LMFDB - Embedded newform 2352.2.h.n.2255.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2352,2,Mod(2255,2352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2352, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1, 0]))

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

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

chi := DirichletCharacter("2352.2255");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2352 = 2^{4} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2352.h (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$18.7808145554$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.8275904784.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} + 5x^{6} - 6x^{5} + 4x^{4} - 18x^{3} + 45x^{2} - 81x + 81 $ x^8 - 3x^7 + 5x^6 - 6x^5 + 4x^4 - 18x^3 + 45x^2 - 81*x + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 336)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2255.1
Root			$-1.37009 + 1.05965i$ of defining polynomial
Character	$\chi$	$=$	2352.2255
Dual form			2352.2.h.n.2255.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.60273 - 0.656712i) q^{3} +0.670944i q^{5} +(2.13746 + 2.10506i) q^{9} +O(q^{10})$	$q+(-1.60273 - 0.656712i) q^{3} +0.670944i q^{5} +(2.13746 + 2.10506i) q^{9} -5.24879 q^{11} +2.00000 q^{13} +(0.440617 - 1.07534i) q^{15} -4.21012i q^{17} -1.31342i q^{19} +3.20545 q^{23} +4.54983 q^{25} +(-2.04334 - 4.77753i) q^{27} +6.22295i q^{29} +1.73205i q^{31} +(8.41238 + 3.44695i) q^{33} -6.27492 q^{37} +(-3.20545 - 1.31342i) q^{39} -9.76212i q^{41} +9.55505i q^{43} +(-1.41238 + 1.43411i) q^{45} -9.61635 q^{47} +(-2.76483 + 6.74766i) q^{51} +11.7750i q^{53} -3.52165i q^{55} +(-0.862541 + 2.10506i) q^{57} +7.57301 q^{59} -3.72508 q^{61} +1.34189i q^{65} -2.98793i q^{67} +(-5.13746 - 2.10506i) q^{69} -4.08668 q^{71} +0.274917 q^{73} +(-7.29214 - 2.98793i) q^{75} -5.19615i q^{79} +(0.137459 + 8.99895i) q^{81} -2.04334 q^{83} +2.82475 q^{85} +(4.08668 - 9.97368i) q^{87} +13.9722i q^{89} +(1.13746 - 2.77600i) q^{93} +0.881234 q^{95} -3.27492 q^{97} +(-11.2191 - 11.0490i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 2 q^{9}+O(q^{10}) $ 8 * q + 2 * q^9	$ 8 q + 2 q^{9} + 16 q^{13} - 24 q^{25} + 22 q^{33} - 20 q^{37} + 34 q^{45} - 22 q^{57} - 60 q^{61} - 26 q^{69} - 28 q^{73} - 14 q^{81} - 68 q^{85} - 6 q^{93} + 4 q^{97}+O(q^{100}) $ 8 * q + 2 * q^9 + 16 * q^13 - 24 * q^25 + 22 * q^33 - 20 * q^37 + 34 * q^45 - 22 * q^57 - 60 * q^61 - 26 * q^69 - 28 * q^73 - 14 * q^81 - 68 * q^85 - 6 * q^93 + 4 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times$.

$n$	$785$	$1471$	$1765$	$2257$
$\chi(n)$	$-1$	$-1$	$1$	$1$

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
$3$	−1.60273	−	0.656712i	−0.925334	−	0.379153i
$3$	−1.60273	−	0.656712i	−0.925334	−	0.379153i
$4$		0			0
$5$			0.670944i			0.300055i	0.988682	+	0.150028i	$0.0479363\pi$
$5$			0.670944i			0.300055i	−0.988682	+	0.150028i	$0.952064\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	2.13746	+	2.10506i	0.712486	+	0.701686i
$10$		0			0
$11$	−5.24879			−1.58257			−0.791285	−	0.611447i	$-0.790588\pi$
$11$	−5.24879			−1.58257			−0.791285	+	0.611447i	$0.790588\pi$
$12$		0			0
$13$	2.00000			0.554700			0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000			0.554700			0.277350	+	0.960769i	$0.410544\pi$
$14$		0			0
$15$	0.440617	−	1.07534i	0.113767	−	0.277651i
$16$		0			0
$17$		−	4.21012i		−	1.02110i	−0.859847	−	0.510552i	$-0.829441\pi$
$17$		−	4.21012i		−	1.02110i	0.859847	−	0.510552i	$-0.170559\pi$
$18$		0			0
$19$		−	1.31342i		−	0.301320i	−0.988586	−	0.150660i	$-0.951860\pi$
$19$		−	1.31342i		−	0.301320i	0.988586	−	0.150660i	$-0.0481399\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	3.20545			0.668383			0.334191	−	0.942505i	$-0.391537\pi$
$23$	3.20545			0.668383			0.334191	+	0.942505i	$0.391537\pi$
$24$		0			0
$25$	4.54983			0.909967
$26$		0			0
$27$	−2.04334	−	4.77753i	−0.393241	−	0.919435i
$28$		0			0
$29$			6.22295i			1.15557i	0.816188	+	0.577786i	$0.196083\pi$
$29$			6.22295i			1.15557i	−0.816188	+	0.577786i	$0.803917\pi$
$30$		0			0
$31$			1.73205i			0.311086i	0.987829	+	0.155543i	$0.0497126\pi$
$31$			1.73205i			0.311086i	−0.987829	+	0.155543i	$0.950287\pi$
$32$		0			0
$33$	8.41238	+	3.44695i	1.46441	+	0.600036i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−6.27492			−1.03159			−0.515795	−	0.856712i	$-0.672503\pi$
$37$	−6.27492			−1.03159			−0.515795	+	0.856712i	$0.672503\pi$
$38$		0			0
$39$	−3.20545	−	1.31342i	−0.513283	−	0.210316i
$40$		0			0
$41$		−	9.76212i		−	1.52459i	−0.647231	−	0.762294i	$-0.724073\pi$
$41$		−	9.76212i		−	1.52459i	0.647231	−	0.762294i	$-0.275927\pi$
$42$		0			0
$43$			9.55505i			1.45713i	0.684976	+	0.728566i	$0.259813\pi$
$43$			9.55505i			1.45713i	−0.684976	+	0.728566i	$0.740187\pi$
$44$		0			0
$45$	−1.41238	+	1.43411i	−0.210545	+	0.213785i
$46$		0			0
$47$	−9.61635			−1.40269			−0.701345	−	0.712822i	$-0.747417\pi$
$47$	−9.61635			−1.40269			−0.701345	+	0.712822i	$0.747417\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	−2.76483	+	6.74766i	−0.387154	+	0.944862i
$52$		0			0
$53$			11.7750i			1.61741i	0.588212	+	0.808707i	$0.299832\pi$
$53$			11.7750i			1.61741i	−0.588212	+	0.808707i	$0.700168\pi$
$54$		0			0
$55$		−	3.52165i		−	0.474859i
$56$		0			0
$57$	−0.862541	+	2.10506i	−0.114246	+	0.278822i
$58$		0			0
$59$	7.57301			0.985922			0.492961	−	0.870051i	$-0.335915\pi$
$59$	7.57301			0.985922			0.492961	+	0.870051i	$0.335915\pi$
$60$		0			0
$61$	−3.72508			−0.476948			−0.238474	−	0.971149i	$-0.576647\pi$
$61$	−3.72508			−0.476948			−0.238474	+	0.971149i	$0.576647\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			1.34189i			0.166441i
