LMFDB - Embedded newform 441.6.c.b.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:	$24$
Twist minimal:	no (minimal twist has level 63)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.25760i q^{2} +4.35762 q^{4} +22.5074 q^{5} -191.154i q^{8} -118.335i q^{10} +236.195i q^{11} +74.9079i q^{13} -865.567 q^{16} -1412.75 q^{17} +2193.97i q^{19} +98.0787 q^{20} +1241.82 q^{22} +1416.49i q^{23} -2618.42 q^{25} +393.836 q^{26} -2670.05i q^{29} +21.5852i q^{31} -1566.12i q^{32} +7427.69i q^{34} +3123.88 q^{37} +11535.0 q^{38} -4302.38i q^{40} -12756.7 q^{41} -16880.8 q^{43} +1029.25i q^{44} +7447.35 q^{46} -5790.06 q^{47} +13766.6i q^{50} +326.420i q^{52} +2470.33i q^{53} +5316.14i q^{55} -14038.0 q^{58} +22383.0 q^{59} +51675.6i q^{61} +113.486 q^{62} -35932.2 q^{64} +1685.98i q^{65} -45656.1 q^{67} -6156.23 q^{68} -77353.6i q^{71} -7283.42i q^{73} -16424.1i q^{74} +9560.47i q^{76} -95570.1 q^{79} -19481.7 q^{80} +67069.4i q^{82} -94774.5 q^{83} -31797.4 q^{85} +88752.4i q^{86} +45149.6 q^{88} +132480. q^{89} +6172.53i q^{92} +30441.8i q^{94} +49380.6i q^{95} +58525.4i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 608 q^{4} + 7288 q^{16} - 5648 q^{22} + 37704 q^{25} - 41096 q^{37} + 2200 q^{43} + 51424 q^{46} - 308600 q^{58} - 327880 q^{64} - 312648 q^{67} - 331512 q^{79} - 284448 q^{85} - 1164616 q^{88}+O(q^{100}) $ 24 * q - 608 * q^4 + 7288 * q^16 - 5648 * q^22 + 37704 * q^25 - 41096 * q^37 + 2200 * q^43 + 51424 * q^46 - 308600 * q^58 - 327880 * q^64 - 312648 * q^67 - 331512 * q^79 - 284448 * q^85 - 1164616 * q^88

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 $

$2$	− 5.25760i	− 0.929422i	−0.885463	−	0.464711i	$-0.846158\pi$
$2$	− 5.25760i	− 0.929422i	0.885463	−	0.464711i	$-0.153842\pi$
$3$	0	0
$3$	0	0
$4$	4.35762	0.136176
$5$	22.5074	0.402625	0.201312	−	0.979527i	$-0.435479\pi$
$5$	22.5074	0.402625	0.201312	+	0.979527i	$0.435479\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	− 191.154i	− 1.05599i
$9$	0	0
$10$	− 118.335i	− 0.374208i
$11$	236.195i	0.588557i	0.955720	+	0.294279i	$0.0950794\pi$
$11$	236.195i	0.588557i	−0.955720	+	0.294279i	$0.904921\pi$
$12$	0	0
$13$	74.9079i	0.122933i	0.998109	+	0.0614666i	$0.0195778\pi$
$13$	74.9079i	0.122933i	−0.998109	+	0.0614666i	$0.980422\pi$
$14$	0	0
$15$	0	0
$16$	−865.567	−0.845281
$17$	−1412.75	−1.18562	−0.592808	−	0.805344i	$-0.701981\pi$
$17$	−1412.75	−1.18562	−0.592808	+	0.805344i	$0.701981\pi$
$18$	0	0
$19$	2193.97i	1.39427i	0.716941	+	0.697134i	$0.245542\pi$
$19$	2193.97i	1.39427i	−0.716941	+	0.697134i	$0.754458\pi$
$20$	98.0787	0.0548277
$21$	0	0
$22$	1241.82	0.547018
$23$	1416.49i	0.558334i	0.960243	+	0.279167i	$0.0900584\pi$
$23$	1416.49i	0.558334i	−0.960243	+	0.279167i	$0.909942\pi$
$24$	0	0
$25$	−2618.42	−0.837893
$26$	393.836	0.114257
$27$	0	0
$28$	0	0
$29$	− 2670.05i	− 0.589555i	−0.955566	−	0.294777i	$-0.904755\pi$
$29$	− 2670.05i	− 0.589555i	0.955566	−	0.294777i	$-0.0952455\pi$
$30$	0	0
$31$	21.5852i	0.00403415i	0.999998	+	0.00201707i	$0.000642055\pi$
$31$	21.5852i	0.00403415i	−0.999998	+	0.00201707i	$0.999358\pi$
$32$	− 1566.12i	− 0.270364i
$33$	0	0
$34$	7427.69i	1.10194i
$35$	0	0
$36$	0	0
$37$	3123.88	0.375137	0.187568	−	0.982252i	$-0.439939\pi$
$37$	3123.88	0.375137	0.187568	+	0.982252i	$0.439939\pi$
$38$	11535.0	1.29586
$39$	0	0
$40$	− 4302.38i	− 0.425166i
$41$	−12756.7	−1.18516	−0.592580	−	0.805511i	$-0.701891\pi$
$41$	−12756.7	−1.18516	−0.592580	+	0.805511i	$0.701891\pi$
$42$	0	0
$43$	−16880.8	−1.39226	−0.696132	−	0.717914i	$-0.745097\pi$
$43$	−16880.8	−1.39226	−0.696132	+	0.717914i	$0.745097\pi$
$44$	1029.25i	0.0801471i
$45$	0	0
$46$	7447.35	0.518928
$47$	−5790.06	−0.382330	−0.191165	−	0.981558i	$-0.561227\pi$
$47$	−5790.06	−0.382330	−0.191165	+	0.981558i	$0.561227\pi$
$48$	0	0
$49$	0	0
$50$	13766.6i	0.778756i
$51$	0	0
$52$	326.420i	0.0167405i
$53$	2470.33i	0.120799i	0.998174	+	0.0603997i	$0.0192375\pi$
$53$	2470.33i	0.120799i	−0.998174	+	0.0603997i	$0.980762\pi$
$54$	0	0
$55$	5316.14i	0.236968i
$56$	0	0
$57$	0	0
$58$	−14038.0	−0.547945
$59$	22383.0	0.837121	0.418561	−	0.908189i	$-0.362535\pi$
$59$	22383.0	0.837121	0.418561	+	0.908189i	$0.362535\pi$
$60$	0	0
$61$	51675.6i	1.77812i	0.457790	+	0.889060i	$0.348641\pi$
$61$	51675.6i	1.77812i	−0.457790	+	0.889060i	$0.651359\pi$
$62$	113.486	0.00374942
$63$	0	0
$64$	−35932.2	−1.09656
$65$	1685.98i	0.0494960i
$66$	0	0
$67$	−45656.1	−1.24254	−0.621271	−	0.783595i	$-0.713384\pi$
$67$	−45656.1	−1.24254	−0.621271	+	0.783595i	$0.713384\pi$
$68$	−6156.23	−0.161452
$69$	0	0
$70$	0	0
$71$	− 77353.6i	− 1.82110i	−0.413396	−	0.910551i	$-0.635657\pi$
$71$	− 77353.6i	− 1.82110i	0.413396	−	0.910551i	$-0.364343\pi$
$72$	0	0
$73$	− 7283.42i	− 0.159966i	−0.996796	−	0.0799831i	$-0.974513\pi$
$73$	− 7283.42i	− 0.159966i	0.996796	−	0.0799831i	$-0.0254867\pi$
$74$	− 16424.1i	− 0.348660i
$75$	0	0
$76$	9560.47i	0.189865i
$77$	0	0
$78$	0	0
$79$	−95570.1	−1.72288	−0.861438	−	0.507862i	$-0.830436\pi$
$79$	−95570.1	−1.72288	−0.861438	+	0.507862i	$0.830436\pi$
