LMFDB - Embedded newform 2288.2.b.c.2287.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2288,2,Mod(2287,2288)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2288, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2288.2287");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2288 = 2^{4} \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2288.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$18.2697719825$
Analytic rank:	$0$
Dimension:	$56$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2287.4
Character	$\chi$	$=$	2288.2287
Dual form			2288.2.b.c.2287.53

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.16710i q^{3} +3.14763i q^{5} +2.40102i q^{7} -7.03050 q^{9} +O(q^{10})$	$q-3.16710i q^{3} +3.14763i q^{5} +2.40102i q^{7} -7.03050 q^{9} +(0.431142 - 3.28848i) q^{11} +(-1.75285 - 3.15080i) q^{13} +9.96885 q^{15} +7.61629i q^{17} -4.05677i q^{19} +7.60427 q^{21} +3.30945i q^{23} -4.90758 q^{25} +12.7650i q^{27} -5.47997i q^{29} +8.31806 q^{31} +(-10.4149 - 1.36547i) q^{33} -7.55754 q^{35} -0.176007i q^{37} +(-9.97888 + 5.55144i) q^{39} +7.90644 q^{41} -1.86073 q^{43} -22.1294i q^{45} +5.82734 q^{47} +1.23508 q^{49} +24.1215 q^{51} +6.88359 q^{53} +(10.3509 + 1.35708i) q^{55} -12.8482 q^{57} +7.02535 q^{59} -1.32967i q^{61} -16.8804i q^{63} +(9.91755 - 5.51732i) q^{65} +3.42363 q^{67} +10.4813 q^{69} +13.0170 q^{71} +3.44558 q^{73} +15.5428i q^{75} +(7.89573 + 1.03518i) q^{77} +3.84227 q^{79} +19.3364 q^{81} +9.97051i q^{83} -23.9733 q^{85} -17.3556 q^{87} +12.3401i q^{89} +(7.56514 - 4.20863i) q^{91} -26.3441i q^{93} +12.7692 q^{95} +11.6764i q^{97} +(-3.03114 + 23.1197i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q - 72 q^{9}+O(q^{10}) $ 56 * q - 72 * q^9	$ 56 q - 72 q^{9} - 72 q^{25} - 8 q^{49} + 72 q^{53} + 8 q^{69} - 48 q^{77} + 88 q^{81}+O(q^{100}) $ 56 * q - 72 * q^9 - 72 * q^25 - 8 * q^49 + 72 * q^53 + 8 * q^69 - 48 * q^77 + 88 * q^81

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2288\mathbb{Z}\right)^\times$.

$n$	$287$	$353$	$1717$	$2081$
$\chi(n)$	$-1$	$-1$	$1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		−	3.16710i		−	1.82852i	−0.405124	−	0.914262i	$-0.632772\pi$
$3$		−	3.16710i		−	1.82852i	0.405124	−	0.914262i	$-0.367228\pi$
$4$		0			0
$5$			3.14763i			1.40766i	0.710367	+	0.703832i	$0.248529\pi$
$5$			3.14763i			1.40766i	−0.710367	+	0.703832i	$0.751471\pi$
$6$		0			0
$7$			2.40102i			0.907502i	0.891129	+	0.453751i	$0.149914\pi$
$7$			2.40102i			0.907502i	−0.891129	+	0.453751i	$0.850086\pi$
$8$		0			0
$9$	−7.03050			−2.34350
$10$		0			0
$11$	0.431142	−	3.28848i	0.129994	−	0.991515i
$11$	0.431142	−	3.28848i	0.129994	−	0.991515i
$12$		0			0
$13$	−1.75285	−	3.15080i	−0.486153	−	0.873874i
$13$	−1.75285	−	3.15080i	−0.486153	−	0.873874i
$14$		0			0
$15$	9.96885			2.57395
$16$		0			0
$17$			7.61629i			1.84722i	0.383330	+	0.923611i	$0.374777\pi$
$17$			7.61629i			1.84722i	−0.383330	+	0.923611i	$0.625223\pi$
$18$		0			0
$19$		−	4.05677i		−	0.930687i	−0.885130	−	0.465343i	$-0.845931\pi$
$19$		−	4.05677i		−	0.930687i	0.885130	−	0.465343i	$-0.154069\pi$
$20$		0			0
$21$	7.60427			1.65939
$22$		0			0
$23$			3.30945i			0.690067i	0.938590	+	0.345034i	$0.112132\pi$
$23$			3.30945i			0.690067i	−0.938590	+	0.345034i	$0.887868\pi$
$24$		0			0
$25$	−4.90758			−0.981516
$26$		0			0
$27$			12.7650i			2.45662i
$28$		0			0
$29$		−	5.47997i		−	1.01760i	−0.860883	−	0.508802i	$-0.830088\pi$
$29$		−	5.47997i		−	1.01760i	0.860883	−	0.508802i	$-0.169912\pi$
$30$		0			0
$31$	8.31806			1.49397			0.746984	−	0.664842i	$-0.231501\pi$
$31$	8.31806			1.49397			0.746984	+	0.664842i	$0.231501\pi$
$32$		0			0
$33$	−10.4149	−	1.36547i	−1.81301	−	0.237697i
$34$		0			0
$35$	−7.55754			−1.27746
$36$		0			0
$37$		−	0.176007i		−	0.0289354i	−0.999895	−	0.0144677i	$-0.995395\pi$
$37$		−	0.176007i		−	0.0289354i	0.999895	−	0.0144677i	$-0.00460537\pi$
$38$		0			0
$39$	−9.97888	+	5.55144i	−1.59790	+	0.888942i
$40$		0			0
$41$	7.90644			1.23478			0.617390	−	0.786658i	$-0.288190\pi$
$41$	7.90644			1.23478			0.617390	+	0.786658i	$0.288190\pi$
$42$		0			0
$43$	−1.86073			−0.283759			−0.141880	−	0.989884i	$-0.545315\pi$
$43$	−1.86073			−0.283759			−0.141880	+	0.989884i	$0.545315\pi$
$44$		0			0
$45$		−	22.1294i		−	3.29886i
$46$		0			0
$47$	5.82734			0.850005			0.425003	−	0.905192i	$-0.360273\pi$
$47$	5.82734			0.850005			0.425003	+	0.905192i	$0.360273\pi$
$48$		0			0
$49$	1.23508			0.176440
$50$		0			0
$51$	24.1215			3.37769
$52$		0			0
$53$	6.88359			0.945534			0.472767	−	0.881187i	$-0.343255\pi$
$53$	6.88359			0.945534			0.472767	+	0.881187i	$0.343255\pi$
$54$		0			0
$55$	10.3509	+	1.35708i	1.39572	+	0.182988i
$56$		0			0
$57$	−12.8482			−1.70178
$58$		0			0
$59$	7.02535			0.914622			0.457311	−	0.889307i	$-0.348813\pi$
$59$	7.02535			0.914622			0.457311	+	0.889307i	$0.348813\pi$
$60$		0			0
$61$		−	1.32967i		−	0.170246i	−0.996370	−	0.0851231i	$-0.972872\pi$
$61$		−	1.32967i		−	0.170246i	0.996370	−	0.0851231i	$-0.0271284\pi$
$62$		0			0
$63$		−	16.8804i		−	2.12673i
$64$		0			0
$65$	9.91755	−	5.51732i	1.23012	−	0.684339i
$66$		0			0
$67$	3.42363			0.418263			0.209132	−	0.977887i	$-0.432936\pi$
$67$	3.42363			0.418263			0.209132	+	0.977887i	$0.432936\pi$
$68$		0			0
$69$	10.4813			1.26180
$70$		0			0
$71$	13.0170			1.54483			0.772417	−	0.635116i	$-0.219048\pi$
$71$	13.0170			1.54483			0.772417	+	0.635116i	$0.219048\pi$
$72$		0			0
$73$	3.44558			0.403274			0.201637	−	0.979460i	$-0.435374\pi$
