LMFDB - Embedded newform 975.2.h.f.649.1

Newspace parameters

Level:	$ N $	$=$	$ 975 = 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	975.h (of order $2$, degree $1$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$7.78541419707$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.1
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	975.649
Dual form			975.2.h.f.649.2

$q$-expansion

$f(q)$	$=$	$q-1.73205 q^{2} -1.00000i q^{3} +1.00000 q^{4} +1.73205i q^{6} +3.46410 q^{7} +1.73205 q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.73205 q^{2} -1.00000i q^{3} +1.00000 q^{4} +1.73205i q^{6} +3.46410 q^{7} +1.73205 q^{8} -1.00000 q^{9} +3.46410i q^{11} -1.00000i q^{12} +(3.46410 - 1.00000i) q^{13} -6.00000 q^{14} -5.00000 q^{16} +6.00000i q^{17} +1.73205 q^{18} +3.46410i q^{19} -3.46410i q^{21} -6.00000i q^{22} -1.73205i q^{24} +(-6.00000 + 1.73205i) q^{26} +1.00000i q^{27} +3.46410 q^{28} -6.00000 q^{29} +3.46410i q^{31} +5.19615 q^{32} +3.46410 q^{33} -10.3923i q^{34} -1.00000 q^{36} -6.92820 q^{37} -6.00000i q^{38} +(-1.00000 - 3.46410i) q^{39} +6.92820i q^{41} +6.00000i q^{42} -4.00000i q^{43} +3.46410i q^{44} -3.46410 q^{47} +5.00000i q^{48} +5.00000 q^{49} +6.00000 q^{51} +(3.46410 - 1.00000i) q^{52} +6.00000i q^{53} -1.73205i q^{54} +6.00000 q^{56} +3.46410 q^{57} +10.3923 q^{58} +10.3923i q^{59} -2.00000 q^{61} -6.00000i q^{62} -3.46410 q^{63} +1.00000 q^{64} -6.00000 q^{66} +10.3923 q^{67} +6.00000i q^{68} -3.46410i q^{71} -1.73205 q^{72} +12.0000 q^{74} +3.46410i q^{76} +12.0000i q^{77} +(1.73205 + 6.00000i) q^{78} +8.00000 q^{79} +1.00000 q^{81} -12.0000i q^{82} -3.46410 q^{83} -3.46410i q^{84} +6.92820i q^{86} +6.00000i q^{87} +6.00000i q^{88} -6.92820i q^{89} +(12.0000 - 3.46410i) q^{91} +3.46410 q^{93} +6.00000 q^{94} -5.19615i q^{96} +13.8564 q^{97} -8.66025 q^{98} -3.46410i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{4} - 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^4 - 4 * q^9	$ 4 q + 4 q^{4} - 4 q^{9} - 24 q^{14} - 20 q^{16} - 24 q^{26} - 24 q^{29} - 4 q^{36} - 4 q^{39} + 20 q^{49} + 24 q^{51} + 24 q^{56} - 8 q^{61} + 4 q^{64} - 24 q^{66} + 48 q^{74} + 32 q^{79} + 4 q^{81} + 48 q^{91} + 24 q^{94}+O(q^{100}) $ 4 * q + 4 * q^4 - 4 * q^9 - 24 * q^14 - 20 * q^16 - 24 * q^26 - 24 * q^29 - 4 * q^36 - 4 * q^39 + 20 * q^49 + 24 * q^51 + 24 * q^56 - 8 * q^61 + 4 * q^64 - 24 * q^66 + 48 * q^74 + 32 * q^79 + 4 * q^81 + 48 * q^91 + 24 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/975\mathbb{Z}\right)^\times$.

$n$	$301$	$326$	$352$
$\chi(n)$	$-1$	$1$	$-1$

Coefficient data

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

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

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

$2$	−1.73205			−1.22474			−0.612372	−	0.790569i	$-0.709785\pi$
$2$	−1.73205			−1.22474			−0.612372	+	0.790569i	$0.709785\pi$
$3$		−	1.00000i		−	0.577350i
$3$		−	1.00000i		−	0.577350i
$4$	1.00000			0.500000
$5$		0			0
$5$		0			0
$6$			1.73205i			0.707107i
$7$	3.46410			1.30931			0.654654	−	0.755929i	$-0.272814\pi$
$7$	3.46410			1.30931			0.654654	+	0.755929i	$0.272814\pi$
$8$	1.73205			0.612372
$9$	−1.00000			−0.333333
$10$		0			0
$11$			3.46410i			1.04447i	0.852803	+	0.522233i	$0.174901\pi$
$11$			3.46410i			1.04447i	−0.852803	+	0.522233i	$0.825099\pi$
$12$		−	1.00000i		−	0.288675i
$13$	3.46410	−	1.00000i	0.960769	−	0.277350i
$13$	3.46410	−	1.00000i	0.960769	−	0.277350i
$14$	−6.00000			−1.60357
$15$		0			0
$16$	−5.00000			−1.25000
$17$			6.00000i			1.45521i	0.685994	+	0.727607i	$0.259367\pi$
$17$			6.00000i			1.45521i	−0.685994	+	0.727607i	$0.740633\pi$
$18$	1.73205			0.408248
$19$			3.46410i			0.794719i	0.917663	+	0.397360i	$0.130073\pi$
$19$			3.46410i			0.794719i	−0.917663	+	0.397360i	$0.869927\pi$
$20$		0			0
$21$		−	3.46410i		−	0.755929i
$22$		−	6.00000i		−	1.27920i
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		−	1.73205i		−	0.353553i
$25$		0			0
$26$	−6.00000	+	1.73205i	−1.17670	+	0.339683i
$27$			1.00000i			0.192450i
$28$	3.46410			0.654654
$29$	−6.00000			−1.11417			−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000			−1.11417			−0.557086	+	0.830455i	$0.688081\pi$
$30$		0			0
$31$			3.46410i			0.622171i	0.950382	+	0.311086i	$0.100693\pi$
$31$			3.46410i			0.622171i	−0.950382	+	0.311086i	$0.899307\pi$
$32$	5.19615			0.918559
$33$	3.46410			0.603023
$34$		−	10.3923i		−	1.78227i
$35$		0			0
$36$	−1.00000			−0.166667
$37$	−6.92820			−1.13899			−0.569495	−	0.821995i	$-0.692861\pi$
$37$	−6.92820			−1.13899			−0.569495	+	0.821995i	$0.692861\pi$
$38$		−	6.00000i		−	0.973329i
$39$	−1.00000	−	3.46410i	−0.160128	−	0.554700i
$40$		0			0
$41$			6.92820i			1.08200i	0.841021	+	0.541002i	$0.181955\pi$
$41$			6.92820i			1.08200i	−0.841021	+	0.541002i	$0.818045\pi$
$42$			6.00000i			0.925820i
