LMFDB - Embedded newform 1323.2.a.u.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1323,2,Mod(1,1323)]

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([0, 0]))

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

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

chi := DirichletCharacter("1323.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1323 = 3^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1323.a (trivial)

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:	yes
Analytic conductor:	$10.5642081874$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 6 $ x^2 - 6
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 189)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.44949$ of defining polynomial
Character	$\chi$	$=$	1323.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.44949 q^{2} +4.00000 q^{4} -2.44949 q^{5} +4.89898 q^{8} +O(q^{10})$	$q+2.44949 q^{2} +4.00000 q^{4} -2.44949 q^{5} +4.89898 q^{8} -6.00000 q^{10} +4.89898 q^{11} +4.00000 q^{13} +4.00000 q^{16} +2.44949 q^{17} +1.00000 q^{19} -9.79796 q^{20} +12.0000 q^{22} +2.44949 q^{23} +1.00000 q^{25} +9.79796 q^{26} -7.34847 q^{29} +7.00000 q^{31} +6.00000 q^{34} +8.00000 q^{37} +2.44949 q^{38} -12.0000 q^{40} -7.34847 q^{41} -1.00000 q^{43} +19.5959 q^{44} +6.00000 q^{46} -2.44949 q^{47} +2.44949 q^{50} +16.0000 q^{52} -2.44949 q^{53} -12.0000 q^{55} -18.0000 q^{58} +9.79796 q^{59} -5.00000 q^{61} +17.1464 q^{62} -8.00000 q^{64} -9.79796 q^{65} +2.00000 q^{67} +9.79796 q^{68} +1.00000 q^{73} +19.5959 q^{74} +4.00000 q^{76} -4.00000 q^{79} -9.79796 q^{80} -18.0000 q^{82} -14.6969 q^{83} -6.00000 q^{85} -2.44949 q^{86} +24.0000 q^{88} -2.44949 q^{89} +9.79796 q^{92} -6.00000 q^{94} -2.44949 q^{95} +1.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 8 q^{4}+O(q^{10}) $ 2 * q + 8 * q^4	$ 2 q + 8 q^{4} - 12 q^{10} + 8 q^{13} + 8 q^{16} + 2 q^{19} + 24 q^{22} + 2 q^{25} + 14 q^{31} + 12 q^{34} + 16 q^{37} - 24 q^{40} - 2 q^{43} + 12 q^{46} + 32 q^{52} - 24 q^{55} - 36 q^{58} - 10 q^{61} - 16 q^{64} + 4 q^{67} + 2 q^{73} + 8 q^{76} - 8 q^{79} - 36 q^{82} - 12 q^{85} + 48 q^{88} - 12 q^{94} + 2 q^{97}+O(q^{100}) $ 2 * q + 8 * q^4 - 12 * q^10 + 8 * q^13 + 8 * q^16 + 2 * q^19 + 24 * q^22 + 2 * q^25 + 14 * q^31 + 12 * q^34 + 16 * q^37 - 24 * q^40 - 2 * q^43 + 12 * q^46 + 32 * q^52 - 24 * q^55 - 36 * q^58 - 10 * q^61 - 16 * q^64 + 4 * q^67 + 2 * q^73 + 8 * q^76 - 8 * q^79 - 36 * q^82 - 12 * q^85 + 48 * q^88 - 12 * q^94 + 2 * q^97

Display 100 coefficients 10 coefficients

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 $

$2$	2.44949	1.73205	0.866025	−	0.500000i	$-0.166667\pi$
$2$	2.44949	1.73205	0.866025	+	0.500000i	$0.166667\pi$
$3$	0	0
$3$	0	0
$4$	4.00000	2.00000
$5$	−2.44949	−1.09545	−0.547723	−	0.836660i	$-0.684505\pi$
$5$	−2.44949	−1.09545	−0.547723	+	0.836660i	$0.684505\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	4.89898	1.73205
$9$	0	0
$10$	−6.00000	−1.89737
$11$	4.89898	1.47710	0.738549	−	0.674200i	$-0.235511\pi$
$11$	4.89898	1.47710	0.738549	+	0.674200i	$0.235511\pi$
$12$	0	0
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	0	0
$15$	0	0
$16$	4.00000	1.00000
$17$	2.44949	0.594089	0.297044	−	0.954864i	$-0.403999\pi$
$17$	2.44949	0.594089	0.297044	+	0.954864i	$0.403999\pi$
$18$	0	0
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	−9.79796	−2.19089
$21$	0	0
$22$	12.0000	2.55841
$23$	2.44949	0.510754	0.255377	−	0.966842i	$-0.417800\pi$
$23$	2.44949	0.510754	0.255377	+	0.966842i	$0.417800\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	9.79796	1.92154
$27$	0	0
$28$	0	0
$29$	−7.34847	−1.36458	−0.682288	−	0.731083i	$-0.739015\pi$
$29$	−7.34847	−1.36458	−0.682288	+	0.731083i	$0.739015\pi$
$30$	0	0
$31$	7.00000	1.25724	0.628619	−	0.777714i	$-0.283621\pi$
$31$	7.00000	1.25724	0.628619	+	0.777714i	$0.283621\pi$
$32$	0	0
$33$	0	0
$34$	6.00000	1.02899
$35$	0	0
$36$	0	0
$37$	8.00000	1.31519	0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000	1.31519	0.657596	+	0.753371i	$0.271573\pi$
$38$	2.44949	0.397360
$39$	0	0
$40$	−12.0000	−1.89737
$41$	−7.34847	−1.14764	−0.573819	−	0.818982i	$-0.694539\pi$
$41$	−7.34847	−1.14764	−0.573819	+	0.818982i	$0.694539\pi$
$42$	0	0
$43$	−1.00000	−0.152499	−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000	−0.152499	−0.0762493	+	0.997089i	$0.524294\pi$
$44$	19.5959	2.95420
$45$	0	0
$46$	6.00000	0.884652
$47$	−2.44949	−0.357295	−0.178647	−	0.983913i	$-0.557172\pi$
$47$	−2.44949	−0.357295	−0.178647	+	0.983913i	$0.557172\pi$
$48$	0	0
$49$	0	0
$50$	2.44949	0.346410
$51$	0	0
$52$	16.0000	2.21880
$53$	−2.44949	−0.336463	−0.168232	−	0.985747i	$-0.553806\pi$
$53$	−2.44949	−0.336463	−0.168232	+	0.985747i	$0.553806\pi$
$54$	0	0
$55$	−12.0000	−1.61808
$56$	0	0
$57$	0	0
$58$	−18.0000	−2.36352
$59$	9.79796	1.27559	0.637793	−	0.770208i	$-0.279848\pi$
