LMFDB - Embedded newform 1216.3.g.e.417.21

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1216,3,Mod(417,1216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1216, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1216.417");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1216 = 2^{6} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1216.g (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:	$33.1336001462$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			417.21
Character	$\chi$	$=$	1216.417
Dual form			1216.3.g.e.417.24

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.08780 q^{3} +2.09363i q^{5} -3.72137 q^{7} -7.81670 q^{9} +O(q^{10})$	$q-1.08780 q^{3} +2.09363i q^{5} -3.72137 q^{7} -7.81670 q^{9} +6.64392i q^{11} -9.32905 q^{13} -2.27744i q^{15} -2.02233 q^{17} +(-17.3333 - 7.78179i) q^{19} +4.04809 q^{21} -7.73929 q^{23} +20.6167 q^{25} +18.2931 q^{27} +35.3394 q^{29} -37.0375i q^{31} -7.22723i q^{33} -7.79115i q^{35} +33.2473 q^{37} +10.1481 q^{39} +57.3302i q^{41} -28.3128i q^{43} -16.3653i q^{45} -73.9638 q^{47} -35.1514 q^{49} +2.19989 q^{51} +59.7645 q^{53} -13.9099 q^{55} +(18.8551 + 8.46500i) q^{57} +60.0313 q^{59} -18.9084i q^{61} +29.0888 q^{63} -19.5316i q^{65} +91.3968 q^{67} +8.41877 q^{69} -83.7992i q^{71} +10.2134 q^{73} -22.4268 q^{75} -24.7245i q^{77} -58.8531i q^{79} +50.4511 q^{81} -5.17199i q^{83} -4.23401i q^{85} -38.4421 q^{87} +145.970i q^{89} +34.7168 q^{91} +40.2893i q^{93} +(16.2922 - 36.2895i) q^{95} -39.3499i q^{97} -51.9335i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q + 264 q^{9}+O(q^{10}) $ 48 * q + 264 * q^9	$ 48 q + 264 q^{9} + 24 q^{17} - 160 q^{25} + 328 q^{49} + 8 q^{57} - 472 q^{73} - 736 q^{81}+O(q^{100}) $ 48 * q + 264 * q^9 + 24 * q^17 - 160 * q^25 + 328 * q^49 + 8 * q^57 - 472 * q^73 - 736 * q^81

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−1.08780			−0.362599			−0.181299	−	0.983428i	$-0.558030\pi$
$3$	−1.08780			−0.362599			−0.181299	+	0.983428i	$0.558030\pi$
$4$		0			0
$5$			2.09363i			0.418725i	0.977838	+	0.209363i	$0.0671389\pi$
$5$			2.09363i			0.418725i	−0.977838	+	0.209363i	$0.932861\pi$
$6$		0			0
$7$	−3.72137			−0.531624			−0.265812	−	0.964025i	$-0.585640\pi$
$7$	−3.72137			−0.531624			−0.265812	+	0.964025i	$0.585640\pi$
$8$		0			0
$9$	−7.81670			−0.868522
$10$		0			0
$11$			6.64392i			0.603993i	0.953309	+	0.301996i	$0.0976530\pi$
$11$			6.64392i			0.603993i	−0.953309	+	0.301996i	$0.902347\pi$
$12$		0			0
$13$	−9.32905			−0.717619			−0.358810	−	0.933411i	$-0.616817\pi$
$13$	−9.32905			−0.717619			−0.358810	+	0.933411i	$0.616817\pi$
$14$		0			0
$15$		−	2.27744i		−	0.151829i
$16$		0			0
$17$	−2.02233			−0.118961			−0.0594804	−	0.998229i	$-0.518944\pi$
$17$	−2.02233			−0.118961			−0.0594804	+	0.998229i	$0.518944\pi$
$18$		0			0
$19$	−17.3333	−	7.78179i	−0.912280	−	0.409568i
$19$	−17.3333	−	7.78179i	−0.912280	−	0.409568i
$20$		0			0
$21$	4.04809			0.192766
$22$		0			0
$23$	−7.73929			−0.336491			−0.168245	−	0.985745i	$-0.553810\pi$
$23$	−7.73929			−0.336491			−0.168245	+	0.985745i	$0.553810\pi$
$24$		0			0
$25$	20.6167			0.824669
$26$		0			0
$27$	18.2931			0.677524
$28$		0			0
$29$	35.3394			1.21860			0.609300	−	0.792939i	$-0.291450\pi$
$29$	35.3394			1.21860			0.609300	+	0.792939i	$0.291450\pi$
$30$		0			0
$31$		−	37.0375i		−	1.19476i	−0.801959	−	0.597379i	$-0.796209\pi$
$31$		−	37.0375i		−	1.19476i	0.801959	−	0.597379i	$-0.203791\pi$
$32$		0			0
$33$		−	7.22723i		−	0.219007i
$34$		0			0
$35$		−	7.79115i		−	0.222604i
$36$		0			0
$37$	33.2473			0.898577			0.449288	−	0.893387i	$-0.351678\pi$
$37$	33.2473			0.898577			0.449288	+	0.893387i	$0.351678\pi$
$38$		0			0
$39$	10.1481			0.260208
$40$		0			0
$41$			57.3302i			1.39830i	0.714977	+	0.699148i	$0.246437\pi$
$41$			57.3302i			1.39830i	−0.714977	+	0.699148i	$0.753563\pi$
$42$		0			0
$43$		−	28.3128i		−	0.658437i	−0.944254	−	0.329219i	$-0.893215\pi$
$43$		−	28.3128i		−	0.658437i	0.944254	−	0.329219i	$-0.106785\pi$
$44$		0			0
$45$		−	16.3653i		−	0.363672i
$46$		0			0
$47$	−73.9638			−1.57370			−0.786849	−	0.617146i	$-0.788289\pi$
$47$	−73.9638			−1.57370			−0.786849	+	0.617146i	$0.788289\pi$
$48$		0			0
$49$	−35.1514			−0.717376
$50$		0			0
$51$	2.19989			0.0431350
$52$		0			0
$53$	59.7645			1.12763			0.563816	−	0.825900i	$-0.309333\pi$
$53$	59.7645			1.12763			0.563816	+	0.825900i	$0.309333\pi$
$54$		0			0
$55$	−13.9099			−0.252907
$56$		0			0
$57$	18.8551	+	8.46500i	0.330792	+	0.148509i
$58$		0			0
$59$	60.0313			1.01748			0.508740	−	0.860920i	$-0.330112\pi$
$59$	60.0313			1.01748			0.508740	+	0.860920i	$0.330112\pi$
$60$		0			0
$61$		−	18.9084i		−	0.309974i	−0.987917	−	0.154987i	$-0.950466\pi$
$61$		−	18.9084i		−	0.309974i	0.987917	−	0.154987i	$-0.0495335\pi$
$62$		0			0
$63$	29.0888			0.461727
$64$		0			0
$65$		−	19.5316i		−	0.300485i
$66$		0			0
$67$	91.3968			1.36413			0.682066	−	0.731291i	$-0.261082\pi$
$67$	91.3968			1.36413			0.682066	+	0.731291i	$0.261082\pi$
$68$		0			0
$69$	8.41877			0.122011
$70$		0			0
$71$		−	83.7992i		−	1.18027i	−0.807304	−	0.590135i	$-0.799074\pi$
$71$		−	83.7992i		−	1.18027i	0.807304	−	0.590135i	$-0.200926\pi$
$72$		0			0
$73$	10.2134			0.139910			0.0699551	−	0.997550i	$-0.477714\pi$
$73$	10.2134			0.139910			0.0699551	+	0.997550i	$0.477714\pi$
$74$		0			0
