LMFDB - Embedded newform 441.4.c.a.440.16

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			440.16
Root			$4.21355 - 2.43270i$ of defining polynomial
Character	$\chi$	$=$	441.440
Dual form			441.4.c.a.440.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.86539i q^{2} -15.6720 q^{4} +12.7643 q^{5} -37.3274i q^{8} +O(q^{10})$	$q+4.86539i q^{2} -15.6720 q^{4} +12.7643 q^{5} -37.3274i q^{8} +62.1035i q^{10} -54.1131i q^{11} +8.85528i q^{13} +56.2362 q^{16} -68.9173 q^{17} -163.849i q^{19} -200.043 q^{20} +263.281 q^{22} -93.9704i q^{23} +37.9283 q^{25} -43.0844 q^{26} -119.620i q^{29} -98.8783i q^{31} -25.0082i q^{32} -335.310i q^{34} +94.1898 q^{37} +797.190 q^{38} -476.460i q^{40} -259.347 q^{41} +5.01418 q^{43} +848.062i q^{44} +457.203 q^{46} +57.3493 q^{47} +184.536i q^{50} -138.780i q^{52} +470.030i q^{53} -690.718i q^{55} +581.998 q^{58} -225.958 q^{59} +427.990i q^{61} +481.082 q^{62} +571.564 q^{64} +113.032i q^{65} +163.853 q^{67} +1080.07 q^{68} -79.8529i q^{71} -769.547i q^{73} +458.270i q^{74} +2567.85i q^{76} +534.817 q^{79} +717.818 q^{80} -1261.82i q^{82} -438.520 q^{83} -879.684 q^{85} +24.3959i q^{86} -2019.90 q^{88} +25.6483 q^{89} +1472.71i q^{92} +279.027i q^{94} -2091.43i q^{95} +1381.00i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	4.86539i	1.72018i	0.510146	+	0.860088i	$0.329591\pi$
$2$	4.86539i	1.72018i	−0.510146	+	0.860088i	$0.670409\pi$
$3$	0	0
$3$	0	0
$4$	−15.6720	−1.95900
$5$	12.7643	1.14168	0.570839	−	0.821062i	$-0.306618\pi$
$5$	12.7643	1.14168	0.570839	+	0.821062i	$0.306618\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	− 37.3274i	− 1.64965i
$9$	0	0
$10$	62.1035i	1.96388i
$11$	− 54.1131i	− 1.48325i	−0.670817	−	0.741623i	$-0.734056\pi$
$11$	− 54.1131i	− 1.48325i	0.670817	−	0.741623i	$-0.265944\pi$
$12$	0	0
$13$	8.85528i	0.188924i	0.995528	+	0.0944620i	$0.0301131\pi$
$13$	8.85528i	0.188924i	−0.995528	+	0.0944620i	$0.969887\pi$
$14$	0	0
$15$	0	0
$16$	56.2362	0.878691
$17$	−68.9173	−0.983230	−0.491615	−	0.870813i	$-0.663593\pi$
$17$	−68.9173	−0.983230	−0.491615	+	0.870813i	$0.663593\pi$
$18$	0	0
$19$	− 163.849i	− 1.97840i	−0.146578	−	0.989199i	$-0.546826\pi$
$19$	− 163.849i	− 1.97840i	0.146578	−	0.989199i	$-0.453174\pi$
$20$	−200.043	−2.23655
$21$	0	0
$22$	263.281	2.55144
$23$	− 93.9704i	− 0.851921i	−0.904742	−	0.425960i	$-0.859936\pi$
$23$	− 93.9704i	− 0.851921i	0.904742	−	0.425960i	$-0.140064\pi$
$24$	0	0
$25$	37.9283	0.303427
$26$	−43.0844	−0.324982
$27$	0	0
$28$	0	0
$29$	− 119.620i	− 0.765961i	−0.923756	−	0.382981i	$-0.874898\pi$
$29$	− 119.620i	− 0.765961i	0.923756	−	0.382981i	$-0.125102\pi$
$30$	0	0
$31$	− 98.8783i	− 0.572873i	−0.958099	−	0.286437i	$-0.907529\pi$
$31$	− 98.8783i	− 0.572873i	0.958099	−	0.286437i	$-0.0924708\pi$
$32$	− 25.0082i	− 0.138152i
$33$	0	0
$34$	− 335.310i	− 1.69133i
$35$	0	0
$36$	0	0
$37$	94.1898	0.418506	0.209253	−	0.977862i	$-0.432897\pi$
$37$	94.1898	0.418506	0.209253	+	0.977862i	$0.432897\pi$
$38$	797.190	3.40319
$39$	0	0
$40$	− 476.460i	− 1.88337i
$41$	−259.347	−0.987883	−0.493941	−	0.869495i	$-0.664444\pi$
$41$	−259.347	−0.987883	−0.493941	+	0.869495i	$0.664444\pi$
$42$	0	0
$43$	5.01418	0.0177827	0.00889133	−	0.999960i	$-0.497170\pi$
$43$	5.01418	0.0177827	0.00889133	+	0.999960i	$0.497170\pi$
$44$	848.062i	2.90568i
$45$	0	0
$46$	457.203	1.46545
$47$	57.3493	0.177984	0.0889921	−	0.996032i	$-0.471635\pi$
$47$	57.3493	0.177984	0.0889921	+	0.996032i	$0.471635\pi$
$48$	0	0
$49$	0	0
$50$	184.536i	0.521947i
$51$	0	0
$52$	− 138.780i	− 0.370103i
$53$	470.030i	1.21818i	0.793101	+	0.609090i	$0.208465\pi$
$53$	470.030i	1.21818i	−0.793101	+	0.609090i	$0.791535\pi$
$54$	0	0
$55$	− 690.718i	− 1.69339i
$56$	0	0
$57$	0	0
$58$	581.998	1.31759
$59$	−225.958	−0.498597	−0.249299	−	0.968427i	$-0.580200\pi$
$59$	−225.958	−0.498597	−0.249299	+	0.968427i	$0.580200\pi$
$60$	0	0
$61$	427.990i	0.898336i	0.893447	+	0.449168i	$0.148280\pi$
$61$	427.990i	0.898336i	−0.893447	+	0.449168i	$0.851720\pi$
$62$	481.082	0.985442
$63$	0	0
$64$	571.564	1.11634
$65$	113.032i	0.215690i
$66$	0	0
$67$	163.853	0.298774	0.149387	−	0.988779i	$-0.452270\pi$
$67$	163.853	0.298774	0.149387	+	0.988779i	$0.452270\pi$
$68$	1080.07	1.92615
$69$	0	0
$70$	0	0
$71$	− 79.8529i	− 0.133476i	−0.997771	−	0.0667380i	$-0.978741\pi$
$71$	− 79.8529i	− 0.133476i	0.997771	−	0.0667380i	$-0.0212592\pi$
$72$	0	0
$73$	− 769.547i	− 1.23382i	−0.787035	−	0.616909i	$-0.788385\pi$
$73$	− 769.547i	− 1.23382i	0.787035	−	0.616909i	$-0.211615\pi$
$74$	458.270i	0.719903i
$75$	0	0
$76$	2567.85i	3.87569i
$77$	0	0
$78$	0	0
$79$	534.817	0.761666	0.380833	−	0.924644i	$-0.375637\pi$
$79$	534.817	0.761666	0.380833	+	0.924644i	$0.375637\pi$
$80$	717.818	1.00318
$81$	0	0
$82$	− 1261.82i	− 1.69933i
$83$	−438.520	−0.579926	−0.289963	−	0.957038i	$-0.593643\pi$
$83$	−438.520	−0.579926	−0.289963	+	0.957038i	$0.593643\pi$
