LMFDB - Embedded newform 1323.2.c.f.1322.3

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:	$16$
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} + 24x^{14} + 212x^{12} + 872x^{10} + 1815x^{8} + 1928x^{6} + 996x^{4} + 200x^{2} + 1 $ x^16 + 24x^14 + 212x^12 + 872x^10 + 1815x^8 + 1928x^6 + 996x^4 + 200*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1322.3
Root			$-0.829521i$ of defining polynomial
Character	$\chi$	$=$	1323.1322
Dual form			1323.2.c.f.1322.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.02428i q^{2} -2.09771 q^{4} -0.182734 q^{5} +0.197790i q^{8} +O(q^{10})$	$q-2.02428i q^{2} -2.09771 q^{4} -0.182734 q^{5} +0.197790i q^{8} +0.369905i q^{10} +3.75918i q^{11} -5.85148i q^{13} -3.79504 q^{16} +4.31383 q^{17} -3.82318i q^{19} +0.383323 q^{20} +7.60963 q^{22} -4.66593i q^{23} -4.96661 q^{25} -11.8450 q^{26} +5.70220i q^{29} -8.38983i q^{31} +8.07779i q^{32} -8.73241i q^{34} -8.44514 q^{37} -7.73919 q^{38} -0.0361430i q^{40} +1.51057 q^{41} -3.00675 q^{43} -7.88566i q^{44} -9.44514 q^{46} +0.591307 q^{47} +10.0538i q^{50} +12.2747i q^{52} -13.0758i q^{53} -0.686930i q^{55} +11.5428 q^{58} -10.0238 q^{59} +1.40824i q^{61} -16.9834 q^{62} +8.76164 q^{64} +1.06927i q^{65} +7.78828 q^{67} -9.04917 q^{68} +5.78012i q^{71} -4.53567i q^{73} +17.0953i q^{74} +8.01992i q^{76} -7.88566 q^{79} +0.693482 q^{80} -3.05781i q^{82} +11.3338 q^{83} -0.788285 q^{85} +6.08650i q^{86} -0.743528 q^{88} -17.6768 q^{89} +9.78775i q^{92} -1.19697i q^{94} +0.698626i q^{95} -2.33590i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{4}+O(q^{10}) $ 16 * q - 16 * q^4	$ 16 q - 16 q^{4} + 48 q^{16} + 64 q^{22} - 16 q^{25} + 32 q^{37} - 16 q^{43} + 16 q^{46} - 32 q^{64} + 48 q^{67} - 64 q^{79} + 64 q^{85} - 176 q^{88}+O(q^{100}) $ 16 * q - 16 * q^4 + 48 * q^16 + 64 * q^22 - 16 * q^25 + 32 * q^37 - 16 * q^43 + 16 * q^46 - 32 * q^64 + 48 * q^67 - 64 * q^79 + 64 * q^85 - 176 * 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 $

$2$	− 2.02428i	− 1.43138i	−0.698417	−	0.715691i	$-0.746112\pi$
$2$	− 2.02428i	− 1.43138i	0.698417	−	0.715691i	$-0.253888\pi$
$3$	0	0
$3$	0	0
$4$	−2.09771	−1.04885
$5$	−0.182734	−0.0817212	−0.0408606	−	0.999165i	$-0.513010\pi$
$5$	−0.182734	−0.0817212	−0.0408606	+	0.999165i	$0.513010\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0.197790i	0.0699293i
$9$	0	0
$10$	0.369905i	0.116974i
$11$	3.75918i	1.13344i	0.823912	+	0.566718i	$0.191787\pi$
$11$	3.75918i	1.13344i	−0.823912	+	0.566718i	$0.808213\pi$
$12$	0	0
$13$	− 5.85148i	− 1.62291i	−0.584415	−	0.811455i	$-0.698676\pi$
$13$	− 5.85148i	− 1.62291i	0.584415	−	0.811455i	$-0.301324\pi$
$14$	0	0
$15$	0	0
$16$	−3.79504	−0.948759
$17$	4.31383	1.04626	0.523129	−	0.852253i	$-0.324765\pi$
$17$	4.31383	1.04626	0.523129	+	0.852253i	$0.324765\pi$
$18$	0	0
$19$	− 3.82318i	− 0.877098i	−0.898707	−	0.438549i	$-0.855493\pi$
$19$	− 3.82318i	− 0.877098i	0.898707	−	0.438549i	$-0.144507\pi$
$20$	0.383323	0.0857136
$21$	0	0
$22$	7.60963	1.62238
$23$	− 4.66593i	− 0.972913i	−0.873705	−	0.486456i	$-0.838289\pi$
$23$	− 4.66593i	− 0.972913i	0.873705	−	0.486456i	$-0.161711\pi$
$24$	0	0
$25$	−4.96661	−0.993322
$26$	−11.8450	−2.32300
$27$	0	0
$28$	0	0
$29$	5.70220i	1.05887i	0.848350	+	0.529436i	$0.177596\pi$
$29$	5.70220i	1.05887i	−0.848350	+	0.529436i	$0.822404\pi$
$30$	0	0
$31$	− 8.38983i	− 1.50686i	−0.657530	−	0.753429i	$-0.728399\pi$
$31$	− 8.38983i	− 1.50686i	0.657530	−	0.753429i	$-0.271601\pi$
$32$	8.07779i	1.42797i
$33$	0	0
$34$	− 8.73241i	− 1.49760i
$35$	0	0
$36$	0	0
$37$	−8.44514	−1.38837	−0.694186	−	0.719796i	$-0.744236\pi$
$37$	−8.44514	−1.38837	−0.694186	+	0.719796i	$0.744236\pi$
$38$	−7.73919	−1.25546
$39$	0	0
$40$	− 0.0361430i	− 0.00571471i
$41$	1.51057	0.235911	0.117956	−	0.993019i	$-0.462366\pi$
$41$	1.51057	0.235911	0.117956	+	0.993019i	$0.462366\pi$
$42$	0	0
$43$	−3.00675	−0.458525	−0.229263	−	0.973365i	$-0.573631\pi$
$43$	−3.00675	−0.458525	−0.229263	+	0.973365i	$0.573631\pi$
$44$	− 7.88566i	− 1.18881i
$45$	0	0
$46$	−9.44514	−1.39261
$47$	0.591307	0.0862511	0.0431255	−	0.999070i	$-0.486268\pi$
$47$	0.591307	0.0862511	0.0431255	+	0.999070i	$0.486268\pi$
$48$	0	0
$49$	0	0
$50$	10.0538i	1.42182i
$51$	0	0
$52$	12.2747i	1.70220i
$53$	− 13.0758i	− 1.79609i	−0.439901	−	0.898046i	$-0.644986\pi$
$53$	− 13.0758i	− 1.79609i	0.439901	−	0.898046i	$-0.355014\pi$
$54$	0	0
$55$	− 0.686930i	− 0.0926257i
$56$	0	0
$57$	0	0
$58$	11.5428	1.51565
$59$	−10.0238	−1.30499	−0.652494	−	0.757794i	$-0.726277\pi$
$59$	−10.0238	−1.30499	−0.652494	+	0.757794i	$0.726277\pi$
$60$	0	0
$61$	1.40824i	0.180306i	0.995928	+	0.0901532i	$0.0287357\pi$
$61$	1.40824i	0.180306i	−0.995928	+	0.0901532i	$0.971264\pi$
$62$	−16.9834	−2.15689
$63$	0	0
$64$	8.76164	1.09521
$65$	1.06927i	0.132626i
$66$	0	0
$67$	7.78828	0.951490	0.475745	−	0.879583i	$-0.342178\pi$
$67$	7.78828	0.951490	0.475745	+	0.879583i	$0.342178\pi$
$68$	−9.04917	−1.09737
