LMFDB - Embedded newform 1323.2.c.d.1322.8

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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 9x^{10} + 59x^{8} - 180x^{6} + 403x^{4} - 198x^{2} + 81 $ x^12 - 9x^10 + 59x^8 - 180x^6 + 403x^4 - 198*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 189)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1322.8
Root			$-1.65604 + 0.956115i$ of defining polynomial
Character	$\chi$	$=$	1323.1322
Dual form			1323.2.c.d.1322.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.656620i q^{2} +1.56885 q^{4} +3.31208 q^{5} +2.34338i q^{8} +O(q^{10})$	$q+0.656620i q^{2} +1.56885 q^{4} +3.31208 q^{5} +2.34338i q^{8} +2.17478i q^{10} -2.34338i q^{11} +1.58003i q^{13} +1.59899 q^{16} +1.13730 q^{17} -4.44938i q^{19} +5.19615 q^{20} +1.53871 q^{22} +9.39331i q^{23} +5.96986 q^{25} -1.03748 q^{26} +3.65662i q^{29} -7.31873i q^{31} +5.73669i q^{32} +0.746774i q^{34} -5.16784 q^{37} +2.92155 q^{38} +7.76146i q^{40} +9.64553 q^{41} -2.16784 q^{43} -3.67641i q^{44} -6.16784 q^{46} -5.58668 q^{47} +3.91993i q^{50} +2.47883i q^{52} -10.0499i q^{53} -7.76146i q^{55} -2.40101 q^{58} +2.17478 q^{59} -4.00665i q^{61} +4.80563 q^{62} -0.568850 q^{64} +5.23317i q^{65} -10.7668 q^{67} +1.78425 q^{68} +6.39331i q^{71} -10.6830i q^{73} -3.39331i q^{74} -6.98041i q^{76} +1.23317 q^{79} +5.29597 q^{80} +6.33345i q^{82} -1.03748 q^{83} +3.76683 q^{85} -1.42345i q^{86} +5.49143 q^{88} +7.47075 q^{89} +14.7367i q^{92} -3.66833i q^{94} -14.7367i q^{95} +13.5524i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 16 q^{4}+O(q^{10}) $ 12 * q - 16 * q^4	$ 12 q - 16 q^{4} + 8 q^{16} - 40 q^{22} + 48 q^{25} - 16 q^{37} + 20 q^{43} - 28 q^{46} - 40 q^{58} + 28 q^{64} - 72 q^{67} + 72 q^{79} - 12 q^{85} + 148 q^{88}+O(q^{100}) $ 12 * q - 16 * q^4 + 8 * q^16 - 40 * q^22 + 48 * q^25 - 16 * q^37 + 20 * q^43 - 28 * q^46 - 40 * q^58 + 28 * q^64 - 72 * q^67 + 72 * q^79 - 12 * q^85 + 148 * 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 $

$2$	0.656620i	0.464301i	0.972680	+	0.232150i	$0.0745761\pi$
$2$	0.656620i	0.464301i	−0.972680	+	0.232150i	$0.925424\pi$
$3$	0	0
$3$	0	0
$4$	1.56885	0.784425
$5$	3.31208	1.48121	0.740603	−	0.671943i	$-0.234540\pi$
$5$	3.31208	1.48121	0.740603	+	0.671943i	$0.234540\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	2.34338i	0.828510i
$9$	0	0
$10$	2.17478i	0.687725i
$11$	− 2.34338i	− 0.706556i	−0.935518	−	0.353278i	$-0.885067\pi$
$11$	− 2.34338i	− 0.706556i	0.935518	−	0.353278i	$-0.114933\pi$
$12$	0	0
$13$	1.58003i	0.438221i	0.975700	+	0.219110i	$0.0703155\pi$
$13$	1.58003i	0.438221i	−0.975700	+	0.219110i	$0.929685\pi$
$14$	0	0
$15$	0	0
$16$	1.59899	0.399747
$17$	1.13730	0.275836	0.137918	−	0.990444i	$-0.455959\pi$
$17$	1.13730	0.275836	0.137918	+	0.990444i	$0.455959\pi$
$18$	0	0
$19$	− 4.44938i	− 1.02076i	−0.859950	−	0.510379i	$-0.829505\pi$
$19$	− 4.44938i	− 1.02076i	0.859950	−	0.510379i	$-0.170495\pi$
$20$	5.19615	1.16190
$21$	0	0
$22$	1.53871	0.328054
$23$	9.39331i	1.95864i	0.202317	+	0.979320i	$0.435153\pi$
$23$	9.39331i	1.95864i	−0.202317	+	0.979320i	$0.564847\pi$
$24$	0	0
$25$	5.96986	1.19397
$26$	−1.03748	−0.203466
$27$	0	0
$28$	0	0
$29$	3.65662i	0.679017i	0.940603	+	0.339509i	$0.110261\pi$
$29$	3.65662i	0.679017i	−0.940603	+	0.339509i	$0.889739\pi$
$30$	0	0
$31$	− 7.31873i	− 1.31448i	−0.753680	−	0.657241i	$-0.771723\pi$
$31$	− 7.31873i	− 1.31448i	0.753680	−	0.657241i	$-0.228277\pi$
$32$	5.73669i	1.01411i
$33$	0	0
$34$	0.746774i	0.128071i
$35$	0	0
$36$	0	0
$37$	−5.16784	−0.849587	−0.424794	−	0.905290i	$-0.639653\pi$
$37$	−5.16784	−0.849587	−0.424794	+	0.905290i	$0.639653\pi$
$38$	2.92155	0.473938
$39$	0	0
$40$	7.76146i	1.22719i
$41$	9.64553	1.50638	0.753189	−	0.657804i	$-0.228514\pi$
$41$	9.64553	1.50638	0.753189	+	0.657804i	$0.228514\pi$
$42$	0	0
$43$	−2.16784	−0.330592	−0.165296	−	0.986244i	$-0.552858\pi$
$43$	−2.16784	−0.330592	−0.165296	+	0.986244i	$0.552858\pi$
$44$	− 3.67641i	− 0.554240i
$45$	0	0
$46$	−6.16784	−0.909398
$47$	−5.58668	−0.814901	−0.407450	−	0.913227i	$-0.633582\pi$
$47$	−5.58668	−0.814901	−0.407450	+	0.913227i	$0.633582\pi$
$48$	0	0
$49$	0	0
$50$	3.91993i	0.554362i
$51$	0	0
$52$	2.47883i	0.343751i
$53$	− 10.0499i	− 1.38046i	−0.723588	−	0.690232i	$-0.757509\pi$
$53$	− 10.0499i	− 1.38046i	0.723588	−	0.690232i	$-0.242491\pi$
$54$	0	0
$55$	− 7.76146i	− 1.04655i
$56$	0	0
$57$	0	0
$58$	−2.40101	−0.315268
$59$	2.17478	0.283132	0.141566	−	0.989929i	$-0.454786\pi$
$59$	2.17478	0.283132	0.141566	+	0.989929i	$0.454786\pi$
$60$	0	0
$61$	− 4.00665i	− 0.512999i	−0.966544	−	0.256500i	$-0.917431\pi$
$61$	− 4.00665i	− 0.512999i	0.966544	−	0.256500i	$-0.0825692\pi$
$62$	4.80563	0.610315
$63$	0	0
$64$	−0.568850	−0.0711062
$65$	5.23317i	0.649095i
$66$	0	0
$67$	−10.7668	−1.31538	−0.657689	−	0.753290i	$-0.728466\pi$
$67$	−10.7668	−1.31538	−0.657689	+	0.753290i	$0.728466\pi$
$68$	1.78425	0.216372
$69$	0	0
$70$	0	0
$71$	6.39331i	0.758746i	0.925244	+	0.379373i	$0.123860\pi$