$66$		0			0
$67$		−	2.98793i		−	0.365034i	−0.983203	−	0.182517i	$-0.941576\pi$
$67$		−	2.98793i		−	0.365034i	0.983203	−	0.182517i	$-0.0584244\pi$
$68$		0			0
$69$	−5.13746	−	2.10506i	−0.618477	−	0.253419i
$70$		0			0
$71$	−4.08668			−0.485000			−0.242500	−	0.970151i	$-0.577967\pi$
$71$	−4.08668			−0.485000			−0.242500	+	0.970151i	$0.577967\pi$
$72$		0			0
$73$	0.274917			0.0321766			0.0160883	−	0.999871i	$-0.494879\pi$
$73$	0.274917			0.0321766			0.0160883	+	0.999871i	$0.494879\pi$
$74$		0			0
$75$	−7.29214	−	2.98793i	−0.842023	−	0.345017i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		−	5.19615i		−	0.584613i	−0.956325	−	0.292306i	$-0.905577\pi$
$79$		−	5.19615i		−	0.584613i	0.956325	−	0.292306i	$-0.0944227\pi$
$80$		0			0
$81$	0.137459	+	8.99895i	0.0152732	+	0.999883i
$82$		0			0
$83$	−2.04334			−0.224286			−0.112143	−	0.993692i	$-0.535771\pi$
$83$	−2.04334			−0.224286			−0.112143	+	0.993692i	$0.535771\pi$
$84$		0			0
$85$	2.82475			0.306387
$86$		0			0
$87$	4.08668	−	9.97368i	0.438139	−	1.06929i
$88$		0			0
$89$			13.9722i			1.48105i	0.672026	+	0.740527i	$0.265424\pi$
$89$			13.9722i			1.48105i	−0.672026	+	0.740527i	$0.734576\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	1.13746	−	2.77600i	0.117949	−	0.287858i
$94$		0			0
$95$	0.881234			0.0904127
$96$		0			0
$97$	−3.27492			−0.332517			−0.166259	−	0.986082i	$-0.553169\pi$
$97$	−3.27492			−0.332517			−0.166259	+	0.986082i	$0.553169\pi$
$98$		0			0
$99$	−11.2191	−	11.0490i	−1.12756	−	1.11047i
$100$		0			0
$101$			2.86823i			0.285399i	0.989766	+	0.142700i	$0.0455783\pi$
$101$			2.86823i			0.285399i	−0.989766	+	0.142700i	$0.954422\pi$
$102$		0			0
$103$			16.9594i			1.67106i	0.549443	+	0.835531i	$0.314840\pi$
$103$			16.9594i			1.67106i	−0.549443	+	0.835531i	$0.685160\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−11.6597			−1.12719			−0.563593	−	0.826053i	$-0.690581\pi$
$107$	−11.6597			−1.12719			−0.563593	+	0.826053i	$0.690581\pi$
$108$		0			0
$109$	14.8248			1.41995			0.709977	−	0.704225i	$-0.248705\pi$
$109$	14.8248			1.41995			0.709977	+	0.704225i	$0.248705\pi$
$110$		0			0
$111$	10.0570	+	4.12081i	0.954565	+	0.391130i
$112$		0			0
$113$			9.76212i			0.918343i	0.888348	+	0.459172i	$0.151854\pi$
$113$			9.76212i			0.918343i	−0.888348	+	0.459172i	$0.848146\pi$
$114$		0			0
$115$			2.15068i			0.200552i
$116$		0			0
$117$	4.27492	+	4.21012i	0.395216	+	0.389225i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	16.5498			1.50453
$122$		0			0
$123$	−6.41090	+	15.6460i	−0.578052	+	1.41075i
$124$		0			0
$125$			6.40740i			0.573095i
$126$		0			0
$127$			0.418627i			0.0371471i	0.999827	+	0.0185736i	$0.00591249\pi$
$127$			0.418627i			0.0371471i	−0.999827	+	0.0185736i	$0.994088\pi$
$128$		0			0
$129$	6.27492	−	15.3141i	0.552476	−	1.34833i
$130$		0			0
$131$	5.24879			0.458589			0.229295	−	0.973357i	$-0.426358\pi$
$131$	5.24879			0.458589			0.229295	+	0.973357i	$0.426358\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	3.20545	−	1.37097i	0.275881	−	0.117994i
$136$		0			0
$137$			15.3141i			1.30837i	0.756333	+	0.654187i	$0.226989\pi$
$137$			15.3141i			1.30837i	−0.756333	+	0.654187i	$0.773011\pi$
$138$		0			0
$139$			13.9715i			1.18505i	0.805553	+	0.592523i	$0.201868\pi$
$139$			13.9715i			1.18505i	−0.805553	+	0.592523i	$0.798132\pi$
$140$		0			0
$141$	15.4124	+	6.31518i	1.29796	+	0.531834i
$142$		0			0
$143$	−10.4976			−0.877852
$144$		0			0
$145$	−4.17525			−0.346736
$146$		0			0
$147$		0			0
$148$		0			0
$149$			13.9722i			1.14465i	0.820027	+	0.572325i	$0.193958\pi$
$149$			13.9722i			1.14465i	−0.820027	+	0.572325i	$0.806042\pi$
$150$		0			0
$151$			7.82300i			0.636627i	0.947986	+	0.318313i	$0.103116\pi$
$151$			7.82300i			0.636627i	−0.947986	+	0.318313i	$0.896884\pi$
$152$		0			0
$153$	8.86254	−	8.99895i	0.716494	−	0.727522i
$154$		0			0
$155$	−1.16211			−0.0933428
$156$		0			0
$157$	−23.3746			−1.86550			−0.932748	−	0.360530i	$-0.882596\pi$
$157$	−23.3746			−1.86550			−0.932748	+	0.360530i	$0.882596\pi$
$158$		0			0
$159$	7.73275	−	18.8720i	0.613247	−	1.49665i
$160$		0			0
$161$		0			0
$162$		0			0
$163$		−	2.15068i		−	0.168454i	−0.996447	−	0.0842270i	$-0.973158\pi$
$163$		−	2.15068i		−	0.168454i	0.996447	−	0.0842270i	$-0.0268421\pi$
$164$		0			0
$165$	−2.31271	+	5.64423i	−0.180044	+	0.439403i
$166$		0			0
$167$	20.9952			1.62466			0.812328	−	0.583201i	$-0.198200\pi$
$167$	20.9952			1.62466			0.812328	+	0.583201i	$0.198200\pi$
$168$		0			0
$169$	−9.00000			−0.692308
$170$		0			0
$171$	2.76483	−	2.80739i	0.211432	−	0.214686i
$172$		0			0
$173$			16.6560i			1.26633i	0.774016	+	0.633167i	$0.218245\pi$
$173$			16.6560i			1.26633i	−0.774016	+	0.633167i	$0.781755\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−12.1375	−	4.97329i	−0.912307	−	0.373815i
$178$		0			0
$179$	−16.0273			−1.19793			−0.598967	−	0.800774i	$-0.704422\pi$
$179$	−16.0273			−1.19793			−0.598967	+	0.800774i	$0.704422\pi$
$180$		0			0
$181$	−15.0997			−1.12235			−0.561175	−	0.827697i	$-0.689650\pi$
$181$	−15.0997			−1.12235			−0.561175	+	0.827697i	$0.689650\pi$
$182$		0			0
$183$	5.97029	+	2.44631i	0.441336	+	0.180836i
$184$		0			0
$185$		−	4.21012i		−	0.309534i
$186$		0			0
$187$			22.0980i			1.61597i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	11.9406			0.863989			0.431995	−	0.901876i	$-0.357810\pi$
$191$	11.9406			0.863989			0.431995	+	0.901876i	$0.357810\pi$
$192$		0			0
$193$	−18.0997			−1.30284			−0.651421	−	0.758716i	$-0.725827\pi$
$193$	−18.0997			−1.30284			−0.651421	+	0.758716i	$0.725827\pi$
$194$		0			0
$195$	0.881234	−	2.15068i	0.0631065	−	0.154013i
$196$		0			0
