LMFDB - Embedded newform 1323.2.c.b.1322.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1323,2,Mod(1322,1323)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1323, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1323.1322");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1323 = 3^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1323.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:	$10.5642081874$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 25 $ x^4 - 5*x^2 + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 189)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1322.1
Root			$-1.93649 - 1.11803i$ of defining polynomial
Character	$\chi$	$=$	1323.1322
Dual form			1323.2.c.b.1322.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.23607i q^{2} -3.00000 q^{4} +2.23607i q^{8} +O(q^{10})$	$q-2.23607i q^{2} -3.00000 q^{4} +2.23607i q^{8} +4.47214i q^{11} -1.73205i q^{13} -1.00000 q^{16} -7.74597 q^{17} -3.46410i q^{19} +10.0000 q^{22} +4.47214i q^{23} -5.00000 q^{25} -3.87298 q^{26} +4.47214i q^{29} +1.73205i q^{31} +6.70820i q^{32} +17.3205i q^{34} +5.00000 q^{37} -7.74597 q^{38} -7.74597 q^{41} -7.00000 q^{43} -13.4164i q^{44} +10.0000 q^{46} -7.74597 q^{47} +11.1803i q^{50} +5.19615i q^{52} +4.47214i q^{53} +10.0000 q^{58} +7.74597 q^{59} +8.66025i q^{61} +3.87298 q^{62} +13.0000 q^{64} -1.00000 q^{67} +23.2379 q^{68} -8.94427i q^{71} -6.92820i q^{73} -11.1803i q^{74} +10.3923i q^{76} +11.0000 q^{79} +17.3205i q^{82} +7.74597 q^{83} +15.6525i q^{86} -10.0000 q^{88} -15.4919 q^{89} -13.4164i q^{92} +17.3205i q^{94} +1.73205i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{4}+O(q^{10}) $ 4 * q - 12 * q^4	$ 4 q - 12 q^{4} - 4 q^{16} + 40 q^{22} - 20 q^{25} + 20 q^{37} - 28 q^{43} + 40 q^{46} + 40 q^{58} + 52 q^{64} - 4 q^{67} + 44 q^{79} - 40 q^{88}+O(q^{100}) $ 4 * q - 12 * q^4 - 4 * q^16 + 40 * q^22 - 20 * q^25 + 20 * q^37 - 28 * q^43 + 40 * q^46 + 40 * q^58 + 52 * q^64 - 4 * q^67 + 44 * q^79 - 40 * q^88

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1323\mathbb{Z}\right)^\times$.

$n$	$785$	$1081$
$\chi(n)$	$-1$	$-1$

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 $

$2$	− 2.23607i	− 1.58114i	−0.612372	−	0.790569i	$-0.709785\pi$
$2$	− 2.23607i	− 1.58114i	0.612372	−	0.790569i	$-0.290215\pi$
$3$	0	0
$3$	0	0
$4$	−3.00000	−1.50000
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	2.23607i	0.790569i
$9$	0	0
$10$	0	0
$11$	4.47214i	1.34840i	0.738549	+	0.674200i	$0.235511\pi$
$11$	4.47214i	1.34840i	−0.738549	+	0.674200i	$0.764489\pi$
$12$	0	0
$13$	− 1.73205i	− 0.480384i	−0.970725	−	0.240192i	$-0.922790\pi$
$13$	− 1.73205i	− 0.480384i	0.970725	−	0.240192i	$-0.0772105\pi$
$14$	0	0
$15$	0	0
$16$	−1.00000	−0.250000
$17$	−7.74597	−1.87867	−0.939336	−	0.342997i	$-0.888558\pi$
$17$	−7.74597	−1.87867	−0.939336	+	0.342997i	$0.888558\pi$
$18$	0	0
$19$	− 3.46410i	− 0.794719i	−0.917663	−	0.397360i	$-0.869927\pi$
$19$	− 3.46410i	− 0.794719i	0.917663	−	0.397360i	$-0.130073\pi$
$20$	0	0
$21$	0	0
$22$	10.0000	2.13201
$23$	4.47214i	0.932505i	0.884652	+	0.466252i	$0.154396\pi$
$23$	4.47214i	0.932505i	−0.884652	+	0.466252i	$0.845604\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	−3.87298	−0.759555
$27$	0	0
$28$	0	0
$29$	4.47214i	0.830455i	0.909718	+	0.415227i	$0.136298\pi$
$29$	4.47214i	0.830455i	−0.909718	+	0.415227i	$0.863702\pi$
$30$	0	0
$31$	1.73205i	0.311086i	0.987829	+	0.155543i	$0.0497126\pi$
$31$	1.73205i	0.311086i	−0.987829	+	0.155543i	$0.950287\pi$
$32$	6.70820i	1.18585i
$33$	0	0
$34$	17.3205i	2.97044i
$35$	0	0
$36$	0	0
$37$	5.00000	0.821995	0.410997	−	0.911636i	$-0.365181\pi$
$37$	5.00000	0.821995	0.410997	+	0.911636i	$0.365181\pi$
$38$	−7.74597	−1.25656
$39$	0	0
$40$	0	0
$41$	−7.74597	−1.20972	−0.604858	−	0.796333i	$-0.706770\pi$
$41$	−7.74597	−1.20972	−0.604858	+	0.796333i	$0.706770\pi$
$42$	0	0
$43$	−7.00000	−1.06749	−0.533745	−	0.845645i	$-0.679216\pi$
$43$	−7.00000	−1.06749	−0.533745	+	0.845645i	$0.679216\pi$
$44$	− 13.4164i	− 2.02260i
$45$	0	0
$46$	10.0000	1.47442
$47$	−7.74597	−1.12987	−0.564933	−	0.825137i	$-0.691098\pi$
$47$	−7.74597	−1.12987	−0.564933	+	0.825137i	$0.691098\pi$
$48$	0	0
$49$	0	0
$50$	11.1803i	1.58114i
$51$	0	0
$52$	5.19615i	0.720577i
$53$	4.47214i	0.614295i	0.951662	+	0.307148i	$0.0993745\pi$
$53$	4.47214i	0.614295i	−0.951662	+	0.307148i	$0.900625\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	10.0000	1.31306
$59$	7.74597	1.00844	0.504219	−	0.863576i	$-0.331780\pi$
$59$	7.74597	1.00844	0.504219	+	0.863576i	$0.331780\pi$
$60$	0	0
$61$	8.66025i	1.10883i	0.832240	+	0.554416i	$0.187058\pi$
$61$	8.66025i	1.10883i	−0.832240	+	0.554416i	$0.812942\pi$
$62$	3.87298	0.491869
$63$	0	0
$64$	13.0000	1.62500