$59$	9.79796	1.27559	0.637793	+	0.770208i	$0.279848\pi$
$60$	0	0
$61$	−5.00000	−0.640184	−0.320092	−	0.947386i	$-0.603714\pi$
$61$	−5.00000	−0.640184	−0.320092	+	0.947386i	$0.603714\pi$
$62$	17.1464	2.17760
$63$	0	0
$64$	−8.00000	−1.00000
$65$	−9.79796	−1.21529
$66$	0	0
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	9.79796	1.18818
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	1.00000	0.117041	0.0585206	−	0.998286i	$-0.481362\pi$
$73$	1.00000	0.117041	0.0585206	+	0.998286i	$0.481362\pi$
$74$	19.5959	2.27798
$75$	0	0
$76$	4.00000	0.458831
$77$	0	0
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	−9.79796	−1.09545
$81$	0	0
$82$	−18.0000	−1.98777
$83$	−14.6969	−1.61320	−0.806599	−	0.591099i	$-0.798694\pi$
$83$	−14.6969	−1.61320	−0.806599	+	0.591099i	$0.798694\pi$
$84$	0	0
$85$	−6.00000	−0.650791
$86$	−2.44949	−0.264135
$87$	0	0
$88$	24.0000	2.55841
$89$	−2.44949	−0.259645	−0.129823	−	0.991537i	$-0.541441\pi$
$89$	−2.44949	−0.259645	−0.129823	+	0.991537i	$0.541441\pi$
$90$	0	0
$91$	0	0
$92$	9.79796	1.02151
$93$	0	0
$94$	−6.00000	−0.618853
$95$	−2.44949	−0.251312
$96$	0	0
$97$	1.00000	0.101535	0.0507673	−	0.998711i	$-0.483833\pi$
$97$	1.00000	0.101535	0.0507673	+	0.998711i	$0.483833\pi$
$98$	0	0
$99$	0	0
$100$	4.00000	0.400000
$101$	−4.89898	−0.487467	−0.243733	−	0.969842i	$-0.578372\pi$
$101$	−4.89898	−0.487467	−0.243733	+	0.969842i	$0.578372\pi$
$102$	0	0
$103$	−2.00000	−0.197066	−0.0985329	−	0.995134i	$-0.531415\pi$
$103$	−2.00000	−0.197066	−0.0985329	+	0.995134i	$0.531415\pi$
$104$	19.5959	1.92154
$105$	0	0
$106$	−6.00000	−0.582772
$107$	−19.5959	−1.89441	−0.947204	−	0.320630i	$-0.896105\pi$
$107$	−19.5959	−1.89441	−0.947204	+	0.320630i	$0.896105\pi$
$108$	0	0
$109$	−1.00000	−0.0957826	−0.0478913	−	0.998853i	$-0.515250\pi$
$109$	−1.00000	−0.0957826	−0.0478913	+	0.998853i	$0.515250\pi$
$110$	−29.3939	−2.80260
$111$	0	0
$112$	0	0
$113$	−14.6969	−1.38257	−0.691286	−	0.722581i	$-0.742955\pi$
$113$	−14.6969	−1.38257	−0.691286	+	0.722581i	$0.742955\pi$
$114$	0	0
$115$	−6.00000	−0.559503
$116$	−29.3939	−2.72915
$117$	0	0
$118$	24.0000	2.20938
$119$	0	0
$120$	0	0
$121$	13.0000	1.18182
$122$	−12.2474	−1.10883
$123$	0	0
$124$	28.0000	2.51447
$125$	9.79796	0.876356
$126$	0	0
$127$	11.0000	0.976092	0.488046	−	0.872818i	$-0.337710\pi$
$127$	11.0000	0.976092	0.488046	+	0.872818i	$0.337710\pi$
$128$	−19.5959	−1.73205
$129$	0	0
$130$	−24.0000	−2.10494
$131$	−2.44949	−0.214013	−0.107006	−	0.994258i	$-0.534127\pi$
$131$	−2.44949	−0.214013	−0.107006	+	0.994258i	$0.534127\pi$
$132$	0	0
$133$	0	0
$134$	4.89898	0.423207
$135$	0	0
$136$	12.0000	1.02899
$137$	−17.1464	−1.46492	−0.732459	−	0.680811i	$-0.761627\pi$
$137$	−17.1464	−1.46492	−0.732459	+	0.680811i	$0.761627\pi$
$138$	0	0
$139$	−14.0000	−1.18746	−0.593732	−	0.804663i	$-0.702346\pi$
$139$	−14.0000	−1.18746	−0.593732	+	0.804663i	$0.702346\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	19.5959	1.63869
$144$	0	0
$145$	18.0000	1.49482
$146$	2.44949	0.202721
$147$	0	0
$148$	32.0000	2.63038
$149$	2.44949	0.200670	0.100335	−	0.994954i	$-0.468009\pi$
$149$	2.44949	0.200670	0.100335	+	0.994954i	$0.468009\pi$
$150$	0	0
$151$	5.00000	0.406894	0.203447	−	0.979086i	$-0.434786\pi$
$151$	5.00000	0.406894	0.203447	+	0.979086i	$0.434786\pi$
$152$	4.89898	0.397360
$153$	0	0
$154$	0	0
$155$	−17.1464	−1.37723
$156$	0	0
$157$	−20.0000	−1.59617	−0.798087	−	0.602542i	$-0.794154\pi$
$157$	−20.0000	−1.59617	−0.798087	+	0.602542i	$0.794154\pi$
$158$	−9.79796	−0.779484
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−1.00000	−0.0783260	−0.0391630	−	0.999233i	$-0.512469\pi$
$163$	−1.00000	−0.0783260	−0.0391630	+	0.999233i	$0.512469\pi$
$164$	−29.3939	−2.29528
$165$	0	0
$166$	−36.0000	−2.79414
$167$	14.6969	1.13728	0.568642	−	0.822585i	$-0.307469\pi$
$167$	14.6969	1.13728	0.568642	+	0.822585i	$0.307469\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	−14.6969	−1.12720
$171$	0	0
$172$	−4.00000	−0.304997
$173$	−2.44949	−0.186231	−0.0931156	−	0.995655i	$-0.529683\pi$
$173$	−2.44949	−0.186231	−0.0931156	+	0.995655i	$0.529683\pi$
$174$	0	0
$175$	0	0
$176$	19.5959	1.47710
$177$	0	0
$178$	−6.00000	−0.449719
$179$	−2.44949	−0.183083	−0.0915417	−	0.995801i	$-0.529179\pi$
$179$	−2.44949	−0.183083	−0.0915417	+	0.995801i	$0.529179\pi$
$180$	0	0
$181$	7.00000	0.520306	0.260153	−	0.965567i	$-0.416227\pi$
$181$	7.00000	0.520306	0.260153	+	0.965567i	$0.416227\pi$
$182$	0	0
$183$	0	0
$184$	12.0000	0.884652
$185$	−19.5959	−1.44072
$186$	0	0
$187$	12.0000	0.877527
$188$	−9.79796	−0.714590
$189$	0	0
$190$	−6.00000	−0.435286
$191$	2.44949	0.177239	0.0886194	−	0.996066i	$-0.471755\pi$
$191$	2.44949	0.177239	0.0886194	+	0.996066i	$0.471755\pi$
$192$	0	0
$193$	20.0000	1.43963	0.719816	−	0.694165i	$-0.244226\pi$
$193$	20.0000	1.43963	0.719816	+	0.694165i	$0.244226\pi$
$194$	2.44949	0.175863
$195$	0	0
$196$	0	0
