LMFDB - Embedded newform 4788.2.a.r.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4788,2,Mod(1,4788)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4788, 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("4788.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4788 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4788.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:	$38.2323724878$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} + 4 $ x^4 - 6*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.28825$ of defining polynomial
Character	$\chi$	$=$	4788.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.70246 q^{5} +1.00000 q^{7} +O(q^{10})$	$q+3.70246 q^{5} +1.00000 q^{7} -4.57649 q^{11} -5.23607 q^{13} -0.874032 q^{17} -1.00000 q^{19} -1.08036 q^{23} +8.70820 q^{25} -4.78282 q^{29} -7.23607 q^{31} +3.70246 q^{35} -6.00000 q^{37} -4.57649 q^{41} +12.1803 q^{43} +10.0270 q^{47} +1.00000 q^{49} -9.35931 q^{53} -16.9443 q^{55} -12.6491 q^{59} -10.9443 q^{61} -19.3863 q^{65} +6.47214 q^{67} +0.206331 q^{71} -12.4721 q^{73} -4.57649 q^{77} -12.9443 q^{79} +16.7642 q^{83} -3.23607 q^{85} -4.57649 q^{89} -5.23607 q^{91} -3.70246 q^{95} +9.41641 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{7}+O(q^{10}) $ 4 * q + 4 * q^7	$ 4 q + 4 q^{7} - 12 q^{13} - 4 q^{19} + 8 q^{25} - 20 q^{31} - 24 q^{37} + 4 q^{43} + 4 q^{49} - 32 q^{55} - 8 q^{61} + 8 q^{67} - 32 q^{73} - 16 q^{79} - 4 q^{85} - 12 q^{91} - 16 q^{97}+O(q^{100}) $ 4 * q + 4 * q^7 - 12 * q^13 - 4 * q^19 + 8 * q^25 - 20 * q^31 - 24 * q^37 + 4 * q^43 + 4 * q^49 - 32 * q^55 - 8 * q^61 + 8 * q^67 - 32 * q^73 - 16 * q^79 - 4 * q^85 - 12 * q^91 - 16 * 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 $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	3.70246	1.65579	0.827895	−	0.560883i	$-0.189538\pi$
$5$	3.70246	1.65579	0.827895	+	0.560883i	$0.189538\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−4.57649	−1.37986	−0.689932	−	0.723874i	$-0.742360\pi$
$11$	−4.57649	−1.37986	−0.689932	+	0.723874i	$0.742360\pi$
$12$	0	0
$13$	−5.23607	−1.45222	−0.726112	−	0.687576i	$-0.758675\pi$
$13$	−5.23607	−1.45222	−0.726112	+	0.687576i	$0.758675\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−0.874032	−0.211984	−0.105992	−	0.994367i	$-0.533802\pi$
$17$	−0.874032	−0.211984	−0.105992	+	0.994367i	$0.533802\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.08036	−0.225271	−0.112636	−	0.993636i	$-0.535929\pi$
$23$	−1.08036	−0.225271	−0.112636	+	0.993636i	$0.535929\pi$
$24$	0	0
$25$	8.70820	1.74164
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−4.78282	−0.888148	−0.444074	−	0.895990i	$-0.646467\pi$
$29$	−4.78282	−0.888148	−0.444074	+	0.895990i	$0.646467\pi$
$30$	0	0
$31$	−7.23607	−1.29964	−0.649818	−	0.760090i	$-0.725155\pi$
$31$	−7.23607	−1.29964	−0.649818	+	0.760090i	$0.725155\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	3.70246	0.625830
$36$	0	0
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−4.57649	−0.714728	−0.357364	−	0.933965i	$-0.616324\pi$
$41$	−4.57649	−0.714728	−0.357364	+	0.933965i	$0.616324\pi$
$42$	0	0
$43$	12.1803	1.85748	0.928742	−	0.370726i	$-0.120891\pi$
$43$	12.1803	1.85748	0.928742	+	0.370726i	$0.120891\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	10.0270	1.46259	0.731295	−	0.682061i	$-0.238916\pi$
$47$	10.0270	1.46259	0.731295	+	0.682061i	$0.238916\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−9.35931	−1.28560	−0.642800	−	0.766034i	$-0.722227\pi$
$53$	−9.35931	−1.28560	−0.642800	+	0.766034i	$0.722227\pi$
$54$	0	0
$55$	−16.9443	−2.28477
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−12.6491	−1.64677	−0.823387	−	0.567480i	$-0.807918\pi$
$59$	−12.6491	−1.64677	−0.823387	+	0.567480i	$0.807918\pi$
$60$	0	0
$61$	−10.9443	−1.40127	−0.700635	−	0.713520i	$-0.747100\pi$
$61$	−10.9443	−1.40127	−0.700635	+	0.713520i	$0.747100\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−19.3863	−2.40458
$66$	0	0
$67$	6.47214	0.790697	0.395349	−	0.918531i	$-0.370624\pi$
$67$	6.47214	0.790697	0.395349	+	0.918531i	$0.370624\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0.206331	0.0244870	0.0122435	−	0.999925i	$-0.496103\pi$
$71$	0.206331	0.0244870	0.0122435	+	0.999925i	$0.496103\pi$
$72$	0	0
$73$	−12.4721	−1.45975	−0.729877	−	0.683579i	$-0.760422\pi$
$73$	−12.4721	−1.45975	−0.729877	+	0.683579i	$0.760422\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−4.57649	−0.521540
