LMFDB - Embedded newform 1216.2.b.d.607.2

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	$9.70980888579$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.49787136.1
Defining polynomial:	$x^{8} + 3 x^{6} + 5 x^{4} + 12 x^{2} + 16$
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			607.2
Root			$-1.09445 + 0.895644i$ of defining polynomial
Character	$\chi$	$=$	1216.607
Dual form			1216.2.b.d.607.7

$q$-expansion

$f(q)$	$=$	$q-2.64575i q^{3} -3.46410i q^{5} +1.00000i q^{7} -4.00000 q^{9} +O(q^{10})$	$q-2.64575i q^{3} -3.46410i q^{5} +1.00000i q^{7} -4.00000 q^{9} -2.64575 q^{13} -9.16515 q^{15} -3.00000 q^{17} +(3.46410 - 2.64575i) q^{19} +2.64575 q^{21} +3.00000i q^{23} -7.00000 q^{25} +2.64575i q^{27} -7.93725 q^{29} +3.46410 q^{35} -10.5830 q^{37} +7.00000i q^{39} -9.16515i q^{41} +10.3923 q^{43} +13.8564i q^{45} +6.00000 q^{49} +7.93725i q^{51} +7.93725 q^{53} +(-7.00000 - 9.16515i) q^{57} -7.93725i q^{59} +6.92820i q^{61} -4.00000i q^{63} +9.16515i q^{65} +13.2288i q^{67} +7.93725 q^{69} +9.16515 q^{71} +7.00000 q^{73} +18.5203i q^{75} -9.16515 q^{79} -5.00000 q^{81} -17.3205 q^{83} +10.3923i q^{85} +21.0000i q^{87} +9.16515i q^{89} -2.64575i q^{91} +(-9.16515 - 12.0000i) q^{95} -9.16515i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8q - 32q^{9} + O(q^{10}) $	$ 8q - 32q^{9} - 24q^{17} - 56q^{25} + 48q^{49} - 56q^{57} + 56q^{73} - 40q^{81} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		−	2.64575i		−	1.52753i	−0.645497	−	0.763763i	$-0.723350\pi$
$3$		−	2.64575i		−	1.52753i	0.645497	−	0.763763i	$-0.276650\pi$
$4$		0			0
$5$		−	3.46410i		−	1.54919i	−0.632456	−	0.774597i	$-0.717953\pi$
$5$		−	3.46410i		−	1.54919i	0.632456	−	0.774597i	$-0.282047\pi$
$6$		0			0
$7$			1.00000i			0.377964i	0.981981	+	0.188982i	$0.0605189\pi$
$7$			1.00000i			0.377964i	−0.981981	+	0.188982i	$0.939481\pi$
$8$		0			0
$9$	−4.00000			−1.33333
$10$		0			0
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$		0			0
$13$	−2.64575			−0.733799			−0.366900	−	0.930261i	$-0.619581\pi$
$13$	−2.64575			−0.733799			−0.366900	+	0.930261i	$0.619581\pi$
$14$		0			0
$15$	−9.16515			−2.36643
$16$		0			0
$17$	−3.00000			−0.727607			−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000			−0.727607			−0.363803	+	0.931476i	$0.618522\pi$
$18$		0			0
$19$	3.46410	−	2.64575i	0.794719	−	0.606977i
$19$	3.46410	−	2.64575i	0.794719	−	0.606977i
$20$		0			0
$21$	2.64575			0.577350
$22$		0			0
$23$			3.00000i			0.625543i	0.949828	+	0.312772i	$0.101257\pi$
$23$			3.00000i			0.625543i	−0.949828	+	0.312772i	$0.898743\pi$
$24$		0			0
$25$	−7.00000			−1.40000
$26$		0			0
$27$			2.64575i			0.509175i
$28$		0			0
$29$	−7.93725			−1.47391			−0.736956	−	0.675941i	$-0.763737\pi$
$29$	−7.93725			−1.47391			−0.736956	+	0.675941i	$0.763737\pi$
$30$		0			0
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	3.46410			0.585540
$36$		0			0
$37$	−10.5830			−1.73984			−0.869918	−	0.493197i	$-0.835828\pi$
$37$	−10.5830			−1.73984			−0.869918	+	0.493197i	$0.835828\pi$
$38$		0			0
$39$			7.00000i			1.12090i
$40$		0			0
$41$		−	9.16515i		−	1.43136i	−0.698430	−	0.715678i	$-0.746118\pi$
$41$		−	9.16515i		−	1.43136i	0.698430	−	0.715678i	$-0.253882\pi$
$42$		0			0
$43$	10.3923			1.58481			0.792406	−	0.609994i	$-0.208828\pi$
$43$	10.3923			1.58481			0.792406	+	0.609994i	$0.208828\pi$
$44$		0			0
$45$			13.8564i			2.06559i
$46$		0			0
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$		0			0
$49$	6.00000			0.857143
$50$		0			0
$51$			7.93725i			1.11144i
$52$		0			0
$53$	7.93725			1.09027			0.545133	−	0.838350i	$-0.316479\pi$
$53$	7.93725			1.09027			0.545133	+	0.838350i	$0.316479\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−7.00000	−	9.16515i	−0.927173	−	1.21395i
$58$		0			0
$59$		−	7.93725i		−	1.03334i	−0.856184	−	0.516671i	$-0.827171\pi$
$59$		−	7.93725i		−	1.03334i	0.856184	−	0.516671i	$-0.172829\pi$
$60$		0			0
$61$			6.92820i			0.887066i	0.896258	+	0.443533i	$0.146275\pi$
$61$			6.92820i			0.887066i	−0.896258	+	0.443533i	$0.853725\pi$
