LMFDB - Embedded newform 441.3.d.a.244.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,3,Mod(244,441)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("441.244");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 441 = 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	441.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$12.0163796583$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 21)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			244.2
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	441.244
Dual form			441.3.d.a.244.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{2} +5.00000 q^{4} +5.19615i q^{5} -3.00000 q^{8} +O(q^{10})$	$q-3.00000 q^{2} +5.00000 q^{4} +5.19615i q^{5} -3.00000 q^{8} -15.5885i q^{10} -15.0000 q^{11} +13.8564i q^{13} -11.0000 q^{16} -10.3923i q^{17} +10.3923i q^{19} +25.9808i q^{20} +45.0000 q^{22} -2.00000 q^{25} -41.5692i q^{26} +9.00000 q^{29} -12.1244i q^{31} +45.0000 q^{32} +31.1769i q^{34} +10.0000 q^{37} -31.1769i q^{38} -15.5885i q^{40} -10.3923i q^{41} -74.0000 q^{43} -75.0000 q^{44} +6.00000 q^{50} +69.2820i q^{52} -33.0000 q^{53} -77.9423i q^{55} -27.0000 q^{58} -15.5885i q^{59} -90.0666i q^{61} +36.3731i q^{62} -91.0000 q^{64} -72.0000 q^{65} -76.0000 q^{67} -51.9615i q^{68} -84.0000 q^{71} -62.3538i q^{73} -30.0000 q^{74} +51.9615i q^{76} -43.0000 q^{79} -57.1577i q^{80} +31.1769i q^{82} -119.512i q^{83} +54.0000 q^{85} +222.000 q^{86} +45.0000 q^{88} -72.7461i q^{89} -54.0000 q^{95} -185.329i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{2} + 10 q^{4} - 6 q^{8}+O(q^{10}) $ 2 * q - 6 * q^2 + 10 * q^4 - 6 * q^8	$ 2 q - 6 q^{2} + 10 q^{4} - 6 q^{8} - 30 q^{11} - 22 q^{16} + 90 q^{22} - 4 q^{25} + 18 q^{29} + 90 q^{32} + 20 q^{37} - 148 q^{43} - 150 q^{44} + 12 q^{50} - 66 q^{53} - 54 q^{58} - 182 q^{64} - 144 q^{65} - 152 q^{67} - 168 q^{71} - 60 q^{74} - 86 q^{79} + 108 q^{85} + 444 q^{86} + 90 q^{88} - 108 q^{95}+O(q^{100}) $ 2 * q - 6 * q^2 + 10 * q^4 - 6 * q^8 - 30 * q^11 - 22 * q^16 + 90 * q^22 - 4 * q^25 + 18 * q^29 + 90 * q^32 + 20 * q^37 - 148 * q^43 - 150 * q^44 + 12 * q^50 - 66 * q^53 - 54 * q^58 - 182 * q^64 - 144 * q^65 - 152 * q^67 - 168 * q^71 - 60 * q^74 - 86 * q^79 + 108 * q^85 + 444 * q^86 + 90 * q^88 - 108 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/441\mathbb{Z}\right)^\times$.

$n$	$199$	$344$
$\chi(n)$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−3.00000	−1.50000	−0.750000	−	0.661438i	$-0.769947\pi$
$2$	−3.00000	−1.50000	−0.750000	+	0.661438i	$0.769947\pi$
$3$	0	0
$3$	0	0
$4$	5.00000	1.25000
$5$	5.19615i	1.03923i	0.854400	+	0.519615i	$0.173925\pi$
$5$	5.19615i	1.03923i	−0.854400	+	0.519615i	$0.826075\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−3.00000	−0.375000
$9$	0	0
$10$	− 15.5885i	− 1.55885i
$11$	−15.0000	−1.36364	−0.681818	−	0.731522i	$-0.738810\pi$
$11$	−15.0000	−1.36364	−0.681818	+	0.731522i	$0.738810\pi$
$12$	0	0
$13$	13.8564i	1.06588i	0.846154	+	0.532939i	$0.178912\pi$
$13$	13.8564i	1.06588i	−0.846154	+	0.532939i	$0.821088\pi$
$14$	0	0
$15$	0	0
$16$	−11.0000	−0.687500
$17$	− 10.3923i	− 0.611312i	−0.952142	−	0.305656i	$-0.901124\pi$
$17$	− 10.3923i	− 0.611312i	0.952142	−	0.305656i	$-0.0988758\pi$
$18$	0	0
$19$	10.3923i	0.546963i	0.961877	+	0.273482i	$0.0881753\pi$
$19$	10.3923i	0.546963i	−0.961877	+	0.273482i	$0.911825\pi$
$20$	25.9808i	1.29904i
$21$	0	0
$22$	45.0000	2.04545
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	−2.00000	−0.0800000
$26$	− 41.5692i	− 1.59882i
$27$	0	0
$28$	0	0
$29$	9.00000	0.310345	0.155172	−	0.987887i	$-0.450407\pi$
$29$	9.00000	0.310345	0.155172	+	0.987887i	$0.450407\pi$
$30$	0	0
$31$	− 12.1244i	− 0.391108i	−0.980693	−	0.195554i	$-0.937349\pi$
$31$	− 12.1244i	− 0.391108i	0.980693	−	0.195554i	$-0.0626505\pi$
$32$	45.0000	1.40625
$33$	0	0
$34$	31.1769i	0.916968i
$35$	0	0
$36$	0	0
$37$	10.0000	0.270270	0.135135	−	0.990827i	$-0.456853\pi$
$37$	10.0000	0.270270	0.135135	+	0.990827i	$0.456853\pi$
$38$	− 31.1769i	− 0.820445i
$39$	0	0
$40$	− 15.5885i	− 0.389711i
$41$	− 10.3923i	− 0.253471i	−0.991937	−	0.126735i	$-0.959550\pi$
$41$	− 10.3923i	− 0.253471i	0.991937	−	0.126735i	$-0.0404499\pi$
$42$	0	0
$43$	−74.0000	−1.72093	−0.860465	−	0.509509i	$-0.829827\pi$
$43$	−74.0000	−1.72093	−0.860465	+	0.509509i	$0.829827\pi$
$44$	−75.0000	−1.70455
$45$	0	0
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	0	0
$50$	6.00000	0.120000
$51$	0	0
$52$	69.2820i	1.33235i
$53$	−33.0000	−0.622642	−0.311321	−	0.950305i	$-0.600771\pi$
$53$	−33.0000	−0.622642	−0.311321	+	0.950305i	$0.600771\pi$
$54$	0	0
$55$	− 77.9423i	− 1.41713i
$56$	0	0
$57$	0	0
$58$	−27.0000	−0.465517
$59$	− 15.5885i	− 0.264211i	−0.991236	−	0.132106i	$-0.957826\pi$
$59$	− 15.5885i	− 0.264211i	0.991236	−	0.132106i	$-0.0421738\pi$
$60$	0	0
$61$	− 90.0666i	− 1.47650i	−0.674526	−	0.738251i	$-0.735652\pi$
$61$	− 90.0666i	− 1.47650i	0.674526	−	0.738251i	$-0.264348\pi$
$62$	36.3731i	0.586662i
$63$	0	0
$64$	−91.0000	−1.42188
$65$	−72.0000	−1.10769
$66$	0	0
$67$	−76.0000	−1.13433	−0.567164	−	0.823605i	$-0.691960\pi$