$78$	0	0
$79$	−12.9443	−1.45634	−0.728172	−	0.685394i	$-0.759630\pi$
$79$	−12.9443	−1.45634	−0.728172	+	0.685394i	$0.759630\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	16.7642	1.84011	0.920057	−	0.391785i	$-0.128142\pi$
$83$	16.7642	1.84011	0.920057	+	0.391785i	$0.128142\pi$
$84$	0	0
$85$	−3.23607	−0.351001
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−4.57649	−0.485107	−0.242554	−	0.970138i	$-0.577985\pi$
$89$	−4.57649	−0.485107	−0.242554	+	0.970138i	$0.577985\pi$
$90$	0	0
$91$	−5.23607	−0.548889
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−3.70246	−0.379864
$96$	0	0
$97$	9.41641	0.956091	0.478046	−	0.878335i	$-0.341345\pi$
$97$	9.41641	0.956091	0.478046	+	0.878335i	$0.341345\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−13.9358	−1.38666	−0.693332	−	0.720618i	$-0.743858\pi$
$101$	−13.9358	−1.38666	−0.693332	+	0.720618i	$0.743858\pi$
$102$	0	0
$103$	4.94427	0.487174	0.243587	−	0.969879i	$-0.421676\pi$
$103$	4.94427	0.487174	0.243587	+	0.969879i	$0.421676\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	13.5231	1.30733	0.653666	−	0.756783i	$-0.273230\pi$
$107$	13.5231	1.30733	0.653666	+	0.756783i	$0.273230\pi$
$108$	0	0
$109$	−10.9443	−1.04827	−0.524136	−	0.851635i	$-0.675611\pi$
$109$	−10.9443	−1.04827	−0.524136	+	0.851635i	$0.675611\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	8.69161	0.817638	0.408819	−	0.912615i	$-0.365941\pi$
$113$	8.69161	0.817638	0.408819	+	0.912615i	$0.365941\pi$
$114$	0	0
$115$	−4.00000	−0.373002
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−0.874032	−0.0801224
$120$	0	0
$121$	9.94427	0.904025
$122$	0	0
$123$	0	0
$124$	0	0
$125$	13.7295	1.22800
$126$	0	0
$127$	−18.4721	−1.63914	−0.819569	−	0.572981i	$-0.805787\pi$
$127$	−18.4721	−1.63914	−0.819569	+	0.572981i	$0.805787\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	9.35931	0.817727	0.408864	−	0.912596i	$-0.365925\pi$
$131$	9.35931	0.817727	0.408864	+	0.912596i	$0.365925\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−18.3060	−1.56398	−0.781992	−	0.623288i	$-0.785796\pi$
$137$	−18.3060	−1.56398	−0.781992	+	0.623288i	$0.785796\pi$
$138$	0	0
$139$	1.52786	0.129592	0.0647959	−	0.997899i	$-0.479360\pi$
$139$	1.52786	0.129592	0.0647959	+	0.997899i	$0.479360\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	23.9628	2.00387
$144$	0	0
$145$	−17.7082	−1.47059
$146$	0	0
$147$	0	0
$148$	0	0
$149$	7.40492	0.606635	0.303317	−	0.952890i	$-0.401906\pi$
$149$	7.40492	0.606635	0.303317	+	0.952890i	$0.401906\pi$
$150$	0	0
$151$	−12.9443	−1.05339	−0.526695	−	0.850054i	$-0.676569\pi$
$151$	−12.9443	−1.05339	−0.526695	+	0.850054i	$0.676569\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−26.7912	−2.15192
$156$	0	0
$157$	0.472136	0.0376806	0.0188403	−	0.999823i	$-0.494003\pi$
$157$	0.472136	0.0376806	0.0188403	+	0.999823i	$0.494003\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−1.08036	−0.0851445
$162$	0	0
$163$	0.763932	0.0598358	0.0299179	−	0.999552i	$-0.490475\pi$
$163$	0.763932	0.0598358	0.0299179	+	0.999552i	$0.490475\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−6.06952	−0.469673	−0.234837	−	0.972035i	$-0.575456\pi$
$167$	−6.06952	−0.469673	−0.234837	+	0.972035i	$0.575456\pi$
$168$	0	0
$169$	14.4164	1.10895
$170$	0	0
$171$	0	0
$172$	0	0
$173$	15.8902	1.20811	0.604055	−	0.796943i	$-0.293551\pi$
$173$	15.8902	1.20811	0.604055	+	0.796943i	$0.293551\pi$
$174$	0	0
$175$	8.70820	0.658278
$176$	0	0
$177$	0	0
$178$	0	0
$179$	23.0888	1.72574	0.862868	−	0.505429i	$-0.168666\pi$
$179$	23.0888	1.72574	0.862868	+	0.505429i	$0.168666\pi$
$180$	0	0
$181$	15.8885	1.18099	0.590493	−	0.807043i	$-0.298933\pi$
$181$	15.8885	1.18099	0.590493	+	0.807043i	$0.298933\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−22.2148	−1.63326
$186$	0	0
$187$	4.00000	0.292509
$188$	0	0
$189$	0	0
$190$	0	0
$191$	4.57649	0.331143	0.165572	−	0.986198i	$-0.447053\pi$
$191$	4.57649	0.331143	0.165572	+	0.986198i	$0.447053\pi$
$192$	0	0
$193$	−5.41641	−0.389882	−0.194941	−	0.980815i	$-0.562451\pi$
$193$	−5.41641	−0.389882	−0.194941	+	0.980815i	$0.562451\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	6.99226	0.498178	0.249089	−	0.968481i	$-0.419869\pi$
$197$	6.99226	0.498178	0.249089	+	0.968481i	$0.419869\pi$
$198$	0	0
$199$	−3.05573	−0.216615	−0.108307	−	0.994117i	$-0.534543\pi$
