LMFDB - Embedded newform 2912.2.h.a.2575.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2912,2,Mod(2575,2912)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2912.2575"); S:= CuspForms(chi, 2); N := Newforms(S);

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

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

Newform invariants

comment:select newform

sage:traces = [48,0,0,0,0,0,0,0,-48,0,4,0,-48] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$23.2524370686$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	no (minimal twist has level 728)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2575.6
Character	$\chi$	$=$	2912.2575
Dual form			2912.2.h.a.2575.43

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.65471i q^{3} -0.263032 q^{5} +(2.63919 + 0.186171i) q^{7} -4.04746 q^{9} +3.24934 q^{11} -1.00000 q^{13} +0.698273i q^{15} +1.08092i q^{17} -3.39215i q^{19} +(0.494230 - 7.00628i) q^{21} +0.355094i q^{23} -4.93081 q^{25} +2.78070i q^{27} +5.01796i q^{29} +8.52830 q^{31} -8.62605i q^{33} +(-0.694192 - 0.0489690i) q^{35} -7.16212i q^{37} +2.65471i q^{39} -10.7436i q^{41} -9.41079 q^{43} +1.06461 q^{45} +7.23645 q^{47} +(6.93068 + 0.982684i) q^{49} +2.86952 q^{51} -7.26724i q^{53} -0.854681 q^{55} -9.00515 q^{57} -2.37654i q^{59} +5.15481 q^{61} +(-10.6820 - 0.753521i) q^{63} +0.263032 q^{65} +0.139168 q^{67} +0.942669 q^{69} -13.0320i q^{71} +13.6589i q^{73} +13.0899i q^{75} +(8.57564 + 0.604934i) q^{77} -5.14057i q^{79} -4.76044 q^{81} -13.8310i q^{83} -0.284317i q^{85} +13.3212 q^{87} +1.56109i q^{89} +(-2.63919 - 0.186171i) q^{91} -22.6401i q^{93} +0.892243i q^{95} -4.36070i q^{97} -13.1516 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 48 q^{9} + 4 q^{11} - 48 q^{13} + 48 q^{25} - 12 q^{35} + 4 q^{43} + 24 q^{45} + 40 q^{51} - 20 q^{63} + 4 q^{67} - 20 q^{77} + 64 q^{81} + 40 q^{87} - 20 q^{99}+O(q^{100}) $ 48 * q - 48 * q^9 + 4 * q^11 - 48 * q^13 + 48 * q^25 - 12 * q^35 + 4 * q^43 + 24 * q^45 + 40 * q^51 - 20 * q^63 + 4 * q^67 - 20 * q^77 + 64 * q^81 + 40 * q^87 - 20 * q^99

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
$3$		−	2.65471i		−	1.53269i	−0.642426	−	0.766347i	$-0.722072\pi$
$3$		−	2.65471i		−	1.53269i	0.642426	−	0.766347i	$-0.277928\pi$
$4$		0			0
$5$	−0.263032			−0.117631			−0.0588157	−	0.998269i	$-0.518732\pi$
$5$	−0.263032			−0.117631			−0.0588157	+	0.998269i	$0.518732\pi$
$6$		0			0
$7$	2.63919	+	0.186171i	0.997521	+	0.0703661i
$7$	2.63919	+	0.186171i	0.997521	+	0.0703661i
$8$		0			0
$9$	−4.04746			−1.34915
$10$		0			0
$11$	3.24934			0.979714			0.489857	−	0.871803i	$-0.337049\pi$
$11$	3.24934			0.979714			0.489857	+	0.871803i	$0.337049\pi$
$12$		0			0
$13$	−1.00000			−0.277350
$13$	−1.00000			−0.277350
$14$		0			0
$15$			0.698273i			0.180293i
$16$		0			0
$17$			1.08092i			0.262162i	0.991372	+	0.131081i	$0.0418447\pi$
$17$			1.08092i			0.262162i	−0.991372	+	0.131081i	$0.958155\pi$
$18$		0			0
$19$		−	3.39215i		−	0.778212i	−0.921193	−	0.389106i	$-0.872784\pi$
$19$		−	3.39215i		−	0.778212i	0.921193	−	0.389106i	$-0.127216\pi$
$20$		0			0
$21$	0.494230	−	7.00628i	0.107850	−	1.52890i
$22$		0			0
$23$			0.355094i			0.0740421i	0.999314	+	0.0370211i	$0.0117869\pi$
$23$			0.355094i			0.0740421i	−0.999314	+	0.0370211i	$0.988213\pi$
$24$		0			0
$25$	−4.93081			−0.986163
$26$		0			0
$27$			2.78070i			0.535146i
$28$		0			0
$29$			5.01796i			0.931811i	0.884834	+	0.465906i	$0.154271\pi$
$29$			5.01796i			0.931811i	−0.884834	+	0.465906i	$0.845729\pi$
$30$		0			0
$31$	8.52830			1.53173			0.765864	−	0.643002i	$-0.222311\pi$
$31$	8.52830			1.53173			0.765864	+	0.643002i	$0.222311\pi$
$32$		0			0
$33$		−	8.62605i		−	1.50160i
$34$		0			0
$35$	−0.694192	−	0.0489690i	−0.117340	−	0.00827727i
$36$		0			0
$37$		−	7.16212i		−	1.17745i	−0.808335	−	0.588723i	$-0.799631\pi$
$37$		−	7.16212i		−	1.17745i	0.808335	−	0.588723i	$-0.200369\pi$
$38$		0			0
$39$			2.65471i			0.425093i
$40$		0			0
$41$		−	10.7436i		−	1.67787i	−0.544232	−	0.838934i	$-0.683179\pi$
$41$		−	10.7436i		−	1.67787i	0.544232	−	0.838934i	$-0.316821\pi$
$42$		0			0
$43$	−9.41079			−1.43513			−0.717566	−	0.696491i	$-0.754744\pi$
$43$	−9.41079			−1.43513			−0.717566	+	0.696491i	$0.754744\pi$
$44$		0			0
$45$	1.06461			0.158703
$46$		0			0
$47$	7.23645			1.05555			0.527773	−	0.849386i	$-0.323027\pi$
$47$	7.23645			1.05555			0.527773	+	0.849386i	$0.323027\pi$
$48$		0			0
$49$	6.93068	+	0.982684i	0.990097	+	0.140383i
$50$		0			0
$51$	2.86952			0.401814
$52$		0			0
$53$		−	7.26724i		−	0.998232i	−0.866535	−	0.499116i	$-0.833658\pi$
$53$		−	7.26724i		−	0.998232i	0.866535	−	0.499116i	$-0.166342\pi$
$54$		0			0
$55$	−0.854681			−0.115245
$56$		0			0
$57$	−9.00515			−1.19276
$58$		0			0
$59$		−	2.37654i		−	0.309399i	−0.987962	−	0.154699i	$-0.950559\pi$
$59$		−	2.37654i		−	0.309399i	0.987962	−	0.154699i	$-0.0494409\pi$
$60$		0			0
$61$	5.15481			0.660005			0.330003	−	0.943980i	$-0.392950\pi$
$61$	5.15481			0.660005			0.330003	+	0.943980i	$0.392950\pi$
$62$		0			0
$63$	−10.6820	−	0.753521i	−1.34581	−	0.0949347i
$64$		0			0
$65$	0.263032			0.0326251
$66$		0			0
$67$	0.139168			0.0170021			0.00850105	−	0.999964i	$-0.497294\pi$
$67$	0.139168			0.0170021			0.00850105	+	0.999964i	$0.497294\pi$
$68$		0			0
$69$	0.942669			0.113484
$70$		0			0
$71$		−	13.0320i		−	1.54662i	−0.634029	−	0.773309i	$-0.718600\pi$
$71$		−	13.0320i		−	1.54662i	0.634029	−	0.773309i	$-0.281400\pi$