$69$	0	0
$70$	0	0
$71$	5.78012i	0.685975i	0.939340	+	0.342987i	$0.111439\pi$
$71$	5.78012i	0.685975i	−0.939340	+	0.342987i	$0.888561\pi$
$72$	0	0
$73$	− 4.53567i	− 0.530860i	−0.964130	−	0.265430i	$-0.914486\pi$
$73$	− 4.53567i	− 0.530860i	0.964130	−	0.265430i	$-0.0855139\pi$
$74$	17.0953i	1.98729i
$75$	0	0
$76$	8.01992i	0.919948i
$77$	0	0
$78$	0	0
$79$	−7.88566	−0.887206	−0.443603	−	0.896223i	$-0.646300\pi$
$79$	−7.88566	−0.887206	−0.443603	+	0.896223i	$0.646300\pi$
$80$	0.693482	0.0775337
$81$	0	0
$82$	− 3.05781i	− 0.337679i
$83$	11.3338	1.24404	0.622022	−	0.783000i	$-0.286311\pi$
$83$	11.3338	1.24404	0.622022	+	0.783000i	$0.286311\pi$
$84$	0	0
$85$	−0.788285	−0.0855015
$86$	6.08650i	0.656325i
$87$	0	0
$88$	−0.743528	−0.0792604
$89$	−17.6768	−1.87374	−0.936870	−	0.349677i	$-0.886291\pi$
$89$	−17.6768	−1.87374	−0.936870	+	0.349677i	$0.886291\pi$
$90$	0	0
$91$	0	0
$92$	9.78775i	1.02044i
$93$	0	0
$94$	− 1.19697i	− 0.123458i
$95$	0.698626i	0.0716775i
$96$	0	0
$97$	− 2.33590i	− 0.237174i	−0.992944	−	0.118587i	$-0.962163\pi$
$97$	− 2.33590i	− 0.237174i	0.992944	−	0.118587i	$-0.0378365\pi$
$98$	0	0
$99$	0	0
$100$	10.4185	1.04185
$101$	−8.42527	−0.838346	−0.419173	−	0.907906i	$-0.637680\pi$
$101$	−8.42527	−0.838346	−0.419173	+	0.907906i	$0.637680\pi$
$102$	0	0
$103$	2.13406i	0.210275i	0.994458	+	0.105138i	$0.0335283\pi$
$103$	2.13406i	0.210275i	−0.994458	+	0.105138i	$0.966472\pi$
$104$	1.15736	0.113489
$105$	0	0
$106$	−26.4690	−2.57089
$107$	− 4.86372i	− 0.470193i	−0.971972	−	0.235097i	$-0.924459\pi$
$107$	− 4.86372i	− 0.470193i	0.971972	−	0.235097i	$-0.0755406\pi$
$108$	0	0
$109$	19.2263	1.84155	0.920775	−	0.390094i	$-0.127558\pi$
$109$	19.2263	1.84155	0.920775	+	0.390094i	$0.127558\pi$
$110$	−1.39054	−0.132583
$111$	0	0
$112$	0	0
$113$	1.99129i	0.187325i	0.995604	+	0.0936623i	$0.0298574\pi$
$113$	1.99129i	0.187325i	−0.995604	+	0.0936623i	$0.970143\pi$
$114$	0	0
$115$	0.852624i	0.0795076i
$116$	− 11.9616i	− 1.11060i
$117$	0	0
$118$	20.2910i	1.86794i
$119$	0	0
$120$	0	0
$121$	−3.13143	−0.284675
$122$	2.85067	0.258087
$123$	0	0
$124$	17.5994i	1.58047i
$125$	1.82124	0.162897
$126$	0	0
$127$	−8.81921	−0.782578	−0.391289	−	0.920268i	$-0.627971\pi$
$127$	−8.81921	−0.782578	−0.391289	+	0.920268i	$0.627971\pi$
$128$	− 1.58043i	− 0.139692i
$129$	0	0
$130$	2.16449	0.189839
$131$	17.5624	1.53443	0.767217	−	0.641388i	$-0.221641\pi$
$131$	17.5624	1.53443	0.767217	+	0.641388i	$0.221641\pi$
$132$	0	0
$133$	0	0
$134$	− 15.7657i	− 1.36195i
$135$	0	0
$136$	0.853233i	0.0731641i
$137$	1.75823i	0.150216i	0.997175	+	0.0751078i	$0.0239301\pi$
$137$	1.75823i	0.150216i	−0.997175	+	0.0751078i	$0.976070\pi$
$138$	0	0
$139$	20.6901i	1.75491i	0.479659	+	0.877455i	$0.340760\pi$
$139$	20.6901i	1.75491i	−0.479659	+	0.877455i	$0.659240\pi$
$140$	0	0
$141$	0	0
$142$	11.7006	0.981892
$143$	21.9968	1.83946
$144$	0	0
$145$	− 1.04199i	− 0.0865323i
$146$	−9.18146	−0.759863
$147$	0	0
$148$	17.7154	1.45620
$149$	− 15.6525i	− 1.28230i	−0.767415	−	0.641150i	$-0.778458\pi$
$149$	− 15.6525i	− 1.28230i	0.767415	−	0.641150i	$-0.221542\pi$
$150$	0	0
$151$	14.9337	1.21529	0.607643	−	0.794210i	$-0.292115\pi$
$151$	14.9337	1.21529	0.607643	+	0.794210i	$0.292115\pi$
$152$	0.756187	0.0613349
$153$	0	0
$154$	0	0
$155$	1.53311i	0.123142i
$156$	0	0
$157$	− 12.4586i	− 0.994308i	−0.867662	−	0.497154i	$-0.834378\pi$
$157$	− 12.4586i	− 0.994308i	0.867662	−	0.497154i	$-0.165622\pi$
$158$	15.9628i	1.26993i
$159$	0	0
$160$	− 1.47609i	− 0.116695i
$161$	0	0
$162$	0	0
$163$	14.0548	1.10085	0.550427	−	0.834883i	$-0.314465\pi$
$163$	14.0548	1.10085	0.550427	+	0.834883i	$0.314465\pi$
$164$	−3.16873	−0.247437
$165$	0	0
$166$	− 22.9427i	− 1.78070i
$167$	4.11935	0.318765	0.159382	−	0.987217i	$-0.449050\pi$
$167$	4.11935	0.318765	0.159382	+	0.987217i	$0.449050\pi$
$168$	0	0
$169$	−21.2398	−1.63383
$170$	1.59571i	0.122385i
$171$	0	0
$172$	6.30729	0.480926
$173$	15.8986	1.20875	0.604376	−	0.796699i	$-0.293423\pi$
$173$	15.8986	1.20875	0.604376	+	0.796699i	$0.293423\pi$
$174$	0	0
$175$	0	0
$176$	− 14.2662i	− 1.07536i
$177$	0	0
$178$	35.7829i	2.68204i
$179$	1.72995i	0.129303i	0.997908	+	0.0646514i	$0.0205935\pi$
$179$	1.72995i	0.129303i	−0.997908	+	0.0646514i	$0.979406\pi$
$180$	0	0
$181$	− 1.61068i	− 0.119721i	−0.998207	−	0.0598606i	$-0.980934\pi$
$181$	− 1.61068i	− 0.119721i	0.998207	−	0.0598606i	$-0.0190656\pi$
$182$	0	0
$183$	0	0
$184$	0.922873	0.0680351
$185$	1.54321	0.113459
$186$	0	0
$187$	16.2165i	1.18587i
$188$	−1.24039	−0.0904648
$189$	0	0
$190$	1.41421	0.102598
$191$	− 6.37117i	− 0.461002i	−0.973072	−	0.230501i	$-0.925964\pi$
$191$	− 6.37117i	− 0.461002i	0.973072	−	0.230501i	$-0.0740364\pi$
$192$	0	0
$193$	−5.85935	−0.421765	−0.210883	−	0.977511i	$-0.567634\pi$
$193$	−5.85935	−0.421765	−0.210883	+	0.977511i	$0.567634\pi$
$194$	−4.72851	−0.339487
$195$	0	0
$196$	0	0
$197$	3.00195i	0.213880i	0.994265	+	0.106940i	$0.0341053\pi$
