LMFDB - Embedded newform 1152.3.m.c.991.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1152,3,Mod(415,1152)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1152, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 1, 0]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1152.415");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1152 = 2^{7} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1152.m (of order $4$, degree $2$, not 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:	$31.3897264543$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 6 x^{14} - 4 x^{13} + 10 x^{12} + 56 x^{11} + 88 x^{10} - 128 x^{9} - 496 x^{8} - 512 x^{7} + \cdots + 65536 $ x^16 - 6x^14 - 4x^13 + 10x^12 + 56x^11 + 88x^10 - 128x^9 - 496x^8 - 512x^7 + 1408x^6 + 3584x^5 + 2560x^4 - 4096x^3 - 24576*x^2 + 65536
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{28} $
Twist minimal:	no (minimal twist has level 48)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			991.6
Root			$-1.96679 + 0.362960i$ of defining polynomial
Character	$\chi$	$=$	1152.991
Dual form			1152.3.m.c.415.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.69930 - 1.69930i) q^{5} +5.74280 q^{7} +O(q^{10})$	$q+(1.69930 - 1.69930i) q^{5} +5.74280 q^{7} +(5.59560 + 5.59560i) q^{11} +(13.5782 + 13.5782i) q^{13} -19.7023 q^{17} +(-21.6943 + 21.6943i) q^{19} +24.9257 q^{23} +19.2247i q^{25} +(1.50581 + 1.50581i) q^{29} +2.20037i q^{31} +(9.75877 - 9.75877i) q^{35} +(-27.6956 + 27.6956i) q^{37} -51.3127i q^{41} +(21.4400 + 21.4400i) q^{43} -76.5216i q^{47} -16.0202 q^{49} +(-56.5145 + 56.5145i) q^{53} +19.0173 q^{55} +(48.0041 + 48.0041i) q^{59} +(51.5587 + 51.5587i) q^{61} +46.1469 q^{65} +(63.4445 - 63.4445i) q^{67} +43.4856 q^{71} +73.9992i q^{73} +(32.1344 + 32.1344i) q^{77} +4.12659i q^{79} +(-38.4428 + 38.4428i) q^{83} +(-33.4803 + 33.4803i) q^{85} +52.9839i q^{89} +(77.9767 + 77.9767i) q^{91} +73.7305i q^{95} +23.1008 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 32 q^{11} - 32 q^{19} - 128 q^{23} + 32 q^{29} - 96 q^{35} + 96 q^{37} + 160 q^{43} + 112 q^{49} - 160 q^{53} + 256 q^{55} + 128 q^{59} + 32 q^{61} + 32 q^{65} + 320 q^{67} + 512 q^{71} + 224 q^{77} + 160 q^{83} - 160 q^{85} - 480 q^{91}+O(q^{100}) $ 16 * q - 32 * q^11 - 32 * q^19 - 128 * q^23 + 32 * q^29 - 96 * q^35 + 96 * q^37 + 160 * q^43 + 112 * q^49 - 160 * q^53 + 256 * q^55 + 128 * q^59 + 32 * q^61 + 32 * q^65 + 320 * q^67 + 512 * q^71 + 224 * q^77 + 160 * q^83 - 160 * q^85 - 480 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times$.

$n$	$127$	$641$	$901$
$\chi(n)$	$-1$	$1$	$e\left(\frac{3}{4}\right)$

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
$5$	1.69930	−	1.69930i	0.339861	−	0.339861i	−0.516454	−	0.856315i	$-0.672748\pi$
$5$	1.69930	−	1.69930i	0.339861	−	0.339861i	0.856315	+	0.516454i	$0.172748\pi$
$6$		0			0
$7$	5.74280			0.820400			0.410200	−	0.911996i	$-0.365459\pi$
$7$	5.74280			0.820400			0.410200	+	0.911996i	$0.365459\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	5.59560	+	5.59560i	0.508691	+	0.508691i	0.914125	−	0.405434i	$-0.132879\pi$
$11$	5.59560	+	5.59560i	0.508691	+	0.508691i	−0.405434	+	0.914125i	$0.632879\pi$
$12$		0			0
$13$	13.5782	+	13.5782i	1.04447	+	1.04447i	0.998964	+	0.0455110i	$0.0144916\pi$
$13$	13.5782	+	13.5782i	1.04447	+	1.04447i	0.0455110	+	0.998964i	$0.485508\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−19.7023			−1.15896			−0.579481	−	0.814986i	$-0.696745\pi$
$17$	−19.7023			−1.15896			−0.579481	+	0.814986i	$0.696745\pi$
$18$		0			0
$19$	−21.6943	+	21.6943i	−1.14181	+	1.14181i	−0.153687	+	0.988120i	$0.549115\pi$
$19$	−21.6943	+	21.6943i	−1.14181	+	1.14181i	−0.988120	+	0.153687i	$0.950885\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	24.9257			1.08373			0.541863	−	0.840467i	$-0.317719\pi$
$23$	24.9257			1.08373			0.541863	+	0.840467i	$0.317719\pi$
$24$		0			0
$25$			19.2247i			0.768989i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1.50581	+	1.50581i	0.0519245	+	0.0519245i	0.732592	−	0.680668i	$-0.238310\pi$
$29$	1.50581	+	1.50581i	0.0519245	+	0.0519245i	−0.680668	+	0.732592i	$0.738310\pi$
$30$		0			0
$31$			2.20037i			0.0709796i	0.999370	+	0.0354898i	$0.0112991\pi$
$31$			2.20037i			0.0709796i	−0.999370	+	0.0354898i	$0.988701\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	9.75877	−	9.75877i	0.278822	−	0.278822i
$36$		0			0
$37$	−27.6956	+	27.6956i	−0.748530	+	0.748530i	−0.974203	−	0.225673i	$-0.927542\pi$
$37$	−27.6956	+	27.6956i	−0.748530	+	0.748530i	0.225673	+	0.974203i	$0.427542\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	51.3127i		−	1.25153i	−0.780012	−	0.625764i	$-0.784787\pi$
$41$		−	51.3127i		−	1.25153i	0.780012	−	0.625764i	$-0.215213\pi$
$42$		0			0
$43$	21.4400	+	21.4400i	0.498606	+	0.498606i	0.911004	−	0.412398i	$-0.135309\pi$
$43$	21.4400	+	21.4400i	0.498606	+	0.498606i	−0.412398	+	0.911004i	$0.635309\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	76.5216i		−	1.62812i	−0.580781	−	0.814060i	$-0.697253\pi$
$47$		−	76.5216i		−	1.62812i	0.580781	−	0.814060i	$-0.302747\pi$
$48$		0			0
$49$	−16.0202			−0.326944
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−56.5145	+	56.5145i	−1.06631	+	1.06631i	−0.0686712	+	0.997639i	$0.521876\pi$
$53$	−56.5145	+	56.5145i	−1.06631	+	1.06631i	−0.997639	+	0.0686712i	$0.978124\pi$
$54$		0			0
$55$	19.0173			0.345768
$56$		0			0
$57$		0			0
$58$		0			0
$59$	48.0041	+	48.0041i	0.813628	+	0.813628i	0.985176	−	0.171547i	$-0.0548767\pi$
$59$	48.0041	+	48.0041i	0.813628	+	0.813628i	−0.171547	+	0.985176i	$0.554877\pi$
$60$		0			0
$61$	51.5587	+	51.5587i	0.845224	+	0.845224i	0.989533	−	0.144308i	$-0.0460957\pi$
$61$	51.5587	+	51.5587i	0.845224	+	0.845224i	−0.144308	+	0.989533i	$0.546096\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	46.1469			0.709952
