LMFDB - Embedded newform 1932.2.a.i.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1932,2,Mod(1,1932)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1932, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("1932.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1932 = 2^{2} \cdot 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1932.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:	$15.4270976705$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1509.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 7x + 4 $ x^3 - x^2 - 7*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.47735$ of defining polynomial
Character	$\chi$	$=$	1932.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +2.47735 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +2.47735 q^{5} +1.00000 q^{7} +1.00000 q^{9} +5.95470 q^{11} -0.137275 q^{13} -2.47735 q^{15} +4.61463 q^{17} +1.00000 q^{19} -1.00000 q^{21} +1.00000 q^{23} +1.13727 q^{25} -1.00000 q^{27} -1.65992 q^{29} -2.61463 q^{31} -5.95470 q^{33} +2.47735 q^{35} -11.5693 q^{37} +0.137275 q^{39} -3.68016 q^{41} +11.0920 q^{43} +2.47735 q^{45} +5.34008 q^{47} +1.00000 q^{49} -4.61463 q^{51} -5.13727 q^{53} +14.7519 q^{55} -1.00000 q^{57} +5.13727 q^{59} +4.52265 q^{61} +1.00000 q^{63} -0.340078 q^{65} +2.47735 q^{67} -1.00000 q^{69} -11.3665 q^{71} -0.340078 q^{73} -1.13727 q^{75} +5.95470 q^{77} -12.5240 q^{79} +1.00000 q^{81} +1.65992 q^{83} +11.4321 q^{85} +1.65992 q^{87} +3.86273 q^{89} -0.137275 q^{91} +2.61463 q^{93} +2.47735 q^{95} -2.88918 q^{97} +5.95470 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} - q^{5} + 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 - q^5 + 3 * q^7 + 3 * q^9	$ 3 q - 3 q^{3} - q^{5} + 3 q^{7} + 3 q^{9} + q^{11} + 3 q^{13} + q^{15} + 2 q^{17} + 3 q^{19} - 3 q^{21} + 3 q^{23} - 3 q^{27} - 10 q^{29} + 4 q^{31} - q^{33} - q^{35} - 6 q^{37} - 3 q^{39} - q^{41} + 13 q^{43} - q^{45} + 11 q^{47} + 3 q^{49} - 2 q^{51} - 12 q^{53} + 29 q^{55} - 3 q^{57} + 12 q^{59} + 22 q^{61} + 3 q^{63} + 4 q^{65} - q^{67} - 3 q^{69} - 7 q^{71} + 4 q^{73} + q^{77} + 8 q^{79} + 3 q^{81} + 10 q^{83} + 9 q^{85} + 10 q^{87} + 15 q^{89} + 3 q^{91} - 4 q^{93} - q^{95} + 10 q^{97} + q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 - q^5 + 3 * q^7 + 3 * q^9 + q^11 + 3 * q^13 + q^15 + 2 * q^17 + 3 * q^19 - 3 * q^21 + 3 * q^23 - 3 * q^27 - 10 * q^29 + 4 * q^31 - q^33 - q^35 - 6 * q^37 - 3 * q^39 - q^41 + 13 * q^43 - q^45 + 11 * q^47 + 3 * q^49 - 2 * q^51 - 12 * q^53 + 29 * q^55 - 3 * q^57 + 12 * q^59 + 22 * q^61 + 3 * q^63 + 4 * q^65 - q^67 - 3 * q^69 - 7 * q^71 + 4 * q^73 + q^77 + 8 * q^79 + 3 * q^81 + 10 * q^83 + 9 * q^85 + 10 * q^87 + 15 * q^89 + 3 * q^91 - 4 * q^93 - q^95 + 10 * q^97 + q^99

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 $

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	2.47735	1.10791	0.553953	−	0.832548i	$-0.313119\pi$
$5$	2.47735	1.10791	0.553953	+	0.832548i	$0.313119\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	5.95470	1.79541	0.897706	−	0.440596i	$-0.145233\pi$
$11$	5.95470	1.79541	0.897706	+	0.440596i	$0.145233\pi$
$12$	0	0
$13$	−0.137275	−0.0380731	−0.0190366	−	0.999819i	$-0.506060\pi$
$13$	−0.137275	−0.0380731	−0.0190366	+	0.999819i	$0.506060\pi$
$14$	0	0
$15$	−2.47735	−0.639650
$16$	0	0
$17$	4.61463	1.11921	0.559606	−	0.828759i	$-0.310953\pi$
$17$	4.61463	1.11921	0.559606	+	0.828759i	$0.310953\pi$
$18$	0	0
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.13727	0.227455
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−1.65992	−0.308240	−0.154120	−	0.988052i	$-0.549254\pi$
$29$	−1.65992	−0.308240	−0.154120	+	0.988052i	$0.549254\pi$
$30$	0	0
$31$	−2.61463	−0.469601	−0.234800	−	0.972044i	$-0.575444\pi$
$31$	−2.61463	−0.469601	−0.234800	+	0.972044i	$0.575444\pi$
$32$	0	0
$33$	−5.95470	−1.03658
$34$	0	0
$35$	2.47735	0.418749
$36$	0	0
$37$	−11.5693	−1.90199	−0.950993	−	0.309212i	$-0.899935\pi$
$37$	−11.5693	−1.90199	−0.950993	+	0.309212i	$0.899935\pi$
$38$	0	0
$39$	0.137275	0.0219815
$40$	0	0
$41$	−3.68016	−0.574744	−0.287372	−	0.957819i	$-0.592782\pi$
$41$	−3.68016	−0.574744	−0.287372	+	0.957819i	$0.592782\pi$
$42$	0	0
$43$	11.0920	1.69151	0.845755	−	0.533571i	$-0.179150\pi$
$43$	11.0920	1.69151	0.845755	+	0.533571i	$0.179150\pi$
$44$	0	0
$45$	2.47735	0.369302
$46$	0	0
$47$	5.34008	0.778930	0.389465	−	0.921041i	$-0.372660\pi$
$47$	5.34008	0.778930	0.389465	+	0.921041i	$0.372660\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−4.61463	−0.646177
$52$	0	0
$53$	−5.13727	−0.705659	−0.352829	−	0.935688i	$-0.614780\pi$
$53$	−5.13727	−0.705659	−0.352829	+	0.935688i	$0.614780\pi$
$54$	0	0
$55$	14.7519	1.98915
$56$	0	0
$57$	−1.00000	−0.132453
$58$	0	0
$59$	5.13727	0.668816	0.334408	−	0.942428i	$-0.391464\pi$