$84$	0	0
$85$	−879.684	−1.12253
$86$	24.3959i	0.0305893i
$87$	0	0
$88$	−2019.90	−2.44684
$89$	25.6483	0.0305474	0.0152737	−	0.999883i	$-0.495138\pi$
$89$	25.6483	0.0305474	0.0152737	+	0.999883i	$0.495138\pi$
$90$	0	0
$91$	0	0
$92$	1472.71i	1.66892i
$93$	0	0
$94$	279.027i	0.306164i
$95$	− 2091.43i	− 2.25869i
$96$	0	0
$97$	1381.00i	1.44555i	0.691081	+	0.722777i	$0.257135\pi$
$97$	1381.00i	1.44555i	−0.691081	+	0.722777i	$0.742865\pi$
$98$	0	0
$99$	0	0
$100$	−594.414	−0.594414
$101$	−713.616	−0.703044	−0.351522	−	0.936180i	$-0.614336\pi$
$101$	−713.616	−0.703044	−0.351522	+	0.936180i	$0.614336\pi$
$102$	0	0
$103$	1792.58i	1.71483i	0.514624	+	0.857416i	$0.327932\pi$
$103$	1792.58i	1.71483i	−0.514624	+	0.857416i	$0.672068\pi$
$104$	330.544	0.311659
$105$	0	0
$106$	−2286.88	−2.09548
$107$	22.5589i	0.0203818i	0.999948	+	0.0101909i	$0.00324391\pi$
$107$	22.5589i	0.0203818i	−0.999948	+	0.0101909i	$0.996756\pi$
$108$	0	0
$109$	952.420	0.836930	0.418465	−	0.908233i	$-0.362568\pi$
$109$	952.420	0.836930	0.418465	+	0.908233i	$0.362568\pi$
$110$	3360.61	2.91292
$111$	0	0
$112$	0	0
$113$	120.145i	0.100020i	0.998749	+	0.0500102i	$0.0159254\pi$
$113$	120.145i	0.100020i	−0.998749	+	0.0500102i	$0.984075\pi$
$114$	0	0
$115$	− 1199.47i	− 0.972619i
$116$	1874.69i	1.50052i
$117$	0	0
$118$	− 1099.37i	− 0.857674i
$119$	0	0
$120$	0	0
$121$	−1597.22	−1.20002
$122$	−2082.34	−1.54530
$123$	0	0
$124$	1549.62i	1.12226i
$125$	−1111.41	−0.795262
$126$	0	0
$127$	884.302	0.617867	0.308934	−	0.951084i	$-0.400028\pi$
$127$	884.302	0.617867	0.308934	+	0.951084i	$0.400028\pi$
$128$	2580.82i	1.78214i
$129$	0	0
$130$	−549.944	−0.371025
$131$	−1606.88	−1.07171	−0.535853	−	0.844311i	$-0.680010\pi$
$131$	−1606.88	−1.07171	−0.535853	+	0.844311i	$0.680010\pi$
$132$	0	0
$133$	0	0
$134$	797.210i	0.513944i
$135$	0	0
$136$	2572.50i	1.62199i
$137$	− 710.483i	− 0.443071i	−0.975152	−	0.221535i	$-0.928893\pi$
$137$	− 710.483i	− 0.443071i	0.975152	−	0.221535i	$-0.0711068\pi$
$138$	0	0
$139$	− 1531.91i	− 0.934782i	−0.884051	−	0.467391i	$-0.845194\pi$
$139$	− 1531.91i	− 0.934782i	0.884051	−	0.467391i	$-0.154806\pi$
$140$	0	0
$141$	0	0
$142$	388.515	0.229602
$143$	479.186	0.280221
$144$	0	0
$145$	− 1526.87i	− 0.874480i
$146$	3744.15	2.12238
$147$	0	0
$148$	−1476.15	−0.819854
$149$	− 2401.07i	− 1.32016i	−0.751197	−	0.660079i	$-0.770523\pi$
$149$	− 2401.07i	− 1.32016i	0.751197	−	0.660079i	$-0.229477\pi$
$150$	0	0
$151$	2467.97	1.33007	0.665035	−	0.746812i	$-0.268416\pi$
$151$	2467.97	1.33007	0.665035	+	0.746812i	$0.268416\pi$
$152$	−6116.06	−3.26367
$153$	0	0
$154$	0	0
$155$	− 1262.12i	− 0.654036i
$156$	0	0
$157$	− 2436.12i	− 1.23837i	−0.785246	−	0.619184i	$-0.787463\pi$
$157$	− 2436.12i	− 1.23837i	0.785246	−	0.619184i	$-0.212537\pi$
$158$	2602.09i	1.31020i
$159$	0	0
$160$	− 319.213i	− 0.157725i
$161$	0	0
$162$	0	0
$163$	3277.01	1.57469	0.787347	−	0.616510i	$-0.211454\pi$
$163$	3277.01	1.57469	0.787347	+	0.616510i	$0.211454\pi$
$164$	4064.49	1.93527
$165$	0	0
$166$	− 2133.57i	− 0.997574i
$167$	365.585	0.169400	0.0847000	−	0.996406i	$-0.473007\pi$
$167$	365.585	0.169400	0.0847000	+	0.996406i	$0.473007\pi$
$168$	0	0
$169$	2118.58	0.964308
$170$	− 4280.01i	− 1.93095i
$171$	0	0
$172$	−78.5823	−0.0348363
$173$	2092.93	0.919785	0.459892	−	0.887975i	$-0.347888\pi$
$173$	2092.93	0.919785	0.459892	+	0.887975i	$0.347888\pi$
$174$	0	0
$175$	0	0
$176$	− 3043.11i	− 1.30331i
$177$	0	0
$178$	124.789i	0.0525469i
$179$	− 1759.78i	− 0.734817i	−0.930060	−	0.367408i	$-0.880245\pi$
$179$	− 1759.78i	− 0.734817i	0.930060	−	0.367408i	$-0.119755\pi$
$180$	0	0
$181$	− 3197.54i	− 1.31310i	−0.754282	−	0.656551i	$-0.772015\pi$
$181$	− 3197.54i	− 1.31310i	0.754282	−	0.656551i	$-0.227985\pi$
$182$	0	0
$183$	0	0
$184$	−3507.67	−1.40537
$185$	1202.27	0.477798
$186$	0	0
$187$	3729.33i	1.45837i
$188$	−898.780	−0.348672
$189$	0	0
$190$	10175.6	3.88535
$191$	549.374i	0.208122i	0.994571	+	0.104061i	$0.0331837\pi$
$191$	549.374i	0.208122i	−0.994571	+	0.104061i	$0.966816\pi$
$192$	0	0
$193$	−704.476	−0.262742	−0.131371	−	0.991333i	$-0.541938\pi$
$193$	−704.476	−0.262742	−0.131371	+	0.991333i	$0.541938\pi$
$194$	−6719.08	−2.48661
$195$	0	0
$196$	0	0
$197$	− 5317.81i	− 1.92324i	−0.274384	−	0.961620i	$-0.588474\pi$
$197$	− 5317.81i	− 1.92324i	0.274384	−	0.961620i	$-0.411526\pi$
$198$	0	0
$199$	2489.30i	0.886743i	0.896338	+	0.443371i	$0.146218\pi$
$199$	2489.30i	0.886743i	−0.896338	+	0.443371i	$0.853782\pi$
$200$	− 1415.77i	− 0.500549i
$201$	0	0
$202$	− 3472.02i	− 1.20936i
$203$	0	0
$204$	0	0
$205$	−3310.39	−1.12784
$206$	−8721.58	−2.94981
$207$	0	0
$208$	497.987i	0.166006i
$209$	−8866.38	−2.93445
$210$	0	0
$211$	−3454.31	−1.12704	−0.563519	−	0.826103i	$-0.690553\pi$
$211$	−3454.31	−1.12704	−0.563519	+	0.826103i	$0.690553\pi$
$212$	− 7366.32i	− 2.38642i
$213$	0	0
$214$	−109.758	−0.0350602
