LMFDB - Embedded newform 2912.2.i.a.337.10

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2912,2,Mod(337,2912)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2912, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2912.337"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2912 = 2^{5} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2912.i (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

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

Self dual:	no
Analytic conductor:	$23.2524370686$
Analytic rank:	$0$
Dimension:	$84$
Twist minimal:	no (minimal twist has level 728)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.10
Character	$\chi$	$=$	2912.337
Dual form			2912.2.i.a.337.9

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.39926i q^{3} +2.48834 q^{5} -1.00000i q^{7} -8.55495 q^{9} +4.88409 q^{11} +(-2.21870 - 2.84207i) q^{13} +8.45851i q^{15} +5.61904 q^{17} +5.49683 q^{19} +3.39926 q^{21} +5.27964 q^{23} +1.19184 q^{25} -18.8827i q^{27} -3.18688i q^{29} +2.54576i q^{31} +16.6023i q^{33} -2.48834i q^{35} +3.21861 q^{37} +(9.66094 - 7.54192i) q^{39} -4.30478i q^{41} +7.45874i q^{43} -21.2876 q^{45} -0.0608216i q^{47} -1.00000 q^{49} +19.1006i q^{51} -2.00045i q^{53} +12.1533 q^{55} +18.6851i q^{57} -0.469731 q^{59} +2.65509i q^{61} +8.55495i q^{63} +(-5.52087 - 7.07205i) q^{65} +1.22805 q^{67} +17.9469i q^{69} +4.67427i q^{71} +4.82978i q^{73} +4.05137i q^{75} -4.88409i q^{77} +4.24575 q^{79} +38.5223 q^{81} -4.65004 q^{83} +13.9821 q^{85} +10.8330 q^{87} -9.17987i q^{89} +(-2.84207 + 2.21870i) q^{91} -8.65371 q^{93} +13.6780 q^{95} +13.3230i q^{97} -41.7832 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 84 q - 84 q^{9} + 8 q^{17} + 24 q^{23} + 92 q^{25} + 24 q^{39} - 84 q^{49} - 32 q^{55} - 24 q^{65} + 40 q^{79} + 84 q^{81} + 48 q^{87} + 16 q^{95}+O(q^{100}) $ 84 * q - 84 * q^9 + 8 * q^17 + 24 * q^23 + 92 * q^25 + 24 * q^39 - 84 * q^49 - 32 * q^55 - 24 * q^65 + 40 * q^79 + 84 * q^81 + 48 * q^87 + 16 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2912\mathbb{Z}\right)^\times$.

$n$	$1093$	$1249$	$2017$	$2367$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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
$3$			3.39926i			1.96256i	0.192581	+	0.981281i	$0.438314\pi$
$3$			3.39926i			1.96256i	−0.192581	+	0.981281i	$0.561686\pi$
$4$		0			0
$5$	2.48834			1.11282			0.556410	−	0.830908i	$-0.312178\pi$
$5$	2.48834			1.11282			0.556410	+	0.830908i	$0.312178\pi$
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$	−8.55495			−2.85165
$10$		0			0
$11$	4.88409			1.47261			0.736304	−	0.676650i	$-0.236569\pi$
$11$	4.88409			1.47261			0.736304	+	0.676650i	$0.236569\pi$
$12$		0			0
$13$	−2.21870	−	2.84207i	−0.615356	−	0.788249i
$13$	−2.21870	−	2.84207i	−0.615356	−	0.788249i
$14$		0			0
$15$			8.45851i			2.18398i
$16$		0			0
$17$	5.61904			1.36282			0.681408	−	0.731904i	$-0.261368\pi$
$17$	5.61904			1.36282			0.681408	+	0.731904i	$0.261368\pi$
$18$		0			0
$19$	5.49683			1.26106			0.630530	−	0.776165i	$-0.282838\pi$
$19$	5.49683			1.26106			0.630530	+	0.776165i	$0.282838\pi$
$20$		0			0
$21$	3.39926			0.741779
$22$		0			0
$23$	5.27964			1.10088			0.550441	−	0.834874i	$-0.314460\pi$
$23$	5.27964			1.10088			0.550441	+	0.834874i	$0.314460\pi$
$24$		0			0
$25$	1.19184			0.238368
$26$		0			0
$27$		−	18.8827i		−	3.63398i
$28$		0			0
$29$		−	3.18688i		−	0.591789i	−0.955221	−	0.295894i	$-0.904382\pi$
$29$		−	3.18688i		−	0.591789i	0.955221	−	0.295894i	$-0.0956176\pi$
$30$		0			0
$31$			2.54576i			0.457233i	0.973517	+	0.228616i	$0.0734201\pi$
$31$			2.54576i			0.457233i	−0.973517	+	0.228616i	$0.926580\pi$
$32$		0			0
$33$			16.6023i			2.89009i
$34$		0			0
$35$		−	2.48834i		−	0.420606i
$36$		0			0
$37$	3.21861			0.529137			0.264568	−	0.964367i	$-0.414770\pi$
$37$	3.21861			0.529137			0.264568	+	0.964367i	$0.414770\pi$
$38$		0			0
$39$	9.66094	−	7.54192i	1.54699	−	1.20767i
$40$		0			0
$41$		−	4.30478i		−	0.672294i	−0.941809	−	0.336147i	$-0.890876\pi$
$41$		−	4.30478i		−	0.672294i	0.941809	−	0.336147i	$-0.109124\pi$
$42$		0			0
$43$			7.45874i			1.13745i	0.822529	+	0.568724i	$0.192563\pi$
$43$			7.45874i			1.13745i	−0.822529	+	0.568724i	$0.807437\pi$
$44$		0			0
$45$	−21.2876			−3.17337
$46$		0			0
$47$		−	0.0608216i		−	0.00887175i	−0.999990	−	0.00443587i	$-0.998588\pi$
$47$		−	0.0608216i		−	0.00887175i	0.999990	−	0.00443587i	$-0.00141199\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$			19.1006i			2.67461i
$52$		0			0
$53$		−	2.00045i		−	0.274782i	−0.990517	−	0.137391i	$-0.956128\pi$
$53$		−	2.00045i		−	0.274782i	0.990517	−	0.137391i	$-0.0438717\pi$
$54$		0			0
$55$	12.1533			1.63875
$56$		0			0
$57$			18.6851i			2.47491i
$58$		0			0
$59$	−0.469731			−0.0611538			−0.0305769	−	0.999532i	$-0.509734\pi$
$59$	−0.469731			−0.0611538			−0.0305769	+	0.999532i	$0.509734\pi$
$60$		0			0
$61$			2.65509i			0.339950i	0.985448	+	0.169975i	$0.0543686\pi$
$61$			2.65509i			0.339950i	−0.985448	+	0.169975i	$0.945631\pi$
$62$		0			0
$63$			8.55495i			1.07782i
$64$		0			0
$65$	−5.52087	−	7.07205i	−0.684780	−	0.877179i
$66$		0			0
$67$	1.22805			0.150030			0.0750150	−	0.997182i	$-0.476100\pi$
$67$	1.22805			0.150030			0.0750150	+	0.997182i	$0.476100\pi$
$68$		0			0
$69$			17.9469i			2.16055i
$70$		0			0
$71$			4.67427i			0.554734i	0.960764	+	0.277367i	$0.0894618\pi$