$199$	−3.05573	−0.216615	−0.108307	+	0.994117i	$0.534543\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−4.78282	−0.335688
$204$	0	0
$205$	−16.9443	−1.18344
$206$	0	0
$207$	0	0
$208$	0	0
$209$	4.57649	0.316563
$210$	0	0
$211$	−11.4164	−0.785938	−0.392969	−	0.919552i	$-0.628552\pi$
$211$	−11.4164	−0.785938	−0.392969	+	0.919552i	$0.628552\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	45.0972	3.07560
$216$	0	0
$217$	−7.23607	−0.491216
$218$	0	0
$219$	0	0
$220$	0	0
$221$	4.57649	0.307848
$222$	0	0
$223$	−0.763932	−0.0511567	−0.0255783	−	0.999673i	$-0.508143\pi$
$223$	−0.763932	−0.0511567	−0.0255783	+	0.999673i	$0.508143\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	16.5579	1.09899	0.549493	−	0.835498i	$-0.314821\pi$
$227$	16.5579	1.09899	0.549493	+	0.835498i	$0.314821\pi$
$228$	0	0
$229$	26.3607	1.74196	0.870981	−	0.491316i	$-0.163484\pi$
$229$	26.3607	1.74196	0.870981	+	0.491316i	$0.163484\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	20.8794	1.36785	0.683926	−	0.729551i	$-0.260271\pi$
$233$	20.8794	1.36785	0.683926	+	0.729551i	$0.260271\pi$
$234$	0	0
$235$	37.1246	2.42174
$236$	0	0
$237$	0	0
$238$	0	0
$239$	23.2951	1.50684	0.753418	−	0.657542i	$-0.228404\pi$
$239$	23.2951	1.50684	0.753418	+	0.657542i	$0.228404\pi$
$240$	0	0
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	3.70246	0.236541
$246$	0	0
$247$	5.23607	0.333163
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−9.61435	−0.606853	−0.303426	−	0.952855i	$-0.598131\pi$
$251$	−9.61435	−0.606853	−0.303426	+	0.952855i	$0.598131\pi$
$252$	0	0
$253$	4.94427	0.310844
$254$	0	0
$255$	0	0
$256$	0	0
$257$	21.1344	1.31833	0.659164	−	0.752000i	$-0.270910\pi$
$257$	21.1344	1.31833	0.659164	+	0.752000i	$0.270910\pi$
$258$	0	0
$259$	−6.00000	−0.372822
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−9.82068	−0.605569	−0.302785	−	0.953059i	$-0.597916\pi$
$263$	−9.82068	−0.605569	−0.302785	+	0.953059i	$0.597916\pi$
$264$	0	0
$265$	−34.6525	−2.12868
$266$	0	0
$267$	0	0
$268$	0	0
$269$	13.3168	0.811940	0.405970	−	0.913886i	$-0.366934\pi$
$269$	13.3168	0.811940	0.405970	+	0.913886i	$0.366934\pi$
$270$	0	0
$271$	−15.4164	−0.936480	−0.468240	−	0.883601i	$-0.655112\pi$
$271$	−15.4164	−0.936480	−0.468240	+	0.883601i	$0.655112\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−39.8530	−2.40323
$276$	0	0
$277$	−13.2361	−0.795278	−0.397639	−	0.917542i	$-0.630170\pi$
$277$	−13.2361	−0.795278	−0.397639	+	0.917542i	$0.630170\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	18.5123	1.10435	0.552175	−	0.833728i	$-0.313798\pi$
$281$	18.5123	1.10435	0.552175	+	0.833728i	$0.313798\pi$
$282$	0	0
$283$	−2.47214	−0.146953	−0.0734766	−	0.997297i	$-0.523409\pi$
$283$	−2.47214	−0.146953	−0.0734766	+	0.997297i	$0.523409\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−4.57649	−0.270142
$288$	0	0
$289$	−16.2361	−0.955063
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−2.41577	−0.141131	−0.0705653	−	0.997507i	$-0.522480\pi$
$293$	−2.41577	−0.141131	−0.0705653	+	0.997507i	$0.522480\pi$
$294$	0	0
$295$	−46.8328	−2.72671
$296$	0	0
$297$	0	0
$298$	0	0
$299$	5.65685	0.327144
$300$	0	0
$301$	12.1803	0.702063
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−40.5207	−2.32021
$306$	0	0
$307$	8.18034	0.466877	0.233438	−	0.972372i	$-0.425002\pi$
$307$	8.18034	0.466877	0.233438	+	0.972372i	$0.425002\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−27.2526	−1.54535	−0.772677	−	0.634799i	$-0.781083\pi$
$311$	−27.2526	−1.54535	−0.772677	+	0.634799i	$0.781083\pi$
$312$	0	0
$313$	−34.3607	−1.94218	−0.971090	−	0.238713i	$-0.923275\pi$
$313$	−34.3607	−1.94218	−0.971090	+	0.238713i	$0.923275\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−24.4242	−1.37180	−0.685900	−	0.727696i	$-0.740591\pi$
$317$	−24.4242	−1.37180	−0.685900	+	0.727696i	$0.740591\pi$
$318$	0	0
$319$	21.8885	1.22552
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0.874032	0.0486324
$324$	0	0
$325$	−45.5967	−2.52925
$326$	0	0
$327$	0	0
$328$	0	0
$329$	10.0270	0.552807
$330$	0	0
$331$	15.4164	0.847362	0.423681	−	0.905811i	$-0.360738\pi$
$331$	15.4164	0.847362	0.423681	+	0.905811i	$0.360738\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	23.9628	1.30923
$336$	0	0
$337$	−35.8885	−1.95497	−0.977487	−	0.210997i	$-0.932329\pi$
$337$	−35.8885	−1.95497	−0.977487	+	0.210997i	$0.932329\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	33.1158	1.79332