$59$	5.13727	0.668816	0.334408	+	0.942428i	$0.391464\pi$
$60$	0	0
$61$	4.52265	0.579066	0.289533	−	0.957168i	$-0.406500\pi$
$61$	4.52265	0.579066	0.289533	+	0.957168i	$0.406500\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	−0.340078	−0.0421814
$66$	0	0
$67$	2.47735	0.302657	0.151328	−	0.988484i	$-0.451645\pi$
$67$	2.47735	0.302657	0.151328	+	0.988484i	$0.451645\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−11.3665	−1.34896	−0.674479	−	0.738294i	$-0.735632\pi$
$71$	−11.3665	−1.34896	−0.674479	+	0.738294i	$0.735632\pi$
$72$	0	0
$73$	−0.340078	−0.0398031	−0.0199015	−	0.999802i	$-0.506335\pi$
$73$	−0.340078	−0.0398031	−0.0199015	+	0.999802i	$0.506335\pi$
$74$	0	0
$75$	−1.13727	−0.131321
$76$	0	0
$77$	5.95470	0.678602
$78$	0	0
$79$	−12.5240	−1.40906	−0.704532	−	0.709672i	$-0.748843\pi$
$79$	−12.5240	−1.40906	−0.704532	+	0.709672i	$0.748843\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	1.65992	0.182200	0.0911001	−	0.995842i	$-0.470962\pi$
$83$	1.65992	0.182200	0.0911001	+	0.995842i	$0.470962\pi$
$84$	0	0
$85$	11.4321	1.23998
$86$	0	0
$87$	1.65992	0.177962
$88$	0	0
$89$	3.86273	0.409448	0.204724	−	0.978820i	$-0.434370\pi$
$89$	3.86273	0.409448	0.204724	+	0.978820i	$0.434370\pi$
$90$	0	0
$91$	−0.137275	−0.0143903
$92$	0	0
$93$	2.61463	0.271124
$94$	0	0
$95$	2.47735	0.254171
$96$	0	0
$97$	−2.88918	−0.293351	−0.146676	−	0.989185i	$-0.546857\pi$
$97$	−2.88918	−0.293351	−0.146676	+	0.989185i	$0.546857\pi$
$98$	0	0
$99$	5.95470	0.598470
$100$	0	0
$101$	−19.3212	−1.92253	−0.961267	−	0.275618i	$-0.911118\pi$
$101$	−19.3212	−1.92253	−0.961267	+	0.275618i	$0.911118\pi$
$102$	0	0
$103$	6.38537	0.629169	0.314585	−	0.949229i	$-0.398135\pi$
$103$	6.38537	0.629169	0.314585	+	0.949229i	$0.398135\pi$
$104$	0	0
$105$	−2.47735	−0.241765
$106$	0	0
$107$	0.908021	0.0877817	0.0438908	−	0.999036i	$-0.486025\pi$
$107$	0.908021	0.0877817	0.0438908	+	0.999036i	$0.486025\pi$
$108$	0	0
$109$	6.75190	0.646715	0.323357	−	0.946277i	$-0.395188\pi$
$109$	6.75190	0.646715	0.323357	+	0.946277i	$0.395188\pi$
$110$	0	0
$111$	11.5693	1.09811
$112$	0	0
$113$	18.6613	1.75551	0.877754	−	0.479111i	$-0.159041\pi$
$113$	18.6613	1.75551	0.877754	+	0.479111i	$0.159041\pi$
$114$	0	0
$115$	2.47735	0.231014
$116$	0	0
$117$	−0.137275	−0.0126910
$118$	0	0
$119$	4.61463	0.423022
$120$	0	0
$121$	24.4585	2.22350
$122$	0	0
$123$	3.68016	0.331828
$124$	0	0
$125$	−9.56933	−0.855907
$126$	0	0
$127$	−4.15751	−0.368919	−0.184460	−	0.982840i	$-0.559053\pi$
$127$	−4.15751	−0.368919	−0.184460	+	0.982840i	$0.559053\pi$
$128$	0	0
$129$	−11.0920	−0.976594
$130$	0	0
$131$	14.2293	1.24322	0.621608	−	0.783329i	$-0.286480\pi$
$131$	14.2293	1.24322	0.621608	+	0.783329i	$0.286480\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	−2.47735	−0.213217
$136$	0	0
$137$	−7.27455	−0.621507	−0.310753	−	0.950491i	$-0.600581\pi$
$137$	−7.27455	−0.621507	−0.310753	+	0.950491i	$0.600581\pi$
$138$	0	0
$139$	15.4321	1.30893	0.654465	−	0.756092i	$-0.272894\pi$
$139$	15.4321	1.30893	0.654465	+	0.756092i	$0.272894\pi$
$140$	0	0
$141$	−5.34008	−0.449716
$142$	0	0
$143$	−0.817430	−0.0683569
$144$	0	0
$145$	−4.11221	−0.341501
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	−15.2495	−1.24929	−0.624643	−	0.780910i	$-0.714756\pi$
$149$	−15.2495	−1.24929	−0.624643	+	0.780910i	$0.714756\pi$
$150$	0	0
$151$	16.5693	1.34839	0.674197	−	0.738552i	$-0.264490\pi$
$151$	16.5693	1.34839	0.674197	+	0.738552i	$0.264490\pi$
$152$	0	0
$153$	4.61463	0.373070
$154$	0	0
$155$	−6.47735	−0.520273
$156$	0	0
$157$	−0.659922	−0.0526675	−0.0263338	−	0.999653i	$-0.508383\pi$
$157$	−0.659922	−0.0526675	−0.0263338	+	0.999653i	$0.508383\pi$
$158$	0	0
$159$	5.13727	0.407412
$160$	0	0
$161$	1.00000	0.0788110
$162$	0	0
$163$	23.3212	1.82666	0.913330	−	0.407220i	$-0.133502\pi$
$163$	23.3212	1.82666	0.913330	+	0.407220i	$0.133502\pi$
$164$	0	0
$165$	−14.7519	−1.14843
$166$	0	0
$167$	9.00000	0.696441	0.348220	−	0.937413i	$-0.386786\pi$
$167$	9.00000	0.696441	0.348220	+	0.937413i	$0.386786\pi$
$168$	0	0
$169$	−12.9812	−0.998550
$170$	0	0
$171$	1.00000	0.0764719
$172$	0	0
$173$	15.1637	1.15288	0.576438	−	0.817141i	$-0.304442\pi$
$173$	15.1637	1.15288	0.576438	+	0.817141i	$0.304442\pi$
$174$	0	0
$175$	1.13727	0.0859699
$176$	0	0
$177$	−5.13727	−0.386141
$178$	0	0
$179$	−16.3868	−1.22480	−0.612402	−	0.790546i	$-0.709797\pi$
$179$	−16.3868	−1.22480	−0.612402	+	0.790546i	$0.709797\pi$
$180$	0	0
$181$	11.0202	0.819127	0.409564	−	0.912282i	$-0.365681\pi$
$181$	11.0202	0.819127	0.409564	+	0.912282i	$0.365681\pi$
$182$	0	0
$183$	−4.52265	−0.334324
$184$	0	0
$185$	−28.6613	−2.10722
$186$	0	0
$187$	27.4787	2.00944
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	8.02023	0.580324	0.290162	−	0.956978i	$-0.406291\pi$
$191$	8.02023	0.580324	0.290162	+	0.956978i	$0.406291\pi$