$71$			4.67427i			0.554734i	−0.960764	+	0.277367i	$0.910538\pi$
$72$		0			0
$73$			4.82978i			0.565283i	0.959226	+	0.282642i	$0.0912107\pi$
$73$			4.82978i			0.565283i	−0.959226	+	0.282642i	$0.908789\pi$
$74$		0			0
$75$			4.05137i			0.467812i
$76$		0			0
$77$		−	4.88409i		−	0.556594i
$78$		0			0
$79$	4.24575			0.477684			0.238842	−	0.971058i	$-0.423232\pi$
$79$	4.24575			0.477684			0.238842	+	0.971058i	$0.423232\pi$
$80$		0			0
$81$	38.5223			4.28026
$82$		0			0
$83$	−4.65004			−0.510408			−0.255204	−	0.966887i	$-0.582143\pi$
$83$	−4.65004			−0.510408			−0.255204	+	0.966887i	$0.582143\pi$
$84$		0			0
$85$	13.9821			1.51657
$86$		0			0
$87$	10.8330			1.16142
$88$		0			0
$89$		−	9.17987i		−	0.973065i	−0.873663	−	0.486532i	$-0.838262\pi$
$89$		−	9.17987i		−	0.973065i	0.873663	−	0.486532i	$-0.161738\pi$
$90$		0			0
$91$	−2.84207	+	2.21870i	−0.297930	+	0.232583i
$92$		0			0
$93$	−8.65371			−0.897348
$94$		0			0
$95$	13.6780			1.40333
$96$		0			0
$97$			13.3230i			1.35275i	0.736558	+	0.676374i	$0.236450\pi$
$97$			13.3230i			1.35275i	−0.736558	+	0.676374i	$0.763550\pi$
$98$		0			0
$99$	−41.7832			−4.19937
$100$		0			0
$101$		−	4.36260i		−	0.434095i	−0.976161	−	0.217047i	$-0.930357\pi$
$101$		−	4.36260i		−	0.434095i	0.976161	−	0.217047i	$-0.0696426\pi$
$102$		0			0
$103$	−18.1093			−1.78437			−0.892183	−	0.451674i	$-0.850827\pi$
$103$	−18.1093			−1.78437			−0.892183	+	0.451674i	$0.850827\pi$
$104$		0			0
$105$	8.45851			0.825466
$106$		0			0
$107$		−	2.62320i		−	0.253594i	−0.991929	−	0.126797i	$-0.959530\pi$
$107$		−	2.62320i		−	0.253594i	0.991929	−	0.126797i	$-0.0404697\pi$
$108$		0			0
$109$	−0.526525			−0.0504320			−0.0252160	−	0.999682i	$-0.508027\pi$
$109$	−0.526525			−0.0504320			−0.0252160	+	0.999682i	$0.508027\pi$
$110$		0			0
$111$			10.9409i			1.03846i
$112$		0			0
$113$	−2.26619			−0.213185			−0.106593	−	0.994303i	$-0.533994\pi$
$113$	−2.26619			−0.213185			−0.106593	+	0.994303i	$0.533994\pi$
$114$		0			0
$115$	13.1375			1.22508
$116$		0			0
$117$	18.9808	+	24.3138i	1.75478	+	2.24781i
$118$		0			0
$119$		−	5.61904i		−	0.515096i
$120$		0			0
$121$	12.8543			1.16858
$122$		0			0
$123$	14.6331			1.31942
$124$		0			0
$125$	−9.47600			−0.847559
$126$		0			0
$127$	4.76216			0.422573			0.211287	−	0.977424i	$-0.432235\pi$
$127$	4.76216			0.422573			0.211287	+	0.977424i	$0.432235\pi$
$128$		0			0
$129$	−25.3542			−2.23231
$130$		0			0
$131$			1.91457i			0.167277i	0.996496	+	0.0836384i	$0.0266541\pi$
$131$			1.91457i			0.167277i	−0.996496	+	0.0836384i	$0.973346\pi$
$132$		0			0
$133$		−	5.49683i		−	0.476636i
$134$		0			0
$135$		−	46.9866i		−	4.04396i
$136$		0			0
$137$		−	13.8145i		−	1.18025i	−0.807311	−	0.590127i	$-0.799078\pi$
$137$		−	13.8145i		−	1.18025i	0.807311	−	0.590127i	$-0.200922\pi$
$138$		0			0
$139$			7.88775i			0.669030i	0.942390	+	0.334515i	$0.108573\pi$
$139$			7.88775i			0.669030i	−0.942390	+	0.334515i	$0.891427\pi$
$140$		0			0
$141$	0.206748			0.0174114
$142$		0			0
$143$	−10.8363	−	13.8809i	−0.906179	−	1.16078i
$144$		0			0
$145$		−	7.93004i		−	0.658554i
$146$		0			0
$147$		−	3.39926i		−	0.280366i
$148$		0			0
$149$	0.0267996			0.00219551			0.00109776	−	0.999999i	$-0.499651\pi$
$149$	0.0267996			0.00219551			0.00109776	+	0.999999i	$0.499651\pi$
$150$		0			0
$151$		−	4.49589i		−	0.365871i	−0.983125	−	0.182935i	$-0.941440\pi$
$151$		−	4.49589i		−	0.365871i	0.983125	−	0.182935i	$-0.0585599\pi$
$152$		0			0
$153$	−48.0706			−3.88628
$154$		0			0
$155$			6.33473i			0.508818i
$156$		0			0
$157$		−	21.7850i		−	1.73863i	−0.494258	−	0.869315i	$-0.664560\pi$
$157$		−	21.7850i		−	1.73863i	0.494258	−	0.869315i	$-0.335440\pi$
$158$		0			0
$159$	6.80003			0.539277
$160$		0			0
$161$		−	5.27964i		−	0.416094i
$162$		0			0
$163$	−14.6923			−1.15079			−0.575393	−	0.817877i	$-0.695151\pi$
$163$	−14.6923			−1.15079			−0.575393	+	0.817877i	$0.695151\pi$
$164$		0			0
$165$			41.3121i			3.21615i
$166$		0			0
$167$			20.2389i			1.56613i	0.621938	+	0.783066i	$0.286345\pi$
$167$			20.2389i			1.56613i	−0.621938	+	0.783066i	$0.713655\pi$
$168$		0			0
$169$	−3.15476	+	12.6114i	−0.242674	+	0.970108i
$170$		0			0
$171$	−47.0251			−3.59610
$172$		0			0
$173$		−	20.8335i		−	1.58394i	−0.610559	−	0.791970i	$-0.709055\pi$
$173$		−	20.8335i		−	1.58394i	0.610559	−	0.791970i	$-0.290945\pi$
$174$		0			0
$175$		−	1.19184i		−	0.0900946i
$176$		0			0
$177$		−	1.59674i		−	0.120018i
$178$		0			0
$179$		−	6.49084i		−	0.485148i	−0.970133	−	0.242574i	$-0.922008\pi$
$179$		−	6.49084i		−	0.485148i	0.970133	−	0.242574i	$-0.0779918\pi$
$180$		0			0
$181$			8.85796i			0.658407i	0.944259	+	0.329204i	$0.106780\pi$
$181$			8.85796i			0.658407i	−0.944259	+	0.329204i	$0.893220\pi$
$182$		0			0
$183$	−9.02534			−0.667172
$184$		0			0
$185$	8.00901			0.588834
$186$		0			0
$187$	27.4439			2.00690
$188$		0			0
$189$	−18.8827			−1.37352
$190$		0			0
$191$	−4.33544			−0.313701			−0.156851	−	0.987622i	$-0.550134\pi$
$191$	−4.33544			−0.313701			−0.156851	+	0.987622i	$0.550134\pi$
$192$		0			0
$193$			9.14057i			0.657952i	0.944338	+	0.328976i	$0.106704\pi$
$193$			9.14057i			0.657952i	−0.944338	+	0.328976i	$0.893296\pi$
$194$		0			0
$195$	24.0397	−	18.7669i	1.72152	−	1.34392i
$196$		0			0
$197$	−21.5709			−1.53686			−0.768431	−	0.639933i	$-0.778962\pi$
$197$	−21.5709			−1.53686			−0.768431	+	0.639933i	$0.778962\pi$
