LMFDB - Embedded newform 441.4.c.a.440.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,4,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, 4, names="a")

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

chi := DirichletCharacter("441.440");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 441 = 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
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:	$26.0198423125$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 48 x^{14} + 1647 x^{12} - 27620 x^{10} + 336765 x^{8} - 1200006 x^{6} + 3242464 x^{4} + \cdots + 810000 $ x^16 - 48x^14 + 1647x^12 - 27620x^10 + 336765x^8 - 1200006x^6 + 3242464x^4 - 1762200*x^2 + 810000
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{8}\cdot 7^{4} $
Twist minimal:	no (minimal twist has level 63)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			440.12
Root			$-1.57646 - 0.910170i$ of defining polynomial
Character	$\chi$	$=$	441.440
Dual form			441.4.c.a.440.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.82034i q^{2} +4.68636 q^{4} +15.0874 q^{5} +23.0935i q^{8} +O(q^{10})$	$q+1.82034i q^{2} +4.68636 q^{4} +15.0874 q^{5} +23.0935i q^{8} +27.4643i q^{10} -9.89034i q^{11} +67.8891i q^{13} -4.54712 q^{16} -70.1373 q^{17} +61.4580i q^{19} +70.7052 q^{20} +18.0038 q^{22} +131.515i q^{23} +102.631 q^{25} -123.581 q^{26} -158.738i q^{29} -76.4814i q^{31} +176.471i q^{32} -127.674i q^{34} +348.682 q^{37} -111.874 q^{38} +348.422i q^{40} +138.909 q^{41} +539.651 q^{43} -46.3497i q^{44} -239.402 q^{46} -223.643 q^{47} +186.823i q^{50} +318.153i q^{52} -530.011i q^{53} -149.220i q^{55} +288.958 q^{58} -542.876 q^{59} +134.197i q^{61} +139.222 q^{62} -357.614 q^{64} +1024.27i q^{65} +320.580 q^{67} -328.689 q^{68} +416.958i q^{71} +545.607i q^{73} +634.719i q^{74} +288.014i q^{76} -322.737 q^{79} -68.6044 q^{80} +252.861i q^{82} +885.170 q^{83} -1058.19 q^{85} +982.349i q^{86} +228.403 q^{88} -1624.62 q^{89} +616.327i q^{92} -407.106i q^{94} +927.244i q^{95} -739.155i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 64 q^{4}+O(q^{10}) $ 16 * q - 64 * q^4	$ 16 q - 64 q^{4} + 376 q^{16} + 528 q^{22} + 40 q^{25} + 2392 q^{37} + 328 q^{43} + 2784 q^{46} + 6744 q^{58} + 5432 q^{64} - 616 q^{67} + 4352 q^{79} - 4608 q^{85} - 1416 q^{88}+O(q^{100}) $ 16 * q - 64 * q^4 + 376 * q^16 + 528 * q^22 + 40 * q^25 + 2392 * q^37 + 328 * q^43 + 2784 * q^46 + 6744 * q^58 + 5432 * q^64 - 616 * q^67 + 4352 * q^79 - 4608 * q^85 - 1416 * 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 $

$2$	1.82034i	0.643587i	0.946810	+	0.321794i	$0.104286\pi$
$2$	1.82034i	0.643587i	−0.946810	+	0.321794i	$0.895714\pi$
$3$	0	0
$3$	0	0
$4$	4.68636	0.585795
$5$	15.0874	1.34946	0.674731	−	0.738064i	$-0.264260\pi$
$5$	15.0874	1.34946	0.674731	+	0.738064i	$0.264260\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	23.0935i	1.02060i
$9$	0	0
$10$	27.4643i	0.868497i
$11$	− 9.89034i	− 0.271096i	−0.990771	−	0.135548i	$-0.956721\pi$
$11$	− 9.89034i	− 0.271096i	0.990771	−	0.135548i	$-0.0432794\pi$
$12$	0	0
$13$	67.8891i	1.44839i	0.689596	+	0.724194i	$0.257788\pi$
$13$	67.8891i	1.44839i	−0.689596	+	0.724194i	$0.742212\pi$
$14$	0	0
$15$	0	0
$16$	−4.54712	−0.0710487
$17$	−70.1373	−1.00064	−0.500318	−	0.865842i	$-0.666783\pi$
$17$	−70.1373	−1.00064	−0.500318	+	0.865842i	$0.666783\pi$
$18$	0	0
$19$	61.4580i	0.742075i	0.928618	+	0.371038i	$0.120998\pi$
$19$	61.4580i	0.742075i	−0.928618	+	0.371038i	$0.879002\pi$
$20$	70.7052	0.790508
$21$	0	0
$22$	18.0038	0.174474
$23$	131.515i	1.19229i	0.802875	+	0.596147i	$0.203303\pi$
$23$	131.515i	1.19229i	−0.802875	+	0.596147i	$0.796697\pi$
$24$	0	0
$25$	102.631	0.821047
$26$	−123.581	−0.932165
$27$	0	0
$28$	0	0
$29$	− 158.738i	− 1.01645i	−0.861225	−	0.508223i	$-0.830302\pi$
$29$	− 158.738i	− 1.01645i	0.861225	−	0.508223i	$-0.169698\pi$
$30$	0	0
$31$	− 76.4814i	− 0.443112i	−0.975148	−	0.221556i	$-0.928886\pi$
$31$	− 76.4814i	− 0.443112i	0.975148	−	0.221556i	$-0.0711136\pi$
$32$	176.471i	0.974872i
$33$	0	0
$34$	− 127.674i	− 0.643996i
$35$	0	0
$36$	0	0
$37$	348.682	1.54927	0.774634	−	0.632410i	$-0.217934\pi$
$37$	348.682	1.54927	0.774634	+	0.632410i	$0.217934\pi$
$38$	−111.874	−0.477590
$39$	0	0
$40$	348.422i	1.37726i
$41$	138.909	0.529120	0.264560	−	0.964369i	$-0.414773\pi$
$41$	138.909	0.529120	0.264560	+	0.964369i	$0.414773\pi$
$42$	0	0
$43$	539.651	1.91386	0.956931	−	0.290316i	$-0.0937604\pi$
$43$	539.651	1.91386	0.956931	+	0.290316i	$0.0937604\pi$
$44$	− 46.3497i	− 0.158806i
$45$	0	0
$46$	−239.402	−0.767346
$47$	−223.643	−0.694078	−0.347039	−	0.937851i	$-0.612813\pi$
$47$	−223.643	−0.694078	−0.347039	+	0.937851i	$0.612813\pi$
$48$	0	0
$49$	0	0
$50$	186.823i	0.528415i
$51$	0	0
$52$	318.153i	0.848459i
$53$	− 530.011i	− 1.37363i	−0.726830	−	0.686817i	$-0.759007\pi$
$53$	− 530.011i	− 1.37363i	0.726830	−	0.686817i	$-0.240993\pi$
$54$	0	0
$55$	− 149.220i	− 0.365833i
$56$	0	0
$57$	0	0
$58$	288.958	0.654172
$59$	−542.876	−1.19791	−0.598953	−	0.800784i	$-0.704417\pi$
$59$	−542.876	−1.19791	−0.598953	+	0.800784i	$0.704417\pi$
$60$	0	0