$215$	64.0026	0.0203021
$216$	0	0
$217$	0	0
$218$	4633.90i	1.43967i
$219$	0	0
$220$	10824.9i	3.31735i
$221$	− 610.282i	− 0.185756i
$222$	0	0
$223$	3896.38i	1.17005i	0.811016	+	0.585024i	$0.198915\pi$
$223$	3896.38i	1.17005i	−0.811016	+	0.585024i	$0.801085\pi$
$224$	0	0
$225$	0	0
$226$	−584.553	−0.172053
$227$	−605.495	−0.177040	−0.0885201	−	0.996074i	$-0.528214\pi$
$227$	−605.495	−0.177040	−0.0885201	+	0.996074i	$0.528214\pi$
$228$	0	0
$229$	− 2208.92i	− 0.637422i	−0.947852	−	0.318711i	$-0.896750\pi$
$229$	− 2208.92i	− 0.637422i	0.947852	−	0.318711i	$-0.103250\pi$
$230$	5835.89	1.67307
$231$	0	0
$232$	−4465.10	−1.26357
$233$	2385.35i	0.670683i	0.942097	+	0.335342i	$0.108852\pi$
$233$	2385.35i	0.670683i	−0.942097	+	0.335342i	$0.891148\pi$
$234$	0	0
$235$	732.026	0.203201
$236$	3541.22	0.976753
$237$	0	0
$238$	0	0
$239$	− 3017.95i	− 0.816798i	−0.912803	−	0.408399i	$-0.866087\pi$
$239$	− 3017.95i	− 0.816798i	0.912803	−	0.408399i	$-0.133913\pi$
$240$	0	0
$241$	2515.49i	0.672353i	0.941799	+	0.336176i	$0.109134\pi$
$241$	2515.49i	0.672353i	−0.941799	+	0.336176i	$0.890866\pi$
$242$	− 7771.12i	− 2.06424i
$243$	0	0
$244$	− 6707.47i	− 1.75984i
$245$	0	0
$246$	0	0
$247$	1450.93	0.373767
$248$	−3690.87	−0.945042
$249$	0	0
$250$	− 5407.45i	− 1.36799i
$251$	1306.11	0.328451	0.164226	−	0.986423i	$-0.447488\pi$
$251$	1306.11	0.328451	0.164226	+	0.986423i	$0.447488\pi$
$252$	0	0
$253$	−5085.03	−1.26361
$254$	4302.48i	1.06284i
$255$	0	0
$256$	−7984.17	−1.94926
$257$	−7471.83	−1.81354	−0.906770	−	0.421626i	$-0.861459\pi$
$257$	−7471.83	−1.81354	−0.906770	+	0.421626i	$0.861459\pi$
$258$	0	0
$259$	0	0
$260$	− 1771.44i	− 0.422538i
$261$	0	0
$262$	− 7818.09i	− 1.84352i
$263$	1536.64i	0.360278i	0.983641	+	0.180139i	$0.0576547\pi$
$263$	1536.64i	0.360278i	−0.983641	+	0.180139i	$0.942345\pi$
$264$	0	0
$265$	5999.62i	1.39077i
$266$	0	0
$267$	0	0
$268$	−2567.91	−0.585300
$269$	−3916.18	−0.887636	−0.443818	−	0.896117i	$-0.646376\pi$
$269$	−3916.18	−0.887636	−0.443818	+	0.896117i	$0.646376\pi$
$270$	0	0
$271$	3599.68i	0.806882i	0.915006	+	0.403441i	$0.132186\pi$
$271$	3599.68i	0.806882i	−0.915006	+	0.403441i	$0.867814\pi$
$272$	−3875.65	−0.863955
$273$	0	0
$274$	3456.78	0.762159
$275$	− 2052.42i	− 0.450056i
$276$	0	0
$277$	284.081	0.0616200	0.0308100	−	0.999525i	$-0.490191\pi$
$277$	284.081	0.0616200	0.0308100	+	0.999525i	$0.490191\pi$
$278$	7453.33	1.60799
$279$	0	0
$280$	0	0
$281$	321.256i	0.0682011i	0.999418	+	0.0341006i	$0.0108566\pi$
$281$	321.256i	0.0682011i	−0.999418	+	0.0341006i	$0.989143\pi$
$282$	0	0
$283$	− 6802.80i	− 1.42892i	−0.699676	−	0.714460i	$-0.746672\pi$
$283$	− 6802.80i	− 1.42892i	0.699676	−	0.714460i	$-0.253328\pi$
$284$	1251.46i	0.261480i
$285$	0	0
$286$	2331.43i	0.482029i
$287$	0	0
$288$	0	0
$289$	−163.403	−0.0332594
$290$	7428.82	1.50426
$291$	0	0
$292$	12060.4i	2.41705i
$293$	3180.05	0.634063	0.317031	−	0.948415i	$-0.397314\pi$
$293$	3180.05	0.634063	0.317031	+	0.948415i	$0.397314\pi$
$294$	0	0
$295$	−2884.20	−0.569237
$296$	− 3515.86i	− 0.690390i
$297$	0	0
$298$	11682.1	2.27090
$299$	832.134	0.160948
$300$	0	0
$301$	0	0
$302$	12007.6i	2.28795i
$303$	0	0
$304$	− 9214.25i	− 1.73840i
$305$	5463.01i	1.02561i
$306$	0	0
$307$	2976.39i	0.553328i	0.960967	+	0.276664i	$0.0892289\pi$
$307$	2976.39i	0.553328i	−0.960967	+	0.276664i	$0.910771\pi$
$308$	0	0
$309$	0	0
$310$	6140.69	1.12506
$311$	4681.79	0.853633	0.426817	−	0.904338i	$-0.359635\pi$
$311$	4681.79	0.853633	0.426817	+	0.904338i	$0.359635\pi$
$312$	0	0
$313$	− 982.047i	− 0.177344i	−0.996061	−	0.0886718i	$-0.971738\pi$
$313$	− 982.047i	− 0.177344i	0.996061	−	0.0886718i	$-0.0282622\pi$
$314$	11852.7	2.13021
$315$	0	0
$316$	−8381.66	−1.49211
$317$	4930.34i	0.873550i	0.899571	+	0.436775i	$0.143879\pi$
$317$	4930.34i	0.873550i	−0.899571	+	0.436775i	$0.856121\pi$
$318$	0	0
$319$	−6473.01	−1.13611
$320$	7295.64	1.27450
$321$	0	0
$322$	0	0
$323$	11292.0i	1.94522i
$324$	0	0
$325$	335.866i	0.0573246i
$326$	15943.9i	2.70875i
$327$	0	0
$328$	9680.75i	1.62966i
$329$	0	0
$330$	0	0
$331$	−4034.51	−0.669959	−0.334980	−	0.942225i	$-0.608729\pi$
$331$	−4034.51	−0.669959	−0.334980	+	0.942225i	$0.608729\pi$
$332$	6872.50	1.13608
$333$	0	0
$334$	1778.71i	0.291398i
$335$	2091.48	0.341104
$336$	0	0
$337$	2771.62	0.448011	0.224006	−	0.974588i	$-0.428087\pi$
$337$	2771.62	0.448011	0.224006	+	0.974588i	$0.428087\pi$
$338$	10307.7i	1.65878i
$339$	0	0
$340$	13786.4	2.19904
$341$	−5350.61	−0.849712
$342$	0	0
$343$	0	0
$344$	− 187.166i	− 0.0293352i
$345$	0	0
$346$	10182.9i	1.58219i
$347$	4322.21i	0.668670i	0.942454	+	0.334335i	$0.108512\pi$
$347$	4322.21i	0.668670i	−0.942454	+	0.334335i	$0.891488\pi$
$348$	0	0
$349$	− 1331.65i	− 0.204245i	−0.994772	−	0.102122i	$-0.967437\pi$
$349$	− 1331.65i	− 0.204245i	0.994772	−	0.102122i	$-0.0325634\pi$
$350$	0	0
$351$	0	0
$352$	−1353.27	−0.204913