$67$	−76.0000	−1.13433	−0.567164	+	0.823605i	$0.691960\pi$
$68$	− 51.9615i	− 0.764140i
$69$	0	0
$70$	0	0
$71$	−84.0000	−1.18310	−0.591549	−	0.806269i	$-0.701483\pi$
$71$	−84.0000	−1.18310	−0.591549	+	0.806269i	$0.701483\pi$
$72$	0	0
$73$	− 62.3538i	− 0.854162i	−0.904213	−	0.427081i	$-0.859542\pi$
$73$	− 62.3538i	− 0.854162i	0.904213	−	0.427081i	$-0.140458\pi$
$74$	−30.0000	−0.405405
$75$	0	0
$76$	51.9615i	0.683704i
$77$	0	0
$78$	0	0
$79$	−43.0000	−0.544304	−0.272152	−	0.962254i	$-0.587735\pi$
$79$	−43.0000	−0.544304	−0.272152	+	0.962254i	$0.587735\pi$
$80$	− 57.1577i	− 0.714471i
$81$	0	0
$82$	31.1769i	0.380206i
$83$	− 119.512i	− 1.43990i	−0.694027	−	0.719949i	$-0.744165\pi$
$83$	− 119.512i	− 1.43990i	0.694027	−	0.719949i	$-0.255835\pi$
$84$	0	0
$85$	54.0000	0.635294
$86$	222.000	2.58140
$87$	0	0
$88$	45.0000	0.511364
$89$	− 72.7461i	− 0.817372i	−0.912675	−	0.408686i	$-0.865987\pi$
$89$	− 72.7461i	− 0.817372i	0.912675	−	0.408686i	$-0.134013\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−54.0000	−0.568421
$96$	0	0
$97$	− 185.329i	− 1.91061i	−0.295618	−	0.955306i	$-0.595525\pi$
$97$	− 185.329i	− 1.91061i	0.295618	−	0.955306i	$-0.404475\pi$
$98$	0	0
$99$	0	0
$100$	−10.0000	−0.100000
$101$	145.492i	1.44052i	0.693706	+	0.720259i	$0.255977\pi$
$101$	145.492i	1.44052i	−0.693706	+	0.720259i	$0.744023\pi$
$102$	0	0
$103$	69.2820i	0.672641i	0.941748	+	0.336321i	$0.109183\pi$
$103$	69.2820i	0.672641i	−0.941748	+	0.336321i	$0.890817\pi$
$104$	− 41.5692i	− 0.399704i
$105$	0	0
$106$	99.0000	0.933962
$107$	−93.0000	−0.869159	−0.434579	−	0.900634i	$-0.643103\pi$
$107$	−93.0000	−0.869159	−0.434579	+	0.900634i	$0.643103\pi$
$108$	0	0
$109$	−8.00000	−0.0733945	−0.0366972	−	0.999326i	$-0.511684\pi$
$109$	−8.00000	−0.0733945	−0.0366972	+	0.999326i	$0.511684\pi$
$110$	233.827i	2.12570i
$111$	0	0
$112$	0	0
$113$	−42.0000	−0.371681	−0.185841	−	0.982580i	$-0.559501\pi$
$113$	−42.0000	−0.371681	−0.185841	+	0.982580i	$0.559501\pi$
$114$	0	0
$115$	0	0
$116$	45.0000	0.387931
$117$	0	0
$118$	46.7654i	0.396317i
$119$	0	0
$120$	0	0
$121$	104.000	0.859504
$122$	270.200i	2.21475i
$123$	0	0
$124$	− 60.6218i	− 0.488885i
$125$	119.512i	0.956092i
$126$	0	0
$127$	35.0000	0.275591	0.137795	−	0.990461i	$-0.455998\pi$
$127$	35.0000	0.275591	0.137795	+	0.990461i	$0.455998\pi$
$128$	93.0000	0.726562
$129$	0	0
$130$	216.000	1.66154
$131$	171.473i	1.30895i	0.756082	+	0.654477i	$0.227111\pi$
$131$	171.473i	1.30895i	−0.756082	+	0.654477i	$0.772889\pi$
$132$	0	0
$133$	0	0
$134$	228.000	1.70149
$135$	0	0
$136$	31.1769i	0.229242i
$137$	96.0000	0.700730	0.350365	−	0.936613i	$-0.386058\pi$
$137$	96.0000	0.700730	0.350365	+	0.936613i	$0.386058\pi$
$138$	0	0
$139$	183.597i	1.32084i	0.750894	+	0.660422i	$0.229623\pi$
$139$	183.597i	1.32084i	−0.750894	+	0.660422i	$0.770377\pi$
$140$	0	0
$141$	0	0
$142$	252.000	1.77465
$143$	− 207.846i	− 1.45347i
$144$	0	0
$145$	46.7654i	0.322520i
$146$	187.061i	1.28124i
$147$	0	0
$148$	50.0000	0.337838
$149$	186.000	1.24832	0.624161	−	0.781296i	$-0.285441\pi$
$149$	186.000	1.24832	0.624161	+	0.781296i	$0.285441\pi$
$150$	0	0
$151$	79.0000	0.523179	0.261589	−	0.965179i	$-0.415753\pi$
$151$	79.0000	0.523179	0.261589	+	0.965179i	$0.415753\pi$
$152$	− 31.1769i	− 0.205111i
$153$	0	0
$154$	0	0
$155$	63.0000	0.406452
$156$	0	0
$157$	− 20.7846i	− 0.132386i	−0.997807	−	0.0661930i	$-0.978915\pi$
$157$	− 20.7846i	− 0.132386i	0.997807	−	0.0661930i	$-0.0210853\pi$
$158$	129.000	0.816456
$159$	0	0
$160$	233.827i	1.46142i
$161$	0	0
$162$	0	0
$163$	−208.000	−1.27607	−0.638037	−	0.770006i	$-0.720253\pi$
$163$	−208.000	−1.27607	−0.638037	+	0.770006i	$0.720253\pi$
$164$	− 51.9615i	− 0.316839i
$165$	0	0
$166$	358.535i	2.15985i
$167$	− 249.415i	− 1.49350i	−0.665102	−	0.746752i	$-0.731612\pi$
$167$	− 249.415i	− 1.49350i	0.665102	−	0.746752i	$-0.268388\pi$
$168$	0	0
$169$	−23.0000	−0.136095
$170$	−162.000	−0.952941
$171$	0	0
$172$	−370.000	−2.15116
$173$	− 228.631i	− 1.32156i	−0.750578	−	0.660782i	$-0.770225\pi$
$173$	− 228.631i	− 1.32156i	0.750578	−	0.660782i	$-0.229775\pi$
$174$	0	0
$175$	0	0
$176$	165.000	0.937500
$177$	0	0
$178$	218.238i	1.22606i
$179$	−90.0000	−0.502793	−0.251397	−	0.967884i	$-0.580890\pi$
$179$	−90.0000	−0.502793	−0.251397	+	0.967884i	$0.580890\pi$
$180$	0	0
$181$	10.3923i	0.0574160i	0.999588	+	0.0287080i	$0.00913930\pi$
$181$	10.3923i	0.0574160i	−0.999588	+	0.0287080i	$0.990861\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	51.9615i	0.280873i
$186$	0	0
$187$	155.885i	0.833607i
$188$	0	0
$189$	0	0
$190$	162.000	0.852632
$191$	312.000	1.63351	0.816754	−	0.576986i	$-0.195771\pi$
$191$	312.000	1.63351	0.816754	+	0.576986i	$0.195771\pi$
$192$	0	0
$193$	−185.000	−0.958549	−0.479275	−	0.877665i	$-0.659100\pi$
$193$	−185.000	−0.958549	−0.479275	+	0.877665i	$0.659100\pi$
$194$	555.988i	2.86592i
$195$	0	0
$196$	0	0
$197$	−330.000	−1.67513	−0.837563	−	0.546340i	$-0.816021\pi$
$197$	−330.000	−1.67513	−0.837563	+	0.546340i	$0.816021\pi$
$198$	0	0
$199$	6.92820i	0.0348151i	0.999848	+	0.0174075i	$0.00554127\pi$
