LMFDB - Embedded newform 3360.2.q.b.2239.35

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3360,2,Mod(2239,3360)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3360, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 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("3360.2239");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3360.q (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:	$26.8297350792$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2239.35
Character	$\chi$	$=$	3360.2239
Dual form			3360.2.q.b.2239.36

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +(-2.00218 - 0.995638i) q^{5} +(1.37846 - 2.25829i) q^{7} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{3} +(-2.00218 - 0.995638i) q^{5} +(1.37846 - 2.25829i) q^{7} -1.00000 q^{9} -3.02613i q^{11} +1.42925 q^{13} +(-0.995638 + 2.00218i) q^{15} +7.83802 q^{17} -0.687358 q^{19} +(-2.25829 - 1.37846i) q^{21} +4.79936 q^{23} +(3.01741 + 3.98688i) q^{25} +1.00000i q^{27} +7.65607 q^{29} +10.5319 q^{31} -3.02613 q^{33} +(-5.00835 + 3.14904i) q^{35} +8.43446i q^{37} -1.42925i q^{39} -5.26200i q^{41} -7.78536 q^{43} +(2.00218 + 0.995638i) q^{45} -5.98497i q^{47} +(-3.19972 - 6.22590i) q^{49} -7.83802i q^{51} +11.6423i q^{53} +(-3.01293 + 6.05883i) q^{55} +0.687358i q^{57} +8.22110 q^{59} -13.7714i q^{61} +(-1.37846 + 2.25829i) q^{63} +(-2.86160 - 1.42301i) q^{65} -2.90155 q^{67} -4.79936i q^{69} +0.334004i q^{71} -2.35424 q^{73} +(3.98688 - 3.01741i) q^{75} +(-6.83386 - 4.17138i) q^{77} -2.84547i q^{79} +1.00000 q^{81} -11.1344i q^{83} +(-15.6931 - 7.80383i) q^{85} -7.65607i q^{87} -1.41353i q^{89} +(1.97015 - 3.22765i) q^{91} -10.5319i q^{93} +(1.37621 + 0.684360i) q^{95} -18.4477 q^{97} +3.02613i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q + 8 q^{5} - 48 q^{9}+O(q^{10}) $ 48 * q + 8 * q^5 - 48 * q^9	$ 48 q + 8 q^{5} - 48 q^{9} + 8 q^{25} + 16 q^{33} - 8 q^{45} - 16 q^{49} - 16 q^{73} - 48 q^{77} + 48 q^{81} - 40 q^{85} - 48 q^{97}+O(q^{100}) $ 48 * q + 8 * q^5 - 48 * q^9 + 8 * q^25 + 16 * q^33 - 8 * q^45 - 16 * q^49 - 16 * q^73 - 48 * q^77 + 48 * q^81 - 40 * q^85 - 48 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3360\mathbb{Z}\right)^\times$.

$n$	$421$	$1121$	$1471$	$1921$	$2017$
$\chi(n)$	$1$	$1$	$-1$	$-1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		−	1.00000i		−	0.577350i
$3$		−	1.00000i		−	0.577350i
$4$		0			0
$5$	−2.00218	−	0.995638i	−0.895400	−	0.445263i
$5$	−2.00218	−	0.995638i	−0.895400	−	0.445263i
$6$		0			0
$7$	1.37846	−	2.25829i	0.521007	−	0.853552i
$7$	1.37846	−	2.25829i	0.521007	−	0.853552i
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$		−	3.02613i		−	0.912411i	−0.889874	−	0.456206i	$-0.849208\pi$
$11$		−	3.02613i		−	0.912411i	0.889874	−	0.456206i	$-0.150792\pi$
$12$		0			0
$13$	1.42925			0.396401			0.198201	−	0.980161i	$-0.436490\pi$
$13$	1.42925			0.396401			0.198201	+	0.980161i	$0.436490\pi$
$14$		0			0
$15$	−0.995638	+	2.00218i	−0.257073	+	0.516959i
$16$		0			0
$17$	7.83802			1.90100			0.950500	−	0.310725i	$-0.100572\pi$
$17$	7.83802			1.90100			0.950500	+	0.310725i	$0.100572\pi$
$18$		0			0
$19$	−0.687358			−0.157691			−0.0788454	−	0.996887i	$-0.525123\pi$
$19$	−0.687358			−0.157691			−0.0788454	+	0.996887i	$0.525123\pi$
$20$		0			0
$21$	−2.25829	−	1.37846i	−0.492799	−	0.300804i
$22$		0			0
$23$	4.79936			1.00074			0.500368	−	0.865813i	$-0.333198\pi$
$23$	4.79936			1.00074			0.500368	+	0.865813i	$0.333198\pi$
$24$		0			0
$25$	3.01741	+	3.98688i	0.603482	+	0.797377i
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$	7.65607			1.42170			0.710849	−	0.703345i	$-0.248311\pi$
$29$	7.65607			1.42170			0.710849	+	0.703345i	$0.248311\pi$
$30$		0			0
$31$	10.5319			1.89159			0.945796	−	0.324761i	$-0.105284\pi$
$31$	10.5319			1.89159			0.945796	+	0.324761i	$0.105284\pi$
$32$		0			0
$33$	−3.02613			−0.526781
$34$		0			0
$35$	−5.00835	+	3.14904i	−0.846565	+	0.532285i
$36$		0			0
$37$			8.43446i			1.38662i	0.720641	+	0.693308i	$0.243847\pi$
$37$			8.43446i			1.38662i	−0.720641	+	0.693308i	$0.756153\pi$
$38$		0			0
$39$		−	1.42925i		−	0.228862i
$40$		0			0
$41$		−	5.26200i		−	0.821786i	−0.911684	−	0.410893i	$-0.865217\pi$
$41$		−	5.26200i		−	0.821786i	0.911684	−	0.410893i	$-0.134783\pi$
$42$		0			0
$43$	−7.78536			−1.18726			−0.593628	−	0.804740i	$-0.702305\pi$
$43$	−7.78536			−1.18726			−0.593628	+	0.804740i	$0.702305\pi$
$44$		0			0
$45$	2.00218	+	0.995638i	0.298467	+	0.148421i
$46$		0			0
$47$		−	5.98497i		−	0.872998i	−0.899705	−	0.436499i	$-0.856218\pi$
$47$		−	5.98497i		−	0.872998i	0.899705	−	0.436499i	$-0.143782\pi$
$48$		0			0
$49$	−3.19972	−	6.22590i	−0.457103	−	0.889414i
$50$		0			0
$51$		−	7.83802i		−	1.09754i
$52$		0			0
$53$			11.6423i			1.59920i	0.600534	+	0.799600i	$0.294955\pi$
$53$			11.6423i			1.59920i	−0.600534	+	0.799600i	$0.705045\pi$
$54$		0			0
$55$	−3.01293	+	6.05883i	−0.406263	+	0.816973i
$56$		0			0
$57$			0.687358i			0.0910428i
$58$		0			0
$59$	8.22110			1.07030			0.535148	−	0.844758i	$-0.320256\pi$
$59$	8.22110			1.07030			0.535148	+	0.844758i	$0.320256\pi$
$60$		0			0
$61$		−	13.7714i		−	1.76325i	−0.471949	−	0.881626i	$-0.656449\pi$
$61$		−	13.7714i		−	1.76325i	0.471949	−	0.881626i	$-0.343551\pi$
$62$		0			0
$63$	−1.37846	+	2.25829i	−0.173669	+	0.284517i
$64$		0			0
$65$	−2.86160	−	1.42301i	−0.354938	−	0.176503i
$66$		0			0
$67$	−2.90155			−0.354480			−0.177240	−	0.984168i	$-0.556717\pi$
$67$	−2.90155			−0.354480			−0.177240	+	0.984168i	$0.556717\pi$
$68$		0			0
$69$		−	4.79936i		−	0.577775i
$70$		0			0
$71$			0.334004i			0.0396390i	0.999804	+	0.0198195i	$0.00630915\pi$
$71$			0.334004i			0.0396390i	−0.999804	+	0.0198195i	$0.993691\pi$
$72$		0			0
