LMFDB - Embedded newform 1216.2.c.i.609.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1216,2,Mod(609,1216)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1216.609"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1216, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1216 = 2^{6} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1216.c (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,24,0,0,0,0,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$9.70980888579$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.2702336256.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 9x^{6} + 56x^{4} + 225x^{2} + 625 $ x^8 + 9x^6 + 56x^4 + 225*x^2 + 625
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			609.1
Root			$-0.656712 - 2.13746i$ of defining polynomial
Character	$\chi$	$=$	1216.609
Dual form			1216.2.c.i.609.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.04547i q^{5} -0.418627 q^{7} +3.00000 q^{9} -5.27492i q^{11} +2.62685i q^{13} +3.27492 q^{17} -1.00000i q^{19} -3.46410 q^{23} -4.27492 q^{25} +2.62685i q^{29} -6.09095 q^{31} +1.27492i q^{35} -9.55505i q^{37} +4.54983 q^{41} -2.72508i q^{43} -9.13642i q^{45} +0.418627 q^{47} -6.82475 q^{49} -10.3923i q^{53} -16.0646 q^{55} +6.54983i q^{59} -3.04547i q^{61} -1.25588 q^{63} +8.00000 q^{65} +6.54983i q^{67} +11.3446 q^{71} -3.27492 q^{73} +2.20822i q^{77} -13.0192 q^{79} +9.00000 q^{81} -17.0997i q^{83} -9.97368i q^{85} -10.0000 q^{89} -1.09967i q^{91} -3.04547 q^{95} +16.5498 q^{97} -15.8248i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 24 q^{9} - 4 q^{17} - 4 q^{25} - 24 q^{41} + 36 q^{49} + 64 q^{65} + 4 q^{73} + 72 q^{81} - 80 q^{89} + 72 q^{97}+O(q^{100}) $ 8 * q + 24 * q^9 - 4 * q^17 - 4 * q^25 - 24 * q^41 + 36 * q^49 + 64 * q^65 + 4 * q^73 + 72 * q^81 - 80 * q^89 + 72 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times$.

$n$	$191$	$705$	$837$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0	1.00000			$0$
$3$	0	0	−1.00000			$\pi$
$4$	0	0
$5$	− 3.04547i	− 1.36198i	−0.732294	−	0.680989i	$-0.761550\pi$
$5$	− 3.04547i	− 1.36198i	0.732294	−	0.680989i	$-0.238450\pi$
$6$	0	0
$7$	−0.418627	−0.158226	−0.0791130	−	0.996866i	$-0.525209\pi$
$7$	−0.418627	−0.158226	−0.0791130	+	0.996866i	$0.525209\pi$
$8$	0	0
$9$	3.00000	1.00000
$10$	0	0
$11$	− 5.27492i	− 1.59045i	−0.606316	−	0.795224i	$-0.707353\pi$
$11$	− 5.27492i	− 1.59045i	0.606316	−	0.795224i	$-0.292647\pi$
$12$	0	0
$13$	2.62685i	0.728557i	0.931290	+	0.364278i	$0.118684\pi$
$13$	2.62685i	0.728557i	−0.931290	+	0.364278i	$0.881316\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.27492	0.794284	0.397142	−	0.917757i	$-0.370002\pi$
$17$	3.27492	0.794284	0.397142	+	0.917757i	$0.370002\pi$
$18$	0	0
$19$	− 1.00000i	− 0.229416i
$19$	− 1.00000i	− 0.229416i
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3.46410	−0.722315	−0.361158	−	0.932505i	$-0.617618\pi$
$23$	−3.46410	−0.722315	−0.361158	+	0.932505i	$0.617618\pi$
$24$	0	0
$25$	−4.27492	−0.854983
$26$	0	0
$27$	0	0
$28$	0	0
$29$	2.62685i	0.487793i	0.969801	+	0.243897i	$0.0784258\pi$
$29$	2.62685i	0.487793i	−0.969801	+	0.243897i	$0.921574\pi$
$30$	0	0
$31$	−6.09095	−1.09397	−0.546983	−	0.837143i	$-0.684224\pi$
$31$	−6.09095	−1.09397	−0.546983	+	0.837143i	$0.684224\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.27492i	0.215500i
$36$	0	0
$37$	− 9.55505i	− 1.57084i	−0.618963	−	0.785420i	$-0.712447\pi$
$37$	− 9.55505i	− 1.57084i	0.618963	−	0.785420i	$-0.287553\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	4.54983	0.710565	0.355282	−	0.934759i	$-0.384385\pi$
$41$	4.54983	0.710565	0.355282	+	0.934759i	$0.384385\pi$
$42$	0	0
$43$	− 2.72508i	− 0.415571i	−0.978174	−	0.207786i	$-0.933374\pi$
$43$	− 2.72508i	− 0.415571i	0.978174	−	0.207786i	$-0.0666256\pi$
$44$	0	0
$45$	− 9.13642i	− 1.36198i
$46$	0	0
$47$	0.418627	0.0610630	0.0305315	−	0.999534i	$-0.490280\pi$
$47$	0.418627	0.0610630	0.0305315	+	0.999534i	$0.490280\pi$
$48$	0	0
$49$	−6.82475	−0.974965
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 10.3923i	− 1.42749i	−0.700404	−	0.713746i	$-0.746997\pi$
$53$	− 10.3923i	− 1.42749i	0.700404	−	0.713746i	$-0.253003\pi$
$54$	0	0
$55$	−16.0646	−2.16615
$56$	0	0
$57$	0	0
$58$	0	0
$59$	6.54983i	0.852716i	0.904555	+	0.426358i	$0.140204\pi$
$59$	6.54983i	0.852716i	−0.904555	+	0.426358i	$0.859796\pi$
$60$	0	0
$61$	− 3.04547i	− 0.389933i	−0.980810	−	0.194967i	$-0.937540\pi$
$61$	− 3.04547i	− 0.389933i	0.980810	−	0.194967i	$-0.0624598\pi$
$62$	0	0
$63$	−1.25588	−0.158226
$64$	0	0
$65$	8.00000	0.992278
$66$	0	0
$67$	6.54983i	0.800190i	0.916474	+	0.400095i	$0.131023\pi$