$62$		0			0
$63$		−	4.00000i		−	0.503953i
$64$		0			0
$65$			9.16515i			1.13680i
$66$		0			0
$67$			13.2288i			1.61615i	0.589080	+	0.808075i	$0.299490\pi$
$67$			13.2288i			1.61615i	−0.589080	+	0.808075i	$0.700510\pi$
$68$		0			0
$69$	7.93725			0.955533
$70$		0			0
$71$	9.16515			1.08770			0.543852	−	0.839181i	$-0.316965\pi$
$71$	9.16515			1.08770			0.543852	+	0.839181i	$0.316965\pi$
$72$		0			0
$73$	7.00000			0.819288			0.409644	−	0.912245i	$-0.365653\pi$
$73$	7.00000			0.819288			0.409644	+	0.912245i	$0.365653\pi$
$74$		0			0
$75$			18.5203i			2.13854i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−9.16515			−1.03116			−0.515580	−	0.856841i	$-0.672424\pi$
$79$	−9.16515			−1.03116			−0.515580	+	0.856841i	$0.672424\pi$
$80$		0			0
$81$	−5.00000			−0.555556
$82$		0			0
$83$	−17.3205			−1.90117			−0.950586	−	0.310460i	$-0.899517\pi$
$83$	−17.3205			−1.90117			−0.950586	+	0.310460i	$0.899517\pi$
$84$		0			0
$85$			10.3923i			1.12720i
$86$		0			0
$87$			21.0000i			2.25144i
$88$		0			0
$89$			9.16515i			0.971504i	0.874097	+	0.485752i	$0.161454\pi$
$89$			9.16515i			0.971504i	−0.874097	+	0.485752i	$0.838546\pi$
$90$		0			0
$91$		−	2.64575i		−	0.277350i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−9.16515	−	12.0000i	−0.940325	−	1.23117i
$96$		0			0
$97$		−	9.16515i		−	0.930580i	−0.885158	−	0.465290i	$-0.845950\pi$
$97$		−	9.16515i		−	0.930580i	0.885158	−	0.465290i	$-0.154050\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	13.8564i		−	1.37876i	−0.724398	−	0.689382i	$-0.757882\pi$
$101$		−	13.8564i		−	1.37876i	0.724398	−	0.689382i	$-0.242118\pi$
$102$		0			0
$103$	9.16515			0.903069			0.451535	−	0.892254i	$-0.350877\pi$
$103$	9.16515			0.903069			0.451535	+	0.892254i	$0.350877\pi$
$104$		0			0
$105$		−	9.16515i		−	0.894427i
$106$		0			0
$107$		−	7.93725i		−	0.767323i	−0.923474	−	0.383662i	$-0.874663\pi$
$107$		−	7.93725i		−	0.767323i	0.923474	−	0.383662i	$-0.125337\pi$
$108$		0			0
$109$	2.64575			0.253417			0.126709	−	0.991940i	$-0.459559\pi$
$109$	2.64575			0.253417			0.126709	+	0.991940i	$0.459559\pi$
$110$		0			0
$111$			28.0000i			2.65764i
$112$		0			0
$113$		−	9.16515i		−	0.862185i	−0.902308	−	0.431092i	$-0.858128\pi$
$113$		−	9.16515i		−	0.862185i	0.902308	−	0.431092i	$-0.141872\pi$
$114$		0			0
$115$	10.3923			0.969087
$116$		0			0
$117$	10.5830			0.978399
$118$		0			0
$119$		−	3.00000i		−	0.275010i
$120$		0			0
$121$	−11.0000			−1.00000
$122$		0			0
$123$	−24.2487			−2.18643
$124$		0			0
$125$			6.92820i			0.619677i
$126$		0			0
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$		0			0
$129$		−	27.4955i		−	2.42084i
$130$		0			0
$131$	3.46410			0.302660			0.151330	−	0.988483i	$-0.451644\pi$
$131$	3.46410			0.302660			0.151330	+	0.988483i	$0.451644\pi$
$132$		0			0
$133$	2.64575	+	3.46410i	0.229416	+	0.300376i
$134$		0			0
$135$	9.16515			0.788811
$136$		0			0
$137$	3.00000			0.256307			0.128154	−	0.991754i	$-0.459095\pi$
$137$	3.00000			0.256307			0.128154	+	0.991754i	$0.459095\pi$
$138$		0			0
$139$	13.8564			1.17529			0.587643	−	0.809121i	$-0.300056\pi$
$139$	13.8564			1.17529			0.587643	+	0.809121i	$0.300056\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$			27.4955i			2.28337i
$146$		0			0
$147$		−	15.8745i		−	1.30931i
$148$		0			0
$149$		−	17.3205i		−	1.41895i	−0.704730	−	0.709476i	$-0.748932\pi$
$149$		−	17.3205i		−	1.41895i	0.704730	−	0.709476i	$-0.251068\pi$
$150$		0			0
$151$	−18.3303			−1.49170			−0.745849	−	0.666115i	$-0.767956\pi$
$151$	−18.3303			−1.49170			−0.745849	+	0.666115i	$0.767956\pi$
$152$		0			0
$153$	12.0000			0.970143
$154$		0			0
$155$		0			0
$156$		0			0
$157$		−	10.3923i		−	0.829396i	−0.909959	−	0.414698i	$-0.863887\pi$
$157$		−	10.3923i		−	0.829396i	0.909959	−	0.414698i	$-0.136113\pi$
$158$		0			0
$159$		−	21.0000i		−	1.66541i
$160$		0			0
$161$	−3.00000			−0.236433
$162$		0			0
$163$	−6.92820			−0.542659			−0.271329	−	0.962487i	$-0.587463\pi$