$353$	11348.5	1.71111	0.855553	−	0.517715i	$-0.173217\pi$
$353$	11348.5	1.71111	0.855553	+	0.517715i	$0.173217\pi$
$354$	0	0
$355$	− 1019.27i	− 0.152386i
$356$	−401.962	−0.0598425
$357$	0	0
$358$	8562.02	1.26401
$359$	− 8368.68i	− 1.23031i	−0.788406	−	0.615156i	$-0.789093\pi$
$359$	− 8368.68i	− 1.23031i	0.788406	−	0.615156i	$-0.210907\pi$
$360$	0	0
$361$	−19987.5	−2.91406
$362$	15557.3	2.25876
$363$	0	0
$364$	0	0
$365$	− 9822.76i	− 1.40862i
$366$	0	0
$367$	− 2715.06i	− 0.386171i	−0.981182	−	0.193086i	$-0.938150\pi$
$367$	− 2715.06i	− 0.386171i	0.981182	−	0.193086i	$-0.0618495\pi$
$368$	− 5284.54i	− 0.748575i
$369$	0	0
$370$	5849.52i	0.821897i
$371$	0	0
$372$	0	0
$373$	6097.11	0.846371	0.423186	−	0.906043i	$-0.360912\pi$
$373$	6097.11	0.846371	0.423186	+	0.906043i	$0.360912\pi$
$374$	−18144.6	−2.50865
$375$	0	0
$376$	− 2140.70i	− 0.293612i
$377$	1059.27	0.144708
$378$	0	0
$379$	−9922.24	−1.34478	−0.672389	−	0.740198i	$-0.734732\pi$
$379$	−9922.24	−1.34478	−0.672389	+	0.740198i	$0.734732\pi$
$380$	32776.9i	4.42479i
$381$	0	0
$382$	−2672.92	−0.358006
$383$	1219.06	0.162640	0.0813202	−	0.996688i	$-0.474086\pi$
$383$	1219.06	0.162640	0.0813202	+	0.996688i	$0.474086\pi$
$384$	0	0
$385$	0	0
$386$	− 3427.55i	− 0.451963i
$387$	0	0
$388$	− 21643.0i	− 2.83185i
$389$	− 13134.3i	− 1.71192i	−0.517045	−	0.855958i	$-0.672968\pi$
$389$	− 13134.3i	− 1.71192i	0.517045	−	0.855958i	$-0.327032\pi$
$390$	0	0
$391$	6476.19i	0.837634i
$392$	0	0
$393$	0	0
$394$	25873.2	3.30831
$395$	6826.58	0.869576
$396$	0	0
$397$	3115.02i	0.393799i	0.980424	+	0.196900i	$0.0630873\pi$
$397$	3115.02i	0.393799i	−0.980424	+	0.196900i	$0.936913\pi$
$398$	−12111.4	−1.52535
$399$	0	0
$400$	2132.94	0.266618
$401$	− 11228.8i	− 1.39836i	−0.714946	−	0.699179i	$-0.753549\pi$
$401$	− 11228.8i	− 1.39836i	0.714946	−	0.699179i	$-0.246451\pi$
$402$	0	0
$403$	875.595	0.108229
$404$	11183.8	1.37727
$405$	0	0
$406$	0	0
$407$	− 5096.90i	− 0.620747i
$408$	0	0
$409$	12400.8i	1.49922i	0.661880	+	0.749610i	$0.269759\pi$
$409$	12400.8i	1.49922i	−0.661880	+	0.749610i	$0.730241\pi$
$410$	− 16106.4i	− 1.94009i
$411$	0	0
$412$	− 28093.3i	− 3.35936i
$413$	0	0
$414$	0	0
$415$	−5597.42	−0.662088
$416$	221.454	0.0261002
$417$	0	0
$418$	− 43138.4i	− 5.04777i
$419$	−8260.19	−0.963095	−0.481547	−	0.876420i	$-0.659925\pi$
$419$	−8260.19	−0.963095	−0.481547	+	0.876420i	$0.659925\pi$
$420$	0	0
$421$	5571.81	0.645020	0.322510	−	0.946566i	$-0.395473\pi$
$421$	5571.81	0.645020	0.322510	+	0.946566i	$0.395473\pi$
$422$	− 16806.6i	− 1.93870i
$423$	0	0
$424$	17545.0	2.00957
$425$	−2613.92	−0.298338
$426$	0	0
$427$	0	0
$428$	− 353.543i	− 0.0399279i
$429$	0	0
$430$	311.398i	0.0349231i
$431$	7037.76i	0.786535i	0.919424	+	0.393268i	$0.128655\pi$
$431$	7037.76i	0.786535i	−0.919424	+	0.393268i	$0.871345\pi$
$432$	0	0
$433$	9212.26i	1.02243i	0.859452	+	0.511216i	$0.170805\pi$
$433$	9212.26i	1.02243i	−0.859452	+	0.511216i	$0.829195\pi$
$434$	0	0
$435$	0	0
$436$	−14926.4	−1.63955
$437$	−15397.0	−1.68544
$438$	0	0
$439$	− 8481.62i	− 0.922109i	−0.887372	−	0.461054i	$-0.847471\pi$
$439$	− 8481.62i	− 0.922109i	0.887372	−	0.461054i	$-0.152529\pi$
$440$	−25782.7	−2.79350
$441$	0	0
$442$	2969.26	0.319532
$443$	13055.0i	1.40014i	0.714076	+	0.700068i	$0.246847\pi$
$443$	13055.0i	1.40014i	−0.714076	+	0.700068i	$0.753153\pi$
$444$	0	0
$445$	327.384	0.0348753
$446$	−18957.4	−2.01269
$447$	0	0
$448$	0	0
$449$	14114.0i	1.48348i	0.670689	+	0.741738i	$0.265998\pi$
$449$	14114.0i	1.48348i	−0.670689	+	0.741738i	$0.734002\pi$
$450$	0	0
$451$	14034.1i	1.46527i
$452$	− 1882.92i	− 0.195940i
$453$	0	0
$454$	− 2945.97i	− 0.304540i
$455$	0	0
$456$	0	0
$457$	−8973.74	−0.918543	−0.459271	−	0.888296i	$-0.651889\pi$
$457$	−8973.74	−0.918543	−0.459271	+	0.888296i	$0.651889\pi$
$458$	10747.3	1.09648
$459$	0	0
$460$	18798.1i	1.90536i
$461$	955.010	0.0964842	0.0482421	−	0.998836i	$-0.484638\pi$
$461$	955.010	0.0964842	0.0482421	+	0.998836i	$0.484638\pi$
$462$	0	0
$463$	12004.5	1.20496	0.602479	−	0.798135i	$-0.294180\pi$
$463$	12004.5	1.20496	0.602479	+	0.798135i	$0.294180\pi$
$464$	− 6726.97i	− 0.673043i
$465$	0	0
$466$	−11605.6	−1.15369
$467$	−5064.91	−0.501876	−0.250938	−	0.968003i	$-0.580739\pi$
$467$	−5064.91	−0.501876	−0.250938	+	0.968003i	$0.580739\pi$
$468$	0	0
$469$	0	0
$470$	3561.59i	0.349540i
$471$	0	0
$472$	8434.43i	0.822513i
$473$	− 271.333i	− 0.0263761i
$474$	0	0
$475$	− 6214.52i	− 0.600299i
$476$	0	0
$477$	0	0
$478$	14683.5	1.40504
$479$	15213.7	1.45121	0.725607	−	0.688109i	$-0.241559\pi$
$479$	15213.7	1.45121	0.725607	+	0.688109i	$0.241559\pi$
$480$	0	0
$481$	834.077i	0.0790658i
$482$	−12238.8	−1.15656
$483$	0	0
$484$	25031.7	2.35084
$485$	17627.5i	1.65036i
$486$	0	0
$487$	15811.8	1.47126	0.735629	−	0.677384i	$-0.236887\pi$
$487$	15811.8	1.47126	0.735629	+	0.677384i	$0.236887\pi$
$488$	15975.8	1.48194
$489$	0	0
$490$	0	0