$73$	−2.35424			−0.275543			−0.137772	−	0.990464i	$-0.543994\pi$
$73$	−2.35424			−0.275543			−0.137772	+	0.990464i	$0.543994\pi$
$74$		0			0
$75$	3.98688	−	3.01741i	0.460366	−	0.348421i
$76$		0			0
$77$	−6.83386	−	4.17138i	−0.778791	−	0.475373i
$78$		0			0
$79$		−	2.84547i		−	0.320141i	−0.987106	−	0.160070i	$-0.948828\pi$
$79$		−	2.84547i		−	0.320141i	0.987106	−	0.160070i	$-0.0511721\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	11.1344i		−	1.22216i	−0.791569	−	0.611080i	$-0.790735\pi$
$83$		−	11.1344i		−	1.22216i	0.791569	−	0.611080i	$-0.209265\pi$
$84$		0			0
$85$	−15.6931	−	7.80383i	−1.70215	−	0.846444i
$86$		0			0
$87$		−	7.65607i		−	0.820817i
$88$		0			0
$89$		−	1.41353i		−	0.149833i	−0.997190	−	0.0749167i	$-0.976131\pi$
$89$		−	1.41353i		−	0.149833i	0.997190	−	0.0749167i	$-0.0238691\pi$
$90$		0			0
$91$	1.97015	−	3.22765i	0.206528	−	0.338349i
$92$		0			0
$93$		−	10.5319i		−	1.09211i
$94$		0			0
$95$	1.37621	+	0.684360i	0.141196	+	0.0702139i
$96$		0			0
$97$	−18.4477			−1.87308			−0.936539	−	0.350563i	$-0.885990\pi$
$97$	−18.4477			−1.87308			−0.936539	+	0.350563i	$0.885990\pi$
$98$		0			0
$99$			3.02613i			0.304137i
$100$		0			0
$101$			13.9082i			1.38392i	0.721936	+	0.691959i	$0.243252\pi$
$101$			13.9082i			1.38392i	−0.721936	+	0.691959i	$0.756748\pi$
$102$		0			0
$103$			14.0564i			1.38502i	0.721410	+	0.692509i	$0.243495\pi$
$103$			14.0564i			1.38502i	−0.721410	+	0.692509i	$0.756505\pi$
$104$		0			0
$105$	3.14904	+	5.00835i	0.307315	+	0.488765i
$106$		0			0
$107$	17.5326			1.69494			0.847470	−	0.530843i	$-0.178125\pi$
$107$	17.5326			1.69494			0.847470	+	0.530843i	$0.178125\pi$
$108$		0			0
$109$	−12.4023			−1.18793			−0.593963	−	0.804492i	$-0.702438\pi$
$109$	−12.4023			−1.18793			−0.593963	+	0.804492i	$0.702438\pi$
$110$		0			0
$111$	8.43446			0.800563
$112$		0			0
$113$			5.75259i			0.541159i	0.962698	+	0.270579i	$0.0872152\pi$
$113$			5.75259i			0.541159i	−0.962698	+	0.270579i	$0.912785\pi$
$114$		0			0
$115$	−9.60916	−	4.77842i	−0.896058	−	0.445590i
$116$		0			0
$117$	−1.42925			−0.132134
$118$		0			0
$119$	10.8044	−	17.7005i	0.990435	−	1.62260i
$120$		0			0
$121$	1.84256			0.167506
$122$		0			0
$123$	−5.26200			−0.474458
$124$		0			0
$125$	−2.07189	−	10.9867i	−0.185316	−	0.982679i
$126$		0			0
$127$	18.4840			1.64019			0.820094	−	0.572229i	$-0.193921\pi$
$127$	18.4840			1.64019			0.820094	+	0.572229i	$0.193921\pi$
$128$		0			0
$129$			7.78536i			0.685462i
$130$		0			0
$131$	0.943045			0.0823943			0.0411971	−	0.999151i	$-0.486883\pi$
$131$	0.943045			0.0823943			0.0411971	+	0.999151i	$0.486883\pi$
$132$		0			0
$133$	−0.947493	+	1.55225i	−0.0821581	+	0.134597i
$134$		0			0
$135$	0.995638	−	2.00218i	0.0856909	−	0.172320i
$136$		0			0
$137$		−	8.14799i		−	0.696129i	−0.937470	−	0.348065i	$-0.886839\pi$
$137$		−	8.14799i		−	0.696129i	0.937470	−	0.348065i	$-0.113161\pi$
$138$		0			0
$139$	2.76975			0.234927			0.117463	−	0.993077i	$-0.462524\pi$
$139$	2.76975			0.234927			0.117463	+	0.993077i	$0.462524\pi$
$140$		0			0
$141$	−5.98497			−0.504026
$142$		0			0
$143$		−	4.32508i		−	0.361681i
$144$		0			0
$145$	−15.3288	−	7.62268i	−1.27299	−	0.633029i
$146$		0			0
$147$	−6.22590	+	3.19972i	−0.513503	+	0.263908i
$148$		0			0
$149$	−10.3965			−0.851714			−0.425857	−	0.904791i	$-0.640027\pi$
$149$	−10.3965			−0.851714			−0.425857	+	0.904791i	$0.640027\pi$
$150$		0			0
$151$		−	7.94743i		−	0.646752i	−0.946270	−	0.323376i	$-0.895182\pi$
$151$		−	7.94743i		−	0.646752i	0.946270	−	0.323376i	$-0.104818\pi$
$152$		0			0
$153$	−7.83802			−0.633667
$154$		0			0
$155$	−21.0868	−	10.4860i	−1.69373	−	0.842256i
$156$		0			0
$157$	−11.1880			−0.892899			−0.446449	−	0.894809i	$-0.647312\pi$
$157$	−11.1880			−0.892899			−0.446449	+	0.894809i	$0.647312\pi$
$158$		0			0
$159$	11.6423			0.923298
$160$		0			0
$161$	6.61570	−	10.8383i	0.521391	−	0.854180i
$162$		0			0
$163$	−18.6598			−1.46154			−0.730772	−	0.682621i	$-0.760840\pi$
$163$	−18.6598			−1.46154			−0.730772	+	0.682621i	$0.760840\pi$
$164$		0			0
$165$	6.05883	+	3.01293i	0.471680	+	0.234556i
$166$		0			0
$167$			13.1475i			1.01738i	0.860949	+	0.508691i	$0.169871\pi$
$167$			13.1475i			1.01738i	−0.860949	+	0.508691i	$0.830129\pi$
$168$		0			0
$169$	−10.9573			−0.842866
$170$		0			0
$171$	0.687358			0.0525636
$172$		0			0
$173$	18.6034			1.41439			0.707196	−	0.707017i	$-0.249960\pi$
$173$	18.6034			1.41439			0.707196	+	0.707017i	$0.249960\pi$
$174$		0			0
$175$	13.1629	−	1.31844i	0.995021	−	0.0996644i
$176$		0			0
$177$		−	8.22110i		−	0.617936i
$178$		0			0
$179$		−	3.33165i		−	0.249019i	−0.992218	−	0.124510i	$-0.960264\pi$
$179$		−	3.33165i		−	0.249019i	0.992218	−	0.124510i	$-0.0397357\pi$
$180$		0			0
$181$			6.65318i			0.494527i	0.968948	+	0.247264i	$0.0795313\pi$
$181$			6.65318i			0.494527i	−0.968948	+	0.247264i	$0.920469\pi$
$182$		0			0
$183$	−13.7714			−1.01801
$184$		0			0
$185$	8.39766	−	16.8873i	0.617409	−	1.24158i
$186$		0			0
$187$		−	23.7188i		−	1.73449i
$188$		0			0
$189$	2.25829	+	1.37846i	0.164266	+	0.100268i
$190$		0			0
$191$		−	11.2423i		−	0.813466i	−0.913547	−	0.406733i	$-0.866668\pi$
$191$		−	11.2423i		−	0.813466i	0.913547	−	0.406733i	$-0.133332\pi$
$192$		0			0
$193$			2.68797i			0.193484i	0.995309	+	0.0967421i	$0.0308422\pi$
$193$			2.68797i			0.193484i	−0.995309	+	0.0967421i	$0.969158\pi$
$194$		0			0
$195$	−1.42301	+	2.86160i	−0.101904	+	0.204923i
$196$		0			0
$197$			8.33609i			0.593922i	0.954890	+	0.296961i	$0.0959731\pi$
$197$			8.33609i			0.593922i	−0.954890	+	0.296961i	$0.904027\pi$
$198$		0			0
$199$	3.62648			0.257075			0.128537	−	0.991705i	$-0.458972\pi$
$199$	3.62648			0.257075			0.128537	+	0.991705i	$0.458972\pi$
$200$		0			0