$198$		0			0
$199$	−18.4078			−1.30490			−0.652448	−	0.757833i	$-0.726258\pi$
$199$	−18.4078			−1.30490			−0.652448	+	0.757833i	$0.726258\pi$
$200$		0			0
$201$			4.17445i			0.294443i
$202$		0			0
$203$	−3.18688			−0.223675
$204$		0			0
$205$		−	10.7118i		−	0.748142i
$206$		0			0
$207$	−45.1671			−3.13933
$208$		0			0
$209$	26.8470			1.85705
$210$		0			0
$211$		−	9.58863i		−	0.660109i	−0.943962	−	0.330054i	$-0.892933\pi$
$211$		−	9.58863i		−	0.660109i	0.943962	−	0.330054i	$-0.107067\pi$
$212$		0			0
$213$	−15.8890			−1.08870
$214$		0			0
$215$			18.5599i			1.26577i
$216$		0			0
$217$	2.54576			0.172818
$218$		0			0
$219$	−16.4177			−1.10940
$220$		0			0
$221$	−12.4669	−	15.9697i	−0.838617	−	1.07424i
$222$		0			0
$223$		−	0.955307i		−	0.0639721i	−0.999488	−	0.0319860i	$-0.989817\pi$
$223$		−	0.955307i		−	0.0639721i	0.999488	−	0.0319860i	$-0.0101832\pi$
$224$		0			0
$225$	−10.1961			−0.679742
$226$		0			0
$227$	1.00765			0.0668803			0.0334401	−	0.999441i	$-0.489354\pi$
$227$	1.00765			0.0668803			0.0334401	+	0.999441i	$0.489354\pi$
$228$		0			0
$229$	18.3736			1.21416			0.607082	−	0.794639i	$-0.292340\pi$
$229$	18.3736			1.21416			0.607082	+	0.794639i	$0.292340\pi$
$230$		0			0
$231$	16.6023			1.09235
$232$		0			0
$233$	23.8268			1.56095			0.780474	−	0.625188i	$-0.214978\pi$
$233$	23.8268			1.56095			0.780474	+	0.625188i	$0.214978\pi$
$234$		0			0
$235$		−	0.151345i		−	0.00987266i
$236$		0			0
$237$			14.4324i			0.937485i
$238$		0			0
$239$			1.27550i			0.0825052i	0.999149	+	0.0412526i	$0.0131348\pi$
$239$			1.27550i			0.0825052i	−0.999149	+	0.0412526i	$0.986865\pi$
$240$		0			0
$241$			8.67754i			0.558970i	0.960150	+	0.279485i	$0.0901637\pi$
$241$			8.67754i			0.558970i	−0.960150	+	0.279485i	$0.909836\pi$
$242$		0			0
$243$			74.2992i			4.76630i
$244$		0			0
$245$	−2.48834			−0.158974
$246$		0			0
$247$	−12.1958	−	15.6224i	−0.776001	−	0.994030i
$248$		0			0
$249$		−	15.8067i		−	1.00171i
$250$		0			0
$251$			21.5737i			1.36172i	0.732415	+	0.680859i	$0.238393\pi$
$251$			21.5737i			1.36172i	−0.732415	+	0.680859i	$0.761607\pi$
$252$		0			0
$253$	25.7863			1.62117
$254$		0			0
$255$			47.5287i			2.97636i
$256$		0			0
$257$	15.2113			0.948858			0.474429	−	0.880294i	$-0.342655\pi$
$257$	15.2113			0.948858			0.474429	+	0.880294i	$0.342655\pi$
$258$		0			0
$259$		−	3.21861i		−	0.199995i
$260$		0			0
$261$			27.2636i			1.68757i
$262$		0			0
$263$	12.2626			0.756146			0.378073	−	0.925776i	$-0.376587\pi$
$263$	12.2626			0.756146			0.378073	+	0.925776i	$0.376587\pi$
$264$		0			0
$265$		−	4.97779i		−	0.305783i
$266$		0			0
$267$	31.2047			1.90970
$268$		0			0
$269$			32.2178i			1.96436i	0.187951	+	0.982178i	$0.439815\pi$
$269$			32.2178i			1.96436i	−0.187951	+	0.982178i	$0.560185\pi$
$270$		0			0
$271$			18.0540i			1.09670i	0.836248	+	0.548351i	$0.184744\pi$
$271$			18.0540i			1.09670i	−0.836248	+	0.548351i	$0.815256\pi$
$272$		0			0
$273$	−7.54192	−	9.66094i	−0.456458	−	0.584707i
$274$		0			0
$275$	5.82105			0.351023
$276$		0			0
$277$		−	21.9458i		−	1.31860i	−0.751881	−	0.659299i	$-0.770853\pi$
$277$		−	21.9458i		−	1.31860i	0.751881	−	0.659299i	$-0.229147\pi$
$278$		0			0
$279$		−	21.7789i		−	1.30387i
$280$		0			0
$281$			4.51449i			0.269312i	0.990892	+	0.134656i	$0.0429929\pi$
$281$			4.51449i			0.269312i	−0.990892	+	0.134656i	$0.957007\pi$
$282$		0			0
$283$			4.07681i			0.242341i	0.992632	+	0.121171i	$0.0386648\pi$
$283$			4.07681i			0.242341i	−0.992632	+	0.121171i	$0.961335\pi$
$284$		0			0
$285$			46.4950i			2.75413i
$286$		0			0
$287$	−4.30478			−0.254103
$288$		0			0
$289$	14.5736			0.857269
$290$		0			0
$291$	−45.2884			−2.65485
$292$		0			0
$293$	−28.4951			−1.66470			−0.832352	−	0.554247i	$-0.813006\pi$
$293$	−28.4951			−1.66470			−0.832352	+	0.554247i	$0.813006\pi$
$294$		0			0
$295$	−1.16885			−0.0680532
$296$		0			0
$297$		−	92.2249i		−	5.35143i
$298$		0			0
$299$	−11.7139	−	15.0051i	−0.677434	−	0.867769i
$300$		0			0
$301$	7.45874			0.429915
$302$		0			0
$303$	14.8296			0.851938
$304$		0			0
$305$			6.60677i			0.378303i
$306$		0			0
$307$	16.6161			0.948329			0.474165	−	0.880436i	$-0.342750\pi$
$307$	16.6161			0.948329			0.474165	+	0.880436i	$0.342750\pi$
$308$		0			0
$309$		−	61.5583i		−	3.50193i
$310$		0			0
$311$	20.3578			1.15439			0.577194	−	0.816607i	$-0.304148\pi$
$311$	20.3578			1.15439			0.577194	+	0.816607i	$0.304148\pi$
$312$		0			0
$313$	−23.6818			−1.33857			−0.669287	−	0.743004i	$-0.733400\pi$
$313$	−23.6818			−1.33857			−0.669287	+	0.743004i	$0.733400\pi$
$314$		0			0
$315$			21.2876i			1.19942i
$316$		0			0
$317$	−12.5799			−0.706556			−0.353278	−	0.935518i	$-0.614933\pi$
$317$	−12.5799			−0.706556			−0.353278	+	0.935518i	$0.614933\pi$
$318$		0			0
$319$		−	15.5650i		−	0.871473i
$320$		0			0
$321$	8.91694			0.497695
$322$		0			0
$323$	30.8869			1.71859
$324$		0			0
$325$	−2.64433	−	3.38729i	−0.146681	−	0.187893i
$326$		0			0
$327$		−	1.78980i		−	0.0989759i
$328$		0			0
$329$	−0.0608216			−0.00335321
$330$		0			0
$331$	−27.4135			−1.50678			−0.753391	−	0.657573i	$-0.771583\pi$
$331$	−27.4135			−1.50678			−0.753391	+	0.657573i	$0.771583\pi$
$332$		0			0
$333$	−27.5351			−1.50891
$334$		0			0
$335$	3.05580			0.166956
$336$		0			0
$337$	−23.9384			−1.30401			−0.652005	−	0.758215i	$-0.726072\pi$
$337$	−23.9384			−1.30401			−0.652005	+	0.758215i	$0.726072\pi$
$338$		0			0