$61$	134.197i	0.281674i	0.990033	+	0.140837i	$0.0449793\pi$
$61$	134.197i	0.281674i	−0.990033	+	0.140837i	$0.955021\pi$
$62$	139.222	0.285181
$63$	0	0
$64$	−357.614	−0.698464
$65$	1024.27i	1.95454i
$66$	0	0
$67$	320.580	0.584553	0.292276	−	0.956334i	$-0.405587\pi$
$67$	320.580	0.584553	0.292276	+	0.956334i	$0.405587\pi$
$68$	−328.689	−0.586167
$69$	0	0
$70$	0	0
$71$	416.958i	0.696955i	0.937317	+	0.348478i	$0.113301\pi$
$71$	416.958i	0.696955i	−0.937317	+	0.348478i	$0.886699\pi$
$72$	0	0
$73$	545.607i	0.874774i	0.899273	+	0.437387i	$0.144096\pi$
$73$	545.607i	0.874774i	−0.899273	+	0.437387i	$0.855904\pi$
$74$	634.719i	0.997089i
$75$	0	0
$76$	288.014i	0.434704i
$77$	0	0
$78$	0	0
$79$	−322.737	−0.459630	−0.229815	−	0.973234i	$-0.573812\pi$
$79$	−322.737	−0.459630	−0.229815	+	0.973234i	$0.573812\pi$
$80$	−68.6044	−0.0958776
$81$	0	0
$82$	252.861i	0.340535i
$83$	885.170	1.17060	0.585301	−	0.810816i	$-0.300976\pi$
$83$	885.170	1.17060	0.585301	+	0.810816i	$0.300976\pi$
$84$	0	0
$85$	−1058.19	−1.35032
$86$	982.349i	1.23174i
$87$	0	0
$88$	228.403	0.276680
$89$	−1624.62	−1.93494	−0.967471	−	0.252984i	$-0.918588\pi$
$89$	−1624.62	−1.93494	−0.967471	+	0.252984i	$0.918588\pi$
$90$	0	0
$91$	0	0
$92$	616.327i	0.698441i
$93$	0	0
$94$	− 407.106i	− 0.446700i
$95$	927.244i	1.00140i
$96$	0	0
$97$	− 739.155i	− 0.773710i	−0.922141	−	0.386855i	$-0.873561\pi$
$97$	− 739.155i	− 0.773710i	0.922141	−	0.386855i	$-0.126439\pi$
$98$	0	0
$99$	0	0
$100$	480.965	0.480965
$101$	−239.516	−0.235967	−0.117984	−	0.993016i	$-0.537643\pi$
$101$	−239.516	−0.235967	−0.117984	+	0.993016i	$0.537643\pi$
$102$	0	0
$103$	51.1361i	0.0489184i	0.999701	+	0.0244592i	$0.00778637\pi$
$103$	51.1361i	0.0489184i	−0.999701	+	0.0244592i	$0.992214\pi$
$104$	−1567.80	−1.47822
$105$	0	0
$106$	964.800	0.884054
$107$	− 1190.99i	− 1.07605i	−0.842929	−	0.538025i	$-0.819171\pi$
$107$	− 1190.99i	− 1.07605i	0.842929	−	0.538025i	$-0.180829\pi$
$108$	0	0
$109$	389.170	0.341979	0.170989	−	0.985273i	$-0.445304\pi$
$109$	389.170	0.341979	0.170989	+	0.985273i	$0.445304\pi$
$110$	271.631	0.235446
$111$	0	0
$112$	0	0
$113$	− 718.545i	− 0.598186i	−0.954224	−	0.299093i	$-0.903316\pi$
$113$	− 718.545i	− 0.598186i	0.954224	−	0.299093i	$-0.0966841\pi$
$114$	0	0
$115$	1984.23i	1.60896i
$116$	− 743.905i	− 0.595430i
$117$	0	0
$118$	− 988.220i	− 0.770958i
$119$	0	0
$120$	0	0
$121$	1233.18	0.926507
$122$	−244.283	−0.181282
$123$	0	0
$124$	− 358.420i	− 0.259573i
$125$	−337.493	−0.241490
$126$	0	0
$127$	−179.456	−0.125387	−0.0626934	−	0.998033i	$-0.519969\pi$
$127$	−179.456	−0.125387	−0.0626934	+	0.998033i	$0.519969\pi$
$128$	760.787i	0.525349i
$129$	0	0
$130$	−1864.53	−1.25792
$131$	2446.87	1.63194	0.815968	−	0.578096i	$-0.196204\pi$
$131$	2446.87	1.63194	0.815968	+	0.578096i	$0.196204\pi$
$132$	0	0
$133$	0	0
$134$	583.564i	0.376211i
$135$	0	0
$136$	− 1619.72i	− 1.02125i
$137$	− 511.557i	− 0.319016i	−0.987197	−	0.159508i	$-0.949009\pi$
$137$	− 511.557i	− 0.319016i	0.987197	−	0.159508i	$-0.0509908\pi$
$138$	0	0
$139$	− 599.427i	− 0.365775i	−0.983134	−	0.182888i	$-0.941456\pi$
$139$	− 599.427i	− 0.365775i	0.983134	−	0.182888i	$-0.0585444\pi$
$140$	0	0
$141$	0	0
$142$	−759.006	−0.448552
$143$	671.447	0.392652
$144$	0	0
$145$	− 2394.95i	− 1.37166i
$146$	−993.191	−0.562994
$147$	0	0
$148$	1634.05	0.907554
$149$	− 2193.50i	− 1.20603i	−0.797729	−	0.603016i	$-0.793966\pi$
$149$	− 2193.50i	− 1.20603i	0.797729	−	0.603016i	$-0.206034\pi$
$150$	0	0
$151$	−717.366	−0.386612	−0.193306	−	0.981139i	$-0.561921\pi$
$151$	−717.366	−0.386612	−0.193306	+	0.981139i	$0.561921\pi$
$152$	−1419.28	−0.757360
$153$	0	0
$154$	0	0
$155$	− 1153.91i	− 0.597963i
$156$	0	0
$157$	− 1802.94i	− 0.916500i	−0.888823	−	0.458250i	$-0.848476\pi$
$157$	− 1802.94i	− 0.916500i	0.888823	−	0.458250i	$-0.151524\pi$
$158$	− 587.492i	− 0.295812i
$159$	0	0
$160$	2662.49i	1.31555i
$161$	0	0
$162$	0	0
$163$	−2907.81	−1.39728	−0.698642	−	0.715472i	$-0.746212\pi$
$163$	−2907.81	−1.39728	−0.698642	+	0.715472i	$0.746212\pi$
$164$	650.977	0.309956
$165$	0	0
$166$	1611.31i	0.753385i
$167$	3491.37	1.61779	0.808893	−	0.587956i	$-0.200067\pi$
$167$	3491.37	1.61779	0.808893	+	0.587956i	$0.200067\pi$
$168$	0	0
$169$	−2411.93	−1.09783
$170$	− 1926.27i	− 0.869048i
$171$	0	0
$172$	2529.00	1.12113
$173$	1754.75	0.771165	0.385583	−	0.922673i	$-0.374001\pi$
$173$	1754.75	0.771165	0.385583	+	0.922673i	$0.374001\pi$
$174$	0	0
$175$	0	0
$176$	44.9726i	0.0192610i
$177$	0	0
$178$	− 2957.37i	− 1.24530i
$179$	− 791.707i	− 0.330586i	−0.986244	−	0.165293i	$-0.947143\pi$
$179$	− 791.707i	− 0.330586i	0.986244	−	0.165293i	$-0.0528571\pi$
$180$	0	0
$181$	2522.19i	1.03576i	0.855452	+	0.517882i	$0.173279\pi$
$181$	2522.19i	1.03576i	−0.855452	+	0.517882i	$0.826721\pi$
$182$	0	0
$183$	0	0
$184$	−3037.14	−1.21685
$185$	5260.71	2.09068
$186$	0	0
$187$	693.682i	0.271268i
$188$	−1048.07	−0.406588
$189$	0	0
$190$	−1687.90	−0.644490
$191$	903.283i	0.342195i	0.985254	+	0.171098i	$0.0547313\pi$
$191$	903.283i	0.342195i	−0.985254	+	0.171098i	$0.945269\pi$
$192$	0	0
$193$	−198.875	−0.0741727	−0.0370863	−	0.999312i	$-0.511808\pi$