$66$		0			0
$67$	63.4445	−	63.4445i	0.946934	−	0.946934i	−0.0517277	−	0.998661i	$-0.516473\pi$
$67$	63.4445	−	63.4445i	0.946934	−	0.946934i	0.998661	+	0.0517277i	$0.0164728\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	43.4856			0.612473			0.306237	−	0.951955i	$-0.400930\pi$
$71$	43.4856			0.612473			0.306237	+	0.951955i	$0.400930\pi$
$72$		0			0
$73$			73.9992i			1.01369i	0.862038	+	0.506844i	$0.169188\pi$
$73$			73.9992i			1.01369i	−0.862038	+	0.506844i	$0.830812\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	32.1344	+	32.1344i	0.417330	+	0.417330i
$78$		0			0
$79$			4.12659i			0.0522354i	0.999659	+	0.0261177i	$0.00831446\pi$
$79$			4.12659i			0.0522354i	−0.999659	+	0.0261177i	$0.991686\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−38.4428	+	38.4428i	−0.463166	+	0.463166i	−0.899692	−	0.436526i	$-0.856209\pi$
$83$	−38.4428	+	38.4428i	−0.463166	+	0.463166i	0.436526	+	0.899692i	$0.356209\pi$
$84$		0			0
$85$	−33.4803	+	33.4803i	−0.393886	+	0.393886i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			52.9839i			0.595325i	0.954671	+	0.297662i	$0.0962070\pi$
$89$			52.9839i			0.595325i	−0.954671	+	0.297662i	$0.903793\pi$
$90$		0			0
$91$	77.9767	+	77.9767i	0.856887	+	0.856887i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			73.7305i			0.776111i
$96$		0			0
$97$	23.1008			0.238153			0.119077	−	0.992885i	$-0.462007\pi$
$97$	23.1008			0.238153			0.119077	+	0.992885i	$0.462007\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	16.1216	−	16.1216i	0.159619	−	0.159619i	−0.622779	−	0.782398i	$-0.713996\pi$
$101$	16.1216	−	16.1216i	0.159619	−	0.159619i	0.782398	+	0.622779i	$0.213996\pi$
$102$		0			0
$103$	98.8380			0.959592			0.479796	−	0.877380i	$-0.340711\pi$
$103$	98.8380			0.959592			0.479796	+	0.877380i	$0.340711\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−15.6655	−	15.6655i	−0.146406	−	0.146406i	0.630104	−	0.776511i	$-0.283012\pi$
$107$	−15.6655	−	15.6655i	−0.146406	−	0.146406i	−0.776511	+	0.630104i	$0.783012\pi$
$108$		0			0
$109$	−84.6938	−	84.6938i	−0.777008	−	0.777008i	0.202313	−	0.979321i	$-0.435154\pi$
$109$	−84.6938	−	84.6938i	−0.777008	−	0.777008i	−0.979321	+	0.202313i	$0.935154\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−63.8537			−0.565077			−0.282538	−	0.959256i	$-0.591176\pi$
$113$	−63.8537			−0.565077			−0.282538	+	0.959256i	$0.591176\pi$
$114$		0			0
$115$	42.3563	−	42.3563i	0.368316	−	0.368316i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−113.147			−0.950812
$120$		0			0
$121$		−	58.3785i		−	0.482467i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	75.1513	+	75.1513i	0.601210	+	0.601210i
$126$		0			0
$127$		−	36.8901i		−	0.290473i	−0.989397	−	0.145237i	$-0.953606\pi$
$127$		−	36.8901i		−	0.290473i	0.989397	−	0.145237i	$-0.0463944\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	40.4136	−	40.4136i	0.308500	−	0.308500i	−0.535827	−	0.844328i	$-0.680000\pi$
$131$	40.4136	−	40.4136i	0.308500	−	0.308500i	0.844328	+	0.535827i	$0.180000\pi$
$132$		0			0
$133$	−124.586	+	124.586i	−0.936738	+	0.936738i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			253.499i			1.85036i	0.379531	+	0.925179i	$0.376085\pi$
$137$			253.499i			1.85036i	−0.379531	+	0.925179i	$0.623915\pi$
$138$		0			0
$139$	67.8065	+	67.8065i	0.487816	+	0.487816i	0.907617	−	0.419800i	$-0.137900\pi$
$139$	67.8065	+	67.8065i	0.487816	+	0.487816i	−0.419800	+	0.907617i	$0.637900\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			151.956i			1.06263i
$144$		0			0
$145$	5.11766			0.0352942
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−43.9337	+	43.9337i	−0.294857	+	0.294857i	−0.838996	−	0.544138i	$-0.816857\pi$
$149$	−43.9337	+	43.9337i	−0.294857	+	0.294857i	0.544138	+	0.838996i	$0.316857\pi$
$150$		0			0
$151$	223.084			1.47738			0.738688	−	0.674047i	$-0.235446\pi$
$151$	223.084			1.47738			0.738688	+	0.674047i	$0.235446\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	3.73909	+	3.73909i	0.0241232	+	0.0241232i
$156$		0			0
$157$	78.8526	+	78.8526i	0.502246	+	0.502246i	0.912135	−	0.409889i	$-0.134433\pi$
$157$	78.8526	+	78.8526i	0.502246	+	0.502246i	−0.409889	+	0.912135i	$0.634433\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	143.143			0.889089
$162$		0			0
$163$	52.2425	−	52.2425i	0.320506	−	0.320506i	−0.528455	−	0.848961i	$-0.677228\pi$
$163$	52.2425	−	52.2425i	0.320506	−	0.320506i	0.848961	+	0.528455i	$0.177228\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	96.5201			0.577965			0.288982	−	0.957334i	$-0.406683\pi$
$167$	96.5201			0.577965			0.288982	+	0.957334i	$0.406683\pi$
$168$		0			0
$169$			199.734i			1.18186i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−46.3076	−	46.3076i	−0.267674	−	0.267674i	0.560488	−	0.828162i	$-0.310614\pi$
$173$	−46.3076	−	46.3076i	−0.267674	−	0.267674i	−0.828162	+	0.560488i	$0.810614\pi$
$174$		0			0
$175$			110.404i			0.630879i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	93.5440	−	93.5440i	0.522592	−	0.522592i	−0.395761	−	0.918353i	$-0.629519\pi$
$179$	93.5440	−	93.5440i	0.522592	−	0.522592i	0.918353	+	0.395761i	$0.129519\pi$
$180$		0			0
$181$	115.810	−	115.810i	0.639836	−	0.639836i	−0.310679	−	0.950515i	$-0.600556\pi$
$181$	115.810	−	115.810i	0.639836	−	0.639836i	0.950515	+	0.310679i	$0.100556\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			94.1266i			0.508792i
$186$		0			0
$187$	−110.246	−	110.246i	−0.589553	−	0.589553i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			35.2964i			0.184798i	0.995722	+	0.0923991i	$0.0294535\pi$
$191$			35.2964i			0.184798i	−0.995722	+	0.0923991i	$0.970546\pi$
$192$		0			0
$193$	−364.339			−1.88777			−0.943884	−	0.330277i	$-0.892858\pi$