$339$		−	7.70336i		−	0.418389i
$340$		0			0
$341$			12.4337i			0.673325i
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$			44.6579i			2.40430i
$346$		0			0
$347$		−	5.17536i		−	0.277828i	−0.990304	−	0.138914i	$-0.955639\pi$
$347$		−	5.17536i		−	0.277828i	0.990304	−	0.138914i	$-0.0443612\pi$
$348$		0			0
$349$	0.707224			0.0378568			0.0189284	−	0.999821i	$-0.493975\pi$
$349$	0.707224			0.0378568			0.0189284	+	0.999821i	$0.493975\pi$
$350$		0			0
$351$	−53.6661	+	41.8950i	−2.86448	+	2.23619i
$352$		0			0
$353$		−	13.1447i		−	0.699620i	−0.936821	−	0.349810i	$-0.886246\pi$
$353$		−	13.1447i		−	0.699620i	0.936821	−	0.349810i	$-0.113754\pi$
$354$		0			0
$355$			11.6312i			0.617319i
$356$		0			0
$357$	19.1006			1.01091
$358$		0			0
$359$			26.7771i			1.41324i	0.707594	+	0.706620i	$0.249781\pi$
$359$			26.7771i			1.41324i	−0.707594	+	0.706620i	$0.750219\pi$
$360$		0			0
$361$	11.2152			0.590272
$362$		0			0
$363$			43.6952i			2.29341i
$364$		0			0
$365$			12.0181i			0.629058i
$366$		0			0
$367$	3.74105			0.195281			0.0976406	−	0.995222i	$-0.468870\pi$
$367$	3.74105			0.195281			0.0976406	+	0.995222i	$0.468870\pi$
$368$		0			0
$369$			36.8272i			1.91715i
$370$		0			0
$371$	−2.00045			−0.103858
$372$		0			0
$373$			1.41745i			0.0733926i	0.999326	+	0.0366963i	$0.0116834\pi$
$373$			1.41745i			0.0733926i	−0.999326	+	0.0366963i	$0.988317\pi$
$374$		0			0
$375$		−	32.2114i		−	1.66339i
$376$		0			0
$377$	−9.05735	+	7.07072i	−0.466477	+	0.364161i
$378$		0			0
$379$	−22.6988			−1.16596			−0.582980	−	0.812487i	$-0.698113\pi$
$379$	−22.6988			−1.16596			−0.582980	+	0.812487i	$0.698113\pi$
$380$		0			0
$381$			16.1878i			0.829326i
$382$		0			0
$383$			12.5483i			0.641190i	0.947216	+	0.320595i	$0.103883\pi$
$383$			12.5483i			0.641190i	−0.947216	+	0.320595i	$0.896117\pi$
$384$		0			0
$385$		−	12.1533i		−	0.619389i
$386$		0			0
$387$		−	63.8092i		−	3.24360i
$388$		0			0
$389$		−	31.8205i		−	1.61336i	−0.590985	−	0.806682i	$-0.701261\pi$
$389$		−	31.8205i		−	1.61336i	0.590985	−	0.806682i	$-0.298739\pi$
$390$		0			0
$391$	29.6665			1.50030
$392$		0			0
$393$	−6.50811			−0.328291
$394$		0			0
$395$	10.5649			0.531576
$396$		0			0
$397$	−30.5830			−1.53491			−0.767457	−	0.641100i	$-0.778478\pi$
$397$	−30.5830			−1.53491			−0.767457	+	0.641100i	$0.778478\pi$
$398$		0			0
$399$	18.6851			0.935427
$400$		0			0
$401$			14.1076i			0.704500i	0.935906	+	0.352250i	$0.114583\pi$
$401$			14.1076i			0.704500i	−0.935906	+	0.352250i	$0.885417\pi$
$402$		0			0
$403$	7.23525	−	5.64828i	0.360413	−	0.281361i
$404$		0			0
$405$	95.8567			4.76316
$406$		0			0
$407$	15.7200			0.779212
$408$		0			0
$409$			18.0342i			0.891731i	0.895100	+	0.445866i	$0.147104\pi$
$409$			18.0342i			0.891731i	−0.895100	+	0.445866i	$0.852896\pi$
$410$		0			0
$411$	46.9591			2.31632
$412$		0			0
$413$			0.469731i			0.0231140i
$414$		0			0
$415$	−11.5709			−0.567992
$416$		0			0
$417$	−26.8125			−1.31301
$418$		0			0
$419$		−	11.8975i		−	0.581229i	−0.956840	−	0.290614i	$-0.906140\pi$
$419$		−	11.8975i		−	0.581229i	0.956840	−	0.290614i	$-0.0938597\pi$
$420$		0			0
$421$	−12.4939			−0.608915			−0.304457	−	0.952526i	$-0.598475\pi$
$421$	−12.4939			−0.608915			−0.304457	+	0.952526i	$0.598475\pi$
$422$		0			0
$423$			0.520326i			0.0252991i
$424$		0			0
$425$	6.69699			0.324852
$426$		0			0
$427$	2.65509			0.128489
$428$		0			0
$429$	47.1849	−	36.8354i	2.27811	−	1.77843i
$430$		0			0
$431$			30.9521i			1.49091i	0.666556	+	0.745455i	$0.267768\pi$
$431$			30.9521i			1.49091i	−0.666556	+	0.745455i	$0.732232\pi$
$432$		0			0
$433$	25.7199			1.23602			0.618010	−	0.786171i	$-0.287939\pi$
$433$	25.7199			1.23602			0.618010	+	0.786171i	$0.287939\pi$
$434$		0			0
$435$	26.9563			1.29245
$436$		0			0
$437$	29.0213			1.38828
$438$		0			0
$439$	−33.4146			−1.59479			−0.797395	−	0.603457i	$-0.793789\pi$
$439$	−33.4146			−1.59479			−0.797395	+	0.603457i	$0.793789\pi$
$440$		0			0
$441$	8.55495			0.407379
$442$		0			0
$443$		−	3.57603i		−	0.169902i	−0.996385	−	0.0849512i	$-0.972927\pi$
$443$		−	3.57603i		−	0.169902i	0.996385	−	0.0849512i	$-0.0270734\pi$
$444$		0			0
$445$		−	22.8426i		−	1.08285i
$446$		0			0
$447$			0.0910988i			0.00430883i
$448$		0			0
$449$		−	32.7121i		−	1.54378i	−0.635758	−	0.771889i	$-0.719312\pi$
$449$		−	32.7121i		−	1.54378i	0.635758	−	0.771889i	$-0.280688\pi$
$450$		0			0
$451$		−	21.0250i		−	0.990026i
$452$		0			0
$453$	15.2827			0.718044
$454$		0			0
$455$	−7.07205	+	5.52087i	−0.331543	+	0.258823i
$456$		0			0
$457$			3.22486i			0.150852i	0.997151	+	0.0754262i	$0.0240317\pi$
$457$			3.22486i			0.150852i	−0.997151	+	0.0754262i	$0.975968\pi$
$458$		0			0
$459$		−	106.103i		−	4.95245i
$460$		0			0
$461$	7.54106			0.351222			0.175611	−	0.984460i	$-0.443810\pi$
$461$	7.54106			0.351222			0.175611	+	0.984460i	$0.443810\pi$
$462$		0			0
$463$			11.3168i			0.525937i	0.964804	+	0.262969i	$0.0847016\pi$
$463$			11.3168i			0.525937i	−0.964804	+	0.262969i	$0.915298\pi$
$464$		0			0
$465$	−21.5334			−0.998586
$466$		0			0
$467$		−	29.2127i		−	1.35180i	−0.736992	−	0.675901i	$-0.763755\pi$
$467$		−	29.2127i		−	1.35180i	0.736992	−	0.675901i	$-0.236245\pi$
$468$		0			0
$469$		−	1.22805i		−	0.0567060i
$470$		0			0
$471$	74.0528			3.41217
$472$		0			0
$473$			36.4292i			1.67501i
$474$		0			0
$475$	6.55134			0.300596
$476$		0			0
$477$			17.1137i			0.783583i