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	0	0
$199$	19.0000	1.34687	0.673437	−	0.739244i	$-0.264817\pi$
$199$	19.0000	1.34687	0.673437	+	0.739244i	$0.264817\pi$
$200$	4.89898	0.346410
$201$	0	0
$202$	−12.0000	−0.844317
$203$	0	0
$204$	0	0
$205$	18.0000	1.25717
$206$	−4.89898	−0.341328
$207$	0	0
$208$	16.0000	1.10940
$209$	4.89898	0.338869
$210$	0	0
$211$	5.00000	0.344214	0.172107	−	0.985078i	$-0.444942\pi$
$211$	5.00000	0.344214	0.172107	+	0.985078i	$0.444942\pi$
$212$	−9.79796	−0.672927
$213$	0	0
$214$	−48.0000	−3.28121
$215$	2.44949	0.167054
$216$	0	0
$217$	0	0
$218$	−2.44949	−0.165900
$219$	0	0
$220$	−48.0000	−3.23616
$221$	9.79796	0.659082
$222$	0	0
$223$	−26.0000	−1.74109	−0.870544	−	0.492090i	$-0.836233\pi$
$223$	−26.0000	−1.74109	−0.870544	+	0.492090i	$0.836233\pi$
$224$	0	0
$225$	0	0
$226$	−36.0000	−2.39468
$227$	17.1464	1.13805	0.569024	−	0.822321i	$-0.307321\pi$
$227$	17.1464	1.13805	0.569024	+	0.822321i	$0.307321\pi$
$228$	0	0
$229$	7.00000	0.462573	0.231287	−	0.972886i	$-0.425707\pi$
$229$	7.00000	0.462573	0.231287	+	0.972886i	$0.425707\pi$
$230$	−14.6969	−0.969087
$231$	0	0
$232$	−36.0000	−2.36352
$233$	2.44949	0.160471	0.0802357	−	0.996776i	$-0.474433\pi$
$233$	2.44949	0.160471	0.0802357	+	0.996776i	$0.474433\pi$
$234$	0	0
$235$	6.00000	0.391397
$236$	39.1918	2.55117
$237$	0	0
$238$	0	0
$239$	7.34847	0.475333	0.237666	−	0.971347i	$-0.423617\pi$
$239$	7.34847	0.475333	0.237666	+	0.971347i	$0.423617\pi$
$240$	0	0
$241$	13.0000	0.837404	0.418702	−	0.908124i	$-0.362485\pi$
$241$	13.0000	0.837404	0.418702	+	0.908124i	$0.362485\pi$
$242$	31.8434	2.04697
$243$	0	0
$244$	−20.0000	−1.28037
$245$	0	0
$246$	0	0
$247$	4.00000	0.254514
$248$	34.2929	2.17760
$249$	0	0
$250$	24.0000	1.51789
$251$	22.0454	1.39149	0.695747	−	0.718287i	$-0.255074\pi$
$251$	22.0454	1.39149	0.695747	+	0.718287i	$0.255074\pi$
$252$	0	0
$253$	12.0000	0.754434
$254$	26.9444	1.69064
$255$	0	0
$256$	−32.0000	−2.00000
$257$	−24.4949	−1.52795	−0.763975	−	0.645246i	$-0.776755\pi$
$257$	−24.4949	−1.52795	−0.763975	+	0.645246i	$0.776755\pi$
$258$	0	0
$259$	0	0
$260$	−39.1918	−2.43057
$261$	0	0
$262$	−6.00000	−0.370681
$263$	4.89898	0.302084	0.151042	−	0.988527i	$-0.451737\pi$
$263$	4.89898	0.302084	0.151042	+	0.988527i	$0.451737\pi$
$264$	0	0
$265$	6.00000	0.368577
$266$	0	0
$267$	0	0
$268$	8.00000	0.488678
$269$	24.4949	1.49348	0.746740	−	0.665116i	$-0.231618\pi$
$269$	24.4949	1.49348	0.746740	+	0.665116i	$0.231618\pi$
$270$	0	0
$271$	1.00000	0.0607457	0.0303728	−	0.999539i	$-0.490331\pi$
$271$	1.00000	0.0607457	0.0303728	+	0.999539i	$0.490331\pi$
$272$	9.79796	0.594089
$273$	0	0
$274$	−42.0000	−2.53731
$275$	4.89898	0.295420
$276$	0	0
$277$	23.0000	1.38194	0.690968	−	0.722885i	$-0.257185\pi$
$277$	23.0000	1.38194	0.690968	+	0.722885i	$0.257185\pi$
$278$	−34.2929	−2.05675
$279$	0	0
$280$	0	0
$281$	14.6969	0.876746	0.438373	−	0.898793i	$-0.355555\pi$
$281$	14.6969	0.876746	0.438373	+	0.898793i	$0.355555\pi$
$282$	0	0
$283$	25.0000	1.48610	0.743048	−	0.669238i	$-0.233379\pi$
$283$	25.0000	1.48610	0.743048	+	0.669238i	$0.233379\pi$
$284$	0	0
$285$	0	0
$286$	48.0000	2.83830
$287$	0	0
$288$	0	0
$289$	−11.0000	−0.647059
$290$	44.0908	2.58910
$291$	0	0
$292$	4.00000	0.234082
$293$	−29.3939	−1.71721	−0.858604	−	0.512639i	$-0.828668\pi$
$293$	−29.3939	−1.71721	−0.858604	+	0.512639i	$0.828668\pi$
$294$	0	0
$295$	−24.0000	−1.39733
$296$	39.1918	2.27798
$297$	0	0
$298$	6.00000	0.347571
$299$	9.79796	0.566631
$300$	0	0
$301$	0	0
$302$	12.2474	0.704761
$303$	0	0
$304$	4.00000	0.229416
$305$	12.2474	0.701287
$306$	0	0
$307$	25.0000	1.42683	0.713413	−	0.700744i	$-0.247149\pi$
$307$	25.0000	1.42683	0.713413	+	0.700744i	$0.247149\pi$
$308$	0	0
$309$	0	0
$310$	−42.0000	−2.38544
$311$	−34.2929	−1.94457	−0.972285	−	0.233800i	$-0.924884\pi$
$311$	−34.2929	−1.94457	−0.972285	+	0.233800i	$0.924884\pi$
$312$	0	0
$313$	−23.0000	−1.30004	−0.650018	−	0.759918i	$-0.725239\pi$
$313$	−23.0000	−1.30004	−0.650018	+	0.759918i	$0.725239\pi$
$314$	−48.9898	−2.76465
$315$	0	0
$316$	−16.0000	−0.900070
$317$	24.4949	1.37577	0.687885	−	0.725819i	$-0.258539\pi$
$317$	24.4949	1.37577	0.687885	+	0.725819i	$0.258539\pi$
$318$	0	0
$319$	−36.0000	−2.01561
$320$	19.5959	1.09545
$321$	0	0
$322$	0	0
$323$	2.44949	0.136293
$324$	0	0
$325$	4.00000	0.221880
$326$	−2.44949	−0.135665
$327$	0	0
$328$	−36.0000	−1.98777
$329$	0	0
$330$	0	0
$331$	−13.0000	−0.714545	−0.357272	−	0.934000i	$-0.616293\pi$
$331$	−13.0000	−0.714545	−0.357272	+	0.934000i	$0.616293\pi$
$332$	−58.7878	−3.22640
$333$	0	0
$334$	36.0000	1.96983
$335$	−4.89898	−0.267660
$336$	0	0
$337$	5.00000	0.272367	0.136184	−	0.990684i	$-0.456516\pi$
$337$	5.00000	0.272367	0.136184	+	0.990684i	$0.456516\pi$
$338$	7.34847	0.399704
$339$	0	0
$340$	−24.0000	−1.30158
$341$	34.2929	1.85706
$342$	0	0
$343$	0	0
$344$	−4.89898	−0.264135
$345$	0	0