$80$	−19481.7	−0.340331
$81$	0	0
$82$	67069.4i	1.10151i
$83$	−94774.5	−1.51007	−0.755034	−	0.655686i	$-0.772380\pi$
$83$	−94774.5	−1.51007	−0.755034	+	0.655686i	$0.772380\pi$
$84$	0	0
$85$	−31797.4	−0.477358
$86$	88752.4i	1.29400i
$87$	0	0
$88$	45149.6	0.621508
$89$	132480.	1.77287	0.886435	−	0.462854i	$-0.153174\pi$
$89$	132480.	1.77287	0.886435	+	0.462854i	$0.153174\pi$
$90$	0	0
$91$	0	0
$92$	6172.53i	0.0760315i
$93$	0	0
$94$	30441.8i	0.355346i
$95$	49380.6i	0.561367i
$96$	0	0
$97$	58525.4i	0.631560i	0.948832	+	0.315780i	$0.102266\pi$
$97$	58525.4i	0.631560i	−0.948832	+	0.315780i	$0.897734\pi$
$98$	0	0
$99$	0	0
$100$	−11410.1	−0.114101
$101$	118202.	1.15298	0.576488	−	0.817106i	$-0.304423\pi$
$101$	118202.	1.15298	0.576488	+	0.817106i	$0.304423\pi$
$102$	0	0
$103$	66859.7i	0.620971i	0.950578	+	0.310486i	$0.100492\pi$
$103$	66859.7i	0.620971i	−0.950578	+	0.310486i	$0.899508\pi$
$104$	14318.9	0.129816
$105$	0	0
$106$	12988.0	0.112274
$107$	119607.i	1.00994i	0.863136	+	0.504971i	$0.168497\pi$
$107$	119607.i	1.00994i	−0.863136	+	0.504971i	$0.831503\pi$
$108$	0	0
$109$	−86210.6	−0.695015	−0.347508	−	0.937677i	$-0.612972\pi$
$109$	−86210.6	−0.695015	−0.347508	+	0.937677i	$0.612972\pi$
$110$	27950.1	0.220243
$111$	0	0
$112$	0	0
$113$	156109.i	1.15009i	0.818122	+	0.575045i	$0.195016\pi$
$113$	156109.i	1.15009i	−0.818122	+	0.575045i	$0.804984\pi$
$114$	0	0
$115$	31881.6i	0.224799i
$116$	− 11635.0i	− 0.0802829i
$117$	0	0
$118$	− 117681.i	− 0.778039i
$119$	0	0
$120$	0	0
$121$	105263.	0.653600
$122$	271690.	1.65262
$123$	0	0
$124$	94.0600i	0 0.000549352i
$125$	−129269.	−0.739982
$126$	0	0
$127$	−44114.5	−0.242702	−0.121351	−	0.992610i	$-0.538723\pi$
$127$	−44114.5	−0.242702	−0.121351	+	0.992610i	$0.538723\pi$
$128$	138801.i	0.748805i
$129$	0	0
$130$	8864.23	0.0460026
$131$	113531.	0.578013	0.289007	−	0.957327i	$-0.406675\pi$
$131$	113531.	0.578013	0.289007	+	0.957327i	$0.406675\pi$
$132$	0	0
$133$	0	0
$134$	240041.i	1.15485i
$135$	0	0
$136$	270053.i	1.25199i
$137$	− 52803.8i	− 0.240361i	−0.992752	−	0.120180i	$-0.961653\pi$
$137$	− 52803.8i	− 0.240361i	0.992752	−	0.120180i	$-0.0383473\pi$
$138$	0	0
$139$	− 225458.i	− 0.989758i	−0.868962	−	0.494879i	$-0.835212\pi$
$139$	− 225458.i	− 0.989758i	0.868962	−	0.494879i	$-0.164788\pi$
$140$	0	0
$141$	0	0
$142$	−406694.	−1.69257
$143$	−17692.9	−0.0723532
$144$	0	0
$145$	− 60095.9i	− 0.237369i
$146$	−38293.4	−0.148676
$147$	0	0
$148$	13612.7	0.0510844
$149$	193552.i	0.714221i	0.934062	+	0.357111i	$0.116238\pi$
$149$	193552.i	0.714221i	−0.934062	+	0.357111i	$0.883762\pi$
$150$	0	0
$151$	111465.	0.397828	0.198914	−	0.980017i	$-0.436259\pi$
$151$	111465.	0.397828	0.198914	+	0.980017i	$0.436259\pi$
$152$	419386.	1.47233
$153$	0	0
$154$	0	0
$155$	485.827i	0.00162425i
$156$	0	0
$157$	557674.i	1.80564i	0.430017	+	0.902821i	$0.358507\pi$
$157$	557674.i	1.80564i	−0.430017	+	0.902821i	$0.641493\pi$
$158$	502469.i	1.60128i
$159$	0	0
$160$	− 35249.2i	− 0.108855i
$161$	0	0
$162$	0	0
$163$	−340189.	−1.00288	−0.501442	−	0.865191i	$-0.667197\pi$
$163$	−340189.	−1.00288	−0.501442	+	0.865191i	$0.667197\pi$
$164$	−55588.6	−0.161390
$165$	0	0
$166$	498287.i	1.40349i
$167$	443163.	1.22962	0.614812	−	0.788674i	$-0.289232\pi$
$167$	443163.	1.22962	0.614812	+	0.788674i	$0.289232\pi$
$168$	0	0
$169$	365682.	0.984887
$170$	167178.i	0.443667i
$171$	0	0
$172$	−73560.0	−0.189592
$173$	480911.	1.22166	0.610829	−	0.791763i	$-0.290836\pi$
$173$	480911.	1.22166	0.610829	+	0.791763i	$0.290836\pi$
$174$	0	0
$175$	0	0
$176$	− 204443.i	− 0.497496i
$177$	0	0
$178$	− 696530.i	− 1.64774i
$179$	− 418177.i	− 0.975501i	−0.872983	−	0.487750i	$-0.837818\pi$
$179$	− 418177.i	− 0.975501i	0.872983	−	0.487750i	$-0.162182\pi$
$180$	0	0
$181$	− 559881.i	− 1.27028i	−0.772397	−	0.635140i	$-0.780942\pi$
$181$	− 559881.i	− 1.27028i	0.772397	−	0.635140i	$-0.219058\pi$
$182$	0	0
$183$	0	0
$184$	270768.	0.589593
$185$	70310.4	0.151039
$186$	0	0
$187$	− 333685.i	− 0.697803i
$188$	−25230.9	−0.0520640
$189$	0	0
$190$	259623.	0.521747
$191$	722299.i	1.43263i	0.697778	+	0.716314i	$0.254172\pi$
$191$	722299.i	1.43263i	−0.697778	+	0.716314i	$0.745828\pi$
$192$	0	0
$193$	−196615.	−0.379947	−0.189973	−	0.981789i	$-0.560840\pi$
$193$	−196615.	−0.379947	−0.189973	+	0.981789i	$0.560840\pi$
$194$	307703.	0.586986
$195$	0	0
$196$	0	0
$197$	396596.i	0.728087i	0.931382	+	0.364043i	$0.118604\pi$
$197$	396596.i	0.728087i	−0.931382	+	0.364043i	$0.881396\pi$
$198$	0	0
$199$	322237.i	0.576824i	0.957506	+	0.288412i	$0.0931273\pi$
$199$	322237.i	0.576824i	−0.957506	+	0.288412i	$0.906873\pi$
$200$	500520.i	0.884804i
$201$	0	0
$202$	− 621457.i	− 1.07160i
$203$	0	0
$204$	0	0
$205$	−287119.	−0.477175
$206$	351522.	0.577144
$207$	0	0
$208$	− 64837.8i	− 0.103913i
$209$	−518204.	−0.820607
$210$	0	0
$211$	−174621.	−0.270016	−0.135008	−	0.990844i	$-0.543106\pi$
$211$	−174621.	−0.270016	−0.135008	+	0.990844i	$0.543106\pi$
$212$	10764.7i	0.0164499i
$213$	0	0
$214$	628845.	0.938662
$215$	−379943.	−0.560560