$163$	−6.92820			−0.542659			−0.271329	+	0.962487i	$0.587463\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−9.16515			−0.709221			−0.354610	−	0.935014i	$-0.615386\pi$
$167$	−9.16515			−0.709221			−0.354610	+	0.935014i	$0.615386\pi$
$168$		0			0
$169$	−6.00000			−0.461538
$170$		0			0
$171$	−13.8564	+	10.5830i	−1.05963	+	0.809303i
$172$		0			0
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$		−	7.00000i		−	0.529150i
$176$		0			0
$177$	−21.0000			−1.57846
$178$		0			0
$179$		−	15.8745i		−	1.18652i	−0.805012	−	0.593258i	$-0.797841\pi$
$179$		−	15.8745i		−	1.18652i	0.805012	−	0.593258i	$-0.202159\pi$
$180$		0			0
$181$	−10.5830			−0.786629			−0.393314	−	0.919404i	$-0.628672\pi$
$181$	−10.5830			−0.786629			−0.393314	+	0.919404i	$0.628672\pi$
$182$		0			0
$183$	18.3303			1.35501
$184$		0			0
$185$			36.6606i			2.69534i
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−2.64575			−0.192450
$190$		0			0
$191$		−	3.00000i		−	0.217072i	−0.994092	−	0.108536i	$-0.965384\pi$
$191$		−	3.00000i		−	0.217072i	0.994092	−	0.108536i	$-0.0346163\pi$
$192$		0			0
$193$			18.3303i			1.31944i	0.751510	+	0.659722i	$0.229326\pi$
$193$			18.3303i			1.31944i	−0.751510	+	0.659722i	$0.770674\pi$
$194$		0			0
$195$	24.2487			1.73649
$196$		0			0
$197$			10.3923i			0.740421i	0.928948	+	0.370211i	$0.120714\pi$
$197$			10.3923i			0.740421i	−0.928948	+	0.370211i	$0.879286\pi$
$198$		0			0
$199$		−	17.0000i		−	1.20510i	−0.798082	−	0.602549i	$-0.794152\pi$
$199$		−	17.0000i		−	1.20510i	0.798082	−	0.602549i	$-0.205848\pi$
$200$		0			0
$201$	35.0000			2.46871
$202$		0			0
$203$		−	7.93725i		−	0.557086i
$204$		0			0
$205$	−31.7490			−2.21745
$206$		0			0
$207$		−	12.0000i		−	0.834058i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		−	2.64575i		−	0.182141i	−0.995844	−	0.0910705i	$-0.970971\pi$
$211$		−	2.64575i		−	0.182141i	0.995844	−	0.0910705i	$-0.0290289\pi$
$212$		0			0
$213$		−	24.2487i		−	1.66149i
$214$		0			0
$215$		−	36.0000i		−	2.45518i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		−	18.5203i		−	1.25148i
$220$		0			0
$221$	7.93725			0.533917
$222$		0			0
$223$	27.4955			1.84123			0.920616	−	0.390469i	$-0.127687\pi$
$223$	27.4955			1.84123			0.920616	+	0.390469i	$0.127687\pi$
$224$		0			0
$225$	28.0000			1.86667
$226$		0			0
$227$		−	7.93725i		−	0.526814i	−0.964685	−	0.263407i	$-0.915154\pi$
$227$		−	7.93725i		−	0.526814i	0.964685	−	0.263407i	$-0.0848462\pi$
$228$		0			0
$229$		−	20.7846i		−	1.37349i	−0.726900	−	0.686743i	$-0.759040\pi$
$229$		−	20.7846i		−	1.37349i	0.726900	−	0.686743i	$-0.240960\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	18.0000			1.17922			0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000			1.17922			0.589610	+	0.807688i	$0.299282\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$			24.2487i			1.57512i
$238$		0			0
$239$			15.0000i			0.970269i	0.874439	+	0.485135i	$0.161229\pi$
$239$			15.0000i			0.970269i	−0.874439	+	0.485135i	$0.838771\pi$
$240$		0			0
$241$		−	9.16515i		−	0.590379i	−0.955439	−	0.295190i	$-0.904617\pi$
$241$		−	9.16515i		−	0.590379i	0.955439	−	0.295190i	$-0.0953828\pi$
$242$		0			0
$243$			21.1660i			1.35780i
$244$		0			0
$245$		−	20.7846i		−	1.32788i
$246$		0			0
$247$	−9.16515	+	7.00000i	−0.583165	+	0.445399i
$248$		0			0
$249$			45.8258i			2.90409i
$250$		0			0
$251$	17.3205			1.09326			0.546630	−	0.837374i	$-0.315910\pi$
$251$	17.3205			1.09326			0.546630	+	0.837374i	$0.315910\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	27.4955			1.72183
$256$		0			0
$257$		−	9.16515i		−	0.571706i	−0.958273	−	0.285853i	$-0.907723\pi$
$257$		−	9.16515i		−	0.571706i	0.958273	−	0.285853i	$-0.0922770\pi$
$258$		0			0
$259$		−	10.5830i		−	0.657596i
$260$		0			0
$261$	31.7490			1.96521
$262$		0			0
$263$		−	12.0000i		−	0.739952i	−0.929041	−	0.369976i	$-0.879366\pi$
$263$		−	12.0000i		−	0.739952i	0.929041	−	0.369976i	$-0.120634\pi$
$264$		0			0
$265$		−	27.4955i		−	1.68903i
$266$		0			0
$267$	24.2487			1.48400
$268$		0			0
$269$	−31.7490			−1.93577			−0.967886	−	0.251390i	$-0.919112\pi$