$346$	−6.00000	−0.322562
$347$	−9.79796	−0.525982	−0.262991	−	0.964798i	$-0.584709\pi$
$347$	−9.79796	−0.525982	−0.262991	+	0.964798i	$0.584709\pi$
$348$	0	0
$349$	−35.0000	−1.87351	−0.936754	−	0.349990i	$-0.886185\pi$
$349$	−35.0000	−1.87351	−0.936754	+	0.349990i	$0.886185\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−4.89898	−0.260746	−0.130373	−	0.991465i	$-0.541618\pi$
$353$	−4.89898	−0.260746	−0.130373	+	0.991465i	$0.541618\pi$
$354$	0	0
$355$	0	0
$356$	−9.79796	−0.519291
$357$	0	0
$358$	−6.00000	−0.317110
$359$	−19.5959	−1.03423	−0.517116	−	0.855915i	$-0.672995\pi$
$359$	−19.5959	−1.03423	−0.517116	+	0.855915i	$0.672995\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	17.1464	0.901196
$363$	0	0
$364$	0	0
$365$	−2.44949	−0.128212
$366$	0	0
$367$	13.0000	0.678594	0.339297	−	0.940679i	$-0.389811\pi$
$367$	13.0000	0.678594	0.339297	+	0.940679i	$0.389811\pi$
$368$	9.79796	0.510754
$369$	0	0
$370$	−48.0000	−2.49540
$371$	0	0
$372$	0	0
$373$	11.0000	0.569558	0.284779	−	0.958593i	$-0.408080\pi$
$373$	11.0000	0.569558	0.284779	+	0.958593i	$0.408080\pi$
$374$	29.3939	1.51992
$375$	0	0
$376$	−12.0000	−0.618853
$377$	−29.3939	−1.51386
$378$	0	0
$379$	−34.0000	−1.74646	−0.873231	−	0.487306i	$-0.837980\pi$
$379$	−34.0000	−1.74646	−0.873231	+	0.487306i	$0.837980\pi$
$380$	−9.79796	−0.502625
$381$	0	0
$382$	6.00000	0.306987
$383$	19.5959	1.00130	0.500652	−	0.865648i	$-0.333094\pi$
$383$	19.5959	1.00130	0.500652	+	0.865648i	$0.333094\pi$
$384$	0	0
$385$	0	0
$386$	48.9898	2.49351
$387$	0	0
$388$	4.00000	0.203069
$389$	−17.1464	−0.869358	−0.434679	−	0.900585i	$-0.643138\pi$
$389$	−17.1464	−0.869358	−0.434679	+	0.900585i	$0.643138\pi$
$390$	0	0
$391$	6.00000	0.303433
$392$	0	0
$393$	0	0
$394$	0	0
$395$	9.79796	0.492989
$396$	0	0
$397$	−17.0000	−0.853206	−0.426603	−	0.904439i	$-0.640290\pi$
$397$	−17.0000	−0.853206	−0.426603	+	0.904439i	$0.640290\pi$
$398$	46.5403	2.33285
$399$	0	0
$400$	4.00000	0.200000
$401$	24.4949	1.22322	0.611608	−	0.791161i	$-0.290523\pi$
$401$	24.4949	1.22322	0.611608	+	0.791161i	$0.290523\pi$
$402$	0	0
$403$	28.0000	1.39478
$404$	−19.5959	−0.974933
$405$	0	0
$406$	0	0
$407$	39.1918	1.94267
$408$	0	0
$409$	−20.0000	−0.988936	−0.494468	−	0.869196i	$-0.664637\pi$
$409$	−20.0000	−0.988936	−0.494468	+	0.869196i	$0.664637\pi$
$410$	44.0908	2.17749
$411$	0	0
$412$	−8.00000	−0.394132
$413$	0	0
$414$	0	0
$415$	36.0000	1.76717
$416$	0	0
$417$	0	0
$418$	12.0000	0.586939
$419$	−7.34847	−0.358996	−0.179498	−	0.983758i	$-0.557447\pi$
$419$	−7.34847	−0.358996	−0.179498	+	0.983758i	$0.557447\pi$
$420$	0	0
$421$	35.0000	1.70580	0.852898	−	0.522078i	$-0.174843\pi$
$421$	35.0000	1.70580	0.852898	+	0.522078i	$0.174843\pi$
$422$	12.2474	0.596196
$423$	0	0
$424$	−12.0000	−0.582772
$425$	2.44949	0.118818
$426$	0	0
$427$	0	0
$428$	−78.3837	−3.78882
$429$	0	0
$430$	6.00000	0.289346
$431$	19.5959	0.943902	0.471951	−	0.881625i	$-0.343550\pi$
$431$	19.5959	0.943902	0.471951	+	0.881625i	$0.343550\pi$
$432$	0	0
$433$	7.00000	0.336399	0.168199	−	0.985753i	$-0.446205\pi$
$433$	7.00000	0.336399	0.168199	+	0.985753i	$0.446205\pi$
$434$	0	0
$435$	0	0
$436$	−4.00000	−0.191565
$437$	2.44949	0.117175
$438$	0	0
$439$	−14.0000	−0.668184	−0.334092	−	0.942541i	$-0.608430\pi$
$439$	−14.0000	−0.668184	−0.334092	+	0.942541i	$0.608430\pi$
$440$	−58.7878	−2.80260
$441$	0	0
$442$	24.0000	1.14156
$443$	2.44949	0.116379	0.0581894	−	0.998306i	$-0.481467\pi$
$443$	2.44949	0.116379	0.0581894	+	0.998306i	$0.481467\pi$
$444$	0	0
$445$	6.00000	0.284427
$446$	−63.6867	−3.01565
$447$	0	0
$448$	0	0
$449$	22.0454	1.04039	0.520194	−	0.854048i	$-0.325860\pi$
$449$	22.0454	1.04039	0.520194	+	0.854048i	$0.325860\pi$
$450$	0	0
$451$	−36.0000	−1.69517
$452$	−58.7878	−2.76514
$453$	0	0
$454$	42.0000	1.97116
$455$	0	0
$456$	0	0
$457$	23.0000	1.07589	0.537947	−	0.842978i	$-0.319200\pi$
$457$	23.0000	1.07589	0.537947	+	0.842978i	$0.319200\pi$
$458$	17.1464	0.801200
$459$	0	0
$460$	−24.0000	−1.11901
$461$	29.3939	1.36901	0.684505	−	0.729008i	$-0.260019\pi$
$461$	29.3939	1.36901	0.684505	+	0.729008i	$0.260019\pi$
$462$	0	0
$463$	23.0000	1.06890	0.534450	−	0.845200i	$-0.320519\pi$
$463$	23.0000	1.06890	0.534450	+	0.845200i	$0.320519\pi$
$464$	−29.3939	−1.36458
$465$	0	0
$466$	6.00000	0.277945
$467$	−24.4949	−1.13349	−0.566744	−	0.823894i	$-0.691797\pi$
$467$	−24.4949	−1.13349	−0.566744	+	0.823894i	$0.691797\pi$
$468$	0	0
$469$	0	0
$470$	14.6969	0.677919
$471$	0	0
$472$	48.0000	2.20938
$473$	−4.89898	−0.225255
$474$	0	0
$475$	1.00000	0.0458831
$476$	0	0
$477$	0	0
$478$	18.0000	0.823301
$479$	−26.9444	−1.23112	−0.615560	−	0.788090i	$-0.711070\pi$
$479$	−26.9444	−1.23112	−0.615560	+	0.788090i	$0.711070\pi$
$480$	0	0
$481$	32.0000	1.45907
$482$	31.8434	1.45043
$483$	0	0
$484$	52.0000	2.36364
$485$	−2.44949	−0.111226
$486$	0	0
$487$	−19.0000	−0.860972	−0.430486	−	0.902597i	$-0.641658\pi$