$73$	3.44558			0.403274			0.201637	+	0.979460i	$0.435374\pi$
$74$		0			0
$75$			15.5428i			1.79473i
$76$		0			0
$77$	7.89573	+	1.03518i	0.899801	+	0.117970i
$78$		0			0
$79$	3.84227			0.432289			0.216145	−	0.976361i	$-0.430652\pi$
$79$	3.84227			0.432289			0.216145	+	0.976361i	$0.430652\pi$
$80$		0			0
$81$	19.3364			2.14849
$82$		0			0
$83$			9.97051i			1.09441i	0.837000	+	0.547203i	$0.184307\pi$
$83$			9.97051i			1.09441i	−0.837000	+	0.547203i	$0.815693\pi$
$84$		0			0
$85$	−23.9733			−2.60027
$86$		0			0
$87$	−17.3556			−1.86071
$88$		0			0
$89$			12.3401i			1.30804i	0.756475	+	0.654022i	$0.226920\pi$
$89$			12.3401i			1.30804i	−0.756475	+	0.654022i	$0.773080\pi$
$90$		0			0
$91$	7.56514	−	4.20863i	0.793042	−	0.441184i
$92$		0			0
$93$		−	26.3441i		−	2.73175i
$94$		0			0
$95$	12.7692			1.31009
$96$		0			0
$97$			11.6764i			1.18556i	0.805366	+	0.592778i	$0.201969\pi$
$97$			11.6764i			1.18556i	−0.805366	+	0.592778i	$0.798031\pi$
$98$		0			0
$99$	−3.03114	+	23.1197i	−0.304641	+	2.32361i
$100$		0			0
$101$		−	10.6065i		−	1.05539i	−0.849434	−	0.527694i	$-0.823057\pi$
$101$		−	10.6065i		−	1.05539i	0.849434	−	0.527694i	$-0.176943\pi$
$102$		0			0
$103$		−	13.0100i		−	1.28192i	−0.767575	−	0.640959i	$-0.778537\pi$
$103$		−	13.0100i		−	1.28192i	0.767575	−	0.640959i	$-0.221463\pi$
$104$		0			0
$105$			23.9354i			2.33586i
$106$		0			0
$107$	−18.8948			−1.82663			−0.913315	−	0.407253i	$-0.866487\pi$
$107$	−18.8948			−1.82663			−0.913315	+	0.407253i	$0.866487\pi$
$108$		0			0
$109$	9.06081			0.867868			0.433934	−	0.900945i	$-0.357125\pi$
$109$	9.06081			0.867868			0.433934	+	0.900945i	$0.357125\pi$
$110$		0			0
$111$	−0.557431			−0.0529090
$112$		0			0
$113$	12.6042			1.18571			0.592854	−	0.805310i	$-0.298001\pi$
$113$	12.6042			1.18571			0.592854	+	0.805310i	$0.298001\pi$
$114$		0			0
$115$	−10.4169			−0.971382
$116$		0			0
$117$	12.3234	+	22.1517i	1.13930	+	2.04792i
$118$		0			0
$119$	−18.2869			−1.67636
$120$		0			0
$121$	−10.6282	−	2.83560i	−0.966203	−	0.257782i
$122$		0			0
$123$		−	25.0405i		−	2.25782i
$124$		0			0
$125$			0.290905i			0.0260193i
$126$		0			0
$127$	4.58638			0.406975			0.203488	−	0.979078i	$-0.434772\pi$
$127$	4.58638			0.406975			0.203488	+	0.979078i	$0.434772\pi$
$128$		0			0
$129$			5.89312i			0.518861i
$130$		0			0
$131$	−3.70022			−0.323290			−0.161645	−	0.986849i	$-0.551680\pi$
$131$	−3.70022			−0.323290			−0.161645	+	0.986849i	$0.551680\pi$
$132$		0			0
$133$	9.74040			0.844600
$134$		0			0
$135$	−40.1794			−3.45809
$136$		0			0
$137$		−	8.88852i		−	0.759398i	−0.925110	−	0.379699i	$-0.876028\pi$
$137$		−	8.88852i		−	0.759398i	0.925110	−	0.379699i	$-0.123972\pi$
$138$		0			0
$139$	13.8384			1.17375			0.586877	−	0.809676i	$-0.300357\pi$
$139$	13.8384			1.17375			0.586877	+	0.809676i	$0.300357\pi$
$140$		0			0
$141$		−	18.4557i		−	1.55425i
$142$		0			0
$143$	−11.1171	+	4.40577i	−0.929656	+	0.368429i
$144$		0			0
$145$	17.2489			1.43245
$146$		0			0
$147$		−	3.91163i		−	0.322625i
$148$		0			0
$149$	−2.56566			−0.210187			−0.105094	−	0.994462i	$-0.533514\pi$
$149$	−2.56566			−0.210187			−0.105094	+	0.994462i	$0.533514\pi$
$150$		0			0
$151$			12.7402i			1.03678i	0.855143	+	0.518392i	$0.173469\pi$
$151$			12.7402i			1.03678i	−0.855143	+	0.518392i	$0.826531\pi$
$152$		0			0
$153$		−	53.5463i		−	4.32896i
$154$		0			0
$155$			26.1822i			2.10300i
$156$		0			0
$157$	−7.84357			−0.625985			−0.312992	−	0.949756i	$-0.601331\pi$
$157$	−7.84357			−0.625985			−0.312992	+	0.949756i	$0.601331\pi$
$158$		0			0
$159$		−	21.8010i		−	1.72893i
$160$		0			0
$161$	−7.94606			−0.626237
$162$		0			0
$163$	−10.6133			−0.831297			−0.415648	−	0.909525i	$-0.636445\pi$
$163$	−10.6133			−0.831297			−0.415648	+	0.909525i	$0.636445\pi$
$164$		0			0
$165$	4.29799	−	32.7824i	0.334598	−	2.55210i
$166$		0			0
$167$			0.0663381i			0.00513340i	0.999997	+	0.00256670i	$0.000817007\pi$
$167$			0.0663381i			0.00513340i	−0.999997	+	0.00256670i	$0.999183\pi$
$168$		0			0
$169$	−6.85504	+	11.0457i	−0.527311	+	0.849672i
$170$		0			0
$171$			28.5211i			2.18106i
$172$		0			0
$173$			13.4209i			1.02037i	0.860063	+	0.510187i	$0.170424\pi$
$173$			13.4209i			1.02037i	−0.860063	+	0.510187i	$0.829576\pi$
$174$		0			0
$175$		−	11.7832i		−	0.890728i
$176$		0			0
$177$		−	22.2499i		−	1.67241i
$178$		0			0
$179$			2.77278i			0.207248i	0.994617	+	0.103624i	$0.0330438\pi$
$179$			2.77278i			0.207248i	−0.994617	+	0.103624i	$0.966956\pi$
$180$		0			0
$181$	−9.99969			−0.743271			−0.371636	−	0.928379i	$-0.621203\pi$
$181$	−9.99969			−0.743271			−0.371636	+	0.928379i	$0.621203\pi$
$182$		0			0
$183$	−4.21118			−0.311299
$184$		0			0
$185$	0.554005			0.0407312
$186$		0			0
$187$	25.0461	+	3.28370i	1.83155	+	0.240128i
$188$		0			0
$189$	−30.6490			−2.22939
$190$		0			0
$191$			7.43268i			0.537810i	0.963167	+	0.268905i	$0.0866618\pi$
$191$			7.43268i			0.537810i	−0.963167	+	0.268905i	$0.913338\pi$
$192$		0			0
$193$	−21.2916			−1.53261			−0.766303	−	0.642480i	$-0.777906\pi$
$193$	−21.2916			−1.53261			−0.766303	+	0.642480i	$0.777906\pi$
$194$		0			0
$195$	−17.4739	−	31.4098i	−1.25133	−	2.24930i
$196$		0			0
$197$	10.1699			0.724578			0.362289	−	0.932066i	$-0.381995\pi$
$197$	10.1699			0.724578			0.362289	+	0.932066i	$0.381995\pi$
$198$		0			0
$199$		−	21.4636i		−	1.52152i	−0.649035	−	0.760759i	$-0.724827\pi$
$199$		−	21.4636i		−	1.52152i	0.649035	−	0.760759i	$-0.275173\pi$
$200$		0			0