$193$	−364.339			−1.88777			−0.943884	+	0.330277i	$0.892858\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	130.582	−	130.582i	0.662851	−	0.662851i	−0.293200	−	0.956051i	$-0.594720\pi$
$197$	130.582	−	130.582i	0.662851	−	0.662851i	0.956051	+	0.293200i	$0.0947203\pi$
$198$		0			0
$199$	12.7493			0.0640670			0.0320335	−	0.999487i	$-0.489802\pi$
$199$	12.7493			0.0640670			0.0320335	+	0.999487i	$0.489802\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	8.64756	+	8.64756i	0.0425988	+	0.0425988i
$204$		0			0
$205$	−87.1958	−	87.1958i	−0.425346	−	0.425346i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−242.786			−1.16165
$210$		0			0
$211$	8.59499	−	8.59499i	0.0407345	−	0.0407345i	−0.686446	−	0.727181i	$-0.740830\pi$
$211$	8.59499	−	8.59499i	0.0407345	−	0.0407345i	0.727181	+	0.686446i	$0.240830\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	72.8663			0.338913
$216$		0			0
$217$			12.6363i			0.0582317i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−267.522	−	267.522i	−1.21051	−	1.21051i
$222$		0			0
$223$			50.5909i			0.226865i	0.993546	+	0.113433i	$0.0361846\pi$
$223$			50.5909i			0.226865i	−0.993546	+	0.113433i	$0.963815\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−31.7175	+	31.7175i	−0.139725	+	0.139725i	−0.773509	−	0.633785i	$-0.781501\pi$
$227$	−31.7175	+	31.7175i	−0.139725	+	0.139725i	0.633785	+	0.773509i	$0.281501\pi$
$228$		0			0
$229$	169.826	−	169.826i	0.741599	−	0.741599i	−0.231287	−	0.972886i	$-0.574294\pi$
$229$	169.826	−	169.826i	0.741599	−	0.741599i	0.972886	+	0.231287i	$0.0742936\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	363.082i		−	1.55829i	−0.626844	−	0.779145i	$-0.715654\pi$
$233$		−	363.082i		−	1.55829i	0.626844	−	0.779145i	$-0.284346\pi$
$234$		0			0
$235$	−130.033	−	130.033i	−0.553334	−	0.553334i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	27.6282i		−	0.115599i	−0.998328	−	0.0577996i	$-0.981592\pi$
$239$		−	27.6282i		−	0.115599i	0.998328	−	0.0577996i	$-0.0184084\pi$
$240$		0			0
$241$	368.121			1.52747			0.763737	−	0.645527i	$-0.223362\pi$
$241$	368.121			1.52747			0.763737	+	0.645527i	$0.223362\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−27.2233	+	27.2233i	−0.111115	+	0.111115i
$246$		0			0
$247$	−589.139			−2.38518
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−329.839	−	329.839i	−1.31410	−	1.31410i	−0.918365	−	0.395734i	$-0.870490\pi$
$251$	−329.839	−	329.839i	−1.31410	−	1.31410i	−0.395734	−	0.918365i	$-0.629510\pi$
$252$		0			0
$253$	139.474	+	139.474i	0.551281	+	0.551281i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−23.6762			−0.0921252			−0.0460626	−	0.998939i	$-0.514667\pi$
$257$	−23.6762			−0.0921252			−0.0460626	+	0.998939i	$0.514667\pi$
$258$		0			0
$259$	−159.050	+	159.050i	−0.614094	+	0.614094i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	243.854			0.927202			0.463601	−	0.886044i	$-0.346557\pi$
$263$	243.854			0.927202			0.463601	+	0.886044i	$0.346557\pi$
$264$		0			0
$265$			192.071i			0.724794i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	234.293	+	234.293i	0.870976	+	0.870976i	0.992579	−	0.121603i	$-0.0388035\pi$
$269$	234.293	+	234.293i	0.870976	+	0.870976i	−0.121603	+	0.992579i	$0.538803\pi$
$270$		0			0
$271$			30.9533i			0.114219i	0.998368	+	0.0571094i	$0.0181884\pi$
$271$			30.9533i			0.114219i	−0.998368	+	0.0571094i	$0.981812\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−107.574	+	107.574i	−0.391178	+	0.391178i
$276$		0			0
$277$	41.4479	−	41.4479i	0.149631	−	0.149631i	−0.628322	−	0.777953i	$-0.716258\pi$
$277$	41.4479	−	41.4479i	0.149631	−	0.149631i	0.777953	+	0.628322i	$0.216258\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			93.3971i			0.332374i	0.986094	+	0.166187i	$0.0531455\pi$
$281$			93.3971i			0.332374i	−0.986094	+	0.166187i	$0.946854\pi$
$282$		0			0
$283$	40.0982	+	40.0982i	0.141690	+	0.141690i	0.774394	−	0.632704i	$-0.218055\pi$
$283$	40.0982	+	40.0982i	0.141690	+	0.141690i	−0.632704	+	0.774394i	$0.718055\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	294.678i		−	1.02675i
$288$		0			0
$289$	99.1824			0.343192
$290$		0			0
$291$		0			0
$292$		0			0
$293$	141.326	−	141.326i	0.482340	−	0.482340i	−0.423538	−	0.905878i	$-0.639212\pi$
$293$	141.326	−	141.326i	0.482340	−	0.482340i	0.905878	+	0.423538i	$0.139212\pi$
$294$		0			0
$295$	163.147			0.553041
$296$		0			0
$297$		0			0
$298$		0			0
$299$	338.445	+	338.445i	1.13192	+	1.13192i
$300$		0			0
$301$	123.126	+	123.126i	0.409056	+	0.409056i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	175.228			0.574517
$306$		0			0
$307$	−285.548	+	285.548i	−0.930125	+	0.930125i	−0.997713	−	0.0675885i	$-0.978470\pi$
$307$	−285.548	+	285.548i	−0.930125	+	0.930125i	0.0675885	+	0.997713i	$0.478470\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−365.454			−1.17509			−0.587547	−	0.809190i	$-0.699906\pi$
$311$	−365.454			−1.17509			−0.587547	+	0.809190i	$0.699906\pi$
$312$		0			0
$313$		−	461.508i		−	1.47447i	−0.675638	−	0.737234i	$-0.736132\pi$
$313$		−	461.508i		−	1.47447i	0.675638	−	0.737234i	$-0.263868\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−319.216	−	319.216i	−1.00699	−	1.00699i	−0.999975	−	0.00701388i	$-0.997767\pi$
$317$	−319.216	−	319.216i	−1.00699	−	1.00699i	−0.00701388	−	0.999975i	$-0.502233\pi$
$318$		0			0
$319$			16.8518i			0.0528270i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	427.429	−	427.429i	1.32331	−	1.32331i
$324$		0			0
$325$	−261.037	+	261.037i	−0.803190	+	0.803190i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	439.448i		−	1.33571i
$330$		0			0
$331$	−85.7864	−	85.7864i	−0.259173	−	0.259173i	0.565544	−	0.824718i	$-0.308666\pi$
$331$	−85.7864	−	85.7864i	−0.259173	−	0.259173i	−0.824718	+	0.565544i	$0.808666\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	215.623i		−	0.643651i