$487$	−19.0000	−0.860972	−0.430486	+	0.902597i	$0.641658\pi$
$488$	−24.4949	−1.10883
$489$	0	0
$490$	0	0
$491$	−14.6969	−0.663264	−0.331632	−	0.943409i	$-0.607599\pi$
$491$	−14.6969	−0.663264	−0.331632	+	0.943409i	$0.607599\pi$
$492$	0	0
$493$	−18.0000	−0.810679
$494$	9.79796	0.440831
$495$	0	0
$496$	28.0000	1.25724
$497$	0	0
$498$	0	0
$499$	−7.00000	−0.313363	−0.156682	−	0.987649i	$-0.550080\pi$
$499$	−7.00000	−0.313363	−0.156682	+	0.987649i	$0.550080\pi$
$500$	39.1918	1.75271
$501$	0	0
$502$	54.0000	2.41014
$503$	22.0454	0.982956	0.491478	−	0.870890i	$-0.336457\pi$
$503$	22.0454	0.982956	0.491478	+	0.870890i	$0.336457\pi$
$504$	0	0
$505$	12.0000	0.533993
$506$	29.3939	1.30672
$507$	0	0
$508$	44.0000	1.95218
$509$	19.5959	0.868574	0.434287	−	0.900775i	$-0.357000\pi$
$509$	19.5959	0.868574	0.434287	+	0.900775i	$0.357000\pi$
$510$	0	0
$511$	0	0
$512$	−39.1918	−1.73205
$513$	0	0
$514$	−60.0000	−2.64649
$515$	4.89898	0.215875
$516$	0	0
$517$	−12.0000	−0.527759
$518$	0	0
$519$	0	0
$520$	−48.0000	−2.10494
$521$	−19.5959	−0.858513	−0.429256	−	0.903183i	$-0.641224\pi$
$521$	−19.5959	−0.858513	−0.429256	+	0.903183i	$0.641224\pi$
$522$	0	0
$523$	−8.00000	−0.349816	−0.174908	−	0.984585i	$-0.555963\pi$
$523$	−8.00000	−0.349816	−0.174908	+	0.984585i	$0.555963\pi$
$524$	−9.79796	−0.428026
$525$	0	0
$526$	12.0000	0.523225
$527$	17.1464	0.746910
$528$	0	0
$529$	−17.0000	−0.739130
$530$	14.6969	0.638394
$531$	0	0
$532$	0	0
$533$	−29.3939	−1.27319
$534$	0	0
$535$	48.0000	2.07522
$536$	9.79796	0.423207
$537$	0	0
$538$	60.0000	2.58678
$539$	0	0
$540$	0	0
$541$	35.0000	1.50477	0.752384	−	0.658725i	$-0.228904\pi$
$541$	35.0000	1.50477	0.752384	+	0.658725i	$0.228904\pi$
$542$	2.44949	0.105215
$543$	0	0
$544$	0	0
$545$	2.44949	0.104925
$546$	0	0
$547$	−37.0000	−1.58201	−0.791003	−	0.611812i	$-0.790441\pi$
$547$	−37.0000	−1.58201	−0.791003	+	0.611812i	$0.790441\pi$
$548$	−68.5857	−2.92984
$549$	0	0
$550$	12.0000	0.511682
$551$	−7.34847	−0.313055
$552$	0	0
$553$	0	0
$554$	56.3383	2.39358
$555$	0	0
$556$	−56.0000	−2.37493
$557$	41.6413	1.76440	0.882200	−	0.470875i	$-0.156062\pi$
$557$	41.6413	1.76440	0.882200	+	0.470875i	$0.156062\pi$
$558$	0	0
$559$	−4.00000	−0.169182
$560$	0	0
$561$	0	0
$562$	36.0000	1.51857
$563$	−12.2474	−0.516168	−0.258084	−	0.966122i	$-0.583091\pi$
$563$	−12.2474	−0.516168	−0.258084	+	0.966122i	$0.583091\pi$
$564$	0	0
$565$	36.0000	1.51453
$566$	61.2372	2.57399
$567$	0	0
$568$	0	0
$569$	−19.5959	−0.821504	−0.410752	−	0.911747i	$-0.634734\pi$
$569$	−19.5959	−0.821504	−0.410752	+	0.911747i	$0.634734\pi$
$570$	0	0
$571$	29.0000	1.21361	0.606806	−	0.794850i	$-0.292450\pi$
$571$	29.0000	1.21361	0.606806	+	0.794850i	$0.292450\pi$
$572$	78.3837	3.27739
$573$	0	0
$574$	0	0
$575$	2.44949	0.102151
$576$	0	0
$577$	−8.00000	−0.333044	−0.166522	−	0.986038i	$-0.553254\pi$
$577$	−8.00000	−0.333044	−0.166522	+	0.986038i	$0.553254\pi$
$578$	−26.9444	−1.12074
$579$	0	0
$580$	72.0000	2.98964
$581$	0	0
$582$	0	0
$583$	−12.0000	−0.496989
$584$	4.89898	0.202721
$585$	0	0
$586$	−72.0000	−2.97429
$587$	7.34847	0.303304	0.151652	−	0.988434i	$-0.451541\pi$
$587$	7.34847	0.303304	0.151652	+	0.988434i	$0.451541\pi$
$588$	0	0
$589$	7.00000	0.288430
$590$	−58.7878	−2.42025
$591$	0	0
$592$	32.0000	1.31519
$593$	−24.4949	−1.00588	−0.502942	−	0.864320i	$-0.667749\pi$
$593$	−24.4949	−1.00588	−0.502942	+	0.864320i	$0.667749\pi$
$594$	0	0
$595$	0	0
$596$	9.79796	0.401340
$597$	0	0
$598$	24.0000	0.981433
$599$	−31.8434	−1.30108	−0.650542	−	0.759470i	$-0.725458\pi$
$599$	−31.8434	−1.30108	−0.650542	+	0.759470i	$0.725458\pi$
$600$	0	0
$601$	−17.0000	−0.693444	−0.346722	−	0.937968i	$-0.612705\pi$
$601$	−17.0000	−0.693444	−0.346722	+	0.937968i	$0.612705\pi$
$602$	0	0
$603$	0	0
$604$	20.0000	0.813788
$605$	−31.8434	−1.29462
$606$	0	0
$607$	−29.0000	−1.17707	−0.588537	−	0.808470i	$-0.700296\pi$
$607$	−29.0000	−1.17707	−0.588537	+	0.808470i	$0.700296\pi$
$608$	0	0
$609$	0	0
$610$	30.0000	1.21466
$611$	−9.79796	−0.396383
$612$	0	0
$613$	17.0000	0.686624	0.343312	−	0.939222i	$-0.388451\pi$
$613$	17.0000	0.686624	0.343312	+	0.939222i	$0.388451\pi$
$614$	61.2372	2.47133
$615$	0	0
$616$	0	0
$617$	29.3939	1.18335	0.591676	−	0.806176i	$-0.298466\pi$
$617$	29.3939	1.18335	0.591676	+	0.806176i	$0.298466\pi$
$618$	0	0
$619$	−14.0000	−0.562708	−0.281354	−	0.959604i	$-0.590783\pi$
$619$	−14.0000	−0.562708	−0.281354	+	0.959604i	$0.590783\pi$
$620$	−68.5857	−2.75447
$621$	0	0
$622$	−84.0000	−3.36809
$623$	0	0
$624$	0	0
$625$	−29.0000	−1.16000
$626$	−56.3383	−2.25173
$627$	0	0
$628$	−80.0000	−3.19235
$629$	19.5959	0.781340
$630$	0	0
$631$	−25.0000	−0.995234	−0.497617	−	0.867397i	$-0.665792\pi$
$631$	−25.0000	−0.995234	−0.497617	+	0.867397i	$0.665792\pi$
$632$	−19.5959	−0.779484
$633$	0	0
$634$	60.0000	2.38290
$635$	−26.9444	−1.06926
$636$	0	0
$637$	0	0