$75$	−22.4268			−0.299024
$76$		0			0
$77$		−	24.7245i		−	0.321097i
$78$		0			0
$79$		−	58.8531i		−	0.744976i	−0.928037	−	0.372488i	$-0.878505\pi$
$79$		−	58.8531i		−	0.744976i	0.928037	−	0.372488i	$-0.121495\pi$
$80$		0			0
$81$	50.4511			0.622853
$82$		0			0
$83$		−	5.17199i		−	0.0623131i	−0.999515	−	0.0311566i	$-0.990081\pi$
$83$		−	5.17199i		−	0.0623131i	0.999515	−	0.0311566i	$-0.00991905\pi$
$84$		0			0
$85$		−	4.23401i		−	0.0498119i
$86$		0			0
$87$	−38.4421			−0.441863
$88$		0			0
$89$			145.970i			1.64011i	0.572284	+	0.820056i	$0.306058\pi$
$89$			145.970i			1.64011i	−0.572284	+	0.820056i	$0.693942\pi$
$90$		0			0
$91$	34.7168			0.381503
$92$		0			0
$93$			40.2893i			0.433218i
$94$		0			0
$95$	16.2922	−	36.2895i	0.171496	−	0.381995i
$96$		0			0
$97$		−	39.3499i		−	0.405669i	−0.979213	−	0.202835i	$-0.934985\pi$
$97$		−	39.3499i		−	0.405669i	0.979213	−	0.202835i	$-0.0650154\pi$
$98$		0			0
$99$		−	51.9335i		−	0.524581i
$100$		0			0
$101$		−	137.581i		−	1.36219i	−0.732195	−	0.681095i	$-0.761504\pi$
$101$		−	137.581i		−	1.36219i	0.732195	−	0.681095i	$-0.238496\pi$
$102$		0			0
$103$		−	6.09975i		−	0.0592209i	−0.999562	−	0.0296104i	$-0.990573\pi$
$103$		−	6.09975i		−	0.0592209i	0.999562	−	0.0296104i	$-0.00942668\pi$
$104$		0			0
$105$			8.47519i			0.0807161i
$106$		0			0
$107$	−23.4079			−0.218765			−0.109383	−	0.994000i	$-0.534887\pi$
$107$	−23.4079			−0.218765			−0.109383	+	0.994000i	$0.534887\pi$
$108$		0			0
$109$	98.8964			0.907306			0.453653	−	0.891178i	$-0.350121\pi$
$109$	98.8964			0.907306			0.453653	+	0.891178i	$0.350121\pi$
$110$		0			0
$111$	−36.1663			−0.325823
$112$		0			0
$113$			51.2397i			0.453449i	0.973959	+	0.226724i	$0.0728017\pi$
$113$			51.2397i			0.453449i	−0.973959	+	0.226724i	$0.927198\pi$
$114$		0			0
$115$		−	16.2032i		−	0.140897i
$116$		0			0
$117$	72.9224			0.623268
$118$		0			0
$119$	7.52584			0.0632424
$120$		0			0
$121$	76.8583			0.635193
$122$		0			0
$123$		−	62.3635i		−	0.507021i
$124$		0			0
$125$			95.5044i			0.764035i
$126$		0			0
$127$		−	138.012i		−	1.08671i	−0.839504	−	0.543354i	$-0.817154\pi$
$127$		−	138.012i		−	1.08671i	0.839504	−	0.543354i	$-0.182846\pi$
$128$		0			0
$129$			30.7986i			0.238749i
$130$		0			0
$131$			11.8847i			0.0907231i	0.998971	+	0.0453616i	$0.0144440\pi$
$131$			11.8847i			0.0907231i	−0.998971	+	0.0453616i	$0.985556\pi$
$132$		0			0
$133$	64.5036	+	28.9589i	0.484989	+	0.217736i
$134$		0			0
$135$			38.2990i			0.283697i
$136$		0			0
$137$	83.3482			0.608381			0.304190	−	0.952611i	$-0.401614\pi$
$137$	83.3482			0.608381			0.304190	+	0.952611i	$0.401614\pi$
$138$		0			0
$139$			206.396i			1.48487i	0.669920	+	0.742433i	$0.266328\pi$
$139$			206.396i			1.48487i	−0.669920	+	0.742433i	$0.733672\pi$
$140$		0			0
$141$	80.4576			0.570621
$142$		0			0
$143$		−	61.9814i		−	0.433437i
$144$		0			0
$145$			73.9876i			0.510259i
$146$		0			0
$147$	38.2376			0.260120
$148$		0			0
$149$		−	219.150i		−	1.47081i	−0.677631	−	0.735403i	$-0.736993\pi$
$149$		−	219.150i		−	1.47081i	0.677631	−	0.735403i	$-0.263007\pi$
$150$		0			0
$151$		−	147.132i		−	0.974383i	−0.873295	−	0.487192i	$-0.838021\pi$
$151$		−	147.132i		−	0.974383i	0.873295	−	0.487192i	$-0.161979\pi$
$152$		0			0
$153$	15.8080			0.103320
$154$		0			0
$155$	77.5427			0.500276
$156$		0			0
$157$		−	247.777i		−	1.57820i	−0.614266	−	0.789099i	$-0.710548\pi$
$157$		−	247.777i		−	1.57820i	0.614266	−	0.789099i	$-0.289452\pi$
$158$		0			0
$159$	−65.0116			−0.408878
$160$		0			0
$161$	28.8007			0.178886
$162$		0			0
$163$		−	176.310i		−	1.08166i	−0.841133	−	0.540828i	$-0.818111\pi$
$163$		−	176.310i		−	1.08166i	0.841133	−	0.540828i	$-0.181889\pi$
$164$		0			0
$165$	15.1311			0.0917038
$166$		0			0
$167$		−	216.583i		−	1.29690i	−0.761257	−	0.648451i	$-0.775417\pi$
$167$		−	216.583i		−	1.29690i	0.761257	−	0.648451i	$-0.224583\pi$
$168$		0			0
$169$	−81.9689			−0.485023
$170$		0			0
$171$	135.489	+	60.8279i	0.792335	+	0.355719i
$172$		0			0
$173$	130.712			0.755563			0.377781	−	0.925895i	$-0.376687\pi$
$173$	130.712			0.755563			0.377781	+	0.925895i	$0.376687\pi$
$174$		0			0
$175$	−76.7224			−0.438413
$176$		0			0
$177$	−65.3018			−0.368937
$178$		0			0
$179$	24.3199			0.135866			0.0679328	−	0.997690i	$-0.478360\pi$
$179$	24.3199			0.135866			0.0679328	+	0.997690i	$0.478360\pi$
$180$		0			0
$181$	98.1544			0.542290			0.271145	−	0.962539i	$-0.412598\pi$
$181$	98.1544			0.542290			0.271145	+	0.962539i	$0.412598\pi$
$182$		0			0
$183$			20.5685i			0.112396i
$184$		0			0
$185$			69.6076i			0.376257i
$186$		0			0
$187$		−	13.4362i		−	0.0718515i
$188$		0			0
$189$	−68.0755			−0.360188
$190$		0			0
$191$	92.1966			0.482705			0.241352	−	0.970438i	$-0.422409\pi$
$191$	92.1966			0.482705			0.241352	+	0.970438i	$0.422409\pi$
$192$		0			0
$193$		−	168.238i		−	0.871697i	−0.900020	−	0.435848i	$-0.856448\pi$
$193$		−	168.238i		−	0.871697i	0.900020	−	0.435848i	$-0.143552\pi$
$194$		0			0
$195$			21.2464i			0.108956i
$196$		0			0
$197$			223.927i			1.13668i	0.822793	+	0.568341i	$0.192415\pi$
$197$			223.927i			1.13668i	−0.822793	+	0.568341i	$0.807585\pi$
$198$		0			0
$199$	−176.637			−0.887621			−0.443811	−	0.896121i	$-0.646374\pi$
$199$	−176.637			−0.887621			−0.443811	+	0.896121i	$0.646374\pi$
$200$		0			0
$201$	−99.4211			−0.494632
$202$		0			0