$72$		0			0
$73$			13.6589i			1.59865i	0.600900	+	0.799324i	$0.294809\pi$
$73$			13.6589i			1.59865i	−0.600900	+	0.799324i	$0.705191\pi$
$74$		0			0
$75$			13.0899i			1.51149i
$76$		0			0
$77$	8.57564	+	0.604934i	0.977285	+	0.0689387i
$78$		0			0
$79$		−	5.14057i		−	0.578360i	−0.957275	−	0.289180i	$-0.906617\pi$
$79$		−	5.14057i		−	0.578360i	0.957275	−	0.289180i	$-0.0933826\pi$
$80$		0			0
$81$	−4.76044			−0.528938
$82$		0			0
$83$		−	13.8310i		−	1.51815i	−0.651004	−	0.759075i	$-0.725652\pi$
$83$		−	13.8310i		−	1.51815i	0.651004	−	0.759075i	$-0.274348\pi$
$84$		0			0
$85$		−	0.284317i		−	0.0308385i
$86$		0			0
$87$	13.3212			1.42818
$88$		0			0
$89$			1.56109i			0.165476i	0.996571	+	0.0827378i	$0.0263664\pi$
$89$			1.56109i			0.165476i	−0.996571	+	0.0827378i	$0.973634\pi$
$90$		0			0
$91$	−2.63919	−	0.186171i	−0.276663	−	0.0195160i
$92$		0			0
$93$		−	22.6401i		−	2.34767i
$94$		0			0
$95$			0.892243i			0.0915422i
$96$		0			0
$97$		−	4.36070i		−	0.442762i	−0.975187	−	0.221381i	$-0.928944\pi$
$97$		−	4.36070i		−	0.442762i	0.975187	−	0.221381i	$-0.0710564\pi$
$98$		0			0
$99$	−13.1516			−1.32178
$100$		0			0
$101$	2.79503			0.278116			0.139058	−	0.990284i	$-0.455593\pi$
$101$	2.79503			0.278116			0.139058	+	0.990284i	$0.455593\pi$
$102$		0			0
$103$	−1.26494			−0.124638			−0.0623191	−	0.998056i	$-0.519850\pi$
$103$	−1.26494			−0.124638			−0.0623191	+	0.998056i	$0.519850\pi$
$104$		0			0
$105$	−0.129998	+	1.84288i	−0.0126865	+	0.179846i
$106$		0			0
$107$	−1.86660			−0.180451			−0.0902255	−	0.995921i	$-0.528759\pi$
$107$	−1.86660			−0.180451			−0.0902255	+	0.995921i	$0.528759\pi$
$108$		0			0
$109$		−	3.91026i		−	0.374535i	−0.982309	−	0.187267i	$-0.940037\pi$
$109$		−	3.91026i		−	0.374535i	0.982309	−	0.187267i	$-0.0599631\pi$
$110$		0			0
$111$	−19.0133			−1.80467
$112$		0			0
$113$	16.9264			1.59230			0.796152	−	0.605097i	$-0.206866\pi$
$113$	16.9264			1.59230			0.796152	+	0.605097i	$0.206866\pi$
$114$		0			0
$115$		−	0.0934010i		−	0.00870969i
$116$		0			0
$117$	4.04746			0.374188
$118$		0			0
$119$	−0.201236	+	2.85276i	−0.0184473	+	0.261512i
$120$		0			0
$121$	−0.441770			−0.0401609
$122$		0			0
$123$	−28.5211			−2.57166
$124$		0			0
$125$	2.61212			0.233635
$126$		0			0
$127$			2.54307i			0.225661i	0.993614	+	0.112831i	$0.0359918\pi$
$127$			2.54307i			0.225661i	−0.993614	+	0.112831i	$0.964008\pi$
$128$		0			0
$129$			24.9829i			2.19962i
$130$		0			0
$131$			3.23734i			0.282848i	0.989949	+	0.141424i	$0.0451681\pi$
$131$			3.23734i			0.282848i	−0.989949	+	0.141424i	$0.954832\pi$
$132$		0			0
$133$	0.631520	−	8.95253i	0.0547597	−	0.776283i
$134$		0			0
$135$		−	0.731413i		−	0.0629500i
$136$		0			0
$137$	−16.6120			−1.41926			−0.709629	−	0.704576i	$-0.751137\pi$
$137$	−16.6120			−1.41926			−0.709629	+	0.704576i	$0.751137\pi$
$138$		0			0
$139$			6.87742i			0.583335i	0.956520	+	0.291668i	$0.0942101\pi$
$139$			6.87742i			0.583335i	−0.956520	+	0.291668i	$0.905790\pi$
$140$		0			0
$141$		−	19.2107i		−	1.61783i
$142$		0			0
$143$	−3.24934			−0.271724
$144$		0			0
$145$		−	1.31988i		−	0.109610i
$146$		0			0
$147$	2.60874	−	18.3989i	0.215165	−	1.51752i
$148$		0			0
$149$			11.8102i			0.967528i	0.875198	+	0.483764i	$0.160731\pi$
$149$			11.8102i			0.967528i	−0.875198	+	0.483764i	$0.839269\pi$
$150$		0			0
$151$			16.2442i			1.32193i	0.750415	+	0.660967i	$0.229854\pi$
$151$			16.2442i			1.32193i	−0.750415	+	0.660967i	$0.770146\pi$
$152$		0			0
$153$		−	4.37498i		−	0.353696i
$154$		0			0
$155$	−2.24322			−0.180180
$156$		0			0
$157$	−7.03312			−0.561304			−0.280652	−	0.959810i	$-0.590551\pi$
$157$	−7.03312			−0.561304			−0.280652	+	0.959810i	$0.590551\pi$
$158$		0			0
$159$	−19.2924			−1.52998
$160$		0			0
$161$	−0.0661082	+	0.937161i	−0.00521006	+	0.0738586i
$162$		0			0
$163$	21.2161			1.66177			0.830886	−	0.556443i	$-0.187834\pi$
$163$	21.2161			1.66177			0.830886	+	0.556443i	$0.187834\pi$
$164$		0			0
$165$			2.26893i			0.176636i
$166$		0			0
$167$	1.53534			0.118808			0.0594041	−	0.998234i	$-0.481080\pi$
$167$	1.53534			0.118808			0.0594041	+	0.998234i	$0.481080\pi$
$168$		0			0
$169$	1.00000			0.0769231
$170$		0			0
$171$			13.7296i			1.04993i
$172$		0			0
$173$	−25.0144			−1.90181			−0.950907	−	0.309478i	$-0.899846\pi$
$173$	−25.0144			−1.90181			−0.950907	+	0.309478i	$0.899846\pi$
$174$		0			0
$175$	−13.0134	−	0.917976i	−0.983718	−	0.0693924i
$176$		0			0
$177$	−6.30901			−0.474214
$178$		0			0
$179$	−9.45764			−0.706897			−0.353449	−	0.935454i	$-0.614991\pi$
$179$	−9.45764			−0.706897			−0.353449	+	0.935454i	$0.614991\pi$
$180$		0			0
$181$	−5.30571			−0.394371			−0.197185	−	0.980366i	$-0.563180\pi$
$181$	−5.30571			−0.394371			−0.197185	+	0.980366i	$0.563180\pi$
$182$		0			0
$183$		−	13.6845i		−	1.01159i
$184$		0			0
$185$			1.88387i			0.138505i
$186$		0			0
$187$			3.51228i			0.256843i
$188$		0			0
$189$	−0.517686	+	7.33880i	−0.0376561	+	0.533820i
$190$		0			0
$191$		−	15.8277i		−	1.14525i	−0.819817	−	0.572625i	$-0.805925\pi$
$191$		−	15.8277i		−	1.14525i	0.819817	−	0.572625i	$-0.194075\pi$
$192$		0			0
$193$	1.78118			0.128212			0.0641062	−	0.997943i	$-0.479580\pi$
$193$	1.78118			0.128212			0.0641062	+	0.997943i	$0.479580\pi$
$194$		0			0
$195$		−	0.698273i		−	0.0500043i
$196$		0			0
$197$			11.6396i			0.829290i	0.909983	+	0.414645i	$0.136094\pi$
$197$			11.6396i			0.829290i	−0.909983	+	0.414645i	$0.863906\pi$
$198$		0			0
$199$	9.67647			0.685947			0.342973	−	0.939345i	$-0.388566\pi$