$65$	0	0
$66$	0	0
$67$	−1.00000	−0.122169	−0.0610847	−	0.998133i	$-0.519456\pi$
$67$	−1.00000	−0.122169	−0.0610847	+	0.998133i	$0.519456\pi$
$68$	23.2379	2.81801
$69$	0	0
$70$	0	0
$71$	− 8.94427i	− 1.06149i	−0.847532	−	0.530745i	$-0.821912\pi$
$71$	− 8.94427i	− 1.06149i	0.847532	−	0.530745i	$-0.178088\pi$
$72$	0	0
$73$	− 6.92820i	− 0.810885i	−0.914121	−	0.405442i	$-0.867117\pi$
$73$	− 6.92820i	− 0.810885i	0.914121	−	0.405442i	$-0.132883\pi$
$74$	− 11.1803i	− 1.29969i
$75$	0	0
$76$	10.3923i	1.19208i
$77$	0	0
$78$	0	0
$79$	11.0000	1.23760	0.618798	−	0.785550i	$-0.287620\pi$
$79$	11.0000	1.23760	0.618798	+	0.785550i	$0.287620\pi$
$80$	0	0
$81$	0	0
$82$	17.3205i	1.91273i
$83$	7.74597	0.850230	0.425115	−	0.905139i	$-0.360234\pi$
$83$	7.74597	0.850230	0.425115	+	0.905139i	$0.360234\pi$
$84$	0	0
$85$	0	0
$86$	15.6525i	1.68785i
$87$	0	0
$88$	−10.0000	−1.06600
$89$	−15.4919	−1.64214	−0.821071	−	0.570826i	$-0.806623\pi$
$89$	−15.4919	−1.64214	−0.821071	+	0.570826i	$0.806623\pi$
$90$	0	0
$91$	0	0
$92$	− 13.4164i	− 1.39876i
$93$	0	0
$94$	17.3205i	1.78647i
$95$	0	0
$96$	0	0
$97$	1.73205i	0.175863i	0.996127	+	0.0879316i	$0.0280257\pi$
$97$	1.73205i	0.175863i	−0.996127	+	0.0879316i	$0.971974\pi$
$98$	0	0
$99$	0	0
$100$	15.0000	1.50000
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	− 5.19615i	− 0.511992i	−0.966678	−	0.255996i	$-0.917597\pi$
$103$	− 5.19615i	− 0.511992i	0.966678	−	0.255996i	$-0.0824034\pi$
$104$	3.87298	0.379777
$105$	0	0
$106$	10.0000	0.971286
$107$	4.47214i	0.432338i	0.976356	+	0.216169i	$0.0693562\pi$
$107$	4.47214i	0.432338i	−0.976356	+	0.216169i	$0.930644\pi$
$108$	0	0
$109$	−1.00000	−0.0957826	−0.0478913	−	0.998853i	$-0.515250\pi$
$109$	−1.00000	−0.0957826	−0.0478913	+	0.998853i	$0.515250\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 8.94427i	− 0.841406i	−0.907198	−	0.420703i	$-0.861783\pi$
$113$	− 8.94427i	− 0.841406i	0.907198	−	0.420703i	$-0.138217\pi$
$114$	0	0
$115$	0	0
$116$	− 13.4164i	− 1.24568i
$117$	0	0
$118$	− 17.3205i	− 1.59448i
$119$	0	0
$120$	0	0
$121$	−9.00000	−0.818182
$122$	19.3649	1.75322
$123$	0	0
$124$	− 5.19615i	− 0.466628i
$125$	0	0
$126$	0	0
$127$	−1.00000	−0.0887357	−0.0443678	−	0.999015i	$-0.514127\pi$
$127$	−1.00000	−0.0887357	−0.0443678	+	0.999015i	$0.514127\pi$
$128$	− 15.6525i	− 1.38350i
$129$	0	0
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	0	0
$134$	2.23607i	0.193167i
$135$	0	0
$136$	− 17.3205i	− 1.48522i
$137$	4.47214i	0.382080i	0.981582	+	0.191040i	$0.0611861\pi$
$137$	4.47214i	0.382080i	−0.981582	+	0.191040i	$0.938814\pi$
$138$	0	0
$139$	8.66025i	0.734553i	0.930112	+	0.367277i	$0.119710\pi$
$139$	8.66025i	0.734553i	−0.930112	+	0.367277i	$0.880290\pi$
$140$	0	0
$141$	0	0
$142$	−20.0000	−1.67836
$143$	7.74597	0.647750
$144$	0	0
$145$	0	0
$146$	−15.4919	−1.28212
$147$	0	0
$148$	−15.0000	−1.23299
$149$	17.8885i	1.46549i	0.680505	+	0.732743i	$0.261760\pi$
$149$	17.8885i	1.46549i	−0.680505	+	0.732743i	$0.738240\pi$
$150$	0	0
$151$	−13.0000	−1.05792	−0.528962	−	0.848645i	$-0.677419\pi$
$151$	−13.0000	−1.05792	−0.528962	+	0.848645i	$0.677419\pi$
$152$	7.74597	0.628281
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 13.8564i	− 1.10586i	−0.833227	−	0.552931i	$-0.813509\pi$
$157$	− 13.8564i	− 1.10586i	0.833227	−	0.552931i	$-0.186491\pi$
$158$	− 24.5967i	− 1.95681i
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−13.0000	−1.01824	−0.509119	−	0.860696i	$-0.670029\pi$
$163$	−13.0000	−1.01824	−0.509119	+	0.860696i	$0.670029\pi$
$164$	23.2379	1.81458
$165$	0	0
$166$	− 17.3205i	− 1.34433i
$167$	−7.74597	−0.599401	−0.299700	−	0.954033i	$-0.596887\pi$
$167$	−7.74597	−0.599401	−0.299700	+	0.954033i	$0.596887\pi$
$168$	0	0
$169$	10.0000	0.769231
$170$	0	0
$171$	0	0
$172$	21.0000	1.60123
$173$	15.4919	1.17783	0.588915	−	0.808195i	$-0.299555\pi$
$173$	15.4919	1.17783	0.588915	+	0.808195i	$0.299555\pi$
$174$	0	0
$175$	0	0
$176$	− 4.47214i	− 0.337100i
$177$	0	0
$178$	34.6410i	2.59645i
$179$	− 8.94427i	− 0.668526i	−0.942480	−	0.334263i	$-0.891513\pi$
$179$	− 8.94427i	− 0.668526i	0.942480	−	0.334263i	$-0.108487\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	−10.0000	−0.737210
$185$	0	0
$186$	0	0
$187$	− 34.6410i	− 2.53320i
$188$	23.2379	1.69480
$189$	0	0
$190$	0	0
$191$	17.8885i	1.29437i	0.762333	+	0.647185i	$0.224054\pi$
$191$	17.8885i	1.29437i	−0.762333	+	0.647185i	$0.775946\pi$
$192$	0	0
$193$	−7.00000	−0.503871	−0.251936	−	0.967744i	$-0.581067\pi$
$193$	−7.00000	−0.503871	−0.251936	+	0.967744i	$0.581067\pi$
$194$	3.87298	0.278064
$195$	0	0
$196$	0	0
$197$	− 8.94427i	− 0.637253i	−0.947880	−	0.318626i	$-0.896778\pi$
$197$	− 8.94427i	− 0.637253i	0.947880	−	0.318626i	$-0.103222\pi$
$198$	0	0