$197$	3.00195i	0.213880i	−0.994265	+	0.106940i	$0.965895\pi$
$198$	0	0
$199$	− 5.87880i	− 0.416737i	−0.978050	−	0.208369i	$-0.933185\pi$
$199$	− 5.87880i	− 0.416737i	0.978050	−	0.208369i	$-0.0668154\pi$
$200$	− 0.982345i	− 0.0694623i
$201$	0	0
$202$	17.0551i	1.19999i
$203$	0	0
$204$	0	0
$205$	−0.276032	−0.0192789
$206$	4.31994	0.300984
$207$	0	0
$208$	22.2066i	1.53975i
$209$	14.3720	0.994134
$210$	0	0
$211$	−3.54285	−0.243900	−0.121950	−	0.992536i	$-0.538915\pi$
$211$	−3.54285	−0.243900	−0.121950	+	0.992536i	$0.538915\pi$
$212$	27.4291i	1.88384i
$213$	0	0
$214$	−9.84552	−0.673026
$215$	0.549436	0.0374712
$216$	0	0
$217$	0	0
$218$	− 38.9195i	− 2.63596i
$219$	0	0
$220$	1.44098i	0.0971508i
$221$	− 25.2423i	− 1.69798i
$222$	0	0
$223$	− 12.1674i	− 0.814792i	−0.913252	−	0.407396i	$-0.866437\pi$
$223$	− 12.1674i	− 0.814792i	0.913252	−	0.407396i	$-0.133563\pi$
$224$	0	0
$225$	0	0
$226$	4.03093	0.268133
$227$	24.2691	1.61080	0.805399	−	0.592733i	$-0.201951\pi$
$227$	24.2691	1.61080	0.805399	+	0.592733i	$0.201951\pi$
$228$	0	0
$229$	− 13.2775i	− 0.877402i	−0.898633	−	0.438701i	$-0.855439\pi$
$229$	− 13.2775i	− 0.877402i	0.898633	−	0.438701i	$-0.144561\pi$
$230$	1.72595	0.113806
$231$	0	0
$232$	−1.12784	−0.0740462
$233$	24.0123i	1.57310i	0.617527	+	0.786549i	$0.288134\pi$
$233$	24.0123i	1.57310i	−0.617527	+	0.786549i	$0.711866\pi$
$234$	0	0
$235$	−0.108052	−0.00704854
$236$	21.0270	1.36874
$237$	0	0
$238$	0	0
$239$	8.37522i	0.541748i	0.962615	+	0.270874i	$0.0873127\pi$
$239$	8.37522i	0.541748i	−0.962615	+	0.270874i	$0.912687\pi$
$240$	0	0
$241$	− 15.9824i	− 1.02952i	−0.857335	−	0.514759i	$-0.827882\pi$
$241$	− 15.9824i	− 1.02952i	0.857335	−	0.514759i	$-0.172118\pi$
$242$	6.33889i	0.407479i
$243$	0	0
$244$	− 2.95407i	− 0.189115i
$245$	0	0
$246$	0	0
$247$	−22.3713	−1.42345
$248$	1.65942	0.105373
$249$	0	0
$250$	− 3.68670i	− 0.233167i
$251$	27.0976	1.71038	0.855192	−	0.518312i	$-0.173439\pi$
$251$	27.0976	1.71038	0.855192	+	0.518312i	$0.173439\pi$
$252$	0	0
$253$	17.5401	1.10273
$254$	17.8525i	1.12017i
$255$	0	0
$256$	14.3241	0.895253
$257$	14.1696	0.883877	0.441939	−	0.897045i	$-0.354291\pi$
$257$	14.1696	0.883877	0.441939	+	0.897045i	$0.354291\pi$
$258$	0	0
$259$	0	0
$260$	− 2.24301i	− 0.139105i
$261$	0	0
$262$	− 35.5512i	− 2.19636i
$263$	− 21.2052i	− 1.30757i	−0.756682	−	0.653783i	$-0.773181\pi$
$263$	− 21.2052i	− 1.30757i	0.756682	−	0.653783i	$-0.226819\pi$
$264$	0	0
$265$	2.38939i	0.146779i
$266$	0	0
$267$	0	0
$268$	−16.3376	−0.997975
$269$	−24.7243	−1.50747	−0.753735	−	0.657178i	$-0.771750\pi$
$269$	−24.7243	−1.50747	−0.753735	+	0.657178i	$0.771750\pi$
$270$	0	0
$271$	− 22.7403i	− 1.38137i	−0.723154	−	0.690687i	$-0.757308\pi$
$271$	− 22.7403i	− 1.38137i	0.723154	−	0.690687i	$-0.242692\pi$
$272$	−16.3712	−0.992647
$273$	0	0
$274$	3.55915	0.215016
$275$	− 18.6704i	− 1.12587i
$276$	0	0
$277$	−9.86890	−0.592965	−0.296482	−	0.955038i	$-0.595814\pi$
$277$	−9.86890	−0.592965	−0.296482	+	0.955038i	$0.595814\pi$
$278$	41.8825	2.51195
$279$	0	0
$280$	0	0
$281$	20.0360i	1.19525i	0.801776	+	0.597624i	$0.203888\pi$
$281$	20.0360i	1.19525i	−0.801776	+	0.597624i	$0.796112\pi$
$282$	0	0
$283$	18.9539i	1.12669i	0.826222	+	0.563345i	$0.190486\pi$
$283$	18.9539i	1.12669i	−0.826222	+	0.563345i	$0.809514\pi$
$284$	− 12.1250i	− 0.719487i
$285$	0	0
$286$	− 44.5276i	− 2.63297i
$287$	0	0
$288$	0	0
$289$	1.60916	0.0946568
$290$	−2.10927	−0.123861
$291$	0	0
$292$	9.51451i	0.556795i
$293$	−24.6199	−1.43831	−0.719154	−	0.694851i	$-0.755470\pi$
$293$	−24.6199	−1.43831	−0.719154	+	0.694851i	$0.755470\pi$
$294$	0	0
$295$	1.83169	0.106645
$296$	− 1.67036i	− 0.0970879i
$297$	0	0
$298$	−31.6850	−1.83546
$299$	−27.3026	−1.57895
$300$	0	0
$301$	0	0
$302$	− 30.2300i	− 1.73954i
$303$	0	0
$304$	14.5091i	0.832154i
$305$	− 0.257333i	− 0.0147348i
$306$	0	0
$307$	− 30.4490i	− 1.73781i	−0.494975	−	0.868907i	$-0.664823\pi$
$307$	− 30.4490i	− 1.73781i	0.494975	−	0.868907i	$-0.335177\pi$
$308$	0	0
$309$	0	0
$310$	3.10344	0.176263
$311$	0.922300	0.0522988	0.0261494	−	0.999658i	$-0.491675\pi$
$311$	0.922300	0.0522988	0.0261494	+	0.999658i	$0.491675\pi$
$312$	0	0
$313$	− 2.43861i	− 0.137839i	−0.997622	−	0.0689193i	$-0.978045\pi$
$313$	− 2.43861i	− 0.137839i	0.997622	−	0.0689193i	$-0.0219551\pi$
$314$	−25.2198	−1.42324
$315$	0	0
$316$	16.5418	0.930550
$317$	20.1189i	1.12999i	0.825095	+	0.564995i	$0.191122\pi$
$317$	20.1189i	1.12999i	−0.825095	+	0.564995i	$0.808878\pi$
$318$	0	0
$319$	−21.4356	−1.20016
$320$	−1.60105	−0.0895015
$321$	0	0
$322$	0	0
$323$	− 16.4926i	− 0.917671i
$324$	0	0
$325$	29.0620i	1.61207i
$326$	− 28.4508i	− 1.57574i
$327$	0	0
$328$	0.298775i	0.0164971i
$329$	0	0
$330$	0	0
$331$	−10.0952	−0.554885	−0.277442	−	0.960742i	$-0.589487\pi$
$331$	−10.0952	−0.554885	−0.277442	+	0.960742i	$0.589487\pi$
$332$	−23.7750	−1.30482
$333$	0	0
$334$	− 8.33871i	− 0.456274i
$335$	−1.42319	−0.0777569
$336$	0	0
$337$	15.7549	0.858224	0.429112	−	0.903251i	$-0.358827\pi$
