LMFDB - Embedded newform 1344.4.p.c.223.29

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1344,4,Mod(223,1344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1344, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1344.223");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1344 = 2^{6} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1344.p (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:	$79.2985670477$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			223.29
Character	$\chi$	$=$	1344.223
Dual form			1344.4.p.c.223.30

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{3} -15.4018 q^{5} +(6.29285 + 17.4184i) q^{7} -9.00000 q^{9} +O(q^{10})$	$q-3.00000i q^{3} -15.4018 q^{5} +(6.29285 + 17.4184i) q^{7} -9.00000 q^{9} +16.8703 q^{11} -9.10290 q^{13} +46.2054i q^{15} +73.9573i q^{17} +151.262i q^{19} +(52.2551 - 18.8786i) q^{21} -0.264643i q^{23} +112.215 q^{25} +27.0000i q^{27} -279.925i q^{29} +147.800 q^{31} -50.6110i q^{33} +(-96.9212 - 268.274i) q^{35} -418.498i q^{37} +27.3087i q^{39} +164.377i q^{41} +266.668 q^{43} +138.616 q^{45} -277.719 q^{47} +(-263.800 + 219.223i) q^{49} +221.872 q^{51} +276.852i q^{53} -259.833 q^{55} +453.785 q^{57} +212.008i q^{59} -174.541 q^{61} +(-56.6357 - 156.765i) q^{63} +140.201 q^{65} -317.501 q^{67} -0.793928 q^{69} +106.093i q^{71} +755.118i q^{73} -336.645i q^{75} +(106.163 + 293.854i) q^{77} +194.202i q^{79} +81.0000 q^{81} -997.764i q^{83} -1139.08i q^{85} -839.776 q^{87} -988.320i q^{89} +(-57.2832 - 158.558i) q^{91} -443.400i q^{93} -2329.70i q^{95} +1021.78i q^{97} -151.833 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 288 q^{9}+O(q^{10}) $ 32 * q - 288 * q^9	$ 32 q - 288 q^{9} - 224 q^{13} - 72 q^{21} + 1120 q^{25} - 752 q^{49} - 672 q^{57} - 544 q^{61} + 1536 q^{65} - 144 q^{69} - 1632 q^{77} + 2592 q^{81}+O(q^{100}) $ 32 * q - 288 * q^9 - 224 * q^13 - 72 * q^21 + 1120 * q^25 - 752 * q^49 - 672 * q^57 - 544 * q^61 + 1536 * q^65 - 144 * q^69 - 1632 * q^77 + 2592 * q^81

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times$.

$n$	$127$	$449$	$577$	$1093$
$\chi(n)$	$-1$	$1$	$-1$	$-1$

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
$3$		−	3.00000i		−	0.577350i
$3$		−	3.00000i		−	0.577350i
$4$		0			0
$5$	−15.4018			−1.37758			−0.688789	−	0.724962i	$-0.741857\pi$
$5$	−15.4018			−1.37758			−0.688789	+	0.724962i	$0.741857\pi$
$6$		0			0
$7$	6.29285	+	17.4184i	0.339782	+	0.940504i
$7$	6.29285	+	17.4184i	0.339782	+	0.940504i
$8$		0			0
$9$	−9.00000			−0.333333
$10$		0			0
$11$	16.8703			0.462418			0.231209	−	0.972904i	$-0.425732\pi$
$11$	16.8703			0.462418			0.231209	+	0.972904i	$0.425732\pi$
$12$		0			0
$13$	−9.10290			−0.194207			−0.0971034	−	0.995274i	$-0.530958\pi$
$13$	−9.10290			−0.194207			−0.0971034	+	0.995274i	$0.530958\pi$
$14$		0			0
$15$			46.2054i			0.795345i
$16$		0			0
$17$			73.9573i			1.05513i	0.849513	+	0.527567i	$0.176896\pi$
$17$			73.9573i			1.05513i	−0.849513	+	0.527567i	$0.823104\pi$
$18$		0			0
$19$			151.262i			1.82641i	0.407499	+	0.913206i	$0.366401\pi$
$19$			151.262i			1.82641i	−0.407499	+	0.913206i	$0.633599\pi$
$20$		0			0
$21$	52.2551	−	18.8786i	0.543000	−	0.196173i
$22$		0			0
$23$		−	0.264643i		−	0.00239921i	−0.999999	−	0.00119960i	$-0.999618\pi$
$23$		−	0.264643i		−	0.00239921i	0.999999	−	0.00119960i	$-0.000381846\pi$
$24$		0			0
$25$	112.215			0.897721
$26$		0			0
$27$			27.0000i			0.192450i
$28$		0			0
$29$		−	279.925i		−	1.79244i	−0.443607	−	0.896221i	$-0.646301\pi$
$29$		−	279.925i		−	1.79244i	0.443607	−	0.896221i	$-0.353699\pi$
$30$		0			0
$31$	147.800			0.856311			0.428156	−	0.903705i	$-0.359164\pi$
$31$	147.800			0.856311			0.428156	+	0.903705i	$0.359164\pi$
$32$		0			0
$33$		−	50.6110i		−	0.266977i
$34$		0			0
$35$	−96.9212	−	268.274i	−0.468076	−	1.29562i
$36$		0			0
$37$		−	418.498i		−	1.85948i	−0.368221	−	0.929738i	$-0.620033\pi$
$37$		−	418.498i		−	1.85948i	0.368221	−	0.929738i	$-0.379967\pi$
$38$		0			0
$39$			27.3087i			0.112125i
$40$		0			0
$41$			164.377i			0.626131i	0.949731	+	0.313066i	$0.101356\pi$
$41$			164.377i			0.626131i	−0.949731	+	0.313066i	$0.898644\pi$
$42$		0			0
$43$	266.668			0.945731			0.472865	−	0.881135i	$-0.343220\pi$
$43$	266.668			0.945731			0.472865	+	0.881135i	$0.343220\pi$
$44$		0			0
$45$	138.616			0.459193
$46$		0			0
$47$	−277.719			−0.861903			−0.430952	−	0.902375i	$-0.641822\pi$
$47$	−277.719			−0.861903			−0.430952	+	0.902375i	$0.641822\pi$
$48$		0			0
$49$	−263.800	+	219.223i	−0.769096	+	0.639133i
$50$		0			0
$51$	221.872			0.609182
$52$		0			0
$53$			276.852i			0.717519i	0.933430	+	0.358760i	$0.116800\pi$
$53$			276.852i			0.717519i	−0.933430	+	0.358760i	$0.883200\pi$
$54$		0			0
$55$	−259.833			−0.637017
$56$		0			0
$57$	453.785			1.05448
$58$		0			0
$59$			212.008i			0.467815i	0.972259	+	0.233908i	$0.0751513\pi$
$59$			212.008i			0.467815i	−0.972259	+	0.233908i	$0.924849\pi$
$60$		0			0
$61$	−174.541			−0.366355			−0.183177	−	0.983080i	$-0.558638\pi$
$61$	−174.541			−0.366355			−0.183177	+	0.983080i	$0.558638\pi$
$62$		0			0
$63$	−56.6357	−	156.765i	−0.113261	−	0.313501i
$64$		0			0
$65$	140.201			0.267535
$66$		0			0
$67$	−317.501			−0.578940			−0.289470	−	0.957187i	$-0.593479\pi$
$67$	−317.501			−0.578940			−0.289470	+	0.957187i	$0.593479\pi$
$68$		0			0
$69$	−0.793928			−0.00138518
$70$		0			0
$71$			106.093i			0.177338i	0.996061	+	0.0886688i	$0.0282613\pi$
$71$			106.093i			0.177338i	−0.996061	+	0.0886688i	$0.971739\pi$