$199$	6.92820i	0.0348151i	−0.999848	+	0.0174075i	$0.994459\pi$
$200$	6.00000	0.0300000
$201$	0	0
$202$	− 436.477i	− 2.16078i
$203$	0	0
$204$	0	0
$205$	54.0000	0.263415
$206$	− 207.846i	− 1.00896i
$207$	0	0
$208$	− 152.420i	− 0.732791i
$209$	− 155.885i	− 0.745859i
$210$	0	0
$211$	−248.000	−1.17536	−0.587678	−	0.809095i	$-0.699958\pi$
$211$	−248.000	−1.17536	−0.587678	+	0.809095i	$0.699958\pi$
$212$	−165.000	−0.778302
$213$	0	0
$214$	279.000	1.30374
$215$	− 384.515i	− 1.78844i
$216$	0	0
$217$	0	0
$218$	24.0000	0.110092
$219$	0	0
$220$	− 389.711i	− 1.77142i
$221$	144.000	0.651584
$222$	0	0
$223$	192.258i	0.862142i	0.902318	+	0.431071i	$0.141864\pi$
$223$	192.258i	0.862142i	−0.902318	+	0.431071i	$0.858136\pi$
$224$	0	0
$225$	0	0
$226$	126.000	0.557522
$227$	88.3346i	0.389139i	0.980889	+	0.194570i	$0.0623310\pi$
$227$	88.3346i	0.389139i	−0.980889	+	0.194570i	$0.937669\pi$
$228$	0	0
$229$	− 329.090i	− 1.43707i	−0.695489	−	0.718536i	$-0.744812\pi$
$229$	− 329.090i	− 1.43707i	0.695489	−	0.718536i	$-0.255188\pi$
$230$	0	0
$231$	0	0
$232$	−27.0000	−0.116379
$233$	−270.000	−1.15880	−0.579399	−	0.815044i	$-0.696713\pi$
$233$	−270.000	−1.15880	−0.579399	+	0.815044i	$0.696713\pi$
$234$	0	0
$235$	0	0
$236$	− 77.9423i	− 0.330264i
$237$	0	0
$238$	0	0
$239$	228.000	0.953975	0.476987	−	0.878910i	$-0.341729\pi$
$239$	228.000	0.953975	0.476987	+	0.878910i	$0.341729\pi$
$240$	0	0
$241$	445.137i	1.84704i	0.383548	+	0.923521i	$0.374702\pi$
$241$	445.137i	1.84704i	−0.383548	+	0.923521i	$0.625298\pi$
$242$	−312.000	−1.28926
$243$	0	0
$244$	− 450.333i	− 1.84563i
$245$	0	0
$246$	0	0
$247$	−144.000	−0.582996
$248$	36.3731i	0.146666i
$249$	0	0
$250$	− 358.535i	− 1.43414i
$251$	− 5.19615i	− 0.0207018i	−0.999946	−	0.0103509i	$-0.996705\pi$
$251$	− 5.19615i	− 0.0207018i	0.999946	−	0.0103509i	$-0.00329485\pi$
$252$	0	0
$253$	0	0
$254$	−105.000	−0.413386
$255$	0	0
$256$	85.0000	0.332031
$257$	114.315i	0.444807i	0.974955	+	0.222403i	$0.0713902\pi$
$257$	114.315i	0.444807i	−0.974955	+	0.222403i	$0.928610\pi$
$258$	0	0
$259$	0	0
$260$	−360.000	−1.38462
$261$	0	0
$262$	− 514.419i	− 1.96343i
$263$	−186.000	−0.707224	−0.353612	−	0.935392i	$-0.615047\pi$
$263$	−186.000	−0.707224	−0.353612	+	0.935392i	$0.615047\pi$
$264$	0	0
$265$	− 171.473i	− 0.647068i
$266$	0	0
$267$	0	0
$268$	−380.000	−1.41791
$269$	337.750i	1.25558i	0.778384	+	0.627788i	$0.216039\pi$
$269$	337.750i	1.25558i	−0.778384	+	0.627788i	$0.783961\pi$
$270$	0	0
$271$	− 91.7987i	− 0.338741i	−0.985552	−	0.169370i	$-0.945827\pi$
$271$	− 91.7987i	− 0.338741i	0.985552	−	0.169370i	$-0.0541734\pi$
$272$	114.315i	0.420277i
$273$	0	0
$274$	−288.000	−1.05109
$275$	30.0000	0.109091
$276$	0	0
$277$	380.000	1.37184	0.685921	−	0.727676i	$-0.259400\pi$
$277$	380.000	1.37184	0.685921	+	0.727676i	$0.259400\pi$
$278$	− 550.792i	− 1.98127i
$279$	0	0
$280$	0	0
$281$	−300.000	−1.06762	−0.533808	−	0.845606i	$-0.679239\pi$
$281$	−300.000	−1.06762	−0.533808	+	0.845606i	$0.679239\pi$
$282$	0	0
$283$	204.382i	0.722198i	0.932528	+	0.361099i	$0.117598\pi$
$283$	204.382i	0.722198i	−0.932528	+	0.361099i	$0.882402\pi$
$284$	−420.000	−1.47887
$285$	0	0
$286$	623.538i	2.18020i
$287$	0	0
$288$	0	0
$289$	181.000	0.626298
$290$	− 140.296i	− 0.483780i
$291$	0	0
$292$	− 311.769i	− 1.06770i
$293$	545.596i	1.86210i	0.364889	+	0.931051i	$0.381107\pi$
$293$	545.596i	1.86210i	−0.364889	+	0.931051i	$0.618893\pi$
$294$	0	0
$295$	81.0000	0.274576
$296$	−30.0000	−0.101351
$297$	0	0
$298$	−558.000	−1.87248
$299$	0	0
$300$	0	0
$301$	0	0
$302$	−237.000	−0.784768
$303$	0	0
$304$	− 114.315i	− 0.376037i
$305$	468.000	1.53443
$306$	0	0
$307$	− 173.205i	− 0.564186i	−0.959387	−	0.282093i	$-0.908971\pi$
$307$	− 173.205i	− 0.564186i	0.959387	−	0.282093i	$-0.0910287\pi$
$308$	0	0
$309$	0	0
$310$	−189.000	−0.609677
$311$	176.669i	0.568068i	0.958814	+	0.284034i	$0.0916729\pi$
$311$	176.669i	0.568068i	−0.958814	+	0.284034i	$0.908327\pi$
$312$	0	0
$313$	213.042i	0.680646i	0.940309	+	0.340323i	$0.110536\pi$
$313$	213.042i	0.680646i	−0.940309	+	0.340323i	$0.889464\pi$
$314$	62.3538i	0.198579i
$315$	0	0
$316$	−215.000	−0.680380
$317$	−117.000	−0.369085	−0.184543	−	0.982825i	$-0.559080\pi$
$317$	−117.000	−0.369085	−0.184543	+	0.982825i	$0.559080\pi$
$318$	0	0
$319$	−135.000	−0.423197
$320$	− 472.850i	− 1.47766i
$321$	0	0
$322$	0	0
$323$	108.000	0.334365
$324$	0	0
$325$	− 27.7128i	− 0.0852702i
$326$	624.000	1.91411
$327$	0	0
$328$	31.1769i	0.0950516i
$329$	0	0
$330$	0	0
$331$	40.0000	0.120846	0.0604230	−	0.998173i	$-0.480755\pi$
$331$	40.0000	0.120846	0.0604230	+	0.998173i	$0.480755\pi$
$332$	− 597.558i	− 1.79987i
$333$	0	0
$334$	748.246i	2.24026i
$335$	− 394.908i	− 1.17883i
$336$	0	0
$337$	91.0000	0.270030	0.135015	−	0.990844i	$-0.456892\pi$
$337$	91.0000	0.270030	0.135015	+	0.990844i	$0.456892\pi$
$338$	69.0000	0.204142
$339$	0	0
$340$	270.000	0.794118
$341$	181.865i	0.533329i
$342$	0	0
$343$	0	0
$344$	222.000	0.645349
$345$	0	0
$346$	685.892i	1.98235i
$347$	−210.000	−0.605187	−0.302594	−	0.953120i	$-0.597853\pi$
$347$	−210.000	−0.605187	−0.302594	+	0.953120i	$0.597853\pi$