$337$	15.7549	0.858224	0.429112	+	0.903251i	$0.358827\pi$
$338$	42.9954i	2.33864i
$339$	0	0
$340$	1.65359	0.0896786
$341$	31.5389	1.70792
$342$	0	0
$343$	0	0
$344$	− 0.594705i	− 0.0320644i
$345$	0	0
$346$	− 32.1833i	− 1.73018i
$347$	− 9.30392i	− 0.499460i	−0.968315	−	0.249730i	$-0.919658\pi$
$347$	− 9.30392i	− 0.499460i	0.968315	−	0.249730i	$-0.0803419\pi$
$348$	0	0
$349$	10.4886i	0.561444i	0.959789	+	0.280722i	$0.0905740\pi$
$349$	10.4886i	0.561444i	−0.959789	+	0.280722i	$0.909426\pi$
$350$	0	0
$351$	0	0
$352$	−30.3659	−1.61851
$353$	−13.7833	−0.733609	−0.366805	−	0.930298i	$-0.619548\pi$
$353$	−13.7833	−0.733609	−0.366805	+	0.930298i	$0.619548\pi$
$354$	0	0
$355$	− 1.05623i	− 0.0560586i
$356$	37.0809	1.96528
$357$	0	0
$358$	3.50191	0.185082
$359$	3.38359i	0.178579i	0.996006	+	0.0892896i	$0.0284597\pi$
$359$	3.38359i	0.178579i	−0.996006	+	0.0892896i	$0.971540\pi$
$360$	0	0
$361$	4.38329	0.230699
$362$	−3.26047	−0.171367
$363$	0	0
$364$	0	0
$365$	0.828821i	0.0433825i
$366$	0	0
$367$	− 11.5605i	− 0.603451i	−0.953395	−	0.301725i	$-0.902437\pi$
$367$	− 11.5605i	− 0.603451i	0.953395	−	0.301725i	$-0.0975626\pi$
$368$	17.7074i	0.923060i
$369$	0	0
$370$	− 3.12390i	− 0.162404i
$371$	0	0
$372$	0	0
$373$	30.7450	1.59192	0.795958	−	0.605351i	$-0.206967\pi$
$373$	30.7450	1.59192	0.795958	+	0.605351i	$0.206967\pi$
$374$	32.8267	1.69743
$375$	0	0
$376$	0.116955i	0.00603148i
$377$	33.3663	1.71845
$378$	0	0
$379$	19.9375	1.02412	0.512060	−	0.858949i	$-0.328882\pi$
$379$	19.9375	1.02412	0.512060	+	0.858949i	$0.328882\pi$
$380$	− 1.46551i	− 0.0751792i
$381$	0	0
$382$	−12.8970	−0.659869
$383$	7.42161	0.379226	0.189613	−	0.981859i	$-0.439277\pi$
$383$	7.42161	0.379226	0.189613	+	0.981859i	$0.439277\pi$
$384$	0	0
$385$	0	0
$386$	11.8610i	0.603707i
$387$	0	0
$388$	4.90003i	0.248761i
$389$	− 8.84563i	− 0.448491i	−0.974533	−	0.224245i	$-0.928008\pi$
$389$	− 8.84563i	− 0.448491i	0.974533	−	0.224245i	$-0.0719917\pi$
$390$	0	0
$391$	− 20.1280i	− 1.01792i
$392$	0	0
$393$	0	0
$394$	6.07680	0.306145
$395$	1.44098	0.0725036
$396$	0	0
$397$	3.50275i	0.175798i	0.996129	+	0.0878991i	$0.0280153\pi$
$397$	3.50275i	0.175798i	−0.996129	+	0.0878991i	$0.971985\pi$
$398$	−11.9003	−0.596510
$399$	0	0
$400$	18.8485	0.942423
$401$	− 8.39145i	− 0.419049i	−0.977803	−	0.209524i	$-0.932808\pi$
$401$	− 8.39145i	− 0.419049i	0.977803	−	0.209524i	$-0.0671916\pi$
$402$	0	0
$403$	−49.0929	−2.44549
$404$	17.6738	0.879303
$405$	0	0
$406$	0	0
$407$	− 31.7468i	− 1.57363i
$408$	0	0
$409$	− 6.52016i	− 0.322401i	−0.986922	−	0.161201i	$-0.948463\pi$
$409$	− 6.52016i	− 0.322401i	0.986922	−	0.161201i	$-0.0515366\pi$
$410$	0.558767i	0.0275955i
$411$	0	0
$412$	− 4.47664i	− 0.220548i
$413$	0	0
$414$	0	0
$415$	−2.07107	−0.101665
$416$	47.2671	2.31746
$417$	0	0
$418$	− 29.0930i	− 1.42298i
$419$	−14.7161	−0.718928	−0.359464	−	0.933159i	$-0.617040\pi$
$419$	−14.7161	−0.718928	−0.359464	+	0.933159i	$0.617040\pi$
$420$	0	0
$421$	1.60916	0.0784259	0.0392129	−	0.999231i	$-0.487515\pi$
$421$	1.60916	0.0784259	0.0392129	+	0.999231i	$0.487515\pi$
$422$	7.17172i	0.349114i
$423$	0	0
$424$	2.58625	0.125600
$425$	−21.4251	−1.03927
$426$	0	0
$427$	0	0
$428$	10.2027i	0.493164i
$429$	0	0
$430$	− 1.11221i	− 0.0536356i
$431$	26.3912i	1.27122i	0.772010	+	0.635611i	$0.219252\pi$
$431$	26.3912i	1.27122i	−0.772010	+	0.635611i	$0.780748\pi$
$432$	0	0
$433$	12.1988i	0.586235i	0.956076	+	0.293118i	$0.0946928\pi$
$433$	12.1988i	0.586235i	−0.956076	+	0.293118i	$0.905307\pi$
$434$	0	0
$435$	0	0
$436$	−40.3313	−1.93152
$437$	−17.8387	−0.853340
$438$	0	0
$439$	16.6812i	0.796150i	0.917353	+	0.398075i	$0.130322\pi$
$439$	16.6812i	0.796150i	−0.917353	+	0.398075i	$0.869678\pi$
$440$	0.135868	0.00647725
$441$	0	0
$442$	−51.0975	−2.43046
$443$	− 17.5206i	− 0.832427i	−0.909267	−	0.416213i	$-0.863357\pi$
$443$	− 17.5206i	− 0.832427i	0.909267	−	0.416213i	$-0.136643\pi$
$444$	0	0
$445$	3.23016	0.153124
$446$	−24.6303	−1.16628
$447$	0	0
$448$	0	0
$449$	37.1832i	1.75478i	0.479774	+	0.877392i	$0.340719\pi$
$449$	37.1832i	1.75478i	−0.479774	+	0.877392i	$0.659281\pi$
$450$	0	0
$451$	5.67850i	0.267390i
$452$	− 4.17714i	− 0.196476i
$453$	0	0
$454$	− 49.1275i	− 2.30567i
$455$	0	0
$456$	0	0
$457$	−8.75703	−0.409637	−0.204818	−	0.978800i	$-0.565660\pi$
$457$	−8.75703	−0.409637	−0.204818	+	0.978800i	$0.565660\pi$
$458$	−26.8774	−1.25590
$459$	0	0
$460$	− 1.78856i	− 0.0833919i
$461$	18.6070	0.866616	0.433308	−	0.901246i	$-0.357346\pi$
$461$	18.6070	0.866616	0.433308	+	0.901246i	$0.357346\pi$
$462$	0	0
$463$	10.3451	0.480778	0.240389	−	0.970677i	$-0.422725\pi$
$463$	10.3451	0.480778	0.240389	+	0.970677i	$0.422725\pi$
$464$	− 21.6400i	− 1.00461i
$465$	0	0
$466$	48.6076	2.25171
$467$	2.99036	0.138377	0.0691886	−	0.997604i	$-0.477959\pi$
$467$	2.99036	0.138377	0.0691886	+	0.997604i	$0.477959\pi$
$468$	0	0
$469$	0	0
$470$	0.218728i	0.0100891i
$471$	0	0