$197$		−	7.07835i		−	0.504311i	−0.967687	−	0.252156i	$-0.918861\pi$
$197$		−	7.07835i		−	0.504311i	0.967687	−	0.252156i	$-0.0811395\pi$
$198$		0			0
$199$			25.5621i			1.81205i	0.423222	+	0.906026i	$0.360899\pi$
$199$			25.5621i			1.81205i	−0.423222	+	0.906026i	$0.639101\pi$
$200$		0			0
$201$	−1.96221	+	4.78883i	−0.138404	+	0.337778i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	6.54983			0.457460
$206$		0			0
$207$	6.85152	+	6.74766i	0.476214	+	0.468995i
$208$		0			0
$209$			6.89389i			0.476860i
$210$		0			0
$211$			15.6460i			1.07712i	0.842589	+	0.538558i	$0.181031\pi$
$211$			15.6460i			1.07712i	−0.842589	+	0.538558i	$0.818969\pi$
$212$		0			0
$213$	6.54983	+	2.68378i	0.448787	+	0.183889i
$214$		0			0
$215$	−6.41090			−0.437220
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−0.440617	−	0.180541i	−0.0297741	−	0.0121999i
$220$		0			0
$221$		−	8.42023i		−	0.566406i
$222$		0			0
$223$		−	11.7633i		−	0.787727i	−0.919169	−	0.393864i	$-0.871138\pi$
$223$		−	11.7633i		−	0.787727i	0.919169	−	0.393864i	$-0.128862\pi$
$224$		0			0
$225$	9.72508	+	9.57767i	0.648339	+	0.638511i
$226$		0			0
$227$	7.01126			0.465354			0.232677	−	0.972554i	$-0.425252\pi$
$227$	7.01126			0.465354			0.232677	+	0.972554i	$0.425252\pi$
$228$		0			0
$229$	16.2749			1.07548			0.537738	−	0.843112i	$-0.319279\pi$
$229$	16.2749			1.07548			0.537738	+	0.843112i	$0.319279\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	15.3141i		−	1.00326i	−0.865082	−	0.501631i	$-0.832734\pi$
$233$		−	15.3141i		−	1.00326i	0.865082	−	0.501631i	$-0.167266\pi$
$234$		0			0
$235$		−	6.45203i		−	0.420884i
$236$		0			0
$237$	−3.41238	+	8.32801i	−0.221658	+	0.540962i
$238$		0			0
$239$	−8.73512			−0.565028			−0.282514	−	0.959263i	$-0.591168\pi$
$239$	−8.73512			−0.565028			−0.282514	+	0.959263i	$0.591168\pi$
$240$		0			0
$241$	7.00000			0.450910			0.225455	−	0.974254i	$-0.427613\pi$
$241$	7.00000			0.450910			0.225455	+	0.974254i	$0.427613\pi$
$242$		0			0
$243$	5.68941	−	14.5131i	0.364976	−	0.931017i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		−	2.62685i		−	0.167142i
$248$		0			0
$249$	3.27492	+	1.34189i	0.207539	+	0.0850387i
$250$		0			0
$251$	−10.7785			−0.680331			−0.340165	−	0.940366i	$-0.610483\pi$
$251$	−10.7785			−0.680331			−0.340165	+	0.940366i	$0.610483\pi$
$252$		0			0
$253$	−16.8248			−1.05776
$254$		0			0
$255$	−4.52730	−	1.85505i	−0.283511	−	0.116168i
$256$		0			0
$257$		−	22.3925i		−	1.39680i	−0.715706	−	0.698402i	$-0.753895\pi$
$257$		−	22.3925i		−	1.39680i	0.715706	−	0.698402i	$-0.246105\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−13.0997	+	13.3013i	−0.810849	+	0.823329i
$262$		0			0
$263$	18.3515			1.13160			0.565800	−	0.824542i	$-0.308567\pi$
$263$	18.3515			1.13160			0.565800	+	0.824542i	$0.308567\pi$
$264$		0			0
$265$	−7.90033			−0.485313
$266$		0			0
$267$	9.17574	−	22.3937i	0.561546	−	1.37047i
$268$		0			0
$269$		−	0.670944i		−	0.0409082i	−0.999791	−	0.0204541i	$-0.993489\pi$
$269$		−	0.670944i		−	0.0409082i	0.999791	−	0.0204541i	$-0.00651119\pi$
$270$		0			0
$271$			20.8422i			1.26607i	0.774123	+	0.633035i	$0.218191\pi$
$271$			20.8422i			1.26607i	−0.774123	+	0.633035i	$0.781809\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−23.8811			−1.44009
$276$		0			0
$277$	5.72508			0.343987			0.171993	−	0.985098i	$-0.444979\pi$
$277$	5.72508			0.343987			0.171993	+	0.985098i	$0.444979\pi$
$278$		0			0
$279$	−3.64607	+	3.70219i	−0.218284	+	0.221644i
$280$		0			0
$281$		−	2.68378i		−	0.160101i	−0.996791	−	0.0800503i	$-0.974492\pi$
$281$		−	2.68378i		−	0.160101i	0.996791	−	0.0800503i	$-0.0255081\pi$
$282$		0			0
$283$		−	18.7490i		−	1.11451i	−0.830340	−	0.557257i	$-0.811854\pi$
$283$		−	18.7490i		−	1.11451i	0.830340	−	0.557257i	$-0.188146\pi$
$284$		0			0
$285$	−1.41238	−	0.578717i	−0.0836619	−	0.0342802i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−0.725083			−0.0426519
$290$		0			0
$291$	5.24879	+	2.15068i	0.307690	+	0.126075i
$292$		0			0
$293$		−	23.0634i		−	1.34738i	−0.739014	−	0.673690i	$-0.764709\pi$
$293$		−	23.0634i		−	1.34738i	0.739014	−	0.673690i	$-0.235291\pi$
$294$		0			0
$295$			5.08106i			0.295831i
$296$		0			0
$297$	10.7251	+	25.0762i	0.622332	+	1.45507i
$298$		0			0
$299$	6.41090			0.370752
$300$		0			0
$301$		0			0
$302$		0			0
$303$	1.88360	−	4.59698i	0.108210	−	0.264090i
$304$		0			0
$305$		−	2.49932i		−	0.143111i
$306$		0			0
$307$			17.3205i			0.988534i	0.869310	+	0.494267i	$0.164563\pi$
$307$			17.3205i			0.988534i	−0.869310	+	0.494267i	$0.835437\pi$
$308$		0			0
$309$	11.1375	−	27.1813i	0.633588	−	1.54629i
$310$		0			0
$311$	4.96792			0.281705			0.140852	−	0.990031i	$-0.455016\pi$
$311$	4.96792			0.281705			0.140852	+	0.990031i	$0.455016\pi$
$312$		0			0
$313$	−19.5498			−1.10502			−0.552511	−	0.833506i	$-0.686330\pi$
$313$	−19.5498			−1.10502			−0.552511	+	0.833506i	$0.686330\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	9.09118i		−	0.510611i	−0.966860	−	0.255306i	$-0.917824\pi$
$317$		−	9.09118i		−	0.510611i	0.966860	−	0.255306i	$-0.0821761\pi$
$318$		0			0
$319$		−	32.6630i		−	1.82878i
$320$		0			0
$321$	18.6873	+	7.65706i	1.04302	+	0.427376i
$322$		0			0
$323$	−5.52967			−0.307679
$324$		0			0
$325$	9.09967			0.504759
$326$		0			0
$327$	−23.7600	−	9.73559i	−1.31393	−	0.538380i
$328$		0			0
$329$		0			0
$330$		0			0
$331$			2.26577i			0.124538i	0.998059	+	0.0622689i	$0.0198336\pi$
$331$			2.26577i			0.124538i	−0.998059	+	0.0622689i	$0.980166\pi$
$332$		0			0
$333$	−13.4124	−	13.2091i	−0.734994	−	0.723852i
$334$		0			0
$335$	2.00473			0.109530
$336$		0			0
$337$	1.82475			0.0994006			0.0497003	−	0.998764i	$-0.484173\pi$
$337$	1.82475			0.0994006			0.0497003	+	0.998764i	$0.484173\pi$