$192$	0	0
$193$	−3.88918	−0.279949	−0.139975	−	0.990155i	$-0.544702\pi$
$193$	−3.88918	−0.279949	−0.139975	+	0.990155i	$0.544702\pi$
$194$	0	0
$195$	0.340078	0.0243535
$196$	0	0
$197$	1.09198	0.0778003	0.0389002	−	0.999243i	$-0.487615\pi$
$197$	1.09198	0.0778003	0.0389002	+	0.999243i	$0.487615\pi$
$198$	0	0
$199$	−25.2306	−1.78855	−0.894276	−	0.447515i	$-0.852309\pi$
$199$	−25.2306	−1.78855	−0.894276	+	0.447515i	$0.852309\pi$
$200$	0	0
$201$	−2.47735	−0.174739
$202$	0	0
$203$	−1.65992	−0.116504
$204$	0	0
$205$	−9.11704	−0.636762
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	5.95470	0.411896
$210$	0	0
$211$	−17.9547	−1.23605	−0.618026	−	0.786157i	$-0.712068\pi$
$211$	−17.9547	−1.23605	−0.618026	+	0.786157i	$0.712068\pi$
$212$	0	0
$213$	11.3665	0.778822
$214$	0	0
$215$	27.4787	1.87403
$216$	0	0
$217$	−2.61463	−0.177492
$218$	0	0
$219$	0.340078	0.0229803
$220$	0	0
$221$	−0.633471	−0.0426119
$222$	0	0
$223$	28.0216	1.87647	0.938233	−	0.346003i	$-0.112461\pi$
$223$	28.0216	1.87647	0.938233	+	0.346003i	$0.112461\pi$
$224$	0	0
$225$	1.13727	0.0758183
$226$	0	0
$227$	−4.47735	−0.297172	−0.148586	−	0.988899i	$-0.547472\pi$
$227$	−4.47735	−0.297172	−0.148586	+	0.988899i	$0.547472\pi$
$228$	0	0
$229$	−4.49759	−0.297209	−0.148604	−	0.988897i	$-0.547478\pi$
$229$	−4.49759	−0.297209	−0.148604	+	0.988897i	$0.547478\pi$
$230$	0	0
$231$	−5.95470	−0.391791
$232$	0	0
$233$	11.7066	0.766925	0.383463	−	0.923556i	$-0.374731\pi$
$233$	11.7066	0.766925	0.383463	+	0.923556i	$0.374731\pi$
$234$	0	0
$235$	13.2293	0.862981
$236$	0	0
$237$	12.5240	0.813524
$238$	0	0
$239$	2.34630	0.151769	0.0758846	−	0.997117i	$-0.475822\pi$
$239$	2.34630	0.151769	0.0758846	+	0.997117i	$0.475822\pi$
$240$	0	0
$241$	1.27455	0.0821009	0.0410505	−	0.999157i	$-0.486930\pi$
$241$	1.27455	0.0821009	0.0410505	+	0.999157i	$0.486930\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	2.47735	0.158272
$246$	0	0
$247$	−0.137275	−0.00873458
$248$	0	0
$249$	−1.65992	−0.105193
$250$	0	0
$251$	−16.1840	−1.02152	−0.510761	−	0.859723i	$-0.670636\pi$
$251$	−16.1840	−1.02152	−0.510761	+	0.859723i	$0.670636\pi$
$252$	0	0
$253$	5.95470	0.374369
$254$	0	0
$255$	−11.4321	−0.715903
$256$	0	0
$257$	−25.1184	−1.56684	−0.783422	−	0.621490i	$-0.786528\pi$
$257$	−25.1184	−1.56684	−0.783422	+	0.621490i	$0.786528\pi$
$258$	0	0
$259$	−11.5693	−0.718883
$260$	0	0
$261$	−1.65992	−0.102747
$262$	0	0
$263$	−18.1387	−1.11848	−0.559239	−	0.829007i	$-0.688907\pi$
$263$	−18.1387	−1.11848	−0.559239	+	0.829007i	$0.688907\pi$
$264$	0	0
$265$	−12.7268	−0.781804
$266$	0	0
$267$	−3.86273	−0.236395
$268$	0	0
$269$	30.1401	1.83767	0.918836	−	0.394640i	$-0.129131\pi$
$269$	30.1401	1.83767	0.918836	+	0.394640i	$0.129131\pi$
$270$	0	0
$271$	−0.209021	−0.0126971	−0.00634856	−	0.999980i	$-0.502021\pi$
$271$	−0.209021	−0.0126971	−0.00634856	+	0.999980i	$0.502021\pi$
$272$	0	0
$273$	0.137275	0.00830824
$274$	0	0
$275$	6.77213	0.408375
$276$	0	0
$277$	−10.7972	−0.648741	−0.324370	−	0.945930i	$-0.605152\pi$
$277$	−10.7972	−0.648741	−0.324370	+	0.945930i	$0.605152\pi$
$278$	0	0
$279$	−2.61463	−0.156534
$280$	0	0
$281$	−26.2745	−1.56741	−0.783704	−	0.621134i	$-0.786672\pi$
$281$	−26.2745	−1.56741	−0.783704	+	0.621134i	$0.786672\pi$
$282$	0	0
$283$	−11.7066	−0.695886	−0.347943	−	0.937516i	$-0.613120\pi$
$283$	−11.7066	−0.695886	−0.347943	+	0.937516i	$0.613120\pi$
$284$	0	0
$285$	−2.47735	−0.146746
$286$	0	0
$287$	−3.68016	−0.217233
$288$	0	0
$289$	4.29478	0.252634
$290$	0	0
$291$	2.88918	0.169367
$292$	0	0
$293$	−22.4585	−1.31204	−0.656020	−	0.754743i	$-0.727761\pi$
$293$	−22.4585	−1.31204	−0.656020	+	0.754743i	$0.727761\pi$
$294$	0	0
$295$	12.7268	0.740985
$296$	0	0
$297$	−5.95470	−0.345527
$298$	0	0
$299$	−0.137275	−0.00793880
$300$	0	0
$301$	11.0920	0.639331
$302$	0	0
$303$	19.3212	1.10998
$304$	0	0
$305$	11.2042	0.641550
$306$	0	0
$307$	11.9345	0.681136	0.340568	−	0.940220i	$-0.389381\pi$
$307$	11.9345	0.681136	0.340568	+	0.940220i	$0.389381\pi$
$308$	0	0
$309$	−6.38537	−0.363251
$310$	0	0
$311$	−7.75190	−0.439570	−0.219785	−	0.975548i	$-0.570536\pi$
$311$	−7.75190	−0.439570	−0.219785	+	0.975548i	$0.570536\pi$
$312$	0	0
$313$	6.93447	0.391960	0.195980	−	0.980608i	$-0.437211\pi$
$313$	6.93447	0.391960	0.195980	+	0.980608i	$0.437211\pi$
$314$	0	0
$315$	2.47735	0.139583
$316$	0	0
$317$	−26.3868	−1.48203	−0.741014	−	0.671489i	$-0.765655\pi$
$317$	−26.3868	−1.48203	−0.741014	+	0.671489i	$0.765655\pi$
$318$	0	0
$319$	−9.88435	−0.553417
$320$	0	0
$321$	−0.908021	−0.0506808
$322$	0	0
$323$	4.61463	0.256765
$324$	0	0
$325$	−0.156119	−0.00865992
$326$	0	0
$327$	−6.75190	−0.373381
$328$	0	0
$329$	5.34008	0.294408
$330$	0	0
$331$	7.61463	0.418538	0.209269	−	0.977858i	$-0.432892\pi$
$331$	7.61463	0.418538	0.209269	+	0.977858i	$0.432892\pi$