$72$		0			0
$73$			755.118i			1.21068i	0.795966	+	0.605342i	$0.206964\pi$
$73$			755.118i			1.21068i	−0.795966	+	0.605342i	$0.793036\pi$
$74$		0			0
$75$		−	336.645i		−	0.518299i
$76$		0			0
$77$	106.163	+	293.854i	0.157121	+	0.434906i
$78$		0			0
$79$			194.202i			0.276575i	0.990392	+	0.138288i	$0.0441598\pi$
$79$			194.202i			0.276575i	−0.990392	+	0.138288i	$0.955840\pi$
$80$		0			0
$81$	81.0000			0.111111
$82$		0			0
$83$		−	997.764i		−	1.31950i	−0.751484	−	0.659752i	$-0.770661\pi$
$83$		−	997.764i		−	1.31950i	0.751484	−	0.659752i	$-0.229339\pi$
$84$		0			0
$85$		−	1139.08i		−	1.45353i
$86$		0			0
$87$	−839.776			−1.03487
$88$		0			0
$89$		−	988.320i		−	1.17710i	−0.808462	−	0.588549i	$-0.799700\pi$
$89$		−	988.320i		−	1.17710i	0.808462	−	0.588549i	$-0.200300\pi$
$90$		0			0
$91$	−57.2832	−	158.558i	−0.0659880	−	0.182652i
$92$		0			0
$93$		−	443.400i		−	0.494391i
$94$		0			0
$95$		−	2329.70i		−	2.51602i
$96$		0			0
$97$			1021.78i			1.06954i	0.844996	+	0.534772i	$0.179602\pi$
$97$			1021.78i			1.06954i	−0.844996	+	0.534772i	$0.820398\pi$
$98$		0			0
$99$	−151.833			−0.154139
$100$		0			0
$101$	−914.669			−0.901118			−0.450559	−	0.892747i	$-0.648775\pi$
$101$	−914.669			−0.901118			−0.450559	+	0.892747i	$0.648775\pi$
$102$		0			0
$103$	469.098			0.448753			0.224377	−	0.974503i	$-0.427965\pi$
$103$	469.098			0.448753			0.224377	+	0.974503i	$0.427965\pi$
$104$		0			0
$105$	−804.823	+	290.763i	−0.748025	+	0.270244i
$106$		0			0
$107$	−1553.58			−1.40364			−0.701822	−	0.712352i	$-0.747630\pi$
$107$	−1553.58			−1.40364			−0.701822	+	0.712352i	$0.747630\pi$
$108$		0			0
$109$			1424.25i			1.25155i	0.780005	+	0.625773i	$0.215216\pi$
$109$			1424.25i			1.25155i	−0.780005	+	0.625773i	$0.784784\pi$
$110$		0			0
$111$	−1255.49			−1.07357
$112$		0			0
$113$	784.755			0.653305			0.326653	−	0.945144i	$-0.394079\pi$
$113$	784.755			0.653305			0.326653	+	0.945144i	$0.394079\pi$
$114$		0			0
$115$			4.07597i			0.00330510i
$116$		0			0
$117$	81.9261			0.0647356
$118$		0			0
$119$	−1288.22	+	465.402i	−0.992358	+	0.358516i
$120$		0			0
$121$	−1046.39			−0.786170
$122$		0			0
$123$	493.131			0.361497
$124$		0			0
$125$	196.910			0.140897
$126$		0			0
$127$		−	2509.27i		−	1.75324i	−0.481182	−	0.876620i	$-0.659793\pi$
$127$		−	2509.27i		−	1.75324i	0.481182	−	0.876620i	$-0.340207\pi$
$128$		0			0
$129$		−	800.003i		−	0.546018i
$130$		0			0
$131$		−	2050.10i		−	1.36731i	−0.729805	−	0.683655i	$-0.760389\pi$
$131$		−	2050.10i		−	1.36731i	0.729805	−	0.683655i	$-0.239611\pi$
$132$		0			0
$133$	−2634.73	+	951.868i	−1.71775	+	0.620582i
$134$		0			0
$135$		−	415.848i		−	0.265115i
$136$		0			0
$137$	−2516.47			−1.56932			−0.784658	−	0.619929i	$-0.787161\pi$
$137$	−2516.47			−1.56932			−0.784658	+	0.619929i	$0.787161\pi$
$138$		0			0
$139$		−	270.678i		−	0.165170i	−0.996584	−	0.0825849i	$-0.973682\pi$
$139$		−	270.678i		−	0.165170i	0.996584	−	0.0825849i	$-0.0263176\pi$
$140$		0			0
$141$			833.157i			0.497620i
$142$		0			0
$143$	−153.569			−0.0898048
$144$		0			0
$145$			4311.35i			2.46923i
$146$		0			0
$147$	657.668	+	791.400i	0.369004	+	0.444038i
$148$		0			0
$149$			238.893i			0.131348i	0.997841	+	0.0656742i	$0.0209198\pi$
$149$			238.893i			0.131348i	−0.997841	+	0.0656742i	$0.979080\pi$
$150$		0			0
$151$			2859.49i			1.54107i	0.637397	+	0.770535i	$0.280011\pi$
$151$			2859.49i			1.54107i	−0.637397	+	0.770535i	$0.719989\pi$
$152$		0			0
$153$		−	665.616i		−	0.351711i
$154$		0			0
$155$	−2276.38			−1.17964
$156$		0			0
$157$	566.367			0.287904			0.143952	−	0.989585i	$-0.454019\pi$
$157$	566.367			0.287904			0.143952	+	0.989585i	$0.454019\pi$
$158$		0			0
$159$	830.555			0.414260
$160$		0			0
$161$	4.60965	−	1.66536i	0.00225647	−	0.000815208i
$162$		0			0
$163$	1330.25			0.639222			0.319611	−	0.947549i	$-0.396448\pi$
$163$	1330.25			0.639222			0.319611	+	0.947549i	$0.396448\pi$
$164$		0			0
$165$			779.500i			0.367782i
$166$		0			0
$167$	−2770.08			−1.28356			−0.641782	−	0.766887i	$-0.721805\pi$
$167$	−2770.08			−1.28356			−0.641782	+	0.766887i	$0.721805\pi$
$168$		0			0
$169$	−2114.14			−0.962284
$170$		0			0
$171$		−	1361.36i		−	0.608804i
$172$		0			0
$173$	−1389.16			−0.610498			−0.305249	−	0.952273i	$-0.598740\pi$
$173$	−1389.16			−0.610498			−0.305249	+	0.952273i	$0.598740\pi$
$174$		0			0
$175$	706.153	+	1954.61i	0.305029	+	0.844310i
$176$		0			0
$177$	636.024			0.270093
$178$		0			0
$179$	2337.71			0.976136			0.488068	−	0.872806i	$-0.337702\pi$
$179$	2337.71			0.976136			0.488068	+	0.872806i	$0.337702\pi$
$180$		0			0
$181$	−3318.66			−1.36284			−0.681420	−	0.731893i	$-0.738637\pi$
$181$	−3318.66			−1.36284			−0.681420	+	0.731893i	$0.738637\pi$
$182$		0			0
$183$			523.622i			0.211515i
$184$		0			0
$185$			6445.62i			2.56157i
$186$		0			0
$187$			1247.69i			0.487913i
$188$		0			0
$189$	−470.296	+	169.907i	−0.181000	+	0.0653911i
$190$		0			0
$191$		−	1614.06i		−	0.611463i	−0.952118	−	0.305731i	$-0.901099\pi$
$191$		−	1614.06i		−	0.611463i	0.952118	−	0.305731i	$-0.0989010\pi$
$192$		0			0
$193$	−1248.70			−0.465719			−0.232859	−	0.972510i	$-0.574808\pi$
$193$	−1248.70			−0.465719			−0.232859	+	0.972510i	$0.574808\pi$
$194$		0			0
$195$		−	420.603i		−	0.154461i
$196$		0			0
$197$		−	4151.12i		−	1.50129i	−0.660703	−	0.750647i	$-0.729742\pi$
$197$		−	4151.12i		−	1.50129i	0.660703	−	0.750647i	$-0.270258\pi$
$198$		0			0
$199$	−2928.99			−1.04337			−0.521685	−	0.853138i	$-0.674696\pi$