$338$		0			0
$339$	6.41090	−	15.6460i	0.348192	−	0.849774i
$340$		0			0
$341$		−	9.09118i		−	0.492315i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	1.41238	−	3.44695i	0.0760398	−	0.185577i
$346$		0			0
$347$	4.96792			0.266692			0.133346	−	0.991070i	$-0.457428\pi$
$347$	4.96792			0.266692			0.133346	+	0.991070i	$0.457428\pi$
$348$		0			0
$349$	20.5498			1.10001			0.550004	−	0.835162i	$-0.314626\pi$
$349$	20.5498			1.10001			0.550004	+	0.835162i	$0.314626\pi$
$350$		0			0
$351$	−4.08668	−	9.55505i	−0.218131	−	0.510011i
$352$		0			0
$353$		−	4.21012i		−	0.224082i	−0.993704	−	0.112041i	$-0.964261\pi$
$353$		−	4.21012i		−	0.224082i	0.993704	−	0.112041i	$-0.0357388\pi$
$354$		0			0
$355$		−	2.74194i		−	0.145527i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	28.2873			1.49295			0.746474	−	0.665415i	$-0.231745\pi$
$359$	28.2873			1.49295			0.746474	+	0.665415i	$0.231745\pi$
$360$		0			0
$361$	17.2749			0.909206
$362$		0			0
$363$	−26.5248	−	10.8685i	−1.39219	−	0.570447i
$364$		0			0
$365$			0.184454i			0.00965476i
$366$		0			0
$367$			5.31124i			0.277244i	0.990345	+	0.138622i	$0.0442674\pi$
$367$			5.31124i			0.277244i	−0.990345	+	0.138622i	$0.955733\pi$
$368$		0			0
$369$	20.5498	−	20.8661i	1.06978	−	1.08625i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	7.17525			0.371520			0.185760	−	0.982595i	$-0.440525\pi$
$373$	7.17525			0.371520			0.185760	+	0.982595i	$0.440525\pi$
$374$		0			0
$375$	4.20782	−	10.2693i	0.217291	−	0.530305i
$376$		0			0
$377$			12.4459i			0.640996i
$378$		0			0
$379$			10.3923i			0.533817i	0.963722	+	0.266908i	$0.0860021\pi$
$379$			10.3923i			0.533817i	−0.963722	+	0.266908i	$0.913998\pi$
$380$		0			0
$381$	0.274917	−	0.670944i	0.0140844	−	0.0343735i
$382$		0			0
$383$	7.85389			0.401315			0.200657	−	0.979661i	$-0.435692\pi$
$383$	7.85389			0.401315			0.200657	+	0.979661i	$0.435692\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−20.1139	+	20.4235i	−1.02245	+	1.03819i
$388$		0			0
$389$		−	11.2885i		−	0.572348i	−0.958178	−	0.286174i	$-0.907616\pi$
$389$		−	11.2885i		−	0.572348i	0.958178	−	0.286174i	$-0.0923835\pi$
$390$		0			0
$391$		−	13.4953i		−	0.682488i
$392$		0			0
$393$	−8.41238	−	3.44695i	−0.424348	−	0.173875i
$394$		0			0
$395$	3.48633			0.175416
$396$		0			0
$397$	−31.3746			−1.57465			−0.787323	−	0.616541i	$-0.788533\pi$
$397$	−31.3746			−1.57465			−0.787323	+	0.616541i	$0.788533\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	1.15743i		−	0.0577995i	−0.999582	−	0.0288997i	$-0.990800\pi$
$401$		−	1.15743i		−	0.0577995i	0.999582	−	0.0288997i	$-0.00920035\pi$
$402$		0			0
$403$			3.46410i			0.172559i
$404$		0			0
$405$	−6.03779	+	0.0922270i	−0.300020	+	0.00458280i
$406$		0			0
$407$	32.9357			1.63256
$408$		0			0
$409$	−31.5498			−1.56004			−0.780019	−	0.625755i	$-0.784791\pi$
$409$	−31.5498			−1.56004			−0.780019	+	0.625755i	$0.784791\pi$
$410$		0			0
$411$	10.0570	−	24.5443i	0.496074	−	1.21068i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		−	1.37097i		−	0.0672982i
$416$		0			0
$417$	9.17525	−	22.3925i	0.449314	−	1.09656i
$418$		0			0
$419$	−34.3787			−1.67951			−0.839755	−	0.542965i	$-0.817302\pi$
$419$	−34.3787			−1.67951			−0.839755	+	0.542965i	$0.817302\pi$
$420$		0			0
$421$	−19.0997			−0.930861			−0.465430	−	0.885084i	$-0.654100\pi$
$421$	−19.0997			−0.930861			−0.465430	+	0.885084i	$0.654100\pi$
$422$		0			0
$423$	−20.5546	−	20.2430i	−0.999397	−	0.984248i
$424$		0			0
$425$		−	19.1553i		−	0.929170i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	16.8248	+	6.89389i	0.812307	+	0.332840i
$430$		0			0
$431$	1.44298			0.0695061			0.0347530	−	0.999396i	$-0.488936\pi$
$431$	1.44298			0.0695061			0.0347530	+	0.999396i	$0.488936\pi$
$432$		0			0
$433$	−2.00000			−0.0961139			−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−2.00000			−0.0961139			−0.0480569	+	0.998845i	$0.515303\pi$
$434$		0			0
$435$	6.69178	+	2.74194i	0.320846	+	0.131466i
$436$		0			0
$437$		−	4.21012i		−	0.201397i
$438$		0			0
$439$		−	16.5408i		−	0.789449i	−0.918799	−	0.394725i	$-0.870840\pi$
$439$		−	16.5408i		−	0.789449i	0.918799	−	0.394725i	$-0.129160\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	5.81054			0.276067			0.138034	−	0.990428i	$-0.455922\pi$
$443$	5.81054			0.276067			0.138034	+	0.990428i	$0.455922\pi$
$444$		0			0
$445$	−9.37459			−0.444398
$446$		0			0
$447$	9.17574	−	22.3937i	0.433997	−	1.05918i
$448$		0			0
$449$		−	15.1297i		−	0.714013i	−0.934102	−	0.357007i	$-0.883797\pi$
$449$		−	15.1297i		−	0.714013i	0.934102	−	0.357007i	$-0.116203\pi$
$450$		0			0
$451$			51.2394i			2.41277i
$452$		0			0
$453$	5.13746	−	12.5381i	0.241379	−	0.589092i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−16.6495			−0.778831			−0.389415	−	0.921062i	$-0.627323\pi$
$457$	−16.6495			−0.778831			−0.389415	+	0.921062i	$0.627323\pi$
$458$		0			0
$459$	−20.1139	+	8.60271i	−0.938838	+	0.401540i
$460$		0			0
$461$			12.4459i			0.579663i	0.957078	+	0.289832i	$0.0935993\pi$
$461$			12.4459i			0.579663i	−0.957078	+	0.289832i	$0.906401\pi$
$462$		0			0
$463$			20.8997i			0.971291i	0.874156	+	0.485646i	$0.161415\pi$
$463$			20.8997i			0.971291i	−0.874156	+	0.485646i	$0.838585\pi$
$464$		0			0
$465$	1.86254	+	0.763171i	0.0863733	+	0.0353912i
$466$		0			0
$467$	7.85389			0.363435			0.181717	−	0.983351i	$-0.441834\pi$
$467$	7.85389			0.363435			0.181717	+	0.983351i	$0.441834\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	37.4630	+	15.3504i	1.72621	+	0.707308i
$472$		0			0
$473$		−	50.1525i		−	2.30601i
$474$		0			0
$475$		−	5.97586i		−	0.274191i
$476$		0			0
$477$	−24.7870	+	25.1685i	−1.13492	+	1.15239i
$478$		0			0