$201$			2.90155i			0.204659i
$202$		0			0
$203$	10.5536	−	17.2896i	0.740715	−	1.21349i
$204$		0			0
$205$	−5.23904	+	10.5354i	−0.365911	+	0.735827i
$206$		0			0
$207$	−4.79936			−0.333578
$208$		0			0
$209$			2.08003i			0.143879i
$210$		0			0
$211$		−	5.85862i		−	0.403324i	−0.979455	−	0.201662i	$-0.935366\pi$
$211$		−	5.85862i		−	0.403324i	0.979455	−	0.201662i	$-0.0646343\pi$
$212$		0			0
$213$	0.334004			0.0228856
$214$		0			0
$215$	15.5876	+	7.75140i	1.06307	+	0.528641i
$216$		0			0
$217$	14.5178	−	23.7841i	0.985534	−	1.61457i
$218$		0			0
$219$			2.35424i			0.159085i
$220$		0			0
$221$	11.2025			0.753559
$222$		0			0
$223$			13.7266i			0.919203i	0.888125	+	0.459602i	$0.152008\pi$
$223$			13.7266i			0.919203i	−0.888125	+	0.459602i	$0.847992\pi$
$224$		0			0
$225$	−3.01741	−	3.98688i	−0.201161	−	0.265792i
$226$		0			0
$227$		−	22.2000i		−	1.47347i	−0.676182	−	0.736734i	$-0.736367\pi$
$227$		−	22.2000i		−	1.47347i	0.676182	−	0.736734i	$-0.263633\pi$
$228$		0			0
$229$		−	0.735173i		−	0.0485816i	−0.999705	−	0.0242908i	$-0.992267\pi$
$229$		−	0.735173i		−	0.0485816i	0.999705	−	0.0242908i	$-0.00773276\pi$
$230$		0			0
$231$	−4.17138	+	6.83386i	−0.274457	+	0.449635i
$232$		0			0
$233$			23.2855i			1.52549i	0.646701	+	0.762743i	$0.276148\pi$
$233$			23.2855i			1.52549i	−0.646701	+	0.762743i	$0.723852\pi$
$234$		0			0
$235$	−5.95887	+	11.9830i	−0.388714	+	0.781683i
$236$		0			0
$237$	−2.84547			−0.184833
$238$		0			0
$239$			10.9277i			0.706857i	0.935462	+	0.353429i	$0.114984\pi$
$239$			10.9277i			0.706857i	−0.935462	+	0.353429i	$0.885016\pi$
$240$		0			0
$241$			3.45084i			0.222288i	0.993804	+	0.111144i	$0.0354515\pi$
$241$			3.45084i			0.222288i	−0.993804	+	0.111144i	$0.964548\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$	0.207657	+	15.6511i	0.0132667	+	0.999912i
$246$		0			0
$247$	−0.982404			−0.0625089
$248$		0			0
$249$	−11.1344			−0.705615
$250$		0			0
$251$	−22.5521			−1.42347			−0.711737	−	0.702446i	$-0.752091\pi$
$251$	−22.5521			−1.42347			−0.711737	+	0.702446i	$0.752091\pi$
$252$		0			0
$253$		−	14.5235i		−	0.913082i
$254$		0			0
$255$	−7.80383	+	15.6931i	−0.488695	+	0.982740i
$256$		0			0
$257$	−19.3060			−1.20427			−0.602136	−	0.798393i	$-0.705684\pi$
$257$	−19.3060			−1.20427			−0.602136	+	0.798393i	$0.705684\pi$
$258$		0			0
$259$	19.0474	+	11.6265i	1.18355	+	0.722437i
$260$		0			0
$261$	−7.65607			−0.473899
$262$		0			0
$263$	21.2937			1.31302			0.656512	−	0.754316i	$-0.272031\pi$
$263$	21.2937			1.31302			0.656512	+	0.754316i	$0.272031\pi$
$264$		0			0
$265$	11.5916	−	23.3100i	0.712064	−	1.43192i
$266$		0			0
$267$	−1.41353			−0.0865063
$268$		0			0
$269$		−	12.8962i		−	0.786292i	−0.919476	−	0.393146i	$-0.871387\pi$
$269$		−	12.8962i		−	0.786292i	0.919476	−	0.393146i	$-0.128613\pi$
$270$		0			0
$271$	5.11215			0.310541			0.155271	−	0.987872i	$-0.450375\pi$
$271$	5.11215			0.310541			0.155271	+	0.987872i	$0.450375\pi$
$272$		0			0
$273$	−3.22765	−	1.97015i	−0.195346	−	0.119239i
$274$		0			0
$275$	12.0648	−	9.13106i	0.727535	−	0.550624i
$276$		0			0
$277$		−	19.2963i		−	1.15940i	−0.814829	−	0.579702i	$-0.803169\pi$
$277$		−	19.2963i		−	1.15940i	0.814829	−	0.579702i	$-0.196831\pi$
$278$		0			0
$279$	−10.5319			−0.630531
$280$		0			0
$281$	−11.4050			−0.680366			−0.340183	−	0.940359i	$-0.610489\pi$
$281$	−11.4050			−0.680366			−0.340183	+	0.940359i	$0.610489\pi$
$282$		0			0
$283$			5.99679i			0.356472i	0.983988	+	0.178236i	$0.0570391\pi$
$283$			5.99679i			0.356472i	−0.983988	+	0.178236i	$0.942961\pi$
$284$		0			0
$285$	0.684360	−	1.37621i	0.0405380	−	0.0815197i
$286$		0			0
$287$	−11.8831	−	7.25343i	−0.701437	−	0.428157i
$288$		0			0
$289$	44.4346			2.61380
$290$		0			0
$291$			18.4477i			1.08142i
$292$		0			0
$293$	−19.5438			−1.14176			−0.570879	−	0.821034i	$-0.693398\pi$
$293$	−19.5438			−1.14176			−0.570879	+	0.821034i	$0.693398\pi$
$294$		0			0
$295$	−16.4601	−	8.18524i	−0.958344	−	0.476563i
$296$		0			0
$297$	3.02613			0.175594
$298$		0			0
$299$	6.85946			0.396693
$300$		0			0
$301$	−10.7318	+	17.5816i	−0.618569	+	1.01338i
$302$		0			0
$303$	13.9082			0.799006
$304$		0			0
$305$	−13.7114	+	27.5728i	−0.785110	+	1.57882i
$306$		0			0
$307$		−	8.16520i		−	0.466013i	−0.972475	−	0.233006i	$-0.925144\pi$
$307$		−	8.16520i		−	0.466013i	0.972475	−	0.233006i	$-0.0748563\pi$
$308$		0			0
$309$	14.0564			0.799640
$310$		0			0
$311$	−19.3639			−1.09803			−0.549014	−	0.835813i	$-0.684997\pi$
$311$	−19.3639			−1.09803			−0.549014	+	0.835813i	$0.684997\pi$
$312$		0			0
$313$	−6.15238			−0.347753			−0.173876	−	0.984767i	$-0.555629\pi$
$313$	−6.15238			−0.347753			−0.173876	+	0.984767i	$0.555629\pi$
$314$		0			0
$315$	5.00835	−	3.14904i	0.282188	−	0.177428i
$316$		0			0
$317$			13.7415i			0.771802i	0.922540	+	0.385901i	$0.126109\pi$
$317$			13.7415i			0.771802i	−0.922540	+	0.385901i	$0.873891\pi$
$318$		0			0
$319$		−	23.1682i		−	1.29717i
$320$		0			0
$321$		−	17.5326i		−	0.978574i
$322$		0			0
$323$	−5.38753			−0.299770
$324$		0			0
$325$	4.31262	+	5.69824i	0.239221	+	0.316081i
$326$		0			0
$327$			12.4023i			0.685850i
$328$		0			0
$329$	−13.5158	−	8.25002i	−0.745150	−	0.454839i
$330$		0			0
$331$			11.8855i			0.653288i	0.945147	+	0.326644i	$0.105918\pi$
$331$			11.8855i			0.653288i	−0.945147	+	0.326644i	$0.894082\pi$
$332$		0			0
$333$		−	8.43446i		−	0.462205i
$334$		0			0
$335$	5.80941	+	2.88889i	0.317402	+	0.157837i
$336$		0			0
$337$		−	28.6523i		−	1.56079i	−0.625288	−	0.780394i	$-0.715018\pi$
$337$		−	28.6523i		−	1.56079i	0.625288	−	0.780394i	$-0.284982\pi$
$338$		0			0
$339$	5.75259			0.312438
$340$		0			0
$341$		−	31.8710i		−	1.72591i
$342$		0			0
$343$	−18.4705	−	1.35624i	−0.997315	−	0.0732303i