$491$	− 18064.2i	− 1.66034i	−0.557512	−	0.830169i	$-0.688244\pi$
$491$	− 18064.2i	− 1.66034i	0.557512	−	0.830169i	$-0.311756\pi$
$492$	0	0
$493$	8243.89i	0.753116i
$494$	7059.34i	0.642945i
$495$	0	0
$496$	− 5560.54i	− 0.503378i
$497$	0	0
$498$	0	0
$499$	−10524.7	−0.944185	−0.472092	−	0.881549i	$-0.656501\pi$
$499$	−10524.7	−0.944185	−0.472092	+	0.881549i	$0.656501\pi$
$500$	17418.1	1.55792
$501$	0	0
$502$	6354.76i	0.564994i
$503$	7790.82	0.690607	0.345304	−	0.938491i	$-0.387776\pi$
$503$	7790.82	0.690607	0.345304	+	0.938491i	$0.387776\pi$
$504$	0	0
$505$	−9108.83	−0.802649
$506$	− 24740.6i	− 2.17363i
$507$	0	0
$508$	−13858.8	−1.21040
$509$	10196.5	0.887919	0.443960	−	0.896047i	$-0.353573\pi$
$509$	10196.5	0.887919	0.443960	+	0.896047i	$0.353573\pi$
$510$	0	0
$511$	0	0
$512$	− 18199.6i	− 1.57093i
$513$	0	0
$514$	− 36353.4i	− 3.11961i
$515$	22881.0i	1.95778i
$516$	0	0
$517$	− 3103.35i	− 0.263994i
$518$	0	0
$519$	0	0
$520$	4219.18	0.355814
$521$	1927.58	0.162090	0.0810449	−	0.996710i	$-0.474174\pi$
$521$	1927.58	0.162090	0.0810449	+	0.996710i	$0.474174\pi$
$522$	0	0
$523$	− 7755.77i	− 0.648444i	−0.945981	−	0.324222i	$-0.894897\pi$
$523$	− 7755.77i	− 0.648444i	0.945981	−	0.324222i	$-0.105103\pi$
$524$	25183.0	2.09948
$525$	0	0
$526$	−7476.33	−0.619741
$527$	6814.43i	0.563266i
$528$	0	0
$529$	3336.56	0.274231
$530$	−29190.5	−2.39236
$531$	0	0
$532$	0	0
$533$	− 2296.59i	− 0.186635i
$534$	0	0
$535$	287.949i	0.0232694i
$536$	− 6116.22i	− 0.492874i
$537$	0	0
$538$	− 19053.8i	− 1.52689i
$539$	0	0
$540$	0	0
$541$	16760.8	1.33199	0.665993	−	0.745958i	$-0.268008\pi$
$541$	16760.8	1.33199	0.665993	+	0.745958i	$0.268008\pi$
$542$	−17513.9	−1.38798
$543$	0	0
$544$	1723.50i	0.135835i
$545$	12157.0	0.955503
$546$	0	0
$547$	5869.79	0.458819	0.229410	−	0.973330i	$-0.426320\pi$
$547$	5869.79	0.458819	0.229410	+	0.973330i	$0.426320\pi$
$548$	11134.7i	0.867977i
$549$	0	0
$550$	9985.82	0.774176
$551$	−19599.6	−1.51538
$552$	0	0
$553$	0	0
$554$	1382.16i	0.105997i
$555$	0	0
$556$	24008.1i	1.83124i
$557$	− 21658.1i	− 1.64755i	−0.566918	−	0.823774i	$-0.691864\pi$
$557$	− 21658.1i	− 1.64755i	0.566918	−	0.823774i	$-0.308136\pi$
$558$	0	0
$559$	44.4019i	0.00335957i
$560$	0	0
$561$	0	0
$562$	−1563.03	−0.117318
$563$	9590.71	0.717940	0.358970	−	0.933349i	$-0.383128\pi$
$563$	9590.71	0.717940	0.358970	+	0.933349i	$0.383128\pi$
$564$	0	0
$565$	1533.57i	0.114191i
$566$	33098.3	2.45799
$567$	0	0
$568$	−2980.70	−0.220189
$569$	− 16634.0i	− 1.22554i	−0.790260	−	0.612772i	$-0.790054\pi$
$569$	− 16634.0i	− 1.22554i	0.790260	−	0.612772i	$-0.209946\pi$
$570$	0	0
$571$	6331.03	0.464002	0.232001	−	0.972716i	$-0.425473\pi$
$571$	6331.03	0.464002	0.232001	+	0.972716i	$0.425473\pi$
$572$	−7509.82	−0.548953
$573$	0	0
$574$	0	0
$575$	− 3564.14i	− 0.258495i
$576$	0	0
$577$	− 9736.36i	− 0.702479i	−0.936286	−	0.351239i	$-0.885760\pi$
$577$	− 9736.36i	− 0.702479i	0.936286	−	0.351239i	$-0.114240\pi$
$578$	− 795.021i	− 0.0572120i
$579$	0	0
$580$	23929.1i	1.71311i
$581$	0	0
$582$	0	0
$583$	25434.7	1.80686
$584$	−28725.2	−2.03537
$585$	0	0
$586$	15472.2i	1.09070i
$587$	−10940.3	−0.769255	−0.384628	−	0.923072i	$-0.625670\pi$
$587$	−10940.3	−0.769255	−0.384628	+	0.923072i	$0.625670\pi$
$588$	0	0
$589$	−16201.1	−1.13337
$590$	− 14032.8i	− 0.979187i
$591$	0	0
$592$	5296.88	0.367737
$593$	24860.7	1.72160	0.860799	−	0.508946i	$-0.169965\pi$
$593$	24860.7	1.72160	0.860799	+	0.508946i	$0.169965\pi$
$594$	0	0
$595$	0	0
$596$	37629.6i	2.58619i
$597$	0	0
$598$	4048.66i	0.276859i
$599$	23333.9i	1.59165i	0.605526	+	0.795825i	$0.292963\pi$
$599$	23333.9i	1.59165i	−0.605526	+	0.795825i	$0.707037\pi$
$600$	0	0
$601$	13012.4i	0.883175i	0.897218	+	0.441587i	$0.145584\pi$
$601$	13012.4i	0.883175i	−0.897218	+	0.441587i	$0.854416\pi$
$602$	0	0
$603$	0	0
$604$	−38678.1	−2.60561
$605$	−20387.5	−1.37003
$606$	0	0
$607$	8493.62i	0.567950i	0.958832	+	0.283975i	$0.0916532\pi$
$607$	8493.62i	0.567950i	−0.958832	+	0.283975i	$0.908347\pi$
$608$	−4097.57	−0.273320
$609$	0	0
$610$	−26579.7	−1.76423
$611$	507.844i	0.0336255i
$612$	0	0
$613$	−9139.58	−0.602193	−0.301097	−	0.953594i	$-0.597353\pi$
$613$	−9139.58	−0.602193	−0.301097	+	0.953594i	$0.597353\pi$
$614$	−14481.3	−0.951822
$615$	0	0
$616$	0	0
$617$	7360.91i	0.480290i	0.970737	+	0.240145i	$0.0771950\pi$
$617$	7360.91i	0.480290i	−0.970737	+	0.240145i	$0.922805\pi$
$618$	0	0
$619$	− 22953.7i	− 1.49045i	−0.666813	−	0.745225i	$-0.732342\pi$
$619$	− 22953.7i	− 1.49045i	0.666813	−	0.745225i	$-0.267658\pi$
$620$	19779.9i	1.28126i
$621$	0	0
$622$	22778.7i	1.46840i
$623$	0	0
$624$	0	0
$625$	−18927.5	−1.21136
$626$	4778.04	0.305062
$627$	0	0
$628$	38179.0i	2.42597i
$629$	−6491.31	−0.411487
$630$	0	0
$631$	21126.0	1.33282	0.666412	−	0.745584i	$-0.267829\pi$
$631$	21126.0	1.33282	0.666412	+	0.745584i	$0.267829\pi$
$632$	− 19963.3i	− 1.25648i