$203$	−131.511			−0.647837
$204$		0			0
$205$	−120.028			−0.585502
$206$		0			0
$207$	60.4957			0.292250
$208$		0			0
$209$	51.7016	−	115.161i	0.247376	−	0.551010i
$210$		0			0
$211$	−63.0357			−0.298747			−0.149374	−	0.988781i	$-0.547726\pi$
$211$	−63.0357			−0.298747			−0.149374	+	0.988781i	$0.547726\pi$
$212$		0			0
$213$			91.1565i			0.427965i
$214$		0			0
$215$	59.2765			0.275704
$216$		0			0
$217$			137.830i			0.635162i
$218$		0			0
$219$	−11.1101			−0.0507313
$220$		0			0
$221$	18.8664			0.0853685
$222$		0			0
$223$			380.926i			1.70819i	0.520117	+	0.854095i	$0.325888\pi$
$223$			380.926i			1.70819i	−0.520117	+	0.854095i	$0.674112\pi$
$224$		0			0
$225$	−161.155			−0.716243
$226$		0			0
$227$	223.454			0.984378			0.492189	−	0.870488i	$-0.336197\pi$
$227$	223.454			0.984378			0.492189	+	0.870488i	$0.336197\pi$
$228$		0			0
$229$			70.1574i			0.306364i	0.988198	+	0.153182i	$0.0489521\pi$
$229$			70.1574i			0.306364i	−0.988198	+	0.153182i	$0.951048\pi$
$230$		0			0
$231$			26.8952i			0.116429i
$232$		0			0
$233$	254.338			1.09158			0.545789	−	0.837923i	$-0.316230\pi$
$233$	254.338			1.09158			0.545789	+	0.837923i	$0.316230\pi$
$234$		0			0
$235$		−	154.853i		−	0.658947i
$236$		0			0
$237$			64.0202i			0.270127i
$238$		0			0
$239$	−78.6199			−0.328954			−0.164477	−	0.986381i	$-0.552594\pi$
$239$	−78.6199			−0.328954			−0.164477	+	0.986381i	$0.552594\pi$
$240$		0			0
$241$		−	273.046i		−	1.13297i	−0.824072	−	0.566485i	$-0.808303\pi$
$241$		−	273.046i		−	1.13297i	0.824072	−	0.566485i	$-0.191697\pi$
$242$		0			0
$243$	−219.519			−0.903370
$244$		0			0
$245$		−	73.5940i		−	0.300384i
$246$		0			0
$247$	161.703	+	72.5967i	0.654669	+	0.293914i
$248$		0			0
$249$			5.62607i			0.0225947i
$250$		0			0
$251$		−	119.607i		−	0.476521i	−0.971201	−	0.238261i	$-0.923423\pi$
$251$		−	119.607i		−	0.476521i	0.971201	−	0.238261i	$-0.0765772\pi$
$252$		0			0
$253$		−	51.4192i		−	0.203238i
$254$		0			0
$255$			4.60574i			0.0180617i
$256$		0			0
$257$		−	151.484i		−	0.589434i	−0.955585	−	0.294717i	$-0.904775\pi$
$257$		−	151.484i		−	0.589434i	0.955585	−	0.294717i	$-0.0952253\pi$
$258$		0			0
$259$	−123.726			−0.477705
$260$		0			0
$261$	−276.238			−1.05838
$262$		0			0
$263$	−60.4679			−0.229916			−0.114958	−	0.993370i	$-0.536673\pi$
$263$	−60.4679			−0.229916			−0.114958	+	0.993370i	$0.536673\pi$
$264$		0			0
$265$			125.125i			0.472168i
$266$		0			0
$267$		−	158.786i		−	0.594702i
$268$		0			0
$269$	−43.9742			−0.163473			−0.0817365	−	0.996654i	$-0.526047\pi$
$269$	−43.9742			−0.163473			−0.0817365	+	0.996654i	$0.526047\pi$
$270$		0			0
$271$	271.639			1.00236			0.501180	−	0.865343i	$-0.332900\pi$
$271$	271.639			1.00236			0.501180	+	0.865343i	$0.332900\pi$
$272$		0			0
$273$	−37.7648			−0.138333
$274$		0			0
$275$			136.976i			0.498094i
$276$		0			0
$277$			202.731i			0.731880i	0.930638	+	0.365940i	$0.119253\pi$
$277$			202.731i			0.731880i	−0.930638	+	0.365940i	$0.880747\pi$
$278$		0			0
$279$			289.511i			1.03767i
$280$		0			0
$281$			352.493i			1.25443i	0.778848	+	0.627213i	$0.215804\pi$
$281$			352.493i			1.25443i	−0.778848	+	0.627213i	$0.784196\pi$
$282$		0			0
$283$			435.032i			1.53722i	0.639720	+	0.768608i	$0.279050\pi$
$283$			435.032i			1.53722i	−0.639720	+	0.768608i	$0.720950\pi$
$284$		0			0
$285$	−17.7226	+	39.4756i	−0.0621844	+	0.138511i
$286$		0			0
$287$		−	213.346i		−	0.743367i
$288$		0			0
$289$	−284.910			−0.985848
$290$		0			0
$291$			42.8047i			0.147095i
$292$		0			0
$293$	−223.976			−0.764425			−0.382212	−	0.924075i	$-0.624838\pi$
$293$	−223.976			−0.764425			−0.382212	+	0.924075i	$0.624838\pi$
$294$		0			0
$295$			125.683i			0.426045i
$296$		0			0
$297$			121.538i			0.409219i
$298$		0			0
$299$	72.2002			0.241472
$300$		0			0
$301$			105.362i			0.350041i
$302$		0			0
$303$			149.660i			0.493928i
$304$		0			0
$305$	39.5871			0.129794
$306$		0			0
$307$	−216.541			−0.705346			−0.352673	−	0.935747i	$-0.614727\pi$
$307$	−216.541			−0.705346			−0.352673	+	0.935747i	$0.614727\pi$
$308$		0			0
$309$			6.63529i			0.0214734i
$310$		0			0
$311$	−353.673			−1.13721			−0.568606	−	0.822610i	$-0.692517\pi$
$311$	−353.673			−1.13721			−0.568606	+	0.822610i	$0.692517\pi$
$312$		0			0
$313$	9.55967			0.0305421			0.0152710	−	0.999883i	$-0.495139\pi$
$313$	9.55967			0.0305421			0.0152710	+	0.999883i	$0.495139\pi$
$314$		0			0
$315$			60.9011i			0.193337i
$316$		0			0
$317$	53.9855			0.170301			0.0851507	−	0.996368i	$-0.472863\pi$
$317$	53.9855			0.170301			0.0851507	+	0.996368i	$0.472863\pi$
$318$		0			0
$319$			234.792i			0.736026i
$320$		0			0
$321$	25.4630			0.0793241
$322$		0			0
$323$	35.0537	+	15.7374i	0.108526	+	0.0487225i
$324$		0			0
$325$	−192.334			−0.591798
$326$		0			0
$327$	−107.579			−0.328988
$328$		0			0
$329$	275.246			0.836615
$330$		0			0
$331$	−221.010			−0.667703			−0.333852	−	0.942626i	$-0.608348\pi$
$331$	−221.010			−0.667703			−0.333852	+	0.942626i	$0.608348\pi$
$332$		0			0
$333$	−259.884			−0.780434
$334$		0			0
$335$			191.351i			0.571196i
$336$		0			0
$337$		−	481.993i		−	1.43025i	−0.698998	−	0.715124i	$-0.746370\pi$
$337$		−	481.993i		−	1.43025i	0.698998	−	0.715124i	$-0.253630\pi$
$338$		0			0
$339$		−	55.7384i		−	0.164420i
$340$		0			0
$341$	246.074			0.721625
$342$		0			0
$343$	313.158			0.912998
$344$		0			0
$345$			17.6258i			0.0510892i