$43$		−	4.00000i		−	0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$		−	4.00000i		−	0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$			3.46410i			0.522233i
$45$		0			0
$46$		0			0
$47$	−3.46410			−0.505291			−0.252646	−	0.967559i	$-0.581301\pi$
$47$	−3.46410			−0.505291			−0.252646	+	0.967559i	$0.581301\pi$
$48$			5.00000i			0.721688i
$49$	5.00000			0.714286
$50$		0			0
$51$	6.00000			0.840168
$52$	3.46410	−	1.00000i	0.480384	−	0.138675i
$53$			6.00000i			0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$			6.00000i			0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$		−	1.73205i		−	0.235702i
$55$		0			0
$56$	6.00000			0.801784
$57$	3.46410			0.458831
$58$	10.3923			1.36458
$59$			10.3923i			1.35296i	0.736460	+	0.676481i	$0.236496\pi$
$59$			10.3923i			1.35296i	−0.736460	+	0.676481i	$0.763504\pi$
$60$		0			0
$61$	−2.00000			−0.256074			−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000			−0.256074			−0.128037	+	0.991769i	$0.540868\pi$
$62$		−	6.00000i		−	0.762001i
$63$	−3.46410			−0.436436
$64$	1.00000			0.125000
$65$		0			0
$66$	−6.00000			−0.738549
$67$	10.3923			1.26962			0.634811	−	0.772667i	$-0.281078\pi$
$67$	10.3923			1.26962			0.634811	+	0.772667i	$0.281078\pi$
$68$			6.00000i			0.727607i
$69$		0			0
$70$		0			0
$71$		−	3.46410i		−	0.411113i	−0.978645	−	0.205557i	$-0.934100\pi$
$71$		−	3.46410i		−	0.411113i	0.978645	−	0.205557i	$-0.0659005\pi$
$72$	−1.73205			−0.204124
$73$		0			0			−	1.00000i	$-0.5\pi$
$73$		0			0				1.00000i	$0.5\pi$
$74$	12.0000			1.39497
$75$		0			0
$76$			3.46410i			0.397360i
$77$			12.0000i			1.36753i
$78$	1.73205	+	6.00000i	0.196116	+	0.679366i
$79$	8.00000			0.900070			0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000			0.900070			0.450035	+	0.893011i	$0.351411\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		−	12.0000i		−	1.32518i
$83$	−3.46410			−0.380235			−0.190117	−	0.981761i	$-0.560887\pi$
$83$	−3.46410			−0.380235			−0.190117	+	0.981761i	$0.560887\pi$
$84$		−	3.46410i		−	0.377964i
$85$		0			0
$86$			6.92820i			0.747087i
$87$			6.00000i			0.643268i
$88$			6.00000i			0.639602i
$89$		−	6.92820i		−	0.734388i	−0.930144	−	0.367194i	$-0.880318\pi$
$89$		−	6.92820i		−	0.734388i	0.930144	−	0.367194i	$-0.119682\pi$
$90$		0			0
$91$	12.0000	−	3.46410i	1.25794	−	0.363137i
$92$		0			0
$93$	3.46410			0.359211
$94$	6.00000			0.618853
$95$		0			0
$96$		−	5.19615i		−	0.530330i
$97$	13.8564			1.40690			0.703452	−	0.710742i	$-0.251641\pi$
$97$	13.8564			1.40690			0.703452	+	0.710742i	$0.251641\pi$
$98$	−8.66025			−0.874818
$99$		−	3.46410i		−	0.348155i
$100$		0			0
$101$	6.00000			0.597022			0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000			0.597022			0.298511	+	0.954406i	$0.403510\pi$
$102$	−10.3923			−1.02899
$103$			8.00000i			0.788263i	0.919054	+	0.394132i	$0.128955\pi$
$103$			8.00000i			0.788263i	−0.919054	+	0.394132i	$0.871045\pi$
$104$	6.00000	−	1.73205i	0.588348	−	0.169842i
$105$		0			0
$106$		−	10.3923i		−	1.00939i
$107$		−	12.0000i		−	1.16008i	−0.814587	−	0.580042i	$-0.803036\pi$
$107$		−	12.0000i		−	1.16008i	0.814587	−	0.580042i	$-0.196964\pi$
$108$			1.00000i			0.0962250i
$109$			6.92820i			0.663602i	0.943349	+	0.331801i	$0.107656\pi$
$109$			6.92820i			0.663602i	−0.943349	+	0.331801i	$0.892344\pi$
$110$		0			0
$111$			6.92820i			0.657596i
$112$	−17.3205			−1.63663
$113$		−	6.00000i		−	0.564433i	−0.959351	−	0.282216i	$-0.908930\pi$
$113$		−	6.00000i		−	0.564433i	0.959351	−	0.282216i	$-0.0910696\pi$
$114$	−6.00000			−0.561951
$115$		0			0
$116$	−6.00000			−0.557086
$117$	−3.46410	+	1.00000i	−0.320256	+	0.0924500i
$118$		−	18.0000i		−	1.65703i
$119$			20.7846i			1.90532i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$	3.46410			0.313625
$123$	6.92820			0.624695
$124$			3.46410i			0.311086i
$125$		0			0
$126$	6.00000			0.534522
$127$			8.00000i			0.709885i	0.934888	+	0.354943i	$0.115500\pi$
$127$			8.00000i			0.709885i	−0.934888	+	0.354943i	$0.884500\pi$
$128$	−12.1244			−1.07165
$129$	−4.00000			−0.352180
$130$		0			0
$131$	−12.0000			−1.04844			−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000			−1.04844			−0.524222	+	0.851581i	$0.675644\pi$
$132$	3.46410			0.301511
$133$			12.0000i			1.04053i
$134$	−18.0000			−1.55496
$135$		0			0
$136$			10.3923i			0.891133i
$137$	20.7846			1.77575			0.887875	−	0.460086i	$-0.152181\pi$
$137$	20.7846			1.77575			0.887875	+	0.460086i	$0.152181\pi$
$138$		0			0
$139$	4.00000			0.339276			0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000			0.339276			0.169638	+	0.985506i	$0.445740\pi$
$140$		0			0
$141$			3.46410i			0.291730i
$142$			6.00000i			0.503509i
$143$	3.46410	+	12.0000i	0.289683	+	1.00349i
$144$	5.00000			0.416667
$145$		0			0
$146$		0			0
$147$		−	5.00000i		−	0.412393i
$148$	−6.92820			−0.569495
$149$			13.8564i			1.13516i	0.823318	+	0.567581i	$0.192120\pi$
$149$			13.8564i			1.13516i	−0.823318	+	0.567581i	$0.807880\pi$