$348$	0	0
$349$	304.841i	0.873470i	0.899590	+	0.436735i	$0.143865\pi$
$349$	304.841i	0.873470i	−0.899590	+	0.436735i	$0.856135\pi$
$350$	0	0
$351$	0	0
$352$	−675.000	−1.91761
$353$	− 394.908i	− 1.11872i	−0.828925	−	0.559359i	$-0.811047\pi$
$353$	− 394.908i	− 1.11872i	0.828925	−	0.559359i	$-0.188953\pi$
$354$	0	0
$355$	− 436.477i	− 1.22951i
$356$	− 363.731i	− 1.02172i
$357$	0	0
$358$	270.000	0.754190
$359$	−492.000	−1.37047	−0.685237	−	0.728320i	$-0.740301\pi$
$359$	−492.000	−1.37047	−0.685237	+	0.728320i	$0.740301\pi$
$360$	0	0
$361$	253.000	0.700831
$362$	− 31.1769i	− 0.0861241i
$363$	0	0
$364$	0	0
$365$	324.000	0.887671
$366$	0	0
$367$	− 327.358i	− 0.891983i	−0.895037	−	0.445991i	$-0.852851\pi$
$367$	− 327.358i	− 0.891983i	0.895037	−	0.445991i	$-0.147149\pi$
$368$	0	0
$369$	0	0
$370$	− 155.885i	− 0.421310i
$371$	0	0
$372$	0	0
$373$	170.000	0.455764	0.227882	−	0.973689i	$-0.426820\pi$
$373$	170.000	0.455764	0.227882	+	0.973689i	$0.426820\pi$
$374$	− 467.654i	− 1.25041i
$375$	0	0
$376$	0	0
$377$	124.708i	0.330790i
$378$	0	0
$379$	82.0000	0.216359	0.108179	−	0.994131i	$-0.465498\pi$
$379$	82.0000	0.216359	0.108179	+	0.994131i	$0.465498\pi$
$380$	−270.000	−0.710526
$381$	0	0
$382$	−936.000	−2.45026
$383$	218.238i	0.569813i	0.958555	+	0.284907i	$0.0919626\pi$
$383$	218.238i	0.569813i	−0.958555	+	0.284907i	$0.908037\pi$
$384$	0	0
$385$	0	0
$386$	555.000	1.43782
$387$	0	0
$388$	− 926.647i	− 2.38827i
$389$	−306.000	−0.786632	−0.393316	−	0.919403i	$-0.628672\pi$
$389$	−306.000	−0.786632	−0.393316	+	0.919403i	$0.628672\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	990.000	2.51269
$395$	− 223.435i	− 0.565657i
$396$	0	0
$397$	256.344i	0.645702i	0.946450	+	0.322851i	$0.104641\pi$
$397$	256.344i	0.645702i	−0.946450	+	0.322851i	$0.895359\pi$
$398$	− 20.7846i	− 0.0522226i
$399$	0	0
$400$	22.0000	0.0550000
$401$	−132.000	−0.329177	−0.164589	−	0.986362i	$-0.552630\pi$
$401$	−132.000	−0.329177	−0.164589	+	0.986362i	$0.552630\pi$
$402$	0	0
$403$	168.000	0.416873
$404$	727.461i	1.80065i
$405$	0	0
$406$	0	0
$407$	−150.000	−0.368550
$408$	0	0
$409$	− 361.999i	− 0.885082i	−0.896748	−	0.442541i	$-0.854077\pi$
$409$	− 361.999i	− 0.885082i	0.896748	−	0.442541i	$-0.145923\pi$
$410$	−162.000	−0.395122
$411$	0	0
$412$	346.410i	0.840801i
$413$	0	0
$414$	0	0
$415$	621.000	1.49639
$416$	623.538i	1.49889i
$417$	0	0
$418$	467.654i	1.11879i
$419$	644.323i	1.53776i	0.639391	+	0.768882i	$0.279187\pi$
$419$	644.323i	1.53776i	−0.639391	+	0.768882i	$0.720813\pi$
$420$	0	0
$421$	752.000	1.78622	0.893112	−	0.449835i	$-0.148517\pi$
$421$	752.000	1.78622	0.893112	+	0.449835i	$0.148517\pi$
$422$	744.000	1.76303
$423$	0	0
$424$	99.0000	0.233491
$425$	20.7846i	0.0489050i
$426$	0	0
$427$	0	0
$428$	−465.000	−1.08645
$429$	0	0
$430$	1153.55i	2.68266i
$431$	−162.000	−0.375870	−0.187935	−	0.982181i	$-0.560179\pi$
$431$	−162.000	−0.375870	−0.187935	+	0.982181i	$0.560179\pi$
$432$	0	0
$433$	339.482i	0.784023i	0.919960	+	0.392011i	$0.128221\pi$
$433$	339.482i	0.784023i	−0.919960	+	0.392011i	$0.871779\pi$
$434$	0	0
$435$	0	0
$436$	−40.0000	−0.0917431
$437$	0	0
$438$	0	0
$439$	− 389.711i	− 0.887725i	−0.896095	−	0.443863i	$-0.853608\pi$
$439$	− 389.711i	− 0.887725i	0.896095	−	0.443863i	$-0.146392\pi$
$440$	233.827i	0.531425i
$441$	0	0
$442$	−432.000	−0.977376
$443$	297.000	0.670429	0.335214	−	0.942142i	$-0.391191\pi$
$443$	297.000	0.670429	0.335214	+	0.942142i	$0.391191\pi$
$444$	0	0
$445$	378.000	0.849438
$446$	− 576.773i	− 1.29321i
$447$	0	0
$448$	0	0
$449$	492.000	1.09577	0.547884	−	0.836554i	$-0.315433\pi$
$449$	492.000	1.09577	0.547884	+	0.836554i	$0.315433\pi$
$450$	0	0
$451$	155.885i	0.345642i
$452$	−210.000	−0.464602
$453$	0	0
$454$	− 265.004i	− 0.583709i
$455$	0	0
$456$	0	0
$457$	−443.000	−0.969365	−0.484683	−	0.874690i	$-0.661065\pi$
$457$	−443.000	−0.969365	−0.484683	+	0.874690i	$0.661065\pi$
$458$	987.269i	2.15561i
$459$	0	0
$460$	0	0
$461$	415.692i	0.901718i	0.892595	+	0.450859i	$0.148882\pi$
$461$	415.692i	0.901718i	−0.892595	+	0.450859i	$0.851118\pi$
$462$	0	0
$463$	−82.0000	−0.177106	−0.0885529	−	0.996071i	$-0.528224\pi$
$463$	−82.0000	−0.177106	−0.0885529	+	0.996071i	$0.528224\pi$
$464$	−99.0000	−0.213362
$465$	0	0
$466$	810.000	1.73820
$467$	270.200i	0.578587i	0.957241	+	0.289293i	$0.0934203\pi$
$467$	270.200i	0.578587i	−0.957241	+	0.289293i	$0.906580\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	46.7654i	0.0990792i
$473$	1110.00	2.34672
$474$	0	0
$475$	− 20.7846i	− 0.0437571i
$476$	0	0
$477$	0	0
$478$	−684.000	−1.43096
$479$	− 342.946i	− 0.715963i	−0.933729	−	0.357981i	$-0.883465\pi$
$479$	− 342.946i	− 0.715963i	0.933729	−	0.357981i	$-0.116535\pi$
$480$	0	0
$481$	138.564i	0.288075i
$482$	− 1335.41i	− 2.77056i
$483$	0	0
$484$	520.000	1.07438
$485$	963.000	1.98557
$486$	0	0
$487$	−317.000	−0.650924	−0.325462	−	0.945555i	$-0.605520\pi$
$487$	−317.000	−0.650924	−0.325462	+	0.945555i	$0.605520\pi$
$488$	270.200i	0.553688i
$489$	0	0
$490$	0	0
$491$	−27.0000	−0.0549898	−0.0274949	−	0.999622i	$-0.508753\pi$