$67$	6.54983i	0.800190i	−0.916474	+	0.400095i	$0.868977\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	11.3446	1.34636	0.673181	−	0.739478i	$-0.264928\pi$
$71$	11.3446	1.34636	0.673181	+	0.739478i	$0.264928\pi$
$72$	0	0
$73$	−3.27492	−0.383300	−0.191650	−	0.981463i	$-0.561384\pi$
$73$	−3.27492	−0.383300	−0.191650	+	0.981463i	$0.561384\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.20822i	0.251650i
$78$	0	0
$79$	−13.0192	−1.46477	−0.732385	−	0.680891i	$-0.761593\pi$
$79$	−13.0192	−1.46477	−0.732385	+	0.680891i	$0.761593\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	− 17.0997i	− 1.87693i	−0.345371	−	0.938466i	$-0.612247\pi$
$83$	− 17.0997i	− 1.87693i	0.345371	−	0.938466i	$-0.387753\pi$
$84$	0	0
$85$	− 9.97368i	− 1.08180i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	− 1.09967i	− 0.115277i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−3.04547	−0.312459
$96$	0	0
$97$	16.5498	1.68038	0.840191	−	0.542291i	$-0.182443\pi$
$97$	16.5498	1.68038	0.840191	+	0.542291i	$0.182443\pi$
$98$	0	0
$99$	− 15.8248i	− 1.59045i
$100$	0	0
$101$	13.8564i	1.37876i	0.724398	+	0.689382i	$0.242118\pi$
$101$	13.8564i	1.37876i	−0.724398	+	0.689382i	$0.757882\pi$
$102$	0	0
$103$	−6.09095	−0.600159	−0.300080	−	0.953914i	$-0.597013\pi$
$103$	−6.09095	−0.600159	−0.300080	+	0.953914i	$0.597013\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 1.45017i	− 0.140193i	−0.997540	−	0.0700964i	$-0.977669\pi$
$107$	− 1.45017i	− 0.140193i	0.997540	−	0.0700964i	$-0.0223307\pi$
$108$	0	0
$109$	3.46410i	0.331801i	0.986143	+	0.165900i	$0.0530530\pi$
$109$	3.46410i	0.331801i	−0.986143	+	0.165900i	$0.946947\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	12.5498	1.18059	0.590295	−	0.807188i	$-0.299012\pi$
$113$	12.5498	1.18059	0.590295	+	0.807188i	$0.299012\pi$
$114$	0	0
$115$	10.5498i	0.983777i
$116$	0	0
$117$	7.88054i	0.728557i
$118$	0	0
$119$	−1.37097	−0.125676
$120$	0	0
$121$	−16.8248	−1.52952
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 2.20822i	− 0.197509i
$126$	0	0
$127$	17.4356	1.54716	0.773579	−	0.633699i	$-0.218464\pi$
$127$	17.4356	1.54716	0.773579	+	0.633699i	$0.218464\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 5.27492i	− 0.460872i	−0.973088	−	0.230436i	$-0.925985\pi$
$131$	− 5.27492i	− 0.460872i	0.973088	−	0.230436i	$-0.0740152\pi$
$132$	0	0
$133$	0.418627i	0.0362995i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−12.7251	−1.08718	−0.543589	−	0.839352i	$-0.682935\pi$
$137$	−12.7251	−1.08718	−0.543589	+	0.839352i	$0.682935\pi$
$138$	0	0
$139$	− 2.72508i	− 0.231139i	−0.993299	−	0.115569i	$-0.963131\pi$
$139$	− 2.72508i	− 0.231139i	0.993299	−	0.115569i	$-0.0368692\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	13.8564	1.15873
$144$	0	0
$145$	8.00000	0.664364
$146$	0	0
$147$	0	0
$148$	0	0
$149$	16.0646i	1.31607i	0.752989	+	0.658033i	$0.228611\pi$
$149$	16.0646i	1.31607i	−0.752989	+	0.658033i	$0.771389\pi$
$150$	0	0
$151$	0.837253	0.0681347	0.0340674	−	0.999420i	$-0.489154\pi$
$151$	0.837253	0.0681347	0.0340674	+	0.999420i	$0.489154\pi$
$152$	0	0
$153$	9.82475	0.794284
$154$	0	0
$155$	18.5498i	1.48996i
$156$	0	0
$157$	− 12.1819i	− 0.972221i	−0.873897	−	0.486111i	$-0.838415\pi$
$157$	− 12.1819i	− 0.972221i	0.873897	−	0.486111i	$-0.161585\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1.45017	0.114289
$162$	0	0
$163$	− 4.00000i	− 0.313304i	−0.987654	−	0.156652i	$-0.949930\pi$
$163$	− 4.00000i	− 0.313304i	0.987654	−	0.156652i	$-0.0500701\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	19.1101	1.47878	0.739392	−	0.673275i	$-0.235113\pi$
$167$	19.1101	1.47878	0.739392	+	0.673275i	$0.235113\pi$
$168$	0	0
$169$	6.09967	0.469205
$170$	0	0
$171$	− 3.00000i	− 0.229416i
$172$	0	0
$173$	8.71780i	0.662802i	0.943490	+	0.331401i	$0.107521\pi$
$173$	8.71780i	0.662802i	−0.943490	+	0.331401i	$0.892479\pi$
$174$	0	0
$175$	1.78959	0.135281
$176$	0	0
$177$	0	0
$178$	0	0
$179$	21.0997i	1.57706i	0.614994	+	0.788532i	$0.289158\pi$
$179$	21.0997i	1.57706i	−0.614994	+	0.788532i	$0.710842\pi$
$180$	0	0
$181$	− 3.46410i	− 0.257485i	−0.991678	−	0.128742i	$-0.958906\pi$
$181$	− 3.46410i	− 0.257485i	0.991678	−	0.128742i	$-0.0410940\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−29.0997	−2.13945
$186$	0	0
$187$	− 17.2749i	− 1.26327i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−11.7633	−0.851161	−0.425580	−	0.904921i	$-0.639930\pi$
$191$	−11.7633	−0.851161	−0.425580	+	0.904921i	$0.639930\pi$
$192$	0	0
$193$	11.0997	0.798972	0.399486	−	0.916739i	$-0.369189\pi$
$193$	11.0997	0.798972	0.399486	+	0.916739i	$0.369189\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 5.25370i	− 0.374310i	−0.982330	−	0.187155i	$-0.940073\pi$