$199$	−2928.99			−1.04337			−0.521685	+	0.853138i	$0.674696\pi$
$200$		0			0
$201$			952.504i			0.334251i
$202$		0			0
$203$	4875.85	−	1761.53i	1.68580	−	0.609040i
$204$		0			0
$205$		−	2531.70i		−	0.862545i
$206$		0			0
$207$			2.38178i			0.000799737i
$208$		0			0
$209$			2551.84i			0.844566i
$210$		0			0
$211$	−3360.79			−1.09652			−0.548261	−	0.836307i	$-0.684710\pi$
$211$	−3360.79			−1.09652			−0.548261	+	0.836307i	$0.684710\pi$
$212$		0			0
$213$	318.280			0.102386
$214$		0			0
$215$	−4107.16			−1.30282
$216$		0			0
$217$	930.083	+	2574.43i	0.290959	+	0.805364i
$218$		0			0
$219$	2265.35			0.698988
$220$		0			0
$221$		−	673.226i		−	0.204914i
$222$		0			0
$223$	−2329.45			−0.699515			−0.349757	−	0.936840i	$-0.613736\pi$
$223$	−2329.45			−0.699515			−0.349757	+	0.936840i	$0.613736\pi$
$224$		0			0
$225$	−1009.94			−0.299240
$226$		0			0
$227$		−	832.888i		−	0.243528i	−0.992559	−	0.121764i	$-0.961145\pi$
$227$		−	832.888i		−	0.243528i	0.992559	−	0.121764i	$-0.0388550\pi$
$228$		0			0
$229$	−1434.36			−0.413908			−0.206954	−	0.978351i	$-0.566355\pi$
$229$	−1434.36			−0.413908			−0.206954	+	0.978351i	$0.566355\pi$
$230$		0			0
$231$	881.562	−	318.488i	0.251093	−	0.0907140i
$232$		0			0
$233$	2388.42			0.671547			0.335773	−	0.941943i	$-0.391002\pi$
$233$	2388.42			0.671547			0.335773	+	0.941943i	$0.391002\pi$
$234$		0			0
$235$	4277.37			1.18734
$236$		0			0
$237$	582.606			0.159681
$238$		0			0
$239$		−	4808.99i		−	1.30154i	−0.759276	−	0.650769i	$-0.774447\pi$
$239$		−	4808.99i		−	1.30154i	0.759276	−	0.650769i	$-0.225553\pi$
$240$		0			0
$241$		−	2269.59i		−	0.606627i	−0.952891	−	0.303313i	$-0.901907\pi$
$241$		−	2269.59i		−	0.606627i	0.952891	−	0.303313i	$-0.0980929\pi$
$242$		0			0
$243$		−	243.000i		−	0.0641500i
$244$		0			0
$245$	4062.99	−	3376.42i	1.05949	−	0.880455i
$246$		0			0
$247$		−	1376.92i		−	0.354702i
$248$		0			0
$249$	−2993.29			−0.761816
$250$		0			0
$251$			2242.07i			0.563818i	0.959441	+	0.281909i	$0.0909676\pi$
$251$			2242.07i			0.563818i	−0.959441	+	0.281909i	$0.909032\pi$
$252$		0			0
$253$		−	4.46461i		−	0.00110944i
$254$		0			0
$255$	−3417.23			−0.839196
$256$		0			0
$257$			787.861i			0.191227i	0.995419	+	0.0956136i	$0.0304813\pi$
$257$			787.861i			0.191227i	−0.995419	+	0.0956136i	$0.969519\pi$
$258$		0			0
$259$	7289.56	−	2633.54i	1.74885	−	0.631817i
$260$		0			0
$261$			2519.33i			0.597481i
$262$		0			0
$263$			6696.97i			1.57016i	0.619392	+	0.785082i	$0.287379\pi$
$263$			6696.97i			1.57016i	−0.619392	+	0.785082i	$0.712621\pi$
$264$		0			0
$265$		−	4264.01i		−	0.988439i
$266$		0			0
$267$	−2964.96			−0.679597
$268$		0			0
$269$	−4374.92			−0.991612			−0.495806	−	0.868433i	$-0.665127\pi$
$269$	−4374.92			−0.991612			−0.495806	+	0.868433i	$0.665127\pi$
$270$		0			0
$271$	−8058.24			−1.80629			−0.903143	−	0.429340i	$-0.858746\pi$
$271$	−8058.24			−1.80629			−0.903143	+	0.429340i	$0.858746\pi$
$272$		0			0
$273$	−475.673	+	171.850i	−0.105454	+	0.0380982i
$274$		0			0
$275$	1893.11			0.415122
$276$		0			0
$277$			4224.44i			0.916324i	0.888869	+	0.458162i	$0.151492\pi$
$277$			4224.44i			0.916324i	−0.888869	+	0.458162i	$0.848508\pi$
$278$		0			0
$279$	−1330.20			−0.285437
$280$		0			0
$281$	3763.77			0.799031			0.399516	−	0.916726i	$-0.369178\pi$
$281$	3763.77			0.799031			0.399516	+	0.916726i	$0.369178\pi$
$282$		0			0
$283$			1604.16i			0.336952i	0.985706	+	0.168476i	$0.0538846\pi$
$283$			1604.16i			0.336952i	−0.985706	+	0.168476i	$0.946115\pi$
$284$		0			0
$285$	−6989.10			−1.45263
$286$		0			0
$287$	−2863.18	+	1034.40i	−0.588879	+	0.212748i
$288$		0			0
$289$	−556.685			−0.113309
$290$		0			0
$291$	3065.33			0.617502
$292$		0			0
$293$	3124.58			0.623004			0.311502	−	0.950245i	$-0.399168\pi$
$293$	3124.58			0.623004			0.311502	+	0.950245i	$0.399168\pi$
$294$		0			0
$295$		−	3265.30i		−	0.644452i
$296$		0			0
$297$			455.499i			0.0889924i
$298$		0			0
$299$			2.40902i			0.000465943i
$300$		0			0
$301$	1678.10	+	4644.92i	0.321342	+	0.889464i
$302$		0			0
$303$			2744.01i			0.520261i
$304$		0			0
$305$	2688.24			0.504683
$306$		0			0
$307$		−	9789.14i		−	1.81986i	−0.414767	−	0.909928i	$-0.636137\pi$
$307$		−	9789.14i		−	1.81986i	0.414767	−	0.909928i	$-0.363863\pi$
$308$		0			0
$309$		−	1407.29i		−	0.259088i
$310$		0			0
$311$	3916.61			0.714117			0.357059	−	0.934082i	$-0.383780\pi$
$311$	3916.61			0.714117			0.357059	+	0.934082i	$0.383780\pi$
$312$		0			0
$313$			5571.72i			1.00617i	0.864236	+	0.503087i	$0.167802\pi$
$313$			5571.72i			1.00617i	−0.864236	+	0.503087i	$0.832198\pi$
$314$		0			0
$315$	872.290	+	2414.47i	0.156025	+	0.431873i
$316$		0			0
$317$		−	10110.5i		−	1.79136i	−0.444700	−	0.895680i	$-0.646690\pi$
$317$		−	10110.5i		−	1.79136i	0.444700	−	0.895680i	$-0.353310\pi$
$318$		0			0
$319$		−	4722.44i		−	0.828858i
$320$		0			0
$321$			4660.73i			0.810395i
$322$		0			0
$323$	−11186.9			−1.92711
$324$		0			0
$325$	−1021.48			−0.174344
$326$		0			0
$327$	4272.75			0.722580
$328$		0			0
$329$	−1747.64	−	4837.41i	−0.292859	−	0.810624i
$330$		0			0
$331$	396.980			0.0659214			0.0329607	−	0.999457i	$-0.489506\pi$
$331$	396.980			0.0659214			0.0329607	+	0.999457i	$0.489506\pi$
$332$		0			0
$333$			3766.48i			0.619825i
$334$		0			0
$335$	4890.09			0.797535
$336$		0			0
$337$	7848.71			1.26868			0.634342	−	0.773053i	$-0.281271\pi$
$337$	7848.71			1.26868			0.634342	+	0.773053i	$0.281271\pi$
$338$		0			0
$339$		−	2354.26i		−	0.377186i
$340$		0			0