$638$	−88.1816	−3.49114
$639$	0	0
$640$	48.0000	1.89737
$641$	26.9444	1.06424	0.532120	−	0.846669i	$-0.321396\pi$
$641$	26.9444	1.06424	0.532120	+	0.846669i	$0.321396\pi$
$642$	0	0
$643$	−5.00000	−0.197181	−0.0985904	−	0.995128i	$-0.531433\pi$
$643$	−5.00000	−0.197181	−0.0985904	+	0.995128i	$0.531433\pi$
$644$	0	0
$645$	0	0
$646$	6.00000	0.236067
$647$	−19.5959	−0.770395	−0.385198	−	0.922834i	$-0.625867\pi$
$647$	−19.5959	−0.770395	−0.385198	+	0.922834i	$0.625867\pi$
$648$	0	0
$649$	48.0000	1.88416
$650$	9.79796	0.384308
$651$	0	0
$652$	−4.00000	−0.156652
$653$	2.44949	0.0958559	0.0479280	−	0.998851i	$-0.484738\pi$
$653$	2.44949	0.0958559	0.0479280	+	0.998851i	$0.484738\pi$
$654$	0	0
$655$	6.00000	0.234439
$656$	−29.3939	−1.14764
$657$	0	0
$658$	0	0
$659$	−7.34847	−0.286256	−0.143128	−	0.989704i	$-0.545716\pi$
$659$	−7.34847	−0.286256	−0.143128	+	0.989704i	$0.545716\pi$
$660$	0	0
$661$	−11.0000	−0.427850	−0.213925	−	0.976850i	$-0.568625\pi$
$661$	−11.0000	−0.427850	−0.213925	+	0.976850i	$0.568625\pi$
$662$	−31.8434	−1.23763
$663$	0	0
$664$	−72.0000	−2.79414
$665$	0	0
$666$	0	0
$667$	−18.0000	−0.696963
$668$	58.7878	2.27457
$669$	0	0
$670$	−12.0000	−0.463600
$671$	−24.4949	−0.945615
$672$	0	0
$673$	17.0000	0.655302	0.327651	−	0.944799i	$-0.393743\pi$
$673$	17.0000	0.655302	0.327651	+	0.944799i	$0.393743\pi$
$674$	12.2474	0.471754
$675$	0	0
$676$	12.0000	0.461538
$677$	19.5959	0.753132	0.376566	−	0.926390i	$-0.377105\pi$
$677$	19.5959	0.753132	0.376566	+	0.926390i	$0.377105\pi$
$678$	0	0
$679$	0	0
$680$	−29.3939	−1.12720
$681$	0	0
$682$	84.0000	3.21653
$683$	−24.4949	−0.937271	−0.468636	−	0.883392i	$-0.655254\pi$
$683$	−24.4949	−0.937271	−0.468636	+	0.883392i	$0.655254\pi$
$684$	0	0
$685$	42.0000	1.60474
$686$	0	0
$687$	0	0
$688$	−4.00000	−0.152499
$689$	−9.79796	−0.373273
$690$	0	0
$691$	−41.0000	−1.55971	−0.779857	−	0.625958i	$-0.784708\pi$
$691$	−41.0000	−1.55971	−0.779857	+	0.625958i	$0.784708\pi$
$692$	−9.79796	−0.372463
$693$	0	0
$694$	−24.0000	−0.911028
$695$	34.2929	1.30080
$696$	0	0
$697$	−18.0000	−0.681799
$698$	−85.7321	−3.24501
$699$	0	0
$700$	0	0
$701$	22.0454	0.832644	0.416322	−	0.909217i	$-0.363319\pi$
$701$	22.0454	0.832644	0.416322	+	0.909217i	$0.363319\pi$
$702$	0	0
$703$	8.00000	0.301726
$704$	−39.1918	−1.47710
$705$	0	0
$706$	−12.0000	−0.451626
$707$	0	0
$708$	0	0
$709$	−49.0000	−1.84023	−0.920117	−	0.391644i	$-0.871906\pi$
$709$	−49.0000	−1.84023	−0.920117	+	0.391644i	$0.871906\pi$
$710$	0	0
$711$	0	0
$712$	−12.0000	−0.449719
$713$	17.1464	0.642139
$714$	0	0
$715$	−48.0000	−1.79510
$716$	−9.79796	−0.366167
$717$	0	0
$718$	−48.0000	−1.79134
$719$	19.5959	0.730804	0.365402	−	0.930850i	$-0.380931\pi$
$719$	19.5959	0.730804	0.365402	+	0.930850i	$0.380931\pi$
$720$	0	0
$721$	0	0
$722$	−44.0908	−1.64089
$723$	0	0
$724$	28.0000	1.04061
$725$	−7.34847	−0.272915
$726$	0	0
$727$	19.0000	0.704671	0.352335	−	0.935874i	$-0.385388\pi$
$727$	19.0000	0.704671	0.352335	+	0.935874i	$0.385388\pi$
$728$	0	0
$729$	0	0
$730$	−6.00000	−0.222070
$731$	−2.44949	−0.0905977
$732$	0	0
$733$	−11.0000	−0.406294	−0.203147	−	0.979148i	$-0.565117\pi$
$733$	−11.0000	−0.406294	−0.203147	+	0.979148i	$0.565117\pi$
$734$	31.8434	1.17536
$735$	0	0
$736$	0	0
$737$	9.79796	0.360912
$738$	0	0
$739$	−19.0000	−0.698926	−0.349463	−	0.936950i	$-0.613636\pi$
$739$	−19.0000	−0.698926	−0.349463	+	0.936950i	$0.613636\pi$
$740$	−78.3837	−2.88144
$741$	0	0
$742$	0	0
$743$	7.34847	0.269589	0.134795	−	0.990874i	$-0.456963\pi$
$743$	7.34847	0.269589	0.134795	+	0.990874i	$0.456963\pi$
$744$	0	0
$745$	−6.00000	−0.219823
$746$	26.9444	0.986504
$747$	0	0
$748$	48.0000	1.75505
$749$	0	0
$750$	0	0
$751$	−43.0000	−1.56909	−0.784546	−	0.620070i	$-0.787104\pi$
$751$	−43.0000	−1.56909	−0.784546	+	0.620070i	$0.787104\pi$
$752$	−9.79796	−0.357295
$753$	0	0
$754$	−72.0000	−2.62209
$755$	−12.2474	−0.445730
$756$	0	0
$757$	47.0000	1.70824	0.854122	−	0.520073i	$-0.174095\pi$
$757$	47.0000	1.70824	0.854122	+	0.520073i	$0.174095\pi$
$758$	−83.2827	−3.02496
$759$	0	0
$760$	−12.0000	−0.435286
$761$	19.5959	0.710351	0.355176	−	0.934800i	$-0.384421\pi$
$761$	19.5959	0.710351	0.355176	+	0.934800i	$0.384421\pi$
$762$	0	0
$763$	0	0
$764$	9.79796	0.354478
$765$	0	0
$766$	48.0000	1.73431
$767$	39.1918	1.41514
$768$	0	0
$769$	−5.00000	−0.180305	−0.0901523	−	0.995928i	$-0.528735\pi$
$769$	−5.00000	−0.180305	−0.0901523	+	0.995928i	$0.528735\pi$
$770$	0	0
$771$	0	0
$772$	80.0000	2.87926
$773$	46.5403	1.67394	0.836969	−	0.547250i	$-0.184325\pi$
$773$	46.5403	1.67394	0.836969	+	0.547250i	$0.184325\pi$
$774$	0	0
$775$	7.00000	0.251447
$776$	4.89898	0.175863
$777$	0	0
$778$	−42.0000	−1.50577
$779$	−7.34847	−0.263286
$780$	0	0
$781$	0	0
$782$	14.6969	0.525561
$783$	0	0
$784$	0	0
$785$	48.9898	1.74852
$786$	0	0
$787$	7.00000	0.249523	0.124762	−	0.992187i	$-0.460183\pi$