$71$	6.39331i	0.758746i	−0.925244	+	0.379373i	$0.876140\pi$
$72$	0	0
$73$	− 10.6830i	− 1.25035i	−0.780484	−	0.625176i	$-0.785027\pi$
$73$	− 10.6830i	− 1.25035i	0.780484	−	0.625176i	$-0.214973\pi$
$74$	− 3.39331i	− 0.394464i
$75$	0	0
$76$	− 6.98041i	− 0.800707i
$77$	0	0
$78$	0	0
$79$	1.23317	0.138743	0.0693714	−	0.997591i	$-0.477901\pi$
$79$	1.23317	0.138743	0.0693714	+	0.997591i	$0.477901\pi$
$80$	5.29597	0.592108
$81$	0	0
$82$	6.33345i	0.699413i
$83$	−1.03748	−0.113878	−0.0569390	−	0.998378i	$-0.518134\pi$
$83$	−1.03748	−0.113878	−0.0569390	+	0.998378i	$0.518134\pi$
$84$	0	0
$85$	3.76683	0.408570
$86$	− 1.42345i	− 0.153494i
$87$	0	0
$88$	5.49143	0.585388
$89$	7.47075	0.791898	0.395949	−	0.918272i	$-0.370416\pi$
$89$	7.47075	0.791898	0.395949	+	0.918272i	$0.370416\pi$
$90$	0	0
$91$	0	0
$92$	14.7367i	1.53641i
$93$	0	0
$94$	− 3.66833i	− 0.378359i
$95$	− 14.7367i	− 1.51195i
$96$	0	0
$97$	13.5524i	1.37603i	0.725695	+	0.688017i	$0.241518\pi$
$97$	13.5524i	1.37603i	−0.725695	+	0.688017i	$0.758482\pi$
$98$	0	0
$99$	0	0
$100$	9.36581	0.936581
$101$	−13.7044	−1.36364	−0.681819	−	0.731521i	$-0.738811\pi$
$101$	−13.7044	−1.36364	−0.681819	+	0.731521i	$0.738811\pi$
$102$	0	0
$103$	− 4.89211i	− 0.482033i	−0.970521	−	0.241017i	$-0.922519\pi$
$103$	− 4.89211i	− 0.482033i	0.970521	−	0.241017i	$-0.0774809\pi$
$104$	−3.70260	−0.363070
$105$	0	0
$106$	6.59899	0.640950
$107$	2.45359i	0.237197i	0.992942	+	0.118599i	$0.0378401\pi$
$107$	2.45359i	0.237197i	−0.992942	+	0.118599i	$0.962160\pi$
$108$	0	0
$109$	4.33568	0.415282	0.207641	−	0.978205i	$-0.433421\pi$
$109$	4.33568	0.415282	0.207641	+	0.978205i	$0.433421\pi$
$110$	5.09633	0.485916
$111$	0	0
$112$	0	0
$113$	7.16013i	0.673569i	0.941582	+	0.336784i	$0.109339\pi$
$113$	7.16013i	0.673569i	−0.941582	+	0.336784i	$0.890661\pi$
$114$	0	0
$115$	31.1114i	2.90115i
$116$	5.73669i	0.532638i
$117$	0	0
$118$	1.42800i	0.131458i
$119$	0	0
$120$	0	0
$121$	5.50857	0.500779
$122$	2.63085	0.238186
$123$	0	0
$124$	− 11.4820i	− 1.03111i
$125$	3.21226	0.287313
$126$	0	0
$127$	−17.7065	−1.57120	−0.785601	−	0.618733i	$-0.787646\pi$
$127$	−17.7065	−1.57120	−0.785601	+	0.618733i	$0.787646\pi$
$128$	11.0999i	0.981098i
$129$	0	0
$130$	−3.43621	−0.301375
$131$	−13.6046	−1.18864	−0.594318	−	0.804230i	$-0.702578\pi$
$131$	−13.6046	−1.18864	−0.594318	+	0.804230i	$0.702578\pi$
$132$	0	0
$133$	0	0
$134$	− 7.06972i	− 0.610731i
$135$	0	0
$136$	2.66513i	0.228533i
$137$	10.1601i	0.868039i	0.900904	+	0.434019i	$0.142905\pi$
$137$	10.1601i	0.868039i	−0.900904	+	0.434019i	$0.857095\pi$
$138$	0	0
$139$	− 3.85463i	− 0.326945i	−0.986548	−	0.163473i	$-0.947730\pi$
$139$	− 3.85463i	− 0.326945i	0.986548	−	0.163473i	$-0.0522695\pi$
$140$	0	0
$141$	0	0
$142$	−4.19798	−0.352286
$143$	3.70260	0.309627
$144$	0	0
$145$	12.1110i	1.00576i
$146$	7.01468	0.580539
$147$	0	0
$148$	−8.10756	−0.666437
$149$	− 5.08007i	− 0.416175i	−0.978110	−	0.208088i	$-0.933276\pi$
$149$	− 5.08007i	− 0.416175i	0.978110	−	0.208088i	$-0.0667239\pi$
$150$	0	0
$151$	18.3055	1.48968	0.744842	−	0.667241i	$-0.232525\pi$
$151$	18.3055	1.48968	0.744842	+	0.667241i	$0.232525\pi$
$152$	10.4266	0.845707
$153$	0	0
$154$	0	0
$155$	− 24.2402i	− 1.94702i
$156$	0	0
$157$	3.31208i	0.264333i	0.991228	+	0.132166i	$0.0421933\pi$
$157$	3.31208i	0.264333i	−0.991228	+	0.132166i	$0.957807\pi$
$158$	0.809727i	0.0644184i
$159$	0	0
$160$	19.0004i	1.50211i
$161$	0	0
$162$	0	0
$163$	−2.29345	−0.179637	−0.0898185	−	0.995958i	$-0.528629\pi$
$163$	−2.29345	−0.179637	−0.0898185	+	0.995958i	$0.528629\pi$
$164$	15.1324	1.18164
$165$	0	0
$166$	− 0.681229i	− 0.0528737i
$167$	2.66513	0.206234	0.103117	−	0.994669i	$-0.467118\pi$
$167$	2.66513	0.206234	0.103117	+	0.994669i	$0.467118\pi$
$168$	0	0
$169$	10.5035	0.807963
$170$	2.47338i	0.189699i
$171$	0	0
$172$	−3.40101	−0.259325
$173$	2.56530	0.195036	0.0975182	−	0.995234i	$-0.468910\pi$
$173$	2.56530	0.195036	0.0975182	+	0.995234i	$0.468910\pi$
$174$	0	0
$175$	0	0
$176$	− 3.74704i	− 0.282444i
$177$	0	0
$178$	4.90545i	0.367679i
$179$	13.8365i	1.03419i	0.855927	+	0.517096i	$0.172987\pi$
$179$	13.8365i	1.03419i	−0.855927	+	0.517096i	$0.827013\pi$
$180$	0	0
$181$	4.74008i	0.352328i	0.984361	+	0.176164i	$0.0563688\pi$
$181$	4.74008i	0.352328i	−0.984361	+	0.176164i	$0.943631\pi$
$182$	0	0
$183$	0	0
$184$	−22.0121	−1.62275
$185$	−17.1163	−1.25841
$186$	0	0
$187$	− 2.66513i	− 0.194893i
$188$	−8.76466	−0.639228
$189$	0	0
$190$	9.67641	0.702001
$191$	− 8.34338i	− 0.603706i	−0.953355	−	0.301853i	$-0.902395\pi$
$191$	− 8.34338i	− 0.603706i	0.953355	−	0.301853i	$-0.0976051\pi$
$192$	0	0
$193$	−17.8442	−1.28446	−0.642229	−	0.766513i	$-0.721990\pi$
$193$	−17.8442	−1.28446	−0.642229	+	0.766513i	$0.721990\pi$
$194$	−8.89876	−0.638893
$195$	0	0
$196$	0	0
$197$	− 15.1102i	− 1.07656i	−0.842767	−	0.538279i	$-0.819075\pi$
$197$	− 15.1102i	− 1.07656i	0.842767	−	0.538279i	$-0.180925\pi$
$198$	0	0
$199$	− 8.66025i	− 0.613909i	−0.951724	−	0.306955i	$-0.900690\pi$
$199$	− 8.66025i	− 0.613909i	0.951724	−	0.306955i	$-0.0993100\pi$