$216$	0	0
$217$	0	0
$218$	453261.i	0.645962i
$219$	0	0
$220$	23165.7i	0.0322692i
$221$	− 105826.i	− 0.145751i
$222$	0	0
$223$	1.43212e6i	1.92849i	0.265018	+	0.964243i	$0.414622\pi$
$223$	1.43212e6i	1.92849i	−0.265018	+	0.964243i	$0.585378\pi$
$224$	0	0
$225$	0	0
$226$	820759.	1.06892
$227$	−128532.	−0.165557	−0.0827784	−	0.996568i	$-0.526379\pi$
$227$	−128532.	−0.165557	−0.0827784	+	0.996568i	$0.526379\pi$
$228$	0	0
$229$	− 670848.i	− 0.845348i	−0.906282	−	0.422674i	$-0.861091\pi$
$229$	− 670848.i	− 0.845348i	0.906282	−	0.422674i	$-0.138909\pi$
$230$	167621.	0.208933
$231$	0	0
$232$	−510390.	−0.622561
$233$	− 1.49323e6i	− 1.80193i	−0.433891	−	0.900966i	$-0.642860\pi$
$233$	− 1.49323e6i	− 1.80193i	0.433891	−	0.900966i	$-0.357140\pi$
$234$	0	0
$235$	−130319.	−0.153936
$236$	97536.6	0.113995
$237$	0	0
$238$	0	0
$239$	− 296079.i	− 0.335284i	−0.985848	−	0.167642i	$-0.946385\pi$
$239$	− 296079.i	− 0.335284i	0.985848	−	0.167642i	$-0.0536153\pi$
$240$	0	0
$241$	626032.i	0.694311i	0.937808	+	0.347155i	$0.112852\pi$
$241$	626032.i	0.694311i	−0.937808	+	0.347155i	$0.887148\pi$
$242$	− 553431.i	− 0.607470i
$243$	0	0
$244$	225183.i	0.242136i
$245$	0	0
$246$	0	0
$247$	−164345.	−0.171402
$248$	4126.09	0.00426000
$249$	0	0
$250$	679647.i	0.687755i
$251$	−594368.	−0.595486	−0.297743	−	0.954646i	$-0.596234\pi$
$251$	−594368.	−0.595486	−0.297743	+	0.954646i	$0.596234\pi$
$252$	0	0
$253$	−334568.	−0.328612
$254$	231937.i	0.225572i
$255$	0	0
$256$	−420067.	−0.400607
$257$	−1.80098e6	−1.70089	−0.850445	−	0.526064i	$-0.823667\pi$
$257$	−1.80098e6	−1.70089	−0.850445	+	0.526064i	$0.823667\pi$
$258$	0	0
$259$	0	0
$260$	7346.87i	0.00674014i
$261$	0	0
$262$	− 596903.i	− 0.537218i
$263$	− 666778.i	− 0.594418i	−0.954812	−	0.297209i	$-0.903944\pi$
$263$	− 666778.i	− 0.594418i	0.954812	−	0.297209i	$-0.0960558\pi$
$264$	0	0
$265$	55600.6i	0.0486368i
$266$	0	0
$267$	0	0
$268$	−198952.	−0.169204
$269$	−390595.	−0.329114	−0.164557	−	0.986368i	$-0.552619\pi$
$269$	−390595.	−0.329114	−0.164557	+	0.986368i	$0.552619\pi$
$270$	0	0
$271$	936917.i	0.774957i	0.921879	+	0.387479i	$0.126654\pi$
$271$	936917.i	0.774957i	−0.921879	+	0.387479i	$0.873346\pi$
$272$	1.22283e6	1.00218
$273$	0	0
$274$	−277621.	−0.223396
$275$	− 618457.i	− 0.493148i
$276$	0	0
$277$	−1.00927e6	−0.790329	−0.395164	−	0.918610i	$-0.629312\pi$
$277$	−1.00927e6	−0.790329	−0.395164	+	0.918610i	$0.629312\pi$
$278$	−1.18537e6	−0.919903
$279$	0	0
$280$	0	0
$281$	− 2.06576e6i	− 1.56068i	−0.625354	−	0.780341i	$-0.715045\pi$
$281$	− 2.06576e6i	− 1.56068i	0.625354	−	0.780341i	$-0.284955\pi$
$282$	0	0
$283$	− 1.35552e6i	− 1.00610i	−0.864258	−	0.503049i	$-0.832211\pi$
$283$	− 1.35552e6i	− 1.00610i	0.864258	−	0.503049i	$-0.167789\pi$
$284$	− 337077.i	− 0.247990i
$285$	0	0
$286$	93022.0i	0.0672467i
$287$	0	0
$288$	0	0
$289$	576013.	0.405684
$290$	−315960.	−0.220616
$291$	0	0
$292$	− 31738.4i	− 0.0217835i
$293$	−2.10771e6	−1.43431	−0.717154	−	0.696915i	$-0.754555\pi$
$293$	−2.10771e6	−1.43431	−0.717154	+	0.696915i	$0.754555\pi$
$294$	0	0
$295$	503784.	0.337046
$296$	− 597141.i	− 0.396139i
$297$	0	0
$298$	1.01762e6	0.663813
$299$	−106106.	−0.0686378
$300$	0	0
$301$	0	0
$302$	− 586037.i	− 0.369750i
$303$	0	0
$304$	− 1.89903e6i	− 1.17855i
$305$	1.16308e6i	0.715916i
$306$	0	0
$307$	2.27569e6i	1.37806i	0.724733	+	0.689029i	$0.241963\pi$
$307$	2.27569e6i	1.37806i	−0.724733	+	0.689029i	$0.758037\pi$
$308$	0	0
$309$	0	0
$310$	2554.28	0.00150961
$311$	−2.82683e6	−1.65729	−0.828644	−	0.559775i	$-0.810887\pi$
$311$	−2.82683e6	−1.65729	−0.828644	+	0.559775i	$0.810887\pi$
$312$	0	0
$313$	− 776899.i	− 0.448233i	−0.974562	−	0.224117i	$-0.928050\pi$
$313$	− 776899.i	− 0.448233i	0.974562	−	0.224117i	$-0.0719496\pi$
$314$	2.93203e6	1.67820
$315$	0	0
$316$	−416458.	−0.234614
$317$	− 1.16819e6i	− 0.652930i	−0.945209	−	0.326465i	$-0.894143\pi$
$317$	− 1.16819e6i	− 0.652930i	0.945209	−	0.326465i	$-0.105857\pi$
$318$	0	0
$319$	630652.	0.346987
$320$	−808740.	−0.441503
$321$	0	0
$322$	0	0
$323$	− 3.09953e6i	− 1.65307i
$324$	0	0
$325$	− 196140.i	− 0.103005i
$326$	1.78858e6i	0.932102i
$327$	0	0
$328$	2.43848e6i	1.25151i
$329$	0	0
$330$	0	0
$331$	2.21079e6	1.10912	0.554558	−	0.832145i	$-0.312887\pi$
$331$	2.21079e6	1.10912	0.554558	+	0.832145i	$0.312887\pi$
$332$	−412991.	−0.205634
$333$	0	0
$334$	− 2.32998e6i	− 1.14284i
$335$	−1.02760e6	−0.500279
$336$	0	0
$337$	−425609.	−0.204144	−0.102072	−	0.994777i	$-0.532547\pi$
$337$	−425609.	−0.204144	−0.102072	+	0.994777i	$0.532547\pi$
$338$	− 1.92261e6i	− 0.915376i
$339$	0	0
$340$	−138561.	−0.0650045
$341$	−5098.31	−0.00237433
$342$	0	0
$343$	0	0
$344$	3.22683e6i	1.47021i
$345$	0	0
$346$	− 2.52844e6i	− 1.13543i
$347$	− 3.48492e6i	− 1.55371i	−0.629680	−	0.776854i	$-0.716814\pi$
$347$	− 3.48492e6i	− 1.55371i	0.629680	−	0.776854i	$-0.283186\pi$
$348$	0	0
$349$	− 2.41916e6i	− 1.06316i	−0.847007	−	0.531582i	$-0.821598\pi$
$349$	− 2.41916e6i	− 1.06316i	0.847007	−	0.531582i	$-0.178402\pi$
$350$	0	0
$351$	0	0
$352$	369908.	0.159125