$346$		0			0
$347$		−	340.238i		−	0.980514i	−0.871578	−	0.490257i	$-0.836903\pi$
$347$		−	340.238i		−	0.980514i	0.871578	−	0.490257i	$-0.163097\pi$
$348$		0			0
$349$			293.467i			0.840880i	0.907320	+	0.420440i	$0.138124\pi$
$349$			293.467i			0.840880i	−0.907320	+	0.420440i	$0.861876\pi$
$350$		0			0
$351$	−170.658			−0.486204
$352$		0			0
$353$	−545.177			−1.54441			−0.772206	−	0.635372i	$-0.780847\pi$
$353$	−545.177			−1.54441			−0.772206	+	0.635372i	$0.780847\pi$
$354$		0			0
$355$	175.444			0.494209
$356$		0			0
$357$	−8.18658			−0.0229316
$358$		0			0
$359$	315.249			0.878130			0.439065	−	0.898455i	$-0.355310\pi$
$359$	315.249			0.878130			0.439065	+	0.898455i	$0.355310\pi$
$360$		0			0
$361$	239.888	+	269.768i	0.664509	+	0.747281i
$362$		0			0
$363$	−83.6062			−0.230320
$364$		0			0
$365$			21.3831i			0.0585840i
$366$		0			0
$367$	111.782			0.304583			0.152292	−	0.988336i	$-0.451335\pi$
$367$	111.782			0.304583			0.152292	+	0.988336i	$0.451335\pi$
$368$		0			0
$369$		−	448.133i		−	1.21445i
$370$		0			0
$371$	−222.406			−0.599476
$372$		0			0
$373$	−633.441			−1.69823			−0.849117	−	0.528205i	$-0.822865\pi$
$373$	−633.441			−1.69823			−0.849117	+	0.528205i	$0.822865\pi$
$374$		0			0
$375$		−	103.889i		−	0.277038i
$376$		0			0
$377$	−329.683			−0.874491
$378$		0			0
$379$	−231.839			−0.611712			−0.305856	−	0.952078i	$-0.598943\pi$
$379$	−231.839			−0.611712			−0.305856	+	0.952078i	$0.598943\pi$
$380$		0			0
$381$			150.129i			0.394039i
$382$		0			0
$383$			180.106i			0.470250i	0.971965	+	0.235125i	$0.0755500\pi$
$383$			180.106i			0.470250i	−0.971965	+	0.235125i	$0.924450\pi$
$384$		0			0
$385$	51.7638			0.134451
$386$		0			0
$387$			221.313i			0.571867i
$388$		0			0
$389$			35.1789i			0.0904342i	0.998977	+	0.0452171i	$0.0143980\pi$
$389$			35.1789i			0.0904342i	−0.998977	+	0.0452171i	$0.985602\pi$
$390$		0			0
$391$	15.6514			0.0400292
$392$		0			0
$393$		−	12.9282i		−	0.0328961i
$394$		0			0
$395$	123.216			0.311940
$396$		0			0
$397$		−	341.490i		−	0.860177i	−0.902787	−	0.430089i	$-0.858482\pi$
$397$		−	341.490i		−	0.860177i	0.902787	−	0.430089i	$-0.141518\pi$
$398$		0			0
$399$	−70.1668	−	31.5014i	−0.175857	−	0.0789508i
$400$		0			0
$401$		−	163.382i		−	0.407438i	−0.979029	−	0.203719i	$-0.934697\pi$
$401$		−	163.382i		−	0.407438i	0.979029	−	0.203719i	$-0.0653028\pi$
$402$		0			0
$403$			345.525i			0.857381i
$404$		0			0
$405$			105.626i			0.260804i
$406$		0			0
$407$			220.893i			0.542734i
$408$		0			0
$409$			266.286i			0.651065i	0.945531	+	0.325533i	$0.105544\pi$
$409$			266.286i			0.651065i	−0.945531	+	0.325533i	$0.894456\pi$
$410$		0			0
$411$	−90.6659			−0.220598
$412$		0			0
$413$	−223.398			−0.540916
$414$		0			0
$415$	10.8282			0.0260921
$416$		0			0
$417$		−	224.517i		−	0.538411i
$418$		0			0
$419$		−	139.535i		−	0.333018i	−0.986040	−	0.166509i	$-0.946750\pi$
$419$		−	139.535i		−	0.333018i	0.986040	−	0.166509i	$-0.0532495\pi$
$420$		0			0
$421$	−210.410			−0.499787			−0.249893	−	0.968273i	$-0.580396\pi$
$421$	−210.410			−0.499787			−0.249893	+	0.968273i	$0.580396\pi$
$422$		0			0
$423$	578.153			1.36679
$424$		0			0
$425$	−41.6939			−0.0981033
$426$		0			0
$427$			70.3650i			0.164789i
$428$		0			0
$429$			67.4232i			0.157164i
$430$		0			0
$431$		−	141.538i		−	0.328395i	−0.986427	−	0.164198i	$-0.947496\pi$
$431$		−	141.538i		−	0.328395i	0.986427	−	0.164198i	$-0.0525035\pi$
$432$		0			0
$433$		−	790.795i		−	1.82632i	−0.407605	−	0.913158i	$-0.633636\pi$
$433$		−	790.795i		−	1.82632i	0.407605	−	0.913158i	$-0.366364\pi$
$434$		0			0
$435$		−	80.4834i		−	0.185019i
$436$		0			0
$437$	134.148	+	60.2255i	0.306974	+	0.137816i
$438$		0			0
$439$			751.697i			1.71229i	0.516732	+	0.856147i	$0.327148\pi$
$439$			751.697i			1.71229i	−0.516732	+	0.856147i	$0.672852\pi$
$440$		0			0
$441$	274.768			0.623057
$442$		0			0
$443$			415.267i			0.937397i	0.883358	+	0.468698i	$0.155277\pi$
$443$			415.267i			0.937397i	−0.883358	+	0.468698i	$0.844723\pi$
$444$		0			0
$445$	−305.607			−0.686756
$446$		0			0
$447$			238.391i			0.533312i
$448$		0			0
$449$		−	768.511i		−	1.71161i	−0.517302	−	0.855803i	$-0.673064\pi$
$449$		−	768.511i		−	1.71161i	0.517302	−	0.855803i	$-0.326936\pi$
$450$		0			0
$451$	−380.897			−0.844561
$452$		0			0
$453$			160.050i			0.353310i
$454$		0			0
$455$			72.6840i			0.159745i
$456$		0			0
$457$	−75.9879			−0.166275			−0.0831377	−	0.996538i	$-0.526494\pi$
$457$	−75.9879			−0.166275			−0.0831377	+	0.996538i	$0.526494\pi$
$458$		0			0
$459$	−36.9948			−0.0805988
$460$		0			0
$461$		−	40.6371i		−	0.0881499i	−0.999028	−	0.0440750i	$-0.985966\pi$
$461$		−	40.6371i		−	0.0881499i	0.999028	−	0.0440750i	$-0.0140340\pi$
$462$		0			0
$463$	14.7651			0.0318901			0.0159450	−	0.999873i	$-0.494924\pi$
$463$	14.7651			0.0318901			0.0159450	+	0.999873i	$0.494924\pi$
$464$		0			0
$465$	−84.3507			−0.181399
$466$		0			0
$467$		−	273.231i		−	0.585078i	−0.956254	−	0.292539i	$-0.905500\pi$
$467$		−	273.231i		−	0.585078i	0.956254	−	0.292539i	$-0.0945001\pi$
$468$		0			0
$469$	−340.121			−0.725204
$470$		0			0
$471$			269.531i			0.572253i
$472$		0			0
$473$	188.108			0.397691
$474$		0			0
$475$	−357.356	−	160.435i	−0.752329	−	0.337758i
$476$		0			0
$477$	−467.161			−0.979374
$478$		0			0
$479$	573.348			1.19697			0.598485	−	0.801134i	$-0.295770\pi$