$344$		0			0
$345$	−4.77842	+	9.60916i	−0.257262	+	0.517340i
$346$		0			0
$347$	−1.40991			−0.0756877			−0.0378439	−	0.999284i	$-0.512049\pi$
$347$	−1.40991			−0.0756877			−0.0378439	+	0.999284i	$0.512049\pi$
$348$		0			0
$349$			10.7353i			0.574646i	0.957834	+	0.287323i	$0.0927653\pi$
$349$			10.7353i			0.574646i	−0.957834	+	0.287323i	$0.907235\pi$
$350$		0			0
$351$			1.42925i			0.0762875i
$352$		0			0
$353$	3.27554			0.174340			0.0871698	−	0.996193i	$-0.472218\pi$
$353$	3.27554			0.174340			0.0871698	+	0.996193i	$0.472218\pi$
$354$		0			0
$355$	0.332547	−	0.668734i	0.0176498	−	0.0354927i
$356$		0			0
$357$	−17.7005	−	10.8044i	−0.936810	−	0.571828i
$358$		0			0
$359$		−	29.7347i		−	1.56934i	−0.619915	−	0.784669i	$-0.712833\pi$
$359$		−	29.7347i		−	1.56934i	0.619915	−	0.784669i	$-0.287167\pi$
$360$		0			0
$361$	−18.5275			−0.975134
$362$		0			0
$363$		−	1.84256i		−	0.0967094i
$364$		0			0
$365$	4.71361	+	2.34397i	0.246721	+	0.122689i
$366$		0			0
$367$			32.1399i			1.67769i	0.544373	+	0.838843i	$0.316768\pi$
$367$			32.1399i			1.67769i	−0.544373	+	0.838843i	$0.683232\pi$
$368$		0			0
$369$			5.26200i			0.273929i
$370$		0			0
$371$	26.2918	+	16.0485i	1.36500	+	0.833195i
$372$		0			0
$373$			29.9611i			1.55133i	0.631145	+	0.775665i	$0.282585\pi$
$373$			29.9611i			1.55133i	−0.631145	+	0.775665i	$0.717415\pi$
$374$		0			0
$375$	−10.9867	+	2.07189i	−0.567350	+	0.106992i
$376$		0			0
$377$	10.9424			0.563563
$378$		0			0
$379$			12.2077i			0.627066i	0.949577	+	0.313533i	$0.101513\pi$
$379$			12.2077i			0.627066i	−0.949577	+	0.313533i	$0.898487\pi$
$380$		0			0
$381$		−	18.4840i		−	0.946963i
$382$		0			0
$383$		−	11.3296i		−	0.578916i	−0.957191	−	0.289458i	$-0.906525\pi$
$383$		−	11.3296i		−	0.578916i	0.957191	−	0.289458i	$-0.0934751\pi$
$384$		0			0
$385$	9.52940	+	15.1559i	0.485663	+	0.772415i
$386$		0			0
$387$	7.78536			0.395752
$388$		0			0
$389$	35.2405			1.78676			0.893382	−	0.449298i	$-0.148326\pi$
$389$	35.2405			1.78676			0.893382	+	0.449298i	$0.148326\pi$
$390$		0			0
$391$	37.6175			1.90240
$392$		0			0
$393$		−	0.943045i		−	0.0475704i
$394$		0			0
$395$	−2.83306	+	5.69714i	−0.142547	+	0.286654i
$396$		0			0
$397$	−8.45286			−0.424237			−0.212118	−	0.977244i	$-0.568036\pi$
$397$	−8.45286			−0.424237			−0.212118	+	0.977244i	$0.568036\pi$
$398$		0			0
$399$	1.55225	+	0.947493i	0.0777098	+	0.0474340i
$400$		0			0
$401$	9.17021			0.457938			0.228969	−	0.973434i	$-0.426465\pi$
$401$	9.17021			0.457938			0.228969	+	0.973434i	$0.426465\pi$
$402$		0			0
$403$	15.0527			0.749830
$404$		0			0
$405$	−2.00218	−	0.995638i	−0.0994889	−	0.0494736i
$406$		0			0
$407$	25.5237			1.26516
$408$		0			0
$409$			18.0631i			0.893163i	0.894743	+	0.446581i	$0.147359\pi$
$409$			18.0631i			0.893163i	−0.894743	+	0.446581i	$0.852641\pi$
$410$		0			0
$411$	−8.14799			−0.401911
$412$		0			0
$413$	11.3324	−	18.5656i	0.557632	−	0.913554i
$414$		0			0
$415$	−11.0858	+	22.2930i	−0.544183	+	1.09432i
$416$		0			0
$417$		−	2.76975i		−	0.135635i
$418$		0			0
$419$	−27.4862			−1.34279			−0.671394	−	0.741100i	$-0.734304\pi$
$419$	−27.4862			−1.34279			−0.671394	+	0.741100i	$0.734304\pi$
$420$		0			0
$421$	0.296733			0.0144619			0.00723094	−	0.999974i	$-0.497698\pi$
$421$	0.296733			0.0144619			0.00723094	+	0.999974i	$0.497698\pi$
$422$		0			0
$423$			5.98497i			0.290999i
$424$		0			0
$425$	23.6505	+	31.2493i	1.14722	+	1.51581i
$426$		0			0
$427$	−31.0998	−	18.9833i	−1.50503	−	0.918667i
$428$		0			0
$429$	−4.32508			−0.208817
$430$		0			0
$431$			12.0735i			0.581560i	0.956790	+	0.290780i	$0.0939148\pi$
$431$			12.0735i			0.581560i	−0.956790	+	0.290780i	$0.906085\pi$
$432$		0			0
$433$	8.76695			0.421313			0.210656	−	0.977560i	$-0.432440\pi$
$433$	8.76695			0.421313			0.210656	+	0.977560i	$0.432440\pi$
$434$		0			0
$435$	−7.62268	+	15.3288i	−0.365479	+	0.734960i
$436$		0			0
$437$	−3.29888			−0.157807
$438$		0			0
$439$	−5.72036			−0.273018			−0.136509	−	0.990639i	$-0.543588\pi$
$439$	−5.72036			−0.273018			−0.136509	+	0.990639i	$0.543588\pi$
$440$		0			0
$441$	3.19972	+	6.22590i	0.152368	+	0.296471i
$442$		0			0
$443$	−9.41697			−0.447414			−0.223707	−	0.974656i	$-0.571816\pi$
$443$	−9.41697			−0.447414			−0.223707	+	0.974656i	$0.571816\pi$
$444$		0			0
$445$	−1.40736	+	2.83012i	−0.0667152	+	0.134161i
$446$		0			0
$447$			10.3965i			0.491737i
$448$		0			0
$449$	−12.0022			−0.566417			−0.283209	−	0.959058i	$-0.591399\pi$
$449$	−12.0022			−0.566417			−0.283209	+	0.959058i	$0.591399\pi$
$450$		0			0
$451$	−15.9235			−0.749807
$452$		0			0
$453$	−7.94743			−0.373403
$454$		0			0
$455$	−7.15816	+	4.50076i	−0.335580	+	0.210999i
$456$		0			0
$457$			5.65605i			0.264579i	0.991211	+	0.132289i	$0.0422328\pi$
$457$			5.65605i			0.264579i	−0.991211	+	0.132289i	$0.957767\pi$
$458$		0			0
$459$			7.83802i			0.365848i
$460$		0			0
$461$		−	2.28937i		−	0.106627i	−0.998578	−	0.0533133i	$-0.983022\pi$
$461$		−	2.28937i		−	0.106627i	0.998578	−	0.0533133i	$-0.0169782\pi$
$462$		0			0
$463$	−36.8790			−1.71391			−0.856956	−	0.515389i	$-0.827647\pi$
$463$	−36.8790			−1.71391			−0.856956	+	0.515389i	$0.827647\pi$
$464$		0			0
$465$	−10.4860	+	21.0868i	−0.486277	+	0.977877i
$466$		0			0
$467$		−	11.9210i		−	0.551640i	−0.961209	−	0.275820i	$-0.911051\pi$
$467$		−	11.9210i		−	0.551640i	0.961209	−	0.275820i	$-0.0889494\pi$
$468$		0			0
$469$	−3.99966	+	6.55253i	−0.184687	+	0.302568i
$470$		0			0
$471$			11.1880i			0.515515i
$472$		0			0
$473$			23.5595i			1.08327i
$474$		0			0
$475$	−2.07404	−	2.74042i	−0.0951636	−	0.125739i
$476$		0			0
$477$		−	11.6423i		−	0.533066i
$478$		0			0
$479$	16.7545			0.765531			0.382765	−	0.923846i	$-0.374972\pi$