$479$	−32.9357			−1.50487			−0.752436	−	0.658665i	$-0.771121\pi$
$479$	−32.9357			−1.50487			−0.752436	+	0.658665i	$0.771121\pi$
$480$		0			0
$481$	−12.5498			−0.572223
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	2.19729i		−	0.0997736i
$486$		0			0
$487$		−	11.2871i		−	0.511467i	−0.966747	−	0.255734i	$-0.917683\pi$
$487$		−	11.2871i		−	0.511467i	0.966747	−	0.255734i	$-0.0823170\pi$
$488$		0			0
$489$	−1.41238	+	3.44695i	−0.0638698	+	0.155876i
$490$		0			0
$491$	6.69178			0.301996			0.150998	−	0.988534i	$-0.451751\pi$
$491$	6.69178			0.301996			0.150998	+	0.988534i	$0.451751\pi$
$492$		0			0
$493$	26.1993			1.17996
$494$		0			0
$495$	7.41327	−	7.52737i	0.333202	−	0.338330i
$496$		0			0
$497$		0			0
$498$		0			0
$499$		−	3.82518i		−	0.171239i	−0.996328	−	0.0856194i	$-0.972713\pi$
$499$		−	3.82518i		−	0.171239i	0.996328	−	0.0856194i	$-0.0272869\pi$
$500$		0			0
$501$	−33.6495	−	13.7878i	−1.50335	−	0.615993i
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$	−1.92442			−0.0856356
$506$		0			0
$507$	14.4245	+	5.91041i	0.640616	+	0.262490i
$508$		0			0
$509$		−	11.7750i		−	0.521916i	−0.965350	−	0.260958i	$-0.915962\pi$
$509$		−	11.7750i		−	0.521916i	0.965350	−	0.260958i	$-0.0840383\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−6.27492	+	2.68378i	−0.277044	+	0.118492i
$514$		0			0
$515$	−11.3788			−0.501411
$516$		0			0
$517$	50.4743			2.21986
$518$		0			0
$519$	10.9382	−	26.6950i	0.480134	−	1.17178i
$520$		0			0
$521$			32.1546i			1.40872i	0.709844	+	0.704359i	$0.248766\pi$
$521$			32.1546i			1.40872i	−0.709844	+	0.704359i	$0.751234\pi$
$522$		0			0
$523$		−	5.72987i		−	0.250550i	−0.992122	−	0.125275i	$-0.960019\pi$
$523$		−	5.72987i		−	0.250550i	0.992122	−	0.125275i	$-0.0399812\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	7.29214			0.317650
$528$		0			0
$529$	−12.7251			−0.553264
$530$		0			0
$531$	16.1870	+	15.9416i	0.702456	+	0.691808i
$532$		0			0
$533$		−	19.5242i		−	0.845689i
$534$		0			0
$535$		−	7.82300i		−	0.338218i
$536$		0			0
$537$	25.6873	+	10.5253i	1.10849	+	0.454200i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−34.2749			−1.47359			−0.736797	−	0.676114i	$-0.763663\pi$
$541$	−34.2749			−1.47359			−0.736797	+	0.676114i	$0.763663\pi$
$542$		0			0
$543$	24.2006	+	9.91613i	1.03855	+	0.425542i
$544$		0			0
$545$			9.94658i			0.426064i
$546$		0			0
$547$			4.30136i			0.183913i	0.995763	+	0.0919563i	$0.0293120\pi$
$547$			4.30136i			0.183913i	−0.995763	+	0.0919563i	$0.970688\pi$
$548$		0			0
$549$	−7.96221	−	7.84152i	−0.339819	−	0.334668i
$550$		0			0
$551$	8.17337			0.348197
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−2.76483	+	6.74766i	−0.117361	+	0.286422i
$556$		0			0
$557$		−	22.8790i		−	0.969413i	−0.874677	−	0.484706i	$-0.838926\pi$
$557$		−	22.8790i		−	0.969413i	0.874677	−	0.484706i	$-0.161074\pi$
$558$		0			0
$559$			19.1101i			0.808271i
$560$		0			0
$561$	14.5120	−	35.4171i	0.612699	−	1.49531i
$562$		0			0
$563$	−30.3307			−1.27828			−0.639142	−	0.769088i	$-0.720711\pi$
$563$	−30.3307			−1.27828			−0.639142	+	0.769088i	$0.720711\pi$
$564$		0			0
$565$	−6.54983			−0.275554
$566$		0			0
$567$		0			0
$568$		0			0
$569$			31.7857i			1.33253i	0.745717	+	0.666263i	$0.232107\pi$
$569$			31.7857i			1.33253i	−0.745717	+	0.666263i	$0.767893\pi$
$570$		0			0
$571$		−	12.6581i		−	0.529724i	−0.964286	−	0.264862i	$-0.914674\pi$
$571$		−	12.6581i		−	0.529724i	0.964286	−	0.264862i	$-0.0853264\pi$
$572$		0			0
$573$	−19.1375	−	7.84152i	−0.799479	−	0.327584i
$574$		0			0
$575$	14.5843			0.608206
$576$		0			0
$577$	−3.90033			−0.162373			−0.0811865	−	0.996699i	$-0.525871\pi$
$577$	−3.90033			−0.162373			−0.0811865	+	0.996699i	$0.525871\pi$
$578$		0			0
$579$	29.0088	+	11.8863i	1.20556	+	0.493977i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		−	61.8043i		−	2.55967i
$584$		0			0
$585$	−2.82475	+	2.86823i	−0.116789	+	0.118587i
$586$		0			0
$587$	2.04334			0.0843378			0.0421689	−	0.999110i	$-0.486573\pi$
$587$	2.04334			0.0843378			0.0421689	+	0.999110i	$0.486573\pi$
$588$		0			0
$589$	2.27492			0.0937363
$590$		0			0
$591$	−4.64843	+	11.3446i	−0.191211	+	0.466656i
$592$		0			0
$593$		−	2.86823i		−	0.117784i	−0.998264	−	0.0588920i	$-0.981243\pi$
$593$		−	2.86823i		−	0.117784i	0.998264	−	0.0588920i	$-0.0187568\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	16.7870	−	40.9691i	0.687045	−	1.67675i
$598$		0			0
$599$	26.5248			1.08378			0.541888	−	0.840451i	$-0.317710\pi$
$599$	26.5248			1.08378			0.541888	+	0.840451i	$0.317710\pi$
$600$		0			0
$601$	−34.9244			−1.42460			−0.712298	−	0.701877i	$-0.752346\pi$
$601$	−34.9244			−1.42460			−0.712298	+	0.701877i	$0.752346\pi$
$602$		0			0
$603$	6.28977	−	6.38658i	0.256139	−	0.260082i
$604$		0			0
$605$			11.1040i			0.451442i
$606$		0			0
$607$			17.2630i			0.700682i	0.936622	+	0.350341i	$0.113934\pi$
$607$			17.2630i			0.700682i	−0.936622	+	0.350341i	$0.886066\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−19.2327			−0.778072
$612$		0			0
$613$	15.1752			0.612923			0.306461	−	0.951883i	$-0.400855\pi$
$613$	15.1752			0.612923			0.306461	+	0.951883i	$0.400855\pi$
$614$		0			0
$615$	−10.4976	−	4.30136i	−0.423304	−	0.173447i
$616$		0			0
$617$		−	23.9188i		−	0.962935i	−0.876464	−	0.481468i	$-0.840104\pi$
$617$		−	23.9188i		−	0.962935i	0.876464	−	0.481468i	$-0.159896\pi$
$618$		0			0
$619$		−	19.5863i		−	0.787239i	−0.919274	−	0.393619i	$-0.871223\pi$
$619$		−	19.5863i		−	0.787239i	0.919274	−	0.393619i	$-0.128777\pi$
$620$		0			0
$621$	−6.54983	−	15.3141i	−0.262836	−	0.614535i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	18.4502			0.738007
$626$		0			0
$627$	4.52730	−	11.0490i	0.180803	−	0.441255i