$201$		−	10.8430i		−	0.764804i
$202$		0			0
$203$	13.1575			0.923478
$204$		0			0
$205$			24.8866i			1.73815i
$206$		0			0
$207$		−	23.2670i		−	1.61717i
$208$		0			0
$209$	−13.3406	−	1.74904i	−0.922790	−	0.120984i
$210$		0			0
$211$	−13.2069			−0.909197			−0.454599	−	0.890696i	$-0.650217\pi$
$211$	−13.2069			−0.909197			−0.454599	+	0.890696i	$0.650217\pi$
$212$		0			0
$213$		−	41.2261i		−	2.82476i
$214$		0			0
$215$		−	5.85690i		−	0.399438i
$216$		0			0
$217$			19.9719i			1.35578i
$218$		0			0
$219$		−	10.9125i		−	0.737397i
$220$		0			0
$221$	23.9974	−	13.3502i	1.61424	−	0.898032i
$222$		0			0
$223$	13.3944			0.896957			0.448479	−	0.893794i	$-0.351966\pi$
$223$	13.3944			0.896957			0.448479	+	0.893794i	$0.351966\pi$
$224$		0			0
$225$	34.5027			2.30018
$226$		0			0
$227$		−	22.4175i		−	1.48790i	−0.668233	−	0.743952i	$-0.732949\pi$
$227$		−	22.4175i		−	1.48790i	0.668233	−	0.743952i	$-0.267051\pi$
$228$		0			0
$229$			0.946511i			0.0625472i	0.999511	+	0.0312736i	$0.00995632\pi$
$229$			0.946511i			0.0625472i	−0.999511	+	0.0312736i	$0.990044\pi$
$230$		0			0
$231$	3.27852	−	25.0065i	0.215711	−	1.64531i
$232$		0			0
$233$		−	12.2684i		−	0.803731i	−0.915699	−	0.401865i	$-0.868362\pi$
$233$		−	12.2684i		−	0.803731i	0.915699	−	0.401865i	$-0.131638\pi$
$234$		0			0
$235$			18.3423i			1.19652i
$236$		0			0
$237$		−	12.1688i		−	0.790451i
$238$		0			0
$239$		−	27.2646i		−	1.76360i	−0.471623	−	0.881800i	$-0.656332\pi$
$239$		−	27.2646i		−	1.76360i	0.471623	−	0.881800i	$-0.343668\pi$
$240$		0			0
$241$	2.99711			0.193061			0.0965306	−	0.995330i	$-0.469225\pi$
$241$	2.99711			0.193061			0.0965306	+	0.995330i	$0.469225\pi$
$242$		0			0
$243$		−	22.9453i		−	1.47194i
$244$		0			0
$245$			3.88759i			0.248369i
$246$		0			0
$247$	−12.7821	+	7.11090i	−0.813303	+	0.452456i
$248$		0			0
$249$	31.5776			2.00115
$250$		0			0
$251$		−	12.1400i		−	0.766272i	−0.923692	−	0.383136i	$-0.874844\pi$
$251$		−	12.1400i		−	0.766272i	0.923692	−	0.383136i	$-0.125156\pi$
$252$		0			0
$253$	10.8831	+	1.42684i	0.684212	+	0.0897047i
$254$		0			0
$255$			75.9257i			4.75465i
$256$		0			0
$257$	−18.4522			−1.15102			−0.575508	−	0.817796i	$-0.695196\pi$
$257$	−18.4522			−1.15102			−0.575508	+	0.817796i	$0.695196\pi$
$258$		0			0
$259$	0.422597			0.0262589
$260$		0			0
$261$			38.5269i			2.38476i
$262$		0			0
$263$	5.18575			0.319767			0.159884	−	0.987136i	$-0.448888\pi$
$263$	5.18575			0.319767			0.159884	+	0.987136i	$0.448888\pi$
$264$		0			0
$265$			21.6670i			1.33099i
$266$		0			0
$267$	39.0822			2.39179
$268$		0			0
$269$	23.2679			1.41867			0.709336	−	0.704871i	$-0.248995\pi$
$269$	23.2679			1.41867			0.709336	+	0.704871i	$0.248995\pi$
$270$		0			0
$271$			12.5380i			0.761629i	0.924651	+	0.380815i	$0.124356\pi$
$271$			12.5380i			0.761629i	−0.924651	+	0.380815i	$0.875644\pi$
$272$		0			0
$273$	−13.3291	−	23.9595i	−0.806716	−	1.45010i
$274$		0			0
$275$	−2.11586	+	16.1385i	−0.127591	+	0.973188i
$276$		0			0
$277$			11.7878i			0.708260i	0.935196	+	0.354130i	$0.115223\pi$
$277$			11.7878i			0.708260i	−0.935196	+	0.354130i	$0.884777\pi$
$278$		0			0
$279$	−58.4801			−3.50111
$280$		0			0
$281$	−28.7291			−1.71383			−0.856917	−	0.515454i	$-0.827623\pi$
$281$	−28.7291			−1.71383			−0.856917	+	0.515454i	$0.827623\pi$
$282$		0			0
$283$	0.806639			0.0479497			0.0239748	−	0.999713i	$-0.492368\pi$
$283$	0.806639			0.0479497			0.0239748	+	0.999713i	$0.492368\pi$
$284$		0			0
$285$		−	40.4413i		−	2.39554i
$286$		0			0
$287$			18.9836i			1.12056i
$288$		0			0
$289$	−41.0079			−2.41223
$290$		0			0
$291$	36.9802			2.16782
$292$		0			0
$293$	−31.7742			−1.85627			−0.928133	−	0.372248i	$-0.878587\pi$
$293$	−31.7742			−1.85627			−0.928133	+	0.372248i	$0.878587\pi$
$294$		0			0
$295$			22.1132i			1.28748i
$296$		0			0
$297$	41.9774	+	5.50351i	2.43577	+	0.319346i
$298$		0			0
$299$	10.4274	−	5.80096i	0.603032	−	0.335478i
$300$		0			0
$301$		−	4.46767i		−	0.257512i
$302$		0			0
$303$	−33.5919			−1.92980
$304$		0			0
$305$	4.18530			0.239649
$306$		0			0
$307$			6.24893i			0.356645i	0.983972	+	0.178323i	$0.0570671\pi$
$307$			6.24893i			0.356645i	−0.983972	+	0.178323i	$0.942933\pi$
$308$		0			0
$309$	−41.2040			−2.34402
$310$		0			0
$311$			19.9514i			1.13134i	0.824631	+	0.565671i	$0.191383\pi$
$311$			19.9514i			1.13134i	−0.824631	+	0.565671i	$0.808617\pi$
$312$		0			0
$313$	−11.2095			−0.633600			−0.316800	−	0.948492i	$-0.602608\pi$
$313$	−11.2095			−0.633600			−0.316800	+	0.948492i	$0.602608\pi$
$314$		0			0
$315$	53.1332			2.99372
$316$		0			0
$317$			26.6582i			1.49727i	0.662981	+	0.748636i	$0.269291\pi$
$317$			26.6582i			1.49727i	−0.662981	+	0.748636i	$0.730709\pi$
$318$		0			0
$319$	−18.0208	−	2.36264i	−1.00897	−	0.132283i
$320$		0			0
$321$			59.8417i			3.34004i
$322$		0			0
$323$	30.8975			1.71919
$324$		0			0
$325$	8.60224	+	15.4628i	0.477167	+	0.857721i
$326$		0			0
$327$		−	28.6965i		−	1.58692i
$328$		0			0
$329$			13.9916i			0.771381i
$330$		0			0
$331$	0.802933			0.0441332			0.0220666	−	0.999757i	$-0.492975\pi$
$331$	0.802933			0.0441332			0.0220666	+	0.999757i	$0.492975\pi$
$332$		0			0
$333$			1.23742i			0.0678100i
$334$		0			0
$335$			10.7763i			0.588774i
$336$		0			0
$337$		−	32.8602i		−	1.79001i	−0.446056	−	0.895005i	$-0.647172\pi$
$337$		−	32.8602i		−	1.79001i	0.446056	−	0.895005i	$-0.352828\pi$
$338$		0			0
$339$		−	39.9189i		−	2.16809i
$340$		0			0
$341$	3.58626	−	27.3538i	0.194207	−	1.48129i
$342$		0			0