$353$	2.60782e6	1.11388	0.556942	−	0.830551i	$-0.311974\pi$
$353$	2.60782e6	1.11388	0.556942	+	0.830551i	$0.311974\pi$
$354$	0	0
$355$	− 1.74103e6i	− 0.733221i
$356$	577299.	0.241421
$357$	0	0
$358$	−2.19861e6	−0.906651
$359$	− 1.19228e6i	− 0.488249i	−0.969744	−	0.244125i	$-0.921499\pi$
$359$	− 1.19228e6i	− 0.488249i	0.969744	−	0.244125i	$-0.0785006\pi$
$360$	0	0
$361$	−2.33740e6	−0.943984
$362$	−2.94363e6	−1.18063
$363$	0	0
$364$	0	0
$365$	− 163931.i	− 0.0644064i
$366$	0	0
$367$	3.22828e6i	1.25114i	0.780168	+	0.625570i	$0.215134\pi$
$367$	3.22828e6i	1.25114i	−0.780168	+	0.625570i	$0.784866\pi$
$368$	− 1.22607e6i	− 0.471949i
$369$	0	0
$370$	− 369664.i	− 0.140379i
$371$	0	0
$372$	0	0
$373$	1.25369e6	0.466570	0.233285	−	0.972408i	$-0.425052\pi$
$373$	1.25369e6	0.466570	0.233285	+	0.972408i	$0.425052\pi$
$374$	−1.75438e6	−0.648553
$375$	0	0
$376$	1.10679e6i	0.403735i
$377$	200008.	0.0724758
$378$	0	0
$379$	3.09861e6	1.10808	0.554038	−	0.832492i	$-0.313086\pi$
$379$	3.09861e6	1.10808	0.554038	+	0.832492i	$0.313086\pi$
$380$	215182.i	0.0764445i
$381$	0	0
$382$	3.79756e6	1.33152
$383$	−4.42075e6	−1.53992	−0.769962	−	0.638090i	$-0.779725\pi$
$383$	−4.42075e6	−1.53992	−0.769962	+	0.638090i	$0.779725\pi$
$384$	0	0
$385$	0	0
$386$	1.03372e6i	0.353131i
$387$	0	0
$388$	255031.i	0.0860030i
$389$	− 755649.i	− 0.253190i	−0.991954	−	0.126595i	$-0.959595\pi$
$389$	− 755649.i	− 0.253190i	0.991954	−	0.126595i	$-0.0404048\pi$
$390$	0	0
$391$	− 2.00115e6i	− 0.661970i
$392$	0	0
$393$	0	0
$394$	2.08515e6	0.676699
$395$	−2.15104e6	−0.693673
$396$	0	0
$397$	− 2.94135e6i	− 0.936635i	−0.883560	−	0.468317i	$-0.844860\pi$
$397$	− 2.94135e6i	− 0.936635i	0.883560	−	0.468317i	$-0.155140\pi$
$398$	1.69420e6	0.536113
$399$	0	0
$400$	2.26642e6	0.708255
$401$	− 5.66628e6i	− 1.75969i	−0.475257	−	0.879847i	$-0.657645\pi$
$401$	− 5.66628e6i	− 1.75969i	0.475257	−	0.879847i	$-0.342355\pi$
$402$	0	0
$403$	−1616.90	−0.000495930 0
$404$	515078.	0.157007
$405$	0	0
$406$	0	0
$407$	737843.i	0.220789i
$408$	0	0
$409$	315989.i	0.0934036i	0.998909	+	0.0467018i	$0.0148711\pi$
$409$	315989.i	0.0934036i	−0.998909	+	0.0467018i	$0.985129\pi$
$410$	1.50956e6i	0.443497i
$411$	0	0
$412$	291349.i	0.0845611i
$413$	0	0
$414$	0	0
$415$	−2.13313e6	−0.607991
$416$	117314.	0.0332367
$417$	0	0
$418$	2.72451e6i	0.762690i
$419$	2.99225e6	0.832651	0.416325	−	0.909216i	$-0.363318\pi$
$419$	2.99225e6	0.832651	0.416325	+	0.909216i	$0.363318\pi$
$420$	0	0
$421$	2.64469e6	0.727227	0.363613	−	0.931550i	$-0.381543\pi$
$421$	2.64469e6	0.727227	0.363613	+	0.931550i	$0.381543\pi$
$422$	918088.i	0.250959i
$423$	0	0
$424$	472212.	0.127562
$425$	3.69917e6	0.993419
$426$	0	0
$427$	0	0
$428$	521200.i	0.137529i
$429$	0	0
$430$	1.99759e6i	0.520996i
$431$	3.72371e6i	0.965569i	0.875739	+	0.482784i	$0.160374\pi$
$431$	3.72371e6i	0.965569i	−0.875739	+	0.482784i	$0.839626\pi$
$432$	0	0
$433$	− 646932.i	− 0.165821i	−0.996557	−	0.0829104i	$-0.973578\pi$
$433$	− 646932.i	− 0.165821i	0.996557	−	0.0829104i	$-0.0264215\pi$
$434$	0	0
$435$	0	0
$436$	−375673.	−0.0946441
$437$	−3.10774e6	−0.778468
$438$	0	0
$439$	− 4.70178e6i	− 1.16440i	−0.813047	−	0.582198i	$-0.802193\pi$
$439$	− 4.70178e6i	− 1.16440i	0.813047	−	0.582198i	$-0.197807\pi$
$440$	1.01620e6	0.250235
$441$	0	0
$442$	−556393.	−0.135465
$443$	7.30285e6i	1.76800i	0.467484	+	0.884002i	$0.345161\pi$
$443$	7.30285e6i	1.76800i	−0.467484	+	0.884002i	$0.654839\pi$
$444$	0	0
$445$	2.98179e6	0.713801
$446$	7.52951e6	1.79238
$447$	0	0
$448$	0	0
$449$	511884.i	0.119827i	0.998204	+	0.0599136i	$0.0190825\pi$
$449$	511884.i	0.119827i	−0.998204	+	0.0599136i	$0.980917\pi$
$450$	0	0
$451$	− 3.01306e6i	− 0.697535i
$452$	680264.i	0.156614i
$453$	0	0
$454$	675771.i	0.153872i
$455$	0	0
$456$	0	0
$457$	2.11982e6	0.474797	0.237399	−	0.971412i	$-0.423705\pi$
$457$	2.11982e6	0.474797	0.237399	+	0.971412i	$0.423705\pi$
$458$	−3.52705e6	−0.785685
$459$	0	0
$460$	138928.i	0.0306122i
$461$	−5.89721e6	−1.29239	−0.646197	−	0.763171i	$-0.723641\pi$
$461$	−5.89721e6	−1.29239	−0.646197	+	0.763171i	$0.723641\pi$
$462$	0	0
$463$	3.96682e6	0.859982	0.429991	−	0.902833i	$-0.358517\pi$
$463$	3.96682e6	0.859982	0.429991	+	0.902833i	$0.358517\pi$
$464$	2.31111e6i	0.498339i
$465$	0	0
$466$	−7.85083e6	−1.67475
$467$	−3.70636e6	−0.786421	−0.393210	−	0.919449i	$-0.628636\pi$
$467$	−3.70636e6	−0.786421	−0.393210	+	0.919449i	$0.628636\pi$
$468$	0	0
$469$	0	0
$470$	685167.i	0.143071i
$471$	0	0
$472$	− 4.27860e6i	− 0.883989i
$473$	− 3.98715e6i	− 0.819427i
$474$	0	0
$475$	− 5.74472e6i	− 1.16825i
$476$	0	0
$477$	0	0
$478$	−1.55667e6	−0.311620
$479$	5.95995e6	1.18687	0.593436	−	0.804881i	$-0.297771\pi$
$479$	5.95995e6	1.18687	0.593436	+	0.804881i	$0.297771\pi$
$480$	0	0
$481$	234003.i	0.0461167i
$482$	3.29143e6	0.645308
$483$	0	0
$484$	458696.	0.0890043
$485$	1.31725e6i	0.254282i
$486$	0	0
$487$	2.71294e6	0.518345	0.259172	−	0.965831i	$-0.416550\pi$
$487$	2.71294e6	0.518345	0.259172	+	0.965831i	$0.416550\pi$
$488$	9.87800e6	1.87767