$628$		0			0
$629$			26.4181i			1.05336i
$630$		0			0
$631$		−	28.3616i		−	1.12906i	−0.825413	−	0.564529i	$-0.809058\pi$
$631$		−	28.3616i		−	1.12906i	0.825413	−	0.564529i	$-0.190942\pi$
$632$		0			0
$633$	10.2749	−	25.0762i	0.408391	−	0.996691i
$634$		0			0
$635$	−0.280875			−0.0111462
$636$		0			0
$637$		0			0
$638$		0			0
$639$	−8.73512	−	8.60271i	−0.345556	−	0.340318i
$640$		0			0
$641$		−	13.9722i		−	0.551870i	−0.961176	−	0.275935i	$-0.911013\pi$
$641$		−	13.9722i		−	0.551870i	0.961176	−	0.275935i	$-0.0889875\pi$
$642$		0			0
$643$		−	2.62685i		−	0.103593i	−0.998658	−	0.0517964i	$-0.983505\pi$
$643$		−	2.62685i		−	0.103593i	0.998658	−	0.0517964i	$-0.0164947\pi$
$644$		0			0
$645$	10.2749	+	4.21012i	0.404574	+	0.165773i
$646$		0			0
$647$	30.6115			1.20346			0.601732	−	0.798698i	$-0.294478\pi$
$647$	30.6115			1.20346			0.601732	+	0.798698i	$0.294478\pi$
$648$		0			0
$649$	−39.7492			−1.56029
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	33.9830i		−	1.32986i	−0.746907	−	0.664928i	$-0.768462\pi$
$653$		−	33.9830i		−	1.32986i	0.746907	−	0.664928i	$-0.231538\pi$
$654$		0			0
$655$			3.52165i			0.137602i
$656$		0			0
$657$	0.587624	+	0.578717i	0.0229254	+	0.0225779i
$658$		0			0
$659$	−33.2552			−1.29544			−0.647720	−	0.761879i	$-0.724277\pi$
$659$	−33.2552			−1.29544			−0.647720	+	0.761879i	$0.724277\pi$
$660$		0			0
$661$	21.7251			0.845008			0.422504	−	0.906361i	$-0.361151\pi$
$661$	21.7251			0.845008			0.422504	+	0.906361i	$0.361151\pi$
$662$		0			0
$663$	−5.52967	+	13.4953i	−0.214755	+	0.524115i
$664$		0			0
$665$		0			0
$666$		0			0
$667$			19.9474i			0.772365i
$668$		0			0
$669$	−7.72508	+	18.8533i	−0.298669	+	0.728911i
$670$		0			0
$671$	19.5522			0.754804
$672$		0			0
$673$	4.72508			0.182139			0.0910693	−	0.995845i	$-0.470972\pi$
$673$	4.72508			0.182139			0.0910693	+	0.995845i	$0.470972\pi$
$674$		0			0
$675$	−9.29687	−	21.7370i	−0.357837	−	0.836656i
$676$		0			0
$677$		−	38.3775i		−	1.47497i	−0.675364	−	0.737484i	$-0.736014\pi$
$677$		−	38.3775i		−	1.47497i	0.675364	−	0.737484i	$-0.263986\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−11.2371	−	4.60438i	−0.430608	−	0.176440i
$682$		0			0
$683$	36.7416			1.40588			0.702938	−	0.711251i	$-0.251871\pi$
$683$	36.7416			1.40588			0.702938	+	0.711251i	$0.251871\pi$
$684$		0			0
$685$	−10.2749			−0.392584
$686$		0			0
$687$	−26.0842	−	10.6879i	−0.995175	−	0.407770i
$688$		0			0
$689$			23.5499i			0.897180i
$690$		0			0
$691$			8.35671i			0.317904i	0.987286	+	0.158952i	$0.0508116\pi$
$691$			8.35671i			0.317904i	−0.987286	+	0.158952i	$0.949188\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−9.37409			−0.355579
$696$		0			0
$697$	−41.0997			−1.55676
$698$		0			0
$699$	−10.0570	+	24.5443i	−0.380390	+	0.928352i
$700$		0			0
$701$			8.90672i			0.336402i	0.985753	+	0.168201i	$0.0537958\pi$
$701$			8.90672i			0.336402i	−0.985753	+	0.168201i	$0.946204\pi$
$702$		0			0
$703$			8.24163i			0.310839i
$704$		0			0
$705$	−4.23713	+	10.3408i	−0.159579	+	0.389458i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	17.3746			0.652516			0.326258	−	0.945281i	$-0.394212\pi$
$709$	17.3746			0.652516			0.326258	+	0.945281i	$0.394212\pi$
$710$		0			0
$711$	10.9382	−	11.1066i	0.410215	−	0.416529i
$712$		0			0
$713$			5.55200i			0.207924i
$714$		0			0
$715$		−	7.04329i		−	0.263404i
$716$		0			0
$717$	14.0000	+	5.73646i	0.522840	+	0.214232i
$718$		0			0
$719$	11.3788			0.424358			0.212179	−	0.977231i	$-0.431944\pi$
$719$	11.3788			0.424358			0.212179	+	0.977231i	$0.431944\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−11.2191	−	4.59698i	−0.417242	−	0.170964i
$724$		0			0
$725$			28.3134i			1.05153i
$726$		0			0
$727$			32.5479i			1.20713i	0.797312	+	0.603567i	$0.206254\pi$
$727$			32.5479i			1.20713i	−0.797312	+	0.603567i	$0.793746\pi$
$728$		0			0
$729$	−18.6495	+	19.5242i	−0.690722	+	0.723120i
$730$		0			0
$731$	40.2279			1.48788
$732$		0			0
$733$	25.7251			0.950178			0.475089	−	0.879938i	$-0.342416\pi$
$733$	25.7251			0.950178			0.475089	+	0.879938i	$0.342416\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			15.6830i			0.577692i
$738$		0			0
$739$			32.7205i			1.20364i	0.798630	+	0.601822i	$0.205558\pi$
$739$			32.7205i			1.20364i	−0.798630	+	0.601822i	$0.794442\pi$
$740$		0			0
$741$	−1.72508	+	4.21012i	−0.0633725	+	0.154663i
$742$		0			0
$743$	17.4702			0.640921			0.320460	−	0.947262i	$-0.396162\pi$
$743$	17.4702			0.640921			0.320460	+	0.947262i	$0.396162\pi$
$744$		0			0
$745$	−9.37459			−0.343458
$746$		0			0
$747$	−4.36756	−	4.30136i	−0.159801	−	0.157378i
$748$		0			0
$749$		0			0
$750$		0			0
$751$			26.0958i			0.952251i	0.879377	+	0.476126i	$0.157959\pi$
$751$			26.0958i			0.952251i	−0.879377	+	0.476126i	$0.842041\pi$
$752$		0			0
$753$	17.2749	+	7.07835i	0.629533	+	0.257949i
$754$		0			0
$755$	−5.24879			−0.191023
$756$		0			0
$757$	15.0997			0.548807			0.274403	−	0.961615i	$-0.411520\pi$
$757$	15.0997			0.548807			0.274403	+	0.961615i	$0.411520\pi$
$758$		0			0
$759$	26.9655	+	11.0490i	0.978784	+	0.401054i
$760$		0			0
$761$			36.1803i			1.31153i	0.754964	+	0.655767i	$0.227654\pi$
$761$			36.1803i			1.31153i	−0.754964	+	0.655767i	$0.772346\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	6.03779	+	5.94627i	0.218297	+	0.214988i
$766$		0			0
$767$	15.1460			0.546891
$768$		0			0
$769$	−6.17525			−0.222685			−0.111343	−	0.993782i	$-0.535515\pi$
$769$	−6.17525			−0.222685			−0.111343	+	0.993782i	$0.535515\pi$
$770$		0			0
$771$	−14.7054	+	35.8890i	−0.529602	+	1.29251i
$772$		0			0
$773$		−	5.92091i		−	0.212960i	−0.994315	−	0.106480i	$-0.966042\pi$
$773$		−	5.92091i		−	0.212960i	0.994315	−	0.106480i	$-0.0339581\pi$