$472$	− 1.98261i	− 0.0912569i
$473$	− 11.3029i	− 0.519709i
$474$	0	0
$475$	18.9882i	0.871240i
$476$	0	0
$477$	0	0
$478$	16.9538	0.775449
$479$	33.0451	1.50987	0.754935	−	0.655799i	$-0.227668\pi$
$479$	33.0451	1.50987	0.754935	+	0.655799i	$0.227668\pi$
$480$	0	0
$481$	49.4166i	2.25320i
$482$	−32.3529	−1.47363
$483$	0	0
$484$	6.56883	0.298583
$485$	0.426848i	0.0193822i
$486$	0	0
$487$	31.0324	1.40621	0.703106	−	0.711085i	$-0.251796\pi$
$487$	31.0324	1.40621	0.703106	+	0.711085i	$0.251796\pi$
$488$	−0.278535	−0.0126087
$489$	0	0
$490$	0	0
$491$	− 21.2518i	− 0.959081i	−0.877520	−	0.479540i	$-0.840803\pi$
$491$	− 21.2518i	− 0.959081i	0.877520	−	0.479540i	$-0.159197\pi$
$492$	0	0
$493$	24.5983i	1.10785i
$494$	45.2857i	2.03750i
$495$	0	0
$496$	31.8397i	1.42964i
$497$	0	0
$498$	0	0
$499$	18.3721	0.822447	0.411223	−	0.911535i	$-0.365102\pi$
$499$	18.3721	0.822447	0.411223	+	0.911535i	$0.365102\pi$
$500$	−3.82043	−0.170855
$501$	0	0
$502$	− 54.8531i	− 2.44821i
$503$	23.9645	1.06853	0.534263	−	0.845318i	$-0.320589\pi$
$503$	23.9645	1.06853	0.534263	+	0.845318i	$0.320589\pi$
$504$	0	0
$505$	1.53959	0.0685106
$506$	− 35.5060i	− 1.57843i
$507$	0	0
$508$	18.5001	0.820811
$509$	6.04285	0.267844	0.133922	−	0.990992i	$-0.457243\pi$
$509$	6.04285	0.267844	0.133922	+	0.990992i	$0.457243\pi$
$510$	0	0
$511$	0	0
$512$	− 32.1568i	− 1.42114i
$513$	0	0
$514$	− 28.6833i	− 1.26517i
$515$	− 0.389966i	− 0.0171839i
$516$	0	0
$517$	2.22283i	0.0977600i
$518$	0	0
$519$	0	0
$520$	−0.211490	−0.00927445
$521$	6.30488	0.276222	0.138111	−	0.990417i	$-0.455897\pi$
$521$	6.30488	0.276222	0.138111	+	0.990417i	$0.455897\pi$
$522$	0	0
$523$	− 16.1419i	− 0.705838i	−0.935654	−	0.352919i	$-0.885189\pi$
$523$	− 16.1419i	− 0.705838i	0.935654	−	0.352919i	$-0.114811\pi$
$524$	−36.8408	−1.60940
$525$	0	0
$526$	−42.9252	−1.87163
$527$	− 36.1923i	− 1.57656i
$528$	0	0
$529$	1.22914	0.0534408
$530$	4.83679	0.210097
$531$	0	0
$532$	0	0
$533$	− 8.83907i	− 0.382862i
$534$	0	0
$535$	0.888767i	0.0384247i
$536$	1.54044i	0.0665371i
$537$	0	0
$538$	50.0490i	2.15777i
$539$	0	0
$540$	0	0
$541$	−8.36093	−0.359465	−0.179732	−	0.983716i	$-0.557523\pi$
$541$	−8.36093	−0.359465	−0.179732	+	0.983716i	$0.557523\pi$
$542$	−46.0327	−1.97727
$543$	0	0
$544$	34.8463i	1.49402i
$545$	−3.51331	−0.150494
$546$	0	0
$547$	13.9403	0.596044	0.298022	−	0.954559i	$-0.403673\pi$
$547$	13.9403	0.596044	0.298022	+	0.954559i	$0.403673\pi$
$548$	− 3.68825i	− 0.157554i
$549$	0	0
$550$	−37.7941	−1.61154
$551$	21.8005	0.928734
$552$	0	0
$553$	0	0
$554$	19.9774i	0.848759i
$555$	0	0
$556$	− 43.4018i	− 1.84065i
$557$	17.3890i	0.736795i	0.929668	+	0.368397i	$0.120093\pi$
$557$	17.3890i	0.736795i	−0.929668	+	0.368397i	$0.879907\pi$
$558$	0	0
$559$	17.5939i	0.744145i
$560$	0	0
$561$	0	0
$562$	40.5585	1.71086
$563$	43.0664	1.81503	0.907516	−	0.420018i	$-0.137976\pi$
$563$	43.0664	1.81503	0.907516	+	0.420018i	$0.137976\pi$
$564$	0	0
$565$	− 0.363876i	− 0.0153084i
$566$	38.3679	1.61272
$567$	0	0
$568$	−1.14325	−0.0479697
$569$	12.6891i	0.531956i	0.963979	+	0.265978i	$0.0856948\pi$
$569$	12.6891i	0.531956i	−0.963979	+	0.265978i	$0.914305\pi$
$570$	0	0
$571$	−1.72005	−0.0719817	−0.0359908	−	0.999352i	$-0.511459\pi$
$571$	−1.72005	−0.0719817	−0.0359908	+	0.999352i	$0.511459\pi$
$572$	−46.1428	−1.92933
$573$	0	0
$574$	0	0
$575$	23.1738i	0.966415i
$576$	0	0
$577$	47.7326i	1.98714i	0.113240	+	0.993568i	$0.463877\pi$
$577$	47.7326i	1.98714i	−0.113240	+	0.993568i	$0.536123\pi$
$578$	− 3.25740i	− 0.135490i
$579$	0	0
$580$	2.18578i	0.0907597i
$581$	0	0
$582$	0	0
$583$	49.1541	2.03575
$584$	0.897110	0.0371227
$585$	0	0
$586$	49.8375i	2.05877i
$587$	−29.1477	−1.20305	−0.601527	−	0.798852i	$-0.705441\pi$
$587$	−29.1477	−1.20305	−0.601527	+	0.798852i	$0.705441\pi$
$588$	0	0
$589$	−32.0758	−1.32166
$590$	− 3.70785i	− 0.152650i
$591$	0	0
$592$	32.0496	1.31723
$593$	−38.8990	−1.59739	−0.798695	−	0.601736i	$-0.794476\pi$
$593$	−38.8990	−1.59739	−0.798695	+	0.601736i	$0.794476\pi$
$594$	0	0
$595$	0	0
$596$	32.8343i	1.34495i
$597$	0	0
$598$	55.2681i	2.26008i
$599$	21.3529i	0.872458i	0.899836	+	0.436229i	$0.143686\pi$
$599$	21.3529i	0.872458i	−0.899836	+	0.436229i	$0.856314\pi$
$600$	0	0
$601$	13.7270i	0.559936i	0.960009	+	0.279968i	$0.0903239\pi$
$601$	13.7270i	0.559936i	−0.960009	+	0.279968i	$0.909676\pi$
$602$	0	0
$603$	0	0
$604$	−31.3265	−1.27466
$605$	0.572219	0.0232640
$606$	0	0
$607$	− 8.63482i	− 0.350476i	−0.984526	−	0.175238i	$-0.943930\pi$
$607$	− 8.63482i	− 0.350476i	0.984526	−	0.175238i	$-0.0560696\pi$
$608$	30.8829	1.25247
$609$	0	0
$610$	−0.520914	−0.0210912
$611$	− 3.46003i	− 0.139978i
$612$	0	0
$613$	−8.12468	−0.328153	−0.164076	−	0.986448i	$-0.552464\pi$
$613$	−8.12468	−0.328153	−0.164076	+	0.986448i	$0.552464\pi$
$614$	−61.6372	−2.48748
$615$	0	0
$616$	0	0
$617$	27.9726i	1.12613i	0.826412	+	0.563067i	$0.190379\pi$
$617$	27.9726i	1.12613i	−0.826412	+	0.563067i	$0.809621\pi$
$618$	0	0