$341$	2493.43			0.395974
$342$		0			0
$343$	−5478.56	−	3215.44i	−0.862432	−	0.506173i
$344$		0			0
$345$	12.2279			0.00190820
$346$		0			0
$347$	−12522.0			−1.93722			−0.968609	−	0.248591i	$-0.920033\pi$
$347$	−12522.0			−1.93722			−0.968609	+	0.248591i	$0.920033\pi$
$348$		0			0
$349$	5091.86			0.780978			0.390489	−	0.920608i	$-0.372306\pi$
$349$	5091.86			0.780978			0.390489	+	0.920608i	$0.372306\pi$
$350$		0			0
$351$		−	245.778i		−	0.0373751i
$352$		0			0
$353$			9690.22i			1.46107i	0.682874	+	0.730536i	$0.260730\pi$
$353$			9690.22i			1.46107i	−0.682874	+	0.730536i	$0.739270\pi$
$354$		0			0
$355$		−	1634.03i		−	0.244296i
$356$		0			0
$357$	1396.21	+	3864.65i	0.206989	+	0.572938i
$358$		0			0
$359$		−	9222.74i		−	1.35587i	−0.735122	−	0.677935i	$-0.762875\pi$
$359$		−	9222.74i		−	1.35587i	0.735122	−	0.677935i	$-0.237125\pi$
$360$		0			0
$361$	−16021.1			−2.33578
$362$		0			0
$363$			3139.18i			0.453895i
$364$		0			0
$365$		−	11630.2i		−	1.66781i
$366$		0			0
$367$	2968.62			0.422236			0.211118	−	0.977461i	$-0.432289\pi$
$367$	2968.62			0.422236			0.211118	+	0.977461i	$0.432289\pi$
$368$		0			0
$369$		−	1479.39i		−	0.208710i
$370$		0			0
$371$	−4822.31	+	1742.19i	−0.674830	+	0.243800i
$372$		0			0
$373$			2883.02i			0.400207i	0.979775	+	0.200103i	$0.0641278\pi$
$373$			2883.02i			0.400207i	−0.979775	+	0.200103i	$0.935872\pi$
$374$		0			0
$375$		−	590.730i		−	0.0813471i
$376$		0			0
$377$			2548.13i			0.348105i
$378$		0			0
$379$	−2727.26			−0.369630			−0.184815	−	0.982773i	$-0.559169\pi$
$379$	−2727.26			−0.369630			−0.184815	+	0.982773i	$0.559169\pi$
$380$		0			0
$381$	−7527.81			−1.01223
$382$		0			0
$383$	10418.3			1.38995			0.694976	−	0.719033i	$-0.255415\pi$
$383$	10418.3			1.38995			0.694976	+	0.719033i	$0.255415\pi$
$384$		0			0
$385$	−1635.09	−	4525.88i	−0.216447	−	0.599117i
$386$		0			0
$387$	−2400.01			−0.315244
$388$		0			0
$389$			9095.84i			1.18555i	0.805369	+	0.592773i	$0.201967\pi$
$389$			9095.84i			1.18555i	−0.805369	+	0.592773i	$0.798033\pi$
$390$		0			0
$391$	19.5723			0.00253149
$392$		0			0
$393$	−6150.29			−0.789417
$394$		0			0
$395$		−	2991.06i		−	0.381004i
$396$		0			0
$397$	−1349.74			−0.170633			−0.0853167	−	0.996354i	$-0.527190\pi$
$397$	−1349.74			−0.170633			−0.0853167	+	0.996354i	$0.527190\pi$
$398$		0			0
$399$	2855.60	+	7904.20i	0.358293	+	0.991742i
$400$		0			0
$401$	1830.81			0.227996			0.113998	−	0.993481i	$-0.463634\pi$
$401$	1830.81			0.227996			0.113998	+	0.993481i	$0.463634\pi$
$402$		0			0
$403$	−1345.41			−0.166301
$404$		0			0
$405$	−1247.54			−0.153064
$406$		0			0
$407$		−	7060.20i		−	0.859855i
$408$		0			0
$409$		−	14709.9i		−	1.77838i	−0.457542	−	0.889188i	$-0.651270\pi$
$409$		−	14709.9i		−	1.77838i	0.457542	−	0.889188i	$-0.348730\pi$
$410$		0			0
$411$			7549.40i			0.906045i
$412$		0			0
$413$	−3692.84	+	1334.14i	−0.439982	+	0.158955i
$414$		0			0
$415$			15367.3i			1.81772i
$416$		0			0
$417$	−812.034			−0.0953608
$418$		0			0
$419$		−	6721.49i		−	0.783690i	−0.920031	−	0.391845i	$-0.871837\pi$
$419$		−	6721.49i		−	0.783690i	0.920031	−	0.391845i	$-0.128163\pi$
$420$		0			0
$421$		−	4541.66i		−	0.525765i	−0.964828	−	0.262882i	$-0.915327\pi$
$421$		−	4541.66i		−	0.525765i	0.964828	−	0.262882i	$-0.0846731\pi$
$422$		0			0
$423$	2499.47			0.287301
$424$		0			0
$425$			8299.13i			0.947216i
$426$		0			0
$427$	−1098.36	−	3040.22i	−0.124481	−	0.344558i
$428$		0			0
$429$			460.707i			0.0518488i
$430$		0			0
$431$		−	10747.7i		−	1.20116i	−0.799564	−	0.600580i	$-0.794936\pi$
$431$		−	10747.7i		−	1.20116i	0.799564	−	0.600580i	$-0.205064\pi$
$432$		0			0
$433$		−	10058.7i		−	1.11638i	−0.829714	−	0.558188i	$-0.811497\pi$
$433$		−	10058.7i		−	1.11638i	0.829714	−	0.558188i	$-0.188503\pi$
$434$		0			0
$435$	12934.1			1.42561
$436$		0			0
$437$	40.0303			0.00438194
$438$		0			0
$439$	2944.00			0.320067			0.160034	−	0.987112i	$-0.448840\pi$
$439$	2944.00			0.320067			0.160034	+	0.987112i	$0.448840\pi$
$440$		0			0
$441$	2374.20	−	1973.00i	0.256365	−	0.213044i
$442$		0			0
$443$	−8667.71			−0.929606			−0.464803	−	0.885414i	$-0.653875\pi$
$443$	−8667.71			−0.929606			−0.464803	+	0.885414i	$0.653875\pi$
$444$		0			0
$445$			15221.9i			1.62154i
$446$		0			0
$447$	716.680			0.0758340
$448$		0			0
$449$	2980.91			0.313314			0.156657	−	0.987653i	$-0.449928\pi$
$449$	2980.91			0.313314			0.156657	+	0.987653i	$0.449928\pi$
$450$		0			0
$451$			2773.10i			0.289534i
$452$		0			0
$453$	8578.46			0.889738
$454$		0			0
$455$	882.263	+	2442.07i	0.0909036	+	0.251618i
$456$		0			0
$457$	5915.60			0.605515			0.302757	−	0.953068i	$-0.402093\pi$
$457$	5915.60			0.605515			0.302757	+	0.953068i	$0.402093\pi$
$458$		0			0
$459$	−1996.85			−0.203061
$460$		0			0
$461$	−5755.48			−0.581474			−0.290737	−	0.956803i	$-0.593901\pi$
$461$	−5755.48			−0.581474			−0.290737	+	0.956803i	$0.593901\pi$
$462$		0			0
$463$			6502.53i			0.652696i	0.945250	+	0.326348i	$0.105818\pi$
$463$			6502.53i			0.652696i	−0.945250	+	0.326348i	$0.894182\pi$
$464$		0			0
$465$			6829.15i			0.681063i
$466$		0			0
$467$		−	4507.81i		−	0.446674i	−0.974741	−	0.223337i	$-0.928305\pi$
$467$		−	4507.81i		−	0.446674i	0.974741	−	0.223337i	$-0.0716950\pi$
$468$		0			0
$469$	−1997.99	−	5530.36i	−0.196713	−	0.544495i
$470$		0			0
$471$		−	1699.10i		−	0.166222i
$472$		0			0
$473$	4498.77			0.437323
$474$		0			0
$475$			16973.9i			1.63961i
$476$		0			0
$477$		−	2491.67i		−	0.239173i
$478$		0			0