$193$	−198.875	−0.0741727	−0.0370863	+	0.999312i	$0.511808\pi$
$194$	1345.51	0.497950
$195$	0	0
$196$	0	0
$197$	− 3220.69i	− 1.16480i	−0.812904	−	0.582398i	$-0.802114\pi$
$197$	− 3220.69i	− 1.16480i	0.812904	−	0.582398i	$-0.197886\pi$
$198$	0	0
$199$	2849.92i	1.01520i	0.861592	+	0.507601i	$0.169468\pi$
$199$	2849.92i	1.01520i	−0.861592	+	0.507601i	$0.830532\pi$
$200$	2370.11i	0.837959i
$201$	0	0
$202$	− 436.000i	− 0.151866i
$203$	0	0
$204$	0	0
$205$	2095.78	0.714027
$206$	−93.0851	−0.0314832
$207$	0	0
$208$	− 308.700i	− 0.102906i
$209$	607.841	0.201173
$210$	0	0
$211$	1204.50	0.392993	0.196496	−	0.980505i	$-0.437044\pi$
$211$	1204.50	0.392993	0.196496	+	0.980505i	$0.437044\pi$
$212$	− 2483.82i	− 0.804668i
$213$	0	0
$214$	2168.01	0.692532
$215$	8141.96	2.58268
$216$	0	0
$217$	0	0
$218$	708.421i	0.220093i
$219$	0	0
$220$	− 699.299i	− 0.214303i
$221$	− 4761.56i	− 1.44931i
$222$	0	0
$223$	3377.73i	1.01430i	0.861857	+	0.507151i	$0.169301\pi$
$223$	3377.73i	1.01430i	−0.861857	+	0.507151i	$0.830699\pi$
$224$	0	0
$225$	0	0
$226$	1308.00	0.384985
$227$	−4523.96	−1.32276	−0.661378	−	0.750052i	$-0.730028\pi$
$227$	−4523.96	−1.32276	−0.661378	+	0.750052i	$0.730028\pi$
$228$	0	0
$229$	− 3913.99i	− 1.12945i	−0.825280	−	0.564724i	$-0.808983\pi$
$229$	− 3913.99i	− 1.12945i	0.825280	−	0.564724i	$-0.191017\pi$
$230$	−3611.96	−1.03550
$231$	0	0
$232$	3665.82	1.03738
$233$	− 4369.16i	− 1.22847i	−0.789124	−	0.614234i	$-0.789465\pi$
$233$	− 4369.16i	− 1.22847i	0.789124	−	0.614234i	$-0.210535\pi$
$234$	0	0
$235$	−3374.20	−0.936632
$236$	−2544.12	−0.701728
$237$	0	0
$238$	0	0
$239$	− 1945.23i	− 0.526471i	−0.964732	−	0.263235i	$-0.915210\pi$
$239$	− 1945.23i	− 0.526471i	0.964732	−	0.263235i	$-0.0847896\pi$
$240$	0	0
$241$	− 4041.23i	− 1.08016i	−0.841614	−	0.540080i	$-0.818394\pi$
$241$	− 4041.23i	− 1.08016i	0.841614	−	0.540080i	$-0.181606\pi$
$242$	2244.81i	0.596288i
$243$	0	0
$244$	628.894i	0.165003i
$245$	0	0
$246$	0	0
$247$	−4172.33	−1.07481
$248$	1766.22	0.452239
$249$	0	0
$250$	− 614.352i	− 0.155420i
$251$	−4415.70	−1.11042	−0.555212	−	0.831709i	$-0.687363\pi$
$251$	−4415.70	−1.11042	−0.555212	+	0.831709i	$0.687363\pi$
$252$	0	0
$253$	1300.73	0.323226
$254$	− 326.671i	− 0.0806974i
$255$	0	0
$256$	−4245.80	−1.03657
$257$	697.600	0.169319	0.0846597	−	0.996410i	$-0.473020\pi$
$257$	697.600	0.169319	0.0846597	+	0.996410i	$0.473020\pi$
$258$	0	0
$259$	0	0
$260$	4800.11i	1.14496i
$261$	0	0
$262$	4454.13i	1.05029i
$263$	797.510i	0.186983i	0.995620	+	0.0934915i	$0.0298028\pi$
$263$	797.510i	0.186983i	−0.995620	+	0.0934915i	$0.970197\pi$
$264$	0	0
$265$	− 7996.51i	− 1.85367i
$266$	0	0
$267$	0	0
$268$	1502.35	0.342428
$269$	−410.703	−0.0930892	−0.0465446	−	0.998916i	$-0.514821\pi$
$269$	−410.703	−0.0930892	−0.0465446	+	0.998916i	$0.514821\pi$
$270$	0	0
$271$	− 3791.37i	− 0.849850i	−0.905228	−	0.424925i	$-0.860300\pi$
$271$	− 3791.37i	− 0.849850i	0.905228	−	0.424925i	$-0.139700\pi$
$272$	318.923	0.0710939
$273$	0	0
$274$	931.207	0.205315
$275$	− 1015.05i	− 0.222582i
$276$	0	0
$277$	3246.62	0.704226	0.352113	−	0.935958i	$-0.385463\pi$
$277$	3246.62	0.704226	0.352113	+	0.935958i	$0.385463\pi$
$278$	1091.16	0.235408
$279$	0	0
$280$	0	0
$281$	− 1599.58i	− 0.339583i	−0.985480	−	0.169791i	$-0.945691\pi$
$281$	− 1599.58i	− 0.339583i	0.985480	−	0.169791i	$-0.0543094\pi$
$282$	0	0
$283$	− 4266.27i	− 0.896125i	−0.894002	−	0.448062i	$-0.852114\pi$
$283$	− 4266.27i	− 0.896125i	0.894002	−	0.448062i	$-0.147886\pi$
$284$	1954.02i	0.408273i
$285$	0	0
$286$	1222.26i	0.252706i
$287$	0	0
$288$	0	0
$289$	6.24157	0.00127042
$290$	4359.63	0.882780
$291$	0	0
$292$	2556.91i	0.512438i
$293$	2926.77	0.583562	0.291781	−	0.956485i	$-0.405752\pi$
$293$	2926.77	0.583562	0.291781	+	0.956485i	$0.405752\pi$
$294$	0	0
$295$	−8190.62	−1.61653
$296$	8052.28i	1.58118i
$297$	0	0
$298$	3992.92	0.776187
$299$	−8928.44	−1.72691
$300$	0	0
$301$	0	0
$302$	− 1305.85i	− 0.248819i
$303$	0	0
$304$	− 279.457i	− 0.0527235i
$305$	2024.68i	0.380108i
$306$	0	0
$307$	− 3571.36i	− 0.663935i	−0.943291	−	0.331968i	$-0.892288\pi$
$307$	− 3571.36i	− 0.663935i	0.943291	−	0.331968i	$-0.107712\pi$
$308$	0	0
$309$	0	0
$310$	2100.51	0.384841
$311$	−2573.41	−0.469212	−0.234606	−	0.972091i	$-0.575380\pi$
$311$	−2573.41	−0.469212	−0.234606	+	0.972091i	$0.575380\pi$
$312$	0	0
$313$	1476.35i	0.266608i	0.991075	+	0.133304i	$0.0425587\pi$
$313$	1476.35i	0.266608i	−0.991075	+	0.133304i	$0.957441\pi$
$314$	3281.97	0.589848
$315$	0	0
$316$	−1512.46	−0.269249
$317$	2526.90i	0.447712i	0.974622	+	0.223856i	$0.0718645\pi$
$317$	2526.90i	0.447712i	−0.974622	+	0.223856i	$0.928136\pi$
$318$	0	0
$319$	−1569.98	−0.275554
$320$	−5395.47	−0.942550
$321$	0	0
$322$	0	0
$323$	− 4310.50i	− 0.742546i
$324$	0	0
$325$	6967.52i	1.18919i
$326$	− 5293.20i	− 0.899274i
$327$	0	0
$328$	3207.89i	0.540019i
$329$	0	0
$330$	0	0
$331$	1475.56	0.245027	0.122513	−	0.992467i	$-0.460905\pi$
$331$	1475.56	0.245027	0.122513	+	0.992467i	$0.460905\pi$
$332$	4148.23	0.685733
$333$	0	0
$334$	6355.48i	1.04119i
$335$	4836.73	0.788832
$336$	0	0
$337$	−6727.28	−1.08741	−0.543706	−	0.839275i	$-0.682979\pi$