$478$		0			0
$479$		−	11.3774i		−	0.519845i	−0.965630	−	0.259922i	$-0.916303\pi$
$479$		−	11.3774i		−	0.519845i	0.965630	−	0.259922i	$-0.0836970\pi$
$480$		0			0
$481$	−7.14113	−	9.14754i	−0.325608	−	0.417092i
$482$		0			0
$483$	17.9469			0.816611
$484$		0			0
$485$			33.1522i			1.50536i
$486$		0			0
$487$		−	0.437374i		−	0.0198193i	−0.999951	−	0.00990966i	$-0.996846\pi$
$487$		−	0.437374i		−	0.0198193i	0.999951	−	0.00990966i	$-0.00315440\pi$
$488$		0			0
$489$		−	49.9428i		−	2.25849i
$490$		0			0
$491$		−	10.5045i		−	0.474061i	−0.971502	−	0.237031i	$-0.923826\pi$
$491$		−	10.5045i		−	0.474061i	0.971502	−	0.237031i	$-0.0761741\pi$
$492$		0			0
$493$		−	17.9072i		−	0.806499i
$494$		0			0
$495$	−103.971			−4.67314
$496$		0			0
$497$	4.67427			0.209670
$498$		0			0
$499$	44.4934			1.99180			0.995899	−	0.0904684i	$-0.0288364\pi$
$499$	44.4934			1.99180			0.995899	+	0.0904684i	$0.0288364\pi$
$500$		0			0
$501$	−68.7972			−3.07363
$502$		0			0
$503$	0.704814			0.0314261			0.0157130	−	0.999877i	$-0.494998\pi$
$503$	0.704814			0.0314261			0.0157130	+	0.999877i	$0.494998\pi$
$504$		0			0
$505$		−	10.8556i		−	0.483069i
$506$		0			0
$507$	−42.8694	−	10.7239i	−1.90390	−	0.476263i
$508$		0			0
$509$	38.6299			1.71224			0.856119	−	0.516779i	$-0.172869\pi$
$509$	38.6299			1.71224			0.856119	+	0.516779i	$0.172869\pi$
$510$		0			0
$511$	4.82978			0.213657
$512$		0			0
$513$		−	103.795i		−	4.58267i
$514$		0			0
$515$	−45.0622			−1.98568
$516$		0			0
$517$		−	0.297058i		−	0.0130646i
$518$		0			0
$519$	70.8184			3.10858
$520$		0			0
$521$	3.38542			0.148318			0.0741589	−	0.997246i	$-0.476373\pi$
$521$	3.38542			0.148318			0.0741589	+	0.997246i	$0.476373\pi$
$522$		0			0
$523$		−	24.9139i		−	1.08941i	−0.838627	−	0.544705i	$-0.816641\pi$
$523$		−	24.9139i		−	1.08941i	0.838627	−	0.544705i	$-0.183359\pi$
$524$		0			0
$525$	4.05137			0.176816
$526$		0			0
$527$			14.3047i			0.623124i
$528$		0			0
$529$	4.87463			0.211940
$530$		0			0
$531$	4.01853			0.174389
$532$		0			0
$533$	−12.2345	+	9.55101i	−0.529935	+	0.413700i
$534$		0			0
$535$		−	6.52742i		−	0.282205i
$536$		0			0
$537$	22.0640			0.952134
$538$		0			0
$539$	−4.88409			−0.210373
$540$		0			0
$541$	16.3916			0.704728			0.352364	−	0.935863i	$-0.385378\pi$
$541$	16.3916			0.704728			0.352364	+	0.935863i	$0.385378\pi$
$542$		0			0
$543$	−30.1105			−1.29217
$544$		0			0
$545$	−1.31017			−0.0561217
$546$		0			0
$547$		−	38.7606i		−	1.65728i	−0.559779	−	0.828642i	$-0.689114\pi$
$547$		−	38.7606i		−	1.65728i	0.559779	−	0.828642i	$-0.310886\pi$
$548$		0			0
$549$		−	22.7142i		−	0.969417i
$550$		0			0
$551$		−	17.5177i		−	0.746281i
$552$		0			0
$553$		−	4.24575i		−	0.180548i
$554$		0			0
$555$			27.2247i			1.15562i
$556$		0			0
$557$	34.2544			1.45141			0.725704	−	0.688007i	$-0.241514\pi$
$557$	34.2544			1.45141			0.725704	+	0.688007i	$0.241514\pi$
$558$		0			0
$559$	21.1983	−	16.5487i	0.896592	−	0.699935i
$560$		0			0
$561$			93.2888i			3.93866i
$562$		0			0
$563$			4.27510i			0.180174i	0.995934	+	0.0900870i	$0.0287145\pi$
$563$			4.27510i			0.180174i	−0.995934	+	0.0900870i	$0.971285\pi$
$564$		0			0
$565$	−5.63905			−0.237237
$566$		0			0
$567$		−	38.5223i		−	1.61779i
$568$		0			0
$569$	−3.19970			−0.134138			−0.0670692	−	0.997748i	$-0.521365\pi$
$569$	−3.19970			−0.134138			−0.0670692	+	0.997748i	$0.521365\pi$
$570$		0			0
$571$		−	0.311853i		−	0.0130506i	−0.999979	−	0.00652532i	$-0.997923\pi$
$571$		−	0.311853i		−	0.0130506i	0.999979	−	0.00652532i	$-0.00207709\pi$
$572$		0			0
$573$		−	14.7373i		−	0.615658i
$574$		0			0
$575$	6.29248			0.262415
$576$		0			0
$577$			13.0169i			0.541902i	0.962593	+	0.270951i	$0.0873382\pi$
$577$			13.0169i			0.541902i	−0.962593	+	0.270951i	$0.912662\pi$
$578$		0			0
$579$	−31.0711			−1.29127
$580$		0			0
$581$			4.65004i			0.192916i
$582$		0			0
$583$		−	9.77036i		−	0.404647i
$584$		0			0
$585$	47.2308	+	60.5010i	1.95275	+	2.50141i
$586$		0			0
$587$	−4.46455			−0.184272			−0.0921358	−	0.995746i	$-0.529369\pi$
$587$	−4.46455			−0.184272			−0.0921358	+	0.995746i	$0.529369\pi$
$588$		0			0
$589$			13.9936i			0.576598i
$590$		0			0
$591$		−	73.3250i		−	3.01619i
$592$		0			0
$593$			40.5531i			1.66532i	0.553787	+	0.832658i	$0.313182\pi$
$593$			40.5531i			1.66532i	−0.553787	+	0.832658i	$0.686818\pi$
$594$		0			0
$595$		−	13.9821i		−	0.573209i
$596$		0			0
$597$		−	62.5730i		−	2.56094i
$598$		0			0
$599$	−28.6311			−1.16984			−0.584918	−	0.811093i	$-0.698873\pi$
$599$	−28.6311			−1.16984			−0.584918	+	0.811093i	$0.698873\pi$
$600$		0			0
$601$	−23.9824			−0.978263			−0.489132	−	0.872210i	$-0.662686\pi$
$601$	−23.9824			−0.978263			−0.489132	+	0.872210i	$0.662686\pi$
$602$		0			0
$603$	−10.5059			−0.427833
$604$		0			0
$605$	31.9860			1.30042
$606$		0			0
$607$	19.2320			0.780604			0.390302	−	0.920687i	$-0.372371\pi$
$607$	19.2320			0.780604			0.390302	+	0.920687i	$0.372371\pi$
$608$		0			0
$609$		−	10.8330i		−	0.438976i
$610$		0			0
$611$	−0.172860	+	0.134945i	−0.00699315	+	0.00545928i
$612$		0			0
$613$	−12.8640			−0.519571			−0.259786	−	0.965666i	$-0.583652\pi$
$613$	−12.8640			−0.519571			−0.259786	+	0.965666i	$0.583652\pi$
$614$		0			0
$615$	36.4121			1.46828
$616$		0			0
$617$			5.54831i			0.223367i	0.993744	+	0.111683i	$0.0356242\pi$
$617$			5.54831i			0.223367i	−0.993744	+	0.111683i	$0.964376\pi$
$618$		0			0