$200$	13.9897i	0.989218i
$201$	0	0
$202$	− 8.99858i	− 0.633138i
$203$	0	0
$204$	0	0
$205$	31.9468	2.23126
$206$	3.21226	0.223809
$207$	0	0
$208$	2.52645i	0.175177i
$209$	−10.4266	−0.721222
$210$	0	0
$211$	−14.5809	−1.00379	−0.501897	−	0.864928i	$-0.667364\pi$
$211$	−14.5809	−1.00379	−0.501897	+	0.864928i	$0.667364\pi$
$212$	− 15.7668i	− 1.08287i
$213$	0	0
$214$	−1.61107	−0.110131
$215$	−7.18005	−0.489675
$216$	0	0
$217$	0	0
$218$	2.84689i	0.192816i
$219$	0	0
$220$	− 12.1766i	− 0.820943i
$221$	1.79696i	0.120877i
$222$	0	0
$223$	− 10.5832i	− 0.708703i	−0.935112	−	0.354351i	$-0.884702\pi$
$223$	− 10.5832i	− 0.708703i	0.935112	−	0.354351i	$-0.115298\pi$
$224$	0	0
$225$	0	0
$226$	−4.70149	−0.312738
$227$	−6.72398	−0.446286	−0.223143	−	0.974786i	$-0.571632\pi$
$227$	−6.72398	−0.446286	−0.223143	+	0.974786i	$0.571632\pi$
$228$	0	0
$229$	12.2630i	0.810364i	0.914236	+	0.405182i	$0.132792\pi$
$229$	12.2630i	0.810364i	−0.914236	+	0.405182i	$0.867208\pi$
$230$	−20.4284	−1.34701
$231$	0	0
$232$	−8.56885	−0.562573
$233$	5.62648i	0.368603i	0.982870	+	0.184302i	$0.0590023\pi$
$233$	5.62648i	0.368603i	−0.982870	+	0.184302i	$0.940998\pi$
$234$	0	0
$235$	−18.5035	−1.20704
$236$	3.41190	0.222096
$237$	0	0
$238$	0	0
$239$	− 15.9595i	− 1.03234i	−0.856488	−	0.516168i	$-0.827358\pi$
$239$	− 15.9595i	− 1.03234i	0.856488	−	0.516168i	$-0.172642\pi$
$240$	0	0
$241$	− 4.58806i	− 0.295543i	−0.989022	−	0.147771i	$-0.952790\pi$
$241$	− 4.58806i	− 0.295543i	0.989022	−	0.147771i	$-0.0472100\pi$
$242$	3.61704i	0.232512i
$243$	0	0
$244$	− 6.28583i	− 0.402409i
$245$	0	0
$246$	0	0
$247$	7.03014	0.447317
$248$	17.1506	1.08906
$249$	0	0
$250$	2.10923i	0.133400i
$251$	−13.8042	−0.871314	−0.435657	−	0.900113i	$-0.643484\pi$
$251$	−13.8042	−0.871314	−0.435657	+	0.900113i	$0.643484\pi$
$252$	0	0
$253$	22.0121	1.38389
$254$	− 11.6265i	− 0.729510i
$255$	0	0
$256$	−8.42609	−0.526631
$257$	8.96430	0.559178	0.279589	−	0.960120i	$-0.409802\pi$
$257$	8.96430	0.559178	0.279589	+	0.960120i	$0.409802\pi$
$258$	0	0
$259$	0	0
$260$	8.21006i	0.509166i
$261$	0	0
$262$	− 8.93303i	− 0.551885i
$263$	− 28.5534i	− 1.76068i	−0.474343	−	0.880340i	$-0.657314\pi$
$263$	− 28.5534i	− 1.76068i	0.474343	−	0.880340i	$-0.342686\pi$
$264$	0	0
$265$	− 33.2861i	− 2.04475i
$266$	0	0
$267$	0	0
$268$	−16.8915	−1.03181
$269$	−23.9313	−1.45912	−0.729559	−	0.683918i	$-0.760275\pi$
$269$	−23.9313	−1.45912	−0.729559	+	0.683918i	$0.760275\pi$
$270$	0	0
$271$	− 14.6032i	− 0.887080i	−0.896255	−	0.443540i	$-0.853722\pi$
$271$	− 14.6032i	− 0.887080i	0.896255	−	0.443540i	$-0.146278\pi$
$272$	1.81853	0.110265
$273$	0	0
$274$	−6.67135	−0.403031
$275$	− 13.9897i	− 0.843608i
$276$	0	0
$277$	22.3830	1.34486	0.672431	−	0.740160i	$-0.265250\pi$
$277$	22.3830	1.34486	0.672431	+	0.740160i	$0.265250\pi$
$278$	2.53103	0.151801
$279$	0	0
$280$	0	0
$281$	− 11.6067i	− 0.692397i	−0.938161	−	0.346199i	$-0.887472\pi$
$281$	− 11.6067i	− 0.692397i	0.938161	−	0.346199i	$-0.112528\pi$
$282$	0	0
$283$	− 14.9937i	− 0.891283i	−0.895211	−	0.445642i	$-0.852976\pi$
$283$	− 14.9937i	− 0.891283i	0.895211	−	0.445642i	$-0.147024\pi$
$284$	10.0301i	0.595179i
$285$	0	0
$286$	2.43121i	0.143760i
$287$	0	0
$288$	0	0
$289$	−15.7065	−0.923915
$290$	−7.95234	−0.466977
$291$	0	0
$292$	− 16.7600i	− 0.980807i
$293$	−16.3352	−0.954314	−0.477157	−	0.878818i	$-0.658333\pi$
$293$	−16.3352	−0.954314	−0.477157	+	0.878818i	$0.658333\pi$
$294$	0	0
$295$	7.20304	0.419377
$296$	− 12.1102i	− 0.703891i
$297$	0	0
$298$	3.33568	0.193231
$299$	−14.8417	−0.858317
$300$	0	0
$301$	0	0
$302$	12.0198i	0.691661i
$303$	0	0
$304$	− 7.11450i	− 0.408045i
$305$	− 13.2703i	− 0.759857i
$306$	0	0
$307$	3.17340i	0.181115i	0.995891	+	0.0905577i	$0.0288650\pi$
$307$	3.17340i	0.181115i	−0.995891	+	0.0905577i	$0.971135\pi$
$308$	0	0
$309$	0	0
$310$	15.9166	0.904003
$311$	−25.1342	−1.42523	−0.712614	−	0.701556i	$-0.752489\pi$
$311$	−25.1342	−1.42523	−0.712614	+	0.701556i	$0.752489\pi$
$312$	0	0
$313$	18.6441i	1.05383i	0.849919	+	0.526914i	$0.176651\pi$
$313$	18.6441i	1.05383i	−0.849919	+	0.526914i	$0.823349\pi$
$314$	−2.17478	−0.122730
$315$	0	0
$316$	1.93466	0.108833
$317$	− 7.98965i	− 0.448744i	−0.974504	−	0.224372i	$-0.927967\pi$
$317$	− 7.98965i	− 0.448744i	0.974504	−	0.224372i	$-0.0720330\pi$
$318$	0	0
$319$	8.56885	0.479763
$320$	−1.88407	−0.105323
$321$	0	0
$322$	0	0
$323$	− 5.06028i	− 0.281561i
$324$	0	0
$325$	9.43254i	0.523223i
$326$	− 1.50593i	− 0.0834056i
$327$	0	0
$328$	22.6031i	1.24805i
$329$	0	0
$330$	0	0
$331$	−8.93466	−0.491094	−0.245547	−	0.969385i	$-0.578967\pi$
$331$	−8.93466	−0.491094	−0.245547	+	0.969385i	$0.578967\pi$
$332$	−1.62765	−0.0893287
$333$	0	0
$334$	1.74998i	0.0957544i
$335$	−35.6606	−1.94835
$336$	0	0
$337$	4.24526	0.231254	0.115627	−	0.993293i	$-0.463112\pi$
$337$	4.24526	0.231254	0.115627	+	0.993293i	$0.463112\pi$
$338$	6.89682i	0.375138i
$339$	0	0
$340$	5.90958	0.320492
$341$	−17.1506	−0.928755
$342$	0	0
$343$	0	0
$344$	− 5.08007i	− 0.273899i
$345$	0	0