$343$			19.7726i			1.06762i
$344$		0			0
$345$			32.9914i			1.77620i
$346$		0			0
$347$	21.1291			1.13427			0.567136	−	0.823624i	$-0.308052\pi$
$347$	21.1291			1.13427			0.567136	+	0.823624i	$0.308052\pi$
$348$		0			0
$349$	20.7852			1.11261			0.556303	−	0.830980i	$-0.312219\pi$
$349$	20.7852			1.11261			0.556303	+	0.830980i	$0.312219\pi$
$350$		0			0
$351$	40.2198	−	22.3750i	2.14677	−	1.19429i
$352$		0			0
$353$			35.0591i			1.86601i	0.359865	+	0.933005i	$0.382823\pi$
$353$			35.0591i			1.86601i	−0.359865	+	0.933005i	$0.617177\pi$
$354$		0			0
$355$			40.9727i			2.17461i
$356$		0			0
$357$			57.9164i			3.06526i
$358$		0			0
$359$			14.3261i			0.756101i	0.925785	+	0.378051i	$0.123406\pi$
$359$			14.3261i			0.756101i	−0.925785	+	0.378051i	$0.876594\pi$
$360$		0			0
$361$	2.54262			0.133822
$362$		0			0
$363$	−8.98063	+	33.6606i	−0.471361	+	1.76673i
$364$		0			0
$365$			10.8454i			0.567674i
$366$		0			0
$367$		−	2.53145i		−	0.132141i	−0.997815	−	0.0660704i	$-0.978954\pi$
$367$		−	2.53145i		−	0.132141i	0.997815	−	0.0660704i	$-0.0210462\pi$
$368$		0			0
$369$	−55.5862			−2.89370
$370$		0			0
$371$			16.5277i			0.858074i
$372$		0			0
$373$			3.00519i			0.155603i	0.996969	+	0.0778015i	$0.0247900\pi$
$373$			3.00519i			0.155603i	−0.996969	+	0.0778015i	$0.975210\pi$
$374$		0			0
$375$	0.921322			0.0475769
$376$		0			0
$377$	−17.2663	+	9.60556i	−0.889258	+	0.494711i
$378$		0			0
$379$	−3.20192			−0.164472			−0.0822359	−	0.996613i	$-0.526206\pi$
$379$	−3.20192			−0.164472			−0.0822359	+	0.996613i	$0.526206\pi$
$380$		0			0
$381$		−	14.5255i		−	0.744164i
$382$		0			0
$383$	19.3667			0.989593			0.494797	−	0.869009i	$-0.335243\pi$
$383$	19.3667			0.989593			0.494797	+	0.869009i	$0.335243\pi$
$384$		0			0
$385$	−3.25837	+	24.8528i	−0.166062	+	1.26662i
$386$		0			0
$387$	13.0819			0.664989
$388$		0			0
$389$	10.7394			0.544509			0.272254	−	0.962225i	$-0.412231\pi$
$389$	10.7394			0.544509			0.272254	+	0.962225i	$0.412231\pi$
$390$		0			0
$391$	−25.2057			−1.27471
$392$		0			0
$393$			11.7190i			0.591143i
$394$		0			0
$395$			12.0941i			0.608518i
$396$		0			0
$397$			23.9310i			1.20106i	0.799601	+	0.600532i	$0.205045\pi$
$397$			23.9310i			1.20106i	−0.799601	+	0.600532i	$0.794955\pi$
$398$		0			0
$399$		−	30.8488i		−	1.54437i
$400$		0			0
$401$		−	30.5109i		−	1.52364i	−0.647788	−	0.761820i	$-0.724306\pi$
$401$		−	30.5109i		−	1.52364i	0.647788	−	0.761820i	$-0.275694\pi$
$402$		0			0
$403$	−14.5803	−	26.2085i	−0.726296	−	1.30554i
$404$		0			0
$405$			60.8638i			3.02435i
$406$		0			0
$407$	−0.578796	−	0.0758839i	−0.0286898	−	0.00376143i
$408$		0			0
$409$	24.4959			1.21124			0.605622	−	0.795753i	$-0.292924\pi$
$409$	24.4959			1.21124			0.605622	+	0.795753i	$0.292924\pi$
$410$		0			0
$411$	−28.1508			−1.38858
$412$		0			0
$413$			16.8680i			0.830021i
$414$		0			0
$415$	−31.3835			−1.54055
$416$		0			0
$417$		−	43.8274i		−	2.14624i
$418$		0			0
$419$		−	4.20863i		−	0.205605i	−0.994702	−	0.102803i	$-0.967219\pi$
$419$		−	4.20863i		−	0.205605i	0.994702	−	0.102803i	$-0.0327810\pi$
$420$		0			0
$421$			7.95145i			0.387530i	0.981048	+	0.193765i	$0.0620699\pi$
$421$			7.95145i			0.387530i	−0.981048	+	0.193765i	$0.937930\pi$
$422$		0			0
$423$	−40.9691			−1.99199
$424$		0			0
$425$		−	37.3776i		−	1.81308i
$426$		0			0
$427$	3.19256			0.154499
$428$		0			0
$429$	13.9535	+	35.2088i	0.673681	+	1.69990i
$430$		0			0
$431$		−	11.8366i		−	0.570151i	−0.958505	−	0.285075i	$-0.907981\pi$
$431$		−	11.8366i		−	0.570151i	0.958505	−	0.285075i	$-0.0920186\pi$
$432$		0			0
$433$	−20.9918			−1.00880			−0.504401	−	0.863470i	$-0.668287\pi$
$433$	−20.9918			−1.00880			−0.504401	+	0.863470i	$0.668287\pi$
$434$		0			0
$435$		−	54.6290i		−	2.61926i
$436$		0			0
$437$	13.4257			0.642236
$438$		0			0
$439$	36.7915			1.75596			0.877981	−	0.478695i	$-0.158890\pi$
$439$	36.7915			1.75596			0.877981	+	0.478695i	$0.158890\pi$
$440$		0			0
$441$	−8.68324			−0.413488
$442$		0			0
$443$		−	1.47331i		−	0.0699990i	−0.999387	−	0.0349995i	$-0.988857\pi$
$443$		−	1.47331i		−	0.0699990i	0.999387	−	0.0349995i	$-0.0111430\pi$
$444$		0			0
$445$	−38.8420			−1.84129
$446$		0			0
$447$			8.12570i			0.384332i
$448$		0			0
$449$		−	3.97757i		−	0.187713i	−0.995586	−	0.0938566i	$-0.970080\pi$
$449$		−	3.97757i		−	0.187713i	0.995586	−	0.0938566i	$-0.0299195\pi$
$450$		0			0
$451$	3.40880	−	26.0002i	0.160514	−	1.22430i
$452$		0			0
$453$	40.3495			1.89579
$454$		0			0
$455$	13.2472	+	23.8123i	0.621039	+	1.11634i
$456$		0			0
$457$	−31.7588			−1.48561			−0.742806	−	0.669507i	$-0.766506\pi$
$457$	−31.7588			−1.48561			−0.742806	+	0.669507i	$0.766506\pi$
$458$		0			0
$459$	−97.2217			−4.53792
$460$		0			0
$461$	−7.57397			−0.352755			−0.176377	−	0.984323i	$-0.556438\pi$
$461$	−7.57397			−0.352755			−0.176377	+	0.984323i	$0.556438\pi$
$462$		0			0
$463$	31.1299			1.44673			0.723364	−	0.690467i	$-0.242595\pi$
$463$	31.1299			1.44673			0.723364	+	0.690467i	$0.242595\pi$
$464$		0			0
$465$	82.9215			3.84539
$466$		0			0
$467$		−	22.1487i		−	1.02492i	−0.858712	−	0.512459i	$-0.828735\pi$
$467$		−	22.1487i		−	1.02492i	0.858712	−	0.512459i	$-0.171265\pi$
$468$		0			0
$469$			8.22023i			0.379575i
$470$		0			0
$471$			24.8413i			1.14463i
$472$		0			0
$473$	−0.802240	+	6.11899i	−0.0368870	+	0.281352i
$474$		0			0
$475$			19.9089i			0.913484i
$476$		0			0
$477$	−48.3951			−2.21586
$478$		0			0
$479$		−	33.3073i		−	1.52185i	−0.648840	−	0.760925i	$-0.724746\pi$