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	18.9737	1.01856	0.509280	−	0.860601i	$-0.329912\pi$
$347$	18.9737	1.01856	0.509280	+	0.860601i	$0.329912\pi$
$348$	0	0
$349$	−13.4164	−0.718164	−0.359082	−	0.933306i	$-0.616910\pi$
$349$	−13.4164	−0.718164	−0.359082	+	0.933306i	$0.616910\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−13.6808	−0.728154	−0.364077	−	0.931369i	$-0.618615\pi$
$353$	−13.6808	−0.728154	−0.364077	+	0.931369i	$0.618615\pi$
$354$	0	0
$355$	0.763932	0.0405453
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−13.7295	−0.724614	−0.362307	−	0.932059i	$-0.618011\pi$
$359$	−13.7295	−0.724614	−0.362307	+	0.932059i	$0.618011\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−46.1776	−2.41704
$366$	0	0
$367$	−24.3607	−1.27162	−0.635809	−	0.771847i	$-0.719333\pi$
$367$	−24.3607	−1.27162	−0.635809	+	0.771847i	$0.719333\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−9.35931	−0.485911
$372$	0	0
$373$	−22.9443	−1.18801	−0.594005	−	0.804462i	$-0.702454\pi$
$373$	−22.9443	−1.18801	−0.594005	+	0.804462i	$0.702454\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	25.0432	1.28979
$378$	0	0
$379$	−7.05573	−0.362428	−0.181214	−	0.983444i	$-0.558003\pi$
$379$	−7.05573	−0.362428	−0.181214	+	0.983444i	$0.558003\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−3.90879	−0.199730	−0.0998649	−	0.995001i	$-0.531841\pi$
$383$	−3.90879	−0.199730	−0.0998649	+	0.995001i	$0.531841\pi$
$384$	0	0
$385$	−16.9443	−0.863560
$386$	0	0
$387$	0	0
$388$	0	0
$389$	0.412662	0.0209228	0.0104614	−	0.999945i	$-0.496670\pi$
$389$	0.412662	0.0209228	0.0104614	+	0.999945i	$0.496670\pi$
$390$	0	0
$391$	0.944272	0.0477539
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−47.9256	−2.41140
$396$	0	0
$397$	31.8885	1.60044	0.800220	−	0.599706i	$-0.204716\pi$
$397$	31.8885	1.60044	0.800220	+	0.599706i	$0.204716\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	3.28980	0.164285	0.0821423	−	0.996621i	$-0.473824\pi$
$401$	3.28980	0.164285	0.0821423	+	0.996621i	$0.473824\pi$
$402$	0	0
$403$	37.8885	1.88736
$404$	0	0
$405$	0	0
$406$	0	0
$407$	27.4589	1.36109
$408$	0	0
$409$	11.7082	0.578933	0.289467	−	0.957188i	$-0.406522\pi$
$409$	11.7082	0.578933	0.289467	+	0.957188i	$0.406522\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−12.6491	−0.622422
$414$	0	0
$415$	62.0689	3.04684
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−1.69936	−0.0830190	−0.0415095	−	0.999138i	$-0.513217\pi$
$419$	−1.69936	−0.0830190	−0.0415095	+	0.999138i	$0.513217\pi$
$420$	0	0
$421$	30.3607	1.47969	0.739844	−	0.672778i	$-0.234899\pi$
$421$	30.3607	1.47969	0.739844	+	0.672778i	$0.234899\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−7.61125	−0.369200
$426$	0	0
$427$	−10.9443	−0.529630
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−21.7534	−1.04782	−0.523912	−	0.851772i	$-0.675528\pi$
$431$	−21.7534	−1.04782	−0.523912	+	0.851772i	$0.675528\pi$
$432$	0	0
$433$	−0.111456	−0.00535624	−0.00267812	−	0.999996i	$-0.500852\pi$
$433$	−0.111456	−0.00535624	−0.00267812	+	0.999996i	$0.500852\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.08036	0.0516808
$438$	0	0
$439$	38.0689	1.81693	0.908464	−	0.417962i	$-0.137256\pi$
$439$	38.0689	1.81693	0.908464	+	0.417962i	$0.137256\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−35.1189	−1.66855	−0.834275	−	0.551349i	$-0.814113\pi$
$443$	−35.1189	−1.66855	−0.834275	+	0.551349i	$0.814113\pi$
$444$	0	0
$445$	−16.9443	−0.803236
$446$	0	0
$447$	0	0
$448$	0	0
$449$	31.5741	1.49007	0.745036	−	0.667024i	$-0.232432\pi$
$449$	31.5741	1.49007	0.745036	+	0.667024i	$0.232432\pi$
$450$	0	0
$451$	20.9443	0.986227
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−19.3863	−0.908845
$456$	0	0
$457$	−26.7639	−1.25196	−0.625982	−	0.779838i	$-0.715302\pi$
$457$	−26.7639	−1.25196	−0.625982	+	0.779838i	$0.715302\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−40.0593	−1.86575	−0.932875	−	0.360200i	$-0.882708\pi$
$461$	−40.0593	−1.86575	−0.932875	+	0.360200i	$0.882708\pi$
$462$	0	0
$463$	7.05573	0.327907	0.163954	−	0.986468i	$-0.447575\pi$
$463$	7.05573	0.327907	0.163954	+	0.986468i	$0.447575\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−21.5958	−0.999333	−0.499666	−	0.866218i	$-0.666544\pi$
$467$	−21.5958	−0.999333	−0.499666	+	0.866218i	$0.666544\pi$
$468$	0	0
$469$	6.47214	0.298855
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−55.7432	−2.56308