$269$	−31.7490			−1.93577			−0.967886	+	0.251390i	$0.919112\pi$
$270$		0			0
$271$		−	1.00000i		−	0.0607457i	−0.999539	−	0.0303728i	$-0.990331\pi$
$271$		−	1.00000i		−	0.0607457i	0.999539	−	0.0303728i	$-0.00966946\pi$
$272$		0			0
$273$	−7.00000			−0.423659
$274$		0			0
$275$		0			0
$276$		0			0
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			18.3303i			1.09349i	0.837298	+	0.546747i	$0.184134\pi$
$281$			18.3303i			1.09349i	−0.837298	+	0.546747i	$0.815866\pi$
$282$		0			0
$283$	−10.3923			−0.617758			−0.308879	−	0.951101i	$-0.599954\pi$
$283$	−10.3923			−0.617758			−0.308879	+	0.951101i	$0.599954\pi$
$284$		0			0
$285$	−31.7490	+	24.2487i	−1.88065	+	1.43637i
$286$		0			0
$287$	9.16515			0.541002
$288$		0			0
$289$	−8.00000			−0.470588
$290$		0			0
$291$	−24.2487			−1.42148
$292$		0			0
$293$	23.8118			1.39110			0.695549	−	0.718479i	$-0.255161\pi$
$293$	23.8118			1.39110			0.695549	+	0.718479i	$0.255161\pi$
$294$		0			0
$295$	−27.4955			−1.60085
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	7.93725i		−	0.459023i
$300$		0			0
$301$			10.3923i			0.599002i
$302$		0			0
$303$	−36.6606			−2.10610
$304$		0			0
$305$	24.0000			1.37424
$306$		0			0
$307$			26.4575i			1.51001i	0.655719	+	0.755005i	$0.272366\pi$
$307$			26.4575i			1.51001i	−0.655719	+	0.755005i	$0.727634\pi$
$308$		0			0
$309$		−	24.2487i		−	1.37946i
$310$		0			0
$311$			3.00000i			0.170114i	0.996376	+	0.0850572i	$0.0271073\pi$
$311$			3.00000i			0.170114i	−0.996376	+	0.0850572i	$0.972893\pi$
$312$		0			0
$313$	−13.0000			−0.734803			−0.367402	−	0.930062i	$-0.619753\pi$
$313$	−13.0000			−0.734803			−0.367402	+	0.930062i	$0.619753\pi$
$314$		0			0
$315$	−13.8564			−0.780720
$316$		0			0
$317$	23.8118			1.33740			0.668701	−	0.743532i	$-0.266851\pi$
$317$	23.8118			1.33740			0.668701	+	0.743532i	$0.266851\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−21.0000			−1.17211
$322$		0			0
$323$	−10.3923	+	7.93725i	−0.578243	+	0.441641i
$324$		0			0
$325$	18.5203			1.02732
$326$		0			0
$327$		−	7.00000i		−	0.387101i
$328$		0			0
$329$		0			0
$330$		0			0
$331$		−	18.5203i		−	1.01797i	−0.860777	−	0.508983i	$-0.830022\pi$
$331$		−	18.5203i		−	1.01797i	0.860777	−	0.508983i	$-0.169978\pi$
$332$		0			0
$333$	42.3320			2.31978
$334$		0			0
$335$	45.8258			2.50373
$336$		0			0
$337$			9.16515i			0.499258i	0.968342	+	0.249629i	$0.0803086\pi$
$337$			9.16515i			0.499258i	−0.968342	+	0.249629i	$0.919691\pi$
$338$		0			0
$339$	−24.2487			−1.31701
$340$		0			0
$341$		0			0
$342$		0			0
$343$			13.0000i			0.701934i
$344$		0			0
$345$		−	27.4955i		−	1.48031i
$346$		0			0
$347$	−20.7846			−1.11578			−0.557888	−	0.829916i	$-0.688388\pi$
$347$	−20.7846			−1.11578			−0.557888	+	0.829916i	$0.688388\pi$
$348$		0			0
$349$			17.3205i			0.927146i	0.886059	+	0.463573i	$0.153433\pi$
$349$			17.3205i			0.927146i	−0.886059	+	0.463573i	$0.846567\pi$
$350$		0			0
$351$		−	7.00000i		−	0.373632i
$352$		0			0
$353$	21.0000			1.11772			0.558859	−	0.829263i	$-0.311239\pi$
$353$	21.0000			1.11772			0.558859	+	0.829263i	$0.311239\pi$
$354$		0			0
$355$		−	31.7490i		−	1.68506i
$356$		0			0
$357$	−7.93725			−0.420084
$358$		0			0
$359$		−	27.0000i		−	1.42501i	−0.701669	−	0.712503i	$-0.747562\pi$
$359$		−	27.0000i		−	1.42501i	0.701669	−	0.712503i	$-0.252438\pi$
$360$		0			0
$361$	5.00000	−	18.3303i	0.263158	−	0.964753i
$362$		0			0
$363$			29.1033i			1.52753i
$364$		0			0
$365$		−	24.2487i		−	1.26924i
$366$		0			0
$367$			28.0000i			1.46159i	0.682598	+	0.730794i	$0.260850\pi$
$367$			28.0000i			1.46159i	−0.682598	+	0.730794i	$0.739150\pi$
$368$		0			0
$369$			36.6606i			1.90847i
$370$		0			0
$371$			7.93725i			0.412082i
$372$		0			0
$373$	13.2288			0.684959			0.342480	−	0.939525i	$-0.388733\pi$
$373$	13.2288			0.684959			0.342480	+	0.939525i	$0.388733\pi$
$374$		0			0
$375$	18.3303			0.946573
$376$		0			0
$377$	21.0000			1.08156
$378$		0			0
$379$		−	13.2288i		−	0.679516i	−0.940513	−	0.339758i	$-0.889655\pi$