$337$	−6727.28	−1.08741	−0.543706	+	0.839275i	$0.682979\pi$
$338$	− 4390.54i	− 0.706549i
$339$	0	0
$340$	−4959.07	−0.791010
$341$	−756.428	−0.120126
$342$	0	0
$343$	0	0
$344$	12462.4i	1.95328i
$345$	0	0
$346$	3194.25i	0.496312i
$347$	538.160i	0.0832563i	0.999133	+	0.0416281i	$0.0132545\pi$
$347$	538.160i	0.0832563i	−0.999133	+	0.0416281i	$0.986746\pi$
$348$	0	0
$349$	− 6975.93i	− 1.06995i	−0.844867	−	0.534976i	$-0.820321\pi$
$349$	− 6975.93i	− 1.06995i	0.844867	−	0.534976i	$-0.179679\pi$
$350$	0	0
$351$	0	0
$352$	1745.36	0.264283
$353$	−8876.80	−1.33843	−0.669213	−	0.743071i	$-0.733369\pi$
$353$	−8876.80	−1.33843	−0.669213	+	0.743071i	$0.733369\pi$
$354$	0	0
$355$	6290.83i	0.940514i
$356$	−7613.57	−1.13348
$357$	0	0
$358$	1441.18	0.212761
$359$	11045.8i	1.62389i	0.583735	+	0.811944i	$0.301591\pi$
$359$	11045.8i	1.62389i	−0.583735	+	0.811944i	$0.698409\pi$
$360$	0	0
$361$	3081.92	0.449324
$362$	−4591.25	−0.666604
$363$	0	0
$364$	0	0
$365$	8231.82i	1.18047i
$366$	0	0
$367$	8326.07i	1.18424i	0.805848	+	0.592122i	$0.201710\pi$
$367$	8326.07i	1.18424i	−0.805848	+	0.592122i	$0.798290\pi$
$368$	− 598.015i	− 0.0847110i
$369$	0	0
$370$	9576.29i	1.34553i
$371$	0	0
$372$	0	0
$373$	−4545.32	−0.630959	−0.315479	−	0.948932i	$-0.602165\pi$
$373$	−4545.32	−0.630959	−0.315479	+	0.948932i	$0.602165\pi$
$374$	−1262.74	−0.174584
$375$	0	0
$376$	− 5164.70i	− 0.708375i
$377$	10776.6	1.47221
$378$	0	0
$379$	11527.2	1.56230	0.781151	−	0.624343i	$-0.214633\pi$
$379$	11527.2	1.56230	0.781151	+	0.624343i	$0.214633\pi$
$380$	4345.40i	0.586617i
$381$	0	0
$382$	−1644.28	−0.220233
$383$	3920.46	0.523044	0.261522	−	0.965197i	$-0.415776\pi$
$383$	3920.46	0.523044	0.261522	+	0.965197i	$0.415776\pi$
$384$	0	0
$385$	0	0
$386$	− 362.020i	− 0.0477366i
$387$	0	0
$388$	− 3463.95i	− 0.453235i
$389$	− 796.437i	− 0.103807i	−0.998652	−	0.0519035i	$-0.983471\pi$
$389$	− 796.437i	− 0.103807i	0.998652	−	0.0519035i	$-0.0165288\pi$
$390$	0	0
$391$	− 9224.11i	− 1.19305i
$392$	0	0
$393$	0	0
$394$	5862.76	0.749648
$395$	−4869.28	−0.620254
$396$	0	0
$397$	− 3855.07i	− 0.487357i	−0.969856	−	0.243678i	$-0.921646\pi$
$397$	− 3855.07i	− 0.487357i	0.969856	−	0.243678i	$-0.0783541\pi$
$398$	−5187.82	−0.653371
$399$	0	0
$400$	−466.675	−0.0583344
$401$	4655.35i	0.579744i	0.957065	+	0.289872i	$0.0936127\pi$
$401$	4655.35i	0.579744i	−0.957065	+	0.289872i	$0.906387\pi$
$402$	0	0
$403$	5192.26	0.641798
$404$	−1122.46	−0.138229
$405$	0	0
$406$	0	0
$407$	− 3448.58i	− 0.420000i
$408$	0	0
$409$	− 9790.50i	− 1.18364i	−0.806070	−	0.591821i	$-0.798409\pi$
$409$	− 9790.50i	− 1.18364i	0.806070	−	0.591821i	$-0.201591\pi$
$410$	3815.03i	0.459539i
$411$	0	0
$412$	239.642i	0.0286561i
$413$	0	0
$414$	0	0
$415$	13354.9	1.57968
$416$	−11980.4	−1.41199
$417$	0	0
$418$	1106.48i	0.129473i
$419$	−3007.46	−0.350654	−0.175327	−	0.984510i	$-0.556098\pi$
$419$	−3007.46	−0.350654	−0.175327	+	0.984510i	$0.556098\pi$
$420$	0	0
$421$	7646.06	0.885145	0.442573	−	0.896733i	$-0.354066\pi$
$421$	7646.06	0.885145	0.442573	+	0.896733i	$0.354066\pi$
$422$	2192.61i	0.252925i
$423$	0	0
$424$	12239.8	1.40193
$425$	−7198.25	−0.821568
$426$	0	0
$427$	0	0
$428$	− 5581.41i	− 0.630345i
$429$	0	0
$430$	14821.1i	1.66218i
$431$	14991.6i	1.67545i	0.546090	+	0.837727i	$0.316116\pi$
$431$	14991.6i	1.67545i	−0.546090	+	0.837727i	$0.683884\pi$
$432$	0	0
$433$	5666.63i	0.628916i	0.949271	+	0.314458i	$0.101823\pi$
$433$	5666.63i	0.628916i	−0.949271	+	0.314458i	$0.898177\pi$
$434$	0	0
$435$	0	0
$436$	1823.79	0.200330
$437$	−8082.65	−0.884772
$438$	0	0
$439$	5531.10i	0.601332i	0.953729	+	0.300666i	$0.0972090\pi$
$439$	5531.10i	0.601332i	−0.953729	+	0.300666i	$0.902791\pi$
$440$	3446.01	0.373368
$441$	0	0
$442$	8667.66	0.932757
$443$	− 403.222i	− 0.0432452i	−0.999766	−	0.0216226i	$-0.993117\pi$
$443$	− 403.222i	− 0.0432452i	0.999766	−	0.0216226i	$-0.00688323\pi$
$444$	0	0
$445$	−24511.4	−2.61113
$446$	−6148.61	−0.652792
$447$	0	0
$448$	0	0
$449$	− 8429.03i	− 0.885948i	−0.896534	−	0.442974i	$-0.853923\pi$
$449$	− 8429.03i	− 0.885948i	0.896534	−	0.442974i	$-0.146077\pi$
$450$	0	0
$451$	− 1373.86i	− 0.143442i
$452$	− 3367.36i	− 0.350415i
$453$	0	0
$454$	− 8235.15i	− 0.851310i
$455$	0	0
$456$	0	0
$457$	685.661	0.0701835	0.0350917	−	0.999384i	$-0.488828\pi$
$457$	685.661	0.0701835	0.0350917	+	0.999384i	$0.488828\pi$
$458$	7124.79	0.726899
$459$	0	0
$460$	9298.80i	0.942519i
$461$	4864.48	0.491456	0.245728	−	0.969339i	$-0.420973\pi$
$461$	4864.48	0.491456	0.245728	+	0.969339i	$0.420973\pi$
$462$	0	0
$463$	−8354.23	−0.838562	−0.419281	−	0.907857i	$-0.637718\pi$
$463$	−8354.23	−0.838562	−0.419281	+	0.907857i	$0.637718\pi$
$464$	721.802i	0.0722173i
$465$	0	0
$466$	7953.36	0.790627
$467$	−1002.94	−0.0993800	−0.0496900	−	0.998765i	$-0.515823\pi$
$467$	−1002.94	−0.0993800	−0.0496900	+	0.998765i	$0.515823\pi$
$468$	0	0
$469$	0	0
$470$	− 6142.19i	− 0.602804i
$471$	0	0
$472$	− 12536.9i	− 1.22258i
$473$	− 5337.34i	− 0.518839i
$474$	0	0
$475$	6307.49i	0.609279i
$476$	0	0
$477$	0	0
$478$	3540.98	0.338830
$479$	6052.76	0.577364	0.288682	−	0.957425i	$-0.406783\pi$