$346$	1.68443i	0.0905556i
$347$	2.62648i	0.140997i	0.997512	+	0.0704985i	$0.0224590\pi$
$347$	2.62648i	0.140997i	−0.997512	+	0.0704985i	$0.977541\pi$
$348$	0	0
$349$	32.4918i	1.73924i	0.493718	+	0.869622i	$0.335637\pi$
$349$	32.4918i	1.73924i	−0.493718	+	0.869622i	$0.664363\pi$
$350$	0	0
$351$	0	0
$352$	13.4432	0.716527
$353$	23.3842	1.24461	0.622307	−	0.782773i	$-0.286195\pi$
$353$	23.3842	1.24461	0.622307	+	0.782773i	$0.286195\pi$
$354$	0	0
$355$	21.1751i	1.12386i
$356$	11.7205	0.621185
$357$	0	0
$358$	−9.08536	−0.480176
$359$	− 21.3504i	− 1.12683i	−0.826174	−	0.563416i	$-0.809487\pi$
$359$	− 21.3504i	− 1.12683i	0.826174	−	0.563416i	$-0.190513\pi$
$360$	0	0
$361$	−0.796965	−0.0419455
$362$	−3.11243	−0.163586
$363$	0	0
$364$	0	0
$365$	− 35.3830i	− 1.85203i
$366$	0	0
$367$	15.5229i	0.810289i	0.914253	+	0.405145i	$0.132779\pi$
$367$	15.5229i	0.810289i	−0.914253	+	0.405145i	$0.867221\pi$
$368$	15.0198i	0.782961i
$369$	0	0
$370$	− 11.2389i	− 0.584283i
$371$	0	0
$372$	0	0
$373$	−23.3528	−1.20916	−0.604582	−	0.796543i	$-0.706660\pi$
$373$	−23.3528	−1.20916	−0.604582	+	0.796543i	$0.706660\pi$
$374$	1.74998	0.0904891
$375$	0	0
$376$	− 13.0917i	− 0.675153i
$377$	−5.77756	−0.297559
$378$	0	0
$379$	2.53871	0.130405	0.0652024	−	0.997872i	$-0.479231\pi$
$379$	2.53871	0.130405	0.0652024	+	0.997872i	$0.479231\pi$
$380$	− 23.1196i	− 1.18601i
$381$	0	0
$382$	5.47843	0.280301
$383$	16.2041	0.827993	0.413996	−	0.910278i	$-0.364133\pi$
$383$	16.2041	0.827993	0.413996	+	0.910278i	$0.364133\pi$
$384$	0	0
$385$	0	0
$386$	− 11.7169i	− 0.596374i
$387$	0	0
$388$	21.2616i	1.07939i
$389$	22.3330i	1.13233i	0.824292	+	0.566165i	$0.191573\pi$
$389$	22.3330i	1.13233i	−0.824292	+	0.566165i	$0.808427\pi$
$390$	0	0
$391$	10.6830i	0.540263i
$392$	0	0
$393$	0	0
$394$	9.92167	0.499847
$395$	4.08437	0.205507
$396$	0	0
$397$	− 32.6914i	− 1.64073i	−0.571837	−	0.820367i	$-0.693769\pi$
$397$	− 32.6914i	− 1.64073i	0.571837	−	0.820367i	$-0.306231\pi$
$398$	5.68650	0.285038
$399$	0	0
$400$	9.54574	0.477287
$401$	28.9270i	1.44454i	0.691609	+	0.722272i	$0.256902\pi$
$401$	28.9270i	1.44454i	−0.691609	+	0.722272i	$0.743098\pi$
$402$	0	0
$403$	11.5638	0.576033
$404$	−21.5001	−1.06967
$405$	0	0
$406$	0	0
$407$	12.1102i	0.600281i
$408$	0	0
$409$	1.05082i	0.0519598i	0.999662	+	0.0259799i	$0.00827059\pi$
$409$	1.05082i	0.0519598i	−0.999662	+	0.0259799i	$0.991729\pi$
$410$	20.9769i	1.03597i
$411$	0	0
$412$	− 7.67498i	− 0.378119i
$413$	0	0
$414$	0	0
$415$	−3.43621	−0.168677
$416$	−9.06412	−0.444405
$417$	0	0
$418$	− 6.84631i	− 0.334864i
$419$	3.12120	0.152480	0.0762402	−	0.997089i	$-0.475708\pi$
$419$	3.12120	0.152480	0.0762402	+	0.997089i	$0.475708\pi$
$420$	0	0
$421$	−8.70655	−0.424331	−0.212166	−	0.977234i	$-0.568052\pi$
$421$	−8.70655	−0.424331	−0.212166	+	0.977234i	$0.568052\pi$
$422$	− 9.57414i	− 0.466062i
$423$	0	0
$424$	23.5508	1.14373
$425$	6.78952	0.329340
$426$	0	0
$427$	0	0
$428$	3.84931i	0.186063i
$429$	0	0
$430$	− 4.71457i	− 0.227357i
$431$	− 27.2831i	− 1.31418i	−0.753812	−	0.657090i	$-0.771787\pi$
$431$	− 27.2831i	− 1.31418i	0.753812	−	0.657090i	$-0.228213\pi$
$432$	0	0
$433$	12.8711i	0.618547i	0.950973	+	0.309274i	$0.100086\pi$
$433$	12.8711i	0.618547i	−0.950973	+	0.309274i	$0.899914\pi$
$434$	0	0
$435$	0	0
$436$	6.80202	0.325758
$437$	41.7944	1.99930
$438$	0	0
$439$	− 18.4445i	− 0.880306i	−0.897923	−	0.440153i	$-0.854924\pi$
$439$	− 18.4445i	− 0.880306i	0.897923	−	0.440153i	$-0.145076\pi$
$440$	18.1880	0.867081
$441$	0	0
$442$	−1.17992	−0.0561233
$443$	− 2.87000i	− 0.136358i	−0.997673	−	0.0681790i	$-0.978281\pi$
$443$	− 2.87000i	− 0.136358i	0.997673	−	0.0681790i	$-0.0217189\pi$
$444$	0	0
$445$	24.7437	1.17296
$446$	6.94914	0.329051
$447$	0	0
$448$	0	0
$449$	− 1.81675i	− 0.0857380i	−0.999081	−	0.0428690i	$-0.986350\pi$
$449$	− 1.81675i	− 0.0857380i	0.999081	−	0.0428690i	$-0.0136498\pi$
$450$	0	0
$451$	− 22.6031i	− 1.06434i
$452$	11.2332i	0.528364i
$453$	0	0
$454$	− 4.41510i	− 0.207211i
$455$	0	0
$456$	0	0
$457$	6.15575	0.287954	0.143977	−	0.989581i	$-0.454011\pi$
$457$	6.15575	0.287954	0.143977	+	0.989581i	$0.454011\pi$
$458$	−8.05216	−0.376253
$459$	0	0
$460$	48.8091i	2.27573i
$461$	17.0507	0.794132	0.397066	−	0.917790i	$-0.370028\pi$
$461$	17.0507	0.794132	0.397066	+	0.917790i	$0.370028\pi$
$462$	0	0
$463$	29.4131	1.36694	0.683471	−	0.729977i	$-0.260469\pi$
$463$	29.4131	1.36694	0.683471	+	0.729977i	$0.260469\pi$
$464$	5.84689i	0.271435i
$465$	0	0
$466$	−3.69446	−0.171143
$467$	−27.1181	−1.25487	−0.627437	−	0.778667i	$-0.715896\pi$
$467$	−27.1181	−1.25487	−0.627437	+	0.778667i	$0.715896\pi$
$468$	0	0
$469$	0	0
$470$	− 12.1498i	− 0.560428i
$471$	0	0
$472$	5.09633i	0.234578i
$473$	5.08007i	0.233582i
$474$	0	0
$475$	− 26.5622i	− 1.21876i
$476$	0	0
$477$	0	0
$478$	10.4793	0.479314
$479$	−21.5569	−0.984960	−0.492480	−	0.870324i	$-0.663910\pi$
$479$	−21.5569	−0.984960	−0.492480	+	0.870324i	$0.663910\pi$
$480$	0	0
$481$	− 8.16532i	− 0.372307i
$482$	3.01261	0.137221
$483$	0	0
