0.994353i	$-0.533845\pi$
$757$	−3.34651e6	−0.212253	−0.106126	+	0.994353i	$0.533845\pi$
$758$	1.72490e7i	1.09041i
$759$	0	0
$760$	−3.56123e6	−0.223649
$761$	−7.21517e6	−0.451632	−0.225816	−	0.974170i	$-0.572505\pi$
$761$	−7.21517e6	−0.451632	−0.225816	+	0.974170i	$0.572505\pi$
$762$	0	0
$763$	0	0
$764$	− 1.30152e7i	− 0.806709i
$765$	0	0
$766$	− 1.29544e7i	− 0.797712i
$767$	− 1.24322e7i	− 0.763061i
$768$	0	0
$769$	2.57995e7i	1.57324i	0.617436	+	0.786621i	$0.288171\pi$
$769$	2.57995e7i	1.57324i	−0.617436	+	0.786621i	$0.711829\pi$
$770$	0	0
$771$	0	0
$772$	4.53434e6	0.273823
$773$	6.39124e6	0.384713	0.192356	−	0.981325i	$-0.438387\pi$
$773$	6.39124e6	0.384713	0.192356	+	0.981325i	$0.438387\pi$
$774$	0	0
$775$	1.50426e6i	0.0899637i
$776$	−1.69319e7	−1.00937
$777$	0	0
$778$	7.37006e6	0.436538
$779$	1.06000e7i	0.625838i
$780$	0	0
$781$	1.32435e7	0.776920
$782$	846197.	0.0494829
$783$	0	0
$784$	0	0
$785$	− 1.76415e6i	− 0.102179i
$786$	0	0
$787$	2.14740e7i	1.23588i	0.786226	+	0.617940i	$0.212032\pi$
$787$	2.14740e7i	1.23588i	−0.786226	+	0.617940i	$0.787968\pi$
$788$	9.60364e6i	0.550960i
$789$	0	0
$790$	− 7.30504e6i	− 0.416442i
$791$	0	0
$792$	0	0
$793$	−2.39406e7	−1.35192
$794$	8.00906e6	0.450848
$795$	0	0
$796$	1.70100e7i	0.951530i
$797$	2.31038e7	1.28836	0.644180	−	0.764874i	$-0.277199\pi$
$797$	2.31038e7	1.28836	0.644180	+	0.764874i	$0.277199\pi$
$798$	0	0
$799$	3.13353e6	0.173647
$800$	1.34359e7i	0.742233i
$801$	0	0
$802$	−2.66530e6	−0.146322
$803$	−2.29982e7	−1.25865
$804$	0	0
$805$	0	0
$806$	2.00497e6i	0.108710i
$807$	0	0
$808$	2.89268e7i	1.55873i
$809$	− 1.60119e6i	− 0.0860146i	−0.999075	−	0.0430073i	$-0.986306\pi$
$809$	− 1.60119e6i	− 0.0860146i	0.999075	−	0.0430073i	$-0.0136939\pi$
$810$	0	0
$811$	7.12295e6i	0.380284i	0.981757	+	0.190142i	$0.0608948\pi$
$811$	7.12295e6i	0.380284i	−0.981757	+	0.190142i	$0.939105\pi$
$812$	0	0
$813$	0	0
$814$	7.23083e6	0.382496
$815$	1.04083e7	0.548892
$816$	0	0
$817$	− 7.98332e6i	− 0.418436i
$818$	2.08464e7	1.08930
$819$	0	0
$820$	−8.48224e6	−0.440530
$821$	750547.i	0.0388616i	0.999811	+	0.0194308i	$0.00618540\pi$
$821$	750547.i	0.0388616i	−0.999811	+	0.0194308i	$0.993815\pi$
$822$	0	0
$823$	9.70928e6	0.499675	0.249837	−	0.968288i	$-0.419623\pi$
$823$	9.70928e6	0.499675	0.249837	+	0.968288i	$0.419623\pi$
$824$	−8.03742e6	−0.412381
$825$	0	0
$826$	0	0
$827$	− 3.48670e6i	− 0.177276i	−0.996064	−	0.0886382i	$-0.971748\pi$
$827$	− 3.48670e6i	− 0.177276i	0.996064	−	0.0886382i	$-0.0282515\pi$
$828$	0	0
$829$	6.14603e6i	0.310605i	0.987867	+	0.155303i	$0.0496352\pi$
$829$	6.14603e6i	0.310605i	−0.987867	+	0.155303i	$0.950365\pi$