$479$	−2320.76			−0.221375			−0.110687	−	0.993855i	$-0.535305\pi$
$479$	−2320.76			−0.221375			−0.110687	+	0.993855i	$0.535305\pi$
$480$		0			0
$481$			3809.54i			0.361123i
$482$		0			0
$483$	−4.99607	−	13.8289i	−0.000470661	−	0.00130277i
$484$		0			0
$485$		−	15737.2i		−	1.47338i
$486$		0			0
$487$		−	8971.64i		−	0.834792i	−0.908725	−	0.417396i	$-0.862943\pi$
$487$		−	8971.64i		−	0.834792i	0.908725	−	0.417396i	$-0.137057\pi$
$488$		0			0
$489$		−	3990.75i		−	0.369055i
$490$		0			0
$491$	−11942.9			−1.09771			−0.548857	−	0.835916i	$-0.684937\pi$
$491$	−11942.9			−1.09771			−0.548857	+	0.835916i	$0.684937\pi$
$492$		0			0
$493$	20702.5			1.89127
$494$		0			0
$495$	2338.50			0.212339
$496$		0			0
$497$	−1847.98	+	667.630i	−0.166787	+	0.0602561i
$498$		0			0
$499$	4865.01			0.436448			0.218224	−	0.975899i	$-0.429974\pi$
$499$	4865.01			0.436448			0.218224	+	0.975899i	$0.429974\pi$
$500$		0			0
$501$			8310.24i			0.741066i
$502$		0			0
$503$	−5302.73			−0.470054			−0.235027	−	0.971989i	$-0.575518\pi$
$503$	−5302.73			−0.470054			−0.235027	+	0.971989i	$0.575518\pi$
$504$		0			0
$505$	14087.5			1.24136
$506$		0			0
$507$			6342.41i			0.555575i
$508$		0			0
$509$	13255.3			1.15428			0.577142	−	0.816644i	$-0.304168\pi$
$509$	13255.3			1.15428			0.577142	+	0.816644i	$0.304168\pi$
$510$		0			0
$511$	−13152.9	+	4751.85i	−1.13865	+	0.411368i
$512$		0			0
$513$	−4084.07			−0.351493
$514$		0			0
$515$	−7224.95			−0.618193
$516$		0			0
$517$	−4685.21			−0.398560
$518$		0			0
$519$			4167.49i			0.352471i
$520$		0			0
$521$		−	17961.3i		−	1.51036i	−0.655517	−	0.755181i	$-0.727549\pi$
$521$		−	17961.3i		−	1.51036i	0.655517	−	0.755181i	$-0.272451\pi$
$522$		0			0
$523$			15183.8i			1.26948i	0.772724	+	0.634742i	$0.218894\pi$
$523$			15183.8i			1.26948i	−0.772724	+	0.634742i	$0.781106\pi$
$524$		0			0
$525$	5863.82	−	2118.46i	0.487463	−	0.176109i
$526$		0			0
$527$			10930.9i			0.903523i
$528$		0			0
$529$	12166.9			0.999994
$530$		0			0
$531$		−	1908.07i		−	0.155938i
$532$		0			0
$533$		−	1496.31i		−	0.121599i
$534$		0			0
$535$	23927.9			1.93363
$536$		0			0
$537$		−	7013.12i		−	0.563572i
$538$		0			0
$539$	−4450.40	+	3698.36i	−0.355644	+	0.295547i
$540$		0			0
$541$		−	1840.34i		−	0.146252i	−0.997323	−	0.0731262i	$-0.976702\pi$
$541$		−	1840.34i		−	0.146252i	0.997323	−	0.0731262i	$-0.0232976\pi$
$542$		0			0
$543$			9955.98i			0.786836i
$544$		0			0
$545$		−	21936.0i		−	1.72410i
$546$		0			0
$547$	17887.3			1.39818			0.699091	−	0.715033i	$-0.253588\pi$
$547$	17887.3			1.39818			0.699091	+	0.715033i	$0.253588\pi$
$548$		0			0
$549$	1570.87			0.122118
$550$		0			0
$551$	42342.0			3.27374
$552$		0			0
$553$	−3382.69	+	1222.09i	−0.260120	+	0.0939753i
$554$		0			0
$555$	19336.8			1.47893
$556$		0			0
$557$		−	12246.4i		−	0.931590i	−0.884893	−	0.465795i	$-0.845768\pi$
$557$		−	12246.4i		−	0.931590i	0.884893	−	0.465795i	$-0.154232\pi$
$558$		0			0
$559$	−2427.45			−0.183667
$560$		0			0
$561$	3743.06			0.281697
$562$		0			0
$563$		−	18024.0i		−	1.34924i	−0.738165	−	0.674621i	$-0.764307\pi$
$563$		−	18024.0i		−	1.34924i	0.738165	−	0.674621i	$-0.235693\pi$
$564$		0			0
$565$	−12086.6			−0.899979
$566$		0			0
$567$	509.721	+	1410.89i	0.0377536	+	0.104500i
$568$		0			0
$569$	21534.4			1.58659			0.793294	−	0.608839i	$-0.208364\pi$
$569$	21534.4			1.58659			0.793294	+	0.608839i	$0.208364\pi$
$570$		0			0
$571$	5542.36			0.406201			0.203100	−	0.979158i	$-0.434898\pi$
$571$	5542.36			0.406201			0.203100	+	0.979158i	$0.434898\pi$
$572$		0			0
$573$	−4842.19			−0.353028
$574$		0			0
$575$		−	29.6969i		−	0.00215382i
$576$		0			0
$577$		−	10857.0i		−	0.783334i	−0.920107	−	0.391667i	$-0.871898\pi$
$577$		−	10857.0i		−	0.783334i	0.920107	−	0.391667i	$-0.128102\pi$
$578$		0			0
$579$			3746.11i			0.268883i
$580$		0			0
$581$	17379.4	−	6278.78i	1.24100	−	0.448344i
$582$		0			0
$583$			4670.58i			0.331794i
$584$		0			0
$585$	−1261.81			−0.0891784
$586$		0			0
$587$			2426.15i			0.170593i	0.996356	+	0.0852963i	$0.0271837\pi$
$587$			2426.15i			0.170593i	−0.996356	+	0.0852963i	$0.972816\pi$
$588$		0			0
$589$			22356.5i			1.56398i
$590$		0			0
$591$	−12453.4			−0.866773
$592$		0			0
$593$		−	6562.03i		−	0.454418i	−0.973846	−	0.227209i	$-0.927040\pi$
$593$		−	6562.03i		−	0.454418i	0.973846	−	0.227209i	$-0.0729601\pi$
$594$		0			0
$595$	19840.8	−	7168.03i	1.36705	−	0.493883i
$596$		0			0
$597$			8786.97i			0.602390i
$598$		0			0
$599$		−	14414.2i		−	0.983222i	−0.870815	−	0.491611i	$-0.836408\pi$
$599$		−	14414.2i		−	0.983222i	0.870815	−	0.491611i	$-0.163592\pi$
$600$		0			0
$601$			206.035i			0.0139839i	0.999976	+	0.00699196i	$0.00222563\pi$
$601$			206.035i			0.0139839i	−0.999976	+	0.00699196i	$0.997774\pi$
$602$		0			0
$603$	2857.51			0.192980
$604$		0			0
$605$	16116.3			1.08301
$606$		0			0
$607$	9273.66			0.620109			0.310055	−	0.950719i	$-0.399653\pi$
$607$	9273.66			0.620109			0.310055	+	0.950719i	$0.399653\pi$
$608$		0			0
$609$	−5284.59	−	14627.5i	−0.351629	−	0.973297i
$610$		0			0
$611$	2528.05			0.167388
$612$		0			0
$613$		−	1780.49i		−	0.117314i	−0.998278	−	0.0586568i	$-0.981318\pi$
$613$		−	1780.49i		−	0.117314i	0.998278	−	0.0586568i	$-0.0186817\pi$
$614$		0			0
$615$	−7595.10			−0.497990
$616$		0			0
$617$	−10288.9			−0.671338			−0.335669	−	0.941980i	$-0.608962\pi$
$617$	−10288.9			−0.671338			−0.335669	+	0.941980i	$0.608962\pi$
$618$		0			0
$619$			28693.8i			1.86317i	0.363525	+	0.931585i	$0.381573\pi$
$619$			28693.8i			1.86317i	−0.363525	+	0.931585i	$0.618427\pi$