$332$	0	0
$333$	−11.5693	−0.633995
$334$	0	0
$335$	6.13727	0.335315
$336$	0	0
$337$	17.0920	0.931059	0.465530	−	0.885032i	$-0.345864\pi$
$337$	17.0920	0.931059	0.465530	+	0.885032i	$0.345864\pi$
$338$	0	0
$339$	−18.6613	−1.01354
$340$	0	0
$341$	−15.5693	−0.843127
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−2.47735	−0.133376
$346$	0	0
$347$	−35.1136	−1.88500	−0.942498	−	0.334211i	$-0.891530\pi$
$347$	−35.1136	−1.88500	−0.942498	+	0.334211i	$0.891530\pi$
$348$	0	0
$349$	24.1651	1.29353	0.646764	−	0.762690i	$-0.276122\pi$
$349$	24.1651	1.29353	0.646764	+	0.762690i	$0.276122\pi$
$350$	0	0
$351$	0.137275	0.00732718
$352$	0	0
$353$	−0.156119	−0.00830938	−0.00415469	−	0.999991i	$-0.501322\pi$
$353$	−0.156119	−0.00830938	−0.00415469	+	0.999991i	$0.501322\pi$
$354$	0	0
$355$	−28.1589	−1.49452
$356$	0	0
$357$	−4.61463	−0.244232
$358$	0	0
$359$	−12.1122	−0.639258	−0.319629	−	0.947543i	$-0.603558\pi$
$359$	−12.1122	−0.639258	−0.319629	+	0.947543i	$0.603558\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	0	0
$363$	−24.4585	−1.28374
$364$	0	0
$365$	−0.842492	−0.0440981
$366$	0	0
$367$	−21.8830	−1.14228	−0.571141	−	0.820852i	$-0.693499\pi$
$367$	−21.8830	−1.14228	−0.571141	+	0.820852i	$0.693499\pi$
$368$	0	0
$369$	−3.68016	−0.191581
$370$	0	0
$371$	−5.13727	−0.266714
$372$	0	0
$373$	6.88918	0.356708	0.178354	−	0.983966i	$-0.442923\pi$
$373$	6.88918	0.356708	0.178354	+	0.983966i	$0.442923\pi$
$374$	0	0
$375$	9.56933	0.494158
$376$	0	0
$377$	0.227865	0.0117357
$378$	0	0
$379$	−29.1387	−1.49675	−0.748376	−	0.663274i	$-0.769166\pi$
$379$	−29.1387	−1.49675	−0.748376	+	0.663274i	$0.769166\pi$
$380$	0	0
$381$	4.15751	0.212996
$382$	0	0
$383$	16.4307	0.839568	0.419784	−	0.907624i	$-0.362106\pi$
$383$	16.4307	0.839568	0.419784	+	0.907624i	$0.362106\pi$
$384$	0	0
$385$	14.7519	0.751827
$386$	0	0
$387$	11.0920	0.563837
$388$	0	0
$389$	−21.3854	−1.08428	−0.542141	−	0.840288i	$-0.682386\pi$
$389$	−21.3854	−1.08428	−0.542141	+	0.840288i	$0.682386\pi$
$390$	0	0
$391$	4.61463	0.233372
$392$	0	0
$393$	−14.2293	−0.717771
$394$	0	0
$395$	−31.0265	−1.56111
$396$	0	0
$397$	−25.4132	−1.27545	−0.637726	−	0.770263i	$-0.720125\pi$
$397$	−25.4132	−1.27545	−0.637726	+	0.770263i	$0.720125\pi$
$398$	0	0
$399$	−1.00000	−0.0500626
$400$	0	0
$401$	7.00000	0.349563	0.174782	−	0.984607i	$-0.444078\pi$
$401$	7.00000	0.349563	0.174782	+	0.984607i	$0.444078\pi$
$402$	0	0
$403$	0.358922	0.0178792
$404$	0	0
$405$	2.47735	0.123101
$406$	0	0
$407$	−68.8920	−3.41485
$408$	0	0
$409$	−6.06553	−0.299921	−0.149961	−	0.988692i	$-0.547915\pi$
$409$	−6.06553	−0.299921	−0.149961	+	0.988692i	$0.547915\pi$
$410$	0	0
$411$	7.27455	0.358827
$412$	0	0
$413$	5.13727	0.252789
$414$	0	0
$415$	4.11221	0.201861
$416$	0	0
$417$	−15.4321	−0.755711
$418$	0	0
$419$	−19.3415	−0.944892	−0.472446	−	0.881359i	$-0.656629\pi$
$419$	−19.3415	−0.944892	−0.472446	+	0.881359i	$0.656629\pi$
$420$	0	0
$421$	−2.54288	−0.123932	−0.0619662	−	0.998078i	$-0.519737\pi$
$421$	−2.54288	−0.123932	−0.0619662	+	0.998078i	$0.519737\pi$
$422$	0	0
$423$	5.34008	0.259643
$424$	0	0
$425$	5.24810	0.254570
$426$	0	0
$427$	4.52265	0.218866
$428$	0	0
$429$	0.817430	0.0394659
$430$	0	0
$431$	2.79720	0.134736	0.0673681	−	0.997728i	$-0.478540\pi$
$431$	2.79720	0.134736	0.0673681	+	0.997728i	$0.478540\pi$
$432$	0	0
$433$	−11.7080	−0.562650	−0.281325	−	0.959612i	$-0.590774\pi$
$433$	−11.7080	−0.562650	−0.281325	+	0.959612i	$0.590774\pi$
$434$	0	0
$435$	4.11221	0.197165
$436$	0	0
$437$	1.00000	0.0478365
$438$	0	0
$439$	17.9749	0.857897	0.428948	−	0.903329i	$-0.358884\pi$
$439$	17.9749	0.857897	0.428948	+	0.903329i	$0.358884\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−7.81882	−0.371483	−0.185742	−	0.982599i	$-0.559469\pi$
$443$	−7.81882	−0.371483	−0.185742	+	0.982599i	$0.559469\pi$
$444$	0	0
$445$	9.56933	0.453630
$446$	0	0
$447$	15.2495	0.721276
$448$	0	0
$449$	−25.0265	−1.18107	−0.590536	−	0.807012i	$-0.701083\pi$
$449$	−25.0265	−1.18107	−0.590536	+	0.807012i	$0.701083\pi$
$450$	0	0
$451$	−21.9142	−1.03190
$452$	0	0
$453$	−16.5693	−0.778495
$454$	0	0
$455$	−0.340078	−0.0159431
$456$	0	0
$457$	26.3868	1.23432	0.617160	−	0.786837i	$-0.288283\pi$
$457$	26.3868	1.23432	0.617160	+	0.786837i	$0.288283\pi$
$458$	0	0
$459$	−4.61463	−0.215392
$460$	0	0
$461$	−19.2481	−0.896473	−0.448237	−	0.893915i	$-0.647948\pi$
$461$	−19.2481	−0.896473	−0.448237	+	0.893915i	$0.647948\pi$
$462$	0	0
$463$	9.04047	0.420146	0.210073	−	0.977686i	$-0.432630\pi$
$463$	9.04047	0.420146	0.210073	+	0.977686i	$0.432630\pi$
$464$	0	0
$465$	6.47735	0.300380
$466$	0	0
$467$	35.2571	1.63150	0.815752	−	0.578402i	$-0.196323\pi$
$467$	35.2571	1.63150	0.815752	+	0.578402i	$0.196323\pi$
$468$	0	0
$469$	2.47735	0.114394
$470$	0	0
$471$	0.659922	0.0304076
$472$	0	0
$473$	66.0495	3.03696