$336$		0			0
$337$	258.256			0.766339			0.383170	−	0.923678i	$-0.374832\pi$
$337$	258.256			0.766339			0.383170	+	0.923678i	$0.374832\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−12.3124	+	12.3124i	−0.0361067	+	0.0361067i
$342$		0			0
$343$	−373.398			−1.08862
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−27.7237	−	27.7237i	−0.0798953	−	0.0798953i	0.666030	−	0.745925i	$-0.267992\pi$
$347$	−27.7237	−	27.7237i	−0.0798953	−	0.0798953i	−0.745925	+	0.666030i	$0.767992\pi$
$348$		0			0
$349$	−321.089	−	321.089i	−0.920027	−	0.920027i	0.0770037	−	0.997031i	$-0.475465\pi$
$349$	−321.089	−	321.089i	−0.920027	−	0.920027i	−0.997031	+	0.0770037i	$0.975465\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	241.363			0.683748			0.341874	−	0.939746i	$-0.388938\pi$
$353$	241.363			0.683748			0.341874	+	0.939746i	$0.388938\pi$
$354$		0			0
$355$	73.8953	−	73.8953i	0.208156	−	0.208156i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−363.821			−1.01343			−0.506714	−	0.862114i	$-0.669140\pi$
$359$	−363.821			−1.01343			−0.506714	+	0.862114i	$0.669140\pi$
$360$		0			0
$361$		−	580.287i		−	1.60744i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	125.747	+	125.747i	0.344513	+	0.344513i
$366$		0			0
$367$		−	411.402i		−	1.12099i	−0.828159	−	0.560493i	$-0.810612\pi$
$367$		−	411.402i		−	1.12099i	0.828159	−	0.560493i	$-0.189388\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−324.551	+	324.551i	−0.874801	+	0.874801i
$372$		0			0
$373$	225.677	−	225.677i	0.605033	−	0.605033i	−0.336611	−	0.941644i	$-0.609281\pi$
$373$	225.677	−	225.677i	0.605033	−	0.605033i	0.941644	+	0.336611i	$0.109281\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			40.8923i			0.108468i
$378$		0			0
$379$	−157.180	−	157.180i	−0.414724	−	0.414724i	0.468656	−	0.883381i	$-0.344738\pi$
$379$	−157.180	−	157.180i	−0.414724	−	0.414724i	−0.883381	+	0.468656i	$0.844738\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			703.356i			1.83644i	0.396072	+	0.918219i	$0.370373\pi$
$383$			703.356i			1.83644i	−0.396072	+	0.918219i	$0.629627\pi$
$384$		0			0
$385$	109.212			0.283668
$386$		0			0
$387$		0			0
$388$		0			0
$389$	10.7401	−	10.7401i	0.0276095	−	0.0276095i	−0.693167	−	0.720777i	$-0.743785\pi$
$389$	10.7401	−	10.7401i	0.0276095	−	0.0276095i	0.720777	+	0.693167i	$0.243785\pi$
$390$		0			0
$391$	−491.095			−1.25600
$392$		0			0
$393$		0			0
$394$		0			0
$395$	7.01234	+	7.01234i	0.0177528	+	0.0177528i
$396$		0			0
$397$	−365.020	−	365.020i	−0.919446	−	0.919446i	0.0775433	−	0.996989i	$-0.475292\pi$
$397$	−365.020	−	365.020i	−0.919446	−	0.919446i	−0.996989	+	0.0775433i	$0.975292\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−341.735			−0.852207			−0.426104	−	0.904674i	$-0.640114\pi$
$401$	−341.735			−0.852207			−0.426104	+	0.904674i	$0.640114\pi$
$402$		0			0
$403$	−29.8770	+	29.8770i	−0.0741364	+	0.0741364i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−309.947			−0.761541
$408$		0			0
$409$		−	368.259i		−	0.900389i	−0.892931	−	0.450194i	$-0.851355\pi$
$409$		−	368.259i		−	0.900389i	0.892931	−	0.450194i	$-0.148645\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	275.678	+	275.678i	0.667501	+	0.667501i
$414$		0			0
$415$			130.652i			0.314824i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−407.140	+	407.140i	−0.971694	+	0.971694i	−0.999610	−	0.0279165i	$-0.991113\pi$
$419$	−407.140	+	407.140i	−0.971694	+	0.971694i	0.0279165	+	0.999610i	$0.491113\pi$
$420$		0			0
$421$	−57.5576	+	57.5576i	−0.136716	+	0.136716i	−0.772153	−	0.635437i	$-0.780820\pi$
$421$	−57.5576	+	57.5576i	−0.136716	+	0.136716i	0.635437	+	0.772153i	$0.280820\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	378.772i		−	0.891229i
$426$		0			0
$427$	296.091	+	296.091i	0.693422	+	0.693422i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	796.565i		−	1.84818i	−0.382177	−	0.924089i	$-0.624826\pi$
$431$		−	796.565i		−	1.84818i	0.382177	−	0.924089i	$-0.375174\pi$
$432$		0			0
$433$	−335.804			−0.775529			−0.387764	−	0.921758i	$-0.626753\pi$
$433$	−335.804			−0.775529			−0.387764	+	0.921758i	$0.626753\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−540.746	+	540.746i	−1.23741	+	1.23741i
$438$		0			0
$439$	285.630			0.650638			0.325319	−	0.945604i	$-0.394528\pi$
$439$	285.630			0.650638			0.325319	+	0.945604i	$0.394528\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−111.596	−	111.596i	−0.251909	−	0.251909i	0.569844	−	0.821753i	$-0.307004\pi$
$443$	−111.596	−	111.596i	−0.251909	−	0.251909i	−0.821753	+	0.569844i	$0.807004\pi$
$444$		0			0
$445$	90.0358	+	90.0358i	0.202328	+	0.202328i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	99.6741			0.221991			0.110996	−	0.993821i	$-0.464596\pi$
$449$	99.6741			0.221991			0.110996	+	0.993821i	$0.464596\pi$
$450$		0			0
$451$	287.125	−	287.125i	0.636641	−	0.636641i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	265.012			0.582445
$456$		0			0
$457$			32.1643i			0.0703813i	0.999381	+	0.0351907i	$0.0112039\pi$
$457$			32.1643i			0.0703813i	−0.999381	+	0.0351907i	$0.988796\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	165.361	+	165.361i	0.358701	+	0.358701i	0.863334	−	0.504633i	$-0.168372\pi$
$461$	165.361	+	165.361i	0.358701	+	0.358701i	−0.504633	+	0.863334i	$0.668372\pi$
$462$		0			0
$463$		−	923.215i		−	1.99398i	−0.0774991	−	0.996992i	$-0.524693\pi$
$463$		−	923.215i		−	1.99398i	0.0774991	−	0.996992i	$-0.475307\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−507.842	+	507.842i	−1.08746	+	1.08746i	−0.0916660	+	0.995790i	$0.529219\pi$
$467$	−507.842	+	507.842i	−1.08746	+	1.08746i	−0.995790	+	0.0916660i	$0.970781\pi$
$468$		0			0
$469$	364.349	−	364.349i	0.776864	−	0.776864i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			239.940i			0.507272i