$787$	7.00000	0.249523	0.124762	+	0.992187i	$0.460183\pi$
$788$	0	0
$789$	0	0
$790$	24.0000	0.853882
$791$	0	0
$792$	0	0
$793$	−20.0000	−0.710221
$794$	−41.6413	−1.47780
$795$	0	0
$796$	76.0000	2.69375
$797$	14.6969	0.520592	0.260296	−	0.965529i	$-0.416180\pi$
$797$	14.6969	0.520592	0.260296	+	0.965529i	$0.416180\pi$
$798$	0	0
$799$	−6.00000	−0.212265
$800$	0	0
$801$	0	0
$802$	60.0000	2.11867
$803$	4.89898	0.172881
$804$	0	0
$805$	0	0
$806$	68.5857	2.41583
$807$	0	0
$808$	−24.0000	−0.844317
$809$	−24.4949	−0.861195	−0.430597	−	0.902544i	$-0.641697\pi$
$809$	−24.4949	−0.861195	−0.430597	+	0.902544i	$0.641697\pi$
$810$	0	0
$811$	−2.00000	−0.0702295	−0.0351147	−	0.999383i	$-0.511180\pi$
$811$	−2.00000	−0.0702295	−0.0351147	+	0.999383i	$0.511180\pi$
$812$	0	0
$813$	0	0
$814$	96.0000	3.36480
$815$	2.44949	0.0858019
$816$	0	0
$817$	−1.00000	−0.0349856
$818$	−48.9898	−1.71289
$819$	0	0
$820$	72.0000	2.51435
$821$	−19.5959	−0.683902	−0.341951	−	0.939718i	$-0.611088\pi$
$821$	−19.5959	−0.683902	−0.341951	+	0.939718i	$0.611088\pi$
$822$	0	0
$823$	−25.0000	−0.871445	−0.435723	−	0.900081i	$-0.643507\pi$
$823$	−25.0000	−0.871445	−0.435723	+	0.900081i	$0.643507\pi$
$824$	−9.79796	−0.341328
$825$	0	0
$826$	0	0
$827$	44.0908	1.53319	0.766594	−	0.642132i	$-0.221950\pi$
$827$	44.0908	1.53319	0.766594	+	0.642132i	$0.221950\pi$
$828$	0	0
$829$	55.0000	1.91023	0.955114	−	0.296237i	$-0.0957318\pi$
$829$	55.0000	1.91023	0.955114	+	0.296237i	$0.0957318\pi$
$830$	88.1816	3.06083
$831$	0	0
$832$	−32.0000	−1.10940
$833$	0	0
$834$	0	0
$835$	−36.0000	−1.24583
$836$	19.5959	0.677739
$837$	0	0
$838$	−18.0000	−0.621800
$839$	−14.6969	−0.507395	−0.253697	−	0.967284i	$-0.581647\pi$
$839$	−14.6969	−0.507395	−0.253697	+	0.967284i	$0.581647\pi$
$840$	0	0
$841$	25.0000	0.862069
$842$	85.7321	2.95452
$843$	0	0
$844$	20.0000	0.688428
$845$	−7.34847	−0.252795
$846$	0	0
$847$	0	0
$848$	−9.79796	−0.336463
$849$	0	0
$850$	6.00000	0.205798
$851$	19.5959	0.671739
$852$	0	0
$853$	19.0000	0.650548	0.325274	−	0.945620i	$-0.394544\pi$
$853$	19.0000	0.650548	0.325274	+	0.945620i	$0.394544\pi$
$854$	0	0
$855$	0	0
$856$	−96.0000	−3.28121
$857$	39.1918	1.33877	0.669384	−	0.742917i	$-0.266558\pi$
$857$	39.1918	1.33877	0.669384	+	0.742917i	$0.266558\pi$
$858$	0	0
$859$	25.0000	0.852989	0.426494	−	0.904490i	$-0.359748\pi$
$859$	25.0000	0.852989	0.426494	+	0.904490i	$0.359748\pi$
$860$	9.79796	0.334108
$861$	0	0
$862$	48.0000	1.63489
$863$	2.44949	0.0833816	0.0416908	−	0.999131i	$-0.486726\pi$
$863$	2.44949	0.0833816	0.0416908	+	0.999131i	$0.486726\pi$
$864$	0	0
$865$	6.00000	0.204006
$866$	17.1464	0.582659
$867$	0	0
$868$	0	0
$869$	−19.5959	−0.664746
$870$	0	0
$871$	8.00000	0.271070
$872$	−4.89898	−0.165900
$873$	0	0
$874$	6.00000	0.202953
$875$	0	0
$876$	0	0
$877$	47.0000	1.58708	0.793539	−	0.608520i	$-0.208236\pi$
$877$	47.0000	1.58708	0.793539	+	0.608520i	$0.208236\pi$
$878$	−34.2929	−1.15733
$879$	0	0
$880$	−48.0000	−1.61808
$881$	22.0454	0.742729	0.371364	−	0.928487i	$-0.378890\pi$
$881$	22.0454	0.742729	0.371364	+	0.928487i	$0.378890\pi$
$882$	0	0
$883$	−7.00000	−0.235569	−0.117784	−	0.993039i	$-0.537579\pi$
$883$	−7.00000	−0.235569	−0.117784	+	0.993039i	$0.537579\pi$
$884$	39.1918	1.31816
$885$	0	0
$886$	6.00000	0.201574
$887$	19.5959	0.657967	0.328983	−	0.944336i	$-0.393294\pi$
$887$	19.5959	0.657967	0.328983	+	0.944336i	$0.393294\pi$
$888$	0	0
$889$	0	0
$890$	14.6969	0.492642
$891$	0	0
$892$	−104.000	−3.48218
$893$	−2.44949	−0.0819690
$894$	0	0
$895$	6.00000	0.200558
$896$	0	0
$897$	0	0
$898$	54.0000	1.80200
$899$	−51.4393	−1.71560
$900$	0	0
$901$	−6.00000	−0.199889
$902$	−88.1816	−2.93613
$903$	0	0
$904$	−72.0000	−2.39468
$905$	−17.1464	−0.569967
$906$	0	0
$907$	32.0000	1.06254	0.531271	−	0.847202i	$-0.321714\pi$
$907$	32.0000	1.06254	0.531271	+	0.847202i	$0.321714\pi$
$908$	68.5857	2.27610
$909$	0	0
$910$	0	0
$911$	−7.34847	−0.243466	−0.121733	−	0.992563i	$-0.538845\pi$
$911$	−7.34847	−0.243466	−0.121733	+	0.992563i	$0.538845\pi$
$912$	0	0
$913$	−72.0000	−2.38285
$914$	56.3383	1.86350
$915$	0	0
$916$	28.0000	0.925146
$917$	0	0
$918$	0	0
$919$	−1.00000	−0.0329870	−0.0164935	−	0.999864i	$-0.505250\pi$
$919$	−1.00000	−0.0329870	−0.0164935	+	0.999864i	$0.505250\pi$
$920$	−29.3939	−0.969087
$921$	0	0
$922$	72.0000	2.37119
$923$	0	0
$924$	0	0
$925$	8.00000	0.263038
$926$	56.3383	1.85139
$927$	0	0
$928$	0	0
$929$	−46.5403	−1.52694	−0.763469	−	0.645845i	$-0.776505\pi$
$929$	−46.5403	−1.52694	−0.763469	+	0.645845i	$0.776505\pi$
$930$	0	0
$931$	0	0
$932$	9.79796	0.320943
$933$	0	0
$934$	−60.0000	−1.96326
$935$	−29.3939	−0.961283
$936$	0	0
$937$	16.0000	0.522697	0.261349	−	0.965244i	$-0.415833\pi$
$937$	16.0000	0.522697	0.261349	+	0.965244i	$0.415833\pi$
$938$	0	0
$939$	0	0
$940$	24.0000	0.782794
$941$	9.79796	0.319404	0.159702	−	0.987165i	$-0.448947\pi$
$941$	9.79796	0.319404	0.159702	+	0.987165i	$0.448947\pi$