$474$	0	0
$475$	1.13727	0.0521817
$476$	0	0
$477$	−5.13727	−0.235220
$478$	0	0
$479$	35.5721	1.62533	0.812666	−	0.582730i	$-0.198016\pi$
$479$	35.5721	1.62533	0.812666	+	0.582730i	$0.198016\pi$
$480$	0	0
$481$	1.58818	0.0724146
$482$	0	0
$483$	−1.00000	−0.0455016
$484$	0	0
$485$	−7.15751	−0.325006
$486$	0	0
$487$	0.614627	0.0278514	0.0139257	−	0.999903i	$-0.495567\pi$
$487$	0.614627	0.0278514	0.0139257	+	0.999903i	$0.495567\pi$
$488$	0	0
$489$	−23.3212	−1.05462
$490$	0	0
$491$	−12.2683	−0.553662	−0.276831	−	0.960919i	$-0.589284\pi$
$491$	−12.2683	−0.553662	−0.276831	+	0.960919i	$0.589284\pi$
$492$	0	0
$493$	−7.65992	−0.344986
$494$	0	0
$495$	14.7519	0.663049
$496$	0	0
$497$	−11.3665	−0.509858
$498$	0	0
$499$	−24.1122	−1.07941	−0.539705	−	0.841854i	$-0.681464\pi$
$499$	−24.1122	−1.07941	−0.539705	+	0.841854i	$0.681464\pi$
$500$	0	0
$501$	−9.00000	−0.402090
$502$	0	0
$503$	22.7519	1.01446	0.507229	−	0.861812i	$-0.330670\pi$
$503$	22.7519	1.01446	0.507229	+	0.861812i	$0.330670\pi$
$504$	0	0
$505$	−47.8655	−2.12999
$506$	0	0
$507$	12.9812	0.576513
$508$	0	0
$509$	−15.7581	−0.698466	−0.349233	−	0.937036i	$-0.613558\pi$
$509$	−15.7581	−0.698466	−0.349233	+	0.937036i	$0.613558\pi$
$510$	0	0
$511$	−0.340078	−0.0150442
$512$	0	0
$513$	−1.00000	−0.0441511
$514$	0	0
$515$	15.8188	0.697060
$516$	0	0
$517$	31.7986	1.39850
$518$	0	0
$519$	−15.1637	−0.665614
$520$	0	0
$521$	9.45090	0.414052	0.207026	−	0.978335i	$-0.433622\pi$
$521$	9.45090	0.414052	0.207026	+	0.978335i	$0.433622\pi$
$522$	0	0
$523$	4.93447	0.215769	0.107885	−	0.994163i	$-0.465592\pi$
$523$	4.93447	0.215769	0.107885	+	0.994163i	$0.465592\pi$
$524$	0	0
$525$	−1.13727	−0.0496347
$526$	0	0
$527$	−12.0655	−0.525583
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	5.13727	0.222939
$532$	0	0
$533$	0.505192	0.0218823
$534$	0	0
$535$	2.24949	0.0972538
$536$	0	0
$537$	16.3868	0.707141
$538$	0	0
$539$	5.95470	0.256487
$540$	0	0
$541$	43.8391	1.88479	0.942394	−	0.334505i	$-0.108569\pi$
$541$	43.8391	1.88479	0.942394	+	0.334505i	$0.108569\pi$
$542$	0	0
$543$	−11.0202	−0.472923
$544$	0	0
$545$	16.7268	0.716499
$546$	0	0
$547$	9.61602	0.411151	0.205576	−	0.978641i	$-0.434093\pi$
$547$	9.61602	0.411151	0.205576	+	0.978641i	$0.434093\pi$
$548$	0	0
$549$	4.52265	0.193022
$550$	0	0
$551$	−1.65992	−0.0707151
$552$	0	0
$553$	−12.5240	−0.532576
$554$	0	0
$555$	28.6613	1.21660
$556$	0	0
$557$	−1.90663	−0.0807866	−0.0403933	−	0.999184i	$-0.512861\pi$
$557$	−1.90663	−0.0807866	−0.0403933	+	0.999184i	$0.512861\pi$
$558$	0	0
$559$	−1.52265	−0.0644011
$560$	0	0
$561$	−27.4787	−1.16015
$562$	0	0
$563$	15.3665	0.647622	0.323811	−	0.946122i	$-0.395036\pi$
$563$	15.3665	0.647622	0.323811	+	0.946122i	$0.395036\pi$
$564$	0	0
$565$	46.2306	1.94494
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	−18.1637	−0.761463	−0.380731	−	0.924686i	$-0.624328\pi$
$569$	−18.1637	−0.761463	−0.380731	+	0.924686i	$0.624328\pi$
$570$	0	0
$571$	−16.4334	−0.687718	−0.343859	−	0.939021i	$-0.611734\pi$
$571$	−16.4334	−0.687718	−0.343859	+	0.939021i	$0.611734\pi$
$572$	0	0
$573$	−8.02023	−0.335050
$574$	0	0
$575$	1.13727	0.0474276
$576$	0	0
$577$	26.3679	1.09771	0.548855	−	0.835917i	$-0.315064\pi$
$577$	26.3679	1.09771	0.548855	+	0.835917i	$0.315064\pi$
$578$	0	0
$579$	3.88918	0.161629
$580$	0	0
$581$	1.65992	0.0688652
$582$	0	0
$583$	−30.5910	−1.26695
$584$	0	0
$585$	−0.340078	−0.0140605
$586$	0	0
$587$	−3.11221	−0.128455	−0.0642274	−	0.997935i	$-0.520458\pi$
$587$	−3.11221	−0.128455	−0.0642274	+	0.997935i	$0.520458\pi$
$588$	0	0
$589$	−2.61463	−0.107734
$590$	0	0
$591$	−1.09198	−0.0449180
$592$	0	0
$593$	10.2745	0.421925	0.210963	−	0.977494i	$-0.432340\pi$
$593$	10.2745	0.421925	0.210963	+	0.977494i	$0.432340\pi$
$594$	0	0
$595$	11.4321	0.468669
$596$	0	0
$597$	25.2306	1.03262
$598$	0	0
$599$	17.9032	0.731505	0.365752	−	0.930712i	$-0.380812\pi$
$599$	17.9032	0.731505	0.365752	+	0.930712i	$0.380812\pi$
$600$	0	0
$601$	−1.06692	−0.0435205	−0.0217602	−	0.999763i	$-0.506927\pi$
$601$	−1.06692	−0.0435205	−0.0217602	+	0.999763i	$0.506927\pi$
$602$	0	0
$603$	2.47735	0.100886
$604$	0	0
$605$	60.5923	2.46343
$606$	0	0
$607$	3.77213	0.153106	0.0765531	−	0.997066i	$-0.475609\pi$
$607$	3.77213	0.153106	0.0765531	+	0.997066i	$0.475609\pi$
$608$	0	0
$609$	1.65992	0.0672634
$610$	0	0
$611$	−0.733057	−0.0296563
$612$	0	0
$613$	−3.97494	−0.160546	−0.0802731	−	0.996773i	$-0.525579\pi$
$613$	−3.97494	−0.160546	−0.0802731	+	0.996773i	$0.525579\pi$
$614$	0	0
$615$	9.11704	0.367635
$616$	0	0
$617$	25.2355	1.01594	0.507971	−	0.861374i	$-0.330396\pi$
$617$	25.2355	1.01594	0.507971	+	0.861374i	$0.330396\pi$
$618$	0	0
$619$	2.23064	0.0896571	0.0448286	−	0.998995i	$-0.485726\pi$
$619$	2.23064	0.0896571	0.0448286	+	0.998995i	$0.485726\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	3.86273	0.154757