$150$		0			0
$151$		−	10.3923i		−	0.845714i	−0.906196	−	0.422857i	$-0.861027\pi$
$151$		−	10.3923i		−	0.845714i	0.906196	−	0.422857i	$-0.138973\pi$
$152$			6.00000i			0.486664i
$153$		−	6.00000i		−	0.485071i
$154$		−	20.7846i		−	1.67487i
$155$		0			0
$156$	−1.00000	−	3.46410i	−0.0800641	−	0.277350i
$157$		−	14.0000i		−	1.11732i	−0.829396	−	0.558661i	$-0.811315\pi$
$157$		−	14.0000i		−	1.11732i	0.829396	−	0.558661i	$-0.188685\pi$
$158$	−13.8564			−1.10236
$159$	6.00000			0.475831
$160$		0			0
$161$		0			0
$162$	−1.73205			−0.136083
$163$	3.46410			0.271329			0.135665	−	0.990755i	$-0.456683\pi$
$163$	3.46410			0.271329			0.135665	+	0.990755i	$0.456683\pi$
$164$			6.92820i			0.541002i
$165$		0			0
$166$	6.00000			0.465690
$167$	−17.3205			−1.34030			−0.670151	−	0.742225i	$-0.733770\pi$
$167$	−17.3205			−1.34030			−0.670151	+	0.742225i	$0.733770\pi$
$168$		−	6.00000i		−	0.462910i
$169$	11.0000	−	6.92820i	0.846154	−	0.532939i
$170$		0			0
$171$		−	3.46410i		−	0.264906i
$172$		−	4.00000i		−	0.304997i
$173$		−	18.0000i		−	1.36851i	−0.729241	−	0.684257i	$-0.760127\pi$
$173$		−	18.0000i		−	1.36851i	0.729241	−	0.684257i	$-0.239873\pi$
$174$		−	10.3923i		−	0.787839i
$175$		0			0
$176$		−	17.3205i		−	1.30558i
$177$	10.3923			0.781133
$178$			12.0000i			0.899438i
$179$	12.0000			0.896922			0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000			0.896922			0.448461	+	0.893802i	$0.351972\pi$
$180$		0			0
$181$	−10.0000			−0.743294			−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000			−0.743294			−0.371647	+	0.928374i	$0.621207\pi$
$182$	−20.7846	+	6.00000i	−1.54066	+	0.444750i
$183$			2.00000i			0.147844i
$184$		0			0
$185$		0			0
$186$	−6.00000			−0.439941
$187$	−20.7846			−1.51992
$188$	−3.46410			−0.252646
$189$			3.46410i			0.251976i
$190$		0			0
$191$	24.0000			1.73658			0.868290	−	0.496058i	$-0.165220\pi$
$191$	24.0000			1.73658			0.868290	+	0.496058i	$0.165220\pi$
$192$		−	1.00000i		−	0.0721688i
$193$		0			0			−	1.00000i	$-0.5\pi$
$193$		0			0				1.00000i	$0.5\pi$
$194$	−24.0000			−1.72310
$195$		0			0
$196$	5.00000			0.357143
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$			6.00000i			0.426401i
$199$	−16.0000			−1.13421			−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000			−1.13421			−0.567105	+	0.823646i	$0.691937\pi$
$200$		0			0
$201$		−	10.3923i		−	0.733017i
$202$	−10.3923			−0.731200
$203$	−20.7846			−1.45879
$204$	6.00000			0.420084
$205$		0			0
$206$		−	13.8564i		−	0.965422i
$207$		0			0
$208$	−17.3205	+	5.00000i	−1.20096	+	0.346688i
$209$	−12.0000			−0.830057
$210$		0			0
$211$	−20.0000			−1.37686			−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000			−1.37686			−0.688428	+	0.725304i	$0.741699\pi$
$212$			6.00000i			0.412082i
$213$	−3.46410			−0.237356
$214$			20.7846i			1.42081i
$215$		0			0
$216$			1.73205i			0.117851i
$217$			12.0000i			0.814613i
$218$		−	12.0000i		−	0.812743i
$219$		0			0
$220$		0			0
$221$	6.00000	+	20.7846i	0.403604	+	1.39812i
$222$		−	12.0000i		−	0.805387i
$223$	−3.46410			−0.231973			−0.115987	−	0.993251i	$-0.537003\pi$
$223$	−3.46410			−0.231973			−0.115987	+	0.993251i	$0.537003\pi$
$224$	18.0000			1.20268
$225$		0			0
$226$			10.3923i			0.691286i
$227$	17.3205			1.14960			0.574801	−	0.818293i	$-0.305079\pi$
$227$	17.3205			1.14960			0.574801	+	0.818293i	$0.305079\pi$
$228$	3.46410			0.229416
$229$		−	6.92820i		−	0.457829i	−0.973447	−	0.228914i	$-0.926482\pi$
$229$		−	6.92820i		−	0.457829i	0.973447	−	0.228914i	$-0.0735176\pi$
$230$		0			0
$231$	12.0000			0.789542
$232$	−10.3923			−0.682288
$233$		−	6.00000i		−	0.393073i	−0.980497	−	0.196537i	$-0.937031\pi$
$233$		−	6.00000i		−	0.393073i	0.980497	−	0.196537i	$-0.0629694\pi$
$234$	6.00000	−	1.73205i	0.392232	−	0.113228i
$235$		0			0
$236$			10.3923i			0.676481i
$237$		−	8.00000i		−	0.519656i
$238$		−	36.0000i		−	2.33353i
$239$		−	10.3923i		−	0.672222i	−0.941822	−	0.336111i	$-0.890888\pi$
$239$		−	10.3923i		−	0.672222i	0.941822	−	0.336111i	$-0.109112\pi$
$240$		0			0
$241$		−	13.8564i		−	0.892570i	−0.894891	−	0.446285i	$-0.852747\pi$
$241$		−	13.8564i		−	0.892570i	0.894891	−	0.446285i	$-0.147253\pi$
$242$	1.73205			0.111340
$243$		−	1.00000i		−	0.0641500i
$244$	−2.00000			−0.128037
$245$		0			0
$246$	−12.0000			−0.765092
$247$	3.46410	+	12.0000i	0.220416	+	0.763542i
$248$			6.00000i			0.381000i
$249$			3.46410i			0.219529i
$250$		0			0
$251$	−12.0000			−0.757433			−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000			−0.757433			−0.378717	+	0.925513i	$0.623635\pi$
$252$	−3.46410			−0.218218
$253$		0			0
$254$		−	13.8564i		−	0.869428i
$255$		0			0
$256$	19.0000			1.18750
$257$		−	18.0000i		−	1.12281i	−0.827541	−	0.561405i	$-0.810261\pi$
$257$		−	18.0000i		−	1.12281i	0.827541	−	0.561405i	$-0.189739\pi$
$258$	6.92820			0.431331
$259$	−24.0000			−1.49129
$260$		0			0
$261$	6.00000			0.371391
$262$	20.7846			1.28408