$474$		0			0
$475$	−417.068	−	417.068i	−0.878037	−	0.878037i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	52.3866i		−	0.109367i	−0.998504	−	0.0546833i	$-0.982585\pi$
$479$		−	52.3866i		−	0.109367i	0.998504	−	0.0546833i	$-0.0174149\pi$
$480$		0			0
$481$	−752.112			−1.56364
$482$		0			0
$483$		0			0
$484$		0			0
$485$	39.2554	−	39.2554i	0.0809389	−	0.0809389i
$486$		0			0
$487$	715.733			1.46968			0.734839	−	0.678241i	$-0.237258\pi$
$487$	715.733			1.46968			0.734839	+	0.678241i	$0.237258\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	22.3258	+	22.3258i	0.0454701	+	0.0454701i	0.729476	−	0.684006i	$-0.239764\pi$
$491$	22.3258	+	22.3258i	0.0454701	+	0.0454701i	−0.684006	+	0.729476i	$0.739764\pi$
$492$		0			0
$493$	−29.6680	−	29.6680i	−0.0601784	−	0.0601784i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	249.729			0.502473
$498$		0			0
$499$	84.0984	−	84.0984i	0.168534	−	0.168534i	−0.617801	−	0.786335i	$-0.711976\pi$
$499$	84.0984	−	84.0984i	0.168534	−	0.168534i	0.786335	+	0.617801i	$0.211976\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	327.870			0.651829			0.325914	−	0.945399i	$-0.394328\pi$
$503$	327.870			0.651829			0.325914	+	0.945399i	$0.394328\pi$
$504$		0			0
$505$		−	54.7909i		−	0.108497i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	34.6224	+	34.6224i	0.0680205	+	0.0680205i	0.740299	−	0.672278i	$-0.234684\pi$
$509$	34.6224	+	34.6224i	0.0680205	+	0.0680205i	−0.672278	+	0.740299i	$0.734684\pi$
$510$		0			0
$511$			424.963i			0.831630i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	167.956	−	167.956i	0.326128	−	0.326128i
$516$		0			0
$517$	428.184	−	428.184i	0.828210	−	0.828210i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			235.719i			0.452436i	0.974077	+	0.226218i	$0.0726362\pi$
$521$			235.719i			0.452436i	−0.974077	+	0.226218i	$0.927364\pi$
$522$		0			0
$523$	−185.851	−	185.851i	−0.355356	−	0.355356i	0.506742	−	0.862098i	$-0.330850\pi$
$523$	−185.851	−	185.851i	−0.355356	−	0.355356i	−0.862098	+	0.506742i	$0.830850\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	43.3524i		−	0.0822626i
$528$		0			0
$529$	92.2900			0.174461
$530$		0			0
$531$		0			0
$532$		0			0
$533$	696.732	−	696.732i	1.30719	−	1.30719i
$534$		0			0
$535$	−53.2408			−0.0995155
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−89.6428	−	89.6428i	−0.166313	−	0.166313i
$540$		0			0
$541$	315.952	+	315.952i	0.584015	+	0.584015i	0.936004	−	0.351989i	$-0.114494\pi$
$541$	315.952	+	315.952i	0.584015	+	0.584015i	−0.351989	+	0.936004i	$0.614494\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−287.841			−0.528149
$546$		0			0
$547$	−550.957	+	550.957i	−1.00723	+	1.00723i	−0.00725954	+	0.999974i	$0.502311\pi$
$547$	−550.957	+	550.957i	−1.00723	+	1.00723i	−0.999974	+	0.00725954i	$0.997689\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−65.3350			−0.118575
$552$		0			0
$553$			23.6982i			0.0428539i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	2.35545	+	2.35545i	0.00422882	+	0.00422882i	0.709218	−	0.704989i	$-0.249048\pi$
$557$	2.35545	+	2.35545i	0.00422882	+	0.00422882i	−0.704989	+	0.709218i	$0.749048\pi$
$558$		0			0
$559$			582.233i			1.04156i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−269.210	+	269.210i	−0.478170	+	0.478170i	−0.904546	−	0.426376i	$-0.859790\pi$
$563$	−269.210	+	269.210i	−0.478170	+	0.478170i	0.426376	+	0.904546i	$0.359790\pi$
$564$		0			0
$565$	−108.507	+	108.507i	−0.192047	+	0.192047i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	342.558i		−	0.602035i	−0.953619	−	0.301018i	$-0.902674\pi$
$569$		−	342.558i		−	0.602035i	0.953619	−	0.301018i	$-0.0973263\pi$
$570$		0			0
$571$	153.948	+	153.948i	0.269610	+	0.269610i	0.828943	−	0.559333i	$-0.188943\pi$
$571$	153.948	+	153.948i	0.269610	+	0.269610i	−0.559333	+	0.828943i	$0.688943\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			479.190i			0.833373i
$576$		0			0
$577$	563.693			0.976938			0.488469	−	0.872581i	$-0.337556\pi$
$577$	563.693			0.976938			0.488469	+	0.872581i	$0.337556\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−220.769	+	220.769i	−0.379981	+	0.379981i
$582$		0			0
$583$	−632.465			−1.08484
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−176.603	−	176.603i	−0.300857	−	0.300857i	0.540492	−	0.841349i	$-0.318238\pi$
$587$	−176.603	−	176.603i	−0.300857	−	0.300857i	−0.841349	+	0.540492i	$0.818238\pi$
$588$		0			0
$589$	−47.7355	−	47.7355i	−0.0810450	−	0.0810450i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	996.597			1.68060			0.840301	−	0.542120i	$-0.182378\pi$
$593$	996.597			1.68060			0.840301	+	0.542120i	$0.182378\pi$
$594$		0			0
$595$	−192.271	+	192.271i	−0.323144	+	0.323144i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−854.031			−1.42576			−0.712880	−	0.701286i	$-0.752610\pi$
$599$	−854.031			−1.42576			−0.712880	+	0.701286i	$0.752610\pi$
$600$		0			0
$601$			345.733i			0.575263i	0.957741	+	0.287631i	$0.0928678\pi$
$601$			345.733i			0.575263i	−0.957741	+	0.287631i	$0.907132\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−99.2029	−	99.2029i	−0.163972	−	0.163972i
$606$		0			0
$607$			526.354i			0.867141i	0.901120	+	0.433570i	$0.142746\pi$
$607$			526.354i			0.867141i	−0.901120	+	0.433570i	$0.857254\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	1039.02	−	1039.02i	1.70053	−	1.70053i
$612$		0			0
$613$	−410.567	+	410.567i	−0.669767	+	0.669767i	−0.957662	−	0.287895i	$-0.907045\pi$
$613$	−410.567	+	410.567i	−0.669767	+	0.669767i	0.287895	+	0.957662i	$0.407045\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			514.755i			0.834287i	0.908841	+	0.417144i	$0.136969\pi$
$617$			514.755i			0.834287i	−0.908841	+	0.417144i	$0.863031\pi$
$618$		0			0
$619$	314.214	+	314.214i	0.507615	+	0.507615i	0.913794	−	0.406179i	$-0.133139\pi$