$491$	−27.0000	−0.0549898	−0.0274949	+	0.999622i	$0.508753\pi$
$492$	0	0
$493$	− 93.5307i	− 0.189718i
$494$	432.000	0.874494
$495$	0	0
$496$	133.368i	0.268887i
$497$	0	0
$498$	0	0
$499$	−446.000	−0.893788	−0.446894	−	0.894587i	$-0.647470\pi$
$499$	−446.000	−0.893788	−0.446894	+	0.894587i	$0.647470\pi$
$500$	597.558i	1.19512i
$501$	0	0
$502$	15.5885i	0.0310527i
$503$	− 488.438i	− 0.971050i	−0.874223	−	0.485525i	$-0.838628\pi$
$503$	− 488.438i	− 0.971050i	0.874223	−	0.485525i	$-0.161372\pi$
$504$	0	0
$505$	−756.000	−1.49703
$506$	0	0
$507$	0	0
$508$	175.000	0.344488
$509$	98.7269i	0.193962i	0.995286	+	0.0969812i	$0.0309187\pi$
$509$	98.7269i	0.193962i	−0.995286	+	0.0969812i	$0.969081\pi$
$510$	0	0
$511$	0	0
$512$	−627.000	−1.22461
$513$	0	0
$514$	− 342.946i	− 0.667210i
$515$	−360.000	−0.699029
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	216.000	0.415385
$521$	904.131i	1.73538i	0.497110	+	0.867688i	$0.334395\pi$
$521$	904.131i	1.73538i	−0.497110	+	0.867688i	$0.665605\pi$
$522$	0	0
$523$	699.749i	1.33795i	0.743284	+	0.668976i	$0.233267\pi$
$523$	699.749i	1.33795i	−0.743284	+	0.668976i	$0.766733\pi$
$524$	857.365i	1.63619i
$525$	0	0
$526$	558.000	1.06084
$527$	−126.000	−0.239089
$528$	0	0
$529$	−529.000	−1.00000
$530$	514.419i	0.970602i
$531$	0	0
$532$	0	0
$533$	144.000	0.270169
$534$	0	0
$535$	− 483.242i	− 0.903256i
$536$	228.000	0.425373
$537$	0	0
$538$	− 1013.25i	− 1.88336i
$539$	0	0
$540$	0	0
$541$	74.0000	0.136784	0.0683919	−	0.997659i	$-0.478213\pi$
$541$	74.0000	0.136784	0.0683919	+	0.997659i	$0.478213\pi$
$542$	275.396i	0.508111i
$543$	0	0
$544$	− 467.654i	− 0.859658i
$545$	− 41.5692i	− 0.0762738i
$546$	0	0
$547$	−934.000	−1.70750	−0.853748	−	0.520687i	$-0.825676\pi$
$547$	−934.000	−1.70750	−0.853748	+	0.520687i	$0.825676\pi$
$548$	480.000	0.875912
$549$	0	0
$550$	−90.0000	−0.163636
$551$	93.5307i	0.169747i
$552$	0	0
$553$	0	0
$554$	−1140.00	−2.05776
$555$	0	0
$556$	917.987i	1.65106i
$557$	−843.000	−1.51346	−0.756732	−	0.653725i	$-0.773205\pi$
$557$	−843.000	−1.51346	−0.756732	+	0.653725i	$0.773205\pi$
$558$	0	0
$559$	− 1025.37i	− 1.83430i
$560$	0	0
$561$	0	0
$562$	900.000	1.60142
$563$	− 950.896i	− 1.68898i	−0.535571	−	0.844490i	$-0.679904\pi$
$563$	− 950.896i	− 1.68898i	0.535571	−	0.844490i	$-0.320096\pi$
$564$	0	0
$565$	− 218.238i	− 0.386263i
$566$	− 613.146i	− 1.08330i
$567$	0	0
$568$	252.000	0.443662
$569$	−222.000	−0.390158	−0.195079	−	0.980788i	$-0.562496\pi$
$569$	−222.000	−0.390158	−0.195079	+	0.980788i	$0.562496\pi$
$570$	0	0
$571$	440.000	0.770578	0.385289	−	0.922796i	$-0.374102\pi$
$571$	440.000	0.770578	0.385289	+	0.922796i	$0.374102\pi$
$572$	− 1039.23i	− 1.81684i
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	656.447i	1.13769i	0.822445	+	0.568845i	$0.192610\pi$
$577$	656.447i	1.13769i	−0.822445	+	0.568845i	$0.807390\pi$
$578$	−543.000	−0.939446
$579$	0	0
$580$	233.827i	0.403150i
$581$	0	0
$582$	0	0
$583$	495.000	0.849057
$584$	187.061i	0.320311i
$585$	0	0
$586$	− 1636.79i	− 2.79315i
$587$	− 1054.82i	− 1.79697i	−0.439008	−	0.898483i	$-0.644670\pi$
$587$	− 1054.82i	− 1.79697i	0.439008	−	0.898483i	$-0.355330\pi$
$588$	0	0
$589$	126.000	0.213922
$590$	−243.000	−0.411864
$591$	0	0
$592$	−110.000	−0.185811
$593$	706.677i	1.19170i	0.803097	+	0.595849i	$0.203184\pi$
$593$	706.677i	1.19170i	−0.803097	+	0.595849i	$0.796816\pi$
$594$	0	0
$595$	0	0
$596$	930.000	1.56040
$597$	0	0
$598$	0	0
$599$	−516.000	−0.861436	−0.430718	−	0.902487i	$-0.641740\pi$
$599$	−516.000	−0.861436	−0.430718	+	0.902487i	$0.641740\pi$
$600$	0	0
$601$	247.683i	0.412119i	0.978540	+	0.206059i	$0.0660640\pi$
$601$	247.683i	0.412119i	−0.978540	+	0.206059i	$0.933936\pi$
$602$	0	0
$603$	0	0
$604$	395.000	0.653974
$605$	540.400i	0.893223i
$606$	0	0
$607$	649.519i	1.07005i	0.844837	+	0.535024i	$0.179697\pi$
$607$	649.519i	1.07005i	−0.844837	+	0.535024i	$0.820303\pi$
$608$	467.654i	0.769167i
$609$	0	0
$610$	−1404.00	−2.30164
$611$	0	0
$612$	0	0
$613$	−892.000	−1.45514	−0.727569	−	0.686034i	$-0.759350\pi$
$613$	−892.000	−1.45514	−0.727569	+	0.686034i	$0.759350\pi$
$614$	519.615i	0.846279i
$615$	0	0
$616$	0	0
$617$	1224.00	1.98379	0.991896	−	0.127050i	$-0.0405510\pi$
$617$	1224.00	1.98379	0.991896	+	0.127050i	$0.0405510\pi$
$618$	0	0
$619$	− 401.836i	− 0.649169i	−0.945857	−	0.324585i	$-0.894775\pi$
$619$	− 401.836i	− 0.649169i	0.945857	−	0.324585i	$-0.105225\pi$
$620$	315.000	0.508065
$621$	0	0
$622$	− 530.008i	− 0.852102i
$623$	0	0
$624$	0	0
$625$	−671.000	−1.07360
$626$	− 639.127i	− 1.02097i
$627$	0	0
$628$	− 103.923i	− 0.165483i
$629$	− 103.923i	− 0.165219i
$630$	0	0
$631$	1115.00	1.76704	0.883518	−	0.468397i	$-0.155168\pi$
$631$	1115.00	1.76704	0.883518	+	0.468397i	$0.155168\pi$
$632$	129.000	0.204114
$633$	0	0
$634$	351.000	0.553628
$635$	181.865i	0.286402i
$636$	0	0
$637$	0	0
$638$	405.000	0.634796
$639$	0	0
$640$	483.242i	0.755066i
$641$	384.000	0.599064	0.299532	−	0.954086i	$-0.403169\pi$
$641$	384.000	0.599064	0.299532	+	0.954086i	$0.403169\pi$
$642$	0	0
$643$	6.92820i	0.0107748i	0.999985	+	0.00538741i	$0.00171487\pi$