$620$		0			0
$621$	7.14535			0.000461728
$622$		0			0
$623$	17214.9	−	6219.35i	1.10706	−	0.399956i
$624$		0			0
$625$	−17059.7			−1.09182
$626$		0			0
$627$	7655.51			0.487610
$628$		0			0
$629$	30951.0			1.96200
$630$		0			0
$631$			22169.2i			1.39864i	0.714808	+	0.699320i	$0.246514\pi$
$631$			22169.2i			1.39864i	−0.714808	+	0.699320i	$0.753486\pi$
$632$		0			0
$633$			10082.4i			0.633077i
$634$		0			0
$635$			38647.2i			2.41523i
$636$		0			0
$637$	2401.34	−	1995.56i	0.149364	−	0.124124i
$638$		0			0
$639$		−	954.841i		−	0.0591125i
$640$		0			0
$641$	15310.8			0.943430			0.471715	−	0.881751i	$-0.343635\pi$
$641$	15310.8			0.943430			0.471715	+	0.881751i	$0.343635\pi$
$642$		0			0
$643$		−	9.01281i		−	0.000552769i	−1.00000		0.000276385i	$-0.999912\pi$
$643$		−	9.01281i		−	0.000552769i	1.00000		0.000276385i	$-8.79760e-5\pi$
$644$		0			0
$645$			12321.5i			0.752182i
$646$		0			0
$647$	−20077.1			−1.21996			−0.609979	−	0.792417i	$-0.708822\pi$
$647$	−20077.1			−1.21996			−0.609979	+	0.792417i	$0.708822\pi$
$648$		0			0
$649$			3576.65i			0.216326i
$650$		0			0
$651$	7723.30	−	2790.25i	0.464977	−	0.167985i
$652$		0			0
$653$			1796.00i			0.107631i	0.998551	+	0.0538154i	$0.0171383\pi$
$653$			1796.00i			0.107631i	−0.998551	+	0.0538154i	$0.982862\pi$
$654$		0			0
$655$			31575.1i			1.88358i
$656$		0			0
$657$		−	6796.06i		−	0.403561i
$658$		0			0
$659$	−12136.5			−0.717405			−0.358703	−	0.933452i	$-0.616781\pi$
$659$	−12136.5			−0.717405			−0.358703	+	0.933452i	$0.616781\pi$
$660$		0			0
$661$	−18080.9			−1.06394			−0.531970	−	0.846763i	$-0.678548\pi$
$661$	−18080.9			−1.06394			−0.531970	+	0.846763i	$0.678548\pi$
$662$		0			0
$663$	−2019.68			−0.118307
$664$		0			0
$665$	40579.6	−	14660.5i	2.36633	−	0.854900i
$666$		0			0
$667$	−74.0802			−0.00430045
$668$		0			0
$669$			6988.36i			0.403865i
$670$		0			0
$671$	−2944.56			−0.169409
$672$		0			0
$673$	−25019.4			−1.43303			−0.716513	−	0.697573i	$-0.754263\pi$
$673$	−25019.4			−1.43303			−0.716513	+	0.697573i	$0.754263\pi$
$674$		0			0
$675$			3029.81i			0.172766i
$676$		0			0
$677$	−28258.5			−1.60423			−0.802114	−	0.597170i	$-0.796292\pi$
$677$	−28258.5			−1.60423			−0.802114	+	0.597170i	$0.796292\pi$
$678$		0			0
$679$	−17797.7	+	6429.89i	−1.00591	+	0.363412i
$680$		0			0
$681$	−2498.67			−0.140601
$682$		0			0
$683$	−14966.8			−0.838491			−0.419245	−	0.907873i	$-0.637705\pi$
$683$	−14966.8			−0.838491			−0.419245	+	0.907873i	$0.637705\pi$
$684$		0			0
$685$	38758.1			2.16185
$686$		0			0
$687$			4303.07i			0.238970i
$688$		0			0
$689$		−	2520.15i		−	0.139347i
$690$		0			0
$691$		−	11414.7i		−	0.628415i	−0.949354	−	0.314208i	$-0.898261\pi$
$691$		−	11414.7i		−	0.628415i	0.949354	−	0.314208i	$-0.101739\pi$
$692$		0			0
$693$	−955.463	−	2644.69i	−0.0523738	−	0.144969i
$694$		0			0
$695$			4168.92i			0.227534i
$696$		0			0
$697$	−12156.9			−0.660653
$698$		0			0
$699$		−	7165.25i		−	0.387718i
$700$		0			0
$701$			26445.0i			1.42484i	0.701753	+	0.712420i	$0.252401\pi$
$701$			26445.0i			1.42484i	−0.701753	+	0.712420i	$0.747599\pi$
$702$		0			0
$703$	63302.7			3.39617
$704$		0			0
$705$		−	12832.1i		−	0.685511i
$706$		0			0
$707$	−5755.88	−	15932.1i	−0.306184	−	0.847506i
$708$		0			0
$709$			4251.03i			0.225177i	0.993642	+	0.112589i	$0.0359143\pi$
$709$			4251.03i			0.225177i	−0.993642	+	0.112589i	$0.964086\pi$
$710$		0			0
$711$		−	1747.82i		−	0.0921918i
$712$		0			0
$713$		−	39.1142i		−	0.00205447i
$714$		0			0
$715$	2365.24			0.123713
$716$		0			0
$717$	−14427.0			−0.751443
$718$		0			0
$719$	14049.0			0.728703			0.364352	−	0.931261i	$-0.381291\pi$
$719$	14049.0			0.728703			0.364352	+	0.931261i	$0.381291\pi$
$720$		0			0
$721$	2951.96	+	8170.92i	0.152478	+	0.422054i
$722$		0			0
$723$	−6808.76			−0.350236
$724$		0			0
$725$		−	31411.9i		−	1.60911i
$726$		0			0
$727$	22710.8			1.15859			0.579295	−	0.815118i	$-0.303328\pi$
$727$	22710.8			1.15859			0.579295	+	0.815118i	$0.303328\pi$
$728$		0			0
$729$	−729.000			−0.0370370
$730$		0			0
$731$			19722.0i			0.997873i
$732$		0			0
$733$	−34196.7			−1.72317			−0.861586	−	0.507611i	$-0.830529\pi$
$733$	−34196.7			−1.72317			−0.861586	+	0.507611i	$0.830529\pi$
$734$		0			0
$735$	−10129.3	−	12189.0i	−0.508331	−	0.611697i
$736$		0			0
$737$	−5356.35			−0.267712
$738$		0			0
$739$	−33629.7			−1.67400			−0.837001	−	0.547201i	$-0.815693\pi$
$739$	−33629.7			−1.67400			−0.837001	+	0.547201i	$0.815693\pi$
$740$		0			0
$741$	−4130.76			−0.204787
$742$		0			0
$743$		−	12955.5i		−	0.639693i	−0.947469	−	0.319846i	$-0.896369\pi$
$743$		−	12955.5i		−	0.639693i	0.947469	−	0.319846i	$-0.103631\pi$
$744$		0			0
$745$		−	3679.39i		−	0.180943i
$746$		0			0
$747$			8979.87i			0.439835i
$748$		0			0
$749$	−9776.43	−	27060.8i	−0.476933	−	1.32013i
$750$		0			0
$751$		−	13497.2i		−	0.655820i	−0.944709	−	0.327910i	$-0.893656\pi$
$751$		−	13497.2i		−	0.655820i	0.944709	−	0.327910i	$-0.106344\pi$
$752$		0			0
$753$	6726.21			0.325520
$754$		0			0
$755$		−	44041.2i		−	2.12295i
$756$		0			0
$757$			35571.1i			1.70786i	0.520386	+	0.853931i	$0.325788\pi$
$757$			35571.1i			1.70786i	−0.520386	+	0.853931i	$0.674212\pi$
$758$		0			0
$759$	−13.3938			−0.000640534
$760$		0			0
$761$			16927.8i			0.806348i	0.915123	+	0.403174i	$0.132093\pi$
$761$			16927.8i			0.806348i	−0.915123	+	0.403174i	$0.867907\pi$
$762$		0			0
$763$	−24808.1	+	8962.60i	−1.17708	+	0.425253i
$764$		0			0
$765$			10251.7i			0.484510i