$624$	0	0
$625$	−29.3930	−1.17572
$626$	0	0
$627$	−5.95470	−0.237808
$628$	0	0
$629$	−53.3882	−2.12872
$630$	0	0
$631$	−6.47113	−0.257612	−0.128806	−	0.991670i	$-0.541114\pi$
$631$	−6.47113	−0.257612	−0.128806	+	0.991670i	$0.541114\pi$
$632$	0	0
$633$	17.9547	0.713635
$634$	0	0
$635$	−10.2996	−0.408728
$636$	0	0
$637$	−0.137275	−0.00543902
$638$	0	0
$639$	−11.3665	−0.449653
$640$	0	0
$641$	−22.7471	−0.898455	−0.449228	−	0.893417i	$-0.648301\pi$
$641$	−22.7471	−0.898455	−0.449228	+	0.893417i	$0.648301\pi$
$642$	0	0
$643$	−19.5707	−0.771794	−0.385897	−	0.922542i	$-0.626108\pi$
$643$	−19.5707	−0.771794	−0.385897	+	0.922542i	$0.626108\pi$
$644$	0	0
$645$	−27.4787	−1.08197
$646$	0	0
$647$	23.9812	0.942797	0.471398	−	0.881920i	$-0.343749\pi$
$647$	23.9812	0.942797	0.471398	+	0.881920i	$0.343749\pi$
$648$	0	0
$649$	30.5910	1.20080
$650$	0	0
$651$	2.61463	0.102475
$652$	0	0
$653$	−15.1198	−0.591684	−0.295842	−	0.955237i	$-0.595600\pi$
$653$	−15.1198	−0.591684	−0.295842	+	0.955237i	$0.595600\pi$
$654$	0	0
$655$	35.2509	1.37737
$656$	0	0
$657$	−0.340078	−0.0132677
$658$	0	0
$659$	−17.5693	−0.684404	−0.342202	−	0.939626i	$-0.611173\pi$
$659$	−17.5693	−0.684404	−0.342202	+	0.939626i	$0.611173\pi$
$660$	0	0
$661$	−43.7331	−1.70102	−0.850509	−	0.525960i	$-0.823706\pi$
$661$	−43.7331	−1.70102	−0.850509	+	0.525960i	$0.823706\pi$
$662$	0	0
$663$	0.633471	0.0246020
$664$	0	0
$665$	2.47735	0.0960676
$666$	0	0
$667$	−1.65992	−0.0642724
$668$	0	0
$669$	−28.0216	−1.08338
$670$	0	0
$671$	26.9310	1.03966
$672$	0	0
$673$	7.09059	0.273322	0.136661	−	0.990618i	$-0.456363\pi$
$673$	7.09059	0.273322	0.136661	+	0.990618i	$0.456363\pi$
$674$	0	0
$675$	−1.13727	−0.0437737
$676$	0	0
$677$	−6.67394	−0.256500	−0.128250	−	0.991742i	$-0.540936\pi$
$677$	−6.67394	−0.256500	−0.128250	+	0.991742i	$0.540936\pi$
$678$	0	0
$679$	−2.88918	−0.110876
$680$	0	0
$681$	4.47735	0.171573
$682$	0	0
$683$	−39.5721	−1.51418	−0.757092	−	0.653308i	$-0.773381\pi$
$683$	−39.5721	−1.51418	−0.757092	+	0.653308i	$0.773381\pi$
$684$	0	0
$685$	−18.0216	−0.688571
$686$	0	0
$687$	4.49759	0.171594
$688$	0	0
$689$	0.705218	0.0268667
$690$	0	0
$691$	24.4146	0.928775	0.464388	−	0.885632i	$-0.346274\pi$
$691$	24.4146	0.928775	0.464388	+	0.885632i	$0.346274\pi$
$692$	0	0
$693$	5.95470	0.226201
$694$	0	0
$695$	38.2306	1.45017
$696$	0	0
$697$	−16.9825	−0.643260
$698$	0	0
$699$	−11.7066	−0.442785
$700$	0	0
$701$	−32.3693	−1.22257	−0.611286	−	0.791410i	$-0.709347\pi$
$701$	−32.3693	−1.22257	−0.611286	+	0.791410i	$0.709347\pi$
$702$	0	0
$703$	−11.5693	−0.436346
$704$	0	0
$705$	−13.2293	−0.498243
$706$	0	0
$707$	−19.3212	−0.726650
$708$	0	0
$709$	−34.5457	−1.29739	−0.648695	−	0.761049i	$-0.724685\pi$
$709$	−34.5457	−1.29739	−0.648695	+	0.761049i	$0.724685\pi$
$710$	0	0
$711$	−12.5240	−0.469688
$712$	0	0
$713$	−2.61463	−0.0979186
$714$	0	0
$715$	−2.02506	−0.0757330
$716$	0	0
$717$	−2.34630	−0.0876240
$718$	0	0
$719$	25.0202	0.933097	0.466549	−	0.884496i	$-0.345497\pi$
$719$	25.0202	0.933097	0.466549	+	0.884496i	$0.345497\pi$
$720$	0	0
$721$	6.38537	0.237804
$722$	0	0
$723$	−1.27455	−0.0474010
$724$	0	0
$725$	−1.88779	−0.0701107
$726$	0	0
$727$	−33.5491	−1.24427	−0.622134	−	0.782911i	$-0.713734\pi$
$727$	−33.5491	−1.24427	−0.622134	+	0.782911i	$0.713734\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	51.1853	1.89316
$732$	0	0
$733$	−4.73306	−0.174819	−0.0874097	−	0.996172i	$-0.527859\pi$
$733$	−4.73306	−0.174819	−0.0874097	+	0.996172i	$0.527859\pi$
$734$	0	0
$735$	−2.47735	−0.0913785
$736$	0	0
$737$	14.7519	0.543393
$738$	0	0
$739$	29.0202	1.06753	0.533763	−	0.845634i	$-0.320777\pi$
$739$	29.0202	1.06753	0.533763	+	0.845634i	$0.320777\pi$
$740$	0	0
$741$	0.137275	0.00504291
$742$	0	0
$743$	−11.2279	−0.411910	−0.205955	−	0.978561i	$-0.566030\pi$
$743$	−11.2279	−0.411910	−0.205955	+	0.978561i	$0.566030\pi$
$744$	0	0
$745$	−37.7784	−1.38409
$746$	0	0
$747$	1.65992	0.0607334
$748$	0	0
$749$	0.908021	0.0331784
$750$	0	0
$751$	19.3010	0.704304	0.352152	−	0.935943i	$-0.385450\pi$
$751$	19.3010	0.704304	0.352152	+	0.935943i	$0.385450\pi$
$752$	0	0
$753$	16.1840	0.589776
$754$	0	0
$755$	41.0481	1.49389
$756$	0	0
$757$	−7.65992	−0.278405	−0.139202	−	0.990264i	$-0.544454\pi$
$757$	−7.65992	−0.278405	−0.139202	+	0.990264i	$0.544454\pi$
$758$	0	0
$759$	−5.95470	−0.216142
$760$	0	0
$761$	22.9345	0.831374	0.415687	−	0.909508i	$-0.363541\pi$
$761$	22.9345	0.831374	0.415687	+	0.909508i	$0.363541\pi$
$762$	0	0
$763$	6.75190	0.244435
$764$	0	0
$765$	11.4321	0.413327
$766$	0	0
$767$	−0.705218	−0.0254639
$768$	0	0
$769$	54.6425	1.97046	0.985229	−	0.171243i	$-0.0547782\pi$
$769$	54.6425	1.97046	0.985229	+	0.171243i	$0.0547782\pi$
$770$	0	0
$771$	25.1184	0.904618
$772$	0	0