$489$	0	0
$490$	0	0
$491$	− 3.15074e6i	− 0.589806i	−0.955527	−	0.294903i	$-0.904713\pi$
$491$	− 3.15074e6i	− 0.589806i	0.955527	−	0.294903i	$-0.0952873\pi$
$492$	0	0
$493$	3.77212e6i	0.698985i
$494$	864063.i	0.159305i
$495$	0	0
$496$	− 18683.4i	− 0.00340999i
$497$	0	0
$498$	0	0
$499$	−1.76799e6	−0.317854	−0.158927	−	0.987290i	$-0.550803\pi$
$499$	−1.76799e6	−0.317854	−0.158927	+	0.987290i	$0.550803\pi$
$500$	−563307.	−0.100767
$501$	0	0
$502$	3.12495e6i	0.553457i
$503$	696922.	0.122819	0.0614093	−	0.998113i	$-0.480440\pi$
$503$	696922.	0.122819	0.0614093	+	0.998113i	$0.480440\pi$
$504$	0	0
$505$	2.66041e6	0.464217
$506$	1.75903e6i	0.305419i
$507$	0	0
$508$	−192234.	−0.0330500
$509$	−4.88569e6	−0.835856	−0.417928	−	0.908480i	$-0.637243\pi$
$509$	−4.88569e6	−0.835856	−0.417928	+	0.908480i	$0.637243\pi$
$510$	0	0
$511$	0	0
$512$	6.65019e6i	1.12114i
$513$	0	0
$514$	9.46884e6i	1.58084i
$515$	1.50484e6i	0.250019i
$516$	0	0
$517$	− 1.36758e6i	− 0.225023i
$518$	0	0
$519$	0	0
$520$	322282.	0.0522670
$521$	9.01349e6	1.45478	0.727392	−	0.686222i	$-0.240732\pi$
$521$	9.01349e6	1.45478	0.727392	+	0.686222i	$0.240732\pi$
$522$	0	0
$523$	1.50399e6i	0.240431i	0.992748	+	0.120216i	$0.0383586\pi$
$523$	1.50399e6i	0.240431i	−0.992748	+	0.120216i	$0.961641\pi$
$524$	494726.	0.0787112
$525$	0	0
$526$	−3.50565e6	−0.552465
$527$	− 30494.5i	− 0.00478295i
$528$	0	0
$529$	4.42990e6	0.688263
$530$	292326.	0.0452041
$531$	0	0
$532$	0	0
$533$	− 955574.i	− 0.145696i
$534$	0	0
$535$	2.69204e6i	0.406628i
$536$	8.72733e6i	1.31211i
$537$	0	0
$538$	2.05359e6i	0.305885i
$539$	0	0
$540$	0	0
$541$	−6.18531e6	−0.908590	−0.454295	−	0.890851i	$-0.650109\pi$
$541$	−6.18531e6	−0.908590	−0.454295	+	0.890851i	$0.650109\pi$
$542$	4.92594e6	0.720262
$543$	0	0
$544$	2.21253e6i	0.320548i
$545$	−1.94038e6	−0.279830
$546$	0	0
$547$	−1.09172e7	−1.56007	−0.780034	−	0.625737i	$-0.784798\pi$
$547$	−1.09172e7	−1.56007	−0.780034	+	0.625737i	$0.784798\pi$
$548$	− 230099.i	− 0.0327312i
$549$	0	0
$550$	−3.25160e6	−0.458343
$551$	5.85800e6	0.821997
$552$	0	0
$553$	0	0
$554$	5.30634e6i	0.734549i
$555$	0	0
$556$	− 982461.i	− 0.134781i
$557$	− 8.14061e6i	− 1.11178i	−0.831256	−	0.555890i	$-0.812378\pi$
$557$	− 8.14061e6i	− 1.11178i	0.831256	−	0.555890i	$-0.187622\pi$
$558$	0	0
$559$	− 1.26450e6i	− 0.171155i
$560$	0	0
$561$	0	0
$562$	−1.08610e7	−1.45053
$563$	8.23186e6	1.09453	0.547264	−	0.836960i	$-0.315669\pi$
$563$	8.23186e6	1.09453	0.547264	+	0.836960i	$0.315669\pi$
$564$	0	0
$565$	3.51361e6i	0.463055i
$566$	−7.12680e6	−0.935090
$567$	0	0
$568$	−1.47864e7	−1.92306
$569$	9.09214e6i	1.17730i	0.808390	+	0.588648i	$0.200339\pi$
$569$	9.09214e6i	1.17730i	−0.808390	+	0.588648i	$0.799661\pi$
$570$	0	0
$571$	2.11391e6	0.271328	0.135664	−	0.990755i	$-0.456683\pi$
$571$	2.11391e6	0.271328	0.135664	+	0.990755i	$0.456683\pi$
$572$	−77098.7	−0.00985274
$573$	0	0
$574$	0	0
$575$	− 3.70896e6i	− 0.467825i
$576$	0	0
$577$	402348.i	0.0503110i	0.999684	+	0.0251555i	$0.00800809\pi$
$577$	402348.i	0.0503110i	−0.999684	+	0.0251555i	$0.991992\pi$
$578$	− 3.02845e6i	− 0.377051i
$579$	0	0
$580$	− 261875.i	− 0.0323239i
$581$	0	0
$582$	0	0
$583$	−583478.	−0.0710973
$584$	−1.39226e6	−0.168922
$585$	0	0
$586$	1.10815e7i	1.33308i
$587$	−1.06195e7	−1.27207	−0.636034	−	0.771661i	$-0.719426\pi$
$587$	−1.06195e7	−1.27207	−0.636034	+	0.771661i	$0.719426\pi$
$588$	0	0
$589$	−47357.2	−0.00562468
$590$	− 2.64869e6i	− 0.313258i
$591$	0	0
$592$	−2.70392e6	−0.317096
$593$	−6.45752e6	−0.754101	−0.377050	−	0.926193i	$-0.623062\pi$
$593$	−6.45752e6	−0.754101	−0.377050	+	0.926193i	$0.623062\pi$
$594$	0	0
$595$	0	0
$596$	843427.i	0.0972594i
$597$	0	0
$598$	557865.i	0.0637935i
$599$	2.81031e6i	0.320027i	0.987115	+	0.160014i	$0.0511538\pi$
$599$	2.81031e6i	0.320027i	−0.987115	+	0.160014i	$0.948846\pi$
$600$	0	0
$601$	− 33651.1i	− 0.00380025i	−0.999998	−	0.00190013i	$-0.999395\pi$
$601$	− 33651.1i	− 0.00380025i	0.999998	−	0.00190013i	$-0.000604829\pi$
$602$	0	0
$603$	0	0
$604$	485721.	0.0541744
$605$	2.36920e6	0.263156
$606$	0	0
$607$	6.83137e6i	0.752552i	0.926508	+	0.376276i	$0.122796\pi$
$607$	6.83137e6i	0.752552i	−0.926508	+	0.376276i	$0.877204\pi$
$608$	3.43601e6	0.376960
$609$	0	0
$610$	6.11504e6	0.665387
$611$	− 433721.i	− 0.0470011i
$612$	0	0
$613$	−1.04849e7	−1.12698	−0.563488	−	0.826124i	$-0.690541\pi$
$613$	−1.04849e7	−1.12698	−0.563488	+	0.826124i	$0.690541\pi$
$614$	1.19647e7	1.28080
$615$	0	0
$616$	0	0
$617$	− 8.92914e6i	− 0.944272i	−0.881526	−	0.472136i	$-0.843483\pi$
$617$	− 8.92914e6i	− 0.944272i	0.881526	−	0.472136i	$-0.156517\pi$
$618$	0	0
$619$	1.37028e6i	0.143742i	0.997414	+	0.0718708i	$0.0228969\pi$
$619$	1.37028e6i	0.143742i	−0.997414	+	0.0718708i	$0.977103\pi$
$620$	2117.05i	0 0.000221183i
$621$	0	0
$622$	1.48623e7i	1.54032i
$623$	0	0
$624$	0	0
$625$	5.27303e6	0.539958
$626$	−4.08463e6	−0.416598
$627$	0	0
$628$	2.43013e6i	0.245884i
$629$	−4.41326e6	−0.444768
$630$	0	0
$631$	8.20153e6	0.820014	0.410007	−	0.912082i	$-0.365526\pi$