$643$	6.92820i	0.0107748i	−0.999985	+	0.00538741i	$0.998285\pi$
$644$	0	0
$645$	0	0
$646$	−324.000	−0.501548
$647$	− 893.738i	− 1.38136i	−0.723162	−	0.690679i	$-0.757312\pi$
$647$	− 893.738i	− 1.38136i	0.723162	−	0.690679i	$-0.242688\pi$
$648$	0	0
$649$	233.827i	0.360288i
$650$	83.1384i	0.127905i
$651$	0	0
$652$	−1040.00	−1.59509
$653$	−75.0000	−0.114855	−0.0574273	−	0.998350i	$-0.518290\pi$
$653$	−75.0000	−0.114855	−0.0574273	+	0.998350i	$0.518290\pi$
$654$	0	0
$655$	−891.000	−1.36031
$656$	114.315i	0.174261i
$657$	0	0
$658$	0	0
$659$	−642.000	−0.974203	−0.487102	−	0.873345i	$-0.661946\pi$
$659$	−642.000	−0.974203	−0.487102	+	0.873345i	$0.661946\pi$
$660$	0	0
$661$	− 280.592i	− 0.424497i	−0.977216	−	0.212248i	$-0.931921\pi$
$661$	− 280.592i	− 0.424497i	0.977216	−	0.212248i	$-0.0680786\pi$
$662$	−120.000	−0.181269
$663$	0	0
$664$	358.535i	0.539962i
$665$	0	0
$666$	0	0
$667$	0	0
$668$	− 1247.08i	− 1.86688i
$669$	0	0
$670$	1184.72i	1.76824i
$671$	1351.00i	2.01341i
$672$	0	0
$673$	13.0000	0.0193165	0.00965825	−	0.999953i	$-0.496926\pi$
$673$	13.0000	0.0193165	0.00965825	+	0.999953i	$0.496926\pi$
$674$	−273.000	−0.405045
$675$	0	0
$676$	−115.000	−0.170118
$677$	− 1262.67i	− 1.86509i	−0.361055	−	0.932544i	$-0.617584\pi$
$677$	− 1262.67i	− 1.86509i	0.361055	−	0.932544i	$-0.382416\pi$
$678$	0	0
$679$	0	0
$680$	−162.000	−0.238235
$681$	0	0
$682$	− 545.596i	− 0.799994i
$683$	969.000	1.41874	0.709370	−	0.704836i	$-0.248979\pi$
$683$	969.000	1.41874	0.709370	+	0.704836i	$0.248979\pi$
$684$	0	0
$685$	498.831i	0.728220i
$686$	0	0
$687$	0	0
$688$	814.000	1.18314
$689$	− 457.261i	− 0.663660i
$690$	0	0
$691$	100.459i	0.145382i	0.997355	+	0.0726910i	$0.0231587\pi$
$691$	100.459i	0.145382i	−0.997355	+	0.0726910i	$0.976841\pi$
$692$	− 1143.15i	− 1.65196i
$693$	0	0
$694$	630.000	0.907781
$695$	−954.000	−1.37266
$696$	0	0
$697$	−108.000	−0.154950
$698$	− 914.523i	− 1.31020i
$699$	0	0
$700$	0	0
$701$	−597.000	−0.851641	−0.425820	−	0.904808i	$-0.640014\pi$
$701$	−597.000	−0.851641	−0.425820	+	0.904808i	$0.640014\pi$
$702$	0	0
$703$	103.923i	0.147828i
$704$	1365.00	1.93892
$705$	0	0
$706$	1184.72i	1.67808i
$707$	0	0
$708$	0	0
$709$	−830.000	−1.17066	−0.585331	−	0.810794i	$-0.699036\pi$
$709$	−830.000	−1.17066	−0.585331	+	0.810794i	$0.699036\pi$
$710$	1309.43i	1.84427i
$711$	0	0
$712$	218.238i	0.306515i
$713$	0	0
$714$	0	0
$715$	1080.00	1.51049
$716$	−450.000	−0.628492
$717$	0	0
$718$	1476.00	2.05571
$719$	342.946i	0.476976i	0.971145	+	0.238488i	$0.0766518\pi$
$719$	342.946i	0.476976i	−0.971145	+	0.238488i	$0.923348\pi$
$720$	0	0
$721$	0	0
$722$	−759.000	−1.05125
$723$	0	0
$724$	51.9615i	0.0717701i
$725$	−18.0000	−0.0248276
$726$	0	0
$727$	− 50.2295i	− 0.0690914i	−0.999403	−	0.0345457i	$-0.989002\pi$
$727$	− 50.2295i	− 0.0690914i	0.999403	−	0.0345457i	$-0.0109984\pi$
$728$	0	0
$729$	0	0
$730$	−972.000	−1.33151
$731$	769.031i	1.05203i
$732$	0	0
$733$	183.597i	0.250474i	0.992127	+	0.125237i	$0.0399691\pi$
$733$	183.597i	0.250474i	−0.992127	+	0.125237i	$0.960031\pi$
$734$	982.073i	1.33797i
$735$	0	0
$736$	0	0
$737$	1140.00	1.54681
$738$	0	0
$739$	−334.000	−0.451962	−0.225981	−	0.974132i	$-0.572559\pi$
$739$	−334.000	−0.451962	−0.225981	+	0.974132i	$0.572559\pi$
$740$	259.808i	0.351091i
$741$	0	0
$742$	0	0
$743$	−84.0000	−0.113055	−0.0565276	−	0.998401i	$-0.518003\pi$
$743$	−84.0000	−0.113055	−0.0565276	+	0.998401i	$0.518003\pi$
$744$	0	0
$745$	966.484i	1.29729i
$746$	−510.000	−0.683646
$747$	0	0
$748$	779.423i	1.04201i
$749$	0	0
$750$	0	0
$751$	−359.000	−0.478029	−0.239015	−	0.971016i	$-0.576824\pi$
$751$	−359.000	−0.478029	−0.239015	+	0.971016i	$0.576824\pi$
$752$	0	0
$753$	0	0
$754$	− 374.123i	− 0.496184i
$755$	410.496i	0.543703i
$756$	0	0
$757$	−80.0000	−0.105680	−0.0528402	−	0.998603i	$-0.516827\pi$
$757$	−80.0000	−0.105680	−0.0528402	+	0.998603i	$0.516827\pi$
$758$	−246.000	−0.324538
$759$	0	0
$760$	162.000	0.213158
$761$	− 810.600i	− 1.06518i	−0.846374	−	0.532589i	$-0.821219\pi$
$761$	− 810.600i	− 1.06518i	0.846374	−	0.532589i	$-0.178781\pi$
$762$	0	0
$763$	0	0
$764$	1560.00	2.04188
$765$	0	0
$766$	− 654.715i	− 0.854720i
$767$	216.000	0.281617
$768$	0	0
$769$	774.227i	1.00680i	0.864054	+	0.503398i	$0.167917\pi$
$769$	774.227i	1.00680i	−0.864054	+	0.503398i	$0.832083\pi$
$770$	0	0
$771$	0	0
$772$	−925.000	−1.19819
$773$	727.461i	0.941088i	0.882376	+	0.470544i	$0.155942\pi$
$773$	727.461i	0.941088i	−0.882376	+	0.470544i	$0.844058\pi$
$774$	0	0
$775$	24.2487i	0.0312887i
$776$	555.988i	0.716480i
$777$	0	0
$778$	918.000	1.17995
$779$	108.000	0.138639
$780$	0	0
$781$	1260.00	1.61332
$782$	0	0
$783$	0	0
$784$	0	0
$785$	108.000	0.137580
$786$	0	0
$787$	− 1427.21i	− 1.81348i	−0.421689	−	0.906741i	$-0.638562\pi$
$787$	− 1427.21i	− 1.81348i	0.421689	−	0.906741i	$-0.361438\pi$
$788$	−1650.00	−2.09391
$789$	0	0
$790$	670.304i	0.848486i
$791$	0	0
$792$	0	0
$793$	1248.00	1.57377
$794$	− 769.031i	− 0.968552i
$795$	0	0
$796$	34.6410i	0.0435189i
$797$	607.950i	0.762798i	0.924411	+	0.381399i	$0.124558\pi$
$797$	607.950i	0.762798i	−0.924411	+	0.381399i	$0.875442\pi$
$798$	0	0