$479$	16.7545			0.765531			0.382765	+	0.923846i	$0.374972\pi$
$480$		0			0
$481$			12.0549i			0.549657i
$482$		0			0
$483$	−10.8383	−	6.61570i	−0.493161	−	0.301025i
$484$		0			0
$485$	36.9355	+	18.3672i	1.67715	+	0.834012i
$486$		0			0
$487$	22.5183			1.02040			0.510200	−	0.860056i	$-0.329571\pi$
$487$	22.5183			1.02040			0.510200	+	0.860056i	$0.329571\pi$
$488$		0			0
$489$			18.6598i			0.843823i
$490$		0			0
$491$		−	5.79625i		−	0.261581i	−0.991410	−	0.130790i	$-0.958248\pi$
$491$		−	5.79625i		−	0.261581i	0.991410	−	0.130790i	$-0.0417515\pi$
$492$		0			0
$493$	60.0085			2.70265
$494$		0			0
$495$	3.01293	−	6.05883i	0.135421	−	0.272324i
$496$		0			0
$497$	0.754277	+	0.460410i	0.0338339	+	0.0206522i
$498$		0			0
$499$			5.93442i			0.265661i	0.991139	+	0.132831i	$0.0424066\pi$
$499$			5.93442i			0.265661i	−0.991139	+	0.132831i	$0.957593\pi$
$500$		0			0
$501$	13.1475			0.587386
$502$		0			0
$503$			35.0359i			1.56217i	0.624423	+	0.781087i	$0.285334\pi$
$503$			35.0359i			1.56217i	−0.624423	+	0.781087i	$0.714666\pi$
$504$		0			0
$505$	13.8475	−	27.8467i	0.616208	−	1.23916i
$506$		0			0
$507$			10.9573i			0.486629i
$508$		0			0
$509$		−	4.75377i		−	0.210707i	−0.994435	−	0.105353i	$-0.966403\pi$
$509$		−	4.75377i		−	0.210707i	0.994435	−	0.105353i	$-0.0335974\pi$
$510$		0			0
$511$	−3.24522	+	5.31655i	−0.143560	+	0.235191i
$512$		0			0
$513$		−	0.687358i		−	0.0303476i
$514$		0			0
$515$	13.9951	−	28.1434i	0.616697	−	1.24014i
$516$		0			0
$517$	−18.1113			−0.796534
$518$		0			0
$519$		−	18.6034i		−	0.816600i
$520$		0			0
$521$			14.0789i			0.616807i	0.951256	+	0.308403i	$0.0997946\pi$
$521$			14.0789i			0.616807i	−0.951256	+	0.308403i	$0.900205\pi$
$522$		0			0
$523$			17.8545i			0.780725i	0.920661	+	0.390362i	$0.127650\pi$
$523$			17.8545i			0.780725i	−0.920661	+	0.390362i	$0.872350\pi$
$524$		0			0
$525$	−1.31844	−	13.1629i	−0.0575413	−	0.574476i
$526$		0			0
$527$	82.5496			3.59592
$528$		0			0
$529$	0.0338423			0.00147140
$530$		0			0
$531$	−8.22110			−0.356766
$532$		0			0
$533$		−	7.52069i		−	0.325757i
$534$		0			0
$535$	−35.1033	−	17.4561i	−1.51765	−	0.754694i
$536$		0			0
$537$	−3.33165			−0.143771
$538$		0			0
$539$	−18.8404	+	9.68275i	−0.811511	+	0.417066i
$540$		0			0
$541$	12.8083			0.550672			0.275336	−	0.961348i	$-0.411211\pi$
$541$	12.8083			0.550672			0.275336	+	0.961348i	$0.411211\pi$
$542$		0			0
$543$	6.65318			0.285515
$544$		0			0
$545$	24.8316	+	12.3482i	1.06367	+	0.528940i
$546$		0			0
$547$	18.6803			0.798713			0.399356	−	0.916796i	$-0.369234\pi$
$547$	18.6803			0.798713			0.399356	+	0.916796i	$0.369234\pi$
$548$		0			0
$549$			13.7714i			0.587750i
$550$		0			0
$551$	−5.26247			−0.224189
$552$		0			0
$553$	−6.42590	−	3.92236i	−0.273257	−	0.166796i
$554$		0			0
$555$	−16.8873	−	8.39766i	−0.716824	−	0.356461i
$556$		0			0
$557$		−	33.8654i		−	1.43492i	−0.696598	−	0.717462i	$-0.745304\pi$
$557$		−	33.8654i		−	1.43492i	0.696598	−	0.717462i	$-0.254696\pi$
$558$		0			0
$559$	−11.1272			−0.470630
$560$		0			0
$561$	−23.7188			−1.00141
$562$		0			0
$563$			10.9610i			0.461950i	0.972960	+	0.230975i	$0.0741915\pi$
$563$			10.9610i			0.461950i	−0.972960	+	0.230975i	$0.925808\pi$
$564$		0			0
$565$	5.72750	−	11.5177i	0.240958	−	0.484553i
$566$		0			0
$567$	1.37846	−	2.25829i	0.0578897	−	0.0948391i
$568$		0			0
$569$	36.5082			1.53050			0.765252	−	0.643731i	$-0.222615\pi$
$569$	36.5082			1.53050			0.765252	+	0.643731i	$0.222615\pi$
$570$		0			0
$571$		−	41.6950i		−	1.74488i	−0.488719	−	0.872441i	$-0.662536\pi$
$571$		−	41.6950i		−	1.74488i	0.488719	−	0.872441i	$-0.337464\pi$
$572$		0			0
$573$	−11.2423			−0.469655
$574$		0			0
$575$	14.4816	+	19.1345i	0.603926	+	0.797963i
$576$		0			0
$577$	−17.7255			−0.737922			−0.368961	−	0.929445i	$-0.620286\pi$
$577$	−17.7255			−0.737922			−0.368961	+	0.929445i	$0.620286\pi$
$578$		0			0
$579$	2.68797			0.111708
$580$		0			0
$581$	−25.1447	−	15.3483i	−1.04318	−	0.636755i
$582$		0			0
$583$	35.2312			1.45913
$584$		0			0
$585$	2.86160	+	1.42301i	0.118313	+	0.0588343i
$586$		0			0
$587$		−	27.3553i		−	1.12907i	−0.825408	−	0.564537i	$-0.809055\pi$
$587$		−	27.3553i		−	1.12907i	0.825408	−	0.564537i	$-0.190945\pi$
$588$		0			0
$589$	−7.23922			−0.298287
$590$		0			0
$591$	8.33609			0.342901
$592$		0			0
$593$	−9.89416			−0.406305			−0.203152	−	0.979147i	$-0.565119\pi$
$593$	−9.89416			−0.406305			−0.203152	+	0.979147i	$0.565119\pi$
$594$		0			0
$595$	−39.2555	+	24.6823i	−1.60932	+	1.01187i
$596$		0			0
$597$		−	3.62648i		−	0.148422i
$598$		0			0
$599$		−	14.8473i		−	0.606645i	−0.952888	−	0.303323i	$-0.901904\pi$
$599$		−	14.8473i		−	0.606645i	0.952888	−	0.303323i	$-0.0980960\pi$
$600$		0			0
$601$		−	1.51144i		−	0.0616530i	−0.999525	−	0.0308265i	$-0.990186\pi$
$601$		−	1.51144i		−	0.0616530i	0.999525	−	0.0308265i	$-0.00981394\pi$
$602$		0			0
$603$	2.90155			0.118160
$604$		0			0
$605$	−3.68913	−	1.83452i	−0.149985	−	0.0745840i
$606$		0			0
$607$			14.1432i			0.574054i	0.957923	+	0.287027i	$0.0926669\pi$
$607$			14.1432i			0.574054i	−0.957923	+	0.287027i	$0.907333\pi$
$608$		0			0
$609$	−17.2896	−	10.5536i	−0.700610	−	0.427652i
$610$		0			0
$611$		−	8.55400i		−	0.346058i
$612$		0			0
$613$		−	13.0761i		−	0.528140i	−0.964503	−	0.264070i	$-0.914935\pi$
$613$		−	13.0761i		−	0.528140i	0.964503	−	0.264070i	$-0.0850650\pi$
$614$		0			0
$615$	10.5354	+	5.23904i	0.424830	+	0.211259i
$616$		0			0
$617$			10.6951i			0.430568i	0.976551	+	0.215284i	$0.0690677\pi$
$617$			10.6951i			0.430568i	−0.976551	+	0.215284i	$0.930932\pi$
$618$		0			0
$619$	5.59677			0.224953			0.112477	−	0.993654i	$-0.464122\pi$
$619$	5.59677			0.224953			0.112477	+	0.993654i	$0.464122\pi$
$620$		0			0
$621$			4.79936i			0.192592i
$622$		0			0