$942$	0	0
$943$	−18.0000	−0.586161
$944$	39.1918	1.27559
$945$	0	0
$946$	−12.0000	−0.390154
$947$	46.5403	1.51236	0.756178	−	0.654366i	$-0.227064\pi$
$947$	46.5403	1.51236	0.756178	+	0.654366i	$0.227064\pi$
$948$	0	0
$949$	4.00000	0.129845
$950$	2.44949	0.0794719
$951$	0	0
$952$	0	0
$953$	−22.0454	−0.714121	−0.357060	−	0.934081i	$-0.616221\pi$
$953$	−22.0454	−0.714121	−0.357060	+	0.934081i	$0.616221\pi$
$954$	0	0
$955$	−6.00000	−0.194155
$956$	29.3939	0.950666
$957$	0	0
$958$	−66.0000	−2.13236
$959$	0	0
$960$	0	0
$961$	18.0000	0.580645
$962$	78.3837	2.52719
$963$	0	0
$964$	52.0000	1.67481
$965$	−48.9898	−1.57704
$966$	0	0
$967$	−22.0000	−0.707472	−0.353736	−	0.935345i	$-0.615089\pi$
$967$	−22.0000	−0.707472	−0.353736	+	0.935345i	$0.615089\pi$
$968$	63.6867	2.04697
$969$	0	0
$970$	−6.00000	−0.192648
$971$	41.6413	1.33633	0.668167	−	0.744011i	$-0.267079\pi$
$971$	41.6413	1.33633	0.668167	+	0.744011i	$0.267079\pi$
$972$	0	0
$973$	0	0
$974$	−46.5403	−1.49125
$975$	0	0
$976$	−20.0000	−0.640184
$977$	−9.79796	−0.313464	−0.156732	−	0.987641i	$-0.550096\pi$
$977$	−9.79796	−0.313464	−0.156732	+	0.987641i	$0.550096\pi$
$978$	0	0
$979$	−12.0000	−0.383522
$980$	0	0
$981$	0	0
$982$	−36.0000	−1.14881
$983$	39.1918	1.25003	0.625013	−	0.780615i	$-0.285094\pi$
$983$	39.1918	1.25003	0.625013	+	0.780615i	$0.285094\pi$
$984$	0	0
$985$	0	0
$986$	−44.0908	−1.40414
$987$	0	0
$988$	16.0000	0.509028
$989$	−2.44949	−0.0778892
$990$	0	0
$991$	−10.0000	−0.317660	−0.158830	−	0.987306i	$-0.550772\pi$
$991$	−10.0000	−0.317660	−0.158830	+	0.987306i	$0.550772\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−46.5403	−1.47543
$996$	0	0
$997$	−41.0000	−1.29848	−0.649242	−	0.760582i	$-0.724914\pi$
$997$	−41.0000	−1.29848	−0.649242	+	0.760582i	$0.724914\pi$
$998$	−17.1464	−0.542761
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1323.2.a.u.1.2		2
3.2	odd	2	inner	1323.2.a.u.1.1		2
7.3	odd	6		189.2.e.d.163.1	yes	4
7.5	odd	6		189.2.e.d.109.1	✓	4
7.6	odd	2		1323.2.a.v.1.2		2
21.5	even	6		189.2.e.d.109.2	yes	4
21.17	even	6		189.2.e.d.163.2	yes	4
21.20	even	2		1323.2.a.v.1.1		2
63.5	even	6		567.2.h.g.298.1		4
63.31	odd	6		567.2.g.g.541.1		4
63.38	even	6		567.2.h.g.352.1		4
63.40	odd	6		567.2.h.g.298.2		4
63.47	even	6		567.2.g.g.109.2		4
63.52	odd	6		567.2.h.g.352.2		4
63.59	even	6		567.2.g.g.541.2		4
63.61	odd	6		567.2.g.g.109.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.2.e.d.109.1	✓	4	7.5	odd	6
189.2.e.d.109.2	yes	4	21.5	even	6
189.2.e.d.163.1	yes	4	7.3	odd	6
189.2.e.d.163.2	yes	4	21.17	even	6
567.2.g.g.109.1		4	63.61	odd	6
567.2.g.g.109.2		4	63.47	even	6
567.2.g.g.541.1		4	63.31	odd	6
567.2.g.g.541.2		4	63.59	even	6
567.2.h.g.298.1		4	63.5	even	6
567.2.h.g.298.2		4	63.40	odd	6
567.2.h.g.352.1		4	63.38	even	6
567.2.h.g.352.2		4	63.52	odd	6
1323.2.a.u.1.1		2	3.2	odd	2	inner
1323.2.a.u.1.2		2	1.1	even	1	trivial
1323.2.a.v.1.1		2	21.20	even	2
1323.2.a.v.1.2		2	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1323 = 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1323.a (trivial)

\(f(q)\)	\(=\)	\(q+2.44949 q^{2} +4.00000 q^{4} -2.44949 q^{5} +4.89898 q^{8} +O(q^{10})\)	\(q+2.44949 q^{2} +4.00000 q^{4} -2.44949 q^{5} +4.89898 q^{8} -6.00000 q^{10} +4.89898 q^{11} +4.00000 q^{13} +4.00000 q^{16} +2.44949 q^{17} +1.00000 q^{19} -9.79796 q^{20} +12.0000 q^{22} +2.44949 q^{23} +1.00000 q^{25} +9.79796 q^{26} -7.34847 q^{29} +7.00000 q^{31} +6.00000 q^{34} +8.00000 q^{37} +2.44949 q^{38} -12.0000 q^{40} -7.34847 q^{41} -1.00000 q^{43} +19.5959 q^{44} +6.00000 q^{46} -2.44949 q^{47} +2.44949 q^{50} +16.0000 q^{52} -2.44949 q^{53} -12.0000 q^{55} -18.0000 q^{58} +9.79796 q^{59} -5.00000 q^{61} +17.1464 q^{62} -8.00000 q^{64} -9.79796 q^{65} +2.00000 q^{67} +9.79796 q^{68} +1.00000 q^{73} +19.5959 q^{74} +4.00000 q^{76} -4.00000 q^{79} -9.79796 q^{80} -18.0000 q^{82} -14.6969 q^{83} -6.00000 q^{85} -2.44949 q^{86} +24.0000 q^{88} -2.44949 q^{89} +9.79796 q^{92} -6.00000 q^{94} -2.44949 q^{95} +1.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 8 q^{4}+O(q^{10}) \) 2 * q + 8 * q^4	\( 2 q + 8 q^{4} - 12 q^{10} + 8 q^{13} + 8 q^{16} + 2 q^{19} + 24 q^{22} + 2 q^{25} + 14 q^{31} + 12 q^{34} + 16 q^{37} - 24 q^{40} - 2 q^{43} + 12 q^{46} + 32 q^{52} - 24 q^{55} - 36 q^{58} - 10 q^{61} - 16 q^{64} + 4 q^{67} + 2 q^{73} + 8 q^{76} - 8 q^{79} - 36 q^{82} - 12 q^{85} + 48 q^{88} - 12 q^{94} + 2 q^{97}+O(q^{100}) \) 2 * q + 8 * q^4 - 12 * q^10 + 8 * q^13 + 8 * q^16 + 2 * q^19 + 24 * q^22 + 2 * q^25 + 14 * q^31 + 12 * q^34 + 16 * q^37 - 24 * q^40 - 2 * q^43 + 12 * q^46 + 32 * q^52 - 24 * q^55 - 36 * q^58 - 10 * q^61 - 16 * q^64 + 4 * q^67 + 2 * q^73 + 8 * q^76 - 8 * q^79 - 36 * q^82 - 12 * q^85 + 48 * q^88 - 12 * q^94 + 2 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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