$479$	573.348			1.19697			0.598485	+	0.801134i	$0.295770\pi$
$480$		0			0
$481$	−310.166			−0.644836
$482$		0			0
$483$	−31.3293			−0.0648640
$484$		0			0
$485$	82.3840			0.169864
$486$		0			0
$487$			101.704i			0.208838i	0.994533	+	0.104419i	$0.0332984\pi$
$487$			101.704i			0.208838i	−0.994533	+	0.104419i	$0.966702\pi$
$488$		0			0
$489$			191.789i			0.392207i
$490$		0			0
$491$			157.940i			0.321670i	0.986981	+	0.160835i	$0.0514187\pi$
$491$			157.940i			0.321670i	−0.986981	+	0.160835i	$0.948581\pi$
$492$		0			0
$493$	−71.4681			−0.144966
$494$		0			0
$495$	108.729			0.219655
$496$		0			0
$497$			311.848i			0.627460i
$498$		0			0
$499$			623.133i			1.24876i	0.781119	+	0.624382i	$0.214649\pi$
$499$			623.133i			1.24876i	−0.781119	+	0.624382i	$0.785351\pi$
$500$		0			0
$501$			235.598i			0.470255i
$502$		0			0
$503$	253.778			0.504528			0.252264	−	0.967658i	$-0.418825\pi$
$503$	253.778			0.504528			0.252264	+	0.967658i	$0.418825\pi$
$504$		0			0
$505$	288.044			0.570383
$506$		0			0
$507$	89.1654			0.175869
$508$		0			0
$509$	523.377			1.02825			0.514123	−	0.857717i	$-0.328118\pi$
$509$	523.377			1.02825			0.514123	+	0.857717i	$0.328118\pi$
$510$		0			0
$511$	−38.0080			−0.0743796
$512$		0			0
$513$	−317.081	−	142.353i	−0.618091	−	0.277492i
$514$		0			0
$515$	12.7706			0.0247973
$516$		0			0
$517$		−	491.410i		−	0.950502i
$518$		0			0
$519$	−142.188			−0.273966
$520$		0			0
$521$			277.898i			0.533394i	0.963780	+	0.266697i	$0.0859323\pi$
$521$			277.898i			0.533394i	−0.963780	+	0.266697i	$0.914068\pi$
$522$		0			0
$523$	24.4757			0.0467987			0.0233994	−	0.999726i	$-0.492551\pi$
$523$	24.4757			0.0467987			0.0233994	+	0.999726i	$0.492551\pi$
$524$		0			0
$525$	83.4583			0.158968
$526$		0			0
$527$			74.9022i			0.142129i
$528$		0			0
$529$	−469.103			−0.886774
$530$		0			0
$531$	−469.247			−0.883704
$532$		0			0
$533$		−	534.836i		−	1.00344i
$534$		0			0
$535$		−	49.0074i		−	0.0916026i
$536$		0			0
$537$	−26.4552			−0.0492647
$538$		0			0
$539$		−	233.543i		−	0.433290i
$540$		0			0
$541$			495.925i			0.916682i	0.888776	+	0.458341i	$0.151556\pi$
$541$			495.925i			0.916682i	−0.888776	+	0.458341i	$0.848444\pi$
$542$		0			0
$543$	−106.772			−0.196634
$544$		0			0
$545$			207.052i			0.379912i
$546$		0			0
$547$	856.793			1.56635			0.783174	−	0.621802i	$-0.213599\pi$
$547$	856.793			1.56635			0.783174	+	0.621802i	$0.213599\pi$
$548$		0			0
$549$			147.801i			0.269219i
$550$		0			0
$551$	−612.549	−	275.004i	−1.11170	−	0.499100i
$552$		0			0
$553$			219.014i			0.396047i
$554$		0			0
$555$		−	75.7189i		−	0.136430i
$556$		0			0
$557$		−	528.791i		−	0.949356i	−0.880159	−	0.474678i	$-0.842564\pi$
$557$		−	528.791i		−	0.949356i	0.880159	−	0.474678i	$-0.157436\pi$
$558$		0			0
$559$			264.131i			0.472507i
$560$		0			0
$561$			14.6159i			0.0260533i
$562$		0			0
$563$	183.852			0.326557			0.163279	−	0.986580i	$-0.447793\pi$
$563$	183.852			0.326557			0.163279	+	0.986580i	$0.447793\pi$
$564$		0			0
$565$	−107.277			−0.189871
$566$		0			0
$567$	−187.747			−0.331123
$568$		0			0
$569$			12.6119i			0.0221650i	0.999939	+	0.0110825i	$0.00352775\pi$
$569$			12.6119i			0.0221650i	−0.999939	+	0.0110825i	$0.996472\pi$
$570$		0			0
$571$		−	62.9795i		−	0.110297i	−0.998478	−	0.0551484i	$-0.982437\pi$
$571$		−	62.9795i		−	0.110297i	0.998478	−	0.0551484i	$-0.0175632\pi$
$572$		0			0
$573$	−100.291			−0.175028
$574$		0			0
$575$	−159.559			−0.277494
$576$		0			0
$577$	333.803			0.578515			0.289257	−	0.957251i	$-0.406592\pi$
$577$	333.803			0.578515			0.289257	+	0.957251i	$0.406592\pi$
$578$		0			0
$579$			183.008i			0.316076i
$580$		0			0
$581$			19.2469i			0.0331271i
$582$		0			0
$583$			397.071i			0.681082i
$584$		0			0
$585$			152.672i			0.260978i
$586$		0			0
$587$			280.459i			0.477784i	0.971046	+	0.238892i	$0.0767843\pi$
$587$			280.459i			0.477784i	−0.971046	+	0.238892i	$0.923216\pi$
$588$		0			0
$589$	−288.218	+	641.983i	−0.489334	+	1.08995i
$590$		0			0
$591$		−	243.586i		−	0.412160i
$592$		0			0
$593$	562.285			0.948204			0.474102	−	0.880470i	$-0.342773\pi$
$593$	562.285			0.948204			0.474102	+	0.880470i	$0.342773\pi$
$594$		0			0
$595$			15.7563i			0.0264812i
$596$		0			0
$597$	192.145			0.321850
$598$		0			0
$599$		−	1179.08i		−	1.96842i	−0.177007	−	0.984210i	$-0.556642\pi$
$599$		−	1179.08i		−	1.96842i	0.177007	−	0.984210i	$-0.443358\pi$
$600$		0			0
$601$		−	38.6269i		−	0.0642711i	−0.999484	−	0.0321356i	$-0.989769\pi$
$601$		−	38.6269i		−	0.0642711i	0.999484	−	0.0321356i	$-0.0102308\pi$
$602$		0			0
$603$	−714.421			−1.18478
$604$		0			0
$605$			160.913i			0.265971i
$606$		0			0
$607$		−	695.079i		−	1.14511i	−0.819868	−	0.572553i	$-0.805953\pi$
$607$		−	695.079i		−	1.14511i	0.819868	−	0.572553i	$-0.194047\pi$
$608$		0			0
$609$	143.057			0.234905
$610$		0			0
$611$	690.012			1.12932
$612$		0			0
$613$		−	84.3445i		−	0.137593i	−0.997631	−	0.0687965i	$-0.978084\pi$
$613$		−	84.3445i		−	0.137593i	0.997631	−	0.0687965i	$-0.0219159\pi$
$614$		0			0
$615$	130.566			0.212302
$616$		0			0
$617$	244.410			0.396126			0.198063	−	0.980189i	$-0.436535\pi$
$617$	244.410			0.396126			0.198063	+	0.980189i	$0.436535\pi$
$618$		0			0
$619$		−	610.103i		−	0.985626i	−0.870135	−	0.492813i	$-0.835969\pi$
$619$		−	610.103i		−	0.985626i	0.870135	−	0.492813i	$-0.164031\pi$