$484$	8.64212	0.392824
$485$	44.8865i	2.03819i
$486$	0	0
$487$	−14.5809	−0.660725	−0.330363	−	0.943854i	$-0.607171\pi$
$487$	−14.5809	−0.660725	−0.330363	+	0.943854i	$0.607171\pi$
$488$	9.38910	0.425025
$489$	0	0
$490$	0	0
$491$	− 0.283102i	− 0.0127762i	−0.999980	−	0.00638811i	$-0.997967\pi$
$491$	− 0.283102i	− 0.0127762i	0.999980	−	0.00638811i	$-0.00203341\pi$
$492$	0	0
$493$	4.15867i	0.187297i
$494$	4.61613i	0.207690i
$495$	0	0
$496$	− 11.7026i	− 0.525461i
$497$	0	0
$498$	0	0
$499$	12.3055	0.550871	0.275436	−	0.961319i	$-0.411178\pi$
$499$	12.3055	0.550871	0.275436	+	0.961319i	$0.411178\pi$
$500$	5.03955	0.225375
$501$	0	0
$502$	− 9.06412i	− 0.404552i
$503$	1.78425	0.0795559	0.0397779	−	0.999209i	$-0.487335\pi$
$503$	1.78425	0.0795559	0.0397779	+	0.999209i	$0.487335\pi$
$504$	0	0
$505$	−45.3900	−2.01983
$506$	14.4536i	0.642540i
$507$	0	0
$508$	−27.7789	−1.23249
$509$	39.0725	1.73186	0.865928	−	0.500168i	$-0.166728\pi$
$509$	39.0725	1.73186	0.865928	+	0.500168i	$0.166728\pi$
$510$	0	0
$511$	0	0
$512$	16.6670i	0.736583i
$513$	0	0
$514$	5.88614i	0.259627i
$515$	− 16.2030i	− 0.713991i
$516$	0	0
$517$	13.0917i	0.575773i
$518$	0	0
$519$	0	0
$520$	−12.2633	−0.537782
$521$	−20.3628	−0.892111	−0.446056	−	0.895005i	$-0.647172\pi$
$521$	−20.3628	−0.892111	−0.446056	+	0.895005i	$0.647172\pi$
$522$	0	0
$523$	− 1.43259i	− 0.0626426i	−0.999509	−	0.0313213i	$-0.990028\pi$
$523$	− 1.43259i	− 0.0626426i	0.999509	−	0.0313213i	$-0.00997151\pi$
$524$	−21.3435	−0.932396
$525$	0	0
$526$	18.7488	0.817485
$527$	− 8.32359i	− 0.362581i
$528$	0	0
$529$	−65.2342	−2.83627
$530$	21.8564	0.949380
$531$	0	0
$532$	0	0
$533$	15.2402i	0.660126i
$534$	0	0
$535$	8.12647i	0.351338i
$536$	− 25.2308i	− 1.08980i
$537$	0	0
$538$	− 15.7138i	− 0.677470i
$539$	0	0
$540$	0	0
$541$	−10.0904	−0.433821	−0.216910	−	0.976192i	$-0.569598\pi$
$541$	−10.0904	−0.433821	−0.216910	+	0.976192i	$0.569598\pi$
$542$	9.58875	0.411872
$543$	0	0
$544$	6.52433i	0.279729i
$545$	14.3601	0.615119
$546$	0	0
$547$	7.30048	0.312146	0.156073	−	0.987746i	$-0.450117\pi$
$547$	7.30048	0.312146	0.156073	+	0.987746i	$0.450117\pi$
$548$	15.9397i	0.680911i
$549$	0	0
$550$	9.18589	0.391688
$551$	16.2697	0.693112
$552$	0	0
$553$	0	0
$554$	14.6971i	0.624420i
$555$	0	0
$556$	− 6.04733i	− 0.256464i
$557$	22.8597i	0.968595i	0.874903	+	0.484297i	$0.160925\pi$
$557$	22.8597i	0.968595i	−0.874903	+	0.484297i	$0.839075\pi$
$558$	0	0
$559$	− 3.42524i	− 0.144872i
$560$	0	0
$561$	0	0
$562$	7.62119	0.321481
$563$	−21.1096	−0.889663	−0.444832	−	0.895614i	$-0.646736\pi$
$563$	−21.1096	−0.889663	−0.444832	+	0.895614i	$0.646736\pi$
$564$	0	0
$565$	23.7149i	0.997694i
$566$	9.84517	0.413824
$567$	0	0
$568$	−14.9819	−0.628629
$569$	26.4536i	1.10899i	0.832186	+	0.554496i	$0.187089\pi$
$569$	26.4536i	1.10899i	−0.832186	+	0.554496i	$0.812911\pi$
$570$	0	0
$571$	12.0301	0.503446	0.251723	−	0.967799i	$-0.419003\pi$
$571$	12.0301	0.503446	0.251723	+	0.967799i	$0.419003\pi$
$572$	5.80883	0.242879
$573$	0	0
$574$	0	0
$575$	56.0767i	2.33856i
$576$	0	0
$577$	− 36.8413i	− 1.53372i	−0.641812	−	0.766862i	$-0.721817\pi$
$577$	− 36.8413i	− 1.53372i	0.641812	−	0.766862i	$-0.278183\pi$
$578$	− 10.3132i	− 0.428974i
$579$	0	0
$580$	19.0004i	0.788947i
$581$	0	0
$582$	0	0
$583$	−23.5508	−0.975374
$584$	25.0343	1.03593
$585$	0	0
$586$	− 10.7260i	− 0.443089i
$587$	28.7712	1.18752	0.593758	−	0.804644i	$-0.297644\pi$
$587$	28.7712	1.18752	0.593758	+	0.804644i	$0.297644\pi$
$588$	0	0
$589$	−32.5638	−1.34177
$590$	4.72966i	0.194717i
$591$	0	0
$592$	−8.26331	−0.339620
$593$	10.5832	0.434599	0.217300	−	0.976105i	$-0.430275\pi$
$593$	10.5832	0.434599	0.217300	+	0.976105i	$0.430275\pi$
$594$	0	0
$595$	0	0
$596$	− 7.96986i	− 0.326458i
$597$	0	0
$598$	− 9.74535i	− 0.398517i
$599$	− 18.1927i	− 0.743333i	−0.928366	−	0.371666i	$-0.878786\pi$
$599$	− 18.1927i	− 0.743333i	0.928366	−	0.371666i	$-0.121214\pi$
$600$	0	0
$601$	− 26.4101i	− 1.07729i	−0.842532	−	0.538646i	$-0.818936\pi$
$601$	− 26.4101i	− 1.07729i	0.842532	−	0.538646i	$-0.181064\pi$
$602$	0	0
$603$	0	0
$604$	28.7186	1.16854
$605$	18.2448	0.741757
$606$	0	0
$607$	24.9776i	1.01381i	0.862003	+	0.506904i	$0.169210\pi$
$607$	24.9776i	1.01381i	−0.862003	+	0.506904i	$0.830790\pi$
$608$	25.5247	1.03516
$609$	0	0
$610$	8.71358	0.352802
$611$	− 8.82710i	− 0.357106i
$612$	0	0
$613$	−14.1126	−0.570003	−0.285002	−	0.958527i	$-0.591994\pi$
$613$	−14.1126	−0.570003	−0.285002	+	0.958527i	$0.591994\pi$
$614$	−2.08372	−0.0840920
$615$	0	0
$616$	0	0
$617$	14.6265i	0.588840i	0.955676	+	0.294420i	$0.0951264\pi$
$617$	14.6265i	0.588840i	−0.955676	+	0.294420i	$0.904874\pi$
$618$	0	0
$619$	− 26.5442i	− 1.06690i	−0.845831	−	0.533452i	$-0.820895\pi$
$619$	− 26.5442i	− 1.06690i	0.845831	−	0.533452i	$-0.179105\pi$
$620$	− 38.0292i	− 1.52729i
$621$	0	0
$622$	− 16.5036i	− 0.661734i
$623$	0	0
$624$	0	0
$625$	−19.2101	−0.768403
$626$	−12.2421	−0.489293
$627$	0	0
$628$	5.19615i	0.207349i
$629$	−5.87738	−0.234347
$630$	0	0
$631$	−8.20304	−0.326558	−0.163279	−	0.986580i	$-0.552207\pi$