$773$	−8.02784	−0.288741	−0.144371	−	0.989524i	$-0.546116\pi$
$773$	−8.02784	−0.288741	−0.144371	+	0.989524i	$0.546116\pi$
$774$	0	0
$775$	−2.97355	−0.106813
$776$	0	0
$777$	11.5693	0.415047
$778$	0	0
$779$	−3.68016	−0.131855
$780$	0	0
$781$	−67.6843	−2.42194
$782$	0	0
$783$	1.65992	0.0593208
$784$	0	0
$785$	−1.63486	−0.0583507
$786$	0	0
$787$	−43.3617	−1.54568	−0.772839	−	0.634602i	$-0.781164\pi$
$787$	−43.3617	−1.54568	−0.772839	+	0.634602i	$0.781164\pi$
$788$	0	0
$789$	18.1387	0.645754
$790$	0	0
$791$	18.6613	0.663520
$792$	0	0
$793$	−0.620845	−0.0220468
$794$	0	0
$795$	12.7268	0.451374
$796$	0	0
$797$	44.8641	1.58917	0.794584	−	0.607154i	$-0.207689\pi$
$797$	44.8641	1.58917	0.794584	+	0.607154i	$0.207689\pi$
$798$	0	0
$799$	24.6425	0.871788
$800$	0	0
$801$	3.86273	0.136483
$802$	0	0
$803$	−2.02506	−0.0714629
$804$	0	0
$805$	2.47735	0.0873152
$806$	0	0
$807$	−30.1401	−1.06098
$808$	0	0
$809$	−5.40422	−0.190002	−0.0950011	−	0.995477i	$-0.530285\pi$
$809$	−5.40422	−0.190002	−0.0950011	+	0.995477i	$0.530285\pi$
$810$	0	0
$811$	−32.7485	−1.14995	−0.574977	−	0.818170i	$-0.694989\pi$
$811$	−32.7485	−1.14995	−0.574977	+	0.818170i	$0.694989\pi$
$812$	0	0
$813$	0.209021	0.00733068
$814$	0	0
$815$	57.7749	2.02377
$816$	0	0
$817$	11.0920	0.388059
$818$	0	0
$819$	−0.137275	−0.00479676
$820$	0	0
$821$	14.9142	0.520511	0.260255	−	0.965540i	$-0.416193\pi$
$821$	14.9142	0.520511	0.260255	+	0.965540i	$0.416193\pi$
$822$	0	0
$823$	−4.00139	−0.139480	−0.0697398	−	0.997565i	$-0.522217\pi$
$823$	−4.00139	−0.139480	−0.0697398	+	0.997565i	$0.522217\pi$
$824$	0	0
$825$	−6.77213	−0.235775
$826$	0	0
$827$	−6.91563	−0.240480	−0.120240	−	0.992745i	$-0.538366\pi$
$827$	−6.91563	−0.240480	−0.120240	+	0.992745i	$0.538366\pi$
$828$	0	0
$829$	15.7128	0.545729	0.272864	−	0.962052i	$-0.412029\pi$
$829$	15.7128	0.545729	0.272864	+	0.962052i	$0.412029\pi$
$830$	0	0
$831$	10.7972	0.374551
$832$	0	0
$833$	4.61463	0.159887
$834$	0	0
$835$	22.2962	0.771591
$836$	0	0
$837$	2.61463	0.0903747
$838$	0	0
$839$	37.9966	1.31179	0.655893	−	0.754853i	$-0.272292\pi$
$839$	37.9966	1.31179	0.655893	+	0.754853i	$0.272292\pi$
$840$	0	0
$841$	−26.2447	−0.904988
$842$	0	0
$843$	26.2745	0.904944
$844$	0	0
$845$	−32.1589	−1.10630
$846$	0	0
$847$	24.4585	0.840404
$848$	0	0
$849$	11.7066	0.401770
$850$	0	0
$851$	−11.5693	−0.396592
$852$	0	0
$853$	21.7659	0.745251	0.372625	−	0.927982i	$-0.378458\pi$
$853$	21.7659	0.745251	0.372625	+	0.927982i	$0.378458\pi$
$854$	0	0
$855$	2.47735	0.0847237
$856$	0	0
$857$	−54.9777	−1.87800	−0.939001	−	0.343913i	$-0.888247\pi$
$857$	−54.9777	−1.87800	−0.939001	+	0.343913i	$0.888247\pi$
$858$	0	0
$859$	−54.4084	−1.85639	−0.928195	−	0.372094i	$-0.878640\pi$
$859$	−54.4084	−1.85639	−0.928195	+	0.372094i	$0.878640\pi$
$860$	0	0
$861$	3.68016	0.125419
$862$	0	0
$863$	7.81604	0.266061	0.133031	−	0.991112i	$-0.457529\pi$
$863$	7.81604	0.266061	0.133031	+	0.991112i	$0.457529\pi$
$864$	0	0
$865$	37.5659	1.27728
$866$	0	0
$867$	−4.29478	−0.145858
$868$	0	0
$869$	−74.5769	−2.52985
$870$	0	0
$871$	−0.340078	−0.0115231
$872$	0	0
$873$	−2.88918	−0.0977838
$874$	0	0
$875$	−9.56933	−0.323502
$876$	0	0
$877$	45.2118	1.52669	0.763347	−	0.645989i	$-0.223555\pi$
$877$	45.2118	1.52669	0.763347	+	0.645989i	$0.223555\pi$
$878$	0	0
$879$	22.4585	0.757507
$880$	0	0
$881$	37.5443	1.26490	0.632449	−	0.774602i	$-0.282050\pi$
$881$	37.5443	1.26490	0.632449	+	0.774602i	$0.282050\pi$
$882$	0	0
$883$	23.5910	0.793899	0.396949	−	0.917840i	$-0.370069\pi$
$883$	23.5910	0.793899	0.396949	+	0.917840i	$0.370069\pi$
$884$	0	0
$885$	−12.7268	−0.427808
$886$	0	0
$887$	42.4118	1.42405	0.712025	−	0.702154i	$-0.247778\pi$
$887$	42.4118	1.42405	0.712025	+	0.702154i	$0.247778\pi$
$888$	0	0
$889$	−4.15751	−0.139438
$890$	0	0
$891$	5.95470	0.199490
$892$	0	0
$893$	5.34008	0.178699
$894$	0	0
$895$	−40.5958	−1.35697
$896$	0	0
$897$	0.137275	0.00458347
$898$	0	0
$899$	4.34008	0.144750
$900$	0	0
$901$	−23.7066	−0.789782
$902$	0	0
$903$	−11.0920	−0.369118
$904$	0	0
$905$	27.3010	0.907516
$906$	0	0
$907$	−40.0745	−1.33065	−0.665326	−	0.746553i	$-0.731708\pi$
$907$	−40.0745	−1.33065	−0.665326	+	0.746553i	$0.731708\pi$
$908$	0	0
$909$	−19.3212	−0.640845
$910$	0	0
$911$	−13.0858	−0.433551	−0.216775	−	0.976222i	$-0.569554\pi$
$911$	−13.0858	−0.433551	−0.216775	+	0.976222i	$0.569554\pi$
$912$	0	0
$913$	9.88435	0.327124
$914$	0	0
$915$	−11.2042	−0.370399
$916$	0	0
$917$	14.2293	0.469891
$918$	0	0
$919$	−49.6599	−1.63813	−0.819065	−	0.573701i	$-0.805507\pi$
$919$	−49.6599	−1.63813	−0.819065	+	0.573701i	$0.805507\pi$
$920$	0	0
$921$	−11.9345	−0.393254
$922$	0	0
$923$	1.56034	0.0513591
$924$	0	0
$925$	−13.1575	−0.432616
$926$	0	0
$927$	6.38537	0.209723
$928$	0	0
$929$	29.3944	0.964398	0.482199	−	0.876062i	$-0.339838\pi$