$199$	9.67647			0.685947			0.342973	+	0.939345i	$0.388566\pi$
$200$		0			0
$201$		−	0.369450i		−	0.0260590i
$202$		0			0
$203$	−0.934199	+	13.2434i	−0.0655679	+	0.929501i
$204$		0			0
$205$			2.82591i			0.197370i
$206$		0			0
$207$		−	1.43723i		−	0.0998942i
$208$		0			0
$209$		−	11.0222i		−	0.762425i
$210$		0			0
$211$	−13.0610			−0.899156			−0.449578	−	0.893241i	$-0.648426\pi$
$211$	−13.0610			−0.899156			−0.449578	+	0.893241i	$0.648426\pi$
$212$		0			0
$213$	−34.5962			−2.37049
$214$		0			0
$215$	2.47534			0.168817
$216$		0			0
$217$	22.5078	+	1.58772i	1.52793	+	0.107782i
$218$		0			0
$219$	36.2602			2.45024
$220$		0			0
$221$		−	1.08092i		−	0.0727105i
$222$		0			0
$223$	12.2491			0.820263			0.410132	−	0.912026i	$-0.365483\pi$
$223$	12.2491			0.820263			0.410132	+	0.912026i	$0.365483\pi$
$224$		0			0
$225$	19.9573			1.33049
$226$		0			0
$227$			4.34401i			0.288322i	0.989554	+	0.144161i	$0.0460483\pi$
$227$			4.34401i			0.288322i	−0.989554	+	0.144161i	$0.953952\pi$
$228$		0			0
$229$	18.3309			1.21134			0.605670	−	0.795716i	$-0.292905\pi$
$229$	18.3309			1.21134			0.605670	+	0.795716i	$0.292905\pi$
$230$		0			0
$231$	1.60592	−	22.7658i	0.105662	−	1.49788i
$232$		0			0
$233$	−16.9908			−1.11310			−0.556552	−	0.830813i	$-0.687876\pi$
$233$	−16.9908			−1.11310			−0.556552	+	0.830813i	$0.687876\pi$
$234$		0			0
$235$	−1.90342			−0.124165
$236$		0			0
$237$	−13.6467			−0.886449
$238$		0			0
$239$			16.2246i			1.04948i	0.851261	+	0.524742i	$0.175838\pi$
$239$			16.2246i			1.04948i	−0.851261	+	0.524742i	$0.824162\pi$
$240$		0			0
$241$		−	28.1758i		−	1.81496i	−0.420091	−	0.907482i	$-0.638002\pi$
$241$		−	28.1758i		−	1.81496i	0.420091	−	0.907482i	$-0.361998\pi$
$242$		0			0
$243$			20.9797i			1.34585i
$244$		0			0
$245$	−1.82299	−	0.258477i	−0.116467	−	0.0165135i
$246$		0			0
$247$			3.39215i			0.215837i
$248$		0			0
$249$	−36.7172			−2.32686
$250$		0			0
$251$			1.48840i			0.0939468i	0.998896	+	0.0469734i	$0.0149576\pi$
$251$			1.48840i			0.0939468i	−0.998896	+	0.0469734i	$0.985042\pi$
$252$		0			0
$253$			1.15382i			0.0725401i
$254$		0			0
$255$	−0.754777			−0.0472659
$256$		0			0
$257$		−	2.71616i		−	0.169430i	−0.996405	−	0.0847148i	$-0.973002\pi$
$257$		−	2.71616i		−	0.169430i	0.996405	−	0.0847148i	$-0.0269979\pi$
$258$		0			0
$259$	1.33338	−	18.9022i	0.0828523	−	1.17453i
$260$		0			0
$261$		−	20.3100i		−	1.25716i
$262$		0			0
$263$		−	4.95465i		−	0.305517i	−0.988264	−	0.152758i	$-0.951184\pi$
$263$		−	4.95465i		−	0.305517i	0.988264	−	0.152758i	$-0.0488156\pi$
$264$		0			0
$265$			1.91152i			0.117423i
$266$		0			0
$267$	4.14424			0.253624
$268$		0			0
$269$	2.73612			0.166824			0.0834120	−	0.996515i	$-0.473418\pi$
$269$	2.73612			0.166824			0.0834120	+	0.996515i	$0.473418\pi$
$270$		0			0
$271$	9.90746			0.601835			0.300918	−	0.953650i	$-0.402707\pi$
$271$	9.90746			0.601835			0.300918	+	0.953650i	$0.402707\pi$
$272$		0			0
$273$	−0.494230	+	7.00628i	−0.0299121	+	0.424039i
$274$		0			0
$275$	−16.0219			−0.966157
$276$		0			0
$277$			25.6730i			1.54254i	0.636506	+	0.771271i	$0.280379\pi$
$277$			25.6730i			1.54254i	−0.636506	+	0.771271i	$0.719621\pi$
$278$		0			0
$279$	−34.5180			−2.06654
$280$		0			0
$281$	−29.3364			−1.75007			−0.875033	−	0.484064i	$-0.839160\pi$
$281$	−29.3364			−1.75007			−0.875033	+	0.484064i	$0.839160\pi$
$282$		0			0
$283$			5.01662i			0.298207i	0.988822	+	0.149104i	$0.0476388\pi$
$283$			5.01662i			0.298207i	−0.988822	+	0.149104i	$0.952361\pi$
$284$		0			0
$285$	2.36864			0.140306
$286$		0			0
$287$	2.00015	−	28.3544i	0.118065	−	1.67371i
$288$		0			0
$289$	15.8316			0.931271
$290$		0			0
$291$	−11.5764			−0.678619
$292$		0			0
$293$	−24.7206			−1.44419			−0.722096	−	0.691793i	$-0.756821\pi$
$293$	−24.7206			−1.44419			−0.722096	+	0.691793i	$0.756821\pi$
$294$		0			0
$295$			0.625105i			0.0363950i
$296$		0			0
$297$			9.03545i			0.524290i
$298$		0			0
$299$		−	0.355094i		−	0.0205356i
$300$		0			0
$301$	−24.8369	−	1.75202i	−1.43157	−	0.100985i
$302$		0			0
$303$		−	7.41997i		−	0.426266i
$304$		0			0
$305$	−1.35588			−0.0776374
$306$		0			0
$307$			25.8768i			1.47687i	0.674327	+	0.738433i	$0.264434\pi$
$307$			25.8768i			1.47687i	−0.674327	+	0.738433i	$0.735566\pi$
$308$		0			0
$309$			3.35804i			0.191032i
$310$		0			0
$311$	8.89137			0.504183			0.252092	−	0.967703i	$-0.418882\pi$
$311$	8.89137			0.504183			0.252092	+	0.967703i	$0.418882\pi$
$312$		0			0
$313$			19.3019i			1.09101i	0.838109	+	0.545503i	$0.183661\pi$
$313$			19.3019i			1.09101i	−0.838109	+	0.545503i	$0.816339\pi$
$314$		0			0
$315$	2.80972	+	0.198200i	0.158310	+	0.0111673i
$316$		0			0
$317$			7.58551i			0.426045i	0.977047	+	0.213022i	$0.0683307\pi$
$317$			7.58551i			0.426045i	−0.977047	+	0.213022i	$0.931669\pi$
$318$		0			0
$319$			16.3051i			0.912908i
$320$		0			0
$321$			4.95527i			0.276576i
$322$		0			0
$323$	3.66664			0.204017
$324$		0			0
$325$	4.93081			0.273512
$326$		0			0
$327$	−10.3806			−0.574047
$328$		0			0
$329$	19.0984	+	1.34722i	1.05293	+	0.0742746i
$330$		0			0
$331$	−27.4209			−1.50719			−0.753594	−	0.657341i	$-0.771681\pi$
$331$	−27.4209			−1.50719			−0.753594	+	0.657341i	$0.771681\pi$
$332$		0			0
$333$			28.9884i			1.58856i
$334$		0			0
$335$	−0.0366057			−0.00199998
$336$		0			0
$337$	−27.7623			−1.51231			−0.756155	−	0.654393i	$-0.772924\pi$
$337$	−27.7623			−1.51231			−0.756155	+	0.654393i	$0.772924\pi$
$338$		0			0
$339$		−	44.9347i		−	2.44052i