$199$	− 22.5167i	− 1.59616i	−0.602549	−	0.798082i	$-0.705848\pi$
$199$	− 22.5167i	− 1.59616i	0.602549	−	0.798082i	$-0.294152\pi$
$200$	− 11.1803i	− 0.790569i
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	−11.6190	−0.809531
$207$	0	0
$208$	1.73205i	0.120096i
$209$	15.4919	1.07160
$210$	0	0
$211$	−19.0000	−1.30801	−0.654007	−	0.756489i	$-0.726913\pi$
$211$	−19.0000	−1.30801	−0.654007	+	0.756489i	$0.726913\pi$
$212$	− 13.4164i	− 0.921443i
$213$	0	0
$214$	10.0000	0.683586
$215$	0	0
$216$	0	0
$217$	0	0
$218$	2.23607i	0.151446i
$219$	0	0
$220$	0	0
$221$	13.4164i	0.902485i
$222$	0	0
$223$	− 24.2487i	− 1.62381i	−0.583787	−	0.811907i	$-0.698430\pi$
$223$	− 24.2487i	− 1.62381i	0.583787	−	0.811907i	$-0.301570\pi$
$224$	0	0
$225$	0	0
$226$	−20.0000	−1.33038
$227$	−23.2379	−1.54235	−0.771177	−	0.636621i	$-0.780332\pi$
$227$	−23.2379	−1.54235	−0.771177	+	0.636621i	$0.780332\pi$
$228$	0	0
$229$	25.9808i	1.71686i	0.512933	+	0.858429i	$0.328559\pi$
$229$	25.9808i	1.71686i	−0.512933	+	0.858429i	$0.671441\pi$
$230$	0	0
$231$	0	0
$232$	−10.0000	−0.656532
$233$	− 8.94427i	− 0.585959i	−0.956119	−	0.292979i	$-0.905353\pi$
$233$	− 8.94427i	− 0.585959i	0.956119	−	0.292979i	$-0.0946467\pi$
$234$	0	0
$235$	0	0
$236$	−23.2379	−1.51266
$237$	0	0
$238$	0	0
$239$	− 8.94427i	− 0.578557i	−0.957245	−	0.289278i	$-0.906585\pi$
$239$	− 8.94427i	− 0.578557i	0.957245	−	0.289278i	$-0.0934153\pi$
$240$	0	0
$241$	− 15.5885i	− 1.00414i	−0.864827	−	0.502070i	$-0.832572\pi$
$241$	− 15.5885i	− 1.00414i	0.864827	−	0.502070i	$-0.167428\pi$
$242$	20.1246i	1.29366i
$243$	0	0
$244$	− 25.9808i	− 1.66325i
$245$	0	0
$246$	0	0
$247$	−6.00000	−0.381771
$248$	−3.87298	−0.245935
$249$	0	0
$250$	0	0
$251$	−23.2379	−1.46676	−0.733382	−	0.679817i	$-0.762059\pi$
$251$	−23.2379	−1.46676	−0.733382	+	0.679817i	$0.762059\pi$
$252$	0	0
$253$	−20.0000	−1.25739
$254$	2.23607i	0.140303i
$255$	0	0
$256$	−9.00000	−0.562500
$257$	−23.2379	−1.44954	−0.724770	−	0.688991i	$-0.758054\pi$
$257$	−23.2379	−1.44954	−0.724770	+	0.688991i	$0.758054\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	4.47214i	0.275764i	0.990449	+	0.137882i	$0.0440294\pi$
$263$	4.47214i	0.275764i	−0.990449	+	0.137882i	$0.955971\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	3.00000	0.183254
$269$	15.4919	0.944560	0.472280	−	0.881449i	$-0.343431\pi$
$269$	15.4919	0.944560	0.472280	+	0.881449i	$0.343431\pi$
$270$	0	0
$271$	1.73205i	0.105215i	0.998615	+	0.0526073i	$0.0167532\pi$
$271$	1.73205i	0.105215i	−0.998615	+	0.0526073i	$0.983247\pi$
$272$	7.74597	0.469668
$273$	0	0
$274$	10.0000	0.604122
$275$	− 22.3607i	− 1.34840i
$276$	0	0
$277$	5.00000	0.300421	0.150210	−	0.988654i	$-0.452005\pi$
$277$	5.00000	0.300421	0.150210	+	0.988654i	$0.452005\pi$
$278$	19.3649	1.16143
$279$	0	0
$280$	0	0
$281$	17.8885i	1.06714i	0.845756	+	0.533571i	$0.179150\pi$
$281$	17.8885i	1.06714i	−0.845756	+	0.533571i	$0.820850\pi$
$282$	0	0
$283$	12.1244i	0.720718i	0.932814	+	0.360359i	$0.117346\pi$
$283$	12.1244i	0.720718i	−0.932814	+	0.360359i	$0.882654\pi$
$284$	26.8328i	1.59223i
$285$	0	0
$286$	− 17.3205i	− 1.02418i
$287$	0	0
$288$	0	0
$289$	43.0000	2.52941
$290$	0	0
$291$	0	0
$292$	20.7846i	1.21633i
$293$	−7.74597	−0.452524	−0.226262	−	0.974066i	$-0.572651\pi$
$293$	−7.74597	−0.452524	−0.226262	+	0.974066i	$0.572651\pi$
$294$	0	0
$295$	0	0
$296$	11.1803i	0.649844i
$297$	0	0
$298$	40.0000	2.31714
$299$	7.74597	0.447961
$300$	0	0
$301$	0	0
$302$	29.0689i	1.67273i
$303$	0	0
$304$	3.46410i	0.198680i
$305$	0	0
$306$	0	0
$307$	5.19615i	0.296560i	0.988945	+	0.148280i	$0.0473737\pi$
$307$	5.19615i	0.296560i	−0.988945	+	0.148280i	$0.952626\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−15.4919	−0.878467	−0.439233	−	0.898373i	$-0.644750\pi$
$311$	−15.4919	−0.878467	−0.439233	+	0.898373i	$0.644750\pi$
$312$	0	0
$313$	13.8564i	0.783210i	0.920133	+	0.391605i	$0.128080\pi$
$313$	13.8564i	0.783210i	−0.920133	+	0.391605i	$0.871920\pi$
$314$	−30.9839	−1.74852
$315$	0	0
$316$	−33.0000	−1.85640
$317$	4.47214i	0.251180i	0.992082	+	0.125590i	$0.0400824\pi$
$317$	4.47214i	0.251180i	−0.992082	+	0.125590i	$0.959918\pi$
$318$	0	0
$319$	−20.0000	−1.11979
$320$	0	0
$321$	0	0
$322$	0	0
$323$	26.8328i	1.49302i
$324$	0	0
$325$	8.66025i	0.480384i
$326$	29.0689i	1.60998i
$327$	0	0
$328$	− 17.3205i	− 0.956365i
$329$	0	0
$330$	0	0
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	−23.2379	−1.27535
$333$	0	0
$334$	17.3205i	0.947736i
$335$	0	0
$336$	0	0
$337$	−10.0000	−0.544735	−0.272367	−	0.962193i	$-0.587807\pi$
$337$	−10.0000	−0.544735	−0.272367	+	0.962193i	$0.587807\pi$
$338$	− 22.3607i	− 1.21626i
$339$	0	0
$340$	0	0
$341$	−7.74597	−0.419468
$342$	0	0
$343$	0	0
$344$	− 15.6525i	− 0.843925i
$345$	0	0
$346$	− 34.6410i	− 1.86231i