$479$	6052.76	0.577364	0.288682	+	0.957425i	$0.406783\pi$
$480$	0	0
$481$	23671.7i	2.24394i
$482$	7356.41	0.695177
$483$	0	0
$484$	5779.13	0.542743
$485$	− 11152.0i	− 1.04409i
$486$	0	0
$487$	15309.4	1.42451	0.712255	−	0.701920i	$-0.247674\pi$
$487$	15309.4	1.42451	0.712255	+	0.701920i	$0.247674\pi$
$488$	−3099.07	−0.287476
$489$	0	0
$490$	0	0
$491$	4291.01i	0.394400i	0.980363	+	0.197200i	$0.0631848\pi$
$491$	4291.01i	0.394400i	−0.980363	+	0.197200i	$0.936815\pi$
$492$	0	0
$493$	11133.5i	1.01709i
$494$	− 7595.06i	− 0.691736i
$495$	0	0
$496$	347.770i	0.0314826i
$497$	0	0
$498$	0	0
$499$	6891.53	0.618251	0.309126	−	0.951021i	$-0.399964\pi$
$499$	6891.53	0.618251	0.309126	+	0.951021i	$0.399964\pi$
$500$	−1581.61	−0.141464
$501$	0	0
$502$	− 8038.07i	− 0.714655i
$503$	−13534.6	−1.19975	−0.599877	−	0.800092i	$-0.704784\pi$
$503$	−13534.6	−1.19975	−0.599877	+	0.800092i	$0.704784\pi$
$504$	0	0
$505$	−3613.68	−0.318429
$506$	2367.77i	0.208024i
$507$	0	0
$508$	−840.995	−0.0734510
$509$	12087.8	1.05262	0.526310	−	0.850293i	$-0.323575\pi$
$509$	12087.8	1.05262	0.526310	+	0.850293i	$0.323575\pi$
$510$	0	0
$511$	0	0
$512$	− 1642.50i	− 0.141776i
$513$	0	0
$514$	1269.87i	0.108972i
$515$	771.513i	0.0660134i
$516$	0	0
$517$	2211.91i	0.188161i
$518$	0	0
$519$	0	0
$520$	−23654.0	−1.99480
$521$	7625.87	0.641258	0.320629	−	0.947205i	$-0.396106\pi$
$521$	7625.87	0.641258	0.320629	+	0.947205i	$0.396106\pi$
$522$	0	0
$523$	− 15390.7i	− 1.28678i	−0.765537	−	0.643392i	$-0.777527\pi$
$523$	− 15390.7i	− 1.28678i	0.765537	−	0.643392i	$-0.222473\pi$
$524$	11466.9	0.955981
$525$	0	0
$526$	−1451.74	−0.120340
$527$	5364.20i	0.443393i
$528$	0	0
$529$	−5129.20	−0.421567
$530$	14556.4	1.19300
$531$	0	0
$532$	0	0
$533$	9430.40i	0.766371i
$534$	0	0
$535$	− 17969.0i	− 1.45209i
$536$	7403.30i	0.596593i
$537$	0	0
$538$	− 747.619i	− 0.0599110i
$539$	0	0
$540$	0	0
$541$	−13700.9	−1.08881	−0.544406	−	0.838822i	$-0.683245\pi$
$541$	−13700.9	−1.08881	−0.544406	+	0.838822i	$0.683245\pi$
$542$	6901.59	0.546953
$543$	0	0
$544$	− 12377.2i	− 0.975491i
$545$	5871.58	0.461487
$546$	0	0
$547$	−6139.00	−0.479863	−0.239931	−	0.970790i	$-0.577125\pi$
$547$	−6139.00	−0.479863	−0.239931	+	0.970790i	$0.577125\pi$
$548$	− 2397.34i	− 0.186878i
$549$	0	0
$550$	1847.74	0.143251
$551$	9755.73	0.754280
$552$	0	0
$553$	0	0
$554$	5909.95i	0.453231i
$555$	0	0
$556$	− 2809.13i	− 0.214269i
$557$	− 22732.7i	− 1.72929i	−0.502382	−	0.864646i	$-0.667543\pi$
$557$	− 22732.7i	− 1.72929i	0.502382	−	0.864646i	$-0.332457\pi$
$558$	0	0
$559$	36636.4i	2.77202i
$560$	0	0
$561$	0	0
$562$	2911.78	0.218551
$563$	9917.61	0.742411	0.371206	−	0.928551i	$-0.378945\pi$
$563$	9917.61	0.742411	0.371206	+	0.928551i	$0.378945\pi$
$564$	0	0
$565$	− 10841.0i	− 0.807230i
$566$	7766.06	0.576735
$567$	0	0
$568$	−9629.02	−0.711311
$569$	− 5137.02i	− 0.378480i	−0.981931	−	0.189240i	$-0.939398\pi$
$569$	− 5137.02i	− 0.378480i	0.981931	−	0.189240i	$-0.0606024\pi$
$570$	0	0
$571$	−18186.0	−1.33286	−0.666429	−	0.745568i	$-0.732178\pi$
$571$	−18186.0	−1.33286	−0.666429	+	0.745568i	$0.732178\pi$
$572$	3146.64	0.230014
$573$	0	0
$574$	0	0
$575$	13497.5i	0.978930i
$576$	0	0
$577$	12398.6i	0.894562i	0.894394	+	0.447281i	$0.147608\pi$
$577$	12398.6i	0.894562i	−0.894394	+	0.447281i	$0.852392\pi$
$578$	11.3618i	0 0.000817626i
$579$	0	0
$580$	− 11223.6i	− 0.803509i
$581$	0	0
$582$	0	0
$583$	−5241.99	−0.372386
$584$	−12600.0	−0.892793
$585$	0	0
$586$	5327.72i	0.375573i
$587$	−18977.6	−1.33439	−0.667195	−	0.744883i	$-0.732505\pi$
$587$	−18977.6	−1.33439	−0.667195	+	0.744883i	$0.732505\pi$
$588$	0	0
$589$	4700.40	0.328822
$590$	− 14909.7i	− 1.04038i
$591$	0	0
$592$	−1585.50	−0.110074
$593$	−10728.9	−0.742972	−0.371486	−	0.928439i	$-0.621152\pi$
$593$	−10728.9	−0.742972	−0.371486	+	0.928439i	$0.621152\pi$
$594$	0	0
$595$	0	0
$596$	− 10279.5i	− 0.706488i
$597$	0	0
$598$	− 16252.8i	− 1.11142i
$599$	− 1821.37i	− 0.124239i	−0.998069	−	0.0621196i	$-0.980214\pi$
$599$	− 1821.37i	− 0.124239i	0.998069	−	0.0621196i	$-0.0197860\pi$
$600$	0	0
$601$	− 18933.3i	− 1.28503i	−0.766273	−	0.642516i	$-0.777891\pi$
$601$	− 18933.3i	− 1.28503i	0.766273	−	0.642516i	$-0.222109\pi$
$602$	0	0
$603$	0	0
$604$	−3361.84	−0.226475
$605$	18605.5	1.25029
$606$	0	0
$607$	− 15384.4i	− 1.02872i	−0.857574	−	0.514360i	$-0.828029\pi$
$607$	− 15384.4i	− 1.02872i	0.857574	−	0.514360i	$-0.171971\pi$
$608$	−10845.5	−0.723428
$609$	0	0
$610$	−3685.61	−0.244633
$611$	− 15182.9i	− 1.00529i
$612$	0	0
$613$	−5507.20	−0.362861	−0.181431	−	0.983404i	$-0.558073\pi$
$613$	−5507.20	−0.362861	−0.181431	+	0.983404i	$0.558073\pi$
$614$	6501.08	0.427300
$615$	0	0
$616$	0	0
$617$	− 18134.0i	− 1.18322i	−0.806224	−	0.591610i	$-0.798493\pi$
$617$	− 18134.0i	− 1.18322i	0.806224	−	0.591610i	$-0.201507\pi$
$618$	0	0
$619$	3635.84i	0.236085i	0.993009	+	0.118043i	$0.0376619\pi$
$619$	3635.84i	0.236085i	−0.993009	+	0.118043i	$0.962338\pi$
$620$	− 5407.64i	− 0.350284i
$621$	0	0
$622$	− 4684.49i	− 0.301979i
$623$	0	0
$624$	0	0
$625$	−17920.8	−1.14693
$626$	−2687.46	−0.171586
$627$	0	0
$628$	− 8449.24i	− 0.536881i