$633$	0	0
$634$	−23988.0	−1.50266
$635$	11287.5	0.705405
$636$	0	0
$637$	0	0
$638$	− 31493.7i	− 1.95431i
$639$	0	0
$640$	32942.4i	2.03463i
$641$	− 17837.2i	− 1.09911i	−0.835458	−	0.549554i	$-0.814798\pi$
$641$	− 17837.2i	− 1.09911i	0.835458	−	0.549554i	$-0.185202\pi$
$642$	0	0
$643$	− 25449.0i	− 1.56082i	−0.625266	−	0.780412i	$-0.715009\pi$
$643$	− 25449.0i	− 1.56082i	0.625266	−	0.780412i	$-0.284991\pi$
$644$	0	0
$645$	0	0
$646$	−54940.2	−3.34612
$647$	−26298.4	−1.59799	−0.798993	−	0.601341i	$-0.794633\pi$
$647$	−26298.4	−1.59799	−0.798993	+	0.601341i	$0.794633\pi$
$648$	0	0
$649$	12227.3i	0.739542i
$650$	−1634.12	−0.0986083
$651$	0	0
$652$	−51357.4	−3.08483
$653$	− 9472.92i	− 0.567694i	−0.958870	−	0.283847i	$-0.908389\pi$
$653$	− 9472.92i	− 0.567694i	0.958870	−	0.283847i	$-0.0916107\pi$
$654$	0	0
$655$	−20510.7	−1.22354
$656$	−14584.7	−0.868043
$657$	0	0
$658$	0	0
$659$	22384.5i	1.32318i	0.749865	+	0.661591i	$0.230118\pi$
$659$	22384.5i	1.32318i	−0.749865	+	0.661591i	$0.769882\pi$
$660$	0	0
$661$	22929.3i	1.34924i	0.738167	+	0.674618i	$0.235692\pi$
$661$	22929.3i	1.34924i	−0.738167	+	0.674618i	$0.764308\pi$
$662$	− 19629.5i	− 1.15245i
$663$	0	0
$664$	16368.8i	0.956677i
$665$	0	0
$666$	0	0
$667$	−11240.7	−0.652538
$668$	−5729.45	−0.331855
$669$	0	0
$670$	10175.9i	0.586758i
$671$	23159.9	1.33245
$672$	0	0
$673$	−4873.86	−0.279158	−0.139579	−	0.990211i	$-0.544575\pi$
$673$	−4873.86	−0.279158	−0.139579	+	0.990211i	$0.544575\pi$
$674$	13485.0i	0.770658i
$675$	0	0
$676$	−33202.5	−1.88908
$677$	16247.4	0.922363	0.461181	−	0.887306i	$-0.347426\pi$
$677$	16247.4	0.922363	0.461181	+	0.887306i	$0.347426\pi$
$678$	0	0
$679$	0	0
$680$	32836.3i	1.85179i
$681$	0	0
$682$	− 26032.8i	− 1.46165i
$683$	− 21692.4i	− 1.21528i	−0.794212	−	0.607641i	$-0.792116\pi$
$683$	− 21692.4i	− 1.21528i	0.794212	−	0.607641i	$-0.207884\pi$
$684$	0	0
$685$	− 9068.85i	− 0.505844i
$686$	0	0
$687$	0	0
$688$	281.978	0.0156255
$689$	−4162.24	−0.230143
$690$	0	0
$691$	− 22516.5i	− 1.23961i	−0.784757	−	0.619803i	$-0.787212\pi$
$691$	− 22516.5i	− 1.23961i	0.784757	−	0.619803i	$-0.212788\pi$
$692$	−32800.5	−1.80186
$693$	0	0
$694$	−21029.2	−1.15023
$695$	− 19553.8i	− 1.06722i
$696$	0	0
$697$	17873.5	0.971316
$698$	6478.99	0.351337
$699$	0	0
$700$	0	0
$701$	33929.0i	1.82807i	0.405631	+	0.914037i	$0.367052\pi$
$701$	33929.0i	1.82807i	−0.405631	+	0.914037i	$0.632948\pi$
$702$	0	0
$703$	− 15432.9i	− 0.827971i
$704$	− 30929.1i	− 1.65580i
$705$	0	0
$706$	55214.9i	2.94340i
$707$	0	0
$708$	0	0
$709$	19187.2	1.01635	0.508175	−	0.861254i	$-0.330320\pi$
$709$	19187.2	1.01635	0.508175	+	0.861254i	$0.330320\pi$
$710$	4959.14	0.262131
$711$	0	0
$712$	− 957.386i	− 0.0503927i
$713$	−9291.63	−0.488043
$714$	0	0
$715$	6116.49	0.319922
$716$	27579.3i	1.43951i
$717$	0	0
$718$	40716.9	2.11635
$719$	13766.9	0.714071	0.357036	−	0.934091i	$-0.383788\pi$
$719$	13766.9	0.714071	0.357036	+	0.934091i	$0.383788\pi$
$720$	0	0
$721$	0	0
$722$	− 97247.2i	− 5.01269i
$723$	0	0
$724$	50111.9i	2.57237i
$725$	− 4536.99i	− 0.232413i
$726$	0	0
$727$	− 12226.1i	− 0.623717i	−0.950129	−	0.311858i	$-0.899049\pi$
$727$	− 12226.1i	− 0.623717i	0.950129	−	0.311858i	$-0.100951\pi$
$728$	0	0
$729$	0	0
$730$	47791.6	2.42308
$731$	−345.564	−0.0174844
$732$	0	0
$733$	2605.89i	0.131311i	0.997842	+	0.0656553i	$0.0209138\pi$
$733$	2605.89i	0.131311i	−0.997842	+	0.0656553i	$0.979086\pi$
$734$	13209.8	0.664282
$735$	0	0
$736$	−2350.03	−0.117695
$737$	− 8866.61i	− 0.443156i
$738$	0	0
$739$	33743.4	1.67967	0.839833	−	0.542845i	$-0.182653\pi$
$739$	33743.4	1.67967	0.839833	+	0.542845i	$0.182653\pi$
$740$	−18842.0	−0.936009
$741$	0	0
$742$	0	0
$743$	14586.9i	0.720244i	0.932905	+	0.360122i	$0.117265\pi$
$743$	14586.9i	0.720244i	−0.932905	+	0.360122i	$0.882735\pi$
$744$	0	0
$745$	− 30648.1i	− 1.50719i
$746$	29664.8i	1.45591i
$747$	0	0
$748$	− 58446.1i	− 2.85695i
$749$	0	0
$750$	0	0
$751$	−7515.24	−0.365160	−0.182580	−	0.983191i	$-0.558445\pi$
$751$	−7515.24	−0.365160	−0.182580	+	0.983191i	$0.558445\pi$
$752$	3225.11	0.156393
$753$	0	0
$754$	5153.75i	0.248924i
$755$	31502.0	1.51851
$756$	0	0
$757$	23917.4	1.14834	0.574169	−	0.818737i	$-0.305325\pi$
$757$	23917.4	1.14834	0.574169	+	0.818737i	$0.305325\pi$
$758$	− 48275.6i	− 2.31326i
$759$	0	0
$760$	−78067.5	−3.72606
$761$	−12198.5	−0.581073	−0.290537	−	0.956864i	$-0.593834\pi$
$761$	−12198.5	−0.581073	−0.290537	+	0.956864i	$0.593834\pi$
$762$	0	0
$763$	0	0
$764$	− 8609.80i	− 0.407712i
$765$	0	0
$766$	5931.22i	0.279770i
$767$	− 2000.92i	− 0.0941969i
$768$	0	0
$769$	− 2013.08i	− 0.0943999i	−0.998885	−	0.0471999i	$-0.984970\pi$
$769$	− 2013.08i	− 0.0943999i	0.998885	−	0.0471999i	$-0.0150298\pi$
$770$	0	0
$771$	0	0
$772$	11040.6	0.514713
$773$	−30278.2	−1.40884	−0.704418	−	0.709785i	$-0.748792\pi$
$773$	−30278.2	−1.40884	−0.704418	+	0.709785i	$0.748792\pi$
$774$	0	0