$830$	5.67832e6i	0.286105i
$831$	0	0
$832$	1.81658e7i	0.909798i
$833$	0	0
$834$	0	0
$835$	−1.50648e7	−0.747736
$836$	−3.62437e6	−0.179356
$837$	0	0
$838$	− 1.55461e7i	− 0.764734i
$839$	3.95822e6	0.194131	0.0970654	−	0.995278i	$-0.469054\pi$
$839$	3.95822e6	0.194131	0.0970654	+	0.995278i	$0.469054\pi$
$840$	0	0
$841$	3.60000e6	0.175514
$842$	2.04579e6i	0.0994443i
$843$	0	0
$844$	−777683.	−0.0375791
$845$	−1.15024e7	−0.554177
$846$	0	0
$847$	0	0
$848$	− 7645.97i	0 0.000365126i
$849$	0	0
$850$	2.75947e6i	0.131002i
$851$	− 5.52245e6i	− 0.261401i
$852$	0	0
$853$	2.02185e7i	0.951430i	0.879599	+	0.475715i	$0.157811\pi$
$853$	2.02185e7i	0.951430i	−0.879599	+	0.475715i	$0.842189\pi$
$854$	0	0
$855$	0	0
$856$	−3.89075e7	−1.81489
$857$	−6.04169e6	−0.281000	−0.140500	−	0.990081i	$-0.544871\pi$
$857$	−6.04169e6	−0.281000	−0.140500	+	0.990081i	$0.544871\pi$
$858$	0	0
$859$	9.28757e6i	0.429457i	0.976674	+	0.214728i	$0.0688866\pi$
$859$	9.28757e6i	0.429457i	−0.976674	+	0.214728i	$0.931113\pi$
$860$	6.38835e6	0.294539
$861$	0	0
$862$	−8.54770e6	−0.391815
$863$	2.90904e7i	1.32960i	0.747020	+	0.664802i	$0.231484\pi$
$863$	2.90904e7i	1.32960i	−0.747020	+	0.664802i	$0.768516\pi$
$864$	0	0
$865$	8.29675e6	0.377023
$866$	−4.44335e6	−0.201334
$867$	0	0
$868$	0	0
$869$	− 1.95421e7i	− 0.877853i
$870$	0	0
$871$	3.17145e7i	1.41649i
$872$	739502.i	0.0329343i
$873$	0	0
$874$	− 1.73987e6i	− 0.0770439i
$875$	0	0
$876$	0	0
$877$	1.29616e7	0.569061	0.284531	−	0.958667i	$-0.408162\pi$
$877$	1.29616e7	0.569061	0.284531	+	0.958667i	$0.408162\pi$
$878$	5.10221e6	0.223369
$879$	0	0
$880$	68535.6i	0.00298339i
$881$	1.23775e7	0.537269	0.268635	−	0.963242i	$-0.413428\pi$
$881$	1.23775e7	0.537269	0.268635	+	0.963242i	$0.413428\pi$
$882$	0	0
$883$	9.58916e6	0.413884	0.206942	−	0.978353i	$-0.433649\pi$
$883$	9.58916e6	0.413884	0.206942	+	0.978353i	$0.433649\pi$
$884$	− 5.85155e6i	− 0.251849i
$885$	0	0
$886$	−5.69314e6	−0.243651
$887$	2.17463e7	0.928060	0.464030	−	0.885820i	$-0.346403\pi$
$887$	2.17463e7	0.928060	0.464030	+	0.885820i	$0.346403\pi$
$888$	0	0
$889$	0	0
$890$	− 9.91537e6i	− 0.419599i
$891$	0	0
$892$	3.82124e6i	0.160802i
$893$	− 6.44287e6i	− 0.270365i
$894$	0	0
$895$	1.94652e7i	0.812273i
$896$	0	0
$897$	0	0
$898$	−2.44159e6	−0.101037
$899$	2.65949e6	0.109749
$900$	0	0
$901$	282897.i	0.0116096i
$902$	1.42627e7	0.583693
$903$	0	0
$904$	4.01145e7	1.63260
$905$	− 6.81537e6i	− 0.276610i
$906$	0	0
$907$	2.20628e7	0.890518	0.445259	−	0.895402i	$-0.353112\pi$
$907$	2.20628e7	0.890518	0.445259	+	0.895402i	$0.353112\pi$
$908$	3.16977e6	0.127589
$909$	0	0
$910$	0	0