$631$	8.20153e6	0.820014	0.410007	+	0.912082i	$0.365526\pi$
$632$	1.82686e7i	1.81933i
$633$	0	0
$634$	−6.14189e6	−0.606847
$635$	−992904.	−0.0977177
$636$	0	0
$637$	0	0
$638$	− 3.31572e6i	− 0.322497i
$639$	0	0
$640$	3.12406e6i	0.301488i
$641$	7.38791e6i	0.710193i	0.934830	+	0.355097i	$0.115552\pi$
$641$	7.38791e6i	0.710193i	−0.934830	+	0.355097i	$0.884448\pi$
$642$	0	0
$643$	− 8.80653e6i	− 0.839996i	−0.907525	−	0.419998i	$-0.862031\pi$
$643$	− 8.80653e6i	− 0.839996i	0.907525	−	0.419998i	$-0.137969\pi$
$644$	0	0
$645$	0	0
$646$	−1.62961e7	−1.53639
$647$	3.50962e6	0.329609	0.164805	−	0.986326i	$-0.447301\pi$
$647$	3.50962e6	0.329609	0.164805	+	0.986326i	$0.447301\pi$
$648$	0	0
$649$	5.28675e6i	0.492694i
$650$	−1.03123e6	−0.0957350
$651$	0	0
$652$	−1.48241e6	−0.136568
$653$	6.83401e6i	0.627181i	0.949558	+	0.313590i	$0.101532\pi$
$653$	6.83401e6i	0.627181i	−0.949558	+	0.313590i	$0.898468\pi$
$654$	0	0
$655$	2.55530e6	0.232722
$656$	1.10417e7	1.00179
$657$	0	0
$658$	0	0
$659$	− 65765.0i	− 0.00589904i	−0.999996	−	0.00294952i	$-0.999061\pi$
$659$	− 65765.0i	− 0.00589904i	0.999996	−	0.00294952i	$-0.000938862\pi$
$660$	0	0
$661$	− 1.74403e7i	− 1.55257i	−0.630384	−	0.776283i	$-0.717103\pi$
$661$	− 1.74403e7i	− 1.55257i	0.630384	−	0.776283i	$-0.282897\pi$
$662$	− 1.16234e7i	− 1.03084i
$663$	0	0
$664$	1.81165e7i	1.59461i
$665$	0	0
$666$	0	0
$667$	3.78210e6	0.329169
$668$	1.93113e6	0.167445
$669$	0	0
$670$	5.40271e6i	0.464970i
$671$	−1.22055e7	−1.04653
$672$	0	0
$673$	474954.	0.0404216	0.0202108	−	0.999796i	$-0.493566\pi$
$673$	474954.	0.0404216	0.0202108	+	0.999796i	$0.493566\pi$
$674$	2.23768e6i	0.189736i
$675$	0	0
$676$	1.59350e6	0.134118
$677$	1.32237e7	1.10887	0.554437	−	0.832226i	$-0.312934\pi$
$677$	1.32237e7	1.10887	0.554437	+	0.832226i	$0.312934\pi$
$678$	0	0
$679$	0	0
$680$	6.07820e6i	0.504084i
$681$	0	0
$682$	26804.9i	0.00220675i
$683$	9.98423e6i	0.818960i	0.912319	+	0.409480i	$0.134290\pi$
$683$	9.98423e6i	0.818960i	−0.912319	+	0.409480i	$0.865710\pi$
$684$	0	0
$685$	− 1.18848e6i	− 0.0967752i
$686$	0	0
$687$	0	0
$688$	1.46115e7	1.17685
$689$	−185047.	−0.0148502
$690$	0	0
$691$	2.82972e6i	0.225449i	0.993626	+	0.112724i	$0.0359577\pi$
$691$	2.82972e6i	0.225449i	−0.993626	+	0.112724i	$0.964042\pi$
$692$	2.09563e6	0.166360
$693$	0	0
$694$	−1.83223e7	−1.44405
$695$	− 5.07448e6i	− 0.398501i
$696$	0	0
$697$	1.80220e7	1.40514
$698$	−1.27190e7	−0.988128
$699$	0	0
$700$	0	0
$701$	− 1.27474e7i	− 0.979772i	−0.871787	−	0.489886i	$-0.837038\pi$
$701$	− 1.27474e7i	− 0.979772i	0.871787	−	0.489886i	$-0.162962\pi$
$702$	0	0
$703$	6.85368e6i	0.523041i
$704$	− 8.48700e6i	− 0.645390i
$705$	0	0
$706$	− 1.37109e7i	− 1.03527i
$707$	0	0
$708$	0	0
$709$	−1.69625e7	−1.26729	−0.633643	−	0.773625i	$-0.718441\pi$
$709$	−1.69625e7	−1.26729	−0.633643	+	0.773625i	$0.718441\pi$
$710$	−9.15364e6	−0.681472
$711$	0	0
$712$	− 2.53242e7i	− 1.87213i
$713$	−30575.2	−0.00225240
$714$	0	0
$715$	−398221.	−0.0291312
$716$	− 1.82226e6i	− 0.132839i
$717$	0	0
$718$	−6.26853e6	−0.453789
$719$	−9.80354e6	−0.707230	−0.353615	−	0.935391i	$-0.615048\pi$
$719$	−9.80354e6	−0.707230	−0.353615	+	0.935391i	$0.615048\pi$
$720$	0	0
$721$	0	0
$722$	1.22891e7i	0.877359i
$723$	0	0
$724$	− 2.43975e6i	− 0.172981i
$725$	6.99130e6i	0.493984i
$726$	0	0
$727$	− 146850.i	− 0.0103047i	−0.999987	−	0.00515236i	$-0.998360\pi$
$727$	− 146850.i	− 0.0103047i	0.999987	−	0.00515236i	$-0.00164006\pi$
$728$	0	0
$729$	0	0
$730$	−861884.	−0.0598607
$731$	2.38484e7	1.65069
$732$	0	0
$733$	852954.i	0.0586362i	0.999570	+	0.0293181i	$0.00933358\pi$
$733$	852954.i	0.0586362i	−0.999570	+	0.0293181i	$0.990666\pi$
$734$	1.69730e7	1.16284
$735$	0	0
$736$	2.21839e6	0.150953
$737$	− 1.07837e7i	− 0.731308i
$738$	0	0
$739$	−1.01622e7	−0.684506	−0.342253	−	0.939608i	$-0.611190\pi$
$739$	−1.01622e7	−0.684506	−0.342253	+	0.939608i	$0.611190\pi$
$740$	306386.	0.0205679
$741$	0	0
$742$	0	0
$743$	3.62229e6i	0.240719i	0.992730	+	0.120360i	$0.0384048\pi$
$743$	3.62229e6i	0.240719i	−0.992730	+	0.120360i	$0.961595\pi$
$744$	0	0
$745$	4.35636e6i	0.287563i
$746$	− 6.59139e6i	− 0.433640i
$747$	0	0
$748$	− 1.45407e6i	− 0.0950236i
$749$	0	0
$750$	0	0
$751$	2.16640e7	1.40165	0.700824	−	0.713335i	$-0.252816\pi$
$751$	2.16640e7	1.40165	0.700824	+	0.713335i	$0.252816\pi$
$752$	5.01169e6	0.323176
$753$	0	0
$754$	− 1.05156e6i	− 0.0673606i
$755$	2.50878e6	0.160175
$756$	0	0
$757$	1.93250e7	1.22569	0.612845	−	0.790204i	$-0.290025\pi$
$757$	1.93250e7	1.22569	0.612845	+	0.790204i	$0.290025\pi$
$758$	− 1.62913e7i	− 1.02987i
$759$	0	0
$760$	9.43928e6	0.592796
$761$	−2.14706e7	−1.34395	−0.671974	−	0.740575i	$-0.734553\pi$
$761$	−2.14706e7	−1.34395	−0.671974	+	0.740575i	$0.734553\pi$
$762$	0	0
$763$	0	0
$764$	3.14750e6i	0.195089i
$765$	0	0
$766$	2.32426e7i	1.43124i
$767$	1.67666e6i	0.102910i
$768$	0	0
$769$	− 2.72925e7i	− 1.66428i	−0.554562	−	0.832142i	$-0.687114\pi$
$769$	− 2.72925e7i	− 1.66428i	0.554562	−	0.832142i	$-0.312886\pi$
$770$	0	0
$771$	0	0
$772$	−856771.	−0.0517394