$775$	− 3750.29i	− 0.173825i
$776$	51549.0	2.38467
$777$	0	0
$778$	63903.5	2.94480
$779$	42493.8i	1.95443i
$780$	0	0
$781$	−4321.08	−0.197978
$782$	−31509.2	−1.44088
$783$	0	0
$784$	0	0
$785$	− 31095.5i	− 1.41382i
$786$	0	0
$787$	− 28682.0i	− 1.29911i	−0.760313	−	0.649557i	$-0.774954\pi$
$787$	− 28682.0i	− 1.29911i	0.760313	−	0.649557i	$-0.225046\pi$
$788$	83340.9i	3.76763i
$789$	0	0
$790$	33214.0i	1.49582i
$791$	0	0
$792$	0	0
$793$	−3789.97	−0.169717
$794$	−15155.8	−0.677404
$795$	0	0
$796$	− 39012.4i	− 1.73713i
$797$	−27108.9	−1.20483	−0.602413	−	0.798184i	$-0.705794\pi$
$797$	−27108.9	−1.20483	−0.602413	+	0.798184i	$0.705794\pi$
$798$	0	0
$799$	−3952.36	−0.174999
$800$	− 948.519i	− 0.0419190i
$801$	0	0
$802$	54632.7	2.40542
$803$	−41642.6	−1.83006
$804$	0	0
$805$	0	0
$806$	4260.11i	0.186174i
$807$	0	0
$808$	26637.4i	1.15978i
$809$	19778.1i	0.859530i	0.902941	+	0.429765i	$0.141404\pi$
$809$	19778.1i	0.859530i	−0.902941	+	0.429765i	$0.858596\pi$
$810$	0	0
$811$	− 44456.6i	− 1.92488i	−0.271487	−	0.962442i	$-0.587516\pi$
$811$	− 44456.6i	− 1.92488i	0.271487	−	0.962442i	$-0.412484\pi$
$812$	0	0
$813$	0	0
$814$	24798.4	1.06779
$815$	41828.8	1.79779
$816$	0	0
$817$	− 821.568i	− 0.0351812i
$818$	−60334.8	−2.57892
$819$	0	0
$820$	51880.6	2.20945
$821$	11499.2i	0.488823i	0.969672	+	0.244411i	$0.0785948\pi$
$821$	11499.2i	0.488823i	−0.969672	+	0.244411i	$0.921405\pi$
$822$	0	0
$823$	1193.10	0.0505332	0.0252666	−	0.999681i	$-0.491957\pi$
$823$	1193.10	0.0505332	0.0252666	+	0.999681i	$0.491957\pi$
$824$	66912.2	2.82888
$825$	0	0
$826$	0	0
$827$	29552.9i	1.24263i	0.783561	+	0.621315i	$0.213401\pi$
$827$	29552.9i	1.24263i	−0.783561	+	0.621315i	$0.786599\pi$
$828$	0	0
$829$	13582.4i	0.569043i	0.958670	+	0.284521i	$0.0918347\pi$
$829$	13582.4i	0.569043i	−0.958670	+	0.284521i	$0.908165\pi$
$830$	− 27233.6i	− 1.13891i
$831$	0	0
$832$	5061.36i	0.210903i
$833$	0	0
$834$	0	0
$835$	4666.45	0.193400
$836$	138954.	5.74860
$837$	0	0
$838$	− 40189.1i	− 1.65669i
$839$	11623.6	0.478297	0.239149	−	0.970983i	$-0.423132\pi$
$839$	11623.6	0.478297	0.239149	+	0.970983i	$0.423132\pi$
$840$	0	0
$841$	10080.1	0.413303
$842$	27109.0i	1.10955i
$843$	0	0
$844$	54136.1	2.20787
$845$	27042.3	1.10093
$846$	0	0
$847$	0	0
$848$	26432.7i	1.07040i
$849$	0	0
$850$	− 12717.7i	− 0.513194i
$851$	− 8851.06i	− 0.356534i
$852$	0	0
$853$	1019.82i	0.0409354i	0.999791	+	0.0204677i	$0.00651553\pi$
$853$	1019.82i	0.0409354i	−0.999791	+	0.0204677i	$0.993484\pi$
$854$	0	0
$855$	0	0
$856$	842.064	0.0336228
$857$	−8598.86	−0.342744	−0.171372	−	0.985206i	$-0.554820\pi$
$857$	−8598.86	−0.342744	−0.171372	+	0.985206i	$0.554820\pi$
$858$	0	0
$859$	37160.7i	1.47603i	0.674786	+	0.738013i	$0.264236\pi$
$859$	37160.7i	1.47603i	−0.674786	+	0.738013i	$0.735764\pi$
$860$	−1003.05	−0.0397718
$861$	0	0
$862$	−34241.4	−1.35298
$863$	12279.4i	0.484350i	0.970233	+	0.242175i	$0.0778608\pi$
$863$	12279.4i	0.484350i	−0.970233	+	0.242175i	$0.922139\pi$
$864$	0	0
$865$	26714.9	1.05010
$866$	−44821.2	−1.75876
$867$	0	0
$868$	0	0
$869$	− 28940.6i	− 1.12974i
$870$	0	0
$871$	1450.97i	0.0564456i
$872$	− 35551.4i	− 1.38064i
$873$	0	0
$874$	− 74912.3i	− 2.89925i
$875$	0	0
$876$	0	0
$877$	31871.5	1.22717	0.613583	−	0.789630i	$-0.289727\pi$
$877$	31871.5	1.22717	0.613583	+	0.789630i	$0.289727\pi$
$878$	41266.4	1.58619
$879$	0	0
$880$	− 38843.3i	− 1.48796i
$881$	−29427.6	−1.12536	−0.562679	−	0.826676i	$-0.690229\pi$
$881$	−29427.6	−1.12536	−0.562679	+	0.826676i	$0.690229\pi$
$882$	0	0
$883$	846.860	0.0322753	0.0161377	−	0.999870i	$-0.494863\pi$
$883$	846.860	0.0322753	0.0161377	+	0.999870i	$0.494863\pi$
$884$	9564.35i	0.363896i
$885$	0	0
$886$	−63517.5	−2.40848
$887$	7880.13	0.298296	0.149148	−	0.988815i	$-0.452347\pi$
$887$	7880.13	0.298296	0.149148	+	0.988815i	$0.452347\pi$
$888$	0	0
$889$	0	0
$890$	1592.85i	0.0599916i
$891$	0	0
$892$	− 61064.2i	− 2.29213i
$893$	− 9396.64i	− 0.352124i
$894$	0	0
$895$	− 22462.4i	− 0.838923i
$896$	0	0
$897$	0	0
$898$	−68670.1	−2.55184
$899$	−11827.8	−0.438799
$900$	0	0
$901$	− 32393.2i	− 1.19775i
$902$	−68281.2	−2.52053
$903$	0	0
$904$	4484.71	0.164999
$905$	− 40814.5i	− 1.49914i
$906$	0	0
$907$	21293.2	0.779524	0.389762	−	0.920916i	$-0.372557\pi$
$907$	21293.2	0.779524	0.389762	+	0.920916i	$0.372557\pi$
$908$	9489.33	0.346822
$909$	0	0
$910$	0	0
$911$	− 14634.3i	− 0.532225i	−0.963942	−	0.266112i	$-0.914261\pi$
$911$	− 14634.3i	− 0.532225i	0.963942	−	0.266112i	$-0.0857393\pi$
$912$	0	0
$913$	23729.7i	0.860173i
$914$	− 43660.8i	− 1.58005i
$915$	0	0
$916$	34618.2i	1.24871i
$917$	0	0
$918$	0	0
$919$	12197.2	0.437812	0.218906	−	0.975746i	$-0.429751\pi$
$919$	12197.2	0.437812	0.218906	+	0.975746i	$0.429751\pi$
$920$	−44773.1	−1.60448
$921$	0	0
$922$	4646.49i	0.165970i
$923$	707.119	0.0252168
$924$	0	0
$925$	3572.46	0.126986
$926$	58406.4i	2.07274i
$927$	0	0