$347$	− 8.94427i	− 0.480154i	−0.970754	−	0.240077i	$-0.922827\pi$
$347$	− 8.94427i	− 0.480154i	0.970754	−	0.240077i	$-0.0771726\pi$
$348$	0	0
$349$	22.5167i	1.20529i	0.798010	+	0.602645i	$0.205886\pi$
$349$	22.5167i	1.20529i	−0.798010	+	0.602645i	$0.794114\pi$
$350$	0	0
$351$	0	0
$352$	−30.0000	−1.59901
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	0	0
$355$	0	0
$356$	46.4758	2.46321
$357$	0	0
$358$	−20.0000	−1.05703
$359$	4.47214i	0.236030i	0.993012	+	0.118015i	$0.0376531\pi$
$359$	4.47214i	0.236030i	−0.993012	+	0.118015i	$0.962347\pi$
$360$	0	0
$361$	7.00000	0.368421
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	10.3923i	0.542474i	0.962513	+	0.271237i	$0.0874327\pi$
$367$	10.3923i	0.542474i	−0.962513	+	0.271237i	$0.912567\pi$
$368$	− 4.47214i	− 0.233126i
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	2.00000	0.103556	0.0517780	−	0.998659i	$-0.483511\pi$
$373$	2.00000	0.103556	0.0517780	+	0.998659i	$0.483511\pi$
$374$	−77.4597	−4.00534
$375$	0	0
$376$	− 17.3205i	− 0.893237i
$377$	7.74597	0.398938
$378$	0	0
$379$	17.0000	0.873231	0.436616	−	0.899648i	$-0.356177\pi$
$379$	17.0000	0.873231	0.436616	+	0.899648i	$0.356177\pi$
$380$	0	0
$381$	0	0
$382$	40.0000	2.04658
$383$	−23.2379	−1.18740	−0.593701	−	0.804686i	$-0.702334\pi$
$383$	−23.2379	−1.18740	−0.593701	+	0.804686i	$0.702334\pi$
$384$	0	0
$385$	0	0
$386$	15.6525i	0.796690i
$387$	0	0
$388$	− 5.19615i	− 0.263795i
$389$	− 8.94427i	− 0.453493i	−0.973954	−	0.226746i	$-0.927191\pi$
$389$	− 8.94427i	− 0.453493i	0.973954	−	0.226746i	$-0.0728088\pi$
$390$	0	0
$391$	− 34.6410i	− 1.75187i
$392$	0	0
$393$	0	0
$394$	−20.0000	−1.00759
$395$	0	0
$396$	0	0
$397$	− 29.4449i	− 1.47780i	−0.673818	−	0.738898i	$-0.735347\pi$
$397$	− 29.4449i	− 1.47780i	0.673818	−	0.738898i	$-0.264653\pi$
$398$	−50.3488	−2.52376
$399$	0	0
$400$	5.00000	0.250000
$401$	− 22.3607i	− 1.11664i	−0.829626	−	0.558320i	$-0.811446\pi$
$401$	− 22.3607i	− 1.11664i	0.829626	−	0.558320i	$-0.188554\pi$
$402$	0	0
$403$	3.00000	0.149441
$404$	0	0
$405$	0	0
$406$	0	0
$407$	22.3607i	1.10838i
$408$	0	0
$409$	22.5167i	1.11338i	0.830721	+	0.556689i	$0.187928\pi$
$409$	22.5167i	1.11338i	−0.830721	+	0.556689i	$0.812072\pi$
$410$	0	0
$411$	0	0
$412$	15.5885i	0.767988i
$413$	0	0
$414$	0	0
$415$	0	0
$416$	11.6190	0.569666
$417$	0	0
$418$	− 34.6410i	− 1.69435i
$419$	15.4919	0.756830	0.378415	−	0.925636i	$-0.376469\pi$
$419$	15.4919	0.756830	0.378415	+	0.925636i	$0.376469\pi$
$420$	0	0
$421$	−34.0000	−1.65706	−0.828529	−	0.559946i	$-0.810822\pi$
$421$	−34.0000	−1.65706	−0.828529	+	0.559946i	$0.810822\pi$
$422$	42.4853i	2.06815i
$423$	0	0
$424$	−10.0000	−0.485643
$425$	38.7298	1.87867
$426$	0	0
$427$	0	0
$428$	− 13.4164i	− 0.648507i
$429$	0	0
$430$	0	0
$431$	31.3050i	1.50791i	0.656928	+	0.753953i	$0.271855\pi$
$431$	31.3050i	1.50791i	−0.656928	+	0.753953i	$0.728145\pi$
$432$	0	0
$433$	− 15.5885i	− 0.749133i	−0.927200	−	0.374567i	$-0.877791\pi$
$433$	− 15.5885i	− 0.749133i	0.927200	−	0.374567i	$-0.122209\pi$
$434$	0	0
$435$	0	0
$436$	3.00000	0.143674
$437$	15.4919	0.741080
$438$	0	0
$439$	3.46410i	0.165333i	0.996577	+	0.0826663i	$0.0263436\pi$
$439$	3.46410i	0.165333i	−0.996577	+	0.0826663i	$0.973656\pi$
$440$	0	0
$441$	0	0
$442$	30.0000	1.42695
$443$	− 35.7771i	− 1.69982i	−0.526927	−	0.849910i	$-0.676656\pi$
$443$	− 35.7771i	− 1.69982i	0.526927	−	0.849910i	$-0.323344\pi$
$444$	0	0
$445$	0	0
$446$	−54.2218	−2.56748
$447$	0	0
$448$	0	0
$449$	− 8.94427i	− 0.422106i	−0.977475	−	0.211053i	$-0.932311\pi$
$449$	− 8.94427i	− 0.422106i	0.977475	−	0.211053i	$-0.0676893\pi$
$450$	0	0
$451$	− 34.6410i	− 1.63118i
$452$	26.8328i	1.26211i
$453$	0	0
$454$	51.9615i	2.43868i
$455$	0	0
$456$	0	0
$457$	11.0000	0.514558	0.257279	−	0.966337i	$-0.417174\pi$
$457$	11.0000	0.514558	0.257279	+	0.966337i	$0.417174\pi$
$458$	58.0948	2.71459
$459$	0	0
$460$	0	0
$461$	7.74597	0.360766	0.180383	−	0.983596i	$-0.442266\pi$
$461$	7.74597	0.360766	0.180383	+	0.983596i	$0.442266\pi$
$462$	0	0
$463$	8.00000	0.371792	0.185896	−	0.982569i	$-0.440481\pi$
$463$	8.00000	0.371792	0.185896	+	0.982569i	$0.440481\pi$
$464$	− 4.47214i	− 0.207614i
$465$	0	0
$466$	−20.0000	−0.926482
$467$	30.9839	1.43376	0.716881	−	0.697195i	$-0.245569\pi$
$467$	30.9839	1.43376	0.716881	+	0.697195i	$0.245569\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	17.3205i	0.797241i
$473$	− 31.3050i	− 1.43940i
$474$	0	0
$475$	17.3205i	0.794719i
$476$	0	0
$477$	0	0
$478$	−20.0000	−0.914779
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	− 8.66025i	− 0.394874i
$482$	−34.8569	−1.58769
$483$	0	0
$484$	27.0000	1.22727
$485$	0	0
$486$	0	0
$487$	−40.0000	−1.81257	−0.906287	−	0.422664i	$-0.861095\pi$
$487$	−40.0000	−1.81257	−0.906287	+	0.422664i	$0.861095\pi$