$197$	− 5.25370i	− 0.374310i	0.982330	−	0.187155i	$-0.0599267\pi$
$198$	0	0
$199$	14.2750	1.01193	0.505965	−	0.862554i	$-0.331136\pi$
$199$	14.2750	1.01193	0.505965	+	0.862554i	$0.331136\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 1.09967i	− 0.0771816i
$204$	0	0
$205$	− 13.8564i	− 0.967773i
$206$	0	0
$207$	−10.3923	−0.722315
$208$	0	0
$209$	−5.27492	−0.364874
$210$	0	0
$211$	26.5498i	1.82777i	0.405978	+	0.913883i	$0.366931\pi$
$211$	26.5498i	1.82777i	−0.405978	+	0.913883i	$0.633069\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−8.29917	−0.565999
$216$	0	0
$217$	2.54983	0.173094
$218$	0	0
$219$	0	0
$220$	0	0
$221$	8.60271i	0.578681i
$222$	0	0
$223$	−17.4356	−1.16757	−0.583787	−	0.811907i	$-0.698430\pi$
$223$	−17.4356	−1.16757	−0.583787	+	0.811907i	$0.698430\pi$
$224$	0	0
$225$	−12.8248	−0.854983
$226$	0	0
$227$	− 4.00000i	− 0.265489i	−0.991150	−	0.132745i	$-0.957621\pi$
$227$	− 4.00000i	− 0.265489i	0.991150	−	0.132745i	$-0.0423790\pi$
$228$	0	0
$229$	17.7391i	1.17224i	0.810226	+	0.586118i	$0.199344\pi$
$229$	17.7391i	1.17224i	−0.810226	+	0.586118i	$0.800656\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−20.3746	−1.33478	−0.667392	−	0.744707i	$-0.732589\pi$
$233$	−20.3746	−1.33478	−0.667392	+	0.744707i	$0.732589\pi$
$234$	0	0
$235$	− 1.27492i	− 0.0831664i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	25.6197	1.65720	0.828600	−	0.559842i	$-0.189138\pi$
$239$	25.6197	1.65720	0.828600	+	0.559842i	$0.189138\pi$
$240$	0	0
$241$	−12.5498	−0.808406	−0.404203	−	0.914669i	$-0.632451\pi$
$241$	−12.5498	−0.808406	−0.404203	+	0.914669i	$0.632451\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	20.7846i	1.32788i
$246$	0	0
$247$	2.62685	0.167142
$248$	0	0
$249$	0	0
$250$	0	0
$251$	10.7251i	0.676961i	0.940973	+	0.338481i	$0.109913\pi$
$251$	10.7251i	0.676961i	−0.940973	+	0.338481i	$0.890087\pi$
$252$	0	0
$253$	18.2728i	1.14880i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	7.45017	0.464729	0.232364	−	0.972629i	$-0.425354\pi$
$257$	7.45017	0.464729	0.232364	+	0.972629i	$0.425354\pi$
$258$	0	0
$259$	4.00000i	0.248548i
$260$	0	0
$261$	7.88054i	0.487793i
$262$	0	0
$263$	−1.25588	−0.0774409	−0.0387204	−	0.999250i	$-0.512328\pi$
$263$	−1.25588	−0.0774409	−0.0387204	+	0.999250i	$0.512328\pi$
$264$	0	0
$265$	−31.6495	−1.94421
$266$	0	0
$267$	0	0
$268$	0	0
$269$	5.13861i	0.313306i	0.987654	+	0.156653i	$0.0500705\pi$
$269$	5.13861i	0.313306i	−0.987654	+	0.156653i	$0.949929\pi$
$270$	0	0
$271$	27.8279	1.69042	0.845212	−	0.534431i	$-0.179474\pi$
$271$	27.8279	1.69042	0.845212	+	0.534431i	$0.179474\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	22.5498i	1.35981i
$276$	0	0
$277$	− 16.0646i	− 0.965230i	−0.875833	−	0.482615i	$-0.839687\pi$
$277$	− 16.0646i	− 0.965230i	0.875833	−	0.482615i	$-0.160313\pi$
$278$	0	0
$279$	−18.2728	−1.09397
$280$	0	0
$281$	11.0997	0.662151	0.331075	−	0.943604i	$-0.392589\pi$
$281$	11.0997	0.662151	0.331075	+	0.943604i	$0.392589\pi$
$282$	0	0
$283$	26.3746i	1.56781i	0.620883	+	0.783903i	$0.286774\pi$
$283$	26.3746i	1.56781i	−0.620883	+	0.783903i	$0.713226\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1.90468	−0.112430
$288$	0	0
$289$	−6.27492	−0.369113
$290$	0	0
$291$	0	0
$292$	0	0
$293$	30.3397i	1.77246i	0.463244	+	0.886231i	$0.346685\pi$
$293$	30.3397i	1.77246i	−0.463244	+	0.886231i	$0.653315\pi$
$294$	0	0
$295$	19.9474	1.16138
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 9.09967i	− 0.526247i
$300$	0	0
$301$	1.14079i	0.0657542i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−9.27492	−0.531080
$306$	0	0
$307$	1.45017i	0.0827653i	0.999143	+	0.0413827i	$0.0131763\pi$
$307$	1.45017i	0.0827653i	−0.999143	+	0.0413827i	$0.986824\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	24.7824	1.40528	0.702641	−	0.711544i	$-0.252004\pi$
$311$	24.7824	1.40528	0.702641	+	0.711544i	$0.252004\pi$
$312$	0	0
$313$	15.0997	0.853484	0.426742	−	0.904373i	$-0.359661\pi$
$313$	15.0997	0.853484	0.426742	+	0.904373i	$0.359661\pi$
$314$	0	0
$315$	3.82475i	0.215500i
$316$	0	0
$317$	− 8.71780i	− 0.489640i	−0.969569	−	0.244820i	$-0.921271\pi$
$317$	− 8.71780i	− 0.489640i	0.969569	−	0.244820i	$-0.0787289\pi$
$318$	0	0
$319$	13.8564	0.775810
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 3.27492i	− 0.182221i
$324$	0	0
$325$	− 11.2296i	− 0.622904i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−0.175248	−0.00966175
$330$	0	0
$331$	− 9.45017i	− 0.519428i	−0.965686	−	0.259714i	$-0.916372\pi$
$331$	− 9.45017i	− 0.519428i	0.965686	−	0.259714i	$-0.0836283\pi$
$332$	0	0
$333$	− 28.6652i	− 1.57084i
$334$	0	0
$335$	19.9474	1.08984
$336$	0	0
$337$	33.6495	1.83301	0.916503	−	0.400029i	$-0.131000\pi$