$340$		0			0
$341$	27.7114			1.50066
$342$		0			0
$343$	18.1085	+	3.88379i	0.977765	+	0.209705i
$344$		0			0
$345$	−0.247952			−0.0133493
$346$		0			0
$347$	22.5651			1.21136			0.605680	−	0.795709i	$-0.292901\pi$
$347$	22.5651			1.21136			0.605680	+	0.795709i	$0.292901\pi$
$348$		0			0
$349$	28.7568			1.53931			0.769657	−	0.638458i	$-0.220427\pi$
$349$	28.7568			1.53931			0.769657	+	0.638458i	$0.220427\pi$
$350$		0			0
$351$		−	2.78070i		−	0.148423i
$352$		0			0
$353$			17.5497i			0.934075i	0.884238	+	0.467037i	$0.154679\pi$
$353$			17.5497i			0.934075i	−0.884238	+	0.467037i	$0.845321\pi$
$354$		0			0
$355$			3.42784i			0.181931i
$356$		0			0
$357$	7.57323	+	0.534223i	0.400818	+	0.0282741i
$358$		0			0
$359$		−	23.7594i		−	1.25397i	−0.779031	−	0.626986i	$-0.784288\pi$
$359$		−	23.7594i		−	1.25397i	0.779031	−	0.626986i	$-0.215712\pi$
$360$		0			0
$361$	7.49334			0.394386
$362$		0			0
$363$			1.17277i			0.0615544i
$364$		0			0
$365$		−	3.59272i		−	0.188051i
$366$		0			0
$367$	19.1414			0.999174			0.499587	−	0.866264i	$-0.333485\pi$
$367$	19.1414			0.999174			0.499587	+	0.866264i	$0.333485\pi$
$368$		0			0
$369$			43.4843i			2.26370i
$370$		0			0
$371$	1.35295	−	19.1796i	0.0702417	−	0.995757i
$372$		0			0
$373$		−	12.7692i		−	0.661162i	−0.943778	−	0.330581i	$-0.892755\pi$
$373$		−	12.7692i		−	0.661162i	0.943778	−	0.330581i	$-0.107245\pi$
$374$		0			0
$375$		−	6.93441i		−	0.358092i
$376$		0			0
$377$		−	5.01796i		−	0.258438i
$378$		0			0
$379$	13.5309			0.695035			0.347517	−	0.937674i	$-0.387025\pi$
$379$	13.5309			0.695035			0.347517	+	0.937674i	$0.387025\pi$
$380$		0			0
$381$	6.75111			0.345870
$382$		0			0
$383$	15.2577			0.779631			0.389816	−	0.920893i	$-0.372539\pi$
$383$	15.2577			0.779631			0.389816	+	0.920893i	$0.372539\pi$
$384$		0			0
$385$	−2.25567	−	0.159117i	−0.114960	−	0.00810936i
$386$		0			0
$387$	38.0898			1.93621
$388$		0			0
$389$			37.0967i			1.88088i	0.339961	+	0.940440i	$0.389586\pi$
$389$			37.0967i			1.88088i	−0.339961	+	0.940440i	$0.610414\pi$
$390$		0			0
$391$	−0.383828			−0.0194110
$392$		0			0
$393$	8.59419			0.433519
$394$		0			0
$395$			1.35214i			0.0680333i
$396$		0			0
$397$	20.1506			1.01133			0.505665	−	0.862730i	$-0.331247\pi$
$397$	20.1506			1.01133			0.505665	+	0.862730i	$0.331247\pi$
$398$		0			0
$399$	−23.7663	−	1.67650i	−1.18980	−	0.0839300i
$400$		0			0
$401$	10.7530			0.536978			0.268489	−	0.963283i	$-0.413476\pi$
$401$	10.7530			0.536978			0.268489	+	0.963283i	$0.413476\pi$
$402$		0			0
$403$	−8.52830			−0.424825
$404$		0			0
$405$	1.25215			0.0622198
$406$		0			0
$407$		−	23.2722i		−	1.15356i
$408$		0			0
$409$		−	26.9923i		−	1.33468i	−0.744751	−	0.667342i	$-0.767432\pi$
$409$		−	26.9923i		−	1.33468i	0.744751	−	0.667342i	$-0.232568\pi$
$410$		0			0
$411$			44.0999i			2.17529i
$412$		0			0
$413$	0.442443	−	6.27214i	0.0217712	−	0.308632i
$414$		0			0
$415$			3.63800i			0.178582i
$416$		0			0
$417$	18.2575			0.894075
$418$		0			0
$419$		−	30.1369i		−	1.47228i	−0.676827	−	0.736142i	$-0.736645\pi$
$419$		−	30.1369i		−	1.47228i	0.676827	−	0.736142i	$-0.263355\pi$
$420$		0			0
$421$		−	4.56828i		−	0.222644i	−0.993784	−	0.111322i	$-0.964491\pi$
$421$		−	4.56828i		−	0.222644i	0.993784	−	0.111322i	$-0.0355086\pi$
$422$		0			0
$423$	−29.2893			−1.42409
$424$		0			0
$425$		−	5.32981i		−	0.258534i
$426$		0			0
$427$	13.6045	+	0.959677i	0.658369	+	0.0464420i
$428$		0			0
$429$			8.62605i			0.416470i
$430$		0			0
$431$		−	40.7375i		−	1.96225i	−0.193362	−	0.981127i	$-0.561939\pi$
$431$		−	40.7375i		−	1.96225i	0.193362	−	0.981127i	$-0.438061\pi$
$432$		0			0
$433$		−	30.2376i		−	1.45313i	−0.687099	−	0.726564i	$-0.741116\pi$
$433$		−	30.2376i		−	1.45313i	0.687099	−	0.726564i	$-0.258884\pi$
$434$		0			0
$435$	−3.50390			−0.167999
$436$		0			0
$437$	1.20453			0.0576205
$438$		0			0
$439$	21.0215			1.00330			0.501650	−	0.865071i	$-0.332727\pi$
$439$	21.0215			1.00330			0.501650	+	0.865071i	$0.332727\pi$
$440$		0			0
$441$	−28.0517	−	3.97737i	−1.33579	−	0.189399i
$442$		0			0
$443$	18.9078			0.898337			0.449168	−	0.893447i	$-0.351720\pi$
$443$	18.9078			0.898337			0.449168	+	0.893447i	$0.351720\pi$
$444$		0			0
$445$		−	0.410618i		−	0.0194651i
$446$		0			0
$447$	31.3526			1.48293
$448$		0			0
$449$	−17.0089			−0.802699			−0.401349	−	0.915925i	$-0.631459\pi$
$449$	−17.0089			−0.802699			−0.401349	+	0.915925i	$0.631459\pi$
$450$		0			0
$451$		−	34.9097i		−	1.64383i
$452$		0			0
$453$	43.1236			2.02612
$454$		0			0
$455$	0.694192	+	0.0489690i	0.0325442	+	0.00229570i
$456$		0			0
$457$	36.5321			1.70890			0.854449	−	0.519535i	$-0.173895\pi$
$457$	36.5321			1.70890			0.854449	+	0.519535i	$0.173895\pi$
$458$		0			0
$459$	−3.00571			−0.140295
$460$		0			0
$461$	14.3221			0.667045			0.333523	−	0.942742i	$-0.391763\pi$
$461$	14.3221			0.667045			0.333523	+	0.942742i	$0.391763\pi$
$462$		0			0
$463$			24.6893i			1.14741i	0.819062	+	0.573706i	$0.194495\pi$
$463$			24.6893i			1.14741i	−0.819062	+	0.573706i	$0.805505\pi$
$464$		0			0
$465$			5.95508i			0.276160i
$466$		0			0
$467$		−	7.78833i		−	0.360401i	−0.983630	−	0.180201i	$-0.942325\pi$
$467$		−	7.78833i		−	0.360401i	0.983630	−	0.180201i	$-0.0576747\pi$
$468$		0			0
$469$	0.367292	+	0.0259091i	0.0169599	+	0.00119637i
$470$		0			0
$471$			18.6709i			0.860308i
$472$		0			0
$473$	−30.5789			−1.40602
$474$		0			0
$475$			16.7260i			0.767444i
$476$		0			0
$477$			29.4139i			1.34677i