$623$	−3.19215	−	1.94848i	−0.127891	−	0.0780643i
$624$		0			0
$625$	−6.79047	+	24.0601i	−0.271619	+	0.962405i
$626$		0			0
$627$	2.08003			0.0830685
$628$		0			0
$629$			66.1095i			2.63596i
$630$		0			0
$631$			42.0010i			1.67203i	0.548704	+	0.836017i	$0.315122\pi$
$631$			42.0010i			1.67203i	−0.548704	+	0.836017i	$0.684878\pi$
$632$		0			0
$633$	−5.85862			−0.232859
$634$		0			0
$635$	−37.0082	−	18.4034i	−1.46862	−	0.730315i
$636$		0			0
$637$	−4.57318	−	8.89834i	−0.181196	−	0.352565i
$638$		0			0
$639$		−	0.334004i		−	0.0132130i
$640$		0			0
$641$	−17.7028			−0.699220			−0.349610	−	0.936895i	$-0.613686\pi$
$641$	−17.7028			−0.699220			−0.349610	+	0.936895i	$0.613686\pi$
$642$		0			0
$643$		−	18.5110i		−	0.730002i	−0.931007	−	0.365001i	$-0.881069\pi$
$643$		−	18.5110i		−	0.730002i	0.931007	−	0.365001i	$-0.118931\pi$
$644$		0			0
$645$	7.75140	−	15.5876i	0.305211	−	0.613763i
$646$		0			0
$647$		−	46.5750i		−	1.83105i	−0.402261	−	0.915525i	$-0.631775\pi$
$647$		−	46.5750i		−	1.83105i	0.402261	−	0.915525i	$-0.368225\pi$
$648$		0			0
$649$		−	24.8781i		−	0.976551i
$650$		0			0
$651$	−23.7841	−	14.5178i	−0.932174	−	0.568998i
$652$		0			0
$653$			29.9906i			1.17362i	0.809724	+	0.586811i	$0.199617\pi$
$653$			29.9906i			1.17362i	−0.809724	+	0.586811i	$0.800383\pi$
$654$		0			0
$655$	−1.88814	−	0.938932i	−0.0737758	−	0.0366871i
$656$		0			0
$657$	2.35424			0.0918477
$658$		0			0
$659$			18.1418i			0.706704i	0.935490	+	0.353352i	$0.114958\pi$
$659$			18.1418i			0.706704i	−0.935490	+	0.353352i	$0.885042\pi$
$660$		0			0
$661$			13.5041i			0.525249i	0.964898	+	0.262624i	$0.0845880\pi$
$661$			13.5041i			0.525249i	−0.964898	+	0.262624i	$0.915412\pi$
$662$		0			0
$663$		−	11.2025i		−	0.435068i
$664$		0			0
$665$	3.44253	−	2.16452i	0.133496	−	0.0839365i
$666$		0			0
$667$	36.7442			1.42274
$668$		0			0
$669$	13.7266			0.530702
$670$		0			0
$671$	−41.6741			−1.60881
$672$		0			0
$673$			26.2053i			1.01014i	0.863078	+	0.505071i	$0.168534\pi$
$673$			26.2053i			1.01014i	−0.863078	+	0.505071i	$0.831466\pi$
$674$		0			0
$675$	−3.98688	+	3.01741i	−0.153455	+	0.116140i
$676$		0			0
$677$	4.84726			0.186296			0.0931478	−	0.995652i	$-0.470307\pi$
$677$	4.84726			0.186296			0.0931478	+	0.995652i	$0.470307\pi$
$678$		0			0
$679$	−25.4293	+	41.6601i	−0.975887	+	1.59877i
$680$		0			0
$681$	−22.2000			−0.850708
$682$		0			0
$683$	12.0377			0.460610			0.230305	−	0.973119i	$-0.426028\pi$
$683$	12.0377			0.460610			0.230305	+	0.973119i	$0.426028\pi$
$684$		0			0
$685$	−8.11244	+	16.3137i	−0.309961	+	0.623314i
$686$		0			0
$687$	−0.735173			−0.0280486
$688$		0			0
$689$			16.6398i			0.633925i
$690$		0			0
$691$	−8.01961			−0.305080			−0.152540	−	0.988297i	$-0.548745\pi$
$691$	−8.01961			−0.305080			−0.152540	+	0.988297i	$0.548745\pi$
$692$		0			0
$693$	6.83386	+	4.17138i	0.259597	+	0.158458i
$694$		0			0
$695$	−5.54552	−	2.75767i	−0.210354	−	0.104604i
$696$		0			0
$697$		−	41.2437i		−	1.56221i
$698$		0			0
$699$	23.2855			0.880740
$700$		0			0
$701$	15.7856			0.596215			0.298108	−	0.954532i	$-0.403645\pi$
$701$	15.7856			0.596215			0.298108	+	0.954532i	$0.403645\pi$
$702$		0			0
$703$		−	5.79749i		−	0.218657i
$704$		0			0
$705$	11.9830	+	5.95887i	0.451305	+	0.224424i
$706$		0			0
$707$	31.4087	+	19.1719i	1.18125	+	0.721032i
$708$		0			0
$709$	−40.7671			−1.53104			−0.765521	−	0.643411i	$-0.777519\pi$
$709$	−40.7671			−1.53104			−0.765521	+	0.643411i	$0.777519\pi$
$710$		0			0
$711$			2.84547i			0.106714i
$712$		0			0
$713$	50.5466			1.89298
$714$		0			0
$715$	−4.30621	+	8.65956i	−0.161043	+	0.323849i
$716$		0			0
$717$	10.9277			0.408104
$718$		0			0
$719$	−39.8360			−1.48563			−0.742816	−	0.669495i	$-0.766511\pi$
$719$	−39.8360			−1.48563			−0.742816	+	0.669495i	$0.766511\pi$
$720$		0			0
$721$	31.7434	+	19.3761i	1.18218	+	0.721604i
$722$		0			0
$723$	3.45084			0.128338
$724$		0			0
$725$	23.1015	+	30.5239i	0.857969	+	1.13363i
$726$		0			0
$727$		−	42.5875i		−	1.57948i	−0.613440	−	0.789742i	$-0.710215\pi$
$727$		−	42.5875i		−	1.57948i	0.613440	−	0.789742i	$-0.289785\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	−61.0218			−2.25697
$732$		0			0
$733$	10.3533			0.382406			0.191203	−	0.981550i	$-0.438761\pi$
$733$	10.3533			0.382406			0.191203	+	0.981550i	$0.438761\pi$
$734$		0			0
$735$	15.6511	−	0.207657i	0.577299	−	0.00765955i
$736$		0			0
$737$			8.78045i			0.323432i
$738$		0			0
$739$			31.3899i			1.15470i	0.816498	+	0.577348i	$0.195912\pi$
$739$			31.3899i			1.15470i	−0.816498	+	0.577348i	$0.804088\pi$
$740$		0			0
$741$			0.982404i			0.0360895i
$742$		0			0
$743$	−6.30073			−0.231151			−0.115576	−	0.993299i	$-0.536871\pi$
$743$	−6.30073			−0.231151			−0.115576	+	0.993299i	$0.536871\pi$
$744$		0			0
$745$	20.8156	+	10.3511i	0.762624	+	0.379236i
$746$		0			0
$747$			11.1344i			0.407387i
$748$		0			0
$749$	24.1679	−	39.5936i	0.883077	−	1.44672i
$750$		0			0
$751$		−	21.1471i		−	0.771670i	−0.922568	−	0.385835i	$-0.873913\pi$
$751$		−	21.1471i		−	0.771670i	0.922568	−	0.385835i	$-0.126087\pi$
$752$		0			0
$753$			22.5521i			0.821844i
$754$		0			0
$755$	−7.91276	+	15.9121i	−0.287975	+	0.579102i
$756$		0			0
$757$		−	24.6252i		−	0.895018i	−0.894279	−	0.447509i	$-0.852311\pi$
$757$		−	24.6252i		−	0.895018i	0.894279	−	0.447509i	$-0.147689\pi$
$758$		0			0
$759$	−14.5235			−0.527168
$760$		0			0
$761$		−	12.8409i		−	0.465482i	−0.972539	−	0.232741i	$-0.925231\pi$
$761$		−	12.8409i		−	0.465482i	0.972539	−	0.232741i	$-0.0747695\pi$
$762$		0			0
$763$	−17.0961	+	28.0080i	−0.618919	+	1.01396i
$764$		0			0
$765$	15.6931	+	7.80383i	0.567385	+	0.282148i
$766$		0			0
$767$	11.7500			0.424267
$768$		0			0
$769$		−	6.22458i		−	0.224464i	−0.993682	−	0.112232i	$-0.964200\pi$