$337$	33.6495	1.83301	0.916503	+	0.400029i	$0.131000\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	32.1293i	1.73990i
$342$	0	0
$343$	5.78741	0.312491
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 7.82475i	− 0.420055i	−0.977696	−	0.210027i	$-0.932645\pi$
$347$	− 7.82475i	− 0.420055i	0.977696	−	0.210027i	$-0.0673553\pi$
$348$	0	0
$349$	12.7156i	0.680651i	0.940308	+	0.340326i	$0.110537\pi$
$349$	12.7156i	0.680651i	−0.940308	+	0.340326i	$0.889463\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−10.0000	−0.532246	−0.266123	−	0.963939i	$-0.585743\pi$
$353$	−10.0000	−0.532246	−0.266123	+	0.963939i	$0.585743\pi$
$354$	0	0
$355$	− 34.5498i	− 1.83371i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	12.6005	0.665030	0.332515	−	0.943098i	$-0.392103\pi$
$359$	12.6005	0.665030	0.332515	+	0.943098i	$0.392103\pi$
$360$	0	0
$361$	−1.00000	−0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	9.97368i	0.522046i
$366$	0	0
$367$	−15.6460	−0.816715	−0.408357	−	0.912822i	$-0.633898\pi$
$367$	−15.6460	−0.816715	−0.408357	+	0.912822i	$0.633898\pi$
$368$	0	0
$369$	13.6495	0.710565
$370$	0	0
$371$	4.35050i	0.225867i
$372$	0	0
$373$	0.952341i	0.0493104i	0.999696	+	0.0246552i	$0.00784878\pi$
$373$	0.952341i	0.0493104i	−0.999696	+	0.0246552i	$0.992151\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−6.90033	−0.355385
$378$	0	0
$379$	− 23.6495i	− 1.21479i	−0.794399	−	0.607397i	$-0.792214\pi$
$379$	− 23.6495i	− 1.21479i	0.794399	−	0.607397i	$-0.207786\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	32.1293	1.64173	0.820864	−	0.571124i	$-0.193492\pi$
$383$	32.1293	1.64173	0.820864	+	0.571124i	$0.193492\pi$
$384$	0	0
$385$	6.72508	0.342742
$386$	0	0
$387$	− 8.17525i	− 0.415571i
$388$	0	0
$389$	− 4.71998i	− 0.239313i	−0.992815	−	0.119656i	$-0.961821\pi$
$389$	− 4.71998i	− 0.239313i	0.992815	−	0.119656i	$-0.0381793\pi$
$390$	0	0
$391$	−11.3446	−0.573723
$392$	0	0
$393$	0	0
$394$	0	0
$395$	39.6495i	1.99498i
$396$	0	0
$397$	32.6630i	1.63931i	0.572859	+	0.819654i	$0.305834\pi$
$397$	32.6630i	1.63931i	−0.572859	+	0.819654i	$0.694166\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	35.0997	1.75279	0.876397	−	0.481590i	$-0.159940\pi$
$401$	35.0997	1.75279	0.876397	+	0.481590i	$0.159940\pi$
$402$	0	0
$403$	− 16.0000i	− 0.797017i
$404$	0	0
$405$	− 27.4093i	− 1.36198i
$406$	0	0
$407$	−50.4021	−2.49834
$408$	0	0
$409$	18.0000	0.890043	0.445021	−	0.895520i	$-0.353196\pi$
$409$	18.0000	0.890043	0.445021	+	0.895520i	$0.353196\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 2.74194i	− 0.134922i
$414$	0	0
$415$	−52.0766	−2.55634
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 12.0000i	− 0.586238i	−0.956076	−	0.293119i	$-0.905307\pi$
$419$	− 12.0000i	− 0.586238i	0.956076	−	0.293119i	$-0.0946933\pi$
$420$	0	0
$421$	− 24.2487i	− 1.18181i	−0.806741	−	0.590905i	$-0.798771\pi$
$421$	− 24.2487i	− 1.18181i	0.806741	−	0.590905i	$-0.201229\pi$
$422$	0	0
$423$	1.25588	0.0610630
$424$	0	0
$425$	−14.0000	−0.679100
$426$	0	0
$427$	1.27492i	0.0616976i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−1.67451	−0.0806582	−0.0403291	−	0.999186i	$-0.512841\pi$
$431$	−1.67451	−0.0806582	−0.0403291	+	0.999186i	$0.512841\pi$
$432$	0	0
$433$	−17.6495	−0.848181	−0.424091	−	0.905620i	$-0.639406\pi$
$433$	−17.6495	−0.848181	−0.424091	+	0.905620i	$0.639406\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3.46410i	0.165710i
$438$	0	0
$439$	31.2920	1.49349	0.746743	−	0.665113i	$-0.231617\pi$
$439$	31.2920	1.49349	0.746743	+	0.665113i	$0.231617\pi$
$440$	0	0
$441$	−20.4743	−0.974965
$442$	0	0
$443$	10.3746i	0.492911i	0.969154	+	0.246456i	$0.0792660\pi$
$443$	10.3746i	0.492911i	−0.969154	+	0.246456i	$0.920734\pi$
$444$	0	0
$445$	30.4547i	1.44369i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−20.5498	−0.969807	−0.484903	−	0.874568i	$-0.661145\pi$
$449$	−20.5498	−0.969807	−0.484903	+	0.874568i	$0.661145\pi$
$450$	0	0
$451$	− 24.0000i	− 1.13012i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−3.34901	−0.157004
$456$	0	0
$457$	−15.2749	−0.714530	−0.357265	−	0.934003i	$-0.616291\pi$
$457$	−15.2749	−0.714530	−0.357265	+	0.934003i	$0.616291\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	12.7156i	0.592225i	0.955153	+	0.296113i	$0.0956904\pi$
$461$	12.7156i	0.592225i	−0.955153	+	0.296113i	$0.904310\pi$
$462$	0	0
$463$	2.93039	0.136187	0.0680933	−	0.997679i	$-0.478308\pi$
$463$	2.93039	0.136187	0.0680933	+	0.997679i	$0.478308\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	8.17525i	0.378305i	0.981948	+	0.189153i	$0.0605741\pi$
$467$	8.17525i	0.378305i	−0.981948	+	0.189153i	$0.939426\pi$
$468$	0	0