$619$	314.214	+	314.214i	0.507615	+	0.507615i	−0.406179	+	0.913794i	$0.633139\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			304.276i			0.488404i
$624$		0			0
$625$	−225.209			−0.360334
$626$		0			0
$627$		0			0
$628$		0			0
$629$	545.669	−	545.669i	0.867518	−	0.867518i
$630$		0			0
$631$	230.081			0.364629			0.182315	−	0.983240i	$-0.441641\pi$
$631$	230.081			0.364629			0.182315	+	0.983240i	$0.441641\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−62.6875	−	62.6875i	−0.0987205	−	0.0987205i
$636$		0			0
$637$	−217.526	−	217.526i	−0.341484	−	0.341484i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−746.825			−1.16509			−0.582547	−	0.812797i	$-0.697944\pi$
$641$	−746.825			−1.16509			−0.582547	+	0.812797i	$0.697944\pi$
$642$		0			0
$643$	548.092	−	548.092i	0.852398	−	0.852398i	−0.138030	−	0.990428i	$-0.544077\pi$
$643$	548.092	−	548.092i	0.852398	−	0.852398i	0.990428	+	0.138030i	$0.0440772\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	1055.00			1.63060			0.815302	−	0.579036i	$-0.196571\pi$
$647$	1055.00			1.63060			0.815302	+	0.579036i	$0.196571\pi$
$648$		0			0
$649$			537.223i			0.827771i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−854.888	−	854.888i	−1.30917	−	1.30917i	−0.922015	−	0.387155i	$-0.873458\pi$
$653$	−854.888	−	854.888i	−1.30917	−	1.30917i	−0.387155	−	0.922015i	$-0.626542\pi$
$654$		0			0
$655$		−	137.350i		−	0.209694i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	768.766	−	768.766i	1.16656	−	1.16656i	0.183556	−	0.983009i	$-0.441239\pi$
$659$	768.766	−	768.766i	1.16656	−	1.16656i	0.983009	−	0.183556i	$-0.0587607\pi$
$660$		0			0
$661$	−312.323	+	312.323i	−0.472500	+	0.472500i	−0.902723	−	0.430223i	$-0.858435\pi$
$661$	−312.323	+	312.323i	−0.472500	+	0.472500i	0.430223	+	0.902723i	$0.358435\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			423.420i			0.636721i
$666$		0			0
$667$	37.5333	+	37.5333i	0.0562719	+	0.0562719i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			577.004i			0.859916i
$672$		0			0
$673$	740.565			1.10039			0.550197	−	0.835035i	$-0.314553\pi$
$673$	740.565			1.10039			0.550197	+	0.835035i	$0.314553\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−547.118	+	547.118i	−0.808151	+	0.808151i	−0.984354	−	0.176203i	$-0.943619\pi$
$677$	−547.118	+	547.118i	−0.808151	+	0.808151i	0.176203	+	0.984354i	$0.443619\pi$
$678$		0			0
$679$	132.664			0.195381
$680$		0			0
$681$		0			0
$682$		0			0
$683$	407.623	+	407.623i	0.596813	+	0.596813i	0.939463	−	0.342650i	$-0.111324\pi$
$683$	407.623	+	407.623i	0.596813	+	0.596813i	−0.342650	+	0.939463i	$0.611324\pi$
$684$		0			0
$685$	430.772	+	430.772i	0.628864	+	0.628864i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−1534.73			−2.22747
$690$		0			0
$691$	17.6037	−	17.6037i	0.0254757	−	0.0254757i	−0.694254	−	0.719730i	$-0.744266\pi$
$691$	17.6037	−	17.6037i	0.0254757	−	0.0254757i	0.719730	+	0.694254i	$0.244266\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	230.448			0.331579
$696$		0			0
$697$			1010.98i			1.45047i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	164.273	+	164.273i	0.234341	+	0.234341i	0.814502	−	0.580161i	$-0.197010\pi$
$701$	164.273	+	164.273i	0.234341	+	0.234341i	−0.580161	+	0.814502i	$0.697010\pi$
$702$		0			0
$703$		−	1201.68i		−	1.70935i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	92.5829	−	92.5829i	0.130952	−	0.130952i
$708$		0			0
$709$	−422.796	+	422.796i	−0.596327	+	0.596327i	−0.939333	−	0.343006i	$-0.888555\pi$
$709$	−422.796	+	422.796i	−0.596327	+	0.596327i	0.343006	+	0.939333i	$0.388555\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			54.8457i			0.0769224i
$714$		0			0
$715$	258.220	+	258.220i	0.361146	+	0.361146i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	1029.00i		−	1.43115i	−0.698534	−	0.715577i	$-0.746164\pi$
$719$		−	1029.00i		−	1.43115i	0.698534	−	0.715577i	$-0.253836\pi$
$720$		0			0
$721$	567.607			0.787250
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−28.9488	+	28.9488i	−0.0399293	+	0.0399293i
$726$		0			0
$727$	475.001			0.653372			0.326686	−	0.945133i	$-0.394068\pi$
$727$	475.001			0.653372			0.326686	+	0.945133i	$0.394068\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−422.419	−	422.419i	−0.577865	−	0.577865i
$732$		0			0
$733$	344.939	+	344.939i	0.470586	+	0.470586i	0.902104	−	0.431519i	$-0.142022\pi$
$733$	344.939	+	344.939i	0.470586	+	0.470586i	−0.431519	+	0.902104i	$0.642022\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	710.021			0.963393
$738$		0			0
$739$	363.340	−	363.340i	0.491665	−	0.491665i	−0.417166	−	0.908831i	$-0.636976\pi$
$739$	363.340	−	363.340i	0.491665	−	0.491665i	0.908831	+	0.417166i	$0.136976\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	271.667			0.365636			0.182818	−	0.983147i	$-0.441478\pi$
$743$	271.667			0.365636			0.182818	+	0.983147i	$0.441478\pi$
$744$		0			0
$745$			149.314i			0.200421i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−89.9637	−	89.9637i	−0.120112	−	0.120112i
$750$		0			0
$751$			1105.27i			1.47173i	0.677128	+	0.735866i	$0.263224\pi$
$751$			1105.27i			1.47173i	−0.677128	+	0.735866i	$0.736776\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	379.087	−	379.087i	0.502102	−	0.502102i
$756$		0			0
$757$	−554.565	+	554.565i	−0.732583	+	0.732583i	−0.971131	−	0.238548i	$-0.923329\pi$
$757$	−554.565	+	554.565i	−0.732583	+	0.732583i	0.238548	+	0.971131i	$0.423329\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	188.496i		−	0.247695i	−0.992301	−	0.123847i	$-0.960477\pi$
$761$		−	188.496i		−	0.247695i	0.992301	−	0.123847i	$-0.0395234\pi$
$762$		0			0
$763$	−486.380	−	486.380i	−0.637457	−	0.637457i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			1303.62i			1.69963i
$768$		0			0