$769$		−	6.22458i		−	0.224464i	0.993682	−	0.112232i	$-0.0358000\pi$
$770$		0			0
$771$			19.3060i			0.695287i
$772$		0			0
$773$	−4.33155			−0.155795			−0.0778975	−	0.996961i	$-0.524821\pi$
$773$	−4.33155			−0.155795			−0.0778975	+	0.996961i	$0.524821\pi$
$774$		0			0
$775$	31.7792	+	41.9896i	1.14154	+	1.50831i
$776$		0			0
$777$	11.6265	−	19.0474i	0.417099	−	0.683322i
$778$		0			0
$779$			3.61688i			0.129588i
$780$		0			0
$781$	1.01074			0.0361670
$782$		0			0
$783$			7.65607i			0.273606i
$784$		0			0
$785$	22.4003	+	11.1392i	0.799501	+	0.397575i
$786$		0			0
$787$		−	5.21560i		−	0.185916i	−0.995670	−	0.0929580i	$-0.970368\pi$
$787$		−	5.21560i		−	0.185916i	0.995670	−	0.0929580i	$-0.0296322\pi$
$788$		0			0
$789$		−	21.2937i		−	0.758075i
$790$		0			0
$791$	12.9910	+	7.92970i	0.461907	+	0.281948i
$792$		0			0
$793$		−	19.6828i		−	0.698955i
$794$		0			0
$795$	−23.3100	−	11.5916i	−0.826721	−	0.411110i
$796$		0			0
$797$	46.6525			1.65252			0.826258	−	0.563292i	$-0.190465\pi$
$797$	46.6525			1.65252			0.826258	+	0.563292i	$0.190465\pi$
$798$		0			0
$799$		−	46.9104i		−	1.65957i
$800$		0			0
$801$			1.41353i			0.0499445i
$802$		0			0
$803$			7.12423i			0.251409i
$804$		0			0
$805$	−24.0368	+	15.1134i	−0.847188	+	0.532677i
$806$		0			0
$807$	−12.8962			−0.453966
$808$		0			0
$809$	20.2826			0.713099			0.356549	−	0.934277i	$-0.383953\pi$
$809$	20.2826			0.713099			0.356549	+	0.934277i	$0.383953\pi$
$810$		0			0
$811$	26.2255			0.920901			0.460451	−	0.887685i	$-0.347688\pi$
$811$	26.2255			0.920901			0.460451	+	0.887685i	$0.347688\pi$
$812$		0			0
$813$		−	5.11215i		−	0.179291i
$814$		0			0
$815$	37.3601	+	18.5784i	1.30867	+	0.650772i
$816$		0			0
$817$	5.35133			0.187219
$818$		0			0
$819$	−1.97015	+	3.22765i	−0.0688427	+	0.112783i
$820$		0			0
$821$	17.4683			0.609647			0.304824	−	0.952409i	$-0.401402\pi$
$821$	17.4683			0.609647			0.304824	+	0.952409i	$0.401402\pi$
$822$		0			0
$823$	6.17963			0.215408			0.107704	−	0.994183i	$-0.465650\pi$
$823$	6.17963			0.215408			0.107704	+	0.994183i	$0.465650\pi$
$824$		0			0
$825$	−9.13106	−	12.0648i	−0.317903	−	0.420043i
$826$		0			0
$827$	5.49540			0.191094			0.0955469	−	0.995425i	$-0.469540\pi$
$827$	5.49540			0.191094			0.0955469	+	0.995425i	$0.469540\pi$
$828$		0			0
$829$			42.8733i			1.48905i	0.667594	+	0.744525i	$0.267324\pi$
$829$			42.8733i			1.48905i	−0.667594	+	0.744525i	$0.732676\pi$
$830$		0			0
$831$	−19.2963			−0.669382
$832$		0			0
$833$	−25.0795	−	48.7987i	−0.868952	−	1.69078i
$834$		0			0
$835$	13.0901	−	26.3236i	0.453003	−	0.910964i
$836$		0			0
$837$			10.5319i			0.364037i
$838$		0			0
$839$	−21.5298			−0.743290			−0.371645	−	0.928375i	$-0.621206\pi$
$839$	−21.5298			−0.743290			−0.371645	+	0.928375i	$0.621206\pi$
$840$		0			0
$841$	29.6155			1.02122
$842$		0			0
$843$			11.4050i			0.392810i
$844$		0			0
$845$	21.9383	+	10.9095i	0.754702	+	0.375297i
$846$		0			0
$847$	2.53989	−	4.16103i	0.0872717	−	0.142975i
$848$		0			0
$849$	5.99679			0.205809
$850$		0			0
$851$			40.4800i			1.38764i
$852$		0			0
$853$	7.54074			0.258190			0.129095	−	0.991632i	$-0.458793\pi$
$853$	7.54074			0.258190			0.129095	+	0.991632i	$0.458793\pi$
$854$		0			0
$855$	−1.37621	−	0.684360i	−0.0470654	−	0.0234046i
$856$		0			0
$857$	−24.4656			−0.835731			−0.417865	−	0.908509i	$-0.637222\pi$
$857$	−24.4656			−0.835731			−0.417865	+	0.908509i	$0.637222\pi$
$858$		0			0
$859$	−25.0179			−0.853600			−0.426800	−	0.904346i	$-0.640359\pi$
$859$	−25.0179			−0.853600			−0.426800	+	0.904346i	$0.640359\pi$
$860$		0			0
$861$	−7.25343	+	11.8831i	−0.247196	+	0.404975i
$862$		0			0
$863$	−13.0318			−0.443608			−0.221804	−	0.975091i	$-0.571194\pi$
$863$	−13.0318			−0.443608			−0.221804	+	0.975091i	$0.571194\pi$
$864$		0			0
$865$	−37.2473	−	18.5223i	−1.26645	−	0.629776i
$866$		0			0
$867$		−	44.4346i		−	1.50908i
$868$		0			0
$869$	−8.61076			−0.292100
$870$		0			0
$871$	−4.14702			−0.140517
$872$		0			0
$873$	18.4477			0.624359
$874$		0			0
$875$	−27.6671	−	10.4657i	−0.935319	−	0.353806i
$876$		0			0
$877$		−	10.2668i		−	0.346685i	−0.984862	−	0.173343i	$-0.944543\pi$
$877$		−	10.2668i		−	0.346685i	0.984862	−	0.173343i	$-0.0554568\pi$
$878$		0			0
$879$			19.5438i			0.659195i
$880$		0			0
$881$			44.4745i			1.49838i	0.662354	+	0.749191i	$0.269558\pi$
$881$			44.4745i			1.49838i	−0.662354	+	0.749191i	$0.730442\pi$
$882$		0			0
$883$	−55.9002			−1.88119			−0.940596	−	0.339529i	$-0.889732\pi$
$883$	−55.9002			−1.88119			−0.940596	+	0.339529i	$0.889732\pi$
$884$		0			0
$885$	−8.18524	+	16.4601i	−0.275144	+	0.553300i
$886$		0			0
$887$			5.65804i			0.189979i	0.995478	+	0.0949893i	$0.0302817\pi$
$887$			5.65804i			0.189979i	−0.995478	+	0.0949893i	$0.969718\pi$
$888$		0			0
$889$	25.4794	−	41.7421i	0.854550	−	1.39999i
$890$		0			0
$891$		−	3.02613i		−	0.101379i
$892$		0			0
$893$			4.11382i			0.137664i
$894$		0			0
$895$	−3.31712	+	6.67054i	−0.110879	+	0.222972i
$896$		0			0
$897$		−	6.85946i		−	0.229031i
$898$		0			0
$899$	80.6333			2.68927
$900$		0			0
$901$			91.2530i			3.04008i
$902$		0			0
$903$	17.5816	+	10.7318i	0.585078	+	0.357131i
$904$		0			0
$905$	6.62416	−	13.3208i	0.220195	−	0.442800i
$906$		0			0
$907$	34.5116			1.14594			0.572970	−	0.819576i	$-0.305791\pi$
$907$	34.5116			1.14594			0.572970	+	0.819576i	$0.305791\pi$
$908$		0			0
$909$		−	13.9082i		−	0.461306i
$910$		0			0
$911$			12.8633i			0.426180i	0.977033	+	0.213090i	$0.0683528\pi$
$911$			12.8633i			0.426180i	−0.977033	+	0.213090i	$0.931647\pi$
$912$		0			0
$913$	−33.6941			−1.11511
$914$		0			0
$915$	27.5728	+	13.7114i	0.911529	+	0.453284i
$916$		0			0
$917$	1.29995	−	2.12967i	0.0429280	−	0.0703278i
$918$		0			0