$631$	−8.20304	−0.326558	−0.163279	+	0.986580i	$0.552207\pi$
$632$	2.88979i	0.114950i
$633$	0	0
$634$	5.24617	0.208352
$635$	−58.6455	−2.32727
$636$	0	0
$637$	0	0
$638$	5.62648i	0.222755i
$639$	0	0
$640$	36.7636i	1.45321i
$641$	− 16.0499i	− 0.633934i	−0.948437	−	0.316967i	$-0.897336\pi$
$641$	− 16.0499i	− 0.633934i	0.948437	−	0.316967i	$-0.102664\pi$
$642$	0	0
$643$	26.4101i	1.04151i	0.853705	+	0.520757i	$0.174350\pi$
$643$	26.4101i	1.04151i	−0.853705	+	0.520757i	$0.825650\pi$
$644$	0	0
$645$	0	0
$646$	3.32268	0.130729
$647$	31.7015	1.24632	0.623158	−	0.782096i	$-0.285849\pi$
$647$	31.7015	1.24632	0.623158	+	0.782096i	$0.285849\pi$
$648$	0	0
$649$	− 5.09633i	− 0.200048i
$650$	−6.19360	−0.242933
$651$	0	0
$652$	−3.59808	−0.140912
$653$	− 13.2901i	− 0.520083i	−0.965597	−	0.260041i	$-0.916264\pi$
$653$	− 13.2901i	− 0.520083i	0.965597	−	0.260041i	$-0.0837362\pi$
$654$	0	0
$655$	−45.0594	−1.76062
$656$	15.4231	0.602171
$657$	0	0
$658$	0	0
$659$	36.2125i	1.41064i	0.708890	+	0.705319i	$0.249196\pi$
$659$	36.2125i	1.41064i	−0.708890	+	0.705319i	$0.750804\pi$
$660$	0	0
$661$	50.7930i	1.97562i	0.155674	+	0.987809i	$0.450245\pi$
$661$	50.7930i	1.97562i	−0.155674	+	0.987809i	$0.549755\pi$
$662$	− 5.86668i	− 0.228015i
$663$	0	0
$664$	− 2.43121i	− 0.0943491i
$665$	0	0
$666$	0	0
$667$	−34.3478	−1.32995
$668$	4.18118	0.161775
$669$	0	0
$670$	− 23.4155i	− 0.904618i
$671$	−9.38910	−0.362462
$672$	0	0
$673$	3.32865	0.128310	0.0641550	−	0.997940i	$-0.479565\pi$
$673$	3.32865	0.128310	0.0641550	+	0.997940i	$0.479565\pi$
$674$	2.78752i	0.107371i
$675$	0	0
$676$	16.4784	0.633786
$677$	19.9380	0.766280	0.383140	−	0.923690i	$-0.374843\pi$
$677$	19.9380	0.766280	0.383140	+	0.923690i	$0.374843\pi$
$678$	0	0
$679$	0	0
$680$	8.82710i	0.338504i
$681$	0	0
$682$	− 11.2614i	− 0.431222i
$683$	28.6636i	1.09678i	0.836221	+	0.548392i	$0.184760\pi$
$683$	28.6636i	1.09678i	−0.836221	+	0.548392i	$0.815240\pi$
$684$	0	0
$685$	33.6512i	1.28574i
$686$	0	0
$687$	0	0
$688$	−3.46635	−0.132153
$689$	15.8792	0.604948
$690$	0	0
$691$	20.3674i	0.774812i	0.921909	+	0.387406i	$0.126629\pi$
$691$	20.3674i	0.774812i	−0.921909	+	0.387406i	$0.873371\pi$
$692$	4.02458	0.152991
$693$	0	0
$694$	−1.72460	−0.0654650
$695$	− 12.7668i	− 0.484273i
$696$	0	0
$697$	10.9699	0.415513
$698$	−21.3347	−0.807532
$699$	0	0
$700$	0	0
$701$	− 49.1172i	− 1.85513i	−0.373659	−	0.927566i	$-0.621897\pi$
$701$	− 49.1172i	− 1.85513i	0.373659	−	0.927566i	$-0.378103\pi$
$702$	0	0
$703$	22.9937i	0.867222i
$704$	1.33303i	0.0502405i
$705$	0	0
$706$	15.3545i	0.577876i
$707$	0	0
$708$	0	0
$709$	23.3658	0.877522	0.438761	−	0.898604i	$-0.355418\pi$
$709$	23.3658	0.877522	0.438761	+	0.898604i	$0.355418\pi$
$710$	−13.9040	−0.521809
$711$	0	0
$712$	17.5068i	0.656095i
$713$	68.7471	2.57460
$714$	0	0
$715$	12.2633	0.458622
$716$	21.7075i	0.811246i
$717$	0	0
$718$	14.0191	0.523189
$719$	12.6669	0.472396	0.236198	−	0.971705i	$-0.424099\pi$
$719$	12.6669	0.472396	0.236198	+	0.971705i	$0.424099\pi$
$720$	0	0
$721$	0	0
$722$	− 0.523303i	− 0.0194753i
$723$	0	0
$724$	7.43648i	0.276374i
$725$	21.8295i	0.810728i
$726$	0	0
$727$	2.70398i	0.100285i	0.998742	+	0.0501426i	$0.0159676\pi$
$727$	2.70398i	0.100285i	−0.998742	+	0.0501426i	$0.984032\pi$
$728$	0	0
$729$	0	0
$730$	23.2332	0.859898
$731$	−2.46548	−0.0911891
$732$	0	0
$733$	− 21.8087i	− 0.805524i	−0.915305	−	0.402762i	$-0.868050\pi$
$733$	− 21.8087i	− 0.805524i	0.915305	−	0.402762i	$-0.131950\pi$
$734$	−10.1927	−0.376218
$735$	0	0
$736$	−53.8865	−1.98628
$737$	25.2308i	0.929387i
$738$	0	0
$739$	32.1267	1.18180	0.590899	−	0.806745i	$-0.298773\pi$
$739$	32.1267	1.18180	0.590899	+	0.806745i	$0.298773\pi$
$740$	−26.8529	−0.987131
$741$	0	0
$742$	0	0
$743$	41.2728i	1.51415i	0.653328	+	0.757075i	$0.273372\pi$
$743$	41.2728i	1.51415i	−0.653328	+	0.757075i	$0.726628\pi$
$744$	0	0
$745$	− 16.8256i	− 0.616442i
$746$	− 15.3339i	− 0.561415i
$747$	0	0
$748$	− 4.18118i	− 0.152879i
$749$	0	0
$750$	0	0
$751$	45.8091	1.67160	0.835798	−	0.549037i	$-0.185005\pi$
$751$	45.8091	1.67160	0.835798	+	0.549037i	$0.185005\pi$
$752$	−8.93303	−0.325754
$753$	0	0
$754$	− 3.79366i	− 0.138157i
$755$	60.6294	2.20653
$756$	0	0
$757$	50.3427	1.82974	0.914868	−	0.403752i	$-0.132294\pi$
$757$	50.3427	1.82974	0.914868	+	0.403752i	$0.132294\pi$
$758$	1.66697i	0.0605471i
$759$	0	0
$760$	34.5337	1.25267
$761$	−30.3646	−1.10072	−0.550358	−	0.834929i	$-0.685509\pi$
$761$	−30.3646	−1.10072	−0.550358	+	0.834929i	$0.685509\pi$
$762$	0	0
$763$	0	0
$764$	− 13.0895i	− 0.473562i
$765$	0	0
$766$	10.6400i	0.384438i
$767$	3.43621i	0.124074i
$768$	0	0
$769$	− 41.3383i	− 1.49070i	−0.666675	−	0.745349i	$-0.732283\pi$
$769$	− 41.3383i	− 1.49070i	0.666675	−	0.745349i	$-0.267717\pi$
$770$	0	0
$771$	0	0
$772$	−27.9949	−1.00756
$773$	−13.9951	−0.503368	−0.251684	−	0.967809i	$-0.580984\pi$
$773$	−13.9951	−0.503368	−0.251684	+	0.967809i	$0.580984\pi$
$774$	0	0
$775$	− 43.6918i	− 1.56946i
$776$	−31.7583	−1.14006
$777$	0	0