$263$		−	24.0000i		−	1.47990i	−0.672660	−	0.739952i	$-0.734848\pi$
$263$		−	24.0000i		−	1.47990i	0.672660	−	0.739952i	$-0.265152\pi$
$264$	6.00000			0.369274
$265$		0			0
$266$		−	20.7846i		−	1.27439i
$267$	−6.92820			−0.423999
$268$	10.3923			0.634811
$269$	−6.00000			−0.365826			−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000			−0.365826			−0.182913	+	0.983129i	$0.558553\pi$
$270$		0			0
$271$		−	10.3923i		−	0.631288i	−0.948878	−	0.315644i	$-0.897780\pi$
$271$		−	10.3923i		−	0.631288i	0.948878	−	0.315644i	$-0.102220\pi$
$272$		−	30.0000i		−	1.81902i
$273$	−3.46410	−	12.0000i	−0.209657	−	0.726273i
$274$	−36.0000			−2.17484
$275$		0			0
$276$		0			0
$277$			10.0000i			0.600842i	0.953807	+	0.300421i	$0.0971271\pi$
$277$			10.0000i			0.600842i	−0.953807	+	0.300421i	$0.902873\pi$
$278$	−6.92820			−0.415526
$279$		−	3.46410i		−	0.207390i
$280$		0			0
$281$		−	6.92820i		−	0.413302i	−0.978415	−	0.206651i	$-0.933744\pi$
$281$		−	6.92820i		−	0.413302i	0.978415	−	0.206651i	$-0.0662565\pi$
$282$		−	6.00000i		−	0.357295i
$283$			4.00000i			0.237775i	0.992908	+	0.118888i	$0.0379328\pi$
$283$			4.00000i			0.237775i	−0.992908	+	0.118888i	$0.962067\pi$
$284$		−	3.46410i		−	0.205557i
$285$		0			0
$286$	−6.00000	−	20.7846i	−0.354787	−	1.22902i
$287$			24.0000i			1.41668i
$288$	−5.19615			−0.306186
$289$	−19.0000			−1.11765
$290$		0			0
$291$		−	13.8564i		−	0.812277i
$292$		0			0
$293$	27.7128			1.61900			0.809500	−	0.587120i	$-0.199738\pi$
$293$	27.7128			1.61900			0.809500	+	0.587120i	$0.199738\pi$
$294$			8.66025i			0.505076i
$295$		0			0
$296$	−12.0000			−0.697486
$297$	−3.46410			−0.201008
$298$		−	24.0000i		−	1.39028i
$299$		0			0
$300$		0			0
$301$		−	13.8564i		−	0.798670i
$302$			18.0000i			1.03578i
$303$		−	6.00000i		−	0.344691i
$304$		−	17.3205i		−	0.993399i
$305$		0			0
$306$			10.3923i			0.594089i
$307$	10.3923			0.593120			0.296560	−	0.955014i	$-0.404160\pi$
$307$	10.3923			0.593120			0.296560	+	0.955014i	$0.404160\pi$
$308$			12.0000i			0.683763i
$309$	8.00000			0.455104
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$	−1.73205	−	6.00000i	−0.0980581	−	0.339683i
$313$			10.0000i			0.565233i	0.959233	+	0.282617i	$0.0912024\pi$
$313$			10.0000i			0.565233i	−0.959233	+	0.282617i	$0.908798\pi$
$314$			24.2487i			1.36843i
$315$		0			0
$316$	8.00000			0.450035
$317$	−13.8564			−0.778253			−0.389127	−	0.921184i	$-0.627223\pi$
$317$	−13.8564			−0.778253			−0.389127	+	0.921184i	$0.627223\pi$
$318$	−10.3923			−0.582772
$319$		−	20.7846i		−	1.16371i
$320$		0			0
$321$	−12.0000			−0.669775
$322$		0			0
$323$	−20.7846			−1.15649
$324$	1.00000			0.0555556
$325$		0			0
$326$	−6.00000			−0.332309
$327$	6.92820			0.383131
$328$			12.0000i			0.662589i
$329$	−12.0000			−0.661581
$330$		0			0
$331$		−	3.46410i		−	0.190404i	−0.995458	−	0.0952021i	$-0.969650\pi$
$331$		−	3.46410i		−	0.190404i	0.995458	−	0.0952021i	$-0.0303497\pi$
$332$	−3.46410			−0.190117
$333$	6.92820			0.379663
$334$	30.0000			1.64153
$335$		0			0
$336$			17.3205i			0.944911i
$337$			14.0000i			0.762629i	0.924445	+	0.381314i	$0.124528\pi$
$337$			14.0000i			0.762629i	−0.924445	+	0.381314i	$0.875472\pi$
$338$	−19.0526	+	12.0000i	−1.03632	+	0.652714i
$339$	−6.00000			−0.325875
$340$		0			0
$341$	−12.0000			−0.649836
$342$			6.00000i			0.324443i
$343$	−6.92820			−0.374088
$344$		−	6.92820i		−	0.373544i
$345$		0			0
$346$			31.1769i			1.67608i
$347$		−	36.0000i		−	1.93258i	−0.257454	−	0.966291i	$-0.582883\pi$
$347$		−	36.0000i		−	1.93258i	0.257454	−	0.966291i	$-0.417117\pi$
$348$			6.00000i			0.321634i
$349$		−	6.92820i		−	0.370858i	−0.982658	−	0.185429i	$-0.940632\pi$
$349$		−	6.92820i		−	0.370858i	0.982658	−	0.185429i	$-0.0593675\pi$
$350$		0			0
$351$	1.00000	+	3.46410i	0.0533761	+	0.184900i
$352$			18.0000i			0.959403i
$353$	34.6410			1.84376			0.921878	−	0.387481i	$-0.126655\pi$
$353$	34.6410			1.84376			0.921878	+	0.387481i	$0.126655\pi$
$354$	−18.0000			−0.956689
$355$		0			0
$356$		−	6.92820i		−	0.367194i
$357$	20.7846			1.10004
$358$	−20.7846			−1.09850
$359$			17.3205i			0.914141i	0.889430	+	0.457071i	$0.151101\pi$
$359$			17.3205i			0.914141i	−0.889430	+	0.457071i	$0.848899\pi$
$360$		0			0
$361$	7.00000			0.368421
$362$	17.3205			0.910346
$363$			1.00000i			0.0524864i
$364$	12.0000	−	3.46410i	0.628971	−	0.181568i
$365$		0			0
$366$		−	3.46410i		−	0.181071i
$367$			16.0000i			0.835193i	0.908633	+	0.417597i	$0.137127\pi$
$367$			16.0000i			0.835193i	−0.908633	+	0.417597i	$0.862873\pi$
$368$		0			0
$369$		−	6.92820i		−	0.360668i
$370$		0			0
$371$			20.7846i			1.07908i
$372$	3.46410			0.179605
$373$			22.0000i			1.13912i	0.821951	+	0.569558i	$0.192886\pi$
$373$			22.0000i			1.13912i	−0.821951	+	0.569558i	$0.807114\pi$
$374$	36.0000			1.86152
$375$		0			0
$376$	−6.00000			−0.309426
$377$	−20.7846	+	6.00000i	−1.07046	+	0.309016i
$378$		−	6.00000i		−	0.308607i