$479$		−	33.3073i		−	1.52185i	0.648840	−	0.760925i	$-0.275254\pi$
$480$		0			0
$481$	−0.554562	+	0.308513i	−0.0252859	+	0.0140670i
$482$		0			0
$483$			25.1659i			1.14509i
$484$		0			0
$485$	−36.7529			−1.66886
$486$		0			0
$487$	6.66337			0.301946			0.150973	−	0.988538i	$-0.451759\pi$
$487$	6.66337			0.301946			0.150973	+	0.988538i	$0.451759\pi$
$488$		0			0
$489$			33.6133i			1.52005i
$490$		0			0
$491$	42.2981			1.90889			0.954443	−	0.298395i	$-0.0964512\pi$
$491$	42.2981			1.90889			0.954443	+	0.298395i	$0.0964512\pi$
$492$		0			0
$493$	41.7371			1.87974
$494$		0			0
$495$	−72.7722	−	9.54091i	−3.27087	−	0.428832i
$496$		0			0
$497$			31.2541i			1.40194i
$498$		0			0
$499$	−23.3965			−1.04737			−0.523687	−	0.851911i	$-0.675444\pi$
$499$	−23.3965			−1.04737			−0.523687	+	0.851911i	$0.675444\pi$
$500$		0			0
$501$	0.210099			0.00938654
$502$		0			0
$503$	12.4808			0.556491			0.278246	−	0.960510i	$-0.410247\pi$
$503$	12.4808			0.556491			0.278246	+	0.960510i	$0.410247\pi$
$504$		0			0
$505$	33.3854			1.48563
$506$		0			0
$507$	34.9829	+	21.7106i	1.55365	+	0.964201i
$508$		0			0
$509$			11.3574i			0.503409i	0.967804	+	0.251705i	$0.0809911\pi$
$509$			11.3574i			0.503409i	−0.967804	+	0.251705i	$0.919009\pi$
$510$		0			0
$511$			8.27291i			0.365972i
$512$		0			0
$513$	51.7845			2.28634
$514$		0			0
$515$	40.9508			1.80451
$516$		0			0
$517$	2.51241	−	19.1631i	0.110496	−	0.842793i
$518$		0			0
$519$	42.5054			1.86578
$520$		0			0
$521$	37.7715			1.65480			0.827400	−	0.561614i	$-0.189819\pi$
$521$	37.7715			1.65480			0.827400	+	0.561614i	$0.189819\pi$
$522$		0			0
$523$	−27.4940			−1.20223			−0.601115	−	0.799162i	$-0.705277\pi$
$523$	−27.4940			−1.20223			−0.601115	+	0.799162i	$0.705277\pi$
$524$		0			0
$525$	−37.3186			−1.62872
$526$		0			0
$527$			63.3528i			2.75969i
$528$		0			0
$529$	12.0476			0.523807
$530$		0			0
$531$	−49.3917			−2.14342
$532$		0			0
$533$	−13.8588	−	24.9116i	−0.600291	−	1.07904i
$534$		0			0
$535$		−	59.4739i		−	2.57128i
$536$		0			0
$537$	8.78167			0.378957
$538$		0			0
$539$	0.532496	−	4.06155i	0.0229362	−	0.174943i
$540$		0			0
$541$	−4.81385			−0.206963			−0.103482	−	0.994631i	$-0.532998\pi$
$541$	−4.81385			−0.206963			−0.103482	+	0.994631i	$0.532998\pi$
$542$		0			0
$543$			31.6700i			1.35909i
$544$		0			0
$545$			28.5201i			1.22167i
$546$		0			0
$547$	12.8409			0.549035			0.274518	−	0.961582i	$-0.411482\pi$
$547$	12.8409			0.549035			0.274518	+	0.961582i	$0.411482\pi$
$548$		0			0
$549$			9.34821i			0.398972i
$550$		0			0
$551$	−22.2310			−0.947071
$552$		0			0
$553$			9.22539i			0.392303i
$554$		0			0
$555$		−	1.75459i		−	0.0744780i
$556$		0			0
$557$	44.3414			1.87881			0.939403	−	0.342815i	$-0.111380\pi$
$557$	44.3414			1.87881			0.939403	+	0.342815i	$0.111380\pi$
$558$		0			0
$559$	3.26158	+	5.86279i	0.137950	+	0.247970i
$560$		0			0
$561$	10.3998	−	79.3232i	0.439080	−	3.34903i
$562$		0			0
$563$	21.8981			0.922895			0.461447	−	0.887168i	$-0.347330\pi$
$563$	21.8981			0.922895			0.461447	+	0.887168i	$0.347330\pi$
$564$		0			0
$565$			39.6735i			1.66908i
$566$		0			0
$567$			46.4271i			1.94976i
$568$		0			0
$569$		−	21.8555i		−	0.916230i	−0.888893	−	0.458115i	$-0.848525\pi$
$569$		−	21.8555i		−	0.916230i	0.888893	−	0.458115i	$-0.151475\pi$
$570$		0			0
$571$	−30.3679			−1.27086			−0.635429	−	0.772159i	$-0.719177\pi$
$571$	−30.3679			−1.27086			−0.635429	+	0.772159i	$0.719177\pi$
$572$		0			0
$573$	23.5400			0.983398
$574$		0			0
$575$		−	16.2414i		−	0.677312i
$576$		0			0
$577$		−	13.9331i		−	0.580041i	−0.957020	−	0.290021i	$-0.906338\pi$
$577$		−	13.9331i		−	0.580041i	0.957020	−	0.290021i	$-0.0936622\pi$
$578$		0			0
$579$			67.4326i			2.80240i
$580$		0			0
$581$	−23.9394			−0.993175
$582$		0			0
$583$	2.96781	−	22.6366i	0.122914	−	0.937511i
$584$		0			0
$585$	−69.7253	+	38.7895i	−2.88278	+	1.60375i
$586$		0			0
$587$	21.9254			0.904958			0.452479	−	0.891775i	$-0.350540\pi$
$587$	21.9254			0.904958			0.452479	+	0.891775i	$0.350540\pi$
$588$		0			0
$589$		−	33.7444i		−	1.39042i
$590$		0			0
$591$		−	32.2092i		−	1.32491i
$592$		0			0
$593$	22.4296			0.921072			0.460536	−	0.887641i	$-0.347657\pi$
$593$	22.4296			0.921072			0.460536	+	0.887641i	$0.347657\pi$
$594$		0			0
$595$		−	57.5604i		−	2.35975i
$596$		0			0
$597$	−67.9774			−2.78213
$598$		0			0
$599$		−	3.36696i		−	0.137570i	−0.997631	−	0.0687852i	$-0.978088\pi$
$599$		−	3.36696i		−	0.137570i	0.997631	−	0.0687852i	$-0.0219123\pi$
$600$		0			0
$601$		−	10.3204i		−	0.420978i	−0.977596	−	0.210489i	$-0.932494\pi$
$601$		−	10.3204i		−	0.420978i	0.977596	−	0.210489i	$-0.0675056\pi$
$602$		0			0
$603$	−24.0698			−0.980200
$604$		0			0
$605$	8.92544	−	33.4538i	0.362871	−	1.36009i
$606$		0			0
$607$	27.5111			1.11664			0.558321	−	0.829625i	$-0.311446\pi$
$607$	27.5111			1.11664			0.558321	+	0.829625i	$0.311446\pi$
$608$		0			0
$609$		−	41.6712i		−	1.68860i
$610$		0			0
$611$	−10.2144	−	18.3608i	−0.413232	−	0.742797i
$612$		0			0
$613$	−4.95289			−0.200045			−0.100023	−	0.994985i	$-0.531892\pi$
$613$	−4.95289			−0.200045			−0.100023	+	0.994985i	$0.531892\pi$
$614$		0			0
$615$	78.8181			3.17825
$616$		0			0
$617$		−	38.5815i		−	1.55323i	−0.629975	−	0.776616i	$-0.716935\pi$
$617$		−	38.5815i		−	1.55323i	0.629975	−	0.776616i	$-0.283065\pi$
$618$		0			0
$619$	−43.6416			−1.75410			−0.877052	−	0.480396i	$-0.840493\pi$
$619$	−43.6416			−1.75410			−0.877052	+	0.480396i	$0.840493\pi$
$620$		0			0
$621$	−42.2450			−1.69523