$478$		0			0
$479$	−21.9355			−1.00226			−0.501129	−	0.865373i	$-0.667082\pi$
$479$	−21.9355			−1.00226			−0.501129	+	0.865373i	$0.667082\pi$
$480$		0			0
$481$			7.16212i			0.326565i
$482$		0			0
$483$	2.48789	+	0.175498i	0.113203	+	0.00798543i
$484$		0			0
$485$			1.14700i			0.0520827i
$486$		0			0
$487$			31.5345i			1.42896i	0.699654	+	0.714482i	$0.253338\pi$
$487$			31.5345i			1.42896i	−0.699654	+	0.714482i	$0.746662\pi$
$488$		0			0
$489$		−	56.3225i		−	2.54699i
$490$		0			0
$491$	4.14288			0.186965			0.0934827	−	0.995621i	$-0.470200\pi$
$491$	4.14288			0.186965			0.0934827	+	0.995621i	$0.470200\pi$
$492$		0			0
$493$	−5.42401			−0.244285
$494$		0			0
$495$	3.45929			0.155483
$496$		0			0
$497$	2.42619	−	34.3941i	0.108830	−	1.54278i
$498$		0			0
$499$	−5.12089			−0.229243			−0.114621	−	0.993409i	$-0.536565\pi$
$499$	−5.12089			−0.229243			−0.114621	+	0.993409i	$0.536565\pi$
$500$		0			0
$501$		−	4.07588i		−	0.182097i
$502$		0			0
$503$	−9.23511			−0.411773			−0.205887	−	0.978576i	$-0.566008\pi$
$503$	−9.23511			−0.411773			−0.205887	+	0.978576i	$0.566008\pi$
$504$		0			0
$505$	−0.735182			−0.0327152
$506$		0			0
$507$		−	2.65471i		−	0.117900i
$508$		0			0
$509$	32.7554			1.45186			0.725928	−	0.687770i	$-0.241410\pi$
$509$	32.7554			1.45186			0.725928	+	0.687770i	$0.241410\pi$
$510$		0			0
$511$	−2.54289	+	36.0483i	−0.112491	+	1.59469i
$512$		0			0
$513$	9.43254			0.416457
$514$		0			0
$515$	0.332720			0.0146614
$516$		0			0
$517$	23.5137			1.03413
$518$		0			0
$519$			66.4060i			2.91490i
$520$		0			0
$521$			9.14797i			0.400780i	0.979716	+	0.200390i	$0.0642209\pi$
$521$			9.14797i			0.400780i	−0.979716	+	0.200390i	$0.935779\pi$
$522$		0			0
$523$		−	4.07725i		−	0.178286i	−0.996019	−	0.0891428i	$-0.971587\pi$
$523$		−	4.07725i		−	0.178286i	0.996019	−	0.0891428i	$-0.0284128\pi$
$524$		0			0
$525$	−2.43696	+	34.5467i	−0.106357	+	1.50774i
$526$		0			0
$527$			9.21841i			0.401560i
$528$		0			0
$529$	22.8739			0.994518
$530$		0			0
$531$			9.61894i			0.417426i
$532$		0			0
$533$			10.7436i			0.465357i
$534$		0			0
$535$	0.490976			0.0212267
$536$		0			0
$537$			25.1073i			1.08346i
$538$		0			0
$539$	22.5202	+	3.19308i	0.970012	+	0.137536i
$540$		0			0
$541$			23.1946i			0.997213i	0.866828	+	0.498606i	$0.166155\pi$
$541$			23.1946i			0.997213i	−0.866828	+	0.498606i	$0.833845\pi$
$542$		0			0
$543$			14.0851i			0.604450i
$544$		0			0
$545$			1.02852i			0.0440571i
$546$		0			0
$547$	29.3397			1.25447			0.627237	−	0.778829i	$-0.284186\pi$
$547$	29.3397			1.25447			0.627237	+	0.778829i	$0.284186\pi$
$548$		0			0
$549$	−20.8639			−0.890449
$550$		0			0
$551$	17.0216			0.725147
$552$		0			0
$553$	0.957027	−	13.5670i	0.0406969	−	0.576926i
$554$		0			0
$555$	5.00111			0.212285
$556$		0			0
$557$		−	8.05913i		−	0.341476i	−0.985316	−	0.170738i	$-0.945385\pi$
$557$		−	8.05913i		−	0.341476i	0.985316	−	0.170738i	$-0.0546152\pi$
$558$		0			0
$559$	9.41079			0.398034
$560$		0			0
$561$	9.32407			0.393662
$562$		0			0
$563$		−	19.8374i		−	0.836045i	−0.908437	−	0.418023i	$-0.862723\pi$
$563$		−	19.8374i		−	0.836045i	0.908437	−	0.418023i	$-0.137277\pi$
$564$		0			0
$565$	−4.45219			−0.187305
$566$		0			0
$567$	−12.5637	−	0.886258i	−0.527627	−	0.0372193i
$568$		0			0
$569$	12.1101			0.507681			0.253840	−	0.967246i	$-0.418306\pi$
$569$	12.1101			0.507681			0.253840	+	0.967246i	$0.418306\pi$
$570$		0			0
$571$	−5.92712			−0.248042			−0.124021	−	0.992280i	$-0.539579\pi$
$571$	−5.92712			−0.248042			−0.124021	+	0.992280i	$0.539579\pi$
$572$		0			0
$573$	−42.0178			−1.75532
$574$		0			0
$575$		−	1.75090i		−	0.0730176i
$576$		0			0
$577$			24.2713i			1.01043i	0.862994	+	0.505215i	$0.168587\pi$
$577$			24.2713i			1.01043i	−0.862994	+	0.505215i	$0.831413\pi$
$578$		0			0
$579$		−	4.72852i		−	0.196510i
$580$		0			0
$581$	2.57493	−	36.5027i	0.106826	−	1.51439i
$582$		0			0
$583$		−	23.6137i		−	0.977981i
$584$		0			0
$585$	−1.06461			−0.0440163
$586$		0			0
$587$			24.6913i			1.01912i	0.860436	+	0.509559i	$0.170191\pi$
$587$			24.6913i			1.01912i	−0.860436	+	0.509559i	$0.829809\pi$
$588$		0			0
$589$		−	28.9293i		−	1.19201i
$590$		0			0
$591$	30.8998			1.27105
$592$		0			0
$593$		−	27.5409i		−	1.13097i	−0.824759	−	0.565484i	$-0.808689\pi$
$593$		−	27.5409i		−	1.13097i	0.824759	−	0.565484i	$-0.191311\pi$
$594$		0			0
$595$	0.0529316	−	0.750366i	0.00216998	−	0.0307620i
$596$		0			0
$597$		−	25.6882i		−	1.05135i
$598$		0			0
$599$			18.1990i			0.743592i	0.928315	+	0.371796i	$0.121258\pi$
$599$			18.1990i			0.743592i	−0.928315	+	0.371796i	$0.878742\pi$
$600$		0			0
$601$			7.13002i			0.290840i	0.989370	+	0.145420i	$0.0464533\pi$
$601$			7.13002i			0.290840i	−0.989370	+	0.145420i	$0.953547\pi$
$602$		0			0
$603$	−0.563277			−0.0229384
$604$		0			0
$605$	0.116200			0.00472419
$606$		0			0
$607$	1.48689			0.0603509			0.0301754	−	0.999545i	$-0.490393\pi$
$607$	1.48689			0.0603509			0.0301754	+	0.999545i	$0.490393\pi$
$608$		0			0
$609$	35.1572	+	2.48002i	1.42464	+	0.100496i
$610$		0			0
$611$	−7.23645			−0.292756
$612$		0			0
$613$			26.8238i			1.08340i	0.840571	+	0.541701i	$0.182220\pi$
$613$			26.8238i			1.08340i	−0.840571	+	0.541701i	$0.817780\pi$
$614$		0			0
$615$	7.50196			0.302508
$616$		0			0
$617$	−29.6648			−1.19426			−0.597131	−	0.802144i	$-0.703693\pi$
$617$	−29.6648			−1.19426			−0.597131	+	0.802144i	$0.703693\pi$
$618$		0			0
$619$		−	37.5289i		−	1.50841i	−0.656637	−	0.754207i	$-0.728022\pi$