$766$		0			0
$767$		−	1929.89i		−	0.0908529i
$768$		0			0
$769$		−	15208.9i		−	0.713197i	−0.934258	−	0.356599i	$-0.883936\pi$
$769$		−	15208.9i		−	0.713197i	0.934258	−	0.356599i	$-0.116064\pi$
$770$		0			0
$771$	2363.58			0.110405
$772$		0			0
$773$	10234.0			0.476187			0.238093	−	0.971242i	$-0.423478\pi$
$773$	10234.0			0.476187			0.238093	+	0.971242i	$0.423478\pi$
$774$		0			0
$775$	16585.4			0.768728
$776$		0			0
$777$	−7900.63	−	21868.7i	−0.364780	−	1.00970i
$778$		0			0
$779$	−24864.0			−1.14357
$780$		0			0
$781$			1789.83i			0.0820041i
$782$		0			0
$783$	7557.99			0.344956
$784$		0			0
$785$	−8723.06			−0.396611
$786$		0			0
$787$			26421.8i			1.19674i	0.801220	+	0.598370i	$0.204185\pi$
$787$			26421.8i			1.19674i	−0.801220	+	0.598370i	$0.795815\pi$
$788$		0			0
$789$	20090.9			0.906534
$790$		0			0
$791$	4938.34	+	13669.2i	0.221981	+	0.614437i
$792$		0			0
$793$	1588.83			0.0711486
$794$		0			0
$795$	−12792.0			−0.570675
$796$		0			0
$797$	−4607.96			−0.204796			−0.102398	−	0.994744i	$-0.532652\pi$
$797$	−4607.96			−0.204796			−0.102398	+	0.994744i	$0.532652\pi$
$798$		0			0
$799$		−	20539.3i		−	0.909424i
$800$		0			0
$801$			8894.88i			0.392366i
$802$		0			0
$803$			12739.1i			0.559842i
$804$		0			0
$805$	−70.9968	+	25.6495i	−0.00310846	+	0.00112301i
$806$		0			0
$807$			13124.8i			0.572507i
$808$		0			0
$809$	−16750.3			−0.727947			−0.363974	−	0.931409i	$-0.618580\pi$
$809$	−16750.3			−0.727947			−0.363974	+	0.931409i	$0.618580\pi$
$810$		0			0
$811$			25091.7i			1.08642i	0.839596	+	0.543211i	$0.182791\pi$
$811$			25091.7i			1.08642i	−0.839596	+	0.543211i	$0.817209\pi$
$812$		0			0
$813$			24174.7i			1.04286i
$814$		0			0
$815$	−20488.2			−0.880578
$816$		0			0
$817$			40336.6i			1.72729i
$818$		0			0
$819$	515.549	+	1427.02i	0.0219960	+	0.0608841i
$820$		0			0
$821$			1843.11i			0.0783495i	0.999232	+	0.0391747i	$0.0124729\pi$
$821$			1843.11i			0.0783495i	−0.999232	+	0.0391747i	$0.987527\pi$
$822$		0			0
$823$		−	3958.61i		−	0.167665i	−0.996480	−	0.0838327i	$-0.973284\pi$
$823$		−	3958.61i		−	0.167665i	0.996480	−	0.0838327i	$-0.0267161\pi$
$824$		0			0
$825$		−	5679.32i		−	0.239671i
$826$		0			0
$827$	655.225			0.0275507			0.0137753	−	0.999905i	$-0.495615\pi$
$827$	655.225			0.0275507			0.0137753	+	0.999905i	$0.495615\pi$
$828$		0			0
$829$	−8380.73			−0.351115			−0.175558	−	0.984469i	$-0.556173\pi$
$829$	−8380.73			−0.351115			−0.175558	+	0.984469i	$0.556173\pi$
$830$		0			0
$831$	12673.3			0.529040
$832$		0			0
$833$	−16213.1	−	19509.9i	−0.674371	−	0.811500i
$834$		0			0
$835$	42664.2			1.76821
$836$		0			0
$837$			3990.60i			0.164797i
$838$		0			0
$839$	−19219.3			−0.790850			−0.395425	−	0.918498i	$-0.629403\pi$
$839$	−19219.3			−0.790850			−0.395425	+	0.918498i	$0.629403\pi$
$840$		0			0
$841$	−53969.2			−2.21285
$842$		0			0
$843$		−	11291.3i		−	0.461321i
$844$		0			0
$845$	32561.5			1.32562
$846$		0			0
$847$	−6584.79	−	18226.4i	−0.267126	−	0.739396i
$848$		0			0
$849$	4812.48			0.194539
$850$		0			0
$851$	−110.752			−0.00446127
$852$		0			0
$853$	−16201.1			−0.650312			−0.325156	−	0.945660i	$-0.605417\pi$
$853$	−16201.1			−0.650312			−0.325156	+	0.945660i	$0.605417\pi$
$854$		0			0
$855$			20967.3i			0.838675i
$856$		0			0
$857$			40564.8i			1.61688i	0.588578	+	0.808440i	$0.299688\pi$
$857$			40564.8i			1.61688i	−0.588578	+	0.808440i	$0.700312\pi$
$858$		0			0
$859$		−	33166.6i		−	1.31738i	−0.752414	−	0.658690i	$-0.771111\pi$
$859$		−	33166.6i		−	1.31738i	0.752414	−	0.658690i	$-0.228889\pi$
$860$		0			0
$861$	3103.20	+	8589.55i	0.122830	+	0.339990i
$862$		0			0
$863$		−	26307.1i		−	1.03767i	−0.854876	−	0.518833i	$-0.826367\pi$
$863$		−	26307.1i		−	1.03767i	0.854876	−	0.518833i	$-0.173633\pi$
$864$		0			0
$865$	21395.6			0.841009
$866$		0			0
$867$			1670.06i			0.0654188i
$868$		0			0
$869$			3276.26i			0.127893i
$870$		0			0
$871$	2890.18			0.112434
$872$		0			0
$873$		−	9196.00i		−	0.356515i
$874$		0			0
$875$	1239.13	+	3429.85i	0.0478744	+	0.132515i
$876$		0			0
$877$			30669.7i			1.18089i	0.807077	+	0.590447i	$0.201048\pi$
$877$			30669.7i			1.18089i	−0.807077	+	0.590447i	$0.798952\pi$
$878$		0			0
$879$		−	9373.75i		−	0.359692i
$880$		0			0
$881$		−	31170.9i		−	1.19202i	−0.802975	−	0.596012i	$-0.796751\pi$
$881$		−	31170.9i		−	1.19202i	0.802975	−	0.596012i	$-0.203249\pi$
$882$		0			0
$883$	21907.1			0.834920			0.417460	−	0.908695i	$-0.362920\pi$
$883$	21907.1			0.834920			0.417460	+	0.908695i	$0.362920\pi$
$884$		0			0
$885$	−9795.91			−0.372075
$886$		0			0
$887$	1265.56			0.0479069			0.0239535	−	0.999713i	$-0.492375\pi$
$887$	1265.56			0.0479069			0.0239535	+	0.999713i	$0.492375\pi$
$888$		0			0
$889$	43707.4	−	15790.5i	1.64893	−	0.595720i
$890$		0			0
$891$	1366.50			0.0513798
$892$		0			0
$893$		−	42008.2i		−	1.57419i
$894$		0			0
$895$	−36004.8			−1.34470
$896$		0			0
$897$	7.22705			0.000269012
$898$		0			0
$899$		−	41372.9i		−	1.53489i
$900$		0			0
$901$	−20475.2			−0.757079
$902$		0			0
$903$	13934.8	−	5034.30i	0.513532	−	0.185527i
$904$		0			0
$905$	51113.3			1.87742
$906$		0			0
$907$	−11720.3			−0.429068			−0.214534	−	0.976716i	$-0.568823\pi$
$907$	−11720.3			−0.429068			−0.214534	+	0.976716i	$0.568823\pi$
$908$		0			0
$909$	8232.02			0.300373
$910$		0			0
$911$			30432.4i			1.10677i	0.832924	+	0.553387i	$0.186665\pi$
$911$			30432.4i			1.10677i	−0.832924	+	0.553387i	$0.813335\pi$
$912$		0			0
$913$		−	16832.6i		−	0.610162i
$914$		0			0