$773$	−2.97937e7	−1.79340	−0.896698	−	0.442643i	$-0.854041\pi$
$773$	−2.97937e7	−1.79340	−0.896698	+	0.442643i	$0.854041\pi$
$774$	0	0
$775$	− 56519.0i	− 0.00338018i
$776$	1.11874e7	0.666919
$777$	0	0
$778$	−3.97290e6	−0.235320
$779$	− 2.79877e7i	− 1.65243i
$780$	0	0
$781$	1.82705e7	1.07182
$782$	−1.05213e7	−0.615249
$783$	0	0
$784$	0	0
$785$	1.25518e7i	0.726996i
$786$	0	0
$787$	7.44295e6i	0.428359i	0.976794	+	0.214180i	$0.0687078\pi$
$787$	7.44295e6i	0.428359i	−0.976794	+	0.214180i	$0.931292\pi$
$788$	1.72821e6i	0.0991476i
$789$	0	0
$790$	1.13093e7i	0.644715i
$791$	0	0
$792$	0	0
$793$	−3.87091e6	−0.218590
$794$	−1.54644e7	−0.870529
$795$	0	0
$796$	1.40419e6i	0.0785493i
$797$	1.32098e7	0.736632	0.368316	−	0.929701i	$-0.379934\pi$
$797$	1.32098e7	0.736632	0.368316	+	0.929701i	$0.379934\pi$
$798$	0	0
$799$	8.17993e6	0.453297
$800$	4.10074e6i	0.226536i
$801$	0	0
$802$	−2.97911e7	−1.63550
$803$	1.72031e6	0.0941493
$804$	0	0
$805$	0	0
$806$	8501.02i	0 0.000460928i
$807$	0	0
$808$	− 2.25947e7i	− 1.21753i
$809$	− 1.19740e7i	− 0.643232i	−0.946870	−	0.321616i	$-0.895774\pi$
$809$	− 1.19740e7i	− 0.643232i	0.946870	−	0.321616i	$-0.104226\pi$
$810$	0	0
$811$	− 4.67363e6i	− 0.249518i	−0.992187	−	0.124759i	$-0.960184\pi$
$811$	− 4.67363e6i	− 0.249518i	0.992187	−	0.124759i	$-0.0398158\pi$
$812$	0	0
$813$	0	0
$814$	3.87929e6	0.205206
$815$	−7.65677e6	−0.403786
$816$	0	0
$817$	− 3.70359e7i	− 1.94119i
$818$	1.66134e6	0.0868114
$819$	0	0
$820$	−1.25116e6	−0.0649796
$821$	2.46844e7i	1.27810i	0.769166	+	0.639049i	$0.220672\pi$
$821$	2.46844e7i	1.27810i	−0.769166	+	0.639049i	$0.779328\pi$
$822$	0	0
$823$	1.29351e7	0.665689	0.332845	−	0.942982i	$-0.391992\pi$
$823$	1.29351e7	0.665689	0.332845	+	0.942982i	$0.391992\pi$
$824$	1.27805e7	0.655737
$825$	0	0
$826$	0	0
$827$	− 3.64047e6i	− 0.185095i	−0.995708	−	0.0925473i	$-0.970499\pi$
$827$	− 3.64047e6i	− 0.185095i	0.995708	−	0.0925473i	$-0.0295009\pi$
$828$	0	0
$829$	− 1.63208e7i	− 0.824814i	−0.911000	−	0.412407i	$-0.864688\pi$
$829$	− 1.63208e7i	− 0.824814i	0.911000	−	0.412407i	$-0.135312\pi$
$830$	1.12151e7i	0.565080i
$831$	0	0
$832$	− 2.69160e6i	− 0.134804i
$833$	0	0
$834$	0	0
$835$	9.97446e6	0.495077
$836$	−2.25813e6	−0.111747
$837$	0	0
$838$	− 1.57321e7i	− 0.773883i
$839$	997803.	0.0489373	0.0244687	−	0.999701i	$-0.492211\pi$
$839$	997803.	0.0489373	0.0244687	+	0.999701i	$0.492211\pi$
$840$	0	0
$841$	1.33820e7	0.652425
$842$	− 1.39047e7i	− 0.675900i
$843$	0	0
$844$	−760931.	−0.0367696
$845$	8.23055e6	0.396540
$846$	0	0
$847$	0	0
$848$	− 2.13823e6i	− 0.102109i
$849$	0	0
$850$	− 1.94488e7i	− 0.923305i
$851$	4.42494e6i	0.209452i
$852$	0	0
$853$	1.11746e7i	0.525846i	0.964817	+	0.262923i	$0.0846865\pi$
$853$	1.11746e7i	0.525846i	−0.964817	+	0.262923i	$0.915313\pi$
$854$	0	0
$855$	0	0
$856$	2.28633e7	1.06648
$857$	−9.95019e6	−0.462785	−0.231392	−	0.972860i	$-0.574328\pi$
$857$	−9.95019e6	−0.462785	−0.231392	+	0.972860i	$0.574328\pi$
$858$	0	0
$859$	− 1.71599e7i	− 0.793471i	−0.917933	−	0.396736i	$-0.870143\pi$
$859$	− 1.71599e7i	− 0.793471i	0.917933	−	0.396736i	$-0.129857\pi$
$860$	−1.65564e6	−0.0763345
$861$	0	0
$862$	1.95778e7	0.897420
$863$	− 3.39319e6i	− 0.155089i	−0.996989	−	0.0775446i	$-0.975292\pi$
$863$	− 3.39319e6i	− 0.155089i	0.996989	−	0.0775446i	$-0.0247080\pi$
$864$	0	0
$865$	1.08241e7	0.491870
$866$	−3.40131e6	−0.154117
$867$	0	0
$868$	0	0
$869$	− 2.25732e7i	− 1.01401i
$870$	0	0
$871$	− 3.42000e6i	− 0.152750i
$872$	1.64795e7i	0.733926i
$873$	0	0
$874$	1.63392e7i	0.723525i
$875$	0	0
$876$	0	0
$877$	1.70035e7	0.746517	0.373259	−	0.927727i	$-0.378240\pi$
$877$	1.70035e7	0.746517	0.373259	+	0.927727i	$0.378240\pi$
$878$	−2.47201e7	−1.08222
$879$	0	0
$880$	− 4.60148e6i	− 0.200304i
$881$	3.88085e7	1.68456	0.842281	−	0.539038i	$-0.181212\pi$
$881$	3.88085e7	1.68456	0.842281	+	0.539038i	$0.181212\pi$
$882$	0	0
$883$	2.10286e7	0.907628	0.453814	−	0.891096i	$-0.350063\pi$
$883$	2.10286e7	0.907628	0.453814	+	0.891096i	$0.350063\pi$
$884$	− 461150.i	− 0.0198478i
$885$	0	0
$886$	3.83955e7	1.64322
$887$	−5.52936e6	−0.235975	−0.117987	−	0.993015i	$-0.537644\pi$
$887$	−5.52936e6	−0.235975	−0.117987	+	0.993015i	$0.537644\pi$
$888$	0	0
$889$	0	0
$890$	− 1.56771e7i	− 0.663422i
$891$	0	0
$892$	6.24062e6i	0.262613i
$893$	− 1.27032e7i	− 0.533071i
$894$	0	0
$895$	− 9.41208e6i	− 0.392761i
$896$	0	0
$897$	0	0
$898$	2.69128e6	0.111370
$899$	57633.5	0.00237835
$900$	0	0
$901$	− 3.48996e6i	− 0.143222i
$902$	−1.58415e7	−0.648304
$903$	0	0
$904$	2.98409e7	1.21448
$905$	− 1.26015e7i	− 0.511446i
$906$	0	0
$907$	−3.61507e7	−1.45915	−0.729573	−	0.683903i	$-0.760281\pi$
$907$	−3.61507e7	−1.45915	−0.729573	+	0.683903i	$0.760281\pi$
$908$	−560094.	−0.0225448
$909$	0	0
$910$	0	0
$911$	1.01876e7i	0.406701i	0.979106	+	0.203350i	$0.0651831\pi$
$911$	1.01876e7i	0.406701i	−0.979106	+	0.203350i	$0.934817\pi$
$912$	0	0
$913$	− 2.23853e7i	− 0.888761i
$914$	− 1.11452e7i	− 0.441287i
$915$	0	0
$916$	− 2.92330e6i	− 0.115116i
$917$	0	0
$918$	0	0