$488$	−19.3649	−0.876609
$489$	0	0
$490$	0	0
$491$	− 22.3607i	− 1.00912i	−0.863376	−	0.504562i	$-0.831654\pi$
$491$	− 22.3607i	− 1.00912i	0.863376	−	0.504562i	$-0.168346\pi$
$492$	0	0
$493$	− 34.6410i	− 1.56015i
$494$	13.4164i	0.603633i
$495$	0	0
$496$	− 1.73205i	− 0.0777714i
$497$	0	0
$498$	0	0
$499$	−31.0000	−1.38775	−0.693875	−	0.720095i	$-0.744098\pi$
$499$	−31.0000	−1.38775	−0.693875	+	0.720095i	$0.744098\pi$
$500$	0	0
$501$	0	0
$502$	51.9615i	2.31916i
$503$	23.2379	1.03613	0.518063	−	0.855342i	$-0.326653\pi$
$503$	23.2379	1.03613	0.518063	+	0.855342i	$0.326653\pi$
$504$	0	0
$505$	0	0
$506$	44.7214i	1.98811i
$507$	0	0
$508$	3.00000	0.133103
$509$	23.2379	1.03000	0.515001	−	0.857190i	$-0.327792\pi$
$509$	23.2379	1.03000	0.515001	+	0.857190i	$0.327792\pi$
$510$	0	0
$511$	0	0
$512$	− 11.1803i	− 0.494106i
$513$	0	0
$514$	51.9615i	2.29192i
$515$	0	0
$516$	0	0
$517$	− 34.6410i	− 1.52351i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−7.74597	−0.339357	−0.169678	−	0.985499i	$-0.554273\pi$
$521$	−7.74597	−0.339357	−0.169678	+	0.985499i	$0.554273\pi$
$522$	0	0
$523$	− 19.0526i	− 0.833110i	−0.909110	−	0.416555i	$-0.863237\pi$
$523$	− 19.0526i	− 0.833110i	0.909110	−	0.416555i	$-0.136763\pi$
$524$	0	0
$525$	0	0
$526$	10.0000	0.436021
$527$	− 13.4164i	− 0.584428i
$528$	0	0
$529$	3.00000	0.130435
$530$	0	0
$531$	0	0
$532$	0	0
$533$	13.4164i	0.581129i
$534$	0	0
$535$	0	0
$536$	− 2.23607i	− 0.0965834i
$537$	0	0
$538$	− 34.6410i	− 1.49348i
$539$	0	0
$540$	0	0
$541$	14.0000	0.601907	0.300954	−	0.953639i	$-0.402695\pi$
$541$	14.0000	0.601907	0.300954	+	0.953639i	$0.402695\pi$
$542$	3.87298	0.166359
$543$	0	0
$544$	− 51.9615i	− 2.22783i
$545$	0	0
$546$	0	0
$547$	11.0000	0.470326	0.235163	−	0.971956i	$-0.424438\pi$
$547$	11.0000	0.470326	0.235163	+	0.971956i	$0.424438\pi$
$548$	− 13.4164i	− 0.573121i
$549$	0	0
$550$	−50.0000	−2.13201
$551$	15.4919	0.659979
$552$	0	0
$553$	0	0
$554$	− 11.1803i	− 0.475007i
$555$	0	0
$556$	− 25.9808i	− 1.10183i
$557$	− 35.7771i	− 1.51592i	−0.652299	−	0.757962i	$-0.726195\pi$
$557$	− 35.7771i	− 1.51592i	0.652299	−	0.757962i	$-0.273805\pi$
$558$	0	0
$559$	12.1244i	0.512806i
$560$	0	0
$561$	0	0
$562$	40.0000	1.68730
$563$	7.74597	0.326454	0.163227	−	0.986589i	$-0.447810\pi$
$563$	7.74597	0.326454	0.163227	+	0.986589i	$0.447810\pi$
$564$	0	0
$565$	0	0
$566$	27.1109	1.13956
$567$	0	0
$568$	20.0000	0.839181
$569$	17.8885i	0.749927i	0.927040	+	0.374963i	$0.122345\pi$
$569$	17.8885i	0.749927i	−0.927040	+	0.374963i	$0.877655\pi$
$570$	0	0
$571$	−16.0000	−0.669579	−0.334790	−	0.942293i	$-0.608665\pi$
$571$	−16.0000	−0.669579	−0.334790	+	0.942293i	$0.608665\pi$
$572$	−23.2379	−0.971625
$573$	0	0
$574$	0	0
$575$	− 22.3607i	− 0.932505i
$576$	0	0
$577$	39.8372i	1.65844i	0.558920	+	0.829222i	$0.311216\pi$
$577$	39.8372i	1.65844i	−0.558920	+	0.829222i	$0.688784\pi$
$578$	− 96.1509i	− 3.99935i
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−20.0000	−0.828315
$584$	15.4919	0.641061
$585$	0	0
$586$	17.3205i	0.715504i
$587$	30.9839	1.27884	0.639421	−	0.768857i	$-0.279174\pi$
$587$	30.9839	1.27884	0.639421	+	0.768857i	$0.279174\pi$
$588$	0	0
$589$	6.00000	0.247226
$590$	0	0
$591$	0	0
$592$	−5.00000	−0.205499
$593$	30.9839	1.27235	0.636177	−	0.771543i	$-0.280515\pi$
$593$	30.9839	1.27235	0.636177	+	0.771543i	$0.280515\pi$
$594$	0	0
$595$	0	0
$596$	− 53.6656i	− 2.19823i
$597$	0	0
$598$	− 17.3205i	− 0.708288i
$599$	− 8.94427i	− 0.365453i	−0.983164	−	0.182727i	$-0.941508\pi$
$599$	− 8.94427i	− 0.365453i	0.983164	−	0.182727i	$-0.0584923\pi$
$600$	0	0
$601$	43.3013i	1.76630i	0.469095	+	0.883148i	$0.344580\pi$
$601$	43.3013i	1.76630i	−0.469095	+	0.883148i	$0.655420\pi$
$602$	0	0
$603$	0	0
$604$	39.0000	1.58689
$605$	0	0
$606$	0	0
$607$	31.1769i	1.26543i	0.774384	+	0.632716i	$0.218060\pi$
$607$	31.1769i	1.26543i	−0.774384	+	0.632716i	$0.781940\pi$
$608$	23.2379	0.942421
$609$	0	0
$610$	0	0
$611$	13.4164i	0.542770i
$612$	0	0
$613$	23.0000	0.928961	0.464481	−	0.885583i	$-0.346241\pi$
$613$	23.0000	0.928961	0.464481	+	0.885583i	$0.346241\pi$
$614$	11.6190	0.468903
$615$	0	0
$616$	0	0
$617$	− 22.3607i	− 0.900207i	−0.892976	−	0.450104i	$-0.851387\pi$
$617$	− 22.3607i	− 0.900207i	0.892976	−	0.450104i	$-0.148613\pi$
$618$	0	0
$619$	− 5.19615i	− 0.208851i	−0.994533	−	0.104425i	$-0.966700\pi$
$619$	− 5.19615i	− 0.208851i	0.994533	−	0.104425i	$-0.0333004\pi$
$620$	0	0
$621$	0	0
$622$	34.6410i	1.38898i
$623$	0	0
$624$	0	0
$625$	25.0000	1.00000
$626$	30.9839	1.23836
$627$	0	0
$628$	41.5692i	1.65879i
$629$	−38.7298	−1.54426
$630$	0	0
$631$	29.0000	1.15447	0.577236	−	0.816577i	$-0.304131\pi$
$631$	29.0000	1.15447	0.577236	+	0.816577i	$0.304131\pi$
$632$	24.5967i	0.978406i