$619$		−	37.5289i		−	1.50841i	0.656637	−	0.754207i	$-0.271978\pi$
$620$		0			0
$621$	−0.987409			−0.0396234
$622$		0			0
$623$	−0.290631	+	4.12003i	−0.0116439	+	0.165065i
$624$		0			0
$625$	23.9670			0.958680
$626$		0			0
$627$	−29.2608			−1.16856
$628$		0			0
$629$	7.74168			0.308681
$630$		0			0
$631$			31.6865i			1.26142i	0.776019	+	0.630709i	$0.217236\pi$
$631$			31.6865i			1.26142i	−0.776019	+	0.630709i	$0.782764\pi$
$632$		0			0
$633$			34.6731i			1.37813i
$634$		0			0
$635$		−	0.668910i		−	0.0265449i
$636$		0			0
$637$	−6.93068	−	0.982684i	−0.274604	−	0.0389353i
$638$		0			0
$639$			52.7467i			2.08663i
$640$		0			0
$641$	−3.11018			−0.122845			−0.0614223	−	0.998112i	$-0.519564\pi$
$641$	−3.11018			−0.122845			−0.0614223	+	0.998112i	$0.519564\pi$
$642$		0			0
$643$			2.89312i			0.114094i	0.998372	+	0.0570468i	$0.0181684\pi$
$643$			2.89312i			0.114094i	−0.998372	+	0.0570468i	$0.981832\pi$
$644$		0			0
$645$		−	6.57130i		−	0.258745i
$646$		0			0
$647$	34.2196			1.34531			0.672656	−	0.739956i	$-0.265154\pi$
$647$	34.2196			1.34531			0.672656	+	0.739956i	$0.265154\pi$
$648$		0			0
$649$		−	7.72218i		−	0.303122i
$650$		0			0
$651$	4.21494	−	59.7517i	0.165197	−	2.34185i
$652$		0			0
$653$			15.8437i			0.620011i	0.950735	+	0.310006i	$0.100331\pi$
$653$			15.8437i			0.620011i	−0.950735	+	0.310006i	$0.899669\pi$
$654$		0			0
$655$		−	0.851525i		−	0.0332718i
$656$		0			0
$657$		−	55.2837i		−	2.15682i
$658$		0			0
$659$	4.94866			0.192773			0.0963863	−	0.995344i	$-0.469272\pi$
$659$	4.94866			0.192773			0.0963863	+	0.995344i	$0.469272\pi$
$660$		0			0
$661$	−28.6115			−1.11286			−0.556430	−	0.830895i	$-0.687829\pi$
$661$	−28.6115			−1.11286			−0.556430	+	0.830895i	$0.687829\pi$
$662$		0			0
$663$	−2.86952			−0.111443
$664$		0			0
$665$	−0.166110	+	2.35480i	−0.00644147	+	0.0913153i
$666$		0			0
$667$	−1.78184			−0.0689933
$668$		0			0
$669$		−	32.5179i		−	1.25721i
$670$		0			0
$671$	16.7497			0.646616
$672$		0			0
$673$	40.1335			1.54703			0.773516	−	0.633777i	$-0.218496\pi$
$673$	40.1335			1.54703			0.773516	+	0.633777i	$0.218496\pi$
$674$		0			0
$675$		−	13.7111i		−	0.527741i
$676$		0			0
$677$	−26.6691			−1.02498			−0.512489	−	0.858694i	$-0.671276\pi$
$677$	−26.6691			−1.02498			−0.512489	+	0.858694i	$0.671276\pi$
$678$		0			0
$679$	0.811837	−	11.5087i	0.0311554	−	0.441664i
$680$		0			0
$681$	11.5321			0.441910
$682$		0			0
$683$	−22.2590			−0.851715			−0.425858	−	0.904790i	$-0.640028\pi$
$683$	−22.2590			−0.851715			−0.425858	+	0.904790i	$0.640028\pi$
$684$		0			0
$685$	4.36948			0.166949
$686$		0			0
$687$		−	48.6632i		−	1.85662i
$688$		0			0
$689$			7.26724i			0.276860i
$690$		0			0
$691$		−	4.33250i		−	0.164816i	−0.996599	−	0.0824080i	$-0.973739\pi$
$691$		−	4.33250i		−	0.164816i	0.996599	−	0.0824080i	$-0.0262611\pi$
$692$		0			0
$693$	−34.7096	−	2.44845i	−1.31851	−	0.0930088i
$694$		0			0
$695$		−	1.80898i		−	0.0686186i
$696$		0			0
$697$	11.6130			0.439873
$698$		0			0
$699$			45.1055i			1.70605i
$700$		0			0
$701$			3.32697i			0.125658i	0.998024	+	0.0628289i	$0.0200122\pi$
$701$			3.32697i			0.125658i	−0.998024	+	0.0628289i	$0.979988\pi$
$702$		0			0
$703$	−24.2950			−0.916302
$704$		0			0
$705$			5.05302i			0.190308i
$706$		0			0
$707$	7.37662	+	0.520354i	0.277426	+	0.0195699i
$708$		0			0
$709$			17.0704i			0.641093i	0.947233	+	0.320546i	$0.103866\pi$
$709$			17.0704i			0.641093i	−0.947233	+	0.320546i	$0.896134\pi$
$710$		0			0
$711$			20.8063i			0.780296i
$712$		0			0
$713$			3.02835i			0.113412i
$714$		0			0
$715$	0.854681			0.0319633
$716$		0			0
$717$	43.0716			1.60854
$718$		0			0
$719$	7.05544			0.263124			0.131562	−	0.991308i	$-0.458001\pi$
$719$	7.05544			0.263124			0.131562	+	0.991308i	$0.458001\pi$
$720$		0			0
$721$	−3.33842	−	0.235495i	−0.124329	−	0.00877031i
$722$		0			0
$723$	−74.7985			−2.78179
$724$		0			0
$725$		−	24.7426i		−	0.918918i
$726$		0			0
$727$	−40.2279			−1.49197			−0.745986	−	0.665962i	$-0.768021\pi$
$727$	−40.2279			−1.49197			−0.745986	+	0.665962i	$0.768021\pi$
$728$		0			0
$729$	41.4135			1.53383
$730$		0			0
$731$		−	10.1723i		−	0.376236i
$732$		0			0
$733$	2.65544			0.0980807			0.0490404	−	0.998797i	$-0.484384\pi$
$733$	2.65544			0.0980807			0.0490404	+	0.998797i	$0.484384\pi$
$734$		0			0
$735$	−0.686181	+	4.83950i	−0.0253102	+	0.178508i
$736$		0			0
$737$	0.452205			0.0166572
$738$		0			0
$739$	39.2616			1.44426			0.722130	−	0.691757i	$-0.243163\pi$
$739$	39.2616			1.44426			0.722130	+	0.691757i	$0.243163\pi$
$740$		0			0
$741$	9.00515			0.330812
$742$		0			0
$743$		−	17.2808i		−	0.633973i	−0.948430	−	0.316986i	$-0.897329\pi$
$743$		−	17.2808i		−	0.633973i	0.948430	−	0.316986i	$-0.102671\pi$
$744$		0			0
$745$		−	3.10646i		−	0.113812i
$746$		0			0
$747$			55.9804i			2.04822i
$748$		0			0
$749$	−4.92632	−	0.347507i	−0.180004	−	0.0126976i
$750$		0			0
$751$			20.6714i			0.754311i	0.926150	+	0.377156i	$0.123098\pi$
$751$			20.6714i			0.754311i	−0.926150	+	0.377156i	$0.876902\pi$
$752$		0			0
$753$	3.95126			0.143992
$754$		0			0
$755$		−	4.27275i		−	0.155501i
$756$		0			0
$757$			30.7999i			1.11944i	0.828681	+	0.559721i	$0.189092\pi$
$757$			30.7999i			1.11944i	−0.828681	+	0.559721i	$0.810908\pi$
$758$		0			0
$759$	3.06305			0.111182
$760$		0			0
$761$		−	0.263604i		−	0.00955563i	−0.999989	−	0.00477781i	$-0.998479\pi$
$761$		−	0.263604i		−	0.00955563i	0.999989	−	0.00477781i	$-0.00152083\pi$
$762$		0			0