$620$		0			0
$621$	−141.576			−0.227981
$622$		0			0
$623$		−	543.207i		−	0.871922i
$624$		0			0
$625$	315.467			0.504748
$626$		0			0
$627$	−56.2408	+	125.272i	−0.0896982	+	0.199796i
$628$		0			0
$629$	−67.2372			−0.106895
$630$		0			0
$631$	1045.91			1.65755			0.828774	−	0.559583i	$-0.189039\pi$
$631$	1045.91			1.65755			0.828774	+	0.559583i	$0.189039\pi$
$632$		0			0
$633$	68.5700			0.108325
$634$		0			0
$635$	288.946			0.455032
$636$		0			0
$637$	327.930			0.514803
$638$		0			0
$639$			655.033i			1.02509i
$640$		0			0
$641$			323.515i			0.504704i	0.967635	+	0.252352i	$0.0812041\pi$
$641$			323.515i			0.504704i	−0.967635	+	0.252352i	$0.918796\pi$
$642$		0			0
$643$		−	1092.63i		−	1.69927i	−0.527368	−	0.849637i	$-0.676821\pi$
$643$		−	1092.63i		−	1.69927i	0.527368	−	0.849637i	$-0.323179\pi$
$644$		0			0
$645$	−64.4807			−0.0999701
$646$		0			0
$647$	−712.381			−1.10105			−0.550526	−	0.834818i	$-0.685573\pi$
$647$	−712.381			−1.10105			−0.550526	+	0.834818i	$0.685573\pi$
$648$		0			0
$649$			398.843i			0.614550i
$650$		0			0
$651$		−	149.931i		−	0.230309i
$652$		0			0
$653$			219.109i			0.335542i	0.985826	+	0.167771i	$0.0536569\pi$
$653$			219.109i			0.335542i	−0.985826	+	0.167771i	$0.946343\pi$
$654$		0			0
$655$	−24.8822			−0.0379881
$656$		0			0
$657$	−79.8354			−0.121515
$658$		0			0
$659$	−469.826			−0.712938			−0.356469	−	0.934307i	$-0.616019\pi$
$659$	−469.826			−0.712938			−0.356469	+	0.934307i	$0.616019\pi$
$660$		0			0
$661$	152.275			0.230371			0.115186	−	0.993344i	$-0.463254\pi$
$661$	152.275			0.230371			0.115186	+	0.993344i	$0.463254\pi$
$662$		0			0
$663$	−20.5229			−0.0309545
$664$		0			0
$665$	−60.6291	+	135.046i	−0.0911715	+	0.203077i
$666$		0			0
$667$	−273.502			−0.410048
$668$		0			0
$669$		−	414.370i		−	0.619387i
$670$		0			0
$671$	125.626			0.187222
$672$		0			0
$673$		−	366.022i		−	0.543866i	−0.962316	−	0.271933i	$-0.912337\pi$
$673$		−	366.022i		−	0.543866i	0.962316	−	0.271933i	$-0.0876630\pi$
$674$		0			0
$675$	377.145			0.558733
$676$		0			0
$677$	1245.19			1.83927			0.919636	−	0.392772i	$-0.128484\pi$
$677$	1245.19			1.83927			0.919636	+	0.392772i	$0.128484\pi$
$678$		0			0
$679$			146.435i			0.215663i
$680$		0			0
$681$	−243.072			−0.356934
$682$		0			0
$683$	1231.24			1.80270			0.901349	−	0.433093i	$-0.142578\pi$
$683$	1231.24			1.80270			0.901349	+	0.433093i	$0.142578\pi$
$684$		0			0
$685$			174.500i			0.254745i
$686$		0			0
$687$		−	76.3169i		−	0.111087i
$688$		0			0
$689$	−557.546			−0.809211
$690$		0			0
$691$			261.648i			0.378651i	0.981914	+	0.189325i	$0.0606301\pi$
$691$			261.648i			0.378651i	−0.981914	+	0.189325i	$0.939370\pi$
$692$		0			0
$693$			193.264i			0.278880i
$694$		0			0
$695$	−432.117			−0.621751
$696$		0			0
$697$		−	115.941i		−	0.166342i
$698$		0			0
$699$	−276.667			−0.395805
$700$		0			0
$701$		−	1082.48i		−	1.54420i	−0.635503	−	0.772098i	$-0.719207\pi$
$701$		−	1082.48i		−	1.54420i	0.635503	−	0.772098i	$-0.280793\pi$
$702$		0			0
$703$	−576.287	−	258.724i	−0.819754	−	0.368028i
$704$		0			0
$705$			168.448i			0.238934i
$706$		0			0
$707$			511.990i			0.724172i
$708$		0			0
$709$			252.694i			0.356409i	0.983993	+	0.178204i	$0.0570288\pi$
$709$			252.694i			0.356409i	−0.983993	+	0.178204i	$0.942971\pi$
$710$		0			0
$711$			460.037i			0.647028i
$712$		0			0
$713$			286.644i			0.402025i
$714$		0			0
$715$	129.766			0.181491
$716$		0			0
$717$	85.5224			0.119278
$718$		0			0
$719$	−735.101			−1.02239			−0.511197	−	0.859464i	$-0.670798\pi$
$719$	−735.101			−1.02239			−0.511197	+	0.859464i	$0.670798\pi$
$720$		0			0
$721$			22.6994i			0.0314832i
$722$		0			0
$723$			297.018i			0.410813i
$724$		0			0
$725$	728.583			1.00494
$726$		0			0
$727$	−113.878			−0.156641			−0.0783205	−	0.996928i	$-0.524956\pi$
$727$	−113.878			−0.156641			−0.0783205	+	0.996928i	$0.524956\pi$
$728$		0			0
$729$	−215.268			−0.295292
$730$		0			0
$731$			57.2579i			0.0783282i
$732$		0			0
$733$		−	645.099i		−	0.880081i	−0.897978	−	0.440040i	$-0.854964\pi$
$733$		−	645.099i		−	0.880081i	0.897978	−	0.440040i	$-0.145036\pi$
$734$		0			0
$735$			80.0553i			0.108919i
$736$		0			0
$737$			607.233i			0.823925i
$738$		0			0
$739$		−	416.784i		−	0.563984i	−0.959417	−	0.281992i	$-0.909005\pi$
$739$		−	416.784i		−	0.563984i	0.959417	−	0.281992i	$-0.0909952\pi$
$740$		0			0
$741$	−175.900	−	78.9704i	−0.237382	−	0.106573i
$742$		0			0
$743$		−	528.231i		−	0.710943i	−0.934687	−	0.355472i	$-0.884320\pi$
$743$		−	528.231i		−	0.710943i	0.934687	−	0.355472i	$-0.115680\pi$
$744$		0			0
$745$	458.818			0.615864
$746$		0			0
$747$			40.4279i			0.0541203i
$748$		0			0
$749$	87.1093			0.116301
$750$		0			0
$751$			321.869i			0.428588i	0.976769	+	0.214294i	$0.0687450\pi$
$751$			321.869i			0.428588i	−0.976769	+	0.214294i	$0.931255\pi$
$752$		0			0
$753$			130.108i			0.172786i
$754$		0			0
$755$	308.039			0.407999
$756$		0			0
$757$			1443.50i			1.90687i	0.301606	+	0.953433i	$0.402477\pi$
$757$			1443.50i			1.90687i	−0.301606	+	0.953433i	$0.597523\pi$
$758$		0			0
$759$			55.9336i			0.0736939i
$760$		0			0
$761$	−371.415			−0.488062			−0.244031	−	0.969767i	$-0.578470\pi$
$761$	−371.415			−0.488062			−0.244031	+	0.969767i	$0.578470\pi$
$762$		0			0
$763$	−368.030			−0.482345
$764$		0			0
$765$			33.0960i			0.0432628i
$766$		0			0