$769$	−593.354			−0.771592			−0.385796	−	0.922584i	$-0.626073\pi$
$769$	−593.354			−0.771592			−0.385796	+	0.922584i	$0.626073\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	514.720	−	514.720i	0.665873	−	0.665873i	−0.290885	−	0.956758i	$-0.593950\pi$
$773$	514.720	−	514.720i	0.665873	−	0.665873i	0.956758	+	0.290885i	$0.0939498\pi$
$774$		0			0
$775$	−42.3015			−0.0545826
$776$		0			0
$777$		0			0
$778$		0			0
$779$	1113.19	+	1113.19i	1.42900	+	1.42900i
$780$		0			0
$781$	243.328	+	243.328i	0.311560	+	0.311560i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	267.989			0.341387
$786$		0			0
$787$	96.1835	−	96.1835i	0.122215	−	0.122215i	−0.643354	−	0.765569i	$-0.722458\pi$
$787$	96.1835	−	96.1835i	0.122215	−	0.122215i	0.765569	+	0.643354i	$0.222458\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−366.699			−0.463589
$792$		0			0
$793$			1400.15i			1.76563i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−664.410	−	664.410i	−0.833639	−	0.833639i	0.154374	−	0.988013i	$-0.450664\pi$
$797$	−664.410	−	664.410i	−0.833639	−	0.833639i	−0.988013	+	0.154374i	$0.950664\pi$
$798$		0			0
$799$			1507.66i			1.88693i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−414.070	+	414.070i	−0.515654	+	0.515654i
$804$		0			0
$805$	243.244	−	243.244i	0.302166	−	0.302166i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			1371.63i			1.69547i	0.530422	+	0.847734i	$0.322034\pi$
$809$			1371.63i			1.69547i	−0.530422	+	0.847734i	$0.677966\pi$
$810$		0			0
$811$	809.783	+	809.783i	0.998500	+	0.998500i	0.999999	−	0.00149916i	$-0.000477196\pi$
$811$	809.783	+	809.783i	0.998500	+	0.998500i	−0.00149916	+	0.999999i	$0.500477\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	177.552i		−	0.217855i
$816$		0			0
$817$	−930.255			−1.13862
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−1144.74	+	1144.74i	−1.39432	+	1.39432i	−0.578980	+	0.815342i	$0.696549\pi$
$821$	−1144.74	+	1144.74i	−1.39432	+	1.39432i	−0.815342	+	0.578980i	$0.803451\pi$
$822$		0			0
$823$	−439.361			−0.533853			−0.266926	−	0.963717i	$-0.586008\pi$
$823$	−439.361			−0.533853			−0.266926	+	0.963717i	$0.586008\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−911.996	−	911.996i	−1.10278	−	1.10278i	−0.994074	−	0.108701i	$-0.965331\pi$
$827$	−911.996	−	911.996i	−1.10278	−	1.10278i	−0.108701	−	0.994074i	$-0.534669\pi$
$828$		0			0
$829$	−470.575	−	470.575i	−0.567642	−	0.567642i	0.363825	−	0.931467i	$-0.381471\pi$
$829$	−470.575	−	470.575i	−0.567642	−	0.567642i	−0.931467	+	0.363825i	$0.881471\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	315.636			0.378915
$834$		0			0
$835$	164.017	−	164.017i	0.196428	−	0.196428i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	1432.91			1.70787			0.853937	−	0.520376i	$-0.174208\pi$
$839$	1432.91			1.70787			0.853937	+	0.520376i	$0.174208\pi$
$840$		0			0
$841$		−	836.465i		−	0.994608i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	339.408	+	339.408i	0.401666	+	0.401666i
$846$		0			0
$847$		−	335.256i		−	0.395816i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−690.332	+	690.332i	−0.811201	+	0.811201i
$852$		0			0
$853$	−211.443	+	211.443i	−0.247881	+	0.247881i	−0.820101	−	0.572219i	$-0.806083\pi$
$853$	−211.443	+	211.443i	−0.247881	+	0.247881i	0.572219	+	0.820101i	$0.306083\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	710.925i		−	0.829551i	−0.909924	−	0.414775i	$-0.863860\pi$
$857$		−	710.925i		−	0.829551i	0.909924	−	0.414775i	$-0.136140\pi$
$858$		0			0
$859$	348.557	+	348.557i	0.405771	+	0.405771i	0.880261	−	0.474490i	$-0.157368\pi$
$859$	348.557	+	348.557i	0.405771	+	0.405771i	−0.474490	+	0.880261i	$0.657368\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	1493.19i		−	1.73023i	−0.501571	−	0.865116i	$-0.667245\pi$
$863$		−	1493.19i		−	1.73023i	0.501571	−	0.865116i	$-0.332755\pi$
$864$		0			0
$865$	−157.382			−0.181944
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−23.0908	+	23.0908i	−0.0265717	+	0.0265717i
$870$		0			0
$871$	1722.92			1.97810
$872$		0			0
$873$		0			0
$874$		0			0
$875$	431.579	+	431.579i	0.493233	+	0.493233i
$876$		0			0
$877$	332.929	+	332.929i	0.379623	+	0.379623i	0.870966	−	0.491343i	$-0.163494\pi$
$877$	332.929	+	332.929i	0.379623	+	0.379623i	−0.491343	+	0.870966i	$0.663494\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	637.491			0.723599			0.361799	−	0.932256i	$-0.382162\pi$
$881$	637.491			0.723599			0.361799	+	0.932256i	$0.382162\pi$
$882$		0			0
$883$	134.646	−	134.646i	0.152487	−	0.152487i	−0.626741	−	0.779228i	$-0.715612\pi$
$883$	134.646	−	134.646i	0.152487	−	0.152487i	0.779228	+	0.626741i	$0.215612\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	3.90891			0.00440688			0.00220344	−	0.999998i	$-0.499299\pi$
$887$	3.90891			0.00440688			0.00220344	+	0.999998i	$0.499299\pi$
$888$		0			0
$889$		−	211.853i		−	0.238304i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	1660.08	+	1660.08i	1.85900	+	1.85900i
$894$		0			0
$895$		−	317.919i		−	0.355217i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−3.31333	+	3.31333i	−0.00368558	+	0.00368558i
$900$		0			0
$901$	1113.47	−	1113.47i	1.23581	−	1.23581i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	393.594i		−	0.434910i
$906$		0			0
$907$	697.707	+	697.707i	0.769247	+	0.769247i	0.977974	−	0.208727i	$-0.0669319\pi$
$907$	697.707	+	697.707i	0.769247	+	0.769247i	−0.208727	+	0.977974i	$0.566932\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			796.856i			0.874705i	0.899290	+	0.437353i	$0.144084\pi$
$911$			796.856i			0.874705i	−0.899290	+	0.437353i	$0.855916\pi$
$912$		0			0
$913$	−430.221			−0.471216
$914$		0			0
$915$		0			0
$916$		0			0
$917$	232.087	−	232.087i	0.253094	−	0.253094i
$918$		0			0
$919$	−420.532			−0.457597			−0.228798	−	0.973474i	$-0.573480\pi$