$619$	15.0292i	0.604074i	0.953296	+	0.302037i	$0.0976666\pi$
$619$	15.0292i	0.604074i	−0.953296	+	0.302037i	$0.902333\pi$
$620$	− 3.21601i	− 0.129158i
$621$	0	0
$622$	− 1.86699i	− 0.0748596i
$623$	0	0
$624$	0	0
$625$	24.5002	0.980010
$626$	−4.93643	−0.197300
$627$	0	0
$628$	26.1346i	1.04288i
$629$	−36.4309	−1.45260
$630$	0	0
$631$	−15.8217	−0.629851	−0.314925	−	0.949116i	$-0.601980\pi$
$631$	−15.8217	−0.629851	−0.314925	+	0.949116i	$0.601980\pi$
$632$	− 1.55971i	− 0.0620417i
$633$	0	0
$634$	40.7263	1.61745
$635$	1.61157	0.0639532
$636$	0	0
$637$	0	0
$638$	43.3916i	1.71789i
$639$	0	0
$640$	0.288799i	0.0114158i
$641$	− 20.3414i	− 0.803436i	−0.915763	−	0.401718i	$-0.868413\pi$
$641$	− 20.3414i	− 0.803436i	0.915763	−	0.401718i	$-0.131587\pi$
$642$	0	0
$643$	31.5684i	1.24493i	0.782646	+	0.622467i	$0.213870\pi$
$643$	31.5684i	1.24493i	−0.782646	+	0.622467i	$0.786130\pi$
$644$	0	0
$645$	0	0
$646$	−33.3856	−1.31354
$647$	−16.4722	−0.647587	−0.323794	−	0.946128i	$-0.604958\pi$
$647$	−16.4722	−0.647587	−0.323794	+	0.946128i	$0.604958\pi$
$648$	0	0
$649$	− 37.6813i	− 1.47912i
$650$	58.8297	2.30749
$651$	0	0
$652$	−29.4828	−1.15464
$653$	26.6236i	1.04186i	0.853598	+	0.520932i	$0.174415\pi$
$653$	26.6236i	1.04186i	−0.853598	+	0.520932i	$0.825585\pi$
$654$	0	0
$655$	−3.20925	−0.125396
$656$	−5.73266	−0.223823
$657$	0	0
$658$	0	0
$659$	35.9723i	1.40128i	0.713515	+	0.700640i	$0.247102\pi$
$659$	35.9723i	1.40128i	−0.713515	+	0.700640i	$0.752898\pi$
$660$	0	0
$661$	− 20.2162i	− 0.786319i	−0.919470	−	0.393159i	$-0.871382\pi$
$661$	− 20.2162i	− 0.786319i	0.919470	−	0.393159i	$-0.128618\pi$
$662$	20.4356i	0.794252i
$663$	0	0
$664$	2.24171i	0.0869951i
$665$	0	0
$666$	0	0
$667$	26.6060	1.03019
$668$	−8.64119	−0.334338
$669$	0	0
$670$	2.88093i	0.111300i
$671$	−5.29382	−0.204366
$672$	0	0
$673$	13.3681	0.515304	0.257652	−	0.966238i	$-0.417051\pi$
$673$	13.3681	0.515304	0.257652	+	0.966238i	$0.417051\pi$
$674$	− 31.8923i	− 1.22845i
$675$	0	0
$676$	44.5550	1.71365
$677$	−22.6406	−0.870147	−0.435074	−	0.900395i	$-0.643278\pi$
$677$	−22.6406	−0.870147	−0.435074	+	0.900395i	$0.643278\pi$
$678$	0	0
$679$	0	0
$680$	− 0.155915i	− 0.00597906i
$681$	0	0
$682$	− 63.8435i	− 2.44469i
$683$	11.0444i	0.422601i	0.977421	+	0.211300i	$0.0677699\pi$
$683$	11.0444i	0.422601i	−0.977421	+	0.211300i	$0.932230\pi$
$684$	0	0
$685$	− 0.321288i	− 0.0122758i
$686$	0	0
$687$	0	0
$688$	11.4107	0.435030
$689$	−76.5125	−2.91490
$690$	0	0
$691$	1.41580i	0.0538595i	0.999637	+	0.0269297i	$0.00857304\pi$
$691$	1.41580i	0.0538595i	−0.999637	+	0.0269297i	$0.991427\pi$
$692$	−33.3507	−1.26780
$693$	0	0
$694$	−18.8337	−0.714919
$695$	− 3.78078i	− 0.143413i
$696$	0	0
$697$	6.51634	0.246824
$698$	21.2320	0.803641
$699$	0	0
$700$	0	0
$701$	− 15.1510i	− 0.572244i	−0.958193	−	0.286122i	$-0.907634\pi$
$701$	− 15.1510i	− 0.572244i	0.958193	−	0.286122i	$-0.0923663\pi$
$702$	0	0
$703$	32.2873i	1.21774i
$704$	32.9366i	1.24134i
$705$	0	0
$706$	27.9012i	1.05008i
$707$	0	0
$708$	0	0
$709$	19.8996	0.747346	0.373673	−	0.927560i	$-0.378098\pi$
$709$	19.8996	0.747346	0.373673	+	0.927560i	$0.378098\pi$
$710$	−2.13810	−0.0802413
$711$	0	0
$712$	− 3.49630i	− 0.131029i
$713$	−39.1463	−1.46604
$714$	0	0
$715$	−4.01956	−0.150323
$716$	− 3.62894i	− 0.135620i
$717$	0	0
$718$	6.84934	0.255615
$719$	0.0892487	0.00332841	0.00166421	−	0.999999i	$-0.499470\pi$
$719$	0.0892487	0.00332841	0.00166421	+	0.999999i	$0.499470\pi$
$720$	0	0
$721$	0	0
$722$	− 8.87300i	− 0.330219i
$723$	0	0
$724$	3.37875i	0.125570i
$725$	− 28.3206i	− 1.05180i
$726$	0	0
$727$	44.4760i	1.64952i	0.565480	+	0.824762i	$0.308691\pi$
$727$	44.4760i	1.64952i	−0.565480	+	0.824762i	$0.691309\pi$
$728$	0	0
$729$	0	0
$730$	1.67777	0.0620969
$731$	−12.9706	−0.479736
$732$	0	0
$733$	19.3132i	0.713349i	0.934229	+	0.356674i	$0.116089\pi$
$733$	19.3132i	0.713349i	−0.934229	+	0.356674i	$0.883911\pi$
$734$	−23.4016	−0.863769
$735$	0	0
$736$	37.6904	1.38929
$737$	29.2776i	1.07845i
$738$	0	0
$739$	12.9429	0.476112	0.238056	−	0.971251i	$-0.423490\pi$
$739$	12.9429	0.476112	0.238056	+	0.971251i	$0.423490\pi$
$740$	−3.23722	−0.119002
$741$	0	0
$742$	0	0
$743$	− 24.8011i	− 0.909866i	−0.890526	−	0.454933i	$-0.849663\pi$
$743$	− 24.8011i	− 0.909866i	0.890526	−	0.454933i	$-0.150337\pi$
$744$	0	0
$745$	2.86024i	0.104791i
$746$	− 62.2365i	− 2.27864i
$747$	0	0
$748$	− 34.0174i	− 1.24380i
$749$	0	0
$750$	0	0
$751$	−36.5408	−1.33339	−0.666697	−	0.745329i	$-0.732293\pi$
$751$	−36.5408	−1.33339	−0.666697	+	0.745329i	$0.732293\pi$
$752$	−2.24403	−0.0818315
$753$	0	0
$754$	− 67.5428i	− 2.45976i
$755$	−2.72889	−0.0993146
$756$	0	0
$757$	−21.0956	−0.766732	−0.383366	−	0.923597i	$-0.625235\pi$
$757$	−21.0956	−0.766732	−0.383366	+	0.923597i	$0.625235\pi$
$758$	− 40.3591i	− 1.46591i
$759$	0	0
$760$	−0.138181	−0.00501236
$761$	−36.5976	−1.32666	−0.663331	−	0.748326i	$-0.730858\pi$
$761$	−36.5976	−1.32666	−0.663331	+	0.748326i	$0.730858\pi$