$778$	−14.6643	−0.525741
$779$	− 42.9166i	− 1.53765i
$780$	0	0
$781$	14.9819	0.536096
$782$	−7.01468	−0.250845
$783$	0	0
$784$	0	0
$785$	10.9699i	0.391531i
$786$	0	0
$787$	13.4868i	0.480753i	0.970680	+	0.240377i	$0.0772709\pi$
$787$	13.4868i	0.480753i	−0.970680	+	0.240377i	$0.922729\pi$
$788$	− 23.7056i	− 0.844478i
$789$	0	0
$790$	2.68188i	0.0954170i
$791$	0	0
$792$	0	0
$793$	6.33062	0.224807
$794$	21.4658	0.761794
$795$	0	0
$796$	− 13.5866i	− 0.481566i
$797$	8.31735	0.294616	0.147308	−	0.989091i	$-0.452939\pi$
$797$	8.31735	0.294616	0.147308	+	0.989091i	$0.452939\pi$
$798$	0	0
$799$	−6.35373	−0.224779
$800$	34.2472i	1.21082i
$801$	0	0
$802$	−18.9940	−0.670703
$803$	−25.0343	−0.883443
$804$	0	0
$805$	0	0
$806$	7.59302i	0.267453i
$807$	0	0
$808$	− 32.1146i	− 1.12979i
$809$	14.5440i	0.511340i	0.966764	+	0.255670i	$0.0822960\pi$
$809$	14.5440i	0.511340i	−0.966764	+	0.255670i	$0.917704\pi$
$810$	0	0
$811$	41.8287i	1.46880i	0.678715	+	0.734401i	$0.262537\pi$
$811$	41.8287i	1.46880i	−0.678715	+	0.734401i	$0.737463\pi$
$812$	0	0
$813$	0	0
$814$	−7.95181	−0.278711
$815$	−7.59609	−0.266079
$816$	0	0
$817$	9.64553i	0.337454i
$818$	−0.689991	−0.0241250
$819$	0	0
$820$	50.1196	1.75025
$821$	46.4828i	1.62226i	0.584865	+	0.811131i	$0.301148\pi$
$821$	46.4828i	1.62226i	−0.584865	+	0.811131i	$0.698852\pi$
$822$	0	0
$823$	7.73669	0.269684	0.134842	−	0.990867i	$-0.456947\pi$
$823$	7.73669	0.269684	0.134842	+	0.990867i	$0.456947\pi$
$824$	11.4641	0.399369
$825$	0	0
$826$	0	0
$827$	20.8898i	0.726409i	0.931709	+	0.363205i	$0.118317\pi$
$827$	20.8898i	0.726409i	−0.931709	+	0.363205i	$0.881683\pi$
$828$	0	0
$829$	47.6062i	1.65343i	0.562619	+	0.826716i	$0.309794\pi$
$829$	47.6062i	1.65343i	−0.562619	+	0.826716i	$0.690206\pi$
$830$	− 2.25628i	− 0.0783168i
$831$	0	0
$832$	− 0.898798i	− 0.0311602i
$833$	0	0
$834$	0	0
$835$	8.82710	0.305475
$836$	−16.3577	−0.565744
$837$	0	0
$838$	2.04944i	0.0707968i
$839$	22.4035	0.773455	0.386727	−	0.922194i	$-0.373605\pi$
$839$	22.4035	0.773455	0.386727	+	0.922194i	$0.373605\pi$
$840$	0	0
$841$	15.6291	0.538935
$842$	− 5.71690i	− 0.197017i
$843$	0	0
$844$	−22.8753	−0.787400
$845$	34.7885	1.19676
$846$	0	0
$847$	0	0
$848$	− 16.0697i	− 0.551836i
$849$	0	0
$850$	4.45814i	0.152913i
$851$	− 48.5431i	− 1.66404i
$852$	0	0
$853$	− 48.4273i	− 1.65812i	−0.559160	−	0.829060i	$-0.688876\pi$
$853$	− 48.4273i	− 1.65812i	0.559160	−	0.829060i	$-0.311124\pi$
$854$	0	0
$855$	0	0
$856$	−5.74968	−0.196520
$857$	49.0655	1.67605	0.838023	−	0.545636i	$-0.183712\pi$
$857$	49.0655	1.67605	0.838023	+	0.545636i	$0.183712\pi$
$858$	0	0
$859$	12.0245i	0.410272i	0.978733	+	0.205136i	$0.0657636\pi$
$859$	12.0245i	0.410272i	−0.978733	+	0.205136i	$0.934236\pi$
$860$	−11.2644	−0.384113
$861$	0	0
$862$	17.9146	0.610175
$863$	− 46.0499i	− 1.56756i	−0.621040	−	0.783779i	$-0.713290\pi$
$863$	− 46.0499i	− 1.56756i	0.621040	−	0.783779i	$-0.286710\pi$
$864$	0	0
$865$	8.49649	0.288889
$866$	−8.45145	−0.287192
$867$	0	0
$868$	0	0
$869$	− 2.88979i	− 0.0980296i
$870$	0	0
$871$	− 17.0119i	− 0.576426i
$872$	10.1601i	0.344066i
$873$	0	0
$874$	27.4430i	0.928275i
$875$	0	0
$876$	0	0
$877$	13.4734	0.454964	0.227482	−	0.973782i	$-0.426951\pi$
$877$	13.4734	0.454964	0.227482	+	0.973782i	$0.426951\pi$
$878$	12.1110	0.408727
$879$	0	0
$880$	− 12.4105i	− 0.418357i
$881$	−25.5247	−0.859949	−0.429974	−	0.902841i	$-0.641477\pi$
$881$	−25.5247	−0.859949	−0.429974	+	0.902841i	$0.641477\pi$
$882$	0	0
$883$	6.45532	0.217239	0.108619	−	0.994083i	$-0.465357\pi$
$883$	6.45532	0.217239	0.108619	+	0.994083i	$0.465357\pi$
$884$	2.81917i	0.0948189i
$885$	0	0
$886$	1.88450	0.0633111
$887$	33.1208	1.11209	0.556043	−	0.831153i	$-0.312319\pi$
$887$	33.1208	1.11209	0.556043	+	0.831153i	$0.312319\pi$
$888$	0	0
$889$	0	0
$890$	16.2472i	0.544608i
$891$	0	0
$892$	− 16.6034i	− 0.555924i
$893$	24.8572i	0.831816i
$894$	0	0
$895$	45.8277i	1.53185i
$896$	0	0
$897$	0	0
$898$	1.19292	0.0398082
$899$	26.7618	0.892556
$900$	0	0
$901$	− 11.4298i	− 0.380781i
$902$	14.8417	0.494174
$903$	0	0
$904$	−16.7789	−0.558058
$905$	15.6995i	0.521870i
$906$	0	0
$907$	6.21512	0.206370	0.103185	−	0.994662i	$-0.467097\pi$
$907$	6.21512	0.206370	0.103185	+	0.994662i	$0.467097\pi$
$908$	−10.5489	−0.350078
$909$	0	0
$910$	0	0
$911$	18.0475i	0.597941i	0.954262	+	0.298970i	$0.0966432\pi$
$911$	18.0475i	0.597941i	−0.954262	+	0.298970i	$0.903357\pi$
$912$	0	0
$913$	2.43121i	0.0804611i
$914$	4.04199i	0.133697i
$915$	0	0
$916$	19.2389i	0.635670i
$917$	0	0
$918$	0	0
$919$	−8.25825	−0.272415	−0.136207	−	0.990680i	$-0.543491\pi$
$919$	−8.25825	−0.272415	−0.136207	+	0.990680i	$0.543491\pi$
$920$	−72.9057	−2.40363
$921$	0	0
$922$	11.1959i	0.368716i
$923$	−10.1016	−0.332498
$924$	0	0
$925$	−30.8513	−1.01438
$926$	19.3132i	0.634672i
$927$	0	0
$928$	−20.9769	−0.688600
$929$	34.7709	1.14080	0.570399	−	0.821368i	$-0.306789\pi$
$929$	34.7709	1.14080	0.570399	+	0.821368i	$0.306789\pi$
$930$	0	0
$931$	0	0
$932$	8.82710i	0.289141i
$933$	0	0
$934$	− 17.8063i	− 0.582639i