$469$	− 2.74194i	− 0.126611i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−14.3746	−0.660944
$474$	0	0
$475$	4.27492i	0.196147i
$476$	0	0
$477$	− 31.1769i	− 1.42749i
$478$	0	0
$479$	17.3205	0.791394	0.395697	−	0.918381i	$-0.370503\pi$
$479$	17.3205	0.791394	0.395697	+	0.918381i	$0.370503\pi$
$480$	0	0
$481$	25.0997	1.14445
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 50.4021i	− 2.28864i
$486$	0	0
$487$	−39.0575	−1.76986	−0.884931	−	0.465722i	$-0.845795\pi$
$487$	−39.0575	−1.76986	−0.884931	+	0.465722i	$0.845795\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 12.0000i	− 0.541552i	−0.962642	−	0.270776i	$-0.912720\pi$
$491$	− 12.0000i	− 0.541552i	0.962642	−	0.270776i	$-0.0872803\pi$
$492$	0	0
$493$	8.60271i	0.387447i
$494$	0	0
$495$	−48.1939	−2.16615
$496$	0	0
$497$	−4.74917	−0.213029
$498$	0	0
$499$	26.7251i	1.19638i	0.801355	+	0.598190i	$0.204113\pi$
$499$	26.7251i	1.19638i	−0.801355	+	0.598190i	$0.795887\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	41.6843	1.85861	0.929306	−	0.369311i	$-0.120406\pi$
$503$	41.6843	1.85861	0.929306	+	0.369311i	$0.120406\pi$
$504$	0	0
$505$	42.1993	1.87785
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 4.30136i	− 0.190654i	−0.995446	−	0.0953271i	$-0.969610\pi$
$509$	− 4.30136i	− 0.190654i	0.995446	−	0.0953271i	$-0.0303897\pi$
$510$	0	0
$511$	1.37097	0.0606480
$512$	0	0
$513$	0	0
$514$	0	0
$515$	18.5498i	0.817403i
$516$	0	0
$517$	− 2.20822i	− 0.0971175i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−27.0997	−1.18726	−0.593629	−	0.804739i	$-0.702305\pi$
$521$	−27.0997	−1.18726	−0.593629	+	0.804739i	$0.702305\pi$
$522$	0	0
$523$	5.45017i	0.238319i	0.992875	+	0.119160i	$0.0380200\pi$
$523$	5.45017i	0.238319i	−0.992875	+	0.119160i	$0.961980\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−19.9474	−0.868920
$528$	0	0
$529$	−11.0000	−0.478261
$530$	0	0
$531$	19.6495i	0.852716i
$532$	0	0
$533$	11.9517i	0.517687i
$534$	0	0
$535$	−4.41644	−0.190939
$536$	0	0
$537$	0	0
$538$	0	0
$539$	36.0000i	1.55063i
$540$	0	0
$541$	− 29.0838i	− 1.25041i	−0.780461	−	0.625205i	$-0.785015\pi$
$541$	− 29.0838i	− 1.25041i	0.780461	−	0.625205i	$-0.214985\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	10.5498	0.451905
$546$	0	0
$547$	− 32.0000i	− 1.36822i	−0.729378	−	0.684111i	$-0.760191\pi$
$547$	− 32.0000i	− 1.36822i	0.729378	−	0.684111i	$-0.239809\pi$
$548$	0	0
$549$	− 9.13642i	− 0.389933i
$550$	0	0
$551$	2.62685	0.111907
$552$	0	0
$553$	5.45017	0.231765
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 29.9210i	− 1.26779i	−0.773417	−	0.633897i	$-0.781454\pi$
$557$	− 29.9210i	− 1.26779i	0.773417	−	0.633897i	$-0.218546\pi$
$558$	0	0
$559$	7.15838	0.302767
$560$	0	0
$561$	0	0
$562$	0	0
$563$	24.0000i	1.01148i	0.862686	+	0.505740i	$0.168780\pi$
$563$	24.0000i	1.01148i	−0.862686	+	0.505740i	$0.831220\pi$
$564$	0	0
$565$	− 38.2202i	− 1.60794i
$566$	0	0
$567$	−3.76764	−0.158226
$568$	0	0
$569$	−19.4502	−0.815393	−0.407697	−	0.913117i	$-0.633668\pi$
$569$	−19.4502	−0.815393	−0.407697	+	0.913117i	$0.633668\pi$
$570$	0	0
$571$	− 41.0997i	− 1.71997i	−0.510321	−	0.859984i	$-0.670474\pi$
$571$	− 41.0997i	− 1.71997i	0.510321	−	0.859984i	$-0.329526\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	14.8087	0.617567
$576$	0	0
$577$	23.2749	0.968947	0.484474	−	0.874806i	$-0.339011\pi$
$577$	23.2749	0.968947	0.484474	+	0.874806i	$0.339011\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	7.15838i	0.296980i
$582$	0	0
$583$	−54.8185	−2.27035
$584$	0	0
$585$	24.0000	0.992278
$586$	0	0
$587$	− 37.2749i	− 1.53850i	−0.638948	−	0.769250i	$-0.720630\pi$
$587$	− 37.2749i	− 1.53850i	0.638948	−	0.769250i	$-0.279370\pi$
$588$	0	0
$589$	6.09095i	0.250973i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	19.0997	0.784329	0.392165	−	0.919895i	$-0.371726\pi$
$593$	19.0997	0.784329	0.392165	+	0.919895i	$0.371726\pi$
$594$	0	0
$595$	4.17525i	0.171168i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	9.43996	0.385706	0.192853	−	0.981228i	$-0.438226\pi$
$599$	9.43996	0.385706	0.192853	+	0.981228i	$0.438226\pi$
$600$	0	0
$601$	11.0997	0.452765	0.226382	−	0.974038i	$-0.427310\pi$
$601$	11.0997	0.452765	0.226382	+	0.974038i	$0.427310\pi$
$602$	0	0
$603$	19.6495i	0.800190i
$604$	0	0
$605$	51.2394i	2.08318i
$606$	0	0
$607$	−29.3873	−1.19279	−0.596397	−	0.802689i	$-0.703402\pi$
$607$	−29.3873	−1.19279	−0.596397	+	0.802689i	$0.703402\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1.09967i	0.0444878i
$612$	0	0
$613$	− 29.9210i	− 1.20850i	−0.796795	−	0.604250i	$-0.793473\pi$
$613$	− 29.9210i	− 1.20850i	0.796795	−	0.604250i	$-0.206527\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	12.7251	0.512293	0.256146	−	0.966638i	$-0.417547\pi$