$915$		−	8064.72i		−	0.291379i
$916$		0			0
$917$	35709.3	−	12900.9i	1.28596	−	0.464588i
$918$		0			0
$919$			14099.5i			0.506092i	0.967454	+	0.253046i	$0.0814324\pi$
$919$			14099.5i			0.506092i	−0.967454	+	0.253046i	$0.918568\pi$
$920$		0			0
$921$	−29367.4			−1.05069
$922$		0			0
$923$		−	965.757i		−	0.0344402i
$924$		0			0
$925$		−	46961.8i		−	1.66929i
$926$		0			0
$927$	−4221.88			−0.149584
$928$		0			0
$929$			44727.9i			1.57963i	0.613347	+	0.789813i	$0.289823\pi$
$929$			44727.9i			1.57963i	−0.613347	+	0.789813i	$0.710177\pi$
$930$		0			0
$931$	−33160.0	−	39902.9i	−1.16732	−	1.40469i
$932$		0			0
$933$		−	11749.8i		−	0.412296i
$934$		0			0
$935$		−	19216.6i		−	0.672138i
$936$		0			0
$937$		−	56196.2i		−	1.95929i	−0.200748	−	0.979643i	$-0.564337\pi$
$937$		−	56196.2i		−	1.95929i	0.200748	−	0.979643i	$-0.435663\pi$
$938$		0			0
$939$	16715.2			0.580915
$940$		0			0
$941$	−4005.38			−0.138758			−0.0693792	−	0.997590i	$-0.522102\pi$
$941$	−4005.38			−0.138758			−0.0693792	+	0.997590i	$0.522102\pi$
$942$		0			0
$943$	43.5012			0.00150222
$944$		0			0
$945$	7243.40	−	2616.87i	0.249342	−	0.0900813i
$946$		0			0
$947$	−10996.3			−0.377332			−0.188666	−	0.982041i	$-0.560416\pi$
$947$	−10996.3			−0.377332			−0.188666	+	0.982041i	$0.560416\pi$
$948$		0			0
$949$		−	6873.76i		−	0.235123i
$950$		0			0
$951$	−30331.4			−1.03424
$952$		0			0
$953$	54721.9			1.86004			0.930020	−	0.367510i	$-0.119790\pi$
$953$	54721.9			1.86004			0.930020	+	0.367510i	$0.119790\pi$
$954$		0			0
$955$			24859.4i			0.842338i
$956$		0			0
$957$	−14167.3			−0.478541
$958$		0			0
$959$	−15835.7	−	43832.8i	−0.533225	−	1.47595i
$960$		0			0
$961$	−7946.19			−0.266731
$962$		0			0
$963$	13982.2			0.467882
$964$		0			0
$965$	19232.3			0.641564
$966$		0			0
$967$			20511.0i			0.682098i	0.940046	+	0.341049i	$0.110782\pi$
$967$			20511.0i			0.682098i	−0.940046	+	0.341049i	$0.889218\pi$
$968$		0			0
$969$			33560.7i			1.11262i
$970$		0			0
$971$			15402.6i			0.509055i	0.967066	+	0.254527i	$0.0819199\pi$
$971$			15402.6i			0.509055i	−0.967066	+	0.254527i	$0.918080\pi$
$972$		0			0
$973$	4714.77	−	1703.34i	0.155343	−	0.0561217i
$974$		0			0
$975$			3064.45i			0.100657i
$976$		0			0
$977$	30243.1			0.990341			0.495171	−	0.868796i	$-0.335106\pi$
$977$	30243.1			0.990341			0.495171	+	0.868796i	$0.335106\pi$
$978$		0			0
$979$		−	16673.3i		−	0.544311i
$980$		0			0
$981$		−	12818.3i		−	0.417182i
$982$		0			0
$983$	31981.0			1.03767			0.518837	−	0.854873i	$-0.326365\pi$
$983$	31981.0			1.03767			0.518837	+	0.854873i	$0.326365\pi$
$984$		0			0
$985$			63934.7i			2.06815i
$986$		0			0
$987$	−14512.2	+	5242.93i	−0.468014	+	0.169082i
$988$		0			0
$989$		−	70.5716i		−	0.00226901i
$990$		0			0
$991$		−	33096.4i		−	1.06089i	−0.847719	−	0.530445i	$-0.822025\pi$
$991$		−	33096.4i		−	1.06089i	0.847719	−	0.530445i	$-0.177975\pi$
$992$		0			0
$993$		−	1190.94i		−	0.0380598i
$994$		0			0
$995$	45111.7			1.43732
$996$		0			0
$997$	59677.3			1.89569			0.947843	−	0.318739i	$-0.103259\pi$
$997$	59677.3			1.89569			0.947843	+	0.318739i	$0.103259\pi$
$998$		0			0
$999$	11299.4			0.357856

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.4.p.c.223.29	yes	32
4.3	odd	2	inner	1344.4.p.c.223.14	yes	32
7.6	odd	2		1344.4.p.d.223.12	yes	32
8.3	odd	2		1344.4.p.d.223.11	yes	32
8.5	even	2		1344.4.p.d.223.10	yes	32
28.27	even	2		1344.4.p.d.223.9	yes	32
56.13	odd	2	inner	1344.4.p.c.223.13	✓	32
56.27	even	2	inner	1344.4.p.c.223.30	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1344.4.p.c.223.13	✓	32	56.13	odd	2	inner
1344.4.p.c.223.14	yes	32	4.3	odd	2	inner
1344.4.p.c.223.29	yes	32	1.1	even	1	trivial
1344.4.p.c.223.30	yes	32	56.27	even	2	inner
1344.4.p.d.223.9	yes	32	28.27	even	2
1344.4.p.d.223.10	yes	32	8.5	even	2
1344.4.p.d.223.11	yes	32	8.3	odd	2
1344.4.p.d.223.12	yes	32	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1344.p (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.00000i q^{3} -15.4018 q^{5} +(6.29285 + 17.4184i) q^{7} -9.00000 q^{9} +O(q^{10})\)	\(q-3.00000i q^{3} -15.4018 q^{5} +(6.29285 + 17.4184i) q^{7} -9.00000 q^{9} +16.8703 q^{11} -9.10290 q^{13} +46.2054i q^{15} +73.9573i q^{17} +151.262i q^{19} +(52.2551 - 18.8786i) q^{21} -0.264643i q^{23} +112.215 q^{25} +27.0000i q^{27} -279.925i q^{29} +147.800 q^{31} -50.6110i q^{33} +(-96.9212 - 268.274i) q^{35} -418.498i q^{37} +27.3087i q^{39} +164.377i q^{41} +266.668 q^{43} +138.616 q^{45} -277.719 q^{47} +(-263.800 + 219.223i) q^{49} +221.872 q^{51} +276.852i q^{53} -259.833 q^{55} +453.785 q^{57} +212.008i q^{59} -174.541 q^{61} +(-56.6357 - 156.765i) q^{63} +140.201 q^{65} -317.501 q^{67} -0.793928 q^{69} +106.093i q^{71} +755.118i q^{73} -336.645i q^{75} +(106.163 + 293.854i) q^{77} +194.202i q^{79} +81.0000 q^{81} -997.764i q^{83} -1139.08i q^{85} -839.776 q^{87} -988.320i q^{89} +(-57.2832 - 158.558i) q^{91} -443.400i q^{93} -2329.70i q^{95} +1021.78i q^{97} -151.833 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 288 q^{9}+O(q^{10}) \) 32 * q - 288 * q^9	\( 32 q - 288 q^{9} - 224 q^{13} - 72 q^{21} + 1120 q^{25} - 752 q^{49} - 672 q^{57} - 544 q^{61} + 1536 q^{65} - 144 q^{69} - 1632 q^{77} + 2592 q^{81}+O(q^{100}) \) 32 * q - 288 * q^9 - 224 * q^13 - 72 * q^21 + 1120 * q^25 - 752 * q^49 - 672 * q^57 - 544 * q^61 + 1536 * q^65 - 144 * q^69 - 1632 * q^77 + 2592 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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