$935$	− 8.82710i	− 0.288677i
$936$	0	0
$937$	− 51.0703i	− 1.66839i	−0.551466	−	0.834197i	$-0.685931\pi$
$937$	− 51.0703i	− 1.66839i	0.551466	−	0.834197i	$-0.314069\pi$
$938$	0	0
$939$	0	0
$940$	−29.0292	−0.946829
$941$	−3.47744	−0.113361	−0.0566807	−	0.998392i	$-0.518052\pi$
$941$	−3.47744	−0.113361	−0.0566807	+	0.998392i	$0.518052\pi$
$942$	0	0
$943$	90.6034i	2.95045i
$944$	3.47744	0.113181
$945$	0	0
$946$	−3.33568	−0.108452
$947$	14.3401i	0.465989i	0.972478	+	0.232995i	$0.0748525\pi$
$947$	14.3401i	0.465989i	−0.972478	+	0.232995i	$0.925148\pi$
$948$	0	0
$949$	16.8794	0.547930
$950$	17.4413	0.565869
$951$	0	0
$952$	0	0
$953$	7.53697i	0.244147i	0.992521	+	0.122073i	$0.0389543\pi$
$953$	7.53697i	0.244147i	−0.992521	+	0.122073i	$0.961046\pi$
$954$	0	0
$955$	− 27.6339i	− 0.894213i
$956$	− 25.0381i	− 0.809789i
$957$	0	0
$958$	− 14.1547i	− 0.457318i
$959$	0	0
$960$	0	0
$961$	−22.5638	−0.727864
$962$	5.36152	0.172862
$963$	0	0
$964$	− 7.19797i	− 0.231831i
$965$	−59.1015	−1.90255
$966$	0	0
$967$	−11.6161	−0.373550	−0.186775	−	0.982403i	$-0.559803\pi$
$967$	−11.6161	−0.373550	−0.186775	+	0.982403i	$0.559803\pi$
$968$	12.9087i	0.414901i
$969$	0	0
$970$	−29.4734	−0.946333
$971$	35.4952	1.13910	0.569548	−	0.821958i	$-0.307118\pi$
$971$	35.4952	1.13910	0.569548	+	0.821958i	$0.307118\pi$
$972$	0	0
$973$	0	0
$974$	− 9.57414i	− 0.306775i
$975$	0	0
$976$	− 6.40659i	− 0.205070i
$977$	32.0094i	1.02407i	0.858964	+	0.512036i	$0.171109\pi$
$977$	32.0094i	1.02407i	−0.858964	+	0.512036i	$0.828891\pi$
$978$	0	0
$979$	− 17.5068i	− 0.559520i
$980$	0	0
$981$	0	0
$982$	0.185891	0.00593201
$983$	−49.4648	−1.57768	−0.788841	−	0.614598i	$-0.789318\pi$
$983$	−49.4648	−1.57768	−0.788841	+	0.614598i	$0.789318\pi$
$984$	0	0
$985$	− 50.0462i	− 1.59460i
$986$	−2.73067	−0.0869623
$987$	0	0
$988$	11.0292	0.350887
$989$	− 20.3632i	− 0.647511i
$990$	0	0
$991$	17.9518	0.570258	0.285129	−	0.958489i	$-0.407964\pi$
$991$	17.9518	0.570258	0.285129	+	0.958489i	$0.407964\pi$
$992$	41.9853	1.33303
$993$	0	0
$994$	0	0
$995$	− 28.6834i	− 0.909326i
$996$	0	0
$997$	− 33.5037i	− 1.06107i	−0.847662	−	0.530537i	$-0.821990\pi$
$997$	− 33.5037i	− 1.06107i	0.847662	−	0.530537i	$-0.178010\pi$
$998$	8.08007i	0.255770i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1323.2.c.d.1322.8		12
3.2	odd	2	inner	1323.2.c.d.1322.5		12
7.4	even	3		189.2.p.d.26.3	✓	12
7.5	odd	6		189.2.p.d.80.4	yes	12
7.6	odd	2	inner	1323.2.c.d.1322.7		12
21.5	even	6		189.2.p.d.80.3	yes	12
21.11	odd	6		189.2.p.d.26.4	yes	12
21.20	even	2	inner	1323.2.c.d.1322.6		12
63.4	even	3		567.2.s.f.26.4		12
63.5	even	6		567.2.i.f.269.3		12
63.11	odd	6		567.2.i.f.215.3		12
63.25	even	3		567.2.i.f.215.4		12
63.32	odd	6		567.2.s.f.26.3		12
63.40	odd	6		567.2.i.f.269.4		12
63.47	even	6		567.2.s.f.458.4		12
63.61	odd	6		567.2.s.f.458.3		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.2.p.d.26.3	✓	12	7.4	even	3
189.2.p.d.26.4	yes	12	21.11	odd	6
189.2.p.d.80.3	yes	12	21.5	even	6
189.2.p.d.80.4	yes	12	7.5	odd	6
567.2.i.f.215.3		12	63.11	odd	6
567.2.i.f.215.4		12	63.25	even	3
567.2.i.f.269.3		12	63.5	even	6
567.2.i.f.269.4		12	63.40	odd	6
567.2.s.f.26.3		12	63.32	odd	6
567.2.s.f.26.4		12	63.4	even	3
567.2.s.f.458.3		12	63.61	odd	6
567.2.s.f.458.4		12	63.47	even	6
1323.2.c.d.1322.5		12	3.2	odd	2	inner
1323.2.c.d.1322.6		12	21.20	even	2	inner
1323.2.c.d.1322.7		12	7.6	odd	2	inner
1323.2.c.d.1322.8		12	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+0.656620i q^{2} +1.56885 q^{4} +3.31208 q^{5} +2.34338i q^{8} +O(q^{10})\)	\(q+0.656620i q^{2} +1.56885 q^{4} +3.31208 q^{5} +2.34338i q^{8} +2.17478i q^{10} -2.34338i q^{11} +1.58003i q^{13} +1.59899 q^{16} +1.13730 q^{17} -4.44938i q^{19} +5.19615 q^{20} +1.53871 q^{22} +9.39331i q^{23} +5.96986 q^{25} -1.03748 q^{26} +3.65662i q^{29} -7.31873i q^{31} +5.73669i q^{32} +0.746774i q^{34} -5.16784 q^{37} +2.92155 q^{38} +7.76146i q^{40} +9.64553 q^{41} -2.16784 q^{43} -3.67641i q^{44} -6.16784 q^{46} -5.58668 q^{47} +3.91993i q^{50} +2.47883i q^{52} -10.0499i q^{53} -7.76146i q^{55} -2.40101 q^{58} +2.17478 q^{59} -4.00665i q^{61} +4.80563 q^{62} -0.568850 q^{64} +5.23317i q^{65} -10.7668 q^{67} +1.78425 q^{68} +6.39331i q^{71} -10.6830i q^{73} -3.39331i q^{74} -6.98041i q^{76} +1.23317 q^{79} +5.29597 q^{80} +6.33345i q^{82} -1.03748 q^{83} +3.76683 q^{85} -1.42345i q^{86} +5.49143 q^{88} +7.47075 q^{89} +14.7367i q^{92} -3.66833i q^{94} -14.7367i q^{95} +13.5524i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 16 q^{4}+O(q^{10}) \) 12 * q - 16 * q^4	\( 12 q - 16 q^{4} + 8 q^{16} - 40 q^{22} + 48 q^{25} - 16 q^{37} + 20 q^{43} - 28 q^{46} - 40 q^{58} + 28 q^{64} - 72 q^{67} + 72 q^{79} - 12 q^{85} + 148 q^{88}+O(q^{100}) \) 12 * q - 16 * q^4 + 8 * q^16 - 40 * q^22 + 48 * q^25 - 16 * q^37 + 20 * q^43 - 28 * q^46 - 40 * q^58 + 28 * q^64 - 72 * q^67 + 72 * q^79 - 12 * q^85 + 148 * q^88

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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