$633$	0	0
$634$	10.0000	0.397151
$635$	0	0
$636$	0	0
$637$	0	0
$638$	44.7214i	1.77054i
$639$	0	0
$640$	0	0
$641$	4.47214i	0.176639i	0.996092	+	0.0883194i	$0.0281496\pi$
$641$	4.47214i	0.176639i	−0.996092	+	0.0883194i	$0.971850\pi$
$642$	0	0
$643$	19.0526i	0.751360i	0.926750	+	0.375680i	$0.122591\pi$
$643$	19.0526i	0.751360i	−0.926750	+	0.375680i	$0.877409\pi$
$644$	0	0
$645$	0	0
$646$	60.0000	2.36067
$647$	15.4919	0.609051	0.304525	−	0.952504i	$-0.401502\pi$
$647$	15.4919	0.609051	0.304525	+	0.952504i	$0.401502\pi$
$648$	0	0
$649$	34.6410i	1.35978i
$650$	19.3649	0.759555
$651$	0	0
$652$	39.0000	1.52736
$653$	− 22.3607i	− 0.875041i	−0.899208	−	0.437521i	$-0.855857\pi$
$653$	− 22.3607i	− 0.875041i	0.899208	−	0.437521i	$-0.144143\pi$
$654$	0	0
$655$	0	0
$656$	7.74597	0.302429
$657$	0	0
$658$	0	0
$659$	17.8885i	0.696839i	0.937339	+	0.348419i	$0.113281\pi$
$659$	17.8885i	0.696839i	−0.937339	+	0.348419i	$0.886719\pi$
$660$	0	0
$661$	27.7128i	1.07790i	0.842337	+	0.538952i	$0.181179\pi$
$661$	27.7128i	1.07790i	−0.842337	+	0.538952i	$0.818821\pi$
$662$	8.94427i	0.347629i
$663$	0	0
$664$	17.3205i	0.672166i
$665$	0	0
$666$	0	0
$667$	−20.0000	−0.774403
$668$	23.2379	0.899101
$669$	0	0
$670$	0	0
$671$	−38.7298	−1.49515
$672$	0	0
$673$	2.00000	0.0770943	0.0385472	−	0.999257i	$-0.487727\pi$
$673$	2.00000	0.0770943	0.0385472	+	0.999257i	$0.487727\pi$
$674$	22.3607i	0.861301i
$675$	0	0
$676$	−30.0000	−1.15385
$677$	−30.9839	−1.19081	−0.595403	−	0.803427i	$-0.703008\pi$
$677$	−30.9839	−1.19081	−0.595403	+	0.803427i	$0.703008\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	17.3205i	0.663237i
$683$	− 35.7771i	− 1.36897i	−0.729026	−	0.684486i	$-0.760027\pi$
$683$	− 35.7771i	− 1.36897i	0.729026	−	0.684486i	$-0.239973\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	7.00000	0.266872
$689$	7.74597	0.295098
$690$	0	0
$691$	− 32.9090i	− 1.25192i	−0.779857	−	0.625958i	$-0.784708\pi$
$691$	− 32.9090i	− 1.25192i	0.779857	−	0.625958i	$-0.215292\pi$
$692$	−46.4758	−1.76674
$693$	0	0
$694$	−20.0000	−0.759190
$695$	0	0
$696$	0	0
$697$	60.0000	2.27266
$698$	50.3488	1.90573
$699$	0	0
$700$	0	0
$701$	− 8.94427i	− 0.337820i	−0.985631	−	0.168910i	$-0.945975\pi$
$701$	− 8.94427i	− 0.337820i	0.985631	−	0.168910i	$-0.0540248\pi$
$702$	0	0
$703$	− 17.3205i	− 0.653255i
$704$	58.1378i	2.19115i
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	11.0000	0.413114	0.206557	−	0.978435i	$-0.433774\pi$
$709$	11.0000	0.413114	0.206557	+	0.978435i	$0.433774\pi$
$710$	0	0
$711$	0	0
$712$	− 34.6410i	− 1.29823i
$713$	−7.74597	−0.290089
$714$	0	0
$715$	0	0
$716$	26.8328i	1.00279i
$717$	0	0
$718$	10.0000	0.373197
$719$	−15.4919	−0.577752	−0.288876	−	0.957367i	$-0.593281\pi$
$719$	−15.4919	−0.577752	−0.288876	+	0.957367i	$0.593281\pi$
$720$	0	0
$721$	0	0
$722$	− 15.6525i	− 0.582525i
$723$	0	0
$724$	0	0
$725$	− 22.3607i	− 0.830455i
$726$	0	0
$727$	− 19.0526i	− 0.706620i	−0.935506	−	0.353310i	$-0.885056\pi$
$727$	− 19.0526i	− 0.706620i	0.935506	−	0.353310i	$-0.114944\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	54.2218	2.00546
$732$	0	0
$733$	36.3731i	1.34347i	0.740792	+	0.671735i	$0.234451\pi$
$733$	36.3731i	1.34347i	−0.740792	+	0.671735i	$0.765549\pi$
$734$	23.2379	0.857727
$735$	0	0
$736$	−30.0000	−1.10581
$737$	− 4.47214i	− 0.164733i
$738$	0	0
$739$	23.0000	0.846069	0.423034	−	0.906114i	$-0.360965\pi$
$739$	23.0000	0.846069	0.423034	+	0.906114i	$0.360965\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	31.3050i	1.14847i	0.818691	+	0.574234i	$0.194700\pi$
$743$	31.3050i	1.14847i	−0.818691	+	0.574234i	$0.805300\pi$
$744$	0	0
$745$	0	0
$746$	− 4.47214i	− 0.163737i
$747$	0	0
$748$	103.923i	3.79980i
$749$	0	0
$750$	0	0
$751$	−16.0000	−0.583848	−0.291924	−	0.956441i	$-0.594295\pi$
$751$	−16.0000	−0.583848	−0.291924	+	0.956441i	$0.594295\pi$
$752$	7.74597	0.282466
$753$	0	0
$754$	− 17.3205i	− 0.630776i
$755$	0	0
$756$	0	0
$757$	−25.0000	−0.908640	−0.454320	−	0.890838i	$-0.650118\pi$
$757$	−25.0000	−0.908640	−0.454320	+	0.890838i	$0.650118\pi$
$758$	− 38.0132i	− 1.38070i
$759$	0	0
$760$	0	0
$761$	0	0		−	1.00000i	$-0.5\pi$
$761$	0	0			1.00000i	$0.5\pi$
$762$	0	0
$763$	0	0
$764$	− 53.6656i	− 1.94155i
$765$	0	0
$766$	51.9615i	1.87745i
$767$	− 13.4164i	− 0.484438i
$768$	0	0
$769$	34.6410i	1.24919i	0.780950	+	0.624593i	$0.214735\pi$
$769$	34.6410i	1.24919i	−0.780950	+	0.624593i	$0.785265\pi$
$770$	0	0
$771$	0	0
$772$	21.0000	0.755807
$773$	−30.9839	−1.11441	−0.557206	−	0.830374i	$-0.688127\pi$
$773$	−30.9839	−1.11441	−0.557206	+	0.830374i	$0.688127\pi$
$774$	0	0
$775$	− 8.66025i	− 0.311086i
$776$	−3.87298	−0.139032
$777$	0	0
$778$	−20.0000	−0.717035
$779$	26.8328i	0.961385i
$780$	0	0
$781$	40.0000	1.43131