$474$	0	0
$475$	−8.70820	−0.399560
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−24.8369	−1.13482	−0.567412	−	0.823434i	$-0.692055\pi$
$479$	−24.8369	−1.13482	−0.567412	+	0.823434i	$0.692055\pi$
$480$	0	0
$481$	31.4164	1.43246
$482$	0	0
$483$	0	0
$484$	0	0
$485$	34.8639	1.58309
$486$	0	0
$487$	−10.8328	−0.490882	−0.245441	−	0.969412i	$-0.578933\pi$
$487$	−10.8328	−0.490882	−0.245441	+	0.969412i	$0.578933\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	11.9814	0.540713	0.270357	−	0.962760i	$-0.412858\pi$
$491$	11.9814	0.540713	0.270357	+	0.962760i	$0.412858\pi$
$492$	0	0
$493$	4.18034	0.188273
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0.206331	0.00925521
$498$	0	0
$499$	11.0557	0.494922	0.247461	−	0.968898i	$-0.420404\pi$
$499$	11.0557	0.494922	0.247461	+	0.968898i	$0.420404\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	19.3376	0.862222	0.431111	−	0.902299i	$-0.358122\pi$
$503$	19.3376	0.862222	0.431111	+	0.902299i	$0.358122\pi$
$504$	0	0
$505$	−51.5967	−2.29603
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−21.1344	−0.936765	−0.468383	−	0.883526i	$-0.655163\pi$
$509$	−21.1344	−0.936765	−0.468383	+	0.883526i	$0.655163\pi$
$510$	0	0
$511$	−12.4721	−0.551735
$512$	0	0
$513$	0	0
$514$	0	0
$515$	18.3060	0.806657
$516$	0	0
$517$	−45.8885	−2.01818
$518$	0	0
$519$	0	0
$520$	0	0
$521$	16.3029	0.714241	0.357121	−	0.934058i	$-0.383758\pi$
$521$	16.3029	0.714241	0.357121	+	0.934058i	$0.383758\pi$
$522$	0	0
$523$	14.2918	0.624937	0.312468	−	0.949928i	$-0.398844\pi$
$523$	14.2918	0.624937	0.312468	+	0.949928i	$0.398844\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	6.32456	0.275502
$528$	0	0
$529$	−21.8328	−0.949253
$530$	0	0
$531$	0	0
$532$	0	0
$533$	23.9628	1.03794
$534$	0	0
$535$	50.0689	2.16467
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−4.57649	−0.197123
$540$	0	0
$541$	−13.4164	−0.576816	−0.288408	−	0.957508i	$-0.593126\pi$
$541$	−13.4164	−0.576816	−0.288408	+	0.957508i	$0.593126\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−40.5207	−1.73572
$546$	0	0
$547$	−24.9443	−1.06654	−0.533270	−	0.845945i	$-0.679037\pi$
$547$	−24.9443	−1.06654	−0.533270	+	0.845945i	$0.679037\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	4.78282	0.203755
$552$	0	0
$553$	−12.9443	−0.550446
$554$	0	0
$555$	0	0
$556$	0	0
$557$	12.2364	0.518475	0.259237	−	0.965814i	$-0.416529\pi$
$557$	12.2364	0.518475	0.259237	+	0.965814i	$0.416529\pi$
$558$	0	0
$559$	−63.7771	−2.69748
$560$	0	0
$561$	0	0
$562$	0	0
$563$	17.8933	0.754113	0.377056	−	0.926190i	$-0.376936\pi$
$563$	17.8933	0.754113	0.377056	+	0.926190i	$0.376936\pi$
$564$	0	0
$565$	32.1803	1.35384
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−15.4288	−0.646810	−0.323405	−	0.946261i	$-0.604828\pi$
$569$	−15.4288	−0.646810	−0.323405	+	0.946261i	$0.604828\pi$
$570$	0	0
$571$	15.0557	0.630063	0.315031	−	0.949081i	$-0.397985\pi$
$571$	15.0557	0.630063	0.315031	+	0.949081i	$0.397985\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−9.40802	−0.392342
$576$	0	0
$577$	14.3607	0.597843	0.298921	−	0.954278i	$-0.403373\pi$
$577$	14.3607	0.597843	0.298921	+	0.954278i	$0.403373\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	16.7642	0.695498
$582$	0	0
$583$	42.8328	1.77395
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−3.44742	−0.142290	−0.0711451	−	0.997466i	$-0.522665\pi$
$587$	−3.44742	−0.142290	−0.0711451	+	0.997466i	$0.522665\pi$
$588$	0	0
$589$	7.23607	0.298157
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−25.9172	−1.06429	−0.532146	−	0.846652i	$-0.678614\pi$
$593$	−25.9172	−1.06429	−0.532146	+	0.846652i	$0.678614\pi$
$594$	0	0
$595$	−3.23607	−0.132666
$596$	0	0
$597$	0	0
$598$	0	0
$599$	19.5927	0.800534	0.400267	−	0.916398i	$-0.368917\pi$
$599$	19.5927	0.800534	0.400267	+	0.916398i	$0.368917\pi$
$600$	0	0
$601$	13.4164	0.547267	0.273633	−	0.961834i	$-0.411775\pi$
$601$	13.4164	0.547267	0.273633	+	0.961834i	$0.411775\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	36.8183	1.49688
$606$	0	0
$607$	−29.8885	−1.21314	−0.606569	−	0.795031i	$-0.707455\pi$
$607$	−29.8885	−1.21314	−0.606569	+	0.795031i	$0.707455\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−52.5021	−2.12401
$612$	0	0
$613$	37.5967	1.51852	0.759259	−	0.650788i	$-0.225561\pi$
$613$	37.5967	1.51852	0.759259	+	0.650788i	$0.225561\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	42.2688	1.70168	0.850839	−	0.525427i	$-0.176094\pi$