$617$	12.7251	0.512293	0.256146	+	0.966638i	$0.417547\pi$
$618$	0	0
$619$	4.00000i	0.160774i	0.996764	+	0.0803868i	$0.0256155\pi$
$619$	4.00000i	0.160774i	−0.996764	+	0.0803868i	$0.974384\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	4.18627	0.167719
$624$	0	0
$625$	−28.0997	−1.12399
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 31.2920i	− 1.24769i
$630$	0	0
$631$	−13.4378	−0.534950	−0.267475	−	0.963565i	$-0.586189\pi$
$631$	−13.4378	−0.534950	−0.267475	+	0.963565i	$0.586189\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 53.0997i	− 2.10720i
$636$	0	0
$637$	− 17.9276i	− 0.710317i
$638$	0	0
$639$	34.0339	1.34636
$640$	0	0
$641$	−12.5498	−0.495689	−0.247844	−	0.968800i	$-0.579722\pi$
$641$	−12.5498	−0.495689	−0.247844	+	0.968800i	$0.579722\pi$
$642$	0	0
$643$	− 44.9244i	− 1.77165i	−0.464023	−	0.885823i	$-0.653595\pi$
$643$	− 44.9244i	− 1.77165i	0.464023	−	0.885823i	$-0.346405\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−25.6197	−1.00721	−0.503607	−	0.863933i	$-0.667994\pi$
$647$	−25.6197	−1.00721	−0.503607	+	0.863933i	$0.667994\pi$
$648$	0	0
$649$	34.5498	1.35620
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 16.0646i	− 0.628657i	−0.949314	−	0.314329i	$-0.898221\pi$
$653$	− 16.0646i	− 0.628657i	0.949314	−	0.314329i	$-0.101779\pi$
$654$	0	0
$655$	−16.0646	−0.627697
$656$	0	0
$657$	−9.82475	−0.383300
$658$	0	0
$659$	− 13.4502i	− 0.523944i	−0.965075	−	0.261972i	$-0.915627\pi$
$659$	− 13.4502i	− 0.523944i	0.965075	−	0.261972i	$-0.0843728\pi$
$660$	0	0
$661$	− 9.55505i	− 0.371648i	−0.982583	−	0.185824i	$-0.940505\pi$
$661$	− 9.55505i	− 0.371648i	0.982583	−	0.185824i	$-0.0594955\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.27492	0.0494392
$666$	0	0
$667$	− 9.09967i	− 0.352341i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−16.0646	−0.620168
$672$	0	0
$673$	−47.0997	−1.81556	−0.907779	−	0.419448i	$-0.862224\pi$
$673$	−47.0997	−1.81556	−0.907779	+	0.419448i	$0.862224\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 0.722166i	− 0.0277551i	−0.999904	−	0.0138775i	$-0.995582\pi$
$677$	− 0.722166i	− 0.0277551i	0.999904	−	0.0138775i	$-0.00441750\pi$
$678$	0	0
$679$	−6.92820	−0.265880
$680$	0	0
$681$	0	0
$682$	0	0
$683$	22.1993i	0.849434i	0.905326	+	0.424717i	$0.139626\pi$
$683$	22.1993i	0.849434i	−0.905326	+	0.424717i	$0.860374\pi$
$684$	0	0
$685$	38.7539i	1.48071i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	27.2990	1.04001
$690$	0	0
$691$	23.4743i	0.893003i	0.894783	+	0.446501i	$0.147330\pi$
$691$	23.4743i	0.893003i	−0.894783	+	0.446501i	$0.852670\pi$
$692$	0	0
$693$	6.62466i	0.251650i
$694$	0	0
$695$	−8.29917	−0.314806
$696$	0	0
$697$	14.9003	0.564390
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 21.0148i	− 0.793717i	−0.917880	−	0.396859i	$-0.870100\pi$
$701$	− 21.0148i	− 0.793717i	0.917880	−	0.396859i	$-0.129900\pi$
$702$	0	0
$703$	−9.55505	−0.360376
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 5.80066i	− 0.218156i
$708$	0	0
$709$	38.2202i	1.43539i	0.696358	+	0.717695i	$0.254803\pi$
$709$	38.2202i	1.43539i	−0.696358	+	0.717695i	$0.745197\pi$
$710$	0	0
$711$	−39.0575	−1.46477
$712$	0	0
$713$	21.0997	0.790189
$714$	0	0
$715$	− 42.1993i	− 1.57817i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−51.6580	−1.92652	−0.963259	−	0.268575i	$-0.913447\pi$
$719$	−51.6580	−1.92652	−0.963259	+	0.268575i	$0.913447\pi$
$720$	0	0
$721$	2.54983	0.0949608
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 11.2296i	− 0.417055i
$726$	0	0
$727$	10.9260	0.405224	0.202612	−	0.979259i	$-0.435057\pi$
$727$	10.9260	0.405224	0.202612	+	0.979259i	$0.435057\pi$
$728$	0	0
$729$	27.0000	1.00000
$730$	0	0
$731$	− 8.92442i	− 0.330082i
$732$	0	0
$733$	− 19.1101i	− 0.705848i	−0.935652	−	0.352924i	$-0.885187\pi$
$733$	− 19.1101i	− 0.705848i	0.935652	−	0.352924i	$-0.114813\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	34.5498	1.27266
$738$	0	0
$739$	7.82475i	0.287838i	0.989589	+	0.143919i	$0.0459705\pi$
$739$	7.82475i	0.287838i	−0.989589	+	0.143919i	$0.954029\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	52.9139	1.94122	0.970611	−	0.240655i	$-0.0773622\pi$
$743$	52.9139	1.94122	0.970611	+	0.240655i	$0.0773622\pi$
$744$	0	0
$745$	48.9244	1.79245
$746$	0	0
$747$	− 51.2990i	− 1.87693i
$748$	0	0
$749$	0.607078i	0.0221822i
$750$	0	0
$751$	36.3155	1.32517	0.662586	−	0.748986i	$-0.269459\pi$
$751$	36.3155	1.32517	0.662586	+	0.748986i	$0.269459\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 2.54983i	− 0.0927980i
$756$	0	0
$757$	3.04547i	0.110690i	0.998467	+	0.0553448i	$0.0176258\pi$