$782$	−77.4597	−2.76995
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 19.0526i	− 0.679150i	−0.940579	−	0.339575i	$-0.889717\pi$
$787$	− 19.0526i	− 0.679150i	0.940579	−	0.339575i	$-0.110283\pi$
$788$	26.8328i	0.955879i
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	15.0000	0.532666
$794$	−65.8407	−2.33660
$795$	0	0
$796$	67.5500i	2.39425i
$797$	38.7298	1.37188	0.685941	−	0.727658i	$-0.259391\pi$
$797$	38.7298	1.37188	0.685941	+	0.727658i	$0.259391\pi$
$798$	0	0
$799$	60.0000	2.12265
$800$	− 33.5410i	− 1.18585i
$801$	0	0
$802$	−50.0000	−1.76556
$803$	30.9839	1.09340
$804$	0	0
$805$	0	0
$806$	− 6.70820i	− 0.236286i
$807$	0	0
$808$	0	0
$809$	4.47214i	0.157232i	0.996905	+	0.0786160i	$0.0250501\pi$
$809$	4.47214i	0.157232i	−0.996905	+	0.0786160i	$0.974950\pi$
$810$	0	0
$811$	10.3923i	0.364923i	0.983213	+	0.182462i	$0.0584065\pi$
$811$	10.3923i	0.364923i	−0.983213	+	0.182462i	$0.941593\pi$
$812$	0	0
$813$	0	0
$814$	50.0000	1.75250
$815$	0	0
$816$	0	0
$817$	24.2487i	0.848355i
$818$	50.3488	1.76040
$819$	0	0
$820$	0	0
$821$	− 22.3607i	− 0.780393i	−0.920732	−	0.390197i	$-0.872407\pi$
$821$	− 22.3607i	− 0.780393i	0.920732	−	0.390197i	$-0.127593\pi$
$822$	0	0
$823$	−43.0000	−1.49889	−0.749443	−	0.662069i	$-0.769679\pi$
$823$	−43.0000	−1.49889	−0.749443	+	0.662069i	$0.769679\pi$
$824$	11.6190	0.404765
$825$	0	0
$826$	0	0
$827$	− 8.94427i	− 0.311023i	−0.987834	−	0.155511i	$-0.950297\pi$
$827$	− 8.94427i	− 0.311023i	0.987834	−	0.155511i	$-0.0497025\pi$
$828$	0	0
$829$	13.8564i	0.481253i	0.970618	+	0.240626i	$0.0773529\pi$
$829$	13.8564i	0.481253i	−0.970618	+	0.240626i	$0.922647\pi$
$830$	0	0
$831$	0	0
$832$	− 22.5167i	− 0.780625i
$833$	0	0
$834$	0	0
$835$	0	0
$836$	−46.4758	−1.60740
$837$	0	0
$838$	− 34.6410i	− 1.19665i
$839$	7.74597	0.267420	0.133710	−	0.991020i	$-0.457311\pi$
$839$	7.74597	0.267420	0.133710	+	0.991020i	$0.457311\pi$
$840$	0	0
$841$	9.00000	0.310345
$842$	76.0263i	2.62004i
$843$	0	0
$844$	57.0000	1.96202
$845$	0	0
$846$	0	0
$847$	0	0
$848$	− 4.47214i	− 0.153574i
$849$	0	0
$850$	− 86.6025i	− 2.97044i
$851$	22.3607i	0.766514i
$852$	0	0
$853$	− 13.8564i	− 0.474434i	−0.971457	−	0.237217i	$-0.923765\pi$
$853$	− 13.8564i	− 0.474434i	0.971457	−	0.237217i	$-0.0762353\pi$
$854$	0	0
$855$	0	0
$856$	−10.0000	−0.341793
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	0	0
$859$	5.19615i	0.177290i	0.996063	+	0.0886452i	$0.0282537\pi$
$859$	5.19615i	0.177290i	−0.996063	+	0.0886452i	$0.971746\pi$
$860$	0	0
$861$	0	0
$862$	70.0000	2.38421
$863$	− 35.7771i	− 1.21787i	−0.793222	−	0.608933i	$-0.791598\pi$
$863$	− 35.7771i	− 1.21787i	0.793222	−	0.608933i	$-0.208402\pi$
$864$	0	0
$865$	0	0
$866$	−34.8569	−1.18448
$867$	0	0
$868$	0	0
$869$	49.1935i	1.66878i
$870$	0	0
$871$	1.73205i	0.0586883i
$872$	− 2.23607i	− 0.0757228i
$873$	0	0
$874$	− 34.6410i	− 1.17175i
$875$	0	0
$876$	0	0
$877$	−25.0000	−0.844190	−0.422095	−	0.906552i	$-0.638705\pi$
$877$	−25.0000	−0.844190	−0.422095	+	0.906552i	$0.638705\pi$
$878$	7.74597	0.261414
$879$	0	0
$880$	0	0
$881$	23.2379	0.782905	0.391452	−	0.920198i	$-0.371973\pi$
$881$	23.2379	0.782905	0.391452	+	0.920198i	$0.371973\pi$
$882$	0	0
$883$	8.00000	0.269221	0.134611	−	0.990899i	$-0.457022\pi$
$883$	8.00000	0.269221	0.134611	+	0.990899i	$0.457022\pi$
$884$	− 40.2492i	− 1.35373i
$885$	0	0
$886$	−80.0000	−2.68765
$887$	23.2379	0.780252	0.390126	−	0.920761i	$-0.372431\pi$
$887$	23.2379	0.780252	0.390126	+	0.920761i	$0.372431\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	72.7461i	2.43572i
$893$	26.8328i	0.897926i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	−20.0000	−0.667409
$899$	−7.74597	−0.258342
$900$	0	0
$901$	− 34.6410i	− 1.15406i
$902$	−77.4597	−2.57912
$903$	0	0
$904$	20.0000	0.665190
$905$	0	0
$906$	0	0
$907$	5.00000	0.166022	0.0830111	−	0.996549i	$-0.473546\pi$
$907$	5.00000	0.166022	0.0830111	+	0.996549i	$0.473546\pi$
$908$	69.7137	2.31353
$909$	0	0
$910$	0	0
$911$	4.47214i	0.148168i	0.997252	+	0.0740842i	$0.0236034\pi$
$911$	4.47214i	0.148168i	−0.997252	+	0.0740842i	$0.976397\pi$
$912$	0	0
$913$	34.6410i	1.14645i
$914$	− 24.5967i	− 0.813588i
$915$	0	0
$916$	− 77.9423i	− 2.57529i
$917$	0	0
$918$	0	0
$919$	−13.0000	−0.428830	−0.214415	−	0.976743i	$-0.568785\pi$
$919$	−13.0000	−0.428830	−0.214415	+	0.976743i	$0.568785\pi$
$920$	0	0
$921$	0	0
$922$	− 17.3205i	− 0.570421i
$923$	−15.4919	−0.509923
$924$	0	0
$925$	−25.0000	−0.821995
$926$	− 17.8885i	− 0.587854i
$927$	0	0
$928$	−30.0000	−0.984798
$929$	−30.9839	−1.01655	−0.508274	−	0.861196i	$-0.669716\pi$
$929$	−30.9839	−1.01655	−0.508274	+	0.861196i	$0.669716\pi$
$930$	0	0
$931$	0	0
$932$	26.8328i	0.878938i
$933$	0	0
$934$	− 69.2820i	− 2.26698i
$935$	0	0
$936$	0	0