$379$			17.3205i			0.889695i	0.895606	+	0.444847i	$0.146742\pi$
$379$			17.3205i			0.889695i	−0.895606	+	0.444847i	$0.853258\pi$
$380$		0			0
$381$	8.00000			0.409852
$382$	−41.5692			−2.12687
$383$	3.46410			0.177007			0.0885037	−	0.996076i	$-0.471792\pi$
$383$	3.46410			0.177007			0.0885037	+	0.996076i	$0.471792\pi$
$384$			12.1244i			0.618718i
$385$		0			0
$386$		0			0
$387$			4.00000i			0.203331i
$388$	13.8564			0.703452
$389$	18.0000			0.912636			0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000			0.912636			0.456318	+	0.889817i	$0.349168\pi$
$390$		0			0
$391$		0			0
$392$	8.66025			0.437409
$393$			12.0000i			0.605320i
$394$		0			0
$395$		0			0
$396$		−	3.46410i		−	0.174078i
$397$	34.6410			1.73858			0.869291	−	0.494300i	$-0.164576\pi$
$397$	34.6410			1.73858			0.869291	+	0.494300i	$0.164576\pi$
$398$	27.7128			1.38912
$399$	12.0000			0.600751
$400$		0			0
$401$		−	6.92820i		−	0.345978i	−0.984924	−	0.172989i	$-0.944657\pi$
$401$		−	6.92820i		−	0.345978i	0.984924	−	0.172989i	$-0.0553425\pi$
$402$			18.0000i			0.897758i
$403$	3.46410	+	12.0000i	0.172559	+	0.597763i
$404$	6.00000			0.298511
$405$		0			0
$406$	36.0000			1.78665
$407$		−	24.0000i		−	1.18964i
$408$	10.3923			0.514496
$409$			27.7128i			1.37031i	0.728397	+	0.685155i	$0.240266\pi$
$409$			27.7128i			1.37031i	−0.728397	+	0.685155i	$0.759734\pi$
$410$		0			0
$411$		−	20.7846i		−	1.02523i
$412$			8.00000i			0.394132i
$413$			36.0000i			1.77144i
$414$		0			0
$415$		0			0
$416$	18.0000	−	5.19615i	0.882523	−	0.254762i
$417$		−	4.00000i		−	0.195881i
$418$	20.7846			1.01661
$419$	12.0000			0.586238			0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000			0.586238			0.293119	+	0.956076i	$0.405307\pi$
$420$		0			0
$421$			34.6410i			1.68830i	0.536107	+	0.844150i	$0.319894\pi$
$421$			34.6410i			1.68830i	−0.536107	+	0.844150i	$0.680106\pi$
$422$	34.6410			1.68630
$423$	3.46410			0.168430
$424$			10.3923i			0.504695i
$425$		0			0
$426$	6.00000			0.290701
$427$	−6.92820			−0.335279
$428$		−	12.0000i		−	0.580042i
$429$	12.0000	−	3.46410i	0.579365	−	0.167248i
$430$		0			0
$431$			24.2487i			1.16802i	0.811747	+	0.584010i	$0.198517\pi$
$431$			24.2487i			1.16802i	−0.811747	+	0.584010i	$0.801483\pi$
$432$		−	5.00000i		−	0.240563i
$433$			34.0000i			1.63394i	0.576683	+	0.816968i	$0.304347\pi$
$433$			34.0000i			1.63394i	−0.576683	+	0.816968i	$0.695653\pi$
$434$		−	20.7846i		−	0.997693i
$435$		0			0
$436$			6.92820i			0.331801i
$437$		0			0
$438$		0			0
$439$	−8.00000			−0.381819			−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000			−0.381819			−0.190910	+	0.981608i	$0.561144\pi$
$440$		0			0
$441$	−5.00000			−0.238095
$442$	−10.3923	−	36.0000i	−0.494312	−	1.71235i
$443$		−	36.0000i		−	1.71041i	−0.518289	−	0.855206i	$-0.673431\pi$
$443$		−	36.0000i		−	1.71041i	0.518289	−	0.855206i	$-0.326569\pi$
$444$			6.92820i			0.328798i
$445$		0			0
$446$	6.00000			0.284108
$447$	13.8564			0.655386
$448$	3.46410			0.163663
$449$			6.92820i			0.326962i	0.986546	+	0.163481i	$0.0522723\pi$
$449$			6.92820i			0.326962i	−0.986546	+	0.163481i	$0.947728\pi$
$450$		0			0
$451$	−24.0000			−1.13012
$452$		−	6.00000i		−	0.282216i
$453$	−10.3923			−0.488273
$454$	−30.0000			−1.40797
$455$		0			0
$456$	6.00000			0.280976
$457$	27.7128			1.29635			0.648175	−	0.761491i	$-0.275532\pi$
$457$	27.7128			1.29635			0.648175	+	0.761491i	$0.275532\pi$
$458$			12.0000i			0.560723i
$459$	−6.00000			−0.280056
$460$		0			0
$461$			13.8564i			0.645357i	0.946509	+	0.322679i	$0.104583\pi$
$461$			13.8564i			0.645357i	−0.946509	+	0.322679i	$0.895417\pi$
$462$	−20.7846			−0.966988
$463$	−17.3205			−0.804952			−0.402476	−	0.915430i	$-0.631850\pi$
$463$	−17.3205			−0.804952			−0.402476	+	0.915430i	$0.631850\pi$
$464$	30.0000			1.39272
$465$		0			0
$466$			10.3923i			0.481414i
$467$		−	12.0000i		−	0.555294i	−0.960683	−	0.277647i	$-0.910445\pi$
$467$		−	12.0000i		−	0.555294i	0.960683	−	0.277647i	$-0.0895545\pi$
$468$	−3.46410	+	1.00000i	−0.160128	+	0.0462250i
$469$	36.0000			1.66233
$470$		0			0
$471$	−14.0000			−0.645086
$472$			18.0000i			0.828517i
$473$	13.8564			0.637118
$474$			13.8564i			0.636446i
$475$		0			0
$476$			20.7846i			0.952661i
$477$		−	6.00000i		−	0.274721i
$478$			18.0000i			0.823301i
$479$		−	10.3923i		−	0.474837i	−0.971408	−	0.237418i	$-0.923699\pi$
$479$		−	10.3923i		−	0.474837i	0.971408	−	0.237418i	$-0.0763012\pi$
$480$		0			0
$481$	−24.0000	+	6.92820i	−1.09431	+	0.315899i
$482$			24.0000i			1.09317i
$483$		0			0
$484$	−1.00000			−0.0454545
$485$		0			0
$486$			1.73205i			0.0785674i
$487$	−38.1051			−1.72671			−0.863354	−	0.504599i	$-0.831640\pi$
$487$	−38.1051			−1.72671			−0.863354	+	0.504599i	$0.831640\pi$
$488$	−3.46410			−0.156813
$489$		−	3.46410i		−	0.156652i
$490$		0			0
$491$	−12.0000			−0.541552			−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000			−0.541552			−0.270776	+	0.962642i	$0.587280\pi$