$379$		−	13.2288i		−	0.679516i	0.940513	−	0.339758i	$-0.110345\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	27.4955			1.40495			0.702476	−	0.711707i	$-0.252078\pi$
$383$	27.4955			1.40495			0.702476	+	0.711707i	$0.252078\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−41.5692			−2.11308
$388$		0			0
$389$		−	20.7846i		−	1.05382i	−0.849921	−	0.526911i	$-0.823350\pi$
$389$		−	20.7846i		−	1.05382i	0.849921	−	0.526911i	$-0.176650\pi$
$390$		0			0
$391$		−	9.00000i		−	0.455150i
$392$		0			0
$393$		−	9.16515i		−	0.462321i
$394$		0			0
$395$			31.7490i			1.59747i
$396$		0			0
$397$		−	20.7846i		−	1.04315i	−0.853206	−	0.521575i	$-0.825345\pi$
$397$		−	20.7846i		−	1.04315i	0.853206	−	0.521575i	$-0.174655\pi$
$398$		0			0
$399$	9.16515	−	7.00000i	0.458831	−	0.350438i
$400$		0			0
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$			17.3205i			0.860663i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		−	18.3303i		−	0.906375i	−0.891415	−	0.453188i	$-0.850287\pi$
$409$		−	18.3303i		−	0.906375i	0.891415	−	0.453188i	$-0.149713\pi$
$410$		0			0
$411$		−	7.93725i		−	0.391516i
$412$		0			0
$413$	7.93725			0.390567
$414$		0			0
$415$			60.0000i			2.94528i
$416$		0			0
$417$		−	36.6606i		−	1.79528i
$418$		0			0
$419$	−31.1769			−1.52309			−0.761546	−	0.648111i	$-0.775559\pi$
$419$	−31.1769			−1.52309			−0.761546	+	0.648111i	$0.775559\pi$
$420$		0			0
$421$	−2.64575			−0.128946			−0.0644730	−	0.997919i	$-0.520537\pi$
$421$	−2.64575			−0.128946			−0.0644730	+	0.997919i	$0.520537\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	21.0000			1.01865
$426$		0			0
$427$	−6.92820			−0.335279
$428$		0			0
$429$		0			0
$430$		0			0
$431$	18.3303			0.882940			0.441470	−	0.897276i	$-0.354457\pi$
$431$	18.3303			0.882940			0.441470	+	0.897276i	$0.354457\pi$
$432$		0			0
$433$		−	27.4955i		−	1.32135i	−0.750673	−	0.660674i	$-0.770271\pi$
$433$		−	27.4955i		−	1.32135i	0.750673	−	0.660674i	$-0.229729\pi$
$434$		0			0
$435$	72.7461			3.48791
$436$		0			0
$437$	7.93725	+	10.3923i	0.379690	+	0.497131i
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	−24.0000			−1.14286
$442$		0			0
$443$	6.92820			0.329169			0.164584	−	0.986363i	$-0.447372\pi$
$443$	6.92820			0.329169			0.164584	+	0.986363i	$0.447372\pi$
$444$		0			0
$445$	31.7490			1.50505
$446$		0			0
$447$	−45.8258			−2.16748
$448$		0			0
$449$			36.6606i			1.73012i	0.501668	+	0.865060i	$0.332720\pi$
$449$			36.6606i			1.73012i	−0.501668	+	0.865060i	$0.667280\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$			48.4974i			2.27861i
$454$		0			0
$455$	−9.16515			−0.429669
$456$		0			0
$457$	−29.0000			−1.35656			−0.678281	−	0.734802i	$-0.737275\pi$
$457$	−29.0000			−1.35656			−0.678281	+	0.734802i	$0.737275\pi$
$458$		0			0
$459$		−	7.93725i		−	0.370479i
$460$		0			0
$461$			13.8564i			0.645357i	0.946509	+	0.322679i	$0.104583\pi$
$461$			13.8564i			0.645357i	−0.946509	+	0.322679i	$0.895417\pi$
$462$		0			0
$463$		−	16.0000i		−	0.743583i	−0.928316	−	0.371792i	$-0.878744\pi$
$463$		−	16.0000i		−	0.743583i	0.928316	−	0.371792i	$-0.121256\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	31.1769			1.44270			0.721348	−	0.692573i	$-0.243523\pi$
$467$	31.1769			1.44270			0.721348	+	0.692573i	$0.243523\pi$
$468$		0			0
$469$	−13.2288			−0.610847
$470$		0			0
$471$	−27.4955			−1.26692
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−24.2487	+	18.5203i	−1.11261	+	0.849768i
$476$		0			0
$477$	−31.7490			−1.45369
$478$		0			0
$479$		−	24.0000i		−	1.09659i	−0.836286	−	0.548294i	$-0.815277\pi$
$479$		−	24.0000i		−	1.09659i	0.836286	−	0.548294i	$-0.184723\pi$
$480$		0			0
$481$	28.0000			1.27669
$482$		0			0
$483$			7.93725i			0.361158i
$484$		0			0
$485$	−31.7490			−1.44165
$486$		0			0
$487$	9.16515			0.415313			0.207656	−	0.978202i	$-0.433416\pi$
$487$	9.16515			0.415313			0.207656	+	0.978202i	$0.433416\pi$
$488$		0			0
$489$			18.3303i			0.828925i
$490$		0			0
$491$	−13.8564			−0.625331			−0.312665	−	0.949863i	$-0.601222\pi$