$928$	−2991.48	−0.105819
$929$	24696.2	0.872181	0.436091	−	0.899903i	$-0.356363\pi$
$929$	24696.2	0.872181	0.436091	+	0.899903i	$0.356363\pi$
$930$	0	0
$931$	0	0
$932$	− 37383.2i	− 1.31387i
$933$	0	0
$934$	− 24642.8i	− 0.863315i
$935$	47602.4i	1.66499i
$936$	0	0
$937$	− 14448.0i	− 0.503730i	−0.967762	−	0.251865i	$-0.918956\pi$
$937$	− 14448.0i	− 0.503730i	0.967762	−	0.251865i	$-0.0810439\pi$
$938$	0	0
$939$	0	0
$940$	−11472.3	−0.398070
$941$	−31160.8	−1.07950	−0.539752	−	0.841824i	$-0.681482\pi$
$941$	−31160.8	−1.07950	−0.539752	+	0.841824i	$0.681482\pi$
$942$	0	0
$943$	24370.9i	0.841598i
$944$	−12707.0	−0.438113
$945$	0	0
$946$	1320.14	0.0453715
$947$	− 10262.4i	− 0.352146i	−0.984377	−	0.176073i	$-0.943660\pi$
$947$	− 10262.4i	− 0.352146i	0.984377	−	0.176073i	$-0.0563395\pi$
$948$	0	0
$949$	6814.55	0.233098
$950$	30236.1	1.03262
$951$	0	0
$952$	0	0
$953$	12686.9i	0.431236i	0.976478	+	0.215618i	$0.0691766\pi$
$953$	12686.9i	0.431236i	−0.976478	+	0.215618i	$0.930823\pi$
$954$	0	0
$955$	7012.40i	0.237608i
$956$	47297.3i	1.60011i
$957$	0	0
$958$	74020.6i	2.49634i
$959$	0	0
$960$	0	0
$961$	20014.1	0.671816
$962$	−4058.11	−0.136007
$963$	0	0
$964$	− 39422.8i	− 1.31714i
$965$	−8992.17	−0.299967
$966$	0	0
$967$	16129.8	0.536401	0.268200	−	0.963363i	$-0.413571\pi$
$967$	16129.8	0.536401	0.268200	+	0.963363i	$0.413571\pi$
$968$	59620.3i	1.97962i
$969$	0	0
$970$	−85764.6	−2.83890
$971$	52764.8	1.74387	0.871937	−	0.489618i	$-0.162864\pi$
$971$	52764.8	1.74387	0.871937	+	0.489618i	$0.162864\pi$
$972$	0	0
$973$	0	0
$974$	76930.8i	2.53082i
$975$	0	0
$976$	24068.5i	0.789360i
$977$	− 53993.2i	− 1.76806i	−0.467430	−	0.884030i	$-0.654820\pi$
$977$	− 53993.2i	− 1.76806i	0.467430	−	0.884030i	$-0.345180\pi$
$978$	0	0
$979$	− 1387.91i	− 0.0453093i
$980$	0	0
$981$	0	0
$982$	87889.4	2.85607
$983$	−52238.4	−1.69496	−0.847480	−	0.530827i	$-0.821881\pi$
$983$	−52238.4	−1.69496	−0.847480	+	0.530827i	$0.821881\pi$
$984$	0	0
$985$	− 67878.3i	− 2.19572i
$986$	−40109.7	−1.29549
$987$	0	0
$988$	−22739.0	−0.732211
$989$	− 471.184i	− 0.0151494i
$990$	0	0
$991$	−9624.42	−0.308506	−0.154253	−	0.988031i	$-0.549297\pi$
$991$	−9624.42	−0.308506	−0.154253	+	0.988031i	$0.549297\pi$
$992$	−2472.77	−0.0791436
$993$	0	0
$994$	0	0
$995$	31774.3i	1.01237i
$996$	0	0
$997$	28894.1i	0.917838i	0.888478	+	0.458919i	$0.151763\pi$
$997$	28894.1i	0.917838i	−0.888478	+	0.458919i	$0.848237\pi$
$998$	− 51206.6i	− 1.62416i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.4.c.a.440.16		16
3.2	odd	2	inner	441.4.c.a.440.1		16
7.2	even	3		63.4.p.a.17.8	yes	16
7.3	odd	6		63.4.p.a.26.1	yes	16
7.4	even	3		441.4.p.c.215.1		16
7.5	odd	6		441.4.p.c.80.8		16
7.6	odd	2	inner	441.4.c.a.440.15		16
21.2	odd	6		63.4.p.a.17.1	✓	16
21.5	even	6		441.4.p.c.80.1		16
21.11	odd	6		441.4.p.c.215.8		16
21.17	even	6		63.4.p.a.26.8	yes	16
21.20	even	2	inner	441.4.c.a.440.2		16
28.3	even	6		1008.4.bt.a.593.7		16
28.23	odd	6		1008.4.bt.a.17.2		16
84.23	even	6		1008.4.bt.a.17.7		16
84.59	odd	6		1008.4.bt.a.593.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.p.a.17.1	✓	16	21.2	odd	6
63.4.p.a.17.8	yes	16	7.2	even	3
63.4.p.a.26.1	yes	16	7.3	odd	6
63.4.p.a.26.8	yes	16	21.17	even	6
441.4.c.a.440.1		16	3.2	odd	2	inner
441.4.c.a.440.2		16	21.20	even	2	inner
441.4.c.a.440.15		16	7.6	odd	2	inner
441.4.c.a.440.16		16	1.1	even	1	trivial
441.4.p.c.80.1		16	21.5	even	6
441.4.p.c.80.8		16	7.5	odd	6
441.4.p.c.215.1		16	7.4	even	3
441.4.p.c.215.8		16	21.11	odd	6
1008.4.bt.a.17.2		16	28.23	odd	6
1008.4.bt.a.17.7		16	84.23	even	6
1008.4.bt.a.593.2		16	84.59	odd	6
1008.4.bt.a.593.7		16	28.3	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)

\(f(q)\)	\(=\)	\(q+4.86539i q^{2} -15.6720 q^{4} +12.7643 q^{5} -37.3274i q^{8} +O(q^{10})\)	\(q+4.86539i q^{2} -15.6720 q^{4} +12.7643 q^{5} -37.3274i q^{8} +62.1035i q^{10} -54.1131i q^{11} +8.85528i q^{13} +56.2362 q^{16} -68.9173 q^{17} -163.849i q^{19} -200.043 q^{20} +263.281 q^{22} -93.9704i q^{23} +37.9283 q^{25} -43.0844 q^{26} -119.620i q^{29} -98.8783i q^{31} -25.0082i q^{32} -335.310i q^{34} +94.1898 q^{37} +797.190 q^{38} -476.460i q^{40} -259.347 q^{41} +5.01418 q^{43} +848.062i q^{44} +457.203 q^{46} +57.3493 q^{47} +184.536i q^{50} -138.780i q^{52} +470.030i q^{53} -690.718i q^{55} +581.998 q^{58} -225.958 q^{59} +427.990i q^{61} +481.082 q^{62} +571.564 q^{64} +113.032i q^{65} +163.853 q^{67} +1080.07 q^{68} -79.8529i q^{71} -769.547i q^{73} +458.270i q^{74} +2567.85i q^{76} +534.817 q^{79} +717.818 q^{80} -1261.82i q^{82} -438.520 q^{83} -879.684 q^{85} +24.3959i q^{86} -2019.90 q^{88} +25.6483 q^{89} +1472.71i q^{92} +279.027i q^{94} -2091.43i q^{95} +1381.00i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 64 q^{4}+O(q^{10}) \) 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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