$929$	29.3944	0.964398	0.482199	+	0.876062i	$0.339838\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	7.75190	0.253786
$934$	0	0
$935$	68.0745	2.22627
$936$	0	0
$937$	44.2697	1.44623	0.723114	−	0.690728i	$-0.242710\pi$
$937$	44.2697	1.44623	0.723114	+	0.690728i	$0.242710\pi$
$938$	0	0
$939$	−6.93447	−0.226298
$940$	0	0
$941$	−15.2697	−0.497779	−0.248889	−	0.968532i	$-0.580066\pi$
$941$	−15.2697	−0.497779	−0.248889	+	0.968532i	$0.580066\pi$
$942$	0	0
$943$	−3.68016	−0.119842
$944$	0	0
$945$	−2.47735	−0.0805883
$946$	0	0
$947$	−25.2293	−0.819841	−0.409920	−	0.912121i	$-0.634443\pi$
$947$	−25.2293	−0.819841	−0.409920	+	0.912121i	$0.634443\pi$
$948$	0	0
$949$	0.0466840	0.00151543
$950$	0	0
$951$	26.3868	0.855649
$952$	0	0
$953$	27.1527	0.879562	0.439781	−	0.898105i	$-0.355056\pi$
$953$	27.1527	0.879562	0.439781	+	0.898105i	$0.355056\pi$
$954$	0	0
$955$	19.8689	0.642944
$956$	0	0
$957$	9.88435	0.319516
$958$	0	0
$959$	−7.27455	−0.234907
$960$	0	0
$961$	−24.1637	−0.779475
$962$	0	0
$963$	0.908021	0.0292606
$964$	0	0
$965$	−9.63486	−0.310157
$966$	0	0
$967$	−58.9575	−1.89594	−0.947972	−	0.318353i	$-0.896870\pi$
$967$	−58.9575	−1.89594	−0.947972	+	0.318353i	$0.896870\pi$
$968$	0	0
$969$	−4.61463	−0.148243
$970$	0	0
$971$	40.2306	1.29106	0.645531	−	0.763734i	$-0.276636\pi$
$971$	40.2306	1.29106	0.645531	+	0.763734i	$0.276636\pi$
$972$	0	0
$973$	15.4321	0.494729
$974$	0	0
$975$	0.156119	0.00499981
$976$	0	0
$977$	−0.0669171	−0.00214087	−0.00107043	−	0.999999i	$-0.500341\pi$
$977$	−0.0669171	−0.00214087	−0.00107043	+	0.999999i	$0.500341\pi$
$978$	0	0
$979$	23.0014	0.735128
$980$	0	0
$981$	6.75190	0.215572
$982$	0	0
$983$	−38.0278	−1.21290	−0.606450	−	0.795122i	$-0.707407\pi$
$983$	−38.0278	−1.21290	−0.606450	+	0.795122i	$0.707407\pi$
$984$	0	0
$985$	2.70522	0.0861954
$986$	0	0
$987$	−5.34008	−0.169977
$988$	0	0
$989$	11.0920	0.352704
$990$	0	0
$991$	34.6565	1.10090	0.550450	−	0.834868i	$-0.314456\pi$
$991$	34.6565	1.10090	0.550450	+	0.834868i	$0.314456\pi$
$992$	0	0
$993$	−7.61463	−0.241643
$994$	0	0
$995$	−62.5052	−1.98155
$996$	0	0
$997$	−21.1637	−0.670262	−0.335131	−	0.942172i	$-0.608781\pi$
$997$	−21.1637	−0.670262	−0.335131	+	0.942172i	$0.608781\pi$
$998$	0	0
$999$	11.5693	0.366037

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1932.2.a.i.1.3	✓	3
3.2	odd	2		5796.2.a.p.1.1		3
4.3	odd	2		7728.2.a.bv.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1932.2.a.i.1.3	✓	3	1.1	even	1	trivial
5796.2.a.p.1.1		3	3.2	odd	2
7728.2.a.bv.1.3		3	4.3	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:	\( N \)	\(=\)	\( 1932 = 2^{2} \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1932.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +2.47735 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +2.47735 q^{5} +1.00000 q^{7} +1.00000 q^{9} +5.95470 q^{11} -0.137275 q^{13} -2.47735 q^{15} +4.61463 q^{17} +1.00000 q^{19} -1.00000 q^{21} +1.00000 q^{23} +1.13727 q^{25} -1.00000 q^{27} -1.65992 q^{29} -2.61463 q^{31} -5.95470 q^{33} +2.47735 q^{35} -11.5693 q^{37} +0.137275 q^{39} -3.68016 q^{41} +11.0920 q^{43} +2.47735 q^{45} +5.34008 q^{47} +1.00000 q^{49} -4.61463 q^{51} -5.13727 q^{53} +14.7519 q^{55} -1.00000 q^{57} +5.13727 q^{59} +4.52265 q^{61} +1.00000 q^{63} -0.340078 q^{65} +2.47735 q^{67} -1.00000 q^{69} -11.3665 q^{71} -0.340078 q^{73} -1.13727 q^{75} +5.95470 q^{77} -12.5240 q^{79} +1.00000 q^{81} +1.65992 q^{83} +11.4321 q^{85} +1.65992 q^{87} +3.86273 q^{89} -0.137275 q^{91} +2.61463 q^{93} +2.47735 q^{95} -2.88918 q^{97} +5.95470 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} - q^{5} + 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 - q^5 + 3 * q^7 + 3 * q^9	\( 3 q - 3 q^{3} - q^{5} + 3 q^{7} + 3 q^{9} + q^{11} + 3 q^{13} + q^{15} + 2 q^{17} + 3 q^{19} - 3 q^{21} + 3 q^{23} - 3 q^{27} - 10 q^{29} + 4 q^{31} - q^{33} - q^{35} - 6 q^{37} - 3 q^{39} - q^{41} + 13 q^{43} - q^{45} + 11 q^{47} + 3 q^{49} - 2 q^{51} - 12 q^{53} + 29 q^{55} - 3 q^{57} + 12 q^{59} + 22 q^{61} + 3 q^{63} + 4 q^{65} - q^{67} - 3 q^{69} - 7 q^{71} + 4 q^{73} + q^{77} + 8 q^{79} + 3 q^{81} + 10 q^{83} + 9 q^{85} + 10 q^{87} + 15 q^{89} + 3 q^{91} - 4 q^{93} - q^{95} + 10 q^{97} + q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 - q^5 + 3 * q^7 + 3 * q^9 + q^11 + 3 * q^13 + q^15 + 2 * q^17 + 3 * q^19 - 3 * q^21 + 3 * q^23 - 3 * q^27 - 10 * q^29 + 4 * q^31 - q^33 - q^35 - 6 * q^37 - 3 * q^39 - q^41 + 13 * q^43 - q^45 + 11 * q^47 + 3 * q^49 - 2 * q^51 - 12 * q^53 + 29 * q^55 - 3 * q^57 + 12 * q^59 + 22 * q^61 + 3 * q^63 + 4 * q^65 - q^67 - 3 * q^69 - 7 * q^71 + 4 * q^73 + q^77 + 8 * q^79 + 3 * q^81 + 10 * q^83 + 9 * q^85 + 10 * q^87 + 15 * q^89 + 3 * q^91 - 4 * q^93 - q^95 + 10 * q^97 + q^99

Introduction

L-functions

Modular forms

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