$491$	−13.8564			−0.625331			−0.312665	+	0.949863i	$0.601222\pi$
$492$		0			0
$493$	23.8118			1.07243
$494$		0			0
$495$		0			0
$496$		0			0
$497$			9.16515i			0.411113i
$498$		0			0
$499$	31.1769			1.39567			0.697835	−	0.716258i	$-0.254147\pi$
$499$	31.1769			1.39567			0.697835	+	0.716258i	$0.254147\pi$
$500$		0			0
$501$			24.2487i			1.08335i
$502$		0			0
$503$			9.00000i			0.401290i	0.979664	+	0.200645i	$0.0643038\pi$
$503$			9.00000i			0.401290i	−0.979664	+	0.200645i	$0.935696\pi$
$504$		0			0
$505$	−48.0000			−2.13597
$506$		0			0
$507$			15.8745i			0.705012i
$508$		0			0
$509$	−31.7490			−1.40725			−0.703625	−	0.710571i	$-0.748437\pi$
$509$	−31.7490			−1.40725			−0.703625	+	0.710571i	$0.748437\pi$
$510$		0			0
$511$			7.00000i			0.309662i
$512$		0			0
$513$	7.00000	+	9.16515i	0.309058	+	0.404651i
$514$		0			0
$515$		−	31.7490i		−	1.39903i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$			9.16515i			0.401533i	0.979639	+	0.200766i	$0.0643432\pi$
$521$			9.16515i			0.401533i	−0.979639	+	0.200766i	$0.935657\pi$
$522$		0			0
$523$			29.1033i			1.27260i	0.771443	+	0.636298i	$0.219535\pi$
$523$			29.1033i			1.27260i	−0.771443	+	0.636298i	$0.780465\pi$
$524$		0			0
$525$	−18.5203			−0.808290
$526$		0			0
$527$		0			0
$528$		0			0
$529$	14.0000			0.608696
$530$		0			0
$531$			31.7490i			1.37779i
$532$		0			0
$533$			24.2487i			1.05033i
$534$		0			0
$535$	−27.4955			−1.18873
$536$		0			0
$537$	−42.0000			−1.81243
$538$		0			0
$539$		0			0
$540$		0			0
$541$		−	17.3205i		−	0.744667i	−0.928099	−	0.372333i	$-0.878558\pi$
$541$		−	17.3205i		−	0.744667i	0.928099	−	0.372333i	$-0.121442\pi$
$542$		0			0
$543$			28.0000i			1.20160i
$544$		0			0
$545$		−	9.16515i		−	0.392592i
$546$		0			0
$547$			5.29150i			0.226248i	0.993581	+	0.113124i	$0.0360858\pi$
$547$			5.29150i			0.226248i	−0.993581	+	0.113124i	$0.963914\pi$
$548$		0			0
$549$		−	27.7128i		−	1.18275i
$550$		0			0
$551$	−27.4955	+	21.0000i	−1.17135	+	0.894630i
$552$		0			0
$553$		−	9.16515i		−	0.389742i
$554$		0			0
$555$	96.9948			4.11720
$556$		0			0
$557$			3.46410i			0.146779i	0.997303	+	0.0733893i	$0.0233816\pi$
$557$			3.46410i			0.146779i	−0.997303	+	0.0733893i	$0.976618\pi$
$558$		0			0
$559$	−27.4955			−1.16293
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	15.8745i		−	0.669031i	−0.942390	−	0.334515i	$-0.891427\pi$
$563$		−	15.8745i		−	0.669031i	0.942390	−	0.334515i	$-0.108573\pi$
$564$		0			0
$565$	−31.7490			−1.33569
$566$		0			0
$567$		−	5.00000i		−	0.209980i
$568$		0			0
$569$		−	9.16515i		−	0.384223i	−0.981373	−	0.192112i	$-0.938466\pi$
$569$		−	9.16515i		−	0.384223i	0.981373	−	0.192112i	$-0.0615335\pi$
$570$		0			0
$571$	3.46410			0.144968			0.0724841	−	0.997370i	$-0.476907\pi$
$571$	3.46410			0.144968			0.0724841	+	0.997370i	$0.476907\pi$
$572$		0			0
$573$	−7.93725			−0.331584
$574$		0			0
$575$		−	21.0000i		−	0.875761i
$576$		0			0
$577$	−7.00000			−0.291414			−0.145707	−	0.989328i	$-0.546546\pi$
$577$	−7.00000			−0.291414			−0.145707	+	0.989328i	$0.546546\pi$
$578$		0			0
$579$	48.4974			2.01548
$580$		0			0
$581$		−	17.3205i		−	0.718576i
$582$		0			0
$583$		0			0
$584$		0			0
$585$		−	36.6606i		−	1.51573i
$586$		0			0
$587$	6.92820			0.285958			0.142979	−	0.989726i	$-0.454332\pi$
$587$	6.92820			0.285958			0.142979	+	0.989726i	$0.454332\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$	27.4955			1.13101
$592$		0			0
$593$	−6.00000			−0.246390			−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000			−0.246390			−0.123195	+	0.992382i	$0.539314\pi$
$594$		0			0
$595$	−10.3923			−0.426043
$596$		0			0
$597$	−44.9778			−1.84082
$598$		0			0
$599$	18.3303			0.748956			0.374478	−	0.927236i	$-0.377822\pi$
$599$	18.3303			0.748956			0.374478	+	0.927236i	$0.377822\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		−	52.9150i		−	2.15487i
$604$		0			0
$605$			38.1051i			1.54919i
$606$		0			0
$607$	9.16515			0.372002			0.186001	−	0.982550i	$-0.440447\pi$
$607$	9.16515			0.372002			0.186001	+	0.982550i	$0.440447\pi$
$608$		0			0