$757$	3.04547i	0.110690i	−0.998467	+	0.0553448i	$0.982374\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	3.27492	0.118716	0.0593578	−	0.998237i	$-0.481095\pi$
$761$	3.27492	0.118716	0.0593578	+	0.998237i	$0.481095\pi$
$762$	0	0
$763$	− 1.45017i	− 0.0524995i
$764$	0	0
$765$	− 29.9210i	− 1.08180i
$766$	0	0
$767$	−17.2054	−0.621252
$768$	0	0
$769$	−13.8248	−0.498533	−0.249267	−	0.968435i	$-0.580190\pi$
$769$	−13.8248	−0.498533	−0.249267	+	0.968435i	$0.580190\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	34.7561i	1.25009i	0.780589	+	0.625045i	$0.214919\pi$
$773$	34.7561i	1.25009i	−0.780589	+	0.625045i	$0.785081\pi$
$774$	0	0
$775$	26.0383	0.935324
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 4.54983i	− 0.163015i
$780$	0	0
$781$	− 59.8421i	− 2.14132i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−37.0997	−1.32414
$786$	0	0
$787$	− 14.1993i	− 0.506152i	−0.967446	−	0.253076i	$-0.918558\pi$
$787$	− 14.1993i	− 0.506152i	0.967446	−	0.253076i	$-0.0814422\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−5.25370	−0.186800
$792$	0	0
$793$	8.00000	0.284088
$794$	0	0
$795$	0	0
$796$	0	0
$797$	24.2487i	0.858933i	0.903083	+	0.429467i	$0.141298\pi$
$797$	24.2487i	0.858933i	−0.903083	+	0.429467i	$0.858702\pi$
$798$	0	0
$799$	1.37097	0.0485014
$800$	0	0
$801$	−30.0000	−1.06000
$802$	0	0
$803$	17.2749i	0.609619i
$804$	0	0
$805$	− 4.41644i	− 0.155659i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−5.82475	−0.204787	−0.102394	−	0.994744i	$-0.532650\pi$
$809$	−5.82475	−0.204787	−0.102394	+	0.994744i	$0.532650\pi$
$810$	0	0
$811$	47.2990i	1.66089i	0.557099	+	0.830446i	$0.311915\pi$
$811$	47.2990i	1.66089i	−0.557099	+	0.830446i	$0.688085\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−12.1819	−0.426713
$816$	0	0
$817$	−2.72508	−0.0953386
$818$	0	0
$819$	− 3.29901i	− 0.115277i
$820$	0	0
$821$	26.5720i	0.927370i	0.886000	+	0.463685i	$0.153473\pi$
$821$	26.5720i	0.927370i	−0.886000	+	0.463685i	$0.846527\pi$
$822$	0	0
$823$	21.4334	0.747122	0.373561	−	0.927606i	$-0.378137\pi$
$823$	21.4334	0.747122	0.373561	+	0.927606i	$0.378137\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	47.6495i	1.65694i	0.560037	+	0.828468i	$0.310787\pi$
$827$	47.6495i	1.65694i	−0.560037	+	0.828468i	$0.689213\pi$
$828$	0	0
$829$	− 25.3161i	− 0.879266i	−0.898178	−	0.439633i	$-0.855109\pi$
$829$	− 25.3161i	− 0.879266i	0.898178	−	0.439633i	$-0.144891\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−22.3505	−0.774399
$834$	0	0
$835$	− 58.1993i	− 2.01407i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	47.6602	1.64541	0.822706	−	0.568467i	$-0.192463\pi$
$839$	47.6602	1.64541	0.822706	+	0.568467i	$0.192463\pi$
$840$	0	0
$841$	22.0997	0.762058
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 18.5764i	− 0.639047i
$846$	0	0
$847$	7.04329	0.242010
$848$	0	0
$849$	0	0
$850$	0	0
$851$	33.0997i	1.13464i
$852$	0	0
$853$	32.9665i	1.12875i	0.825518	+	0.564376i	$0.190883\pi$
$853$	32.9665i	1.12875i	−0.825518	+	0.564376i	$0.809117\pi$
$854$	0	0
$855$	−9.13642	−0.312459
$856$	0	0
$857$	−16.1993	−0.553359	−0.276679	−	0.960962i	$-0.589234\pi$
$857$	−16.1993	−0.553359	−0.276679	+	0.960962i	$0.589234\pi$
$858$	0	0
$859$	55.4743i	1.89276i	0.323060	+	0.946379i	$0.395289\pi$
$859$	55.4743i	1.89276i	−0.323060	+	0.946379i	$0.604711\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	41.7994	1.42287	0.711434	−	0.702753i	$-0.248046\pi$
$863$	41.7994	1.42287	0.711434	+	0.702753i	$0.248046\pi$
$864$	0	0
$865$	26.5498	0.902721
$866$	0	0
$867$	0	0
$868$	0	0
$869$	68.6750i	2.32964i
$870$	0	0
$871$	−17.2054	−0.582983
$872$	0	0
$873$	49.6495	1.68038
$874$	0	0
$875$	0.924421i	0.0312511i
$876$	0	0
$877$	− 31.4071i	− 1.06054i	−0.847828	−	0.530271i	$-0.822090\pi$
$877$	− 31.4071i	− 1.06054i	0.847828	−	0.530271i	$-0.177910\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1.82475	0.0614774	0.0307387	−	0.999527i	$-0.490214\pi$
$881$	1.82475	0.0614774	0.0307387	+	0.999527i	$0.490214\pi$
$882$	0	0
$883$	13.6254i	0.458532i	0.973364	+	0.229266i	$0.0736325\pi$
$883$	13.6254i	0.458532i	−0.973364	+	0.229266i	$0.926367\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−0.837253	−0.0281122	−0.0140561	−	0.999901i	$-0.504474\pi$
$887$	−0.837253	−0.0281122	−0.0140561	+	0.999901i	$0.504474\pi$
$888$	0	0
$889$	−7.29901	−0.244801
$890$	0	0
$891$	− 47.4743i	− 1.59045i
$892$	0	0
$893$	− 0.418627i	− 0.0140088i
$894$	0	0
$895$	64.2585	2.14793
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 16.0000i	− 0.533630i
$900$	0	0
$901$	− 34.0339i	− 1.13383i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−10.5498	−0.350688
$906$	0	0
$907$	23.6495i	0.785269i	0.919695	+	0.392634i	$0.128436\pi$
$907$	23.6495i	0.785269i	−0.919695	+	0.392634i	$0.871564\pi$