$799$	0	0
$800$	−90.0000	−0.112500
$801$	0	0
$802$	396.000	0.493766
$803$	935.307i	1.16477i
$804$	0	0
$805$	0	0
$806$	−504.000	−0.625310
$807$	0	0
$808$	− 436.477i	− 0.540194i
$809$	168.000	0.207664	0.103832	−	0.994595i	$-0.466890\pi$
$809$	168.000	0.207664	0.103832	+	0.994595i	$0.466890\pi$
$810$	0	0
$811$	− 353.338i	− 0.435682i	−0.975984	−	0.217841i	$-0.930099\pi$
$811$	− 353.338i	− 0.435682i	0.975984	−	0.217841i	$-0.0699015\pi$
$812$	0	0
$813$	0	0
$814$	450.000	0.552826
$815$	− 1080.80i	− 1.32613i
$816$	0	0
$817$	− 769.031i	− 0.941286i
$818$	1086.00i	1.32762i
$819$	0	0
$820$	270.000	0.329268
$821$	−285.000	−0.347138	−0.173569	−	0.984822i	$-0.555530\pi$
$821$	−285.000	−0.347138	−0.173569	+	0.984822i	$0.555530\pi$
$822$	0	0
$823$	−274.000	−0.332928	−0.166464	−	0.986048i	$-0.553235\pi$
$823$	−274.000	−0.332928	−0.166464	+	0.986048i	$0.553235\pi$
$824$	− 207.846i	− 0.252240i
$825$	0	0
$826$	0	0
$827$	−429.000	−0.518742	−0.259371	−	0.965778i	$-0.583515\pi$
$827$	−429.000	−0.518742	−0.259371	+	0.965778i	$0.583515\pi$
$828$	0	0
$829$	− 945.700i	− 1.14077i	−0.821377	−	0.570386i	$-0.806794\pi$
$829$	− 945.700i	− 1.14077i	0.821377	−	0.570386i	$-0.193206\pi$
$830$	−1863.00	−2.24458
$831$	0	0
$832$	− 1260.93i	− 1.51554i
$833$	0	0
$834$	0	0
$835$	1296.00	1.55210
$836$	− 779.423i	− 0.932324i
$837$	0	0
$838$	− 1932.97i	− 2.30665i
$839$	259.808i	0.309663i	0.987941	+	0.154832i	$0.0494835\pi$
$839$	259.808i	0.309663i	−0.987941	+	0.154832i	$0.950516\pi$
$840$	0	0
$841$	−760.000	−0.903686
$842$	−2256.00	−2.67933
$843$	0	0
$844$	−1240.00	−1.46919
$845$	− 119.512i	− 0.141434i
$846$	0	0
$847$	0	0
$848$	363.000	0.428066
$849$	0	0
$850$	− 62.3538i	− 0.0733574i
$851$	0	0
$852$	0	0
$853$	− 997.661i	− 1.16959i	−0.811181	−	0.584796i	$-0.801175\pi$
$853$	− 997.661i	− 1.16959i	0.811181	−	0.584796i	$-0.198825\pi$
$854$	0	0
$855$	0	0
$856$	279.000	0.325935
$857$	− 436.477i	− 0.509308i	−0.967032	−	0.254654i	$-0.918038\pi$
$857$	− 436.477i	− 0.509308i	0.967032	−	0.254654i	$-0.0819615\pi$
$858$	0	0
$859$	453.797i	0.528286i	0.964484	+	0.264143i	$0.0850890\pi$
$859$	453.797i	0.528286i	−0.964484	+	0.264143i	$0.914911\pi$
$860$	− 1922.58i	− 2.23555i
$861$	0	0
$862$	486.000	0.563805
$863$	390.000	0.451912	0.225956	−	0.974138i	$-0.427449\pi$
$863$	390.000	0.451912	0.225956	+	0.974138i	$0.427449\pi$
$864$	0	0
$865$	1188.00	1.37341
$866$	− 1018.45i	− 1.17603i
$867$	0	0
$868$	0	0
$869$	645.000	0.742232
$870$	0	0
$871$	− 1053.09i	− 1.20905i
$872$	24.0000	0.0275229
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−544.000	−0.620296	−0.310148	−	0.950688i	$-0.600379\pi$
$877$	−544.000	−0.620296	−0.310148	+	0.950688i	$0.600379\pi$
$878$	1169.13i	1.33159i
$879$	0	0
$880$	857.365i	0.974279i
$881$	− 187.061i	− 0.212329i	−0.994349	−	0.106164i	$-0.966143\pi$
$881$	− 187.061i	− 0.212329i	0.994349	−	0.106164i	$-0.0338569\pi$
$882$	0	0
$883$	322.000	0.364666	0.182333	−	0.983237i	$-0.441635\pi$
$883$	322.000	0.364666	0.182333	+	0.983237i	$0.441635\pi$
$884$	720.000	0.814480
$885$	0	0
$886$	−891.000	−1.00564
$887$	706.677i	0.796704i	0.917233	+	0.398352i	$0.130418\pi$
$887$	706.677i	0.796704i	−0.917233	+	0.398352i	$0.869582\pi$
$888$	0	0
$889$	0	0
$890$	−1134.00	−1.27416
$891$	0	0
$892$	961.288i	1.07768i
$893$	0	0
$894$	0	0
$895$	− 467.654i	− 0.522518i
$896$	0	0
$897$	0	0
$898$	−1476.00	−1.64365
$899$	− 109.119i	− 0.121378i
$900$	0	0
$901$	342.946i	0.380628i
$902$	− 467.654i	− 0.518463i
$903$	0	0
$904$	126.000	0.139381
$905$	−54.0000	−0.0596685
$906$	0	0
$907$	−1100.00	−1.21279	−0.606395	−	0.795164i	$-0.707385\pi$
$907$	−1100.00	−1.21279	−0.606395	+	0.795164i	$0.707385\pi$
$908$	441.673i	0.486424i
$909$	0	0
$910$	0	0
$911$	−900.000	−0.987925	−0.493963	−	0.869483i	$-0.664452\pi$
$911$	−900.000	−0.987925	−0.493963	+	0.869483i	$0.664452\pi$
$912$	0	0
$913$	1792.67i	1.96350i
$914$	1329.00	1.45405
$915$	0	0
$916$	− 1645.45i	− 1.79634i
$917$	0	0
$918$	0	0
$919$	−1718.00	−1.86942	−0.934712	−	0.355407i	$-0.884342\pi$
$919$	−1718.00	−1.86942	−0.934712	+	0.355407i	$0.884342\pi$
$920$	0	0
$921$	0	0
$922$	− 1247.08i	− 1.35258i
$923$	− 1163.94i	− 1.26104i
$924$	0	0
$925$	−20.0000	−0.0216216
$926$	246.000	0.265659
$927$	0	0
$928$	405.000	0.436422
$929$	− 1486.10i	− 1.59968i	−0.600216	−	0.799838i	$-0.704919\pi$
$929$	− 1486.10i	− 1.59968i	0.600216	−	0.799838i	$-0.295081\pi$
$930$	0	0
$931$	0	0
$932$	−1350.00	−1.44850
$933$	0	0
$934$	− 810.600i	− 0.867880i
$935$	−810.000	−0.866310
$936$	0	0
$937$	957.824i	1.02222i	0.859514	+	0.511112i	$0.170766\pi$
$937$	957.824i	1.02222i	−0.859514	+	0.511112i	$0.829234\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 358.535i	− 0.381014i	−0.981686	−	0.190507i	$-0.938987\pi$
$941$	− 358.535i	− 0.381014i	0.981686	−	0.190507i	$-0.0610132\pi$
$942$	0	0
$943$	0	0
$944$	171.473i	0.181645i
$945$	0	0
$946$	−3330.00	−3.52008
$947$	−162.000	−0.171067	−0.0855333	−	0.996335i	$-0.527259\pi$
$947$	−162.000	−0.171067	−0.0855333	+	0.996335i	$0.527259\pi$
$948$	0	0
$949$	864.000	0.910432
$950$	62.3538i	0.0656356i
$951$	0	0
$952$	0	0