$619$	47.5803			1.91242			0.956208	−	0.292690i	$-0.0945502\pi$
$619$	47.5803			1.91242			0.956208	+	0.292690i	$0.0945502\pi$
$620$		0			0
$621$		−	99.6940i		−	4.00058i
$622$		0			0
$623$	−9.17987			−0.367784
$624$		0			0
$625$	−29.5387			−1.18155
$626$		0			0
$627$			91.2600i			3.64457i
$628$		0			0
$629$	18.0855			0.721117
$630$		0			0
$631$		−	40.7801i		−	1.62343i	−0.584053	−	0.811715i	$-0.698534\pi$
$631$		−	40.7801i		−	1.62343i	0.584053	−	0.811715i	$-0.301466\pi$
$632$		0			0
$633$	32.5942			1.29550
$634$		0			0
$635$	11.8499			0.470248
$636$		0			0
$637$	2.21870	+	2.84207i	0.0879080	+	0.112607i
$638$		0			0
$639$		−	39.9882i		−	1.58191i
$640$		0			0
$641$	28.1404			1.11148			0.555740	−	0.831356i	$-0.312435\pi$
$641$	28.1404			1.11148			0.555740	+	0.831356i	$0.312435\pi$
$642$		0			0
$643$	16.4631			0.649240			0.324620	−	0.945845i	$-0.394764\pi$
$643$	16.4631			0.649240			0.324620	+	0.945845i	$0.394764\pi$
$644$		0			0
$645$	−63.0898			−2.48416
$646$		0			0
$647$	−8.16848			−0.321136			−0.160568	−	0.987025i	$-0.551333\pi$
$647$	−8.16848			−0.321136			−0.160568	+	0.987025i	$0.551333\pi$
$648$		0			0
$649$	−2.29421			−0.0900557
$650$		0			0
$651$			8.65371i			0.339166i
$652$		0			0
$653$			20.4490i			0.800231i	0.916465	+	0.400115i	$0.131030\pi$
$653$			20.4490i			0.800231i	−0.916465	+	0.400115i	$0.868970\pi$
$654$		0			0
$655$			4.76410i			0.186149i
$656$		0			0
$657$		−	41.3186i		−	1.61199i
$658$		0			0
$659$			8.24531i			0.321192i	0.987020	+	0.160596i	$0.0513415\pi$
$659$			8.24531i			0.321192i	−0.987020	+	0.160596i	$0.948658\pi$
$660$		0			0
$661$	−29.5133			−1.14793			−0.573967	−	0.818878i	$-0.694596\pi$
$661$	−29.5133			−1.14793			−0.573967	+	0.818878i	$0.694596\pi$
$662$		0			0
$663$	54.2852	−	42.3783i	2.10826	−	1.64584i
$664$		0			0
$665$		−	13.6780i		−	0.530410i
$666$		0			0
$667$		−	16.8256i		−	0.651489i
$668$		0			0
$669$	3.24733			0.125549
$670$		0			0
$671$			12.9677i			0.500613i
$672$		0			0
$673$	29.8930			1.15229			0.576145	−	0.817348i	$-0.304556\pi$
$673$	29.8930			1.15229			0.576145	+	0.817348i	$0.304556\pi$
$674$		0			0
$675$		−	22.5052i		−	0.866224i
$676$		0			0
$677$			4.73631i			0.182031i	0.995849	+	0.0910156i	$0.0290113\pi$
$677$			4.73631i			0.182031i	−0.995849	+	0.0910156i	$0.970989\pi$
$678$		0			0
$679$	13.3230			0.511291
$680$		0			0
$681$			3.42527i			0.131257i
$682$		0			0
$683$	15.3781			0.588426			0.294213	−	0.955740i	$-0.404942\pi$
$683$	15.3781			0.588426			0.294213	+	0.955740i	$0.404942\pi$
$684$		0			0
$685$		−	34.3752i		−	1.31341i
$686$		0			0
$687$			62.4567i			2.38287i
$688$		0			0
$689$	−5.68541	+	4.43838i	−0.216597	+	0.169089i
$690$		0			0
$691$	36.6063			1.39257			0.696285	−	0.717766i	$-0.254835\pi$
$691$	36.6063			1.39257			0.696285	+	0.717766i	$0.254835\pi$
$692$		0			0
$693$			41.7832i			1.58721i
$694$		0			0
$695$			19.6274i			0.744510i
$696$		0			0
$697$		−	24.1887i		−	0.916214i
$698$		0			0
$699$			80.9936i			3.06346i
$700$		0			0
$701$		−	30.7609i		−	1.16182i	−0.813966	−	0.580912i	$-0.802696\pi$
$701$		−	30.7609i		−	1.16182i	0.813966	−	0.580912i	$-0.197304\pi$
$702$		0			0
$703$	17.6922			0.667273
$704$		0			0
$705$	0.514460			0.0193757
$706$		0			0
$707$	−4.36260			−0.164072
$708$		0			0
$709$	31.7436			1.19216			0.596079	−	0.802926i	$-0.296725\pi$
$709$	31.7436			1.19216			0.596079	+	0.802926i	$0.296725\pi$
$710$		0			0
$711$	−36.3222			−1.36219
$712$		0			0
$713$			13.4407i			0.503359i
$714$		0			0
$715$	−26.9645	−	34.5405i	−1.00841	−	1.29174i
$716$		0			0
$717$	−4.33575			−0.161922
$718$		0			0
$719$	−16.7479			−0.624591			−0.312296	−	0.949985i	$-0.601098\pi$
$719$	−16.7479			−0.624591			−0.312296	+	0.949985i	$0.601098\pi$
$720$		0			0
$721$			18.1093i			0.674427i
$722$		0			0
$723$	−29.4972			−1.09701
$724$		0			0
$725$		−	3.79825i		−	0.141063i
$726$		0			0
$727$	−34.5134			−1.28003			−0.640016	−	0.768362i	$-0.721072\pi$
$727$	−34.5134			−1.28003			−0.640016	+	0.768362i	$0.721072\pi$
$728$		0			0
$729$	−136.995			−5.07390
$730$		0			0
$731$			41.9109i			1.55013i
$732$		0			0
$733$	−24.9136			−0.920206			−0.460103	−	0.887865i	$-0.652188\pi$
$733$	−24.9136			−0.920206			−0.460103	+	0.887865i	$0.652188\pi$
$734$		0			0
$735$		−	8.45851i		−	0.311997i
$736$		0			0
$737$	5.99790			0.220935
$738$		0			0
$739$	−19.1504			−0.704458			−0.352229	−	0.935914i	$-0.614576\pi$
$739$	−19.1504			−0.704458			−0.352229	+	0.935914i	$0.614576\pi$
$740$		0			0
$741$	53.1046	−	41.4567i	1.95084	−	1.52295i
$742$		0			0
$743$			32.5014i			1.19236i	0.802851	+	0.596180i	$0.203315\pi$
$743$			32.5014i			1.19236i	−0.802851	+	0.596180i	$0.796685\pi$
$744$		0			0
$745$	0.0666866			0.00244321
$746$		0			0
$747$	39.7809			1.45551
$748$		0			0
$749$	−2.62320			−0.0958497
$750$		0			0
$751$	11.7010			0.426977			0.213489	−	0.976946i	$-0.431517\pi$
$751$	11.7010			0.426977			0.213489	+	0.976946i	$0.431517\pi$
$752$		0			0
$753$	−73.3344			−2.67246
$754$		0			0
$755$		−	11.1873i		−	0.407148i
$756$		0			0
$757$			45.3763i			1.64923i	0.565694	+	0.824615i	$0.308608\pi$
$757$			45.3763i			1.64923i	−0.565694	+	0.824615i	$0.691392\pi$
$758$		0			0
$759$			87.6541i			3.18164i
$760$		0			0
$761$		−	6.62817i		−	0.240271i	−0.992758	−	0.120135i	$-0.961667\pi$
$761$		−	6.62817i		−	0.240271i	0.992758	−	0.120135i	$-0.0383329\pi$
$762$		0			0
$763$			0.526525i			0.0190615i