$629$	−24455.6	−1.55025
$630$	0	0
$631$	−5912.59	−0.373021	−0.186511	−	0.982453i	$-0.559718\pi$
$631$	−5912.59	−0.373021	−0.186511	+	0.982453i	$0.559718\pi$
$632$	− 7453.13i	− 0.469098i
$633$	0	0
$634$	−4599.81	−0.288142
$635$	−2707.53	−0.169205
$636$	0	0
$637$	0	0
$638$	− 2857.89i	− 0.177343i
$639$	0	0
$640$	11478.3i	0.708939i
$641$	27466.6i	1.69245i	0.532822	+	0.846227i	$0.321132\pi$
$641$	27466.6i	1.69245i	−0.532822	+	0.846227i	$0.678868\pi$
$642$	0	0
$643$	− 28474.0i	− 1.74635i	−0.487403	−	0.873177i	$-0.662056\pi$
$643$	− 28474.0i	− 1.74635i	0.487403	−	0.873177i	$-0.337944\pi$
$644$	0	0
$645$	0	0
$646$	7846.57	0.477894
$647$	−1323.36	−0.0804122	−0.0402061	−	0.999191i	$-0.512801\pi$
$647$	−1323.36	−0.0804122	−0.0402061	+	0.999191i	$0.512801\pi$
$648$	0	0
$649$	5369.24i	0.324747i
$650$	−12683.3	−0.765351
$651$	0	0
$652$	−13627.0	−0.818522
$653$	3846.75i	0.230528i	0.993335	+	0.115264i	$0.0367714\pi$
$653$	3846.75i	0.230528i	−0.993335	+	0.115264i	$0.963229\pi$
$654$	0	0
$655$	36917.0	2.20224
$656$	−631.635	−0.0375933
$657$	0	0
$658$	0	0
$659$	− 6796.84i	− 0.401771i	−0.979615	−	0.200886i	$-0.935618\pi$
$659$	− 6796.84i	− 0.401771i	0.979615	−	0.200886i	$-0.0643819\pi$
$660$	0	0
$661$	− 31064.4i	− 1.82793i	−0.405790	−	0.913966i	$-0.633004\pi$
$661$	− 31064.4i	− 1.82793i	0.405790	−	0.913966i	$-0.366996\pi$
$662$	2686.02i	0.157696i
$663$	0	0
$664$	20441.7i	1.19471i
$665$	0	0
$666$	0	0
$667$	20876.5	1.21190
$668$	16361.8	0.947691
$669$	0	0
$670$	8804.49i	0.507682i
$671$	1327.25	0.0763605
$672$	0	0
$673$	15508.2	0.888259	0.444129	−	0.895963i	$-0.353513\pi$
$673$	15508.2	0.888259	0.444129	+	0.895963i	$0.353513\pi$
$674$	− 12245.9i	− 0.699845i
$675$	0	0
$676$	−11303.2	−0.643103
$677$	30674.8	1.74140	0.870701	−	0.491813i	$-0.163666\pi$
$677$	30674.8	1.74140	0.870701	+	0.491813i	$0.163666\pi$
$678$	0	0
$679$	0	0
$680$	− 24437.4i	− 1.37813i
$681$	0	0
$682$	− 1376.96i	− 0.0773114i
$683$	17658.7i	0.989296i	0.869093	+	0.494648i	$0.164703\pi$
$683$	17658.7i	0.989296i	−0.869093	+	0.494648i	$0.835297\pi$
$684$	0	0
$685$	− 7718.08i	− 0.430500i
$686$	0	0
$687$	0	0
$688$	−2453.86	−0.135977
$689$	35982.0	1.98956
$690$	0	0
$691$	− 13166.2i	− 0.724843i	−0.932014	−	0.362422i	$-0.881950\pi$
$691$	− 13166.2i	− 0.724843i	0.932014	−	0.362422i	$-0.118050\pi$
$692$	8223.42	0.451745
$693$	0	0
$694$	−979.634	−0.0535827
$695$	− 9043.82i	− 0.493599i
$696$	0	0
$697$	−9742.69	−0.529456
$698$	12698.6	0.688607
$699$	0	0
$700$	0	0
$701$	25910.0i	1.39602i	0.716090	+	0.698008i	$0.245930\pi$
$701$	25910.0i	1.39602i	−0.716090	+	0.698008i	$0.754070\pi$
$702$	0	0
$703$	21429.3i	1.14967i
$704$	3536.92i	0.189350i
$705$	0	0
$706$	− 16158.8i	− 0.861394i
$707$	0	0
$708$	0	0
$709$	6208.49	0.328864	0.164432	−	0.986388i	$-0.447421\pi$
$709$	6208.49	0.328864	0.164432	+	0.986388i	$0.447421\pi$
$710$	−11451.5	−0.605303
$711$	0	0
$712$	− 37518.2i	− 1.97480i
$713$	10058.5	0.528320
$714$	0	0
$715$	10130.4	0.529868
$716$	− 3710.23i	− 0.193656i
$717$	0	0
$718$	−20107.1	−1.04511
$719$	−28758.8	−1.49169	−0.745843	−	0.666122i	$-0.767953\pi$
$719$	−28758.8	−1.49169	−0.745843	+	0.666122i	$0.767953\pi$
$720$	0	0
$721$	0	0
$722$	5610.13i	0.289180i
$723$	0	0
$724$	11819.9i	0.606745i
$725$	− 16291.4i	− 0.834550i
$726$	0	0
$727$	35275.7i	1.79959i	0.436312	+	0.899795i	$0.356284\pi$
$727$	35275.7i	1.79959i	−0.436312	+	0.899795i	$0.643716\pi$
$728$	0	0
$729$	0	0
$730$	−14984.7	−0.759738
$731$	−37849.7	−1.91508
$732$	0	0
$733$	7950.28i	0.400615i	0.979733	+	0.200307i	$0.0641940\pi$
$733$	7950.28i	0.400615i	−0.979733	+	0.200307i	$0.935806\pi$
$734$	−15156.3	−0.762164
$735$	0	0
$736$	−23208.5	−1.16233
$737$	− 3170.64i	− 0.158470i
$738$	0	0
$739$	33353.6	1.66026	0.830130	−	0.557570i	$-0.188266\pi$
$739$	33353.6	1.66026	0.830130	+	0.557570i	$0.188266\pi$
$740$	24653.6	1.22471
$741$	0	0
$742$	0	0
$743$	32933.6i	1.62613i	0.582171	+	0.813066i	$0.302203\pi$
$743$	32933.6i	1.62613i	−0.582171	+	0.813066i	$0.697797\pi$
$744$	0	0
$745$	− 33094.3i	− 1.62749i
$746$	− 8274.02i	− 0.406077i
$747$	0	0
$748$	3250.85i	0.158907i
$749$	0	0
$750$	0	0
$751$	−39636.6	−1.92591	−0.962956	−	0.269660i	$-0.913089\pi$
$751$	−39636.6	−1.92591	−0.962956	+	0.269660i	$0.913089\pi$
$752$	1016.93	0.0493134
$753$	0	0
$754$	19617.1i	0.947496i
$755$	−10823.2	−0.521718
$756$	0	0
$757$	−3996.51	−0.191883	−0.0959417	−	0.995387i	$-0.530586\pi$
$757$	−3996.51	−0.191883	−0.0959417	+	0.995387i	$0.530586\pi$
$758$	20983.4i	1.00548i
$759$	0	0
$760$	−21413.3	−1.02203
$761$	−26235.7	−1.24973	−0.624863	−	0.780734i	$-0.714845\pi$
$761$	−26235.7	−1.24973	−0.624863	+	0.780734i	$0.714845\pi$
$762$	0	0
$763$	0	0
$764$	4233.11i	0.200456i
$765$	0	0
$766$	7136.57i	0.336625i
$767$	− 36855.4i	− 1.73503i
$768$	0	0
$769$	36456.9i	1.70958i	0.518971	+	0.854792i	$0.326315\pi$
$769$	36456.9i	1.70958i	−0.518971	+	0.854792i	$0.673685\pi$
$770$	0	0
$771$	0	0
$772$	−932.000	−0.0434500
$773$	9465.47	0.440426	0.220213	−	0.975452i	$-0.429325\pi$
$773$	9465.47	0.440426	0.220213	+	0.975452i	$0.429325\pi$
$774$	0	0
$775$	− 7849.36i	− 0.363816i
$776$	17069.7	0.789646
$777$	0	0