$762$	0	0
$763$	0	0
$764$	13.3649i	0.483524i
$765$	0	0
$766$	− 15.0234i	− 0.542818i
$767$	58.6541i	2.11788i
$768$	0	0
$769$	− 13.0218i	− 0.469579i	−0.972046	−	0.234789i	$-0.924560\pi$
$769$	− 13.0218i	− 0.469579i	0.972046	−	0.234789i	$-0.0754401\pi$
$770$	0	0
$771$	0	0
$772$	12.2912	0.442371
$773$	8.16207	0.293569	0.146785	−	0.989168i	$-0.453108\pi$
$773$	8.16207	0.293569	0.146785	+	0.989168i	$0.453108\pi$
$774$	0	0
$775$	41.6690i	1.49679i
$776$	0.462017	0.0165854
$777$	0	0
$778$	−17.9060	−0.641962
$779$	− 5.77518i	− 0.206917i
$780$	0	0
$781$	−21.7285	−0.777508
$782$	−40.7448	−1.45703
$783$	0	0
$784$	0	0
$785$	2.27662i	0.0812561i
$786$	0	0
$787$	9.73361i	0.346966i	0.984837	+	0.173483i	$0.0555021\pi$
$787$	9.73361i	0.346966i	−0.984837	+	0.173483i	$0.944498\pi$
$788$	− 6.29723i	− 0.224329i
$789$	0	0
$790$	− 2.91695i	− 0.103780i
$791$	0	0
$792$	0	0
$793$	8.24028	0.292621
$794$	7.09056	0.251634
$795$	0	0
$796$	12.3320i	0.437097i
$797$	4.26956	0.151235	0.0756177	−	0.997137i	$-0.475907\pi$
$797$	4.26956	0.151235	0.0756177	+	0.997137i	$0.475907\pi$
$798$	0	0
$799$	2.55080	0.0902409
$800$	− 40.1192i	− 1.41843i
$801$	0	0
$802$	−16.9866	−0.599819
$803$	17.0504	0.601695
$804$	0	0
$805$	0	0
$806$	99.3778i	3.50043i
$807$	0	0
$808$	− 1.66643i	− 0.0586250i
$809$	− 22.3281i	− 0.785015i	−0.919749	−	0.392507i	$-0.871608\pi$
$809$	− 22.3281i	− 0.785015i	0.919749	−	0.392507i	$-0.128392\pi$
$810$	0	0
$811$	− 42.9872i	− 1.50948i	−0.656021	−	0.754742i	$-0.727762\pi$
$811$	− 42.9872i	− 1.50948i	0.656021	−	0.754742i	$-0.272238\pi$
$812$	0	0
$813$	0	0
$814$	−64.2644	−2.25247
$815$	−2.56829	−0.0899631
$816$	0	0
$817$	11.4954i	0.402171i
$818$	−13.1986	−0.461479
$819$	0	0
$820$	0.579036	0.0202208
$821$	24.5917i	0.858258i	0.903243	+	0.429129i	$0.141179\pi$
$821$	24.5917i	0.858258i	−0.903243	+	0.429129i	$0.858821\pi$
$822$	0	0
$823$	−43.2051	−1.50604	−0.753018	−	0.658000i	$-0.771403\pi$
$823$	−43.2051	−1.50604	−0.753018	+	0.658000i	$0.771403\pi$
$824$	−0.422096	−0.0147044
$825$	0	0
$826$	0	0
$827$	− 24.9819i	− 0.868707i	−0.900743	−	0.434353i	$-0.856977\pi$
$827$	− 24.9819i	− 0.868707i	0.900743	−	0.434353i	$-0.143023\pi$
$828$	0	0
$829$	− 35.1819i	− 1.22192i	−0.791662	−	0.610959i	$-0.790784\pi$
$829$	− 35.1819i	− 1.22192i	0.791662	−	0.610959i	$-0.209216\pi$
$830$	4.19242i	0.145521i
$831$	0	0
$832$	− 51.2686i	− 1.77742i
$833$	0	0
$834$	0	0
$835$	−0.752745	−0.0260498
$836$	−30.1483	−1.04270
$837$	0	0
$838$	29.7895i	1.02906i
$839$	52.3211	1.80632	0.903162	−	0.429300i	$-0.141240\pi$
$839$	52.3211	1.80632	0.903162	+	0.429300i	$0.141240\pi$
$840$	0	0
$841$	−3.51508	−0.121210
$842$	− 3.25740i	− 0.112257i
$843$	0	0
$844$	7.43186	0.255815
$845$	3.88124	0.133519
$846$	0	0
$847$	0	0
$848$	49.6229i	1.70406i
$849$	0	0
$850$	43.3704i	1.48759i
$851$	39.4044i	1.35077i
$852$	0	0
$853$	10.8051i	0.369958i	0.982742	+	0.184979i	$0.0592218\pi$
$853$	10.8051i	0.369958i	−0.982742	+	0.184979i	$0.940778\pi$
$854$	0	0
$855$	0	0
$856$	0.961994	0.0328803
$857$	−20.0424	−0.684636	−0.342318	−	0.939584i	$-0.611212\pi$
$857$	−20.0424	−0.684636	−0.342318	+	0.939584i	$0.611212\pi$
$858$	0	0
$859$	− 21.6664i	− 0.739249i	−0.929181	−	0.369624i	$-0.879486\pi$
$859$	− 21.6664i	− 0.739249i	0.929181	−	0.369624i	$-0.120514\pi$
$860$	−1.15256	−0.0393019
$861$	0	0
$862$	53.4232	1.81960
$863$	− 22.7759i	− 0.775299i	−0.921807	−	0.387650i	$-0.873287\pi$
$863$	− 22.7759i	− 0.775299i	0.921807	−	0.387650i	$-0.126713\pi$
$864$	0	0
$865$	−2.90522	−0.0987806
$866$	24.6937	0.839127
$867$	0	0
$868$	0	0
$869$	− 29.6436i	− 1.00559i
$870$	0	0
$871$	− 45.5730i	− 1.54418i
$872$	3.80278i	0.128778i
$873$	0	0
$874$	36.1105i	1.22146i
$875$	0	0
$876$	0	0
$877$	−33.2038	−1.12121	−0.560606	−	0.828083i	$-0.689432\pi$
$877$	−33.2038	−1.12121	−0.560606	+	0.828083i	$0.689432\pi$
$878$	33.7674	1.13959
$879$	0	0
$880$	2.60692i	0.0878794i
$881$	−36.9399	−1.24454	−0.622269	−	0.782804i	$-0.713789\pi$
$881$	−36.9399	−1.24454	−0.622269	+	0.782804i	$0.713789\pi$
$882$	0	0
$883$	47.1433	1.58650	0.793249	−	0.608898i	$-0.208388\pi$
$883$	47.1433	1.58650	0.793249	+	0.608898i	$0.208388\pi$
$884$	52.9510i	1.78094i
$885$	0	0
$886$	−35.4665	−1.19152
$887$	−13.9168	−0.467280	−0.233640	−	0.972323i	$-0.575064\pi$
$887$	−13.9168	−0.467280	−0.233640	+	0.972323i	$0.575064\pi$
$888$	0	0
$889$	0	0
$890$	− 6.53875i	− 0.219179i
$891$	0	0
$892$	25.5237i	0.854598i
$893$	− 2.26068i	− 0.0756506i
$894$	0	0
$895$	− 0.316121i	− 0.0105668i
$896$	0	0
$897$	0	0
$898$	75.2692	2.51177
$899$	47.8405	1.59557
$900$	0	0
$901$	− 56.4066i	− 1.87918i
$902$	11.4949	0.382737
$903$	0	0
$904$	−0.393857	−0.0130995
$905$	0.294327i	0.00978375i
$906$	0	0
$907$	10.0138	0.332504	0.166252	−	0.986083i	$-0.446834\pi$
$907$	10.0138	0.332504	0.166252	+	0.986083i	$0.446834\pi$
$908$	−50.9095	−1.68949
$909$	0	0
$910$	0	0
$911$	− 12.5259i	− 0.415003i	−0.978235	−	0.207501i	$-0.933467\pi$