$617$	42.2688	1.70168	0.850839	+	0.525427i	$0.176094\pi$
$618$	0	0
$619$	−1.52786	−0.0614100	−0.0307050	−	0.999528i	$-0.509775\pi$
$619$	−1.52786	−0.0614100	−0.0307050	+	0.999528i	$0.509775\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−4.57649	−0.183353
$624$	0	0
$625$	7.29180	0.291672
$626$	0	0
$627$	0	0
$628$	0	0
$629$	5.24419	0.209100
$630$	0	0
$631$	24.1803	0.962604	0.481302	−	0.876555i	$-0.340164\pi$
$631$	24.1803	0.962604	0.481302	+	0.876555i	$0.340164\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−68.3923	−2.71407
$636$	0	0
$637$	−5.23607	−0.207461
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−3.03476	−0.119866	−0.0599329	−	0.998202i	$-0.519089\pi$
$641$	−3.03476	−0.119866	−0.0599329	+	0.998202i	$0.519089\pi$
$642$	0	0
$643$	13.5279	0.533487	0.266743	−	0.963768i	$-0.414052\pi$
$643$	13.5279	0.533487	0.266743	+	0.963768i	$0.414052\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	36.9759	1.45367	0.726836	−	0.686811i	$-0.240990\pi$
$647$	36.9759	1.45367	0.726836	+	0.686811i	$0.240990\pi$
$648$	0	0
$649$	57.8885	2.27232
$650$	0	0
$651$	0	0
$652$	0	0
$653$	12.6491	0.494998	0.247499	−	0.968888i	$-0.420391\pi$
$653$	12.6491	0.494998	0.247499	+	0.968888i	$0.420391\pi$
$654$	0	0
$655$	34.6525	1.35398
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−22.0084	−0.857326	−0.428663	−	0.903464i	$-0.641015\pi$
$659$	−22.0084	−0.857326	−0.428663	+	0.903464i	$0.641015\pi$
$660$	0	0
$661$	34.1803	1.32946	0.664731	−	0.747083i	$-0.268546\pi$
$661$	34.1803	1.32946	0.664731	+	0.747083i	$0.268546\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−3.70246	−0.143575
$666$	0	0
$667$	5.16718	0.200074
$668$	0	0
$669$	0	0
$670$	0	0
$671$	50.0864	1.93356
$672$	0	0
$673$	19.3050	0.744151	0.372076	−	0.928202i	$-0.378646\pi$
$673$	19.3050	0.744151	0.372076	+	0.928202i	$0.378646\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	33.2734	1.27880	0.639401	−	0.768874i	$-0.279182\pi$
$677$	33.2734	1.27880	0.639401	+	0.768874i	$0.279182\pi$
$678$	0	0
$679$	9.41641	0.361369
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0.461370	0.0176538	0.00882692	−	0.999961i	$-0.497190\pi$
$683$	0.461370	0.0176538	0.00882692	+	0.999961i	$0.497190\pi$
$684$	0	0
$685$	−67.7771	−2.58963
$686$	0	0
$687$	0	0
$688$	0	0
$689$	49.0060	1.86698
$690$	0	0
$691$	−50.2492	−1.91157	−0.955785	−	0.294065i	$-0.904992\pi$
$691$	−50.2492	−1.91157	−0.955785	+	0.294065i	$0.904992\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	5.65685	0.214577
$696$	0	0
$697$	4.00000	0.151511
$698$	0	0
$699$	0	0
$700$	0	0
$701$	6.99226	0.264094	0.132047	−	0.991243i	$-0.457845\pi$
$701$	6.99226	0.264094	0.132047	+	0.991243i	$0.457845\pi$
$702$	0	0
$703$	6.00000	0.226294
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−13.9358	−0.524110
$708$	0	0
$709$	4.29180	0.161182	0.0805909	−	0.996747i	$-0.474319\pi$
$709$	4.29180	0.161182	0.0805909	+	0.996747i	$0.474319\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	7.81758	0.292771
$714$	0	0
$715$	88.7214	3.31799
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−1.03165	−0.0384742	−0.0192371	−	0.999815i	$-0.506124\pi$
$719$	−1.03165	−0.0384742	−0.0192371	+	0.999815i	$0.506124\pi$
$720$	0	0
$721$	4.94427	0.184134
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−41.6498	−1.54683
$726$	0	0
$727$	−2.11146	−0.0783096	−0.0391548	−	0.999233i	$-0.512467\pi$
$727$	−2.11146	−0.0783096	−0.0391548	+	0.999233i	$0.512467\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−10.6460	−0.393757
$732$	0	0
$733$	10.5836	0.390914	0.195457	−	0.980712i	$-0.437381\pi$
$733$	10.5836	0.390914	0.195457	+	0.980712i	$0.437381\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−29.6197	−1.09105
$738$	0	0
$739$	−22.0689	−0.811817	−0.405909	−	0.913914i	$-0.633045\pi$
$739$	−22.0689	−0.811817	−0.405909	+	0.913914i	$0.633045\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−25.2495	−0.926315	−0.463157	−	0.886276i	$-0.653284\pi$
$743$	−25.2495	−0.926315	−0.463157	+	0.886276i	$0.653284\pi$
$744$	0	0
$745$	27.4164	1.00446
$746$	0	0
$747$	0	0
$748$	0	0
$749$	13.5231	0.494125
$750$	0	0
$751$	3.05573	0.111505	0.0557526	−	0.998445i	$-0.482244\pi$
$751$	3.05573	0.111505	0.0557526	+	0.998445i	$0.482244\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−47.9256	−1.74419
$756$	0	0