$774$		0			0
$775$			7.88054i			0.283078i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−12.8218			−0.459389
$780$		0			0
$781$	21.4502			0.767547
$782$		0			0
$783$	29.7303	−	12.7156i	1.06247	−	0.454419i
$784$		0			0
$785$		−	15.6830i		−	0.559751i
$786$		0			0
$787$			1.42851i			0.0509209i	0.999676	+	0.0254605i	$0.00810520\pi$
$787$			1.42851i			0.0509209i	−0.999676	+	0.0254605i	$0.991895\pi$
$788$		0			0
$789$	−29.4124	−	12.0516i	−1.04711	−	0.429050i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−7.45017			−0.264563
$794$		0			0
$795$	12.6621	+	5.18824i	0.449077	+	0.184008i
$796$		0			0
$797$			0.855398i			0.0302997i	0.999885	+	0.0151499i	$0.00482254\pi$
$797$			0.855398i			0.0302997i	−0.999885	+	0.0151499i	$0.995177\pi$
$798$		0			0
$799$			40.4860i			1.43229i
$800$		0			0
$801$	−29.4124	+	29.8651i	−1.03924	+	1.05523i
$802$		0			0
$803$	−1.44298			−0.0509218
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−0.440617	+	1.07534i	−0.0155104	+	0.0378537i
$808$		0			0
$809$			23.3655i			0.821486i	0.911751	+	0.410743i	$0.134731\pi$
$809$			23.3655i			0.821486i	−0.911751	+	0.410743i	$0.865269\pi$
$810$		0			0
$811$			21.7370i			0.763288i	0.924309	+	0.381644i	$0.124642\pi$
$811$			21.7370i			0.763288i	−0.924309	+	0.381644i	$0.875358\pi$
$812$		0			0
$813$	13.6873	−	33.4043i	0.480034	−	1.17154i
$814$		0			0
$815$	1.44298			0.0505455
$816$		0			0
$817$	12.5498			0.439063
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	13.4857i		−	0.470656i	−0.971916	−	0.235328i	$-0.924384\pi$
$821$		−	13.4857i		−	0.470656i	0.971916	−	0.235328i	$-0.0756164\pi$
$822$		0			0
$823$			6.45203i			0.224904i	0.993657	+	0.112452i	$0.0358704\pi$
$823$			6.45203i			0.224904i	−0.993657	+	0.112452i	$0.964130\pi$
$824$		0			0
$825$	38.2749	+	15.6830i	1.33256	+	0.546013i
$826$		0			0
$827$	27.6870			0.962770			0.481385	−	0.876509i	$-0.340134\pi$
$827$	27.6870			0.962770			0.481385	+	0.876509i	$0.340134\pi$
$828$		0			0
$829$	−27.3746			−0.950759			−0.475379	−	0.879781i	$-0.657689\pi$
$829$	−27.3746			−0.950759			−0.475379	+	0.879781i	$0.657689\pi$
$830$		0			0
$831$	−9.17574	−	3.75973i	−0.318303	−	0.130424i
$832$		0			0
$833$		0			0
$834$		0			0
$835$			14.0866i			0.487486i
$836$		0			0
$837$	8.27492	−	3.53917i	0.286023	−	0.122332i
$838$		0			0
$839$	8.17337			0.282176			0.141088	−	0.989997i	$-0.454940\pi$
$839$	8.17337			0.282176			0.141088	+	0.989997i	$0.454940\pi$
$840$		0			0
$841$	−9.72508			−0.335348
$842$		0			0
$843$	−1.76247	+	4.30136i	−0.0607026	+	0.148147i
$844$		0			0
$845$		−	6.03849i		−	0.207731i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−12.3127	+	30.0495i	−0.422571	+	1.03130i
$850$		0			0
$851$	−20.1139			−0.689497
$852$		0			0
$853$	23.4502			0.802918			0.401459	−	0.915877i	$-0.368503\pi$
$853$	23.4502			0.802918			0.401459	+	0.915877i	$0.368503\pi$
$854$		0			0
$855$	1.88360	+	1.85505i	0.0644178	+	0.0634413i
$856$		0			0
$857$		−	19.3398i		−	0.660635i	−0.943870	−	0.330317i	$-0.892844\pi$
$857$		−	19.3398i		−	0.660635i	0.943870	−	0.330317i	$-0.107156\pi$
$858$		0			0
$859$			7.51946i			0.256561i	0.991738	+	0.128280i	$0.0409457\pi$
$859$			7.51946i			0.256561i	−0.991738	+	0.128280i	$0.959054\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	3.20545			0.109115			0.0545574	−	0.998511i	$-0.482625\pi$
$863$	3.20545			0.109115			0.0545574	+	0.998511i	$0.482625\pi$
$864$		0			0
$865$	−11.1752			−0.379970
$866$		0			0
$867$	1.16211	+	0.476171i	0.0394673	+	0.0161716i
$868$		0			0
$869$			27.2735i			0.925191i
$870$		0			0
$871$		−	5.97586i		−	0.202484i
$872$		0			0
$873$	−7.00000	−	6.89389i	−0.236914	−	0.233323i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	10.8248			0.365526			0.182763	−	0.983157i	$-0.441496\pi$
$877$	10.8248			0.365526			0.182763	+	0.983157i	$0.441496\pi$
$878$		0			0
$879$	−15.1460	+	36.9643i	−0.510863	+	1.24678i
$880$		0			0
$881$		−	44.4160i		−	1.49641i	−0.663465	−	0.748207i	$-0.730915\pi$
$881$		−	44.4160i		−	1.49641i	0.663465	−	0.748207i	$-0.269085\pi$
$882$		0			0
$883$		−	38.1051i		−	1.28234i	−0.767399	−	0.641170i	$-0.778449\pi$
$883$		−	38.1051i		−	1.28234i	0.767399	−	0.641170i	$-0.221551\pi$
$884$		0			0
$885$	3.33680	−	8.14355i	0.112165	−	0.273743i
$886$		0			0
$887$	−1.44298			−0.0484507			−0.0242253	−	0.999707i	$-0.507712\pi$
$887$	−1.44298			−0.0484507			−0.0242253	+	0.999707i	$0.507712\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−0.721492	−	47.2336i	−0.0241709	−	1.58239i
$892$		0			0
$893$			12.6304i			0.422659i
$894$		0			0
$895$		−	10.7534i		−	0.359446i
$896$		0			0
$897$	−10.2749	−	4.21012i	−0.343070	−	0.140572i
$898$		0			0
$899$	−10.7785			−0.359482
$900$		0			0
$901$	49.5739			1.65155
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	10.1310i		−	0.336767i
$906$		0			0
$907$		−	4.05536i		−	0.134656i	−0.997731	−	0.0673280i	$-0.978553\pi$
$907$		−	4.05536i		−	0.134656i	0.997731	−	0.0673280i	$-0.0214474\pi$
$908$		0			0
$909$	−6.03779	+	6.13072i	−0.200261	+	0.203343i
$910$		0			0
$911$	−37.9037			−1.25580			−0.627902	−	0.778292i	$-0.716086\pi$
$911$	−37.9037			−1.25580			−0.627902	+	0.778292i	$0.716086\pi$
$912$		0			0
$913$	10.7251			0.354948
$914$		0			0
$915$	−1.64133	+	4.00573i	−0.0542608	+	0.132425i
$916$		0			0
$917$		0			0
$918$		0			0
$919$			39.4185i			1.30030i	0.759807	+	0.650149i	$0.225293\pi$
$919$			39.4185i			1.30030i	−0.759807	+	0.650149i	$0.774707\pi$
$920$		0			0
$921$	11.3746	−	27.7600i	0.374805	−	0.914724i
$922$		0			0
$923$	−8.17337			−0.269030
$924$		0			0
$925$	−28.5498			−0.938713
$926$		0			0
$927$	−35.7006	+	36.2501i	−1.17256	+	1.19061i
$928$		0			0
$929$			26.4181i			0.866751i	0.901213	+	0.433375i	$0.142678\pi$