$919$	−420.532			−0.457597			−0.228798	+	0.973474i	$0.573480\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	590.455	+	590.455i	0.639713	+	0.639713i
$924$		0			0
$925$	−532.441	−	532.441i	−0.575612	−	0.575612i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−1178.38			−1.26843			−0.634217	−	0.773155i	$-0.718677\pi$
$929$	−1178.38			−1.26843			−0.634217	+	0.773155i	$0.718677\pi$
$930$		0			0
$931$	347.548	−	347.548i	0.373306	−	0.373306i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−374.684			−0.400732
$936$		0			0
$937$			405.962i			0.433257i	0.976254	+	0.216629i	$0.0695061\pi$
$937$			405.962i			0.433257i	−0.976254	+	0.216629i	$0.930494\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	429.859	+	429.859i	0.456810	+	0.456810i	0.897607	−	0.440797i	$-0.145304\pi$
$941$	429.859	+	429.859i	0.456810	+	0.456810i	−0.440797	+	0.897607i	$0.645304\pi$
$942$		0			0
$943$		−	1279.00i		−	1.35631i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−661.681	+	661.681i	−0.698713	+	0.698713i	−0.964133	−	0.265420i	$-0.914489\pi$
$947$	−661.681	+	661.681i	−0.698713	+	0.698713i	0.265420	+	0.964133i	$0.414489\pi$
$948$		0			0
$949$	−1004.77	+	1004.77i	−1.05877	+	1.05877i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	31.1854i		−	0.0327234i	−0.999866	−	0.0163617i	$-0.994792\pi$
$953$		−	31.1854i		−	0.0327234i	0.999866	−	0.0163617i	$-0.00520833\pi$
$954$		0			0
$955$	59.9794	+	59.9794i	0.0628056	+	0.0628056i
$956$		0			0
$957$		0			0
$958$		0			0
$959$			1455.79i			1.51803i
$960$		0			0
$961$	956.158			0.994962
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−619.123	+	619.123i	−0.641578	+	0.641578i
$966$		0			0
$967$	1312.55			1.35734			0.678672	−	0.734442i	$-0.262556\pi$
$967$	1312.55			1.35734			0.678672	+	0.734442i	$0.262556\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−318.758	−	318.758i	−0.328278	−	0.328278i	0.523653	−	0.851931i	$-0.324569\pi$
$971$	−318.758	−	318.758i	−0.328278	−	0.328278i	−0.851931	+	0.523653i	$0.824569\pi$
$972$		0			0
$973$	389.399	+	389.399i	0.400205	+	0.400205i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	531.045			0.543546			0.271773	−	0.962361i	$-0.412390\pi$
$977$	531.045			0.543546			0.271773	+	0.962361i	$0.412390\pi$
$978$		0			0
$979$	−296.477	+	296.477i	−0.302836	+	0.302836i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−1796.17			−1.82723			−0.913614	−	0.406582i	$-0.866721\pi$
$983$	−1796.17			−1.82723			−0.913614	+	0.406582i	$0.866721\pi$
$984$		0			0
$985$		−	443.796i		−	0.450554i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	534.408	+	534.408i	0.540352	+	0.540352i
$990$		0			0
$991$			506.064i			0.510660i	0.966854	+	0.255330i	$0.0821841\pi$
$991$			506.064i			0.510660i	−0.966854	+	0.255330i	$0.917816\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	21.6650	−	21.6650i	0.0217739	−	0.0217739i
$996$		0			0
$997$	249.068	−	249.068i	0.249817	−	0.249817i	−0.571078	−	0.820896i	$-0.693475\pi$
$997$	249.068	−	249.068i	0.249817	−	0.249817i	0.820896	+	0.571078i	$0.193475\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1152.3.m.c.991.6		16
3.2	odd	2		384.3.l.b.223.2		16
4.3	odd	2		1152.3.m.f.991.6		16
8.3	odd	2		144.3.m.c.19.1		16
8.5	even	2		576.3.m.c.559.3		16
12.11	even	2		384.3.l.a.223.6		16
16.3	odd	4		576.3.m.c.271.3		16
16.5	even	4		1152.3.m.f.415.6		16
16.11	odd	4	inner	1152.3.m.c.415.6		16
16.13	even	4		144.3.m.c.91.1		16
24.5	odd	2		192.3.l.a.175.7		16
24.11	even	2		48.3.l.a.19.8	✓	16
48.5	odd	4		384.3.l.a.31.6		16
48.11	even	4		384.3.l.b.31.2		16
48.29	odd	4		48.3.l.a.43.8	yes	16
48.35	even	4		192.3.l.a.79.7		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.3.l.a.19.8	✓	16	24.11	even	2
48.3.l.a.43.8	yes	16	48.29	odd	4
144.3.m.c.19.1		16	8.3	odd	2
144.3.m.c.91.1		16	16.13	even	4
192.3.l.a.79.7		16	48.35	even	4
192.3.l.a.175.7		16	24.5	odd	2
384.3.l.a.31.6		16	48.5	odd	4
384.3.l.a.223.6		16	12.11	even	2
384.3.l.b.31.2		16	48.11	even	4
384.3.l.b.223.2		16	3.2	odd	2
576.3.m.c.271.3		16	16.3	odd	4
576.3.m.c.559.3		16	8.5	even	2
1152.3.m.c.415.6		16	16.11	odd	4	inner
1152.3.m.c.991.6		16	1.1	even	1	trivial
1152.3.m.f.415.6		16	16.5	even	4
1152.3.m.f.991.6		16	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1152 = 2^{7} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1152.m (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.69930 - 1.69930i) q^{5} +5.74280 q^{7} +O(q^{10})\)	\(q+(1.69930 - 1.69930i) q^{5} +5.74280 q^{7} +(5.59560 + 5.59560i) q^{11} +(13.5782 + 13.5782i) q^{13} -19.7023 q^{17} +(-21.6943 + 21.6943i) q^{19} +24.9257 q^{23} +19.2247i q^{25} +(1.50581 + 1.50581i) q^{29} +2.20037i q^{31} +(9.75877 - 9.75877i) q^{35} +(-27.6956 + 27.6956i) q^{37} -51.3127i q^{41} +(21.4400 + 21.4400i) q^{43} -76.5216i q^{47} -16.0202 q^{49} +(-56.5145 + 56.5145i) q^{53} +19.0173 q^{55} +(48.0041 + 48.0041i) q^{59} +(51.5587 + 51.5587i) q^{61} +46.1469 q^{65} +(63.4445 - 63.4445i) q^{67} +43.4856 q^{71} +73.9992i q^{73} +(32.1344 + 32.1344i) q^{77} +4.12659i q^{79} +(-38.4428 + 38.4428i) q^{83} +(-33.4803 + 33.4803i) q^{85} +52.9839i q^{89} +(77.9767 + 77.9767i) q^{91} +73.7305i q^{95} +23.1008 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 32 q^{11} - 32 q^{19} - 128 q^{23} + 32 q^{29} - 96 q^{35} + 96 q^{37} + 160 q^{43} + 112 q^{49} - 160 q^{53} + 256 q^{55} + 128 q^{59} + 32 q^{61} + 32 q^{65} + 320 q^{67} + 512 q^{71} + 224 q^{77} + 160 q^{83} - 160 q^{85} - 480 q^{91}+O(q^{100}) \) 16 * q - 32 * q^11 - 32 * q^19 - 128 * q^23 + 32 * q^29 - 96 * q^35 + 96 * q^37 + 160 * q^43 + 112 * q^49 - 160 * q^53 + 256 * q^55 + 128 * q^59 + 32 * q^61 + 32 * q^65 + 320 * q^67 + 512 * q^71 + 224 * q^77 + 160 * q^83 - 160 * q^85 - 480 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more