$757$	−14.5836	−0.530050	−0.265025	−	0.964242i	$-0.585380\pi$
$757$	−14.5836	−0.530050	−0.265025	+	0.964242i	$0.585380\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−15.9389	−0.577785	−0.288892	−	0.957362i	$-0.593287\pi$
$761$	−15.9389	−0.577785	−0.288892	+	0.957362i	$0.593287\pi$
$762$	0	0
$763$	−10.9443	−0.396209
$764$	0	0
$765$	0	0
$766$	0	0
$767$	66.2316	2.39148
$768$	0	0
$769$	19.3050	0.696154	0.348077	−	0.937466i	$-0.386835\pi$
$769$	19.3050	0.696154	0.348077	+	0.937466i	$0.386835\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	26.3786	0.948772	0.474386	−	0.880317i	$-0.342670\pi$
$773$	26.3786	0.948772	0.474386	+	0.880317i	$0.342670\pi$
$774$	0	0
$775$	−63.0132	−2.26350
$776$	0	0
$777$	0	0
$778$	0	0
$779$	4.57649	0.163970
$780$	0	0
$781$	−0.944272	−0.0337887
$782$	0	0
$783$	0	0
$784$	0	0
$785$	1.74806	0.0623911
$786$	0	0
$787$	−19.2361	−0.685692	−0.342846	−	0.939392i	$-0.611391\pi$
$787$	−19.2361	−0.685692	−0.342846	+	0.939392i	$0.611391\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	8.69161	0.309038
$792$	0	0
$793$	57.3050	2.03496
$794$	0	0
$795$	0	0
$796$	0	0
$797$	5.40182	0.191342	0.0956711	−	0.995413i	$-0.469500\pi$
$797$	5.40182	0.191342	0.0956711	+	0.995413i	$0.469500\pi$
$798$	0	0
$799$	−8.76393	−0.310046
$800$	0	0
$801$	0	0
$802$	0	0
$803$	57.0786	2.01426
$804$	0	0
$805$	−4.00000	−0.140981
$806$	0	0
$807$	0	0
$808$	0	0
$809$	37.4373	1.31622	0.658112	−	0.752920i	$-0.271355\pi$
$809$	37.4373	1.31622	0.658112	+	0.752920i	$0.271355\pi$
$810$	0	0
$811$	−0.944272	−0.0331579	−0.0165789	−	0.999863i	$-0.505277\pi$
$811$	−0.944272	−0.0331579	−0.0165789	+	0.999863i	$0.505277\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	2.82843	0.0990755
$816$	0	0
$817$	−12.1803	−0.426136
$818$	0	0
$819$	0	0
$820$	0	0
$821$	0.922740	0.0322039	0.0161019	−	0.999870i	$-0.494874\pi$
$821$	0.922740	0.0322039	0.0161019	+	0.999870i	$0.494874\pi$
$822$	0	0
$823$	25.3475	0.883559	0.441780	−	0.897124i	$-0.354347\pi$
$823$	25.3475	0.883559	0.441780	+	0.897124i	$0.354347\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−6.27585	−0.218233	−0.109116	−	0.994029i	$-0.534802\pi$
$827$	−6.27585	−0.218233	−0.109116	+	0.994029i	$0.534802\pi$
$828$	0	0
$829$	−2.58359	−0.0897319	−0.0448659	−	0.998993i	$-0.514286\pi$
$829$	−2.58359	−0.0897319	−0.0448659	+	0.998993i	$0.514286\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−0.874032	−0.0302834
$834$	0	0
$835$	−22.4721	−0.777680
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−36.6119	−1.26398	−0.631992	−	0.774975i	$-0.717762\pi$
$839$	−36.6119	−1.26398	−0.631992	+	0.774975i	$0.717762\pi$
$840$	0	0
$841$	−6.12461	−0.211194
$842$	0	0
$843$	0	0
$844$	0	0
$845$	53.3762	1.83620
$846$	0	0
$847$	9.94427	0.341689
$848$	0	0
$849$	0	0
$850$	0	0
$851$	6.48218	0.222206
$852$	0	0
$853$	15.5279	0.531664	0.265832	−	0.964019i	$-0.414353\pi$
$853$	15.5279	0.531664	0.265832	+	0.964019i	$0.414353\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−46.8453	−1.60020	−0.800102	−	0.599864i	$-0.795221\pi$
$857$	−46.8453	−1.60020	−0.800102	+	0.599864i	$0.795221\pi$
$858$	0	0
$859$	8.58359	0.292868	0.146434	−	0.989220i	$-0.453220\pi$
$859$	8.58359	0.292868	0.146434	+	0.989220i	$0.453220\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	46.2263	1.57356	0.786780	−	0.617233i	$-0.211746\pi$
$863$	46.2263	1.57356	0.786780	+	0.617233i	$0.211746\pi$
$864$	0	0
$865$	58.8328	2.00038
$866$	0	0
$867$	0	0
$868$	0	0
$869$	59.2393	2.00956
$870$	0	0
$871$	−33.8885	−1.14827
$872$	0	0
$873$	0	0
$874$	0	0
$875$	13.7295	0.464141
$876$	0	0
$877$	−24.8328	−0.838545	−0.419272	−	0.907861i	$-0.637715\pi$
$877$	−24.8328	−0.838545	−0.419272	+	0.907861i	$0.637715\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−58.1103	−1.95778	−0.978892	−	0.204376i	$-0.934483\pi$
$881$	−58.1103	−1.95778	−0.978892	+	0.204376i	$0.934483\pi$
$882$	0	0
$883$	24.0000	0.807664	0.403832	−	0.914833i	$-0.367678\pi$
$883$	24.0000	0.807664	0.403832	+	0.914833i	$0.367678\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−20.0540	−0.673348	−0.336674	−	0.941621i	$-0.609302\pi$
$887$	−20.0540	−0.673348	−0.336674	+	0.941621i	$0.609302\pi$
$888$	0	0
$889$	−18.4721	−0.619536
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−10.0270	−0.335541
$894$	0	0
$895$	85.4853	2.85746
$896$	0	0
$897$	0	0
$898$	0	0
$899$	34.6088	1.15427
$900$	0	0
$901$	8.18034	0.272527
$902$	0	0
$903$	0	0
$904$	0	0
$905$	58.8267	1.95547