$919$	−919083.	−0.0358977	−0.0179488	−	0.999839i	$-0.505714\pi$
$919$	−919083.	−0.0358977	−0.0179488	+	0.999839i	$0.505714\pi$
$920$	6.09428e6	0.237385
$921$	0	0
$922$	3.10052e7i	1.20118i
$923$	5.79439e6	0.223874
$924$	0	0
$925$	−8.17961e6	−0.314324
$926$	− 2.08559e7i	− 0.799286i
$927$	0	0
$928$	−4.18160e6	−0.159394
$929$	−3.27601e7	−1.24539	−0.622695	−	0.782465i	$-0.713962\pi$
$929$	−3.27601e7	−1.24539	−0.622695	+	0.782465i	$0.713962\pi$
$930$	0	0
$931$	0	0
$932$	− 6.50694e6i	− 0.245379i
$933$	0	0
$934$	1.94865e7i	0.730916i
$935$	− 7.51039e6i	− 0.280953i
$936$	0	0
$937$	1.54558e7i	0.575101i	0.957766	+	0.287550i	$0.0928409\pi$
$937$	1.54558e7i	0.575101i	−0.957766	+	0.287550i	$0.907159\pi$
$938$	0	0
$939$	0	0
$940$	−567882.	−0.0209623
$941$	−2.76892e7	−1.01938	−0.509691	−	0.860358i	$-0.670240\pi$
$941$	−2.76892e7	−1.01938	−0.509691	+	0.860358i	$0.670240\pi$
$942$	0	0
$943$	− 1.80697e7i	− 0.661716i
$944$	−1.93740e7	−0.707603
$945$	0	0
$946$	−2.09629e7	−0.761593
$947$	− 5.98922e6i	− 0.217018i	−0.994095	−	0.108509i	$-0.965392\pi$
$947$	− 5.98922e6i	− 0.217018i	0.994095	−	0.108509i	$-0.0346076\pi$
$948$	0	0
$949$	545586.	0.0196652
$950$	−3.02035e7	−1.08579
$951$	0	0
$952$	0	0
$953$	4.28775e7i	1.52932i	0.644436	+	0.764658i	$0.277092\pi$
$953$	4.28775e7i	1.52932i	−0.644436	+	0.764658i	$0.722908\pi$
$954$	0	0
$955$	1.62571e7i	0.576812i
$956$	− 1.29020e6i	− 0.0456575i
$957$	0	0
$958$	− 3.13351e7i	− 1.10310i
$959$	0	0
$960$	0	0
$961$	2.86287e7	0.999984
$962$	1.23029e6	0.0428619
$963$	0	0
$964$	2.72801e6i	0.0945482i
$965$	−4.42529e6	−0.152976
$966$	0	0
$967$	−3.89106e7	−1.33814	−0.669071	−	0.743199i	$-0.733308\pi$
$967$	−3.89106e7	−1.33814	−0.669071	+	0.743199i	$0.733308\pi$
$968$	− 2.01214e7i	− 0.690193i
$969$	0	0
$970$	6.92560e6	0.236335
$971$	−3.85822e7	−1.31322	−0.656612	−	0.754228i	$-0.728011\pi$
$971$	−3.85822e7	−1.31322	−0.656612	+	0.754228i	$0.728011\pi$
$972$	0	0
$973$	0	0
$974$	− 1.42636e7i	− 0.481761i
$975$	0	0
$976$	− 4.47287e7i	− 1.50301i
$977$	− 3.55372e7i	− 1.19109i	−0.803320	−	0.595547i	$-0.796935\pi$
$977$	− 3.55372e7i	− 1.19109i	0.803320	−	0.595547i	$-0.203065\pi$
$978$	0	0
$979$	3.12912e7i	1.04344i
$980$	0	0
$981$	0	0
$982$	−1.65653e7	−0.548178
$983$	1.15580e7	0.381505	0.190753	−	0.981638i	$-0.438907\pi$
$983$	1.15580e7	0.381505	0.190753	+	0.981638i	$0.438907\pi$
$984$	0	0
$985$	8.92636e6i	0.293146i
$986$	1.98323e7	0.649652
$987$	0	0
$988$	−716155.	−0.0233407
$989$	− 2.39115e7i	− 0.777348i
$990$	0	0
$991$	−1.85258e7	−0.599230	−0.299615	−	0.954060i	$-0.596858\pi$
$991$	−1.85258e7	−0.599230	−0.299615	+	0.954060i	$0.596858\pi$
$992$	33804.9	0.00109069
$993$	0	0
$994$	0	0
$995$	7.25273e6i	0.232244i
$996$	0	0
$997$	3.17014e7i	1.01004i	0.863107	+	0.505022i	$0.168515\pi$
$997$	3.17014e7i	1.01004i	−0.863107	+	0.505022i	$0.831485\pi$
$998$	9.29537e6i	0.295420i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.6.c.b.440.15		24
3.2	odd	2	inner	441.6.c.b.440.10		24
7.2	even	3		63.6.p.b.17.5	✓	24
7.3	odd	6		63.6.p.b.26.8	yes	24
7.6	odd	2	inner	441.6.c.b.440.9		24
21.2	odd	6		63.6.p.b.17.8	yes	24
21.17	even	6		63.6.p.b.26.5	yes	24
21.20	even	2	inner	441.6.c.b.440.16		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.6.p.b.17.5	✓	24	7.2	even	3
63.6.p.b.17.8	yes	24	21.2	odd	6
63.6.p.b.26.5	yes	24	21.17	even	6
63.6.p.b.26.8	yes	24	7.3	odd	6
441.6.c.b.440.9		24	7.6	odd	2	inner
441.6.c.b.440.10		24	3.2	odd	2	inner
441.6.c.b.440.15		24	1.1	even	1	trivial
441.6.c.b.440.16		24	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)

\(f(q)\)	\(=\)	\(q-5.25760i q^{2} +4.35762 q^{4} +22.5074 q^{5} -191.154i q^{8} -118.335i q^{10} +236.195i q^{11} +74.9079i q^{13} -865.567 q^{16} -1412.75 q^{17} +2193.97i q^{19} +98.0787 q^{20} +1241.82 q^{22} +1416.49i q^{23} -2618.42 q^{25} +393.836 q^{26} -2670.05i q^{29} +21.5852i q^{31} -1566.12i q^{32} +7427.69i q^{34} +3123.88 q^{37} +11535.0 q^{38} -4302.38i q^{40} -12756.7 q^{41} -16880.8 q^{43} +1029.25i q^{44} +7447.35 q^{46} -5790.06 q^{47} +13766.6i q^{50} +326.420i q^{52} +2470.33i q^{53} +5316.14i q^{55} -14038.0 q^{58} +22383.0 q^{59} +51675.6i q^{61} +113.486 q^{62} -35932.2 q^{64} +1685.98i q^{65} -45656.1 q^{67} -6156.23 q^{68} -77353.6i q^{71} -7283.42i q^{73} -16424.1i q^{74} +9560.47i q^{76} -95570.1 q^{79} -19481.7 q^{80} +67069.4i q^{82} -94774.5 q^{83} -31797.4 q^{85} +88752.4i q^{86} +45149.6 q^{88} +132480. q^{89} +6172.53i q^{92} +30441.8i q^{94} +49380.6i q^{95} +58525.4i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 608 q^{4} + 7288 q^{16} - 5648 q^{22} + 37704 q^{25} - 41096 q^{37} + 2200 q^{43} + 51424 q^{46} - 308600 q^{58} - 327880 q^{64} - 312648 q^{67} - 331512 q^{79} - 284448 q^{85} - 1164616 q^{88}+O(q^{100}) \) 24 * q - 608 * q^4 + 7288 * q^16 - 5648 * q^22 + 37704 * q^25 - 41096 * q^37 + 2200 * q^43 + 51424 * q^46 - 308600 * q^58 - 327880 * q^64 - 312648 * q^67 - 331512 * q^79 - 284448 * q^85 - 1164616 * q^88

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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