$622$		0			0
$623$	−29.6288			−1.18705
$624$		0			0
$625$	−25.4536			−1.01814
$626$		0			0
$627$	−5.53939	+	42.2510i	−0.221222	+	1.68734i
$628$		0			0
$629$	1.34052			0.0534501
$630$		0			0
$631$	30.6340			1.21952			0.609759	−	0.792587i	$-0.291266\pi$
$631$	30.6340			1.21952			0.609759	+	0.792587i	$0.291266\pi$
$632$		0			0
$633$			41.8274i			1.66249i
$634$		0			0
$635$			14.4362i			0.572884i
$636$		0			0
$637$	−2.16491	−	3.89150i	−0.0857770	−	0.154187i
$638$		0			0
$639$	−91.5159			−3.62031
$640$		0			0
$641$	−4.60612			−0.181931			−0.0909654	−	0.995854i	$-0.528995\pi$
$641$	−4.60612			−0.181931			−0.0909654	+	0.995854i	$0.528995\pi$
$642$		0			0
$643$	−14.7114			−0.580163			−0.290081	−	0.957002i	$-0.593682\pi$
$643$	−14.7114			−0.580163			−0.290081	+	0.957002i	$0.593682\pi$
$644$		0			0
$645$	−18.5494			−0.730381
$646$		0			0
$647$			15.3846i			0.604829i	0.953176	+	0.302415i	$0.0977927\pi$
$647$			15.3846i			0.604829i	−0.953176	+	0.302415i	$0.902207\pi$
$648$		0			0
$649$	3.02892	−	23.1027i	0.118896	−	0.906862i
$650$		0			0
$651$	63.2528			2.47907
$652$		0			0
$653$	12.4775			0.488282			0.244141	−	0.969740i	$-0.421494\pi$
$653$	12.4775			0.488282			0.244141	+	0.969740i	$0.421494\pi$
$654$		0			0
$655$		−	11.6469i		−	0.455083i
$656$		0			0
$657$	−24.2241			−0.945073
$658$		0			0
$659$	−4.08353			−0.159072			−0.0795358	−	0.996832i	$-0.525344\pi$
$659$	−4.08353			−0.159072			−0.0795358	+	0.996832i	$0.525344\pi$
$660$		0			0
$661$			29.2873i			1.13914i	0.821942	+	0.569572i	$0.192891\pi$
$661$			29.2873i			1.13914i	−0.821942	+	0.569572i	$0.807109\pi$
$662$		0			0
$663$	−42.2814	−	76.0021i	−1.64207	−	2.95168i
$664$		0			0
$665$			30.6592i			1.18891i
$666$		0			0
$667$	18.1357			0.702216
$668$		0			0
$669$		−	42.4214i		−	1.64011i
$670$		0			0
$671$	−4.37258	−	0.573274i	−0.168802	−	0.0221310i
$672$		0			0
$673$			5.21622i			0.201070i	0.994934	+	0.100535i	$0.0320555\pi$
$673$			5.21622i			0.201070i	−0.994934	+	0.100535i	$0.967944\pi$
$674$		0			0
$675$		−	62.6451i		−	2.41121i
$676$		0			0
$677$			6.93533i			0.266546i	0.991079	+	0.133273i	$0.0425488\pi$
$677$			6.93533i			0.266546i	−0.991079	+	0.133273i	$0.957451\pi$
$678$		0			0
$679$	−28.0352			−1.07589
$680$		0			0
$681$	−70.9985			−2.72067
$682$		0			0
$683$	−11.0680			−0.423506			−0.211753	−	0.977323i	$-0.567917\pi$
$683$	−11.0680			−0.423506			−0.211753	+	0.977323i	$0.567917\pi$
$684$		0			0
$685$	27.9778			1.06898
$686$		0			0
$687$	2.99769			0.114369
$688$		0			0
$689$	−12.0659	−	21.6888i	−0.459674	−	0.826278i
$690$		0			0
$691$	−31.7462			−1.20768			−0.603840	−	0.797105i	$-0.706364\pi$
$691$	−31.7462			−1.20768			−0.603840	+	0.797105i	$0.706364\pi$
$692$		0			0
$693$	−55.5109	−	7.27784i	−2.10868	−	0.276462i
$694$		0			0
$695$			43.5580i			1.65225i
$696$		0			0
$697$			60.2178i			2.28091i
$698$		0			0
$699$	−38.8552			−1.46964
$700$		0			0
$701$			40.3176i			1.52277i	0.648298	+	0.761386i	$0.275481\pi$
$701$			40.3176i			1.52277i	−0.648298	+	0.761386i	$0.724519\pi$
$702$		0			0
$703$	−0.714019			−0.0269298
$704$		0			0
$705$	58.0919			2.18787
$706$		0			0
$707$	25.4665			0.957767
$708$		0			0
$709$			37.0879i			1.39287i	0.717622	+	0.696433i	$0.245231\pi$
$709$			37.0879i			1.39287i	−0.717622	+	0.696433i	$0.754769\pi$
$710$		0			0
$711$	−27.0131			−1.01307
$712$		0			0
$713$			27.5282i			1.03094i
$714$		0			0
$715$	−13.8677	−	34.9924i	−0.518624	−	1.30864i
$716$		0			0
$717$	−86.3496			−3.22479
$718$		0			0
$719$		−	36.4723i		−	1.36019i	−0.733126	−	0.680093i	$-0.761939\pi$
$719$		−	36.4723i		−	1.36019i	0.733126	−	0.680093i	$-0.238061\pi$
$720$		0			0
$721$	31.2374			1.16334
$722$		0			0
$723$		−	9.49215i		−	0.353017i
$724$		0			0
$725$			26.8934i			0.998796i
$726$		0			0
$727$		−	29.3799i		−	1.08964i	−0.838554	−	0.544819i	$-0.816598\pi$
$727$		−	29.3799i		−	1.08964i	0.838554	−	0.544819i	$-0.183402\pi$
$728$		0			0
$729$	−14.6607			−0.542989
$730$		0			0
$731$		−	14.1719i		−	0.524167i
$732$		0			0
$733$	−32.5173			−1.20105			−0.600527	−	0.799604i	$-0.705043\pi$
$733$	−32.5173			−1.20105			−0.600527	+	0.799604i	$0.705043\pi$
$734$		0			0
$735$	12.3124			0.454148
$736$		0			0
$737$	1.47607	−	11.2586i	0.0543718	−	0.414714i
$738$		0			0
$739$			7.53612i			0.277221i	0.990347	+	0.138610i	$0.0442636\pi$
$739$			7.53612i			0.277221i	−0.990347	+	0.138610i	$0.955736\pi$
$740$		0			0
$741$	22.5209	+	40.4820i	0.827326	+	1.48714i
$742$		0			0
$743$			18.5437i			0.680303i	0.940371	+	0.340151i	$0.110478\pi$
$743$			18.5437i			0.680303i	−0.940371	+	0.340151i	$0.889522\pi$
$744$		0			0
$745$		−	8.07576i		−	0.295873i
$746$		0			0
$747$		−	70.0976i		−	2.56474i
$748$		0			0
$749$		−	45.3669i		−	1.65767i
$750$		0			0
$751$		−	21.6794i		−	0.791094i	−0.918446	−	0.395547i	$-0.870555\pi$
$751$		−	21.6794i		−	0.791094i	0.918446	−	0.395547i	$-0.129445\pi$
$752$		0			0
$753$	−38.4486			−1.40115
$754$		0			0
$755$	−40.1015			−1.45944
$756$		0			0
$757$	−30.0403			−1.09183			−0.545916	−	0.837840i	$-0.683818\pi$
$757$	−30.0403			−1.09183			−0.545916	+	0.837840i	$0.683818\pi$
$758$		0			0
$759$	4.51894	−	34.4677i	0.164027	−	1.25110i
$760$		0			0
$761$	15.3057			0.554831			0.277415	−	0.960750i	$-0.410522\pi$
$761$	15.3057			0.554831			0.277415	+	0.960750i	$0.410522\pi$
$762$		0			0
$763$			21.7552i			0.787592i
$764$		0			0
$765$	168.544			6.09372
$766$		0			0
$767$	−12.3144	−	22.1354i	−0.444646	−	0.799265i
$768$		0			0