$908$	0	0
$909$	41.5692i	1.37876i
$910$	0	0
$911$	−34.0339	−1.12759	−0.563797	−	0.825913i	$-0.690660\pi$
$911$	−34.0339	−1.12759	−0.563797	+	0.825913i	$0.690660\pi$
$912$	0	0
$913$	−90.1993	−2.98516
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2.20822i	0.0729219i
$918$	0	0
$919$	−0.115088	−0.00379639	−0.00189820	−	0.999998i	$-0.500604\pi$
$919$	−0.115088	−0.00379639	−0.00189820	+	0.999998i	$0.500604\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	29.8007i	0.980901i
$924$	0	0
$925$	40.8471i	1.34304i
$926$	0	0
$927$	−18.2728	−0.600159
$928$	0	0
$929$	−7.09967	−0.232933	−0.116466	−	0.993195i	$-0.537157\pi$
$929$	−7.09967	−0.232933	−0.116466	+	0.993195i	$0.537157\pi$
$930$	0	0
$931$	6.82475i	0.223672i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−52.6103	−1.72054
$936$	0	0
$937$	−14.9244	−0.487560	−0.243780	−	0.969831i	$-0.578387\pi$
$937$	−14.9244	−0.487560	−0.243780	+	0.969831i	$0.578387\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 31.1769i	− 1.01634i	−0.861257	−	0.508169i	$-0.830322\pi$
$941$	− 31.1769i	− 1.01634i	0.861257	−	0.508169i	$-0.169678\pi$
$942$	0	0
$943$	−15.7611	−0.513252
$944$	0	0
$945$	0	0
$946$	0	0
$947$	30.1993i	0.981347i	0.871344	+	0.490673i	$0.163249\pi$
$947$	30.1993i	0.981347i	−0.871344	+	0.490673i	$0.836751\pi$
$948$	0	0
$949$	− 8.60271i	− 0.279256i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−45.2990	−1.46738	−0.733689	−	0.679485i	$-0.762203\pi$
$953$	−45.2990	−1.46738	−0.733689	+	0.679485i	$0.762203\pi$
$954$	0	0
$955$	35.8248i	1.15926i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	5.32706	0.172020
$960$	0	0
$961$	6.09967	0.196764
$962$	0	0
$963$	− 4.35050i	− 0.140193i
$964$	0	0
$965$	− 33.8038i	− 1.08818i
$966$	0	0
$967$	−55.5407	−1.78607	−0.893034	−	0.449988i	$-0.851428\pi$
$967$	−55.5407	−1.78607	−0.893034	+	0.449988i	$0.851428\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 44.0000i	− 1.41203i	−0.708198	−	0.706014i	$-0.750492\pi$
$971$	− 44.0000i	− 1.41203i	0.708198	−	0.706014i	$-0.249508\pi$
$972$	0	0
$973$	1.14079i	0.0365721i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−28.1993	−0.902177	−0.451088	−	0.892479i	$-0.648964\pi$
$977$	−28.1993	−0.902177	−0.451088	+	0.892479i	$0.648964\pi$
$978$	0	0
$979$	52.7492i	1.68587i
$980$	0	0
$981$	10.3923i	0.331801i
$982$	0	0
$983$	22.6893	0.723676	0.361838	−	0.932241i	$-0.382149\pi$
$983$	22.6893	0.723676	0.361838	+	0.932241i	$0.382149\pi$
$984$	0	0
$985$	−16.0000	−0.509802
$986$	0	0
$987$	0	0
$988$	0	0
$989$	9.43996i	0.300173i
$990$	0	0
$991$	−47.0531	−1.49469	−0.747345	−	0.664436i	$-0.768672\pi$
$991$	−47.0531	−1.49469	−0.747345	+	0.664436i	$0.768672\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 43.4743i	− 1.37823i
$996$	0	0
$997$	− 42.1029i	− 1.33341i	−0.745320	−	0.666707i	$-0.767703\pi$
$997$	− 42.1029i	− 1.33341i	0.745320	−	0.666707i	$-0.232297\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.2.c.i.609.1	✓	8
4.3	odd	2	inner	1216.2.c.i.609.2	yes	8
8.3	odd	2	inner	1216.2.c.i.609.8	yes	8
8.5	even	2	inner	1216.2.c.i.609.7	yes	8
16.3	odd	4		4864.2.a.bk.1.1		4
16.5	even	4		4864.2.a.bk.1.4		4
16.11	odd	4		4864.2.a.bj.1.4		4
16.13	even	4		4864.2.a.bj.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1216.2.c.i.609.1	✓	8	1.1	even	1	trivial
1216.2.c.i.609.2	yes	8	4.3	odd	2	inner
1216.2.c.i.609.7	yes	8	8.5	even	2	inner
1216.2.c.i.609.8	yes	8	8.3	odd	2	inner
4864.2.a.bj.1.1		4	16.13	even	4
4864.2.a.bj.1.4		4	16.11	odd	4
4864.2.a.bk.1.1		4	16.3	odd	4
4864.2.a.bk.1.4		4	16.5	even	4

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1216 = 2^{6} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1216.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.04547i q^{5} -0.418627 q^{7} +3.00000 q^{9} -5.27492i q^{11} +2.62685i q^{13} +3.27492 q^{17} -1.00000i q^{19} -3.46410 q^{23} -4.27492 q^{25} +2.62685i q^{29} -6.09095 q^{31} +1.27492i q^{35} -9.55505i q^{37} +4.54983 q^{41} -2.72508i q^{43} -9.13642i q^{45} +0.418627 q^{47} -6.82475 q^{49} -10.3923i q^{53} -16.0646 q^{55} +6.54983i q^{59} -3.04547i q^{61} -1.25588 q^{63} +8.00000 q^{65} +6.54983i q^{67} +11.3446 q^{71} -3.27492 q^{73} +2.20822i q^{77} -13.0192 q^{79} +9.00000 q^{81} -17.0997i q^{83} -9.97368i q^{85} -10.0000 q^{89} -1.09967i q^{91} -3.04547 q^{95} +16.5498 q^{97} -15.8248i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 24 q^{9} - 4 q^{17} - 4 q^{25} - 24 q^{41} + 36 q^{49} + 64 q^{65} + 4 q^{73} + 72 q^{81} - 80 q^{89} + 72 q^{97}+O(q^{100}) \) 8 * q + 24 * q^9 - 4 * q^17 - 4 * q^25 - 24 * q^41 + 36 * q^49 + 64 * q^65 + 4 * q^73 + 72 * q^81 - 80 * q^89 + 72 * q^97

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