$609$	−21.0000			−0.850963
$610$		0			0
$611$		0			0
$612$		0			0
$613$		−	3.46410i		−	0.139914i	−0.997550	−	0.0699569i	$-0.977714\pi$
$613$		−	3.46410i		−	0.139914i	0.997550	−	0.0699569i	$-0.0222862\pi$
$614$		0			0
$615$			84.0000i			3.38721i
$616$		0			0
$617$	6.00000			0.241551			0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000			0.241551			0.120775	+	0.992680i	$0.461462\pi$
$618$		0			0
$619$	−13.8564			−0.556936			−0.278468	−	0.960446i	$-0.589827\pi$
$619$	−13.8564			−0.556936			−0.278468	+	0.960446i	$0.589827\pi$
$620$		0			0
$621$	−7.93725			−0.318511
$622$		0			0
$623$	−9.16515			−0.367194
$624$		0			0
$625$	−11.0000			−0.440000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	31.7490			1.26592
$630$		0			0
$631$		−	20.0000i		−	0.796187i	−0.917345	−	0.398094i	$-0.869672\pi$
$631$		−	20.0000i		−	0.796187i	0.917345	−	0.398094i	$-0.130328\pi$
$632$		0			0
$633$	−7.00000			−0.278225
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−15.8745			−0.628971
$638$		0			0
$639$	−36.6606			−1.45027
$640$		0			0
$641$			9.16515i			0.362002i	0.983483	+	0.181001i	$0.0579337\pi$
$641$			9.16515i			0.362002i	−0.983483	+	0.181001i	$0.942066\pi$
$642$		0			0
$643$	−31.1769			−1.22950			−0.614749	−	0.788723i	$-0.710743\pi$
$643$	−31.1769			−1.22950			−0.614749	+	0.788723i	$0.710743\pi$
$644$		0			0
$645$	−95.2470			−3.75035
$646$		0			0
$647$			3.00000i			0.117942i	0.998260	+	0.0589711i	$0.0187820\pi$
$647$			3.00000i			0.117942i	−0.998260	+	0.0589711i	$0.981218\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	6.92820i		−	0.271122i	−0.990769	−	0.135561i	$-0.956716\pi$
$653$		−	6.92820i		−	0.271122i	0.990769	−	0.135561i	$-0.0432836\pi$
$654$		0			0
$655$		−	12.0000i		−	0.468879i
$656$		0			0
$657$	−28.0000			−1.09238
$658$		0			0
$659$			7.93725i			0.309192i	0.987978	+	0.154596i	$0.0494075\pi$
$659$			7.93725i			0.309192i	−0.987978	+	0.154596i	$0.950592\pi$
$660$		0			0
$661$	44.9778			1.74943			0.874716	−	0.484635i	$-0.161048\pi$
$661$	44.9778			1.74943			0.874716	+	0.484635i	$0.161048\pi$
$662$		0			0
$663$		−	21.0000i		−	0.815572i
$664$		0			0
$665$	12.0000	−	9.16515i	0.465340	−	0.355409i
$666$		0			0
$667$		−	23.8118i		−	0.921995i
$668$		0			0
$669$		−	72.7461i		−	2.81253i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		−	9.16515i		−	0.353291i	−0.984275	−	0.176645i	$-0.943475\pi$
$673$		−	9.16515i		−	0.353291i	0.984275	−	0.176645i	$-0.0565246\pi$
$674$		0			0
$675$		−	18.5203i		−	0.712845i
$676$		0			0
$677$	23.8118			0.915160			0.457580	−	0.889168i	$-0.348716\pi$
$677$	23.8118			0.915160			0.457580	+	0.889168i	$0.348716\pi$
$678$		0			0
$679$	9.16515			0.351726
$680$		0			0
$681$	−21.0000			−0.804722
$682$		0			0
$683$			15.8745i			0.607421i	0.952764	+	0.303711i	$0.0982256\pi$
$683$			15.8745i			0.607421i	−0.952764	+	0.303711i	$0.901774\pi$
$684$		0			0
$685$		−	10.3923i		−	0.397070i
$686$		0			0
$687$	−54.9909			−2.09803
$688$		0			0
$689$	−21.0000			−0.800036
$690$		0			0
$691$	−20.7846			−0.790684			−0.395342	−	0.918534i	$-0.629374\pi$
$691$	−20.7846			−0.790684			−0.395342	+	0.918534i	$0.629374\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	48.0000i		−	1.82074i
$696$		0			0
$697$			27.4955i			1.04146i
$698$		0			0
$699$		−	47.6235i		−	1.80129i
$700$		0			0
$701$			13.8564i			0.523349i	0.965156	+	0.261675i	$0.0842747\pi$
$701$			13.8564i			0.523349i	−0.965156	+	0.261675i	$0.915725\pi$
$702$		0			0
$703$	−36.6606	+	28.0000i	−1.38268	+	1.05604i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	13.8564			0.521124
$708$		0			0
$709$		−	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
$711$	36.6606			1.37488
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$	39.6863			1.48211
$718$		0			0
$719$		−	15.0000i		−	0.559406i	−0.960087	−	0.279703i	$-0.909764\pi$
$719$		−	15.0000i		−	0.559406i	0.960087	−	0.279703i	$-0.0902359\pi$
$720$		0			0
$721$			9.16515i			0.341328i
$722$		0			0
$723$	−24.2487			−0.901819
$724$		0			0
$725$	55.5608			2.06348