$769$	19.6236			0.707646			0.353823	−	0.935312i	$-0.384882\pi$
$769$	19.6236			0.707646			0.353823	+	0.935312i	$0.384882\pi$
$770$		0			0
$771$			58.4399i			2.10466i
$772$		0			0
$773$		−	28.5109i		−	1.02547i	−0.858548	−	0.512733i	$-0.828633\pi$
$773$		−	28.5109i		−	1.02547i	0.858548	−	0.512733i	$-0.171367\pi$
$774$		0			0
$775$	−40.8215			−1.46635
$776$		0			0
$777$		−	1.33840i		−	0.0480150i
$778$		0			0
$779$		−	32.0746i		−	1.14919i
$780$		0			0
$781$	5.61217	−	42.8062i	0.200819	−	1.53172i
$782$		0			0
$783$	69.9516			2.49987
$784$		0			0
$785$		−	24.6887i		−	0.881176i
$786$		0			0
$787$			9.37824i			0.334298i	0.985932	+	0.167149i	$0.0534561\pi$
$787$			9.37824i			0.334298i	−0.985932	+	0.167149i	$0.946544\pi$
$788$		0			0
$789$		−	16.4238i		−	0.584702i
$790$		0			0
$791$			30.2631i			1.07603i
$792$		0			0
$793$	−4.18951	+	2.33070i	−0.148774	+	0.0827657i
$794$		0			0
$795$	68.6215			2.43375
$796$		0			0
$797$	51.9397			1.83980			0.919899	−	0.392156i	$-0.128270\pi$
$797$	51.9397			1.83980			0.919899	+	0.392156i	$0.128270\pi$
$798$		0			0
$799$			44.3827i			1.57015i
$800$		0			0
$801$		−	86.7568i		−	3.06540i
$802$		0			0
$803$	1.48553	−	11.3307i	0.0524233	−	0.399852i
$804$		0			0
$805$		−	25.0113i		−	0.881531i
$806$		0			0
$807$		−	73.6918i		−	2.59407i
$808$		0			0
$809$			21.3060i			0.749081i	0.927211	+	0.374540i	$0.122199\pi$
$809$			21.3060i			0.749081i	−0.927211	+	0.374540i	$0.877801\pi$
$810$		0			0
$811$			8.59083i			0.301665i	0.988559	+	0.150832i	$0.0481954\pi$
$811$			8.59083i			0.301665i	−0.988559	+	0.150832i	$0.951805\pi$
$812$		0			0
$813$	39.7090			1.39266
$814$		0			0
$815$		−	33.4067i		−	1.17019i
$816$		0			0
$817$			7.54857i			0.264091i
$818$		0			0
$819$	−53.1867	+	29.5888i	−1.85849	+	1.03392i
$820$		0			0
$821$	−16.8970			−0.589709			−0.294854	−	0.955542i	$-0.595271\pi$
$821$	−16.8970			−0.589709			−0.294854	+	0.955542i	$0.595271\pi$
$822$		0			0
$823$		−	40.0055i		−	1.39450i	−0.716827	−	0.697251i	$-0.754406\pi$
$823$		−	40.0055i		−	1.39450i	0.716827	−	0.697251i	$-0.245594\pi$
$824$		0			0
$825$	51.1121	+	6.70114i	1.77950	+	0.233304i
$826$		0			0
$827$		−	9.72118i		−	0.338038i	−0.985613	−	0.169019i	$-0.945940\pi$
$827$		−	9.72118i		−	0.338038i	0.985613	−	0.169019i	$-0.0540600\pi$
$828$		0			0
$829$	−27.2849			−0.947643			−0.473821	−	0.880621i	$-0.657126\pi$
$829$	−27.2849			−0.947643			−0.473821	+	0.880621i	$0.657126\pi$
$830$		0			0
$831$	37.3331			1.29507
$832$		0			0
$833$			9.40676i			0.325925i
$834$		0			0
$835$	−0.208808			−0.00722610
$836$		0			0
$837$			106.180i			3.67011i
$838$		0			0
$839$	−8.51563			−0.293992			−0.146996	−	0.989137i	$-0.546960\pi$
$839$	−8.51563			−0.293992			−0.146996	+	0.989137i	$0.546960\pi$
$840$		0			0
$841$	−1.03008			−0.0355199
$842$		0			0
$843$			90.9878i			3.13379i
$844$		0			0
$845$	−34.7679	−	21.5771i	−1.19605	−	0.742276i
$846$		0			0
$847$	6.80836	−	25.5186i	0.233938	−	0.876831i
$848$		0			0
$849$		−	2.55470i		−	0.0876771i
$850$		0			0
$851$	0.582485			0.0199673
$852$		0			0
$853$	−9.30552			−0.318615			−0.159307	−	0.987229i	$-0.550926\pi$
$853$	−9.30552			−0.318615			−0.159307	+	0.987229i	$0.550926\pi$
$854$		0			0
$855$	−89.7739			−3.07020
$856$		0			0
$857$			16.4438i			0.561711i	0.959750	+	0.280856i	$0.0906182\pi$
$857$			16.4438i			0.561711i	−0.959750	+	0.280856i	$0.909382\pi$
$858$		0			0
$859$			18.3696i			0.626764i	0.949627	+	0.313382i	$0.101462\pi$
$859$			18.3696i			0.626764i	−0.949627	+	0.313382i	$0.898538\pi$
$860$		0			0
$861$	60.1228			2.04898
$862$		0			0
$863$	−36.4153			−1.23959			−0.619796	−	0.784763i	$-0.712785\pi$
$863$	−36.4153			−1.23959			−0.619796	+	0.784763i	$0.712785\pi$
$864$		0			0
$865$	−42.2442			−1.43634
$866$		0			0
$867$			129.876i			4.41082i
$868$		0			0
$869$	1.65656	−	12.6352i	0.0561951	−	0.428621i
$870$		0			0
$871$	−6.00111	−	10.7872i	−0.203340	−	0.365509i
$872$		0			0
$873$		−	82.0906i		−	2.77835i
$874$		0			0
$875$	−0.698469			−0.0236126
$876$		0			0
$877$	−19.7726			−0.667675			−0.333837	−	0.942631i	$-0.608344\pi$
$877$	−19.7726			−0.667675			−0.333837	+	0.942631i	$0.608344\pi$
$878$		0			0
$879$			100.632i			3.39423i
$880$		0			0
$881$	−32.6771			−1.10092			−0.550460	−	0.834862i	$-0.685548\pi$
$881$	−32.6771			−1.10092			−0.550460	+	0.834862i	$0.685548\pi$
$882$		0			0
$883$			16.0280i			0.539384i	0.962947	+	0.269692i	$0.0869219\pi$
$883$			16.0280i			0.539384i	−0.962947	+	0.269692i	$0.913078\pi$
$884$		0			0
$885$	70.0346			2.35419
$886$		0			0
$887$	−41.4162			−1.39062			−0.695309	−	0.718710i	$-0.744733\pi$
$887$	−41.4162			−1.39062			−0.695309	+	0.718710i	$0.744733\pi$
$888$		0			0
$889$			11.0120i			0.369331i
$890$		0			0
$891$	8.33672	−	63.5873i	0.279291	−	2.13026i
$892$		0			0
$893$		−	23.6402i		−	0.791088i
$894$		0			0
$895$	−8.72770			−0.291735
$896$		0			0
$897$	−18.3722	−	33.0246i	−0.613430	−	1.10266i
$898$		0			0
$899$		−	45.5827i		−	1.52027i
$900$		0			0
$901$			52.4275i			1.74661i
$902$		0			0
$903$	−14.1495			−0.470867
$904$		0			0
$905$		−	31.4753i		−	1.04628i
$906$		0			0
$907$			9.04068i			0.300191i	0.988672	+	0.150096i	$0.0479581\pi$
$907$			9.04068i			0.300191i	−0.988672	+	0.150096i	$0.952042\pi$
$908$		0			0
$909$			74.5691i			2.47330i
$910$		0			0
$911$		−	23.7676i		−	0.787456i	−0.919227	−	0.393728i	$-0.871185\pi$
$911$		−	23.7676i		−	0.787456i	0.919227	−	0.393728i	$-0.128815\pi$
$912$		0			0
$913$	32.7878	+	4.29870i	1.08512	+	0.142266i
$914$		0			0
$915$		−	13.2552i		−	0.438204i