$953$	954.000	1.00105	0.500525	−	0.865722i	$-0.333140\pi$
$953$	954.000	1.00105	0.500525	+	0.865722i	$0.333140\pi$
$954$	0	0
$955$	1621.20i	1.69759i
$956$	1140.00	1.19247
$957$	0	0
$958$	1028.84i	1.07394i
$959$	0	0
$960$	0	0
$961$	814.000	0.847034
$962$	− 415.692i	− 0.432112i
$963$	0	0
$964$	2225.69i	2.30880i
$965$	− 961.288i	− 0.996154i
$966$	0	0
$967$	−751.000	−0.776629	−0.388314	−	0.921527i	$-0.626943\pi$
$967$	−751.000	−0.776629	−0.388314	+	0.921527i	$0.626943\pi$
$968$	−312.000	−0.322314
$969$	0	0
$970$	−2889.00	−2.97835
$971$	− 285.788i	− 0.294324i	−0.989112	−	0.147162i	$-0.952986\pi$
$971$	− 285.788i	− 0.294324i	0.989112	−	0.147162i	$-0.0470138\pi$
$972$	0	0
$973$	0	0
$974$	951.000	0.976386
$975$	0	0
$976$	990.733i	1.01510i
$977$	18.0000	0.0184237	0.00921187	−	0.999958i	$-0.497068\pi$
$977$	18.0000	0.0184237	0.00921187	+	0.999958i	$0.497068\pi$
$978$	0	0
$979$	1091.19i	1.11460i
$980$	0	0
$981$	0	0
$982$	81.0000	0.0824847
$983$	987.269i	1.00434i	0.864768	+	0.502171i	$0.167465\pi$
$983$	987.269i	1.00434i	−0.864768	+	0.502171i	$0.832535\pi$
$984$	0	0
$985$	− 1714.73i	− 1.74084i
$986$	280.592i	0.284576i
$987$	0	0
$988$	−720.000	−0.728745
$989$	0	0
$990$	0	0
$991$	703.000	0.709384	0.354692	−	0.934983i	$-0.384586\pi$
$991$	703.000	0.709384	0.354692	+	0.934983i	$0.384586\pi$
$992$	− 545.596i	− 0.549996i
$993$	0	0
$994$	0	0
$995$	−36.0000	−0.0361809
$996$	0	0
$997$	214.774i	0.215421i	0.994182	+	0.107710i	$0.0343519\pi$
$997$	214.774i	0.215421i	−0.994182	+	0.107710i	$0.965648\pi$
$998$	1338.00	1.34068
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.3.d.a.244.2		2
3.2	odd	2		147.3.d.c.97.1		2
7.2	even	3		441.3.m.g.325.1		2
7.3	odd	6		441.3.m.g.19.1		2
7.4	even	3		63.3.m.d.19.1		2
7.5	odd	6		63.3.m.d.10.1		2
7.6	odd	2	inner	441.3.d.a.244.1		2
12.11	even	2		2352.3.f.a.97.2		2
21.2	odd	6		147.3.f.a.31.1		2
21.5	even	6		21.3.f.a.10.1	✓	2
21.11	odd	6		21.3.f.a.19.1	yes	2
21.17	even	6		147.3.f.a.19.1		2
21.20	even	2		147.3.d.c.97.2		2
28.11	odd	6		1008.3.cg.a.145.1		2
28.19	even	6		1008.3.cg.a.577.1		2
84.11	even	6		336.3.bh.d.145.1		2
84.47	odd	6		336.3.bh.d.241.1		2
84.83	odd	2		2352.3.f.a.97.1		2
105.32	even	12		525.3.s.e.124.1		4
105.47	odd	12		525.3.s.e.199.2		4
105.53	even	12		525.3.s.e.124.2		4
105.68	odd	12		525.3.s.e.199.1		4
105.74	odd	6		525.3.o.h.376.1		2
105.89	even	6		525.3.o.h.451.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.3.f.a.10.1	✓	2	21.5	even	6
21.3.f.a.19.1	yes	2	21.11	odd	6
63.3.m.d.10.1		2	7.5	odd	6
63.3.m.d.19.1		2	7.4	even	3
147.3.d.c.97.1		2	3.2	odd	2
147.3.d.c.97.2		2	21.20	even	2
147.3.f.a.19.1		2	21.17	even	6
147.3.f.a.31.1		2	21.2	odd	6
336.3.bh.d.145.1		2	84.11	even	6
336.3.bh.d.241.1		2	84.47	odd	6
441.3.d.a.244.1		2	7.6	odd	2	inner
441.3.d.a.244.2		2	1.1	even	1	trivial
441.3.m.g.19.1		2	7.3	odd	6
441.3.m.g.325.1		2	7.2	even	3
525.3.o.h.376.1		2	105.74	odd	6
525.3.o.h.451.1		2	105.89	even	6
525.3.s.e.124.1		4	105.32	even	12
525.3.s.e.124.2		4	105.53	even	12
525.3.s.e.199.1		4	105.68	odd	12
525.3.s.e.199.2		4	105.47	odd	12
1008.3.cg.a.145.1		2	28.11	odd	6
1008.3.cg.a.577.1		2	28.19	even	6
2352.3.f.a.97.1		2	84.83	odd	2
2352.3.f.a.97.2		2	12.11	even	2

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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	441.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.00000 q^{2} +5.00000 q^{4} +5.19615i q^{5} -3.00000 q^{8} +O(q^{10})\)	\(q-3.00000 q^{2} +5.00000 q^{4} +5.19615i q^{5} -3.00000 q^{8} -15.5885i q^{10} -15.0000 q^{11} +13.8564i q^{13} -11.0000 q^{16} -10.3923i q^{17} +10.3923i q^{19} +25.9808i q^{20} +45.0000 q^{22} -2.00000 q^{25} -41.5692i q^{26} +9.00000 q^{29} -12.1244i q^{31} +45.0000 q^{32} +31.1769i q^{34} +10.0000 q^{37} -31.1769i q^{38} -15.5885i q^{40} -10.3923i q^{41} -74.0000 q^{43} -75.0000 q^{44} +6.00000 q^{50} +69.2820i q^{52} -33.0000 q^{53} -77.9423i q^{55} -27.0000 q^{58} -15.5885i q^{59} -90.0666i q^{61} +36.3731i q^{62} -91.0000 q^{64} -72.0000 q^{65} -76.0000 q^{67} -51.9615i q^{68} -84.0000 q^{71} -62.3538i q^{73} -30.0000 q^{74} +51.9615i q^{76} -43.0000 q^{79} -57.1577i q^{80} +31.1769i q^{82} -119.512i q^{83} +54.0000 q^{85} +222.000 q^{86} +45.0000 q^{88} -72.7461i q^{89} -54.0000 q^{95} -185.329i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{2} + 10 q^{4} - 6 q^{8}+O(q^{10}) \) 2 * q - 6 * q^2 + 10 * q^4 - 6 * q^8	\( 2 q - 6 q^{2} + 10 q^{4} - 6 q^{8} - 30 q^{11} - 22 q^{16} + 90 q^{22} - 4 q^{25} + 18 q^{29} + 90 q^{32} + 20 q^{37} - 148 q^{43} - 150 q^{44} + 12 q^{50} - 66 q^{53} - 54 q^{58} - 182 q^{64} - 144 q^{65} - 152 q^{67} - 168 q^{71} - 60 q^{74} - 86 q^{79} + 108 q^{85} + 444 q^{86} + 90 q^{88} - 108 q^{95}+O(q^{100}) \) 2 * q - 6 * q^2 + 10 * q^4 - 6 * q^8 - 30 * q^11 - 22 * q^16 + 90 * q^22 - 4 * q^25 + 18 * q^29 + 90 * q^32 + 20 * q^37 - 148 * q^43 - 150 * q^44 + 12 * q^50 - 66 * q^53 - 54 * q^58 - 182 * q^64 - 144 * q^65 - 152 * q^67 - 168 * q^71 - 60 * q^74 - 86 * q^79 + 108 * q^85 + 444 * q^86 + 90 * q^88 - 108 * q^95

Introduction

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