$492$	6.92820			0.312348
$493$		−	36.0000i		−	1.62136i
$494$	−6.00000	−	20.7846i	−0.269953	−	0.935144i
$495$		0			0
$496$		−	17.3205i		−	0.777714i
$497$		−	12.0000i		−	0.538274i
$498$		−	6.00000i		−	0.268866i
$499$		−	10.3923i		−	0.465223i	−0.972570	−	0.232612i	$-0.925273\pi$
$499$		−	10.3923i		−	0.465223i	0.972570	−	0.232612i	$-0.0747271\pi$
$500$		0			0
$501$			17.3205i			0.773823i
$502$	20.7846			0.927663
$503$			24.0000i			1.07011i	0.844818	+	0.535054i	$0.179709\pi$
$503$			24.0000i			1.07011i	−0.844818	+	0.535054i	$0.820291\pi$
$504$	−6.00000			−0.267261
$505$		0			0
$506$		0			0
$507$	−6.92820	−	11.0000i	−0.307692	−	0.488527i
$508$			8.00000i			0.354943i
$509$		−	41.5692i		−	1.84252i	−0.388943	−	0.921262i	$-0.627160\pi$
$509$		−	41.5692i		−	1.84252i	0.388943	−	0.921262i	$-0.372840\pi$
$510$		0			0
$511$		0			0
$512$	−8.66025			−0.382733
$513$	−3.46410			−0.152944
$514$			31.1769i			1.37515i
$515$		0			0
$516$	−4.00000			−0.176090
$517$		−	12.0000i		−	0.527759i
$518$	41.5692			1.82645
$519$	−18.0000			−0.790112
$520$		0			0
$521$	18.0000			0.788594			0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000			0.788594			0.394297	+	0.918983i	$0.370988\pi$
$522$	−10.3923			−0.454859
$523$		−	4.00000i		−	0.174908i	−0.996169	−	0.0874539i	$-0.972127\pi$
$523$		−	4.00000i		−	0.174908i	0.996169	−	0.0874539i	$-0.0278730\pi$
$524$	−12.0000			−0.524222
$525$		0			0
$526$			41.5692i			1.81250i
$527$	−20.7846			−0.905392
$528$	−17.3205			−0.753778
$529$	23.0000			1.00000
$530$		0			0
$531$		−	10.3923i		−	0.450988i
$532$			12.0000i			0.520266i
$533$	6.92820	+	24.0000i	0.300094	+	1.03956i
$534$	12.0000			0.519291
$535$		0			0
$536$	18.0000			0.777482
$537$		−	12.0000i		−	0.517838i
$538$	10.3923			0.448044
$539$			17.3205i			0.746047i
$540$		0			0
$541$		−	6.92820i		−	0.297867i	−0.988847	−	0.148933i	$-0.952416\pi$
$541$		−	6.92820i		−	0.297867i	0.988847	−	0.148933i	$-0.0475840\pi$
$542$			18.0000i			0.773166i
$543$			10.0000i			0.429141i
$544$			31.1769i			1.33670i
$545$		0			0
$546$	6.00000	+	20.7846i	0.256776	+	0.889499i
$547$		−	20.0000i		−	0.855138i	−0.903983	−	0.427569i	$-0.859370\pi$
$547$		−	20.0000i		−	0.855138i	0.903983	−	0.427569i	$-0.140630\pi$
$548$	20.7846			0.887875
$549$	2.00000			0.0853579
$550$		0			0
$551$		−	20.7846i		−	0.885454i
$552$		0			0
$553$	27.7128			1.17847
$554$		−	17.3205i		−	0.735878i
$555$		0			0
$556$	4.00000			0.169638
$557$	−13.8564			−0.587115			−0.293557	−	0.955941i	$-0.594839\pi$
$557$	−13.8564			−0.587115			−0.293557	+	0.955941i	$0.594839\pi$
$558$			6.00000i			0.254000i
$559$	−4.00000	−	13.8564i	−0.169182	−	0.586064i
$560$		0			0
$561$			20.7846i			0.877527i
$562$			12.0000i			0.506189i
$563$		−	12.0000i		−	0.505740i	−0.967500	−	0.252870i	$-0.918626\pi$
$563$		−	12.0000i		−	0.505740i	0.967500	−	0.252870i	$-0.0813744\pi$
$564$			3.46410i			0.145865i
$565$		0			0
$566$		−	6.92820i		−	0.291214i
$567$	3.46410			0.145479
$568$		−	6.00000i		−	0.251754i
$569$	6.00000			0.251533			0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000			0.251533			0.125767	+	0.992060i	$0.459861\pi$
$570$		0			0
$571$	4.00000			0.167395			0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000			0.167395			0.0836974	+	0.996491i	$0.473327\pi$
$572$	3.46410	+	12.0000i	0.144841	+	0.501745i
$573$		−	24.0000i		−	1.00261i
$574$		−	41.5692i		−	1.73507i
$575$		0			0
$576$	−1.00000			−0.0416667
$577$		0			0			−	1.00000i	$-0.5\pi$
$577$		0			0				1.00000i	$0.5\pi$
$578$	32.9090			1.36883
$579$		0			0
$580$		0			0
$581$	−12.0000			−0.497844
$582$			24.0000i			0.994832i
$583$	−20.7846			−0.860811
$584$		0			0
$585$		0			0
$586$	−48.0000			−1.98286
$587$	−10.3923			−0.428936			−0.214468	−	0.976731i	$-0.568802\pi$
$587$	−10.3923			−0.428936			−0.214468	+	0.976731i	$0.568802\pi$
$588$		−	5.00000i		−	0.206197i
$589$	−12.0000			−0.494451
$590$		0			0
$591$		0			0
$592$	34.6410			1.42374
$593$	6.92820			0.284507			0.142254	−	0.989830i	$-0.454565\pi$
$593$	6.92820			0.284507			0.142254	+	0.989830i	$0.454565\pi$
$594$	6.00000			0.246183
$595$		0			0
$596$			13.8564i			0.567581i
$597$			16.0000i			0.654836i
$598$		0			0
$599$	−24.0000			−0.980613			−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000			−0.980613			−0.490307	+	0.871550i	$0.663115\pi$
$600$		0			0
$601$	10.0000			0.407909			0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000			0.407909			0.203954	+	0.978980i	$0.434621\pi$
$602$			24.0000i			0.978167i
$603$	−10.3923			−0.423207
$604$		−	10.3923i		−	0.422857i
$605$		0			0
$606$			10.3923i			0.422159i
$607$		−	32.0000i		−	1.29884i	−0.760430	−	0.649420i	$-0.775012\pi$
$607$		−	32.0000i		−	1.29884i	0.760430	−	0.649420i	$-0.224988\pi$
$608$			18.0000i			0.729996i
$609$			20.7846i			0.842235i
$610$		0			0
$611$	−12.0000	+	3.46410i	−0.485468	+	0.140143i
$612$		−	6.00000i		−	0.242536i