$911$	− 1.88200e7i	− 0.751318i	−0.926758	−	0.375659i	$-0.877416\pi$
$911$	− 1.88200e7i	− 0.751318i	0.926758	−	0.375659i	$-0.122584\pi$
$912$	0	0
$913$	1.51904e7i	0.603104i
$914$	− 1.01851e7i	− 0.403272i
$915$	0	0
$916$	6.58102e6i	0.259152i
$917$	0	0
$918$	0	0
$919$	−3.53461e7	−1.38055	−0.690276	−	0.723546i	$-0.742511\pi$
$919$	−3.53461e7	−1.38055	−0.690276	+	0.723546i	$0.742511\pi$
$920$	3.65964e6	0.142551
$921$	0	0
$922$	3.14497e7i	1.21840i
$923$	4.39631e7	1.69857
$924$	0	0
$925$	1.80089e7	0.692041
$926$	− 4.77480e6i	− 0.182990i
$927$	0	0
$928$	2.37543e7	0.905466
$929$	3.39364e7	1.29011	0.645054	−	0.764137i	$-0.276835\pi$
$929$	3.39364e7	1.29011	0.645054	+	0.764137i	$0.276835\pi$
$930$	0	0
$931$	0	0
$932$	− 7.34011e6i	− 0.276798i
$933$	0	0
$934$	− 1.05849e7i	− 0.397027i
$935$	− 2.53578e6i	− 0.0948600i
$936$	0	0
$937$	− 3.18043e7i	− 1.18341i	−0.806153	−	0.591707i	$-0.798454\pi$
$937$	− 3.18043e7i	− 1.18341i	0.806153	−	0.591707i	$-0.201546\pi$
$938$	0	0
$939$	0	0
$940$	5.15566e6	0.190311
$941$	1.00099e7	0.368516	0.184258	−	0.982878i	$-0.441012\pi$
$941$	1.00099e7	0.368516	0.184258	+	0.982878i	$0.441012\pi$
$942$	0	0
$943$	− 1.08929e7i	− 0.398900i
$944$	−128580.	−0.00469615
$945$	0	0
$946$	−1.07418e7	−0.390257
$947$	3.18248e7i	1.15316i	0.817039	+	0.576582i	$0.195614\pi$
$947$	3.18248e7i	1.15316i	−0.817039	+	0.576582i	$0.804386\pi$
$948$	0	0
$949$	−7.63447e7	−2.75178
$950$	5.67377e6	0.203968
$951$	0	0
$952$	0	0
$953$	− 2.33312e7i	− 0.832157i	−0.909329	−	0.416078i	$-0.863404\pi$
$953$	− 2.33312e7i	− 0.832157i	0.909329	−	0.416078i	$-0.136596\pi$
$954$	0	0
$955$	− 1.87230e7i	− 0.664304i
$956$	− 1.59516e7i	− 0.564495i
$957$	0	0
$958$	2.56132e7i	0.901676i
$959$	0	0
$960$	0	0
$961$	2.82109e7	0.985391
$962$	2.40034e7	0.836247
$963$	0	0
$964$	− 1.54404e7i	− 0.535140i
$965$	6.52288e6	0.225487
$966$	0	0
$967$	−5.47163e6	−0.188170	−0.0940850	−	0.995564i	$-0.529993\pi$
$967$	−5.47163e6	−0.188170	−0.0940850	+	0.995564i	$0.529993\pi$
$968$	− 1.64143e7i	− 0.563033i
$969$	0	0
$970$	−9.26649e6	−0.316217
$971$	−3.91175e7	−1.33144	−0.665722	−	0.746200i	$-0.731876\pi$
$971$	−3.91175e7	−1.33144	−0.665722	+	0.746200i	$0.731876\pi$
$972$	0	0
$973$	0	0
$974$	− 1.69704e7i	− 0.573184i
$975$	0	0
$976$	247605.i	0.00832021i
$977$	2.41269e7i	0.808659i	0.914613	+	0.404329i	$0.132495\pi$
$977$	2.41269e7i	0.808659i	−0.914613	+	0.404329i	$0.867505\pi$
$978$	0	0
$979$	− 2.65251e7i	− 0.884506i
$980$	0	0
$981$	0	0
$982$	2.54014e7	0.840578
$983$	4.47040e6	0.147558	0.0737789	−	0.997275i	$-0.476494\pi$
$983$	4.47040e6	0.147558	0.0737789	+	0.997275i	$0.476494\pi$
$984$	0	0
$985$	1.38153e7i	0.453702i