$906$	0	0
$907$	3.05573	0.101464	0.0507319	−	0.998712i	$-0.483845\pi$
$907$	3.05573	0.101464	0.0507319	+	0.998712i	$0.483845\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	8.12133	0.269072	0.134536	−	0.990909i	$-0.457046\pi$
$911$	8.12133	0.269072	0.134536	+	0.990909i	$0.457046\pi$
$912$	0	0
$913$	−76.7214	−2.53911
$914$	0	0
$915$	0	0
$916$	0	0
$917$	9.35931	0.309072
$918$	0	0
$919$	20.9443	0.690888	0.345444	−	0.938439i	$-0.387728\pi$
$919$	20.9443	0.690888	0.345444	+	0.938439i	$0.387728\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−1.08036	−0.0355606
$924$	0	0
$925$	−52.2492	−1.71794
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−50.1351	−1.64488	−0.822439	−	0.568853i	$-0.807387\pi$
$929$	−50.1351	−1.64488	−0.822439	+	0.568853i	$0.807387\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	0	0
$933$	0	0
$934$	0	0
$935$	14.8098	0.484333
$936$	0	0
$937$	−36.8328	−1.20328	−0.601638	−	0.798769i	$-0.705485\pi$
$937$	−36.8328	−1.20328	−0.601638	+	0.798769i	$0.705485\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	3.65375	0.119109	0.0595544	−	0.998225i	$-0.481032\pi$
$941$	3.65375	0.119109	0.0595544	+	0.998225i	$0.481032\pi$
$942$	0	0
$943$	4.94427	0.161008
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−12.3941	−0.402753	−0.201377	−	0.979514i	$-0.564542\pi$
$947$	−12.3941	−0.402753	−0.201377	+	0.979514i	$0.564542\pi$
$948$	0	0
$949$	65.3050	2.11989
$950$	0	0
$951$	0	0
$952$	0	0
$953$	24.5818	0.796283	0.398141	−	0.917324i	$-0.369655\pi$
$953$	24.5818	0.796283	0.398141	+	0.917324i	$0.369655\pi$
$954$	0	0
$955$	16.9443	0.548304
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−18.3060	−0.591130
$960$	0	0
$961$	21.3607	0.689054
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−20.0540	−0.645562
$966$	0	0
$967$	−19.2361	−0.618590	−0.309295	−	0.950966i	$-0.600093\pi$
$967$	−19.2361	−0.618590	−0.309295	+	0.950966i	$0.600093\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	5.65685	0.181537	0.0907685	−	0.995872i	$-0.471068\pi$
$971$	5.65685	0.181537	0.0907685	+	0.995872i	$0.471068\pi$
$972$	0	0
$973$	1.52786	0.0489811
$974$	0	0
$975$	0	0
$976$	0	0
$977$	7.61125	0.243505	0.121753	−	0.992560i	$-0.461149\pi$
$977$	7.61125	0.243505	0.121753	+	0.992560i	$0.461149\pi$
$978$	0	0
$979$	20.9443	0.669382
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−32.6057	−1.03996	−0.519981	−	0.854178i	$-0.674061\pi$
$983$	−32.6057	−1.03996	−0.519981	+	0.854178i	$0.674061\pi$
$984$	0	0
$985$	25.8885	0.824878
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−13.1592	−0.418438
$990$	0	0
$991$	36.0000	1.14358	0.571789	−	0.820401i	$-0.306250\pi$
$991$	36.0000	1.14358	0.571789	+	0.820401i	$0.306250\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−11.3137	−0.358669
$996$	0	0
$997$	−36.4721	−1.15508	−0.577542	−	0.816361i	$-0.695988\pi$
$997$	−36.4721	−1.15508	−0.577542	+	0.816361i	$0.695988\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4788.2.a.r.1.4	yes	4
3.2	odd	2	inner	4788.2.a.r.1.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4788.2.a.r.1.1	✓	4	3.2	odd	2	inner
4788.2.a.r.1.4	yes	4	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:	\( N \)	\(=\)	\( 4788 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4788.a (trivial)

\(f(q)\)	\(=\)	\(q+3.70246 q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q+3.70246 q^{5} +1.00000 q^{7} -4.57649 q^{11} -5.23607 q^{13} -0.874032 q^{17} -1.00000 q^{19} -1.08036 q^{23} +8.70820 q^{25} -4.78282 q^{29} -7.23607 q^{31} +3.70246 q^{35} -6.00000 q^{37} -4.57649 q^{41} +12.1803 q^{43} +10.0270 q^{47} +1.00000 q^{49} -9.35931 q^{53} -16.9443 q^{55} -12.6491 q^{59} -10.9443 q^{61} -19.3863 q^{65} +6.47214 q^{67} +0.206331 q^{71} -12.4721 q^{73} -4.57649 q^{77} -12.9443 q^{79} +16.7642 q^{83} -3.23607 q^{85} -4.57649 q^{89} -5.23607 q^{91} -3.70246 q^{95} +9.41641 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{7}+O(q^{10}) \) 4 * q + 4 * q^7	\( 4 q + 4 q^{7} - 12 q^{13} - 4 q^{19} + 8 q^{25} - 20 q^{31} - 24 q^{37} + 4 q^{43} + 4 q^{49} - 32 q^{55} - 8 q^{61} + 8 q^{67} - 32 q^{73} - 16 q^{79} - 4 q^{85} - 12 q^{91} - 16 q^{97}+O(q^{100}) \) 4 * q + 4 * q^7 - 12 * q^13 - 4 * q^19 + 8 * q^25 - 20 * q^31 - 24 * q^37 + 4 * q^43 + 4 * q^49 - 32 * q^55 - 8 * q^61 + 8 * q^67 - 32 * q^73 - 16 * q^79 - 4 * q^85 - 12 * q^91 - 16 * q^97

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