$767$	−560.035			−0.730163
$768$		0			0
$769$	−642.327			−0.835276			−0.417638	−	0.908614i	$-0.637142\pi$
$769$	−642.327			−0.835276			−0.417638	+	0.908614i	$0.637142\pi$
$770$		0			0
$771$			164.784i			0.213728i
$772$		0			0
$773$	838.133			1.08426			0.542130	−	0.840295i	$-0.317618\pi$
$773$	838.133			1.08426			0.542130	+	0.840295i	$0.317618\pi$
$774$		0			0
$775$		−	763.592i		−	0.985280i
$776$		0			0
$777$	134.588			0.173215
$778$		0			0
$779$	446.131	−	993.722i	0.572697	−	1.27564i
$780$		0			0
$781$	556.755			0.712875
$782$		0			0
$783$	646.469			0.825631
$784$		0			0
$785$	518.753			0.660832
$786$		0			0
$787$	−637.161			−0.809608			−0.404804	−	0.914404i	$-0.632660\pi$
$787$	−637.161			−0.809608			−0.404804	+	0.914404i	$0.632660\pi$
$788$		0			0
$789$	65.7768			0.0833673
$790$		0			0
$791$		−	190.682i		−	0.241064i
$792$		0			0
$793$			176.397i			0.222443i
$794$		0			0
$795$		−	136.110i		−	0.171208i
$796$		0			0
$797$	1233.71			1.54795			0.773973	−	0.633218i	$-0.218267\pi$
$797$	1233.71			1.54795			0.773973	+	0.633218i	$0.218267\pi$
$798$		0			0
$799$	149.579			0.187208
$800$		0			0
$801$		−	1141.00i		−	1.42447i
$802$		0			0
$803$			67.8573i			0.0845047i
$804$		0			0
$805$			60.2980i			0.0749043i
$806$		0			0
$807$	47.8350			0.0592751
$808$		0			0
$809$	296.868			0.366956			0.183478	−	0.983024i	$-0.441264\pi$
$809$	296.868			0.366956			0.183478	+	0.983024i	$0.441264\pi$
$810$		0			0
$811$	802.739			0.989814			0.494907	−	0.868946i	$-0.335202\pi$
$811$	802.739			0.989814			0.494907	+	0.868946i	$0.335202\pi$
$812$		0			0
$813$	−295.488			−0.363454
$814$		0			0
$815$	369.127			0.452917
$816$		0			0
$817$	−220.324	+	490.755i	−0.269675	+	0.600679i
$818$		0			0
$819$	−271.371			−0.331344
$820$		0			0
$821$		−	1590.22i		−	1.93693i	−0.249148	−	0.968465i	$-0.580151\pi$
$821$		−	1590.22i		−	1.93693i	0.249148	−	0.968465i	$-0.419849\pi$
$822$		0			0
$823$	−283.504			−0.344476			−0.172238	−	0.985055i	$-0.555100\pi$
$823$	−283.504			−0.344476			−0.172238	+	0.985055i	$0.555100\pi$
$824$		0			0
$825$		−	149.002i		−	0.180608i
$826$		0			0
$827$	174.686			0.211228			0.105614	−	0.994407i	$-0.466319\pi$
$827$	174.686			0.211228			0.105614	+	0.994407i	$0.466319\pi$
$828$		0			0
$829$	−1044.05			−1.25940			−0.629702	−	0.776837i	$-0.716823\pi$
$829$	−1044.05			−1.25940			−0.629702	+	0.776837i	$0.716823\pi$
$830$		0			0
$831$		−	220.530i		−	0.265379i
$832$		0			0
$833$	71.0879			0.0853397
$834$		0			0
$835$	453.443			0.543046
$836$		0			0
$837$		−	677.533i		−	0.809477i
$838$		0			0
$839$		−	1115.25i		−	1.32926i	−0.747175	−	0.664628i	$-0.768590\pi$
$839$		−	1115.25i		−	1.32926i	0.747175	−	0.664628i	$-0.231410\pi$
$840$		0			0
$841$	407.875			0.484988
$842$		0			0
$843$		−	383.441i		−	0.454853i
$844$		0			0
$845$		−	171.612i		−	0.203091i
$846$		0			0
$847$	−286.018			−0.337683
$848$		0			0
$849$		−	473.227i		−	0.557393i
$850$		0			0
$851$	−257.311			−0.302363
$852$		0			0
$853$		−	812.448i		−	0.952459i	−0.879321	−	0.476230i	$-0.842003\pi$
$853$		−	812.448i		−	0.952459i	0.879321	−	0.476230i	$-0.157997\pi$
$854$		0			0
$855$	−127.351	+	283.664i	−0.148948	+	0.331771i
$856$		0			0
$857$			619.864i			0.723295i	0.932315	+	0.361648i	$0.117786\pi$
$857$			619.864i			0.723295i	−0.932315	+	0.361648i	$0.882214\pi$
$858$		0			0
$859$			1060.49i			1.23457i	0.786740	+	0.617284i	$0.211767\pi$
$859$			1060.49i			1.23457i	−0.786740	+	0.617284i	$0.788233\pi$
$860$		0			0
$861$			232.078i			0.269544i
$862$		0			0
$863$		−	1211.79i		−	1.40416i	−0.712096	−	0.702082i	$-0.752254\pi$
$863$		−	1211.79i		−	1.40416i	0.712096	−	0.702082i	$-0.247746\pi$
$864$		0			0
$865$			273.663i			0.316373i
$866$		0			0
$867$	309.924			0.357467
$868$		0			0
$869$	391.015			0.449960
$870$		0			0
$871$	−852.645			−0.978927
$872$		0			0
$873$			307.586i			0.352333i
$874$		0			0
$875$		−	355.407i		−	0.406179i
$876$		0			0
$877$	505.369			0.576247			0.288124	−	0.957593i	$-0.406969\pi$
$877$	505.369			0.576247			0.288124	+	0.957593i	$0.406969\pi$
$878$		0			0
$879$	243.641			0.277180
$880$		0			0
$881$	−381.023			−0.432489			−0.216245	−	0.976339i	$-0.569381\pi$
$881$	−381.023			−0.432489			−0.216245	+	0.976339i	$0.569381\pi$
$882$		0			0
$883$		−	45.1768i		−	0.0511629i	−0.999673	−	0.0255815i	$-0.991856\pi$
$883$		−	45.1768i		−	0.0511629i	0.999673	−	0.0255815i	$-0.00814372\pi$
$884$		0			0
$885$		−	136.718i		−	0.154483i
$886$		0			0
$887$		−	1667.14i		−	1.87952i	−0.341829	−	0.939762i	$-0.611046\pi$
$887$		−	1667.14i		−	1.87952i	0.341829	−	0.939762i	$-0.388954\pi$
$888$		0			0
$889$			513.593i			0.577720i
$890$		0			0
$891$			335.193i			0.376199i
$892$		0			0
$893$	1282.04	+	575.571i	1.43565	+	0.644536i
$894$		0			0
$895$			50.9169i			0.0568904i
$896$		0			0
$897$	−78.5391			−0.0875575
$898$		0			0
$899$		−	1308.88i		−	1.45593i
$900$		0			0
$901$	−120.864			−0.134144
$902$		0			0
$903$		−	114.613i		−	0.126924i
$904$		0			0
$905$			205.499i			0.227070i
$906$		0			0
$907$	967.353			1.06654			0.533271	−	0.845945i	$-0.320963\pi$
$907$	967.353			1.06654			0.533271	+	0.845945i	$0.320963\pi$
$908$		0			0
$909$			1075.43i			1.18309i
$910$		0			0
$911$		−	759.283i		−	0.833461i	−0.909030	−	0.416730i	$-0.863176\pi$
$911$		−	759.283i		−	0.833461i	0.909030	−	0.416730i	$-0.136824\pi$
$912$		0			0
$913$	34.3623			0.0376367
$914$		0			0