$986$	4.87869e6	0.159812
$987$	0	0
$988$	−1.20314e7	−0.392124
$989$	8.20393e6i	0.266705i
$990$	0	0
$991$	2.39265e6	0.0773918	0.0386959	−	0.999251i	$-0.487680\pi$
$991$	2.39265e6	0.0773918	0.0386959	+	0.999251i	$0.487680\pi$
$992$	−3.73566e6	−0.120528
$993$	0	0
$994$	0	0
$995$	2.44698e7i	0.783561i
$996$	0	0
$997$	− 1.76025e7i	− 0.560837i	−0.959878	−	0.280418i	$-0.909527\pi$
$997$	− 1.76025e7i	− 0.560837i	0.959878	−	0.280418i	$-0.0904732\pi$
$998$	− 2.16120e7i	− 0.686860i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.6.c.c.440.15	✓	40
3.2	odd	2	inner	441.6.c.c.440.26	yes	40
7.6	odd	2	inner	441.6.c.c.440.25	yes	40
21.20	even	2	inner	441.6.c.c.440.16	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.6.c.c.440.15	✓	40	1.1	even	1	trivial
441.6.c.c.440.16	yes	40	21.20	even	2	inner
441.6.c.c.440.25	yes	40	7.6	odd	2	inner
441.6.c.c.440.26	yes	40	3.2	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:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	441.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.51435i q^{2} +19.6494 q^{4} +28.2666 q^{5} -181.514i q^{8} +O(q^{10})\)	\(q-3.51435i q^{2} +19.6494 q^{4} +28.2666 q^{5} -181.514i q^{8} -99.3387i q^{10} -265.746i q^{11} -882.168i q^{13} -9.12380 q^{16} +337.576 q^{17} -694.092i q^{19} +555.420 q^{20} -933.925 q^{22} +713.272i q^{23} -2326.00 q^{25} -3100.25 q^{26} +4112.32i q^{29} -646.714i q^{31} -5776.38i q^{32} -1186.36i q^{34} -7742.41 q^{37} -2439.28 q^{38} -5130.78i q^{40} -15271.7 q^{41} +11501.8 q^{43} -5221.74i q^{44} +2506.69 q^{46} +9282.44 q^{47} +8174.37i q^{50} -17334.0i q^{52} +838.025i q^{53} -7511.74i q^{55} +14452.1 q^{58} +14092.8 q^{59} -27138.3i q^{61} -2272.78 q^{62} -20592.2 q^{64} -24935.9i q^{65} -35950.6 q^{67} +6633.15 q^{68} +49835.3i q^{71} -86542.1i q^{73} +27209.5i q^{74} -13638.5i q^{76} +73536.7 q^{79} -257.899 q^{80} +53670.2i q^{82} -57161.3 q^{83} +9542.12 q^{85} -40421.4i q^{86} -48236.6 q^{88} +99813.8 q^{89} +14015.3i q^{92} -32621.7i q^{94} -19619.6i q^{95} -93281.8i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 640 q^{4}+O(q^{10}) \) 40 * q - 640 * q^4	\( 40 q - 640 q^{4} + 13584 q^{16} - 16208 q^{22} + 14792 q^{25} + 52448 q^{37} - 43168 q^{43} + 58192 q^{46} + 383664 q^{58} - 278432 q^{64} + 228480 q^{67} + 154592 q^{79} - 357824 q^{85} + 1796656 q^{88}+O(q^{100}) \) 40 * q - 640 * q^4 + 13584 * q^16 - 16208 * q^22 + 14792 * q^25 + 52448 * q^37 - 43168 * q^43 + 58192 * q^46 + 383664 * q^58 - 278432 * q^64 + 228480 * q^67 + 154592 * q^79 - 357824 * q^85 + 1796656 * q^88

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more