$764$		0			0
$765$	−119.616			−4.32473
$766$		0			0
$767$	1.04219	+	1.33501i	0.0376314	+	0.0482045i
$768$		0			0
$769$			33.5163i			1.20863i	0.796746	+	0.604314i	$0.206553\pi$
$769$			33.5163i			1.20863i	−0.796746	+	0.604314i	$0.793447\pi$
$770$		0			0
$771$			51.7073i			1.86219i
$772$		0			0
$773$	−4.22323			−0.151899			−0.0759496	−	0.997112i	$-0.524199\pi$
$773$	−4.22323			−0.151899			−0.0759496	+	0.997112i	$0.524199\pi$
$774$		0			0
$775$			3.03414i			0.108990i
$776$		0			0
$777$	10.9409			0.392503
$778$		0			0
$779$		−	23.6627i		−	0.847803i
$780$		0			0
$781$			22.8296i			0.816906i
$782$		0			0
$783$	−60.1769			−2.15055
$784$		0			0
$785$		−	54.2085i		−	1.93478i
$786$		0			0
$787$	16.4820			0.587519			0.293759	−	0.955879i	$-0.405094\pi$
$787$	16.4820			0.587519			0.293759	+	0.955879i	$0.405094\pi$
$788$		0			0
$789$			41.6838i			1.48398i
$790$		0			0
$791$			2.26619i			0.0805764i
$792$		0			0
$793$	7.54596	−	5.89084i	0.267965	−	0.209190i
$794$		0			0
$795$	16.9208			0.600118
$796$		0			0
$797$		−	21.4662i		−	0.760371i	−0.924910	−	0.380185i	$-0.875860\pi$
$797$		−	21.4662i		−	0.760371i	0.924910	−	0.380185i	$-0.124140\pi$
$798$		0			0
$799$		−	0.341759i		−	0.0120906i
$800$		0			0
$801$			78.5334i			2.77484i
$802$		0			0
$803$			23.5891i			0.832441i
$804$		0			0
$805$		−	13.1375i		−	0.463038i
$806$		0			0
$807$	−109.517			−3.85517
$808$		0			0
$809$	26.7118			0.939137			0.469569	−	0.882896i	$-0.344409\pi$
$809$	26.7118			0.939137			0.469569	+	0.882896i	$0.344409\pi$
$810$		0			0
$811$	−28.1149			−0.987247			−0.493624	−	0.869676i	$-0.664328\pi$
$811$	−28.1149			−0.987247			−0.493624	+	0.869676i	$0.664328\pi$
$812$		0			0
$813$	−61.3702			−2.15235
$814$		0			0
$815$	−36.5593			−1.28062
$816$		0			0
$817$			40.9994i			1.43439i
$818$		0			0
$819$	24.3138	−	18.9808i	0.849593	−	0.663245i
$820$		0			0
$821$	16.9544			0.591713			0.295856	−	0.955232i	$-0.404395\pi$
$821$	16.9544			0.591713			0.295856	+	0.955232i	$0.404395\pi$
$822$		0			0
$823$	−27.1070			−0.944892			−0.472446	−	0.881360i	$-0.656629\pi$
$823$	−27.1070			−0.944892			−0.472446	+	0.881360i	$0.656629\pi$
$824$		0			0
$825$			19.7872i			0.688904i
$826$		0			0
$827$	−55.9469			−1.94547			−0.972733	−	0.231928i	$-0.925497\pi$
$827$	−55.9469			−1.94547			−0.972733	+	0.231928i	$0.925497\pi$
$828$		0			0
$829$		−	8.15802i		−	0.283340i	−0.989914	−	0.141670i	$-0.954753\pi$
$829$		−	8.15802i		−	0.283340i	0.989914	−	0.141670i	$-0.0452471\pi$
$830$		0			0
$831$	74.5995			2.58783
$832$		0			0
$833$	−5.61904			−0.194688
$834$		0			0
$835$			50.3613i			1.74282i
$836$		0			0
$837$	48.0709			1.66157
$838$		0			0
$839$		−	12.7688i		−	0.440827i	−0.975407	−	0.220413i	$-0.929259\pi$
$839$		−	12.7688i		−	0.440827i	0.975407	−	0.220413i	$-0.0707406\pi$
$840$		0			0
$841$	18.8438			0.649786
$842$		0			0
$843$	−15.3459			−0.528541
$844$		0			0
$845$	−7.85013	+	31.3815i	−0.270053	+	1.07956i
$846$		0			0
$847$		−	12.8543i		−	0.441681i
$848$		0			0
$849$	−13.8581			−0.475610
$850$		0			0
$851$	16.9931			0.582517
$852$		0			0
$853$	32.3618			1.10805			0.554024	−	0.832501i	$-0.313092\pi$
$853$	32.3618			1.10805			0.554024	+	0.832501i	$0.313092\pi$
$854$		0			0
$855$	−117.015			−4.00181
$856$		0			0
$857$	−24.0898			−0.822892			−0.411446	−	0.911434i	$-0.634976\pi$
$857$	−24.0898			−0.822892			−0.411446	+	0.911434i	$0.634976\pi$
$858$		0			0
$859$		−	13.5553i		−	0.462499i	−0.972894	−	0.231250i	$-0.925719\pi$
$859$		−	13.5553i		−	0.462499i	0.972894	−	0.231250i	$-0.0742814\pi$
$860$		0			0
$861$		−	14.6331i		−	0.498694i
$862$		0			0
$863$		−	40.3911i		−	1.37493i	−0.726219	−	0.687464i	$-0.758724\pi$
$863$		−	40.3911i		−	1.37493i	0.726219	−	0.687464i	$-0.241276\pi$
$864$		0			0
$865$		−	51.8408i		−	1.76264i
$866$		0			0
$867$			49.5393i			1.68244i
$868$		0			0
$869$	20.7366			0.703442
$870$		0			0
$871$	−2.72467	−	3.49020i	−0.0923218	−	0.118261i
$872$		0			0
$873$		−	113.978i		−	3.85756i
$874$		0			0
$875$			9.47600i			0.320347i
$876$		0			0
$877$	−32.2942			−1.09050			−0.545249	−	0.838274i	$-0.683565\pi$
$877$	−32.2942			−1.09050			−0.545249	+	0.838274i	$0.683565\pi$
$878$		0			0
$879$		−	96.8623i		−	3.26709i
$880$		0			0
$881$	−23.2019			−0.781693			−0.390847	−	0.920456i	$-0.627818\pi$
$881$	−23.2019			−0.781693			−0.390847	+	0.920456i	$0.627818\pi$
$882$		0			0
$883$		−	28.3774i		−	0.954975i	−0.878638	−	0.477488i	$-0.841548\pi$
$883$		−	28.3774i		−	0.954975i	0.878638	−	0.477488i	$-0.158452\pi$
$884$		0			0
$885$		−	3.97323i		−	0.133559i
$886$		0			0
$887$	22.3269			0.749663			0.374832	−	0.927093i	$-0.377701\pi$
$887$	22.3269			0.749663			0.374832	+	0.927093i	$0.377701\pi$
$888$		0			0
$889$		−	4.76216i		−	0.159718i
$890$		0			0
$891$	188.147			6.30315
$892$		0			0
$893$		−	0.334326i		−	0.0111878i
$894$		0			0
$895$		−	16.1514i		−	0.539883i
$896$		0			0
$897$	51.0063	−	39.8187i	1.70305	−	1.32951i
$898$		0			0
$899$	8.11304			0.270585
$900$		0			0
$901$		−	11.2406i		−	0.374478i
$902$		0			0
$903$			25.3542i			0.843734i
$904$		0			0
$905$			22.0416i			0.732689i
$906$		0			0
$907$		−	6.22187i		−	0.206594i	−0.994651	−	0.103297i	$-0.967061\pi$
$907$		−	6.22187i		−	0.206594i	0.994651	−	0.103297i	$-0.0329392\pi$
$908$		0			0
$909$			37.3218i			1.23789i
$910$		0			0
$911$	−7.49278			−0.248247			−0.124123	−	0.992267i	$-0.539612\pi$
$911$	−7.49278			−0.248247			−0.124123	+	0.992267i	$0.539612\pi$