$778$	1449.79	0.0668089
$779$	8537.06i	0.392647i
$780$	0	0
$781$	4123.86	0.188941
$782$	16791.0	0.767833
$783$	0	0
$784$	0	0
$785$	− 27201.8i	− 1.23678i
$786$	0	0
$787$	25017.6i	1.13314i	0.824013	+	0.566571i	$0.191730\pi$
$787$	25017.6i	1.13314i	−0.824013	+	0.566571i	$0.808270\pi$
$788$	− 15093.3i	− 0.682332i
$789$	0	0
$790$	− 8863.75i	− 0.399187i
$791$	0	0
$792$	0	0
$793$	−9110.48	−0.407973
$794$	7017.54	0.313657
$795$	0	0
$796$	13355.7i	0.594701i
$797$	38893.5	1.72858	0.864290	−	0.502994i	$-0.167768\pi$
$797$	38893.5	1.72858	0.864290	+	0.502994i	$0.167768\pi$
$798$	0	0
$799$	15685.7	0.694519
$800$	18111.3i	0.800415i
$801$	0	0
$802$	−8474.32	−0.373116
$803$	5396.24	0.237147
$804$	0	0
$805$	0	0
$806$	9451.67i	0.413053i
$807$	0	0
$808$	− 5531.26i	− 0.240828i
$809$	− 7011.64i	− 0.304717i	−0.988325	−	0.152359i	$-0.951313\pi$
$809$	− 7011.64i	− 0.304717i	0.988325	−	0.152359i	$-0.0486869\pi$
$810$	0	0
$811$	− 5013.82i	− 0.217089i	−0.994092	−	0.108544i	$-0.965381\pi$
$811$	− 5013.82i	− 0.217089i	0.994092	−	0.108544i	$-0.0346189\pi$
$812$	0	0
$813$	0	0
$814$	6277.59	0.270306
$815$	−43871.4	−1.88558
$816$	0	0
$817$	33165.9i	1.42023i
$818$	17822.0	0.761776
$819$	0	0
$820$	9821.58	0.418274
$821$	696.609i	0.0296124i	0.999890	+	0.0148062i	$0.00471314\pi$
$821$	696.609i	0.0296124i	−0.999890	+	0.0148062i	$0.995287\pi$
$822$	0	0
$823$	6413.15	0.271626	0.135813	−	0.990734i	$-0.456635\pi$
$823$	6413.15	0.271626	0.135813	+	0.990734i	$0.456635\pi$
$824$	−1180.91	−0.0499260
$825$	0	0
$826$	0	0
$827$	− 16718.9i	− 0.702989i	−0.936190	−	0.351494i	$-0.885674\pi$
$827$	− 16718.9i	− 0.702989i	0.936190	−	0.351494i	$-0.114326\pi$
$828$	0	0
$829$	− 14557.2i	− 0.609882i	−0.952371	−	0.304941i	$-0.901363\pi$
$829$	− 14557.2i	− 0.609882i	0.952371	−	0.304941i	$-0.0986367\pi$
$830$	24310.5i	1.01666i
$831$	0	0
$832$	− 24278.1i	− 1.01165i
$833$	0	0
$834$	0	0
$835$	52675.8	2.18314
$836$	2848.56	0.117846
$837$	0	0
$838$	− 5474.61i	− 0.225677i
$839$	−19467.0	−0.801045	−0.400523	−	0.916287i	$-0.631171\pi$
$839$	−19467.0	−0.801045	−0.400523	+	0.916287i	$0.631171\pi$
$840$	0	0
$841$	−808.829	−0.0331637
$842$	13918.4i	0.569668i
$843$	0	0
$844$	5644.74	0.230213
$845$	−36389.9	−1.48148
$846$	0	0
$847$	0	0
$848$	2410.02i	0.0975950i
$849$	0	0
$850$	− 13103.3i	− 0.528751i
$851$	45856.9i	1.84718i
$852$	0	0
$853$	22345.3i	0.896938i	0.893798	+	0.448469i	$0.148031\pi$
$853$	22345.3i	0.896938i	−0.893798	+	0.448469i	$0.851969\pi$
$854$	0	0
$855$	0	0
$856$	27504.1	1.09821
$857$	48079.7	1.91642	0.958209	−	0.286068i	$-0.0923483\pi$
$857$	48079.7	1.91642	0.958209	+	0.286068i	$0.0923483\pi$
$858$	0	0
$859$	26110.7i	1.03712i	0.855041	+	0.518560i	$0.173532\pi$
$859$	26110.7i	1.03712i	−0.855041	+	0.518560i	$0.826468\pi$
$860$	38156.2	1.51292
$861$	0	0
$862$	−27289.8	−1.07830
$863$	− 2928.55i	− 0.115514i	−0.998331	−	0.0577572i	$-0.981605\pi$
$863$	− 2928.55i	− 0.115514i	0.998331	−	0.0577572i	$-0.0183949\pi$
$864$	0	0
$865$	26474.8	1.04066
$866$	−10315.2	−0.404763
$867$	0	0
$868$	0	0
$869$	3191.98i	0.124604i
$870$	0	0
$871$	21763.9i	0.846660i
$872$	8987.29i	0.349023i
$873$	0	0
$874$	− 14713.2i	− 0.569428i
$875$	0	0
$876$	0	0
$877$	13090.6	0.504033	0.252016	−	0.967723i	$-0.418906\pi$
$877$	13090.6	0.504033	0.252016	+	0.967723i	$0.418906\pi$
$878$	−10068.5	−0.387010
$879$	0	0
$880$	678.521i	0.0259920i
$881$	−7888.79	−0.301680	−0.150840	−	0.988558i	$-0.548198\pi$
$881$	−7888.79	−0.301680	−0.150840	+	0.988558i	$0.548198\pi$
$882$	0	0
$883$	45061.9	1.71739	0.858694	−	0.512489i	$-0.171276\pi$
$883$	45061.9	1.71739	0.858694	+	0.512489i	$0.171276\pi$
$884$	− 22314.4i	− 0.848998i
$885$	0	0
$886$	734.000	0.0278321
$887$	−3546.58	−0.134253	−0.0671266	−	0.997744i	$-0.521383\pi$
$887$	−3546.58	−0.134253	−0.0671266	+	0.997744i	$0.521383\pi$
$888$	0	0
$889$	0	0
$890$	− 44619.1i	− 1.68049i
$891$	0	0
$892$	15829.2i	0.594173i
$893$	− 13744.6i	− 0.515058i
$894$	0	0
$895$	− 11944.8i	− 0.446114i
$896$	0	0
$897$	0	0
$898$	15343.7	0.570185
$899$	−12140.5	−0.450400
$900$	0	0
$901$	37173.5i	1.37451i
$902$	2500.89	0.0923175
$903$	0	0
$904$	16593.7	0.610508
$905$	38053.4i	1.39772i
$906$	0	0
$907$	−14679.5	−0.537405	−0.268703	−	0.963223i	$-0.586595\pi$
$907$	−14679.5	−0.537405	−0.268703	+	0.963223i	$0.586595\pi$
$908$	−21200.9	−0.774864
$909$	0	0
$910$	0	0
$911$	12355.2i	0.449336i	0.974435	+	0.224668i	$0.0721297\pi$
$911$	12355.2i	0.449336i	−0.974435	+	0.224668i	$0.927870\pi$
$912$	0	0
$913$	− 8754.64i	− 0.317345i
$914$	1248.14i	0.0451692i
$915$	0	0
$916$	− 18342.4i	− 0.661626i
$917$	0	0
$918$	0	0
$919$	−3070.36	−0.110209	−0.0551044	−	0.998481i	$-0.517549\pi$
$919$	−3070.36	−0.110209	−0.0551044	+	0.998481i	$0.517549\pi$
$920$	−45822.7	−1.64210
$921$	0	0
$922$	8855.01i	0.316295i
$923$	−28306.9	−1.00946
$924$	0	0
$925$	35785.5	1.27202
$926$	− 15207.5i	− 0.539688i
$927$	0	0
$928$	28012.6	0.990905
$929$	−30528.7	−1.07816	−0.539082	−	0.842254i	$-0.681229\pi$
$929$	−30528.7	−1.07816	−0.539082	+	0.842254i	$0.681229\pi$
$930$	0	0
$931$	0	0
$932$	− 20475.5i	− 0.719631i
$933$	0	0
$934$	− 1825.69i	− 0.0639597i
$935$	10465.9i	0.366065i
$936$	0	0