$916$		0			0
$917$		−	8.88432i		−	0.293386i
$918$		0			0
$919$	−30.1506			−0.994577			−0.497289	−	0.867585i	$-0.665671\pi$
$919$	−30.1506			−0.994577			−0.497289	+	0.867585i	$0.665671\pi$
$920$		0			0
$921$	19.7910			0.652134
$922$		0			0
$923$	−22.8168	−	41.0139i	−0.751025	−	1.34999i
$924$		0			0
$925$			0.863768i			0.0284005i
$926$		0			0
$927$			91.4670i			3.00417i
$928$		0			0
$929$		−	34.6339i		−	1.13630i	−0.822924	−	0.568151i	$-0.807659\pi$
$929$		−	34.6339i		−	1.13630i	0.822924	−	0.568151i	$-0.192341\pi$
$930$		0			0
$931$		−	5.01045i		−	0.164211i
$932$		0			0
$933$	63.1881			2.06869
$934$		0			0
$935$	−10.3359	+	78.8357i	−0.338020	+	2.57820i
$936$		0			0
$937$			37.1025i			1.21209i	0.795432	+	0.606043i	$0.207244\pi$
$937$			37.1025i			1.21209i	−0.795432	+	0.606043i	$0.792756\pi$
$938$		0			0
$939$			35.5017i			1.15855i
$940$		0			0
$941$	−30.7678			−1.00300			−0.501501	−	0.865157i	$-0.667219\pi$
$941$	−30.7678			−1.00300			−0.501501	+	0.865157i	$0.667219\pi$
$942$		0			0
$943$			26.1660i			0.852081i
$944$		0			0
$945$		−	96.4717i		−	3.13822i
$946$		0			0
$947$	14.3578			0.466565			0.233283	−	0.972409i	$-0.425053\pi$
$947$	14.3578			0.466565			0.233283	+	0.972409i	$0.425053\pi$
$948$		0			0
$949$	−6.03958	−	10.8563i	−0.196053	−	0.352411i
$950$		0			0
$951$	84.4290			2.73780
$952$		0			0
$953$			40.6661i			1.31730i	0.752448	+	0.658652i	$0.228873\pi$
$953$			40.6661i			1.31730i	−0.752448	+	0.658652i	$0.771127\pi$
$954$		0			0
$955$	−23.3953			−0.757056
$956$		0			0
$957$	−7.48272	+	57.0736i	−0.241882	+	1.84493i
$958$		0			0
$959$	21.3416			0.689155
$960$		0			0
$961$	38.1901			1.23194
$962$		0			0
$963$	132.840			4.28071
$964$		0			0
$965$		−	67.0182i		−	2.15739i
$966$		0			0
$967$		−	36.4249i		−	1.17135i	−0.810548	−	0.585673i	$-0.800830\pi$
$967$		−	36.4249i		−	1.17135i	0.810548	−	0.585673i	$-0.199170\pi$
$968$		0			0
$969$		−	97.8555i		−	3.14357i
$970$		0			0
$971$		−	13.7828i		−	0.442311i	−0.975239	−	0.221155i	$-0.929017\pi$
$971$		−	13.7828i		−	0.442311i	0.975239	−	0.221155i	$-0.0709828\pi$
$972$		0			0
$973$			33.2262i			1.06518i
$974$		0			0
$975$	48.9721	−	27.2441i	1.56836	−	0.872510i
$976$		0			0
$977$			22.8272i			0.730308i	0.930947	+	0.365154i	$0.118984\pi$
$977$			22.8272i			0.730308i	−0.930947	+	0.365154i	$0.881016\pi$
$978$		0			0
$979$	40.5801	+	5.32032i	1.29695	+	0.170038i
$980$		0			0
$981$	−63.7020			−2.03385
$982$		0			0
$983$	−2.41325			−0.0769708			−0.0384854	−	0.999259i	$-0.512253\pi$
$983$	−2.41325			−0.0769708			−0.0384854	+	0.999259i	$0.512253\pi$
$984$		0			0
$985$			32.0112i			1.01996i
$986$		0			0
$987$	44.3127			1.41049
$988$		0			0
$989$		−	6.15800i		−	0.195813i
$990$		0			0
$991$			43.0107i			1.36628i	0.730287	+	0.683140i	$0.239386\pi$
$991$			43.0107i			1.36628i	−0.730287	+	0.683140i	$0.760614\pi$
$992$		0			0
$993$		−	2.54297i		−	0.0806986i
$994$		0			0
$995$	67.5596			2.14178
$996$		0			0
$997$			50.6738i			1.60486i	0.596749	+	0.802428i	$0.296459\pi$
$997$			50.6738i			1.60486i	−0.596749	+	0.802428i	$0.703541\pi$
$998$		0			0
$999$	2.24672			0.0710831

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2288.2.b.c.2287.4	yes	56
4.3	odd	2	inner	2288.2.b.c.2287.55	yes	56
11.10	odd	2	inner	2288.2.b.c.2287.3	yes	56
13.12	even	2	inner	2288.2.b.c.2287.1	✓	56
44.43	even	2	inner	2288.2.b.c.2287.56	yes	56
52.51	odd	2	inner	2288.2.b.c.2287.54	yes	56
143.142	odd	2	inner	2288.2.b.c.2287.2	yes	56
572.571	even	2	inner	2288.2.b.c.2287.53	yes	56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2288.2.b.c.2287.1	✓	56	13.12	even	2	inner
2288.2.b.c.2287.2	yes	56	143.142	odd	2	inner
2288.2.b.c.2287.3	yes	56	11.10	odd	2	inner
2288.2.b.c.2287.4	yes	56	1.1	even	1	trivial
2288.2.b.c.2287.53	yes	56	572.571	even	2	inner
2288.2.b.c.2287.54	yes	56	52.51	odd	2	inner
2288.2.b.c.2287.55	yes	56	4.3	odd	2	inner
2288.2.b.c.2287.56	yes	56	44.43	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2288 = 2^{4} \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2288.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.16710i q^{3} +3.14763i q^{5} +2.40102i q^{7} -7.03050 q^{9} +O(q^{10})\)	\(q-3.16710i q^{3} +3.14763i q^{5} +2.40102i q^{7} -7.03050 q^{9} +(0.431142 - 3.28848i) q^{11} +(-1.75285 - 3.15080i) q^{13} +9.96885 q^{15} +7.61629i q^{17} -4.05677i q^{19} +7.60427 q^{21} +3.30945i q^{23} -4.90758 q^{25} +12.7650i q^{27} -5.47997i q^{29} +8.31806 q^{31} +(-10.4149 - 1.36547i) q^{33} -7.55754 q^{35} -0.176007i q^{37} +(-9.97888 + 5.55144i) q^{39} +7.90644 q^{41} -1.86073 q^{43} -22.1294i q^{45} +5.82734 q^{47} +1.23508 q^{49} +24.1215 q^{51} +6.88359 q^{53} +(10.3509 + 1.35708i) q^{55} -12.8482 q^{57} +7.02535 q^{59} -1.32967i q^{61} -16.8804i q^{63} +(9.91755 - 5.51732i) q^{65} +3.42363 q^{67} +10.4813 q^{69} +13.0170 q^{71} +3.44558 q^{73} +15.5428i q^{75} +(7.89573 + 1.03518i) q^{77} +3.84227 q^{79} +19.3364 q^{81} +9.97051i q^{83} -23.9733 q^{85} -17.3556 q^{87} +12.3401i q^{89} +(7.56514 - 4.20863i) q^{91} -26.3441i q^{93} +12.7692 q^{95} +11.6764i q^{97} +(-3.03114 + 23.1197i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 72 q^{9}+O(q^{10}) \) 56 * q - 72 * q^9	\( 56 q - 72 q^{9} - 72 q^{25} - 8 q^{49} + 72 q^{53} + 8 q^{69} - 48 q^{77} + 88 q^{81}+O(q^{100}) \) 56 * q - 72 * q^9 - 72 * q^25 - 8 * q^49 + 72 * q^53 + 8 * q^69 - 48 * q^77 + 88 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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