$929$			26.4181i			0.866751i	−0.901213	+	0.433375i	$0.857322\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	−7.96221	−	3.26249i	−0.260671	−	0.106809i
$934$		0			0
$935$	−14.8265			−0.484880
$936$		0			0
$937$	−49.8248			−1.62770			−0.813852	−	0.581072i	$-0.802633\pi$
$937$	−49.8248			−1.62770			−0.813852	+	0.581072i	$0.802633\pi$
$938$		0			0
$939$	31.3330	+	12.8386i	1.02251	+	0.418972i
$940$		0			0
$941$		−	20.1952i		−	0.658344i	−0.944270	−	0.329172i	$-0.893230\pi$
$941$		−	20.1952i		−	0.658344i	0.944270	−	0.329172i	$-0.106770\pi$
$942$		0			0
$943$		−	31.2920i		−	1.01901i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	24.2006			0.786415			0.393207	−	0.919450i	$-0.371365\pi$
$947$	24.2006			0.786415			0.393207	+	0.919450i	$0.371365\pi$
$948$		0			0
$949$	0.549834			0.0178484
$950$		0			0
$951$	−5.97029	+	14.5707i	−0.193600	+	0.472486i
$952$		0			0
$953$			50.5214i			1.63655i	0.574828	+	0.818274i	$0.305069\pi$
$953$			50.5214i			1.63655i	−0.574828	+	0.818274i	$0.694931\pi$
$954$		0			0
$955$			8.01145i			0.259244i
$956$		0			0
$957$	−21.4502	+	52.3498i	−0.693385	+	1.69223i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	28.0000			0.903226
$962$		0			0
$963$	−24.9221	−	24.5443i	−0.803104	−	0.790930i
$964$		0			0
$965$		−	12.1439i		−	0.390925i
$966$		0			0
$967$		−	9.02134i		−	0.290107i	−0.989424	−	0.145053i	$-0.953665\pi$
$967$		−	9.02134i		−	0.290107i	0.989424	−	0.145053i	$-0.0463354\pi$
$968$		0			0
$969$	8.86254	+	3.63140i	0.284706	+	0.116657i
$970$		0			0
$971$	−7.01126			−0.225002			−0.112501	−	0.993652i	$-0.535886\pi$
$971$	−7.01126			−0.225002			−0.112501	+	0.993652i	$0.535886\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	−14.5843	−	5.97586i	−0.467071	−	0.191381i
$976$		0			0
$977$			40.2059i			1.28630i	0.765740	+	0.643151i	$0.222373\pi$
$977$			40.2059i			1.28630i	−0.765740	+	0.643151i	$0.777627\pi$
$978$		0			0
$979$		−	73.3374i		−	2.34387i
$980$		0			0
$981$	31.6873	+	31.2070i	1.01170	+	0.996362i
$982$		0			0
$983$	−32.3740			−1.03257			−0.516285	−	0.856417i	$-0.672685\pi$
$983$	−32.3740			−1.03257			−0.516285	+	0.856417i	$0.672685\pi$
$984$		0			0
$985$	4.74917			0.151321
$986$		0			0
$987$		0			0
$988$		0			0
$989$			30.6283i			0.973922i
$990$		0			0
$991$		−	15.7035i		−	0.498840i	−0.968395	−	0.249420i	$-0.919760\pi$
$991$		−	15.7035i		−	0.498840i	0.968395	−	0.249420i	$-0.0802399\pi$
$992$		0			0
$993$	1.48796	−	3.63140i	0.0472188	−	0.115239i
$994$		0			0
$995$	−17.1508			−0.543716
$996$		0			0
$997$	13.7251			0.434678			0.217339	−	0.976096i	$-0.430262\pi$
$997$	13.7251			0.434678			0.217339	+	0.976096i	$0.430262\pi$
$998$		0			0
$999$	12.8218	+	29.9786i	0.405664	+	0.948480i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.2.h.n.2255.1		8
3.2	odd	2	inner	2352.2.h.n.2255.7		8
4.3	odd	2	inner	2352.2.h.n.2255.8		8
7.2	even	3		336.2.bj.e.95.4	yes	8
7.4	even	3		336.2.bj.g.191.2	yes	8
7.6	odd	2		2352.2.h.m.2255.8		8
12.11	even	2	inner	2352.2.h.n.2255.2		8
21.2	odd	6		336.2.bj.e.95.3	✓	8
21.11	odd	6		336.2.bj.g.191.1	yes	8
21.20	even	2		2352.2.h.m.2255.2		8
28.11	odd	6		336.2.bj.e.191.3	yes	8
28.23	odd	6		336.2.bj.g.95.1	yes	8
28.27	even	2		2352.2.h.m.2255.1		8
84.11	even	6		336.2.bj.e.191.4	yes	8
84.23	even	6		336.2.bj.g.95.2	yes	8
84.83	odd	2		2352.2.h.m.2255.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.2.bj.e.95.3	✓	8	21.2	odd	6
336.2.bj.e.95.4	yes	8	7.2	even	3
336.2.bj.e.191.3	yes	8	28.11	odd	6
336.2.bj.e.191.4	yes	8	84.11	even	6
336.2.bj.g.95.1	yes	8	28.23	odd	6
336.2.bj.g.95.2	yes	8	84.23	even	6
336.2.bj.g.191.1	yes	8	21.11	odd	6
336.2.bj.g.191.2	yes	8	7.4	even	3
2352.2.h.m.2255.1		8	28.27	even	2
2352.2.h.m.2255.2		8	21.20	even	2
2352.2.h.m.2255.7		8	84.83	odd	2
2352.2.h.m.2255.8		8	7.6	odd	2
2352.2.h.n.2255.1		8	1.1	even	1	trivial
2352.2.h.n.2255.2		8	12.11	even	2	inner
2352.2.h.n.2255.7		8	3.2	odd	2	inner
2352.2.h.n.2255.8		8	4.3	odd	2	inner

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:	\( N \)	\(=\)	\( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2352.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.60273 - 0.656712i) q^{3} +0.670944i q^{5} +(2.13746 + 2.10506i) q^{9} +O(q^{10})\)	\(q+(-1.60273 - 0.656712i) q^{3} +0.670944i q^{5} +(2.13746 + 2.10506i) q^{9} -5.24879 q^{11} +2.00000 q^{13} +(0.440617 - 1.07534i) q^{15} -4.21012i q^{17} -1.31342i q^{19} +3.20545 q^{23} +4.54983 q^{25} +(-2.04334 - 4.77753i) q^{27} +6.22295i q^{29} +1.73205i q^{31} +(8.41238 + 3.44695i) q^{33} -6.27492 q^{37} +(-3.20545 - 1.31342i) q^{39} -9.76212i q^{41} +9.55505i q^{43} +(-1.41238 + 1.43411i) q^{45} -9.61635 q^{47} +(-2.76483 + 6.74766i) q^{51} +11.7750i q^{53} -3.52165i q^{55} +(-0.862541 + 2.10506i) q^{57} +7.57301 q^{59} -3.72508 q^{61} +1.34189i q^{65} -2.98793i q^{67} +(-5.13746 - 2.10506i) q^{69} -4.08668 q^{71} +0.274917 q^{73} +(-7.29214 - 2.98793i) q^{75} -5.19615i q^{79} +(0.137459 + 8.99895i) q^{81} -2.04334 q^{83} +2.82475 q^{85} +(4.08668 - 9.97368i) q^{87} +13.9722i q^{89} +(1.13746 - 2.77600i) q^{93} +0.881234 q^{95} -3.27492 q^{97} +(-11.2191 - 11.0490i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{9}+O(q^{10}) \) 8 * q + 2 * q^9	\( 8 q + 2 q^{9} + 16 q^{13} - 24 q^{25} + 22 q^{33} - 20 q^{37} + 34 q^{45} - 22 q^{57} - 60 q^{61} - 26 q^{69} - 28 q^{73} - 14 q^{81} - 68 q^{85} - 6 q^{93} + 4 q^{97}+O(q^{100}) \) 8 * q + 2 * q^9 + 16 * q^13 - 24 * q^25 + 22 * q^33 - 20 * q^37 + 34 * q^45 - 22 * q^57 - 60 * q^61 - 26 * q^69 - 28 * q^73 - 14 * q^81 - 68 * q^85 - 6 * q^93 + 4 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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