$911$	− 12.5259i	− 0.415003i	0.978235	−	0.207501i	$-0.0665331\pi$
$912$	0	0
$913$	42.6057i	1.41004i
$914$	17.7267i	0.586346i
$915$	0	0
$916$	27.8523i	0.920267i
$917$	0	0
$918$	0	0
$919$	26.2963	0.867433	0.433717	−	0.901049i	$-0.357202\pi$
$919$	26.2963	0.867433	0.433717	+	0.901049i	$0.357202\pi$
$920$	−0.168640	−0.00555991
$921$	0	0
$922$	− 37.6659i	− 1.24046i
$923$	33.8223	1.11327
$924$	0	0
$925$	41.9437	1.37910
$926$	− 20.9414i	− 0.688176i
$927$	0	0
$928$	−46.0612	−1.51203
$929$	11.2533	0.369209	0.184604	−	0.982813i	$-0.440900\pi$
$929$	11.2533	0.369209	0.184604	+	0.982813i	$0.440900\pi$
$930$	0	0
$931$	0	0
$932$	− 50.3708i	− 1.64995i
$933$	0	0
$934$	− 6.05332i	− 0.198071i
$935$	− 2.96330i	− 0.0969104i
$936$	0	0
$937$	− 47.1730i	− 1.54107i	−0.637395	−	0.770537i	$-0.719988\pi$
$937$	− 47.1730i	− 1.54107i	0.637395	−	0.770537i	$-0.280012\pi$
$938$	0	0
$939$	0	0
$940$	0.226662	0.00739289
$941$	28.0488	0.914363	0.457182	−	0.889373i	$-0.348859\pi$
$941$	28.0488	0.914363	0.457182	+	0.889373i	$0.348859\pi$
$942$	0	0
$943$	− 7.04820i	− 0.229521i
$944$	38.0407	1.23812
$945$	0	0
$946$	−22.8803	−0.743902
$947$	− 10.3030i	− 0.334801i	−0.985889	−	0.167401i	$-0.946463\pi$
$947$	− 10.3030i	− 0.334801i	0.985889	−	0.167401i	$-0.0535374\pi$
$948$	0	0
$949$	−26.5404	−0.861537
$950$	38.4375	1.24708
$951$	0	0
$952$	0	0
$953$	− 11.2683i	− 0.365016i	−0.983204	−	0.182508i	$-0.941578\pi$
$953$	− 11.2683i	− 0.365016i	0.983204	−	0.182508i	$-0.0584216\pi$
$954$	0	0
$955$	1.16423i	0.0376736i
$956$	− 17.5688i	− 0.568215i
$957$	0	0
$958$	− 66.8926i	− 2.16120i
$959$	0	0
$960$	0	0
$961$	−39.3892	−1.27062
$962$	100.033	3.22519
$963$	0	0
$964$	33.5264i	1.07981i
$965$	1.07070	0.0344672
$966$	0	0
$967$	−4.41322	−0.141920	−0.0709599	−	0.997479i	$-0.522606\pi$
$967$	−4.41322	−0.141920	−0.0709599	+	0.997479i	$0.522606\pi$
$968$	− 0.619366i	− 0.0199072i
$969$	0	0
$970$	0.864060	0.0277433
$971$	−19.9880	−0.641446	−0.320723	−	0.947173i	$-0.603926\pi$
$971$	−19.9880	−0.641446	−0.320723	+	0.947173i	$0.603926\pi$
$972$	0	0
$973$	0	0
$974$	− 62.8183i	− 2.01283i
$975$	0	0
$976$	− 5.34431i	− 0.171067i
$977$	30.2532i	0.967887i	0.875099	+	0.483943i	$0.160796\pi$
$977$	30.2532i	0.967887i	−0.875099	+	0.483943i	$0.839204\pi$
$978$	0	0
$979$	− 66.4504i	− 2.12376i
$980$	0	0
$981$	0	0
$982$	−43.0196	−1.37281
$983$	−17.4455	−0.556424	−0.278212	−	0.960520i	$-0.589742\pi$
$983$	−17.4455	−0.556424	−0.278212	+	0.960520i	$0.589742\pi$
$984$	0	0
$985$	− 0.548560i	− 0.0174786i
$986$	49.7939	1.58576
$987$	0	0
$988$	46.9284	1.49299
$989$	14.0293i	0.446105i
$990$	0	0
$991$	54.4158	1.72858	0.864288	−	0.502997i	$-0.167769\pi$
$991$	54.4158	1.72858	0.864288	+	0.502997i	$0.167769\pi$
$992$	67.7713	2.15174
$993$	0	0
$994$	0	0
$995$	1.07426i	0.0340563i
$996$	0	0
$997$	− 41.6479i	− 1.31900i	−0.751703	−	0.659502i	$-0.770767\pi$
$997$	− 41.6479i	− 1.31900i	0.751703	−	0.659502i	$-0.229233\pi$
$998$	− 37.1902i	− 1.17724i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1323.2.c.f.1322.3	✓	16
3.2	odd	2	inner	1323.2.c.f.1322.14	yes	16
7.6	odd	2	inner	1323.2.c.f.1322.4	yes	16
21.20	even	2	inner	1323.2.c.f.1322.13	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1323.2.c.f.1322.3	✓	16	1.1	even	1	trivial
1323.2.c.f.1322.4	yes	16	7.6	odd	2	inner
1323.2.c.f.1322.13	yes	16	21.20	even	2	inner
1323.2.c.f.1322.14	yes	16	3.2	odd	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)

\(f(q)\)	\(=\)	\(q-2.02428i q^{2} -2.09771 q^{4} -0.182734 q^{5} +0.197790i q^{8} +O(q^{10})\)	\(q-2.02428i q^{2} -2.09771 q^{4} -0.182734 q^{5} +0.197790i q^{8} +0.369905i q^{10} +3.75918i q^{11} -5.85148i q^{13} -3.79504 q^{16} +4.31383 q^{17} -3.82318i q^{19} +0.383323 q^{20} +7.60963 q^{22} -4.66593i q^{23} -4.96661 q^{25} -11.8450 q^{26} +5.70220i q^{29} -8.38983i q^{31} +8.07779i q^{32} -8.73241i q^{34} -8.44514 q^{37} -7.73919 q^{38} -0.0361430i q^{40} +1.51057 q^{41} -3.00675 q^{43} -7.88566i q^{44} -9.44514 q^{46} +0.591307 q^{47} +10.0538i q^{50} +12.2747i q^{52} -13.0758i q^{53} -0.686930i q^{55} +11.5428 q^{58} -10.0238 q^{59} +1.40824i q^{61} -16.9834 q^{62} +8.76164 q^{64} +1.06927i q^{65} +7.78828 q^{67} -9.04917 q^{68} +5.78012i q^{71} -4.53567i q^{73} +17.0953i q^{74} +8.01992i q^{76} -7.88566 q^{79} +0.693482 q^{80} -3.05781i q^{82} +11.3338 q^{83} -0.788285 q^{85} +6.08650i q^{86} -0.743528 q^{88} -17.6768 q^{89} +9.78775i q^{92} -1.19697i q^{94} +0.698626i q^{95} -2.33590i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{4}+O(q^{10}) \) 16 * q - 16 * q^4	\( 16 q - 16 q^{4} + 48 q^{16} + 64 q^{22} - 16 q^{25} + 32 q^{37} - 16 q^{43} + 16 q^{46} - 32 q^{64} + 48 q^{67} - 64 q^{79} + 64 q^{85} - 176 q^{88}+O(q^{100}) \) 16 * q - 16 * q^4 + 48 * q^16 + 64 * q^22 - 16 * q^25 + 32 * q^37 - 16 * q^43 + 16 * q^46 - 32 * q^64 + 48 * q^67 - 64 * q^79 + 64 * q^85 - 176 * q^88

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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