$763$	0.727977	−	10.3199i	0.0263546	−	0.373606i
$764$		0			0
$765$			1.15076i			0.0416058i
$766$		0			0
$767$			2.37654i			0.0858118i
$768$		0			0
$769$		−	15.8573i		−	0.571827i	−0.958255	−	0.285914i	$-0.907703\pi$
$769$		−	15.8573i		−	0.571827i	0.958255	−	0.285914i	$-0.0922971\pi$
$770$		0			0
$771$	−7.21061			−0.259684
$772$		0			0
$773$	42.5274			1.52961			0.764803	−	0.644264i	$-0.222836\pi$
$773$	42.5274			1.52961			0.764803	+	0.644264i	$0.222836\pi$
$774$		0			0
$775$	−42.0515			−1.51053
$776$		0			0
$777$	−50.1798	−	3.53973i	−1.80019	−	0.126987i
$778$		0			0
$779$	−36.4439			−1.30574
$780$		0			0
$781$		−	42.3456i		−	1.51524i
$782$		0			0
$783$	−13.9534			−0.498655
$784$		0			0
$785$	1.84994			0.0660271
$786$		0			0
$787$			31.7124i			1.13042i	0.824946	+	0.565212i	$0.191206\pi$
$787$			31.7124i			1.13042i	−0.824946	+	0.565212i	$0.808794\pi$
$788$		0			0
$789$	−13.1531			−0.468264
$790$		0			0
$791$	44.6721	+	3.15121i	1.58836	+	0.112044i
$792$		0			0
$793$	−5.15481			−0.183053
$794$		0			0
$795$	5.07451			0.179974
$796$		0			0
$797$	−29.4187			−1.04206			−0.521032	−	0.853537i	$-0.674453\pi$
$797$	−29.4187			−1.04206			−0.521032	+	0.853537i	$0.674453\pi$
$798$		0			0
$799$			7.82203i			0.276723i
$800$		0			0
$801$		−	6.31846i		−	0.223252i
$802$		0			0
$803$			44.3823i			1.56622i
$804$		0			0
$805$	0.0173886	−	0.246503i	0.000612867	−	0.00868810i
$806$		0			0
$807$		−	7.26359i		−	0.255690i
$808$		0			0
$809$	15.9671			0.561372			0.280686	−	0.959800i	$-0.409438\pi$
$809$	15.9671			0.561372			0.280686	+	0.959800i	$0.409438\pi$
$810$		0			0
$811$		−	34.9759i		−	1.22817i	−0.789240	−	0.614085i	$-0.789525\pi$
$811$		−	34.9759i		−	1.22817i	0.789240	−	0.614085i	$-0.210475\pi$
$812$		0			0
$813$		−	26.3014i		−	0.922430i
$814$		0			0
$815$	−5.58051			−0.195477
$816$		0			0
$817$			31.9228i			1.11684i
$818$		0			0
$819$	10.6820	+	0.753521i	0.373260	+	0.0263301i
$820$		0			0
$821$			15.2396i			0.531866i	0.963992	+	0.265933i	$0.0856800\pi$
$821$			15.2396i			0.531866i	−0.963992	+	0.265933i	$0.914320\pi$
$822$		0			0
$823$		−	3.15705i		−	0.110048i	−0.998485	−	0.0550240i	$-0.982476\pi$
$823$		−	3.15705i		−	0.110048i	0.998485	−	0.0550240i	$-0.0175235\pi$
$824$		0			0
$825$			42.5334i			1.48082i
$826$		0			0
$827$	7.84543			0.272812			0.136406	−	0.990653i	$-0.456445\pi$
$827$	7.84543			0.272812			0.136406	+	0.990653i	$0.456445\pi$
$828$		0			0
$829$	−17.3837			−0.603759			−0.301880	−	0.953346i	$-0.597614\pi$
$829$	−17.3837			−0.603759			−0.301880	+	0.953346i	$0.597614\pi$
$830$		0			0
$831$	68.1543			2.36425
$832$		0			0
$833$	−1.06220	+	7.49151i	−0.0368031	+	0.259565i
$834$		0			0
$835$	−0.403844			−0.0139756
$836$		0			0
$837$			23.7147i			0.819698i
$838$		0			0
$839$	−29.8626			−1.03097			−0.515485	−	0.856899i	$-0.672388\pi$
$839$	−29.8626			−1.03097			−0.515485	+	0.856899i	$0.672388\pi$
$840$		0			0
$841$	3.82011			0.131728
$842$		0			0
$843$			77.8796i			2.68232i
$844$		0			0
$845$	−0.263032			−0.00904858
$846$		0			0
$847$	−1.16592	−	0.0822449i	−0.0400614	−	0.00282597i
$848$		0			0
$849$	13.3177			0.457061
$850$		0			0
$851$	2.54322			0.0871806
$852$		0			0
$853$	−43.9029			−1.50321			−0.751603	−	0.659616i	$-0.770719\pi$
$853$	−43.9029			−1.50321			−0.751603	+	0.659616i	$0.770719\pi$
$854$		0			0
$855$		−	3.61132i		−	0.123505i
$856$		0			0
$857$			46.9619i			1.60419i	0.597198	+	0.802093i	$0.296280\pi$
$857$			46.9619i			1.60419i	−0.597198	+	0.802093i	$0.703720\pi$
$858$		0			0
$859$			10.5370i			0.359519i	0.983711	+	0.179759i	$0.0575319\pi$
$859$			10.5370i			0.359519i	−0.983711	+	0.179759i	$0.942468\pi$
$860$		0			0
$861$	−75.2727	−	5.30981i	−2.56529	−	0.180958i
$862$		0			0
$863$		−	36.1116i		−	1.22925i	−0.788818	−	0.614627i	$-0.789307\pi$
$863$		−	36.1116i		−	1.22925i	0.788818	−	0.614627i	$-0.210693\pi$
$864$		0			0
$865$	6.57960			0.223713
$866$		0			0
$867$		−	42.0283i		−	1.42735i
$868$		0			0
$869$		−	16.7035i		−	0.566627i
$870$		0			0
$871$	−0.139168			−0.00471553
$872$		0			0
$873$			17.6498i			0.597354i
$874$		0			0
$875$	6.89389	+	0.486302i	0.233056	+	0.0164400i
$876$		0			0
$877$			46.3298i			1.56445i	0.622999	+	0.782223i	$0.285914\pi$
$877$			46.3298i			1.56445i	−0.622999	+	0.782223i	$0.714086\pi$
$878$		0			0
$879$			65.6259i			2.21351i
$880$		0			0
$881$		−	13.7892i		−	0.464572i	−0.972648	−	0.232286i	$-0.925380\pi$
$881$		−	13.7892i		−	0.464572i	0.972648	−	0.232286i	$-0.0746205\pi$
$882$		0			0
$883$	−2.51811			−0.0847410			−0.0423705	−	0.999102i	$-0.513491\pi$
$883$	−2.51811			−0.0847410			−0.0423705	+	0.999102i	$0.513491\pi$
$884$		0			0
$885$	1.65947			0.0557825
$886$		0			0
$887$	−44.1632			−1.48286			−0.741428	−	0.671032i	$-0.765851\pi$
$887$	−44.1632			−1.48286			−0.741428	+	0.671032i	$0.765851\pi$
$888$		0			0
$889$	−0.473447	+	6.71166i	−0.0158789	+	0.225102i
$890$		0			0
$891$	−15.4683			−0.518208
$892$		0			0
$893$		−	24.5471i		−	0.821438i
$894$		0			0
$895$	2.48766			0.0831534
$896$		0			0
$897$	−0.942669			−0.0314748
$898$		0			0
$899$			42.7947i			1.42728i
$900$		0			0
$901$	7.85530			0.261698
$902$		0			0
$903$	−4.65109	+	65.9346i	−0.154779	+	2.19417i
$904$		0			0
$905$	1.39557			0.0463904
$906$		0			0
$907$	6.25761			0.207781			0.103890	−	0.994589i	$-0.466871\pi$
$907$	6.25761			0.207781			0.103890	+	0.994589i	$0.466871\pi$
$908$		0			0
$909$	−11.3128			−0.375221
$910$		0			0
$911$			38.5455i			1.27707i	0.769593	+	0.638535i	$0.220459\pi$