$613$	−20.7846			−0.839482			−0.419741	−	0.907644i	$-0.637879\pi$
$613$	−20.7846			−0.839482			−0.419741	+	0.907644i	$0.637879\pi$
$614$	−18.0000			−0.726421
$615$		0			0
$616$			20.7846i			0.837436i
$617$	6.92820			0.278919			0.139459	−	0.990228i	$-0.455464\pi$
$617$	6.92820			0.278919			0.139459	+	0.990228i	$0.455464\pi$
$618$	−13.8564			−0.557386
$619$			31.1769i			1.25311i	0.779379	+	0.626553i	$0.215535\pi$
$619$			31.1769i			1.25311i	−0.779379	+	0.626553i	$0.784465\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	24.0000i		−	0.961540i
$624$	5.00000	+	17.3205i	0.200160	+	0.693375i
$625$		0			0
$626$		−	17.3205i		−	0.692267i
$627$			12.0000i			0.479234i
$628$		−	14.0000i		−	0.558661i
$629$		−	41.5692i		−	1.65747i
$630$		0			0
$631$		−	38.1051i		−	1.51694i	−0.651707	−	0.758470i	$-0.725947\pi$
$631$		−	38.1051i		−	1.51694i	0.651707	−	0.758470i	$-0.274053\pi$
$632$	13.8564			0.551178
$633$			20.0000i			0.794929i
$634$	24.0000			0.953162
$635$		0			0
$636$	6.00000			0.237915
$637$	17.3205	−	5.00000i	0.686264	−	0.198107i
$638$			36.0000i			1.42525i
$639$			3.46410i			0.137038i
$640$		0			0
$641$	−6.00000			−0.236986			−0.118493	−	0.992955i	$-0.537806\pi$
$641$	−6.00000			−0.236986			−0.118493	+	0.992955i	$0.537806\pi$
$642$	20.7846			0.820303
$643$	−10.3923			−0.409832			−0.204916	−	0.978780i	$-0.565692\pi$
$643$	−10.3923			−0.409832			−0.204916	+	0.978780i	$0.565692\pi$
$644$		0			0
$645$		0			0
$646$	36.0000			1.41640
$647$			24.0000i			0.943537i	0.881722	+	0.471769i	$0.156384\pi$
$647$			24.0000i			0.943537i	−0.881722	+	0.471769i	$0.843616\pi$
$648$	1.73205			0.0680414
$649$	−36.0000			−1.41312
$650$		0			0
$651$	12.0000			0.470317
$652$	3.46410			0.135665
$653$			6.00000i			0.234798i	0.993085	+	0.117399i	$0.0374557\pi$
$653$			6.00000i			0.234798i	−0.993085	+	0.117399i	$0.962544\pi$
$654$	−12.0000			−0.469237
$655$		0			0
$656$		−	34.6410i		−	1.35250i
$657$		0			0
$658$	20.7846			0.810268
$659$	−12.0000			−0.467454			−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000			−0.467454			−0.233727	+	0.972302i	$0.575092\pi$
$660$		0			0
$661$			20.7846i			0.808428i	0.914665	+	0.404214i	$0.132455\pi$
$661$			20.7846i			0.808428i	−0.914665	+	0.404214i	$0.867545\pi$
$662$			6.00000i			0.233197i
$663$	20.7846	−	6.00000i	0.807207	−	0.233021i
$664$	−6.00000			−0.232845
$665$		0			0
$666$	−12.0000			−0.464991
$667$		0			0
$668$	−17.3205			−0.670151
$669$			3.46410i			0.133930i
$670$		0			0
$671$		−	6.92820i		−	0.267460i
$672$		−	18.0000i		−	0.694365i
$673$		−	46.0000i		−	1.77317i	−0.462566	−	0.886585i	$-0.653071\pi$
$673$		−	46.0000i		−	1.77317i	0.462566	−	0.886585i	$-0.346929\pi$
$674$		−	24.2487i		−	0.934025i
$675$		0			0
$676$	11.0000	−	6.92820i	0.423077	−	0.266469i
$677$		−	6.00000i		−	0.230599i	−0.993331	−	0.115299i	$-0.963217\pi$
$677$		−	6.00000i		−	0.230599i	0.993331	−	0.115299i	$-0.0367827\pi$
$678$	10.3923			0.399114
$679$	48.0000			1.84207
$680$		0			0
$681$		−	17.3205i		−	0.663723i
$682$	20.7846			0.795884
$683$	−31.1769			−1.19295			−0.596476	−	0.802631i	$-0.703433\pi$
$683$	−31.1769			−1.19295			−0.596476	+	0.802631i	$0.703433\pi$
$684$		−	3.46410i		−	0.132453i
$685$		0			0
$686$	12.0000			0.458162
$687$	−6.92820			−0.264327
$688$			20.0000i			0.762493i
$689$	6.00000	+	20.7846i	0.228582	+	0.791831i
$690$		0			0
$691$		−	45.0333i		−	1.71315i	−0.516024	−	0.856574i	$-0.672588\pi$
$691$		−	45.0333i		−	1.71315i	0.516024	−	0.856574i	$-0.327412\pi$
$692$		−	18.0000i		−	0.684257i
$693$		−	12.0000i		−	0.455842i
$694$			62.3538i			2.36692i
$695$		0			0
$696$			10.3923i			0.393919i
$697$	−41.5692			−1.57455
$698$			12.0000i			0.454207i
$699$	−6.00000			−0.226941
$700$		0			0
$701$	−42.0000			−1.58632			−0.793159	−	0.609015i	$-0.791565\pi$
$701$	−42.0000			−1.58632			−0.793159	+	0.609015i	$0.791565\pi$
$702$	−1.73205	−	6.00000i	−0.0653720	−	0.226455i
$703$		−	24.0000i		−	0.905177i
$704$			3.46410i			0.130558i
$705$		0			0
$706$	−60.0000			−2.25813
$707$	20.7846			0.781686
$708$	10.3923			0.390567
$709$		−	6.92820i		−	0.260194i	−0.991501	−	0.130097i	$-0.958471\pi$
$709$		−	6.92820i		−	0.260194i	0.991501	−	0.130097i	$-0.0415289\pi$
$710$		0			0
$711$	−8.00000			−0.300023
$712$		−	12.0000i		−	0.449719i
$713$		0			0
$714$	−36.0000			−1.34727
$715$		0			0
$716$	12.0000			0.448461
$717$	−10.3923			−0.388108
$718$		−	30.0000i		−	1.11959i
$719$	−24.0000			−0.895049			−0.447524	−	0.894272i	$-0.647694\pi$
$719$	−24.0000			−0.895049			−0.447524	+	0.894272i	$0.647694\pi$
$720$		0			0
$721$			27.7128i			1.03208i
$722$	−12.1244			−0.451222
$723$	−13.8564			−0.515325
$724$	−10.0000			−0.371647
$725$		0			0
$726$		−	1.73205i		−	0.0642824i
$727$		−	16.0000i		−	0.593407i	−0.954970	−	0.296704i	$-0.904113\pi$
$727$		−	16.0000i		−	0.593407i	0.954970	−	0.296704i	$-0.0958873\pi$
$728$	20.7846	−	6.00000i	0.770329	−	0.222375i