$937$	− 18235.1i	− 0.635769i	−0.948129	−	0.317885i	$-0.897028\pi$
$937$	− 18235.1i	− 0.635769i	0.948129	−	0.317885i	$-0.102972\pi$
$938$	0	0
$939$	0	0
$940$	−15812.7	−0.548674
$941$	−13372.3	−0.463258	−0.231629	−	0.972804i	$-0.574406\pi$
$941$	−13372.3	−0.463258	−0.231629	+	0.972804i	$0.574406\pi$
$942$	0	0
$943$	18268.6i	0.630867i
$944$	2468.52	0.0851098
$945$	0	0
$946$	9715.77	0.333919
$947$	5117.10i	0.175590i	0.996139	+	0.0877948i	$0.0279820\pi$
$947$	5117.10i	0.175590i	−0.996139	+	0.0877948i	$0.972018\pi$
$948$	0	0
$949$	−37040.8	−1.26701
$950$	−11481.8	−0.392124
$951$	0	0
$952$	0	0
$953$	35456.7i	1.20520i	0.798044	+	0.602599i	$0.205868\pi$
$953$	35456.7i	1.20520i	−0.798044	+	0.602599i	$0.794132\pi$
$954$	0	0
$955$	13628.2i	0.461779i
$956$	− 9116.05i	− 0.308404i
$957$	0	0
$958$	11018.1i	0.371584i
$959$	0	0
$960$	0	0
$961$	23941.6	0.803652
$962$	−43090.5	−1.44417
$963$	0	0
$964$	− 18938.7i	− 0.632752i
$965$	−3000.51	−0.100093
$966$	0	0
$967$	−2804.92	−0.0932784	−0.0466392	−	0.998912i	$-0.514851\pi$
$967$	−2804.92	−0.0932784	−0.0466392	+	0.998912i	$0.514851\pi$
$968$	28478.5i	0.945591i
$969$	0	0
$970$	20300.4	0.671964
$971$	27367.5	0.904496	0.452248	−	0.891892i	$-0.350622\pi$
$971$	27367.5	0.904496	0.452248	+	0.891892i	$0.350622\pi$
$972$	0	0
$973$	0	0
$974$	27868.4i	0.916797i
$975$	0	0
$976$	− 610.208i	− 0.0200126i
$977$	− 24096.8i	− 0.789075i	−0.918880	−	0.394537i	$-0.870905\pi$
$977$	− 24096.8i	− 0.789075i	0.918880	−	0.394537i	$-0.129095\pi$
$978$	0	0
$979$	16068.1i	0.524554i
$980$	0	0
$981$	0	0
$982$	−7811.09	−0.253831
$983$	−9310.72	−0.302102	−0.151051	−	0.988526i	$-0.548266\pi$
$983$	−9310.72	−0.302102	−0.151051	+	0.988526i	$0.548266\pi$
$984$	0	0
$985$	− 48592.0i	− 1.57185i
$986$	−20266.7	−0.654588
$987$	0	0
$988$	−19553.0	−0.629621
$989$	70972.3i	2.28189i
$990$	0	0
$991$	−24190.5	−0.775414	−0.387707	−	0.921783i	$-0.626733\pi$
$991$	−24190.5	−0.775414	−0.387707	+	0.921783i	$0.626733\pi$
$992$	13496.7	0.431977
$993$	0	0
$994$	0	0
$995$	42998.0i	1.36998i
$996$	0	0
$997$	48866.7i	1.55228i	0.630560	+	0.776140i	$0.282825\pi$
$997$	48866.7i	1.55228i	−0.630560	+	0.776140i	$0.717175\pi$
$998$	12544.9i	0.397899i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.4.c.a.440.12		16
3.2	odd	2	inner	441.4.c.a.440.5		16
7.2	even	3		441.4.p.c.80.6		16
7.3	odd	6		441.4.p.c.215.3		16
7.4	even	3		63.4.p.a.26.3	yes	16
7.5	odd	6		63.4.p.a.17.6	yes	16
7.6	odd	2	inner	441.4.c.a.440.11		16
21.2	odd	6		441.4.p.c.80.3		16
21.5	even	6		63.4.p.a.17.3	✓	16
21.11	odd	6		63.4.p.a.26.6	yes	16
21.17	even	6		441.4.p.c.215.6		16
21.20	even	2	inner	441.4.c.a.440.6		16
28.11	odd	6		1008.4.bt.a.593.1		16
28.19	even	6		1008.4.bt.a.17.8		16
84.11	even	6		1008.4.bt.a.593.8		16
84.47	odd	6		1008.4.bt.a.17.1		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.p.a.17.3	✓	16	21.5	even	6
63.4.p.a.17.6	yes	16	7.5	odd	6
63.4.p.a.26.3	yes	16	7.4	even	3
63.4.p.a.26.6	yes	16	21.11	odd	6
441.4.c.a.440.5		16	3.2	odd	2	inner
441.4.c.a.440.6		16	21.20	even	2	inner
441.4.c.a.440.11		16	7.6	odd	2	inner
441.4.c.a.440.12		16	1.1	even	1	trivial
441.4.p.c.80.3		16	21.2	odd	6
441.4.p.c.80.6		16	7.2	even	3
441.4.p.c.215.3		16	7.3	odd	6
441.4.p.c.215.6		16	21.17	even	6
1008.4.bt.a.17.1		16	84.47	odd	6
1008.4.bt.a.17.8		16	28.19	even	6
1008.4.bt.a.593.1		16	28.11	odd	6
1008.4.bt.a.593.8		16	84.11	even	6

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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	441.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.82034i q^{2} +4.68636 q^{4} +15.0874 q^{5} +23.0935i q^{8} +O(q^{10})\)	\(q+1.82034i q^{2} +4.68636 q^{4} +15.0874 q^{5} +23.0935i q^{8} +27.4643i q^{10} -9.89034i q^{11} +67.8891i q^{13} -4.54712 q^{16} -70.1373 q^{17} +61.4580i q^{19} +70.7052 q^{20} +18.0038 q^{22} +131.515i q^{23} +102.631 q^{25} -123.581 q^{26} -158.738i q^{29} -76.4814i q^{31} +176.471i q^{32} -127.674i q^{34} +348.682 q^{37} -111.874 q^{38} +348.422i q^{40} +138.909 q^{41} +539.651 q^{43} -46.3497i q^{44} -239.402 q^{46} -223.643 q^{47} +186.823i q^{50} +318.153i q^{52} -530.011i q^{53} -149.220i q^{55} +288.958 q^{58} -542.876 q^{59} +134.197i q^{61} +139.222 q^{62} -357.614 q^{64} +1024.27i q^{65} +320.580 q^{67} -328.689 q^{68} +416.958i q^{71} +545.607i q^{73} +634.719i q^{74} +288.014i q^{76} -322.737 q^{79} -68.6044 q^{80} +252.861i q^{82} +885.170 q^{83} -1058.19 q^{85} +982.349i q^{86} +228.403 q^{88} -1624.62 q^{89} +616.327i q^{92} -407.106i q^{94} +927.244i q^{95} -739.155i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 64 q^{4}+O(q^{10}) \) 16 * q - 64 * q^4	\( 16 q - 64 q^{4} + 376 q^{16} + 528 q^{22} + 40 q^{25} + 2392 q^{37} + 328 q^{43} + 2784 q^{46} + 6744 q^{58} + 5432 q^{64} - 616 q^{67} + 4352 q^{79} - 4608 q^{85} - 1416 q^{88}+O(q^{100}) \) 16 * q - 64 * q^4 + 376 * q^16 + 528 * q^22 + 40 * q^25 + 2392 * q^37 + 328 * q^43 + 2784 * q^46 + 6744 * q^58 + 5432 * q^64 - 616 * q^67 + 4352 * q^79 - 4608 * q^85 - 1416 * q^88

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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