$912$		0			0
$913$	−22.7112			−0.751631
$914$		0			0
$915$	−22.4581			−0.742442
$916$		0			0
$917$	1.91457			0.0632247
$918$		0			0
$919$	32.3931			1.06855			0.534274	−	0.845311i	$-0.320585\pi$
$919$	32.3931			1.06855			0.534274	+	0.845311i	$0.320585\pi$
$920$		0			0
$921$			56.4823i			1.86115i
$922$		0			0
$923$	13.2846	−	10.3708i	0.437269	−	0.341359i
$924$		0			0
$925$	3.83607			0.126129
$926$		0			0
$927$	154.925			5.08839
$928$		0			0
$929$		−	27.9010i		−	0.915402i	−0.889106	−	0.457701i	$-0.848673\pi$
$929$		−	27.9010i		−	0.915402i	0.889106	−	0.457701i	$-0.151327\pi$
$930$		0			0
$931$	−5.49683			−0.180151
$932$		0			0
$933$			69.2015i			2.26556i
$934$		0			0
$935$	68.2897			2.23331
$936$		0			0
$937$	−33.7720			−1.10328			−0.551642	−	0.834081i	$-0.685998\pi$
$937$	−33.7720			−1.10328			−0.551642	+	0.834081i	$0.685998\pi$
$938$		0			0
$939$		−	80.5005i		−	2.62703i
$940$		0			0
$941$	13.5402			0.441397			0.220699	−	0.975342i	$-0.429166\pi$
$941$	13.5402			0.441397			0.220699	+	0.975342i	$0.429166\pi$
$942$		0			0
$943$		−	22.7277i		−	0.740116i
$944$		0			0
$945$	−46.9866			−1.52847
$946$		0			0
$947$	−23.6453			−0.768370			−0.384185	−	0.923256i	$-0.625517\pi$
$947$	−23.6453			−0.768370			−0.384185	+	0.923256i	$0.625517\pi$
$948$		0			0
$949$	13.7266	−	10.7158i	0.445584	−	0.347850i
$950$		0			0
$951$		−	42.7622i		−	1.38666i
$952$		0			0
$953$	−33.5262			−1.08602			−0.543010	−	0.839726i	$-0.682715\pi$
$953$	−33.5262			−1.08602			−0.543010	+	0.839726i	$0.682715\pi$
$954$		0			0
$955$	−10.7880			−0.349093
$956$		0			0
$957$	52.9095			1.71032
$958$		0			0
$959$	−13.8145			−0.446094
$960$		0			0
$961$	24.5191			0.790938
$962$		0			0
$963$			22.4414i			0.723163i
$964$		0			0
$965$			22.7448i			0.732182i
$966$		0			0
$967$		−	11.8782i		−	0.381978i	−0.981592	−	0.190989i	$-0.938830\pi$
$967$		−	11.8782i		−	0.381978i	0.981592	−	0.190989i	$-0.0611695\pi$
$968$		0			0
$969$			104.993i			3.37285i
$970$		0			0
$971$		−	14.2021i		−	0.455767i	−0.973688	−	0.227884i	$-0.926819\pi$
$971$		−	14.2021i		−	0.455767i	0.973688	−	0.227884i	$-0.0731806\pi$
$972$		0			0
$973$	7.88775			0.252870
$974$		0			0
$975$	11.5143	−	8.98876i	0.368752	−	0.287871i
$976$		0			0
$977$			4.25156i			0.136019i	0.997685	+	0.0680097i	$0.0216649\pi$
$977$			4.25156i			0.136019i	−0.997685	+	0.0680097i	$0.978335\pi$
$978$		0			0
$979$		−	44.8353i		−	1.43294i
$980$		0			0
$981$	4.50440			0.143814
$982$		0			0
$983$			55.0157i			1.75473i	0.479824	+	0.877365i	$0.340701\pi$
$983$			55.0157i			1.75473i	−0.479824	+	0.877365i	$0.659299\pi$
$984$		0			0
$985$	−53.6757			−1.71025
$986$		0			0
$987$		−	0.206748i		−	0.00658087i
$988$		0			0
$989$			39.3795i			1.25219i
$990$		0			0
$991$	−2.83231			−0.0899714			−0.0449857	−	0.998988i	$-0.514324\pi$
$991$	−2.83231			−0.0899714			−0.0449857	+	0.998988i	$0.514324\pi$
$992$		0			0
$993$		−	93.1855i		−	2.95715i
$994$		0			0
$995$	−45.8050			−1.45211
$996$		0			0
$997$		−	30.7479i		−	0.973796i	−0.873459	−	0.486898i	$-0.838128\pi$
$997$		−	30.7479i		−	0.973796i	0.873459	−	0.486898i	$-0.161872\pi$
$998$		0			0
$999$		−	60.7762i		−	1.92287i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2912.2.i.a.337.10		84
4.3	odd	2		728.2.i.a.701.26	yes	84
8.3	odd	2		728.2.i.a.701.60	yes	84
8.5	even	2	inner	2912.2.i.a.337.75		84
13.12	even	2	inner	2912.2.i.a.337.76		84
52.51	odd	2		728.2.i.a.701.59	yes	84
104.51	odd	2		728.2.i.a.701.25	✓	84
104.77	even	2	inner	2912.2.i.a.337.9		84

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
728.2.i.a.701.25	✓	84	104.51	odd	2
728.2.i.a.701.26	yes	84	4.3	odd	2
728.2.i.a.701.59	yes	84	52.51	odd	2
728.2.i.a.701.60	yes	84	8.3	odd	2
2912.2.i.a.337.9		84	104.77	even	2	inner
2912.2.i.a.337.10		84	1.1	even	1	trivial
2912.2.i.a.337.75		84	8.5	even	2	inner
2912.2.i.a.337.76		84	13.12	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2912 = 2^{5} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2912.i (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.39926i q^{3} +2.48834 q^{5} -1.00000i q^{7} -8.55495 q^{9} +4.88409 q^{11} +(-2.21870 - 2.84207i) q^{13} +8.45851i q^{15} +5.61904 q^{17} +5.49683 q^{19} +3.39926 q^{21} +5.27964 q^{23} +1.19184 q^{25} -18.8827i q^{27} -3.18688i q^{29} +2.54576i q^{31} +16.6023i q^{33} -2.48834i q^{35} +3.21861 q^{37} +(9.66094 - 7.54192i) q^{39} -4.30478i q^{41} +7.45874i q^{43} -21.2876 q^{45} -0.0608216i q^{47} -1.00000 q^{49} +19.1006i q^{51} -2.00045i q^{53} +12.1533 q^{55} +18.6851i q^{57} -0.469731 q^{59} +2.65509i q^{61} +8.55495i q^{63} +(-5.52087 - 7.07205i) q^{65} +1.22805 q^{67} +17.9469i q^{69} +4.67427i q^{71} +4.82978i q^{73} +4.05137i q^{75} -4.88409i q^{77} +4.24575 q^{79} +38.5223 q^{81} -4.65004 q^{83} +13.9821 q^{85} +10.8330 q^{87} -9.17987i q^{89} +(-2.84207 + 2.21870i) q^{91} -8.65371 q^{93} +13.6780 q^{95} +13.3230i q^{97} -41.7832 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q - 84 q^{9} + 8 q^{17} + 24 q^{23} + 92 q^{25} + 24 q^{39} - 84 q^{49} - 32 q^{55} - 24 q^{65} + 40 q^{79} + 84 q^{81} + 48 q^{87} + 16 q^{95}+O(q^{100}) \) 84 * q - 84 * q^9 + 8 * q^17 + 24 * q^23 + 92 * q^25 + 24 * q^39 - 84 * q^49 - 32 * q^55 - 24 * q^65 + 40 * q^79 + 84 * q^81 + 48 * q^87 + 16 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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