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	24.0000			0.887672
$732$			2.00000i			0.0739221i
$733$	34.6410			1.27950			0.639748	−	0.768585i	$-0.279039\pi$
$733$	34.6410			1.27950			0.639748	+	0.768585i	$0.279039\pi$
$734$		−	27.7128i		−	1.02290i
$735$		0			0
$736$		0			0
$737$			36.0000i			1.32608i
$738$			12.0000i			0.441726i
$739$		−	38.1051i		−	1.40172i	−0.713299	−	0.700860i	$-0.752800\pi$
$739$		−	38.1051i		−	1.40172i	0.713299	−	0.700860i	$-0.247200\pi$
$740$		0			0
$741$	12.0000	−	3.46410i	0.440831	−	0.127257i
$742$		−	36.0000i		−	1.32160i
$743$	3.46410			0.127086			0.0635428	−	0.997979i	$-0.479760\pi$
$743$	3.46410			0.127086			0.0635428	+	0.997979i	$0.479760\pi$
$744$	6.00000			0.219971
$745$		0			0
$746$		−	38.1051i		−	1.39513i
$747$	3.46410			0.126745
$748$	−20.7846			−0.759961
$749$		−	41.5692i		−	1.51891i
$750$		0			0
$751$	32.0000			1.16770			0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000			1.16770			0.583848	+	0.811863i	$0.301546\pi$
$752$	17.3205			0.631614
$753$			12.0000i			0.437304i
$754$	36.0000	−	10.3923i	1.31104	−	0.378465i
$755$		0			0
$756$			3.46410i			0.125988i
$757$		−	22.0000i		−	0.799604i	−0.916602	−	0.399802i	$-0.869079\pi$
$757$		−	22.0000i		−	0.799604i	0.916602	−	0.399802i	$-0.130921\pi$
$758$		−	30.0000i		−	1.08965i
$759$		0			0
$760$		0			0
$761$			48.4974i			1.75803i	0.476794	+	0.879015i	$0.341799\pi$
$761$			48.4974i			1.75803i	−0.476794	+	0.879015i	$0.658201\pi$
$762$	−13.8564			−0.501965
$763$			24.0000i			0.868858i
$764$	24.0000			0.868290
$765$		0			0
$766$	−6.00000			−0.216789
$767$	10.3923	+	36.0000i	0.375244	+	1.29988i
$768$		−	19.0000i		−	0.685603i
$769$			27.7128i			0.999350i	0.866213	+	0.499675i	$0.166547\pi$
$769$			27.7128i			0.999350i	−0.866213	+	0.499675i	$0.833453\pi$
$770$		0			0
$771$	−18.0000			−0.648254
$772$		0			0
$773$	−13.8564			−0.498380			−0.249190	−	0.968455i	$-0.580164\pi$
$773$	−13.8564			−0.498380			−0.249190	+	0.968455i	$0.580164\pi$
$774$		−	6.92820i		−	0.249029i
$775$		0			0
$776$	24.0000			0.861550
$777$			24.0000i			0.860995i
$778$	−31.1769			−1.11775
$779$	−24.0000			−0.859889
$780$		0			0
$781$	12.0000			0.429394
$782$		0			0
$783$		−	6.00000i		−	0.214423i
$784$	−25.0000			−0.892857
$785$		0			0
$786$		−	20.7846i		−	0.741362i
$787$	10.3923			0.370446			0.185223	−	0.982697i	$-0.440699\pi$
$787$	10.3923			0.370446			0.185223	+	0.982697i	$0.440699\pi$
$788$		0			0
$789$	−24.0000			−0.854423
$790$		0			0
$791$		−	20.7846i		−	0.739016i
$792$		−	6.00000i		−	0.213201i
$793$	−6.92820	+	2.00000i	−0.246028	+	0.0710221i
$794$	−60.0000			−2.12932
$795$		0			0

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:	\( N \)	\(=\)	\( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	975.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.73205 q^{2} -1.00000i q^{3} +1.00000 q^{4} +1.73205i q^{6} +3.46410 q^{7} +1.73205 q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.73205 q^{2} -1.00000i q^{3} +1.00000 q^{4} +1.73205i q^{6} +3.46410 q^{7} +1.73205 q^{8} -1.00000 q^{9} +3.46410i q^{11} -1.00000i q^{12} +(3.46410 - 1.00000i) q^{13} -6.00000 q^{14} -5.00000 q^{16} +6.00000i q^{17} +1.73205 q^{18} +3.46410i q^{19} -3.46410i q^{21} -6.00000i q^{22} -1.73205i q^{24} +(-6.00000 + 1.73205i) q^{26} +1.00000i q^{27} +3.46410 q^{28} -6.00000 q^{29} +3.46410i q^{31} +5.19615 q^{32} +3.46410 q^{33} -10.3923i q^{34} -1.00000 q^{36} -6.92820 q^{37} -6.00000i q^{38} +(-1.00000 - 3.46410i) q^{39} +6.92820i q^{41} +6.00000i q^{42} -4.00000i q^{43} +3.46410i q^{44} -3.46410 q^{47} +5.00000i q^{48} +5.00000 q^{49} +6.00000 q^{51} +(3.46410 - 1.00000i) q^{52} +6.00000i q^{53} -1.73205i q^{54} +6.00000 q^{56} +3.46410 q^{57} +10.3923 q^{58} +10.3923i q^{59} -2.00000 q^{61} -6.00000i q^{62} -3.46410 q^{63} +1.00000 q^{64} -6.00000 q^{66} +10.3923 q^{67} +6.00000i q^{68} -3.46410i q^{71} -1.73205 q^{72} +12.0000 q^{74} +3.46410i q^{76} +12.0000i q^{77} +(1.73205 + 6.00000i) q^{78} +8.00000 q^{79} +1.00000 q^{81} -12.0000i q^{82} -3.46410 q^{83} -3.46410i q^{84} +6.92820i q^{86} +6.00000i q^{87} +6.00000i q^{88} -6.92820i q^{89} +(12.0000 - 3.46410i) q^{91} +3.46410 q^{93} +6.00000 q^{94} -5.19615i q^{96} +13.8564 q^{97} -8.66025 q^{98} -3.46410i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{4} - 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^4 - 4 * q^9	\( 4 q + 4 q^{4} - 4 q^{9} - 24 q^{14} - 20 q^{16} - 24 q^{26} - 24 q^{29} - 4 q^{36} - 4 q^{39} + 20 q^{49} + 24 q^{51} + 24 q^{56} - 8 q^{61} + 4 q^{64} - 24 q^{66} + 48 q^{74} + 32 q^{79} + 4 q^{81} + 48 q^{91} + 24 q^{94}+O(q^{100}) \) 4 * q + 4 * q^4 - 4 * q^9 - 24 * q^14 - 20 * q^16 - 24 * q^26 - 24 * q^29 - 4 * q^36 - 4 * q^39 + 20 * q^49 + 24 * q^51 + 24 * q^56 - 8 * q^61 + 4 * q^64 - 24 * q^66 + 48 * q^74 + 32 * q^79 + 4 * q^81 + 48 * q^91 + 24 * q^94

Introduction

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