$911$			38.5455i			1.27707i	−0.769593	+	0.638535i	$0.779541\pi$
$912$		0			0
$913$		−	44.9417i		−	1.48735i
$914$		0			0
$915$			3.59946i			0.118994i
$916$		0			0
$917$	−0.602700	+	8.54397i	−0.0199029	+	0.282147i
$918$		0			0
$919$			8.55676i			0.282262i	0.989991	+	0.141131i	$0.0450738\pi$
$919$			8.55676i			0.282262i	−0.989991	+	0.141131i	$0.954926\pi$
$920$		0			0
$921$	68.6952			2.26358
$922$		0			0
$923$			13.0320i			0.428955i
$924$		0			0
$925$			35.3151i			1.16115i
$926$		0			0
$927$	5.11979			0.168156
$928$		0			0
$929$			38.4316i			1.26090i	0.776230	+	0.630449i	$0.217129\pi$
$929$			38.4316i			1.26090i	−0.776230	+	0.630449i	$0.782871\pi$
$930$		0			0
$931$	3.33341	−	23.5099i	0.109248	−	0.770505i
$932$		0			0
$933$		−	23.6040i		−	0.772759i
$934$		0			0
$935$		−	0.923842i		−	0.0302129i
$936$		0			0
$937$		−	42.8173i		−	1.39878i	−0.714740	−	0.699390i	$-0.753455\pi$
$937$		−	42.8173i		−	1.39878i	0.714740	−	0.699390i	$-0.246545\pi$
$938$		0			0
$939$	51.2408			1.67218
$940$		0			0
$941$	53.0715			1.73008			0.865040	−	0.501702i	$-0.167293\pi$
$941$	53.0715			1.73008			0.865040	+	0.501702i	$0.167293\pi$
$942$		0			0
$943$	3.81498			0.124233
$944$		0			0
$945$	0.136168	−	1.93034i	0.00442955	−	0.0627940i
$946$		0			0
$947$	21.3987			0.695365			0.347682	−	0.937612i	$-0.386969\pi$
$947$	21.3987			0.695365			0.347682	+	0.937612i	$0.386969\pi$
$948$		0			0
$949$		−	13.6589i		−	0.443385i
$950$		0			0
$951$	20.1373			0.652997
$952$		0			0
$953$	−54.3274			−1.75984			−0.879919	−	0.475124i	$-0.842403\pi$
$953$	−54.3274			−1.75984			−0.879919	+	0.475124i	$0.842403\pi$
$954$		0			0
$955$			4.16319i			0.134718i
$956$		0			0
$957$	43.2851			1.39921
$958$		0			0
$959$	−43.8422	−	3.09267i	−1.41574	−	0.0998677i
$960$		0			0
$961$	41.7320			1.34619
$962$		0			0
$963$	7.55499			0.243456
$964$		0			0
$965$	−0.468508			−0.0150818
$966$		0			0
$967$			46.1350i			1.48360i	0.670620	+	0.741801i	$0.266028\pi$
$967$			46.1350i			1.48360i	−0.670620	+	0.741801i	$0.733972\pi$
$968$		0			0
$969$		−	9.73385i		−	0.312696i
$970$		0			0
$971$		−	26.8761i		−	0.862496i	−0.902233	−	0.431248i	$-0.858073\pi$
$971$		−	26.8761i		−	0.862496i	0.902233	−	0.431248i	$-0.141927\pi$
$972$		0			0
$973$	−1.28038	+	18.1508i	−0.0410470	+	0.581889i
$974$		0			0
$975$		−	13.0899i		−	0.419211i
$976$		0			0
$977$	4.48165			0.143381			0.0716903	−	0.997427i	$-0.477161\pi$
$977$	4.48165			0.143381			0.0716903	+	0.997427i	$0.477161\pi$
$978$		0			0
$979$			5.07253i			0.162119i
$980$		0			0
$981$			15.8266i			0.505305i
$982$		0			0
$983$	−14.8553			−0.473809			−0.236905	−	0.971533i	$-0.576133\pi$
$983$	−14.8553			−0.473809			−0.236905	+	0.971533i	$0.576133\pi$
$984$		0			0
$985$		−	3.06160i		−	0.0975506i
$986$		0			0
$987$	3.57647	−	50.7006i	0.113840	−	1.61382i
$988$		0			0
$989$		−	3.34171i		−	0.106260i
$990$		0			0
$991$		−	0.597856i		−	0.0189915i	−0.999955	−	0.00949576i	$-0.996977\pi$
$991$		−	0.597856i		−	0.0189915i	0.999955	−	0.00949576i	$-0.00302264\pi$
$992$		0			0
$993$			72.7943i			2.31006i
$994$		0			0
$995$	−2.54522			−0.0806890
$996$		0			0
$997$	26.8353			0.849884			0.424942	−	0.905221i	$-0.360294\pi$
$997$	26.8353			0.849884			0.424942	+	0.905221i	$0.360294\pi$
$998$		0			0
$999$	19.9157			0.630105

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2912.2.h.a.2575.6		48
4.3	odd	2		728.2.h.a.27.33	✓	48
7.6	odd	2		2912.2.h.b.2575.43		48
8.3	odd	2		2912.2.h.b.2575.6		48
8.5	even	2		728.2.h.b.27.34	yes	48
28.27	even	2		728.2.h.b.27.33	yes	48
56.13	odd	2		728.2.h.a.27.34	yes	48
56.27	even	2	inner	2912.2.h.a.2575.43		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
728.2.h.a.27.33	✓	48	4.3	odd	2
728.2.h.a.27.34	yes	48	56.13	odd	2
728.2.h.b.27.33	yes	48	28.27	even	2
728.2.h.b.27.34	yes	48	8.5	even	2
2912.2.h.a.2575.6		48	1.1	even	1	trivial
2912.2.h.a.2575.43		48	56.27	even	2	inner
2912.2.h.b.2575.6		48	8.3	odd	2
2912.2.h.b.2575.43		48	7.6	odd	2

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.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.65471i q^{3} -0.263032 q^{5} +(2.63919 + 0.186171i) q^{7} -4.04746 q^{9} +3.24934 q^{11} -1.00000 q^{13} +0.698273i q^{15} +1.08092i q^{17} -3.39215i q^{19} +(0.494230 - 7.00628i) q^{21} +0.355094i q^{23} -4.93081 q^{25} +2.78070i q^{27} +5.01796i q^{29} +8.52830 q^{31} -8.62605i q^{33} +(-0.694192 - 0.0489690i) q^{35} -7.16212i q^{37} +2.65471i q^{39} -10.7436i q^{41} -9.41079 q^{43} +1.06461 q^{45} +7.23645 q^{47} +(6.93068 + 0.982684i) q^{49} +2.86952 q^{51} -7.26724i q^{53} -0.854681 q^{55} -9.00515 q^{57} -2.37654i q^{59} +5.15481 q^{61} +(-10.6820 - 0.753521i) q^{63} +0.263032 q^{65} +0.139168 q^{67} +0.942669 q^{69} -13.0320i q^{71} +13.6589i q^{73} +13.0899i q^{75} +(8.57564 + 0.604934i) q^{77} -5.14057i q^{79} -4.76044 q^{81} -13.8310i q^{83} -0.284317i q^{85} +13.3212 q^{87} +1.56109i q^{89} +(-2.63919 - 0.186171i) q^{91} -22.6401i q^{93} +0.892243i q^{95} -4.36070i q^{97} -13.1516 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 48 q^{9} + 4 q^{11} - 48 q^{13} + 48 q^{25} - 12 q^{35} + 4 q^{43} + 24 q^{45} + 40 q^{51} - 20 q^{63} + 4 q^{67} - 20 q^{77} + 64 q^{81} + 40 q^{87} - 20 q^{99}+O(q^{100}) \) 48 * q - 48 * q^9 + 4 * q^11 - 48 * q^13 + 48 * q^25 - 12 * q^35 + 4 * q^43 + 24 * q^45 + 40 * q^51 - 20 * q^63 + 4 * q^67 - 20 * q^77 + 64 * q^81 + 40 * q^87 - 20 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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