$937$	46.7654i	1.52776i	0.645359	+	0.763879i	$0.276708\pi$
$937$	46.7654i	1.52776i	−0.645359	+	0.763879i	$0.723292\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−38.7298	−1.26256	−0.631278	−	0.775556i	$-0.717469\pi$
$941$	−38.7298	−1.26256	−0.631278	+	0.775556i	$0.717469\pi$
$942$	0	0
$943$	− 34.6410i	− 1.12807i
$944$	−7.74597	−0.252110
$945$	0	0
$946$	−70.0000	−2.27590
$947$	4.47214i	0.145325i	0.997357	+	0.0726624i	$0.0231496\pi$
$947$	4.47214i	0.145325i	−0.997357	+	0.0726624i	$0.976850\pi$
$948$	0	0
$949$	−12.0000	−0.389536
$950$	38.7298	1.25656
$951$	0	0
$952$	0	0
$953$	58.1378i	1.88327i	0.336640	+	0.941634i	$0.390710\pi$
$953$	58.1378i	1.88327i	−0.336640	+	0.941634i	$0.609290\pi$
$954$	0	0
$955$	0	0
$956$	26.8328i	0.867835i
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	28.0000	0.903226
$962$	−19.3649	−0.624350
$963$	0	0
$964$	46.7654i	1.50621i
$965$	0	0
$966$	0	0
$967$	−25.0000	−0.803946	−0.401973	−	0.915652i	$-0.631675\pi$
$967$	−25.0000	−0.803946	−0.401973	+	0.915652i	$0.631675\pi$
$968$	− 20.1246i	− 0.646830i
$969$	0	0
$970$	0	0
$971$	−61.9677	−1.98864	−0.994320	−	0.106436i	$-0.966056\pi$
$971$	−61.9677	−1.98864	−0.994320	+	0.106436i	$0.966056\pi$
$972$	0	0
$973$	0	0
$974$	89.4427i	2.86593i
$975$	0	0
$976$	− 8.66025i	− 0.277208i
$977$	58.1378i	1.85999i	0.367570	+	0.929996i	$0.380190\pi$
$977$	58.1378i	1.85999i	−0.367570	+	0.929996i	$0.619810\pi$
$978$	0	0
$979$	− 69.2820i	− 2.21426i
$980$	0	0
$981$	0	0
$982$	−50.0000	−1.59556
$983$	−46.4758	−1.48235	−0.741174	−	0.671313i	$-0.765731\pi$
$983$	−46.4758	−1.48235	−0.741174	+	0.671313i	$0.765731\pi$
$984$	0	0
$985$	0	0
$986$	−77.4597	−2.46682
$987$	0	0
$988$	18.0000	0.572656
$989$	− 31.3050i	− 0.995440i
$990$	0	0
$991$	−49.0000	−1.55654	−0.778268	−	0.627932i	$-0.783902\pi$
$991$	−49.0000	−1.55654	−0.778268	+	0.627932i	$0.783902\pi$
$992$	−11.6190	−0.368902
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	46.7654i	1.48107i	0.672015	+	0.740537i	$0.265429\pi$
$997$	46.7654i	1.48107i	−0.672015	+	0.740537i	$0.734571\pi$
$998$	69.3181i	2.19423i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1323.2.c.b.1322.1		4
3.2	odd	2	inner	1323.2.c.b.1322.3		4
7.2	even	3		189.2.p.c.80.1	yes	4
7.3	odd	6		189.2.p.c.26.2	yes	4
7.6	odd	2	inner	1323.2.c.b.1322.2		4
21.2	odd	6		189.2.p.c.80.2	yes	4
21.17	even	6		189.2.p.c.26.1	✓	4
21.20	even	2	inner	1323.2.c.b.1322.4		4
63.2	odd	6		567.2.s.e.458.1		4
63.16	even	3		567.2.s.e.458.2		4
63.23	odd	6		567.2.i.c.269.2		4
63.31	odd	6		567.2.s.e.26.1		4
63.38	even	6		567.2.i.c.215.2		4
63.52	odd	6		567.2.i.c.215.1		4
63.58	even	3		567.2.i.c.269.1		4
63.59	even	6		567.2.s.e.26.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.2.p.c.26.1	✓	4	21.17	even	6
189.2.p.c.26.2	yes	4	7.3	odd	6
189.2.p.c.80.1	yes	4	7.2	even	3
189.2.p.c.80.2	yes	4	21.2	odd	6
567.2.i.c.215.1		4	63.52	odd	6
567.2.i.c.215.2		4	63.38	even	6
567.2.i.c.269.1		4	63.58	even	3
567.2.i.c.269.2		4	63.23	odd	6
567.2.s.e.26.1		4	63.31	odd	6
567.2.s.e.26.2		4	63.59	even	6
567.2.s.e.458.1		4	63.2	odd	6
567.2.s.e.458.2		4	63.16	even	3
1323.2.c.b.1322.1		4	1.1	even	1	trivial
1323.2.c.b.1322.2		4	7.6	odd	2	inner
1323.2.c.b.1322.3		4	3.2	odd	2	inner
1323.2.c.b.1322.4		4	21.20	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1323 = 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1323.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.23607i q^{2} -3.00000 q^{4} +2.23607i q^{8} +O(q^{10})\)	\(q-2.23607i q^{2} -3.00000 q^{4} +2.23607i q^{8} +4.47214i q^{11} -1.73205i q^{13} -1.00000 q^{16} -7.74597 q^{17} -3.46410i q^{19} +10.0000 q^{22} +4.47214i q^{23} -5.00000 q^{25} -3.87298 q^{26} +4.47214i q^{29} +1.73205i q^{31} +6.70820i q^{32} +17.3205i q^{34} +5.00000 q^{37} -7.74597 q^{38} -7.74597 q^{41} -7.00000 q^{43} -13.4164i q^{44} +10.0000 q^{46} -7.74597 q^{47} +11.1803i q^{50} +5.19615i q^{52} +4.47214i q^{53} +10.0000 q^{58} +7.74597 q^{59} +8.66025i q^{61} +3.87298 q^{62} +13.0000 q^{64} -1.00000 q^{67} +23.2379 q^{68} -8.94427i q^{71} -6.92820i q^{73} -11.1803i q^{74} +10.3923i q^{76} +11.0000 q^{79} +17.3205i q^{82} +7.74597 q^{83} +15.6525i q^{86} -10.0000 q^{88} -15.4919 q^{89} -13.4164i q^{92} +17.3205i q^{94} +1.73205i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{4}+O(q^{10}) \) 4 * q - 12 * q^4	\( 4 q - 12 q^{4} - 4 q^{16} + 40 q^{22} - 20 q^{25} + 20 q^{37} - 28 q^{43} + 40 q^{46} + 40 q^{58} + 52 q^{64} - 4 q^{67} + 44 q^{79} - 40 q^{88}+O(q^{100}) \) 4 * q - 12 * q^4 - 4 * q^16 + 40 * q^22 - 20 * q^25 + 20 * q^37 - 28 * q^43 + 40 * q^46 + 40 * q^58 + 52 * q^64 - 4 * q^67 + 44 * q^79 - 40 * q^88

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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