$726$		0			0
$727$			47.0000i			1.74313i	0.490277	+	0.871567i	$0.336896\pi$
$727$			47.0000i			1.74313i	−0.490277	+	0.871567i	$0.663104\pi$
$728$		0			0
$729$	41.0000			1.51852
$730$		0			0
$731$	−31.1769			−1.15312
$732$		0			0
$733$			31.1769i			1.15155i	0.817610	+	0.575773i	$0.195299\pi$
$733$			31.1769i			1.15155i	−0.817610	+	0.575773i	$0.804701\pi$
$734$		0			0
$735$	−54.9909			−2.02837
$736$		0			0
$737$		0			0
$738$		0			0
$739$	20.7846			0.764574			0.382287	−	0.924044i	$-0.375137\pi$
$739$	20.7846			0.764574			0.382287	+	0.924044i	$0.375137\pi$
$740$		0			0
$741$	18.5203	+	24.2487i	0.680359	+	0.890799i
$742$		0			0
$743$	−18.3303			−0.672474			−0.336237	−	0.941777i	$-0.609154\pi$
$743$	−18.3303			−0.672474			−0.336237	+	0.941777i	$0.609154\pi$
$744$		0			0
$745$	−60.0000			−2.19823
$746$		0			0
$747$	69.2820			2.53490
$748$		0			0
$749$	7.93725			0.290021
$750$		0			0
$751$	45.8258			1.67221			0.836103	−	0.548573i	$-0.184829\pi$
$751$	45.8258			1.67221			0.836103	+	0.548573i	$0.184829\pi$
$752$		0			0
$753$		−	45.8258i		−	1.66998i
$754$		0			0
$755$			63.4980i			2.31093i
$756$		0			0
$757$			48.4974i			1.76267i	0.472493	+	0.881334i	$0.343354\pi$
$757$			48.4974i			1.76267i	−0.472493	+	0.881334i	$0.656646\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	15.0000			0.543750			0.271875	−	0.962333i	$-0.412356\pi$
$761$	15.0000			0.543750			0.271875	+	0.962333i	$0.412356\pi$
$762$		0			0
$763$			2.64575i			0.0957826i
$764$		0			0
$765$		−	41.5692i		−	1.50294i
$766$		0			0
$767$			21.0000i			0.758266i
$768$		0			0
$769$	−5.00000			−0.180305			−0.0901523	−	0.995928i	$-0.528735\pi$
$769$	−5.00000			−0.180305			−0.0901523	+	0.995928i	$0.528735\pi$
$770$		0			0
$771$	−24.2487			−0.873296
$772$		0			0
$773$	−39.6863			−1.42742			−0.713708	−	0.700443i	$-0.752986\pi$
$773$	−39.6863			−1.42742			−0.713708	+	0.700443i	$0.752986\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−28.0000			−1.00449
$778$		0			0
$779$	−24.2487	−	31.7490i	−0.868800	−	1.13753i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		−	21.0000i		−	0.750479i
$784$		0			0
$785$	−36.0000			−1.28490
$786$		0			0
$787$			2.64575i			0.0943108i	0.998888	+	0.0471554i	$0.0150156\pi$
$787$			2.64575i			0.0943108i	−0.998888	+	0.0471554i	$0.984984\pi$
$788$		0			0
$789$	−31.7490			−1.13029
$790$		0			0
$791$	9.16515			0.325875
$792$		0			0
$793$		−	18.3303i		−	0.650928i
$794$		0			0
$795$	−72.7461			−2.58004
$796$		0			0
$797$	23.8118			0.843456			0.421728	−	0.906722i	$-0.361424\pi$
$797$	23.8118			0.843456			0.421728	+	0.906722i	$0.361424\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		−	36.6606i		−	1.29534i
$802$		0			0
$803$		0			0
$804$		0			0
$805$			10.3923i			0.366281i
$806$		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:	\( N \)	\(=\)	\( 1216 = 2^{6} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1216.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.64575i q^{3} -3.46410i q^{5} +1.00000i q^{7} -4.00000 q^{9} +O(q^{10})\)	\(q-2.64575i q^{3} -3.46410i q^{5} +1.00000i q^{7} -4.00000 q^{9} -2.64575 q^{13} -9.16515 q^{15} -3.00000 q^{17} +(3.46410 - 2.64575i) q^{19} +2.64575 q^{21} +3.00000i q^{23} -7.00000 q^{25} +2.64575i q^{27} -7.93725 q^{29} +3.46410 q^{35} -10.5830 q^{37} +7.00000i q^{39} -9.16515i q^{41} +10.3923 q^{43} +13.8564i q^{45} +6.00000 q^{49} +7.93725i q^{51} +7.93725 q^{53} +(-7.00000 - 9.16515i) q^{57} -7.93725i q^{59} +6.92820i q^{61} -4.00000i q^{63} +9.16515i q^{65} +13.2288i q^{67} +7.93725 q^{69} +9.16515 q^{71} +7.00000 q^{73} +18.5203i q^{75} -9.16515 q^{79} -5.00000 q^{81} -17.3205 q^{83} +10.3923i q^{85} +21.0000i q^{87} +9.16515i q^{89} -2.64575i q^{91} +(-9.16515 - 12.0000i) q^{95} -9.16515i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8q - 32q^{9} + O(q^{10}) \)	\( 8q - 32q^{9} - 24q^{17} - 56q^{25} + 48q^{49} - 56q^{57} + 56q^{73} - 40q^{81} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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