$915$	−43.0627			−0.0470631
$916$		0			0
$917$		−	44.2274i		−	0.0482306i
$918$		0			0
$919$	−1580.32			−1.71961			−0.859804	−	0.510624i	$-0.829414\pi$
$919$	−1580.32			−1.71961			−0.859804	+	0.510624i	$0.829414\pi$
$920$		0			0
$921$	235.553			0.255758
$922$		0			0
$923$			781.767i			0.846985i
$924$		0			0
$925$	685.451			0.741029
$926$		0			0
$927$			47.6799i			0.0514347i
$928$		0			0
$929$	1629.32			1.75384			0.876921	−	0.480634i	$-0.159594\pi$
$929$	1629.32			1.75384			0.876921	+	0.480634i	$0.159594\pi$
$930$		0			0
$931$	609.291	+	273.541i	0.654448	+	0.293814i
$932$		0			0
$933$	384.724			0.412352
$934$		0			0
$935$	28.1304			0.0300860
$936$		0			0
$937$	794.022			0.847409			0.423705	−	0.905800i	$-0.360729\pi$
$937$	794.022			0.847409			0.423705	+	0.905800i	$0.360729\pi$
$938$		0			0
$939$	−10.3990			−0.0110745
$940$		0			0
$941$	802.627			0.852951			0.426475	−	0.904499i	$-0.359755\pi$
$941$	802.627			0.852951			0.426475	+	0.904499i	$0.359755\pi$
$942$		0			0
$943$		−	443.695i		−	0.470514i
$944$		0			0
$945$		−	142.525i		−	0.150820i
$946$		0			0
$947$		−	1386.35i		−	1.46394i	−0.681336	−	0.731971i	$-0.738601\pi$
$947$		−	1386.35i		−	1.46394i	0.681336	−	0.731971i	$-0.261399\pi$
$948$		0			0
$949$	−95.2817			−0.100402
$950$		0			0
$951$	−58.7253			−0.0617511
$952$		0			0
$953$			431.148i			0.452411i	0.974080	+	0.226205i	$0.0726321\pi$
$953$			431.148i			0.452411i	−0.974080	+	0.226205i	$0.927368\pi$
$954$		0			0
$955$			193.025i			0.202121i
$956$		0			0
$957$		−	255.406i		−	0.266882i
$958$		0			0
$959$	−310.169			−0.323430
$960$		0			0
$961$	−410.777			−0.427448
$962$		0			0
$963$	182.972			0.190003
$964$		0			0
$965$	352.227			0.365002
$966$		0			0
$967$	1368.36			1.41505			0.707527	−	0.706686i	$-0.249811\pi$
$967$	1368.36			1.41505			0.707527	+	0.706686i	$0.249811\pi$
$968$		0			0
$969$	−38.1313	−	17.1191i	−0.0393512	−	0.0176667i
$970$		0			0
$971$	87.5512			0.0901660			0.0450830	−	0.998983i	$-0.485645\pi$
$971$	87.5512			0.0901660			0.0450830	+	0.998983i	$0.485645\pi$
$972$		0			0
$973$		−	768.076i		−	0.789390i
$974$		0			0
$975$	209.221			0.214585
$976$		0			0
$977$			1674.52i			1.71394i	0.515368	+	0.856969i	$0.327655\pi$
$977$			1674.52i			1.71394i	−0.515368	+	0.856969i	$0.672345\pi$
$978$		0			0
$979$	−969.812			−0.990615
$980$		0			0
$981$	−773.043			−0.788015
$982$		0			0
$983$			413.238i			0.420385i	0.977660	+	0.210192i	$0.0674090\pi$
$983$			413.238i			0.420385i	−0.977660	+	0.210192i	$0.932591\pi$
$984$		0			0
$985$	−468.819			−0.475958
$986$		0			0
$987$	−299.412			−0.303356
$988$		0			0
$989$			219.121i			0.221558i
$990$		0			0
$991$			435.659i			0.439615i	0.975543	+	0.219808i	$0.0705429\pi$
$991$			435.659i			0.439615i	−0.975543	+	0.219808i	$0.929457\pi$
$992$		0			0
$993$	240.414			0.242108
$994$		0			0
$995$		−	369.811i		−	0.371670i
$996$		0			0
$997$		−	569.052i		−	0.570764i	−0.958414	−	0.285382i	$-0.907880\pi$
$997$		−	569.052i		−	0.570764i	0.958414	−	0.285382i	$-0.0921204\pi$
$998$		0			0
$999$	608.199			0.608807

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.3.g.e.417.21	✓	48
4.3	odd	2	inner	1216.3.g.e.417.25	yes	48
8.3	odd	2	inner	1216.3.g.e.417.22	yes	48
8.5	even	2	inner	1216.3.g.e.417.26	yes	48
19.18	odd	2	inner	1216.3.g.e.417.27	yes	48
76.75	even	2	inner	1216.3.g.e.417.23	yes	48
152.37	odd	2	inner	1216.3.g.e.417.24	yes	48
152.75	even	2	inner	1216.3.g.e.417.28	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1216.3.g.e.417.21	✓	48	1.1	even	1	trivial
1216.3.g.e.417.22	yes	48	8.3	odd	2	inner
1216.3.g.e.417.23	yes	48	76.75	even	2	inner
1216.3.g.e.417.24	yes	48	152.37	odd	2	inner
1216.3.g.e.417.25	yes	48	4.3	odd	2	inner
1216.3.g.e.417.26	yes	48	8.5	even	2	inner
1216.3.g.e.417.27	yes	48	19.18	odd	2	inner
1216.3.g.e.417.28	yes	48	152.75	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:	\( N \)	\(=\)	\( 1216 = 2^{6} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1216.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.08780 q^{3} +2.09363i q^{5} -3.72137 q^{7} -7.81670 q^{9} +O(q^{10})\)	\(q-1.08780 q^{3} +2.09363i q^{5} -3.72137 q^{7} -7.81670 q^{9} +6.64392i q^{11} -9.32905 q^{13} -2.27744i q^{15} -2.02233 q^{17} +(-17.3333 - 7.78179i) q^{19} +4.04809 q^{21} -7.73929 q^{23} +20.6167 q^{25} +18.2931 q^{27} +35.3394 q^{29} -37.0375i q^{31} -7.22723i q^{33} -7.79115i q^{35} +33.2473 q^{37} +10.1481 q^{39} +57.3302i q^{41} -28.3128i q^{43} -16.3653i q^{45} -73.9638 q^{47} -35.1514 q^{49} +2.19989 q^{51} +59.7645 q^{53} -13.9099 q^{55} +(18.8551 + 8.46500i) q^{57} +60.0313 q^{59} -18.9084i q^{61} +29.0888 q^{63} -19.5316i q^{65} +91.3968 q^{67} +8.41877 q^{69} -83.7992i q^{71} +10.2134 q^{73} -22.4268 q^{75} -24.7245i q^{77} -58.8531i q^{79} +50.4511 q^{81} -5.17199i q^{83} -4.23401i q^{85} -38.4421 q^{87} +145.970i q^{89} +34.7168 q^{91} +40.2893i q^{93} +(16.2922 - 36.2895i) q^{95} -39.3499i q^{97} -51.9335i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 264 q^{9}+O(q^{10}) \) 48 * q + 264 * q^9	\( 48 q + 264 q^{9} + 24 q^{17} - 160 q^{25} + 328 q^{49} + 8 q^{57} - 472 q^{73} - 736 q^{81}+O(q^{100}) \) 48 * q + 264 * q^9 + 24 * q^17 - 160 * q^25 + 328 * q^49 + 8 * q^57 - 472 * q^73 - 736 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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