$919$		−	22.1853i		−	0.731826i	−0.930649	−	0.365913i	$-0.880757\pi$
$919$		−	22.1853i		−	0.731826i	0.930649	−	0.365913i	$-0.119243\pi$
$920$		0			0
$921$	−8.16520			−0.269053
$922$		0			0
$923$			0.477374i			0.0157129i
$924$		0			0
$925$	−33.6272	+	25.4502i	−1.10566	+	0.836798i
$926$		0			0
$927$		−	14.0564i		−	0.461672i
$928$		0			0
$929$			41.9552i			1.37650i	0.725471	+	0.688252i	$0.241622\pi$
$929$			41.9552i			1.37650i	−0.725471	+	0.688252i	$0.758378\pi$
$930$		0			0
$931$	2.19935	+	4.27942i	0.0720809	+	0.140252i
$932$		0			0
$933$			19.3639i			0.633947i
$934$		0			0
$935$	−23.6154	+	47.4893i	−0.772305	+	1.55307i
$936$		0			0
$937$	1.44648			0.0472544			0.0236272	−	0.999721i	$-0.492479\pi$
$937$	1.44648			0.0472544			0.0236272	+	0.999721i	$0.492479\pi$
$938$		0			0
$939$			6.15238i			0.200775i
$940$		0			0
$941$		−	25.8740i		−	0.843468i	−0.906720	−	0.421734i	$-0.861422\pi$
$941$		−	25.8740i		−	0.843468i	0.906720	−	0.421734i	$-0.138578\pi$
$942$		0			0
$943$		−	25.2542i		−	0.822390i
$944$		0			0
$945$	−3.14904	−	5.00835i	−0.102438	−	0.162922i
$946$		0			0
$947$	35.6409			1.15817			0.579087	−	0.815266i	$-0.303409\pi$
$947$	35.6409			1.15817			0.579087	+	0.815266i	$0.303409\pi$
$948$		0			0
$949$	−3.36479			−0.109226
$950$		0			0
$951$	13.7415			0.445600
$952$		0			0
$953$		−	35.2180i		−	1.14082i	−0.821359	−	0.570412i	$-0.806784\pi$
$953$		−	35.2180i		−	1.14082i	0.821359	−	0.570412i	$-0.193216\pi$
$954$		0			0
$955$	−11.1933	+	22.5091i	−0.362206	+	0.728377i
$956$		0			0
$957$	−23.1682			−0.748923
$958$		0			0
$959$	−18.4005	−	11.2316i	−0.594183	−	0.362689i
$960$		0			0
$961$	79.9218			2.57812
$962$		0			0
$963$	−17.5326			−0.564980
$964$		0			0
$965$	2.67624	−	5.38178i	0.0861513	−	0.173246i
$966$		0			0
$967$	−34.0127			−1.09377			−0.546887	−	0.837206i	$-0.684187\pi$
$967$	−34.0127			−1.09377			−0.546887	+	0.837206i	$0.684187\pi$
$968$		0			0
$969$			5.38753i			0.173072i
$970$		0			0
$971$	40.1445			1.28830			0.644149	−	0.764900i	$-0.277212\pi$
$971$	40.1445			1.28830			0.644149	+	0.764900i	$0.277212\pi$
$972$		0			0
$973$	3.81798	−	6.25489i	0.122399	−	0.200522i
$974$		0			0
$975$	5.69824	−	4.31262i	0.182490	−	0.138114i
$976$		0			0
$977$			55.5239i			1.77637i	0.459489	+	0.888183i	$0.348033\pi$
$977$			55.5239i			1.77637i	−0.459489	+	0.888183i	$0.651967\pi$
$978$		0			0
$979$	−4.27751			−0.136710
$980$		0			0
$981$	12.4023			0.395976
$982$		0			0
$983$			20.3806i			0.650040i	0.945707	+	0.325020i	$0.105371\pi$
$983$			20.3806i			0.650040i	−0.945707	+	0.325020i	$0.894629\pi$
$984$		0			0
$985$	8.29973	−	16.6903i	0.264451	−	0.531798i
$986$		0			0
$987$	−8.25002	+	13.5158i	−0.262601	+	0.430212i
$988$		0			0
$989$	−37.3647			−1.18813
$990$		0			0
$991$			19.0564i			0.605346i	0.953094	+	0.302673i	$0.0978790\pi$
$991$			19.0564i			0.605346i	−0.953094	+	0.302673i	$0.902121\pi$
$992$		0			0
$993$	11.8855			0.377176
$994$		0			0
$995$	−7.26086	−	3.61067i	−0.230185	−	0.114466i
$996$		0			0
$997$	−21.5548			−0.682648			−0.341324	−	0.939946i	$-0.610875\pi$
$997$	−21.5548			−0.682648			−0.341324	+	0.939946i	$0.610875\pi$
$998$		0			0
$999$	−8.43446			−0.266854

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3360.2.q.b.2239.35	yes	48
4.3	odd	2	inner	3360.2.q.b.2239.30	yes	48
5.4	even	2		3360.2.q.a.2239.14	yes	48
7.6	odd	2		3360.2.q.a.2239.20	yes	48
20.19	odd	2		3360.2.q.a.2239.19	yes	48
28.27	even	2		3360.2.q.a.2239.13	✓	48
35.34	odd	2	inner	3360.2.q.b.2239.29	yes	48
140.139	even	2	inner	3360.2.q.b.2239.36	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3360.2.q.a.2239.13	✓	48	28.27	even	2
3360.2.q.a.2239.14	yes	48	5.4	even	2
3360.2.q.a.2239.19	yes	48	20.19	odd	2
3360.2.q.a.2239.20	yes	48	7.6	odd	2
3360.2.q.b.2239.29	yes	48	35.34	odd	2	inner
3360.2.q.b.2239.30	yes	48	4.3	odd	2	inner
3360.2.q.b.2239.35	yes	48	1.1	even	1	trivial
3360.2.q.b.2239.36	yes	48	140.139	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:	\( N \)	\(=\)	\( 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3360.q (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +(-2.00218 - 0.995638i) q^{5} +(1.37846 - 2.25829i) q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +(-2.00218 - 0.995638i) q^{5} +(1.37846 - 2.25829i) q^{7} -1.00000 q^{9} -3.02613i q^{11} +1.42925 q^{13} +(-0.995638 + 2.00218i) q^{15} +7.83802 q^{17} -0.687358 q^{19} +(-2.25829 - 1.37846i) q^{21} +4.79936 q^{23} +(3.01741 + 3.98688i) q^{25} +1.00000i q^{27} +7.65607 q^{29} +10.5319 q^{31} -3.02613 q^{33} +(-5.00835 + 3.14904i) q^{35} +8.43446i q^{37} -1.42925i q^{39} -5.26200i q^{41} -7.78536 q^{43} +(2.00218 + 0.995638i) q^{45} -5.98497i q^{47} +(-3.19972 - 6.22590i) q^{49} -7.83802i q^{51} +11.6423i q^{53} +(-3.01293 + 6.05883i) q^{55} +0.687358i q^{57} +8.22110 q^{59} -13.7714i q^{61} +(-1.37846 + 2.25829i) q^{63} +(-2.86160 - 1.42301i) q^{65} -2.90155 q^{67} -4.79936i q^{69} +0.334004i q^{71} -2.35424 q^{73} +(3.98688 - 3.01741i) q^{75} +(-6.83386 - 4.17138i) q^{77} -2.84547i q^{79} +1.00000 q^{81} -11.1344i q^{83} +(-15.6931 - 7.80383i) q^{85} -7.65607i q^{87} -1.41353i q^{89} +(1.97015 - 3.22765i) q^{91} -10.5319i q^{93} +(1.37621 + 0.684360i) q^{95} -18.4477 q^{97} +3.02613i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 8 q^{5} - 48 q^{9}+O(q^{10}) \) 48 * q + 8 * q^5 - 48 * q^9	\( 48 q + 8 q^{5} - 48 q^{9} + 8 q^{25} + 16 q^{33} - 8 q^{45} - 16 q^{49} - 16 q^{73} - 48 q^{77} + 48 q^{81} - 40 q^{85} - 48 q^{97}+O(q^{100}) \) 48 * q + 8 * q^5 - 48 * q^9 + 8 * q^25 + 16 * q^33 - 8 * q^45 - 16 * q^49 - 16 * q^73 - 48 * q^77 + 48 * q^81 - 40 * q^85 - 48 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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