LMFDB - Embedded newform 1344.4.p.d.223.17

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			223.17
Character	$\chi$	$=$	1344.223
Dual form			1344.4.p.d.223.18

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{3} -1.64166 q^{5} +(15.2747 - 10.4729i) q^{7} -9.00000 q^{9} +O(q^{10})$	$q-3.00000i q^{3} -1.64166 q^{5} +(15.2747 - 10.4729i) q^{7} -9.00000 q^{9} +20.3470 q^{11} -13.0999 q^{13} +4.92497i q^{15} +23.9396i q^{17} -87.7589i q^{19} +(-31.4188 - 45.8242i) q^{21} -73.6175i q^{23} -122.305 q^{25} +27.0000i q^{27} +58.9537i q^{29} -124.909 q^{31} -61.0410i q^{33} +(-25.0759 + 17.1930i) q^{35} -56.5972i q^{37} +39.2998i q^{39} +135.651i q^{41} +259.929 q^{43} +14.7749 q^{45} +217.682 q^{47} +(123.635 - 319.943i) q^{49} +71.8188 q^{51} -529.342i q^{53} -33.4028 q^{55} -263.277 q^{57} -685.329i q^{59} -149.916 q^{61} +(-137.473 + 94.2565i) q^{63} +21.5056 q^{65} -409.156 q^{67} -220.853 q^{69} +885.869i q^{71} -269.426i q^{73} +366.915i q^{75} +(310.795 - 213.093i) q^{77} +902.587i q^{79} +81.0000 q^{81} -623.963i q^{83} -39.3006i q^{85} +176.861 q^{87} -986.131i q^{89} +(-200.098 + 137.195i) q^{91} +374.726i q^{93} +144.070i q^{95} -179.437i q^{97} -183.123 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−1.64166			−0.146834			−0.0734171	−	0.997301i	$-0.523390\pi$
$5$	−1.64166			−0.146834			−0.0734171	+	0.997301i	$0.523390\pi$
$6$		0			0
$7$	15.2747	−	10.4729i	0.824758	−	0.565486i
$7$	15.2747	−	10.4729i	0.824758	−	0.565486i
$8$		0			0
$9$	−9.00000			−0.333333
$10$		0			0
$11$	20.3470			0.557714			0.278857	−	0.960333i	$-0.410044\pi$
$11$	20.3470			0.557714			0.278857	+	0.960333i	$0.410044\pi$
$12$		0			0
$13$	−13.0999			−0.279482			−0.139741	−	0.990188i	$-0.544627\pi$
$13$	−13.0999			−0.279482			−0.139741	+	0.990188i	$0.544627\pi$
$14$		0			0
$15$			4.92497i			0.0847748i
$16$		0			0
$17$			23.9396i			0.341542i	0.985311	+	0.170771i	$0.0546258\pi$
$17$			23.9396i			0.341542i	−0.985311	+	0.170771i	$0.945374\pi$
$18$		0			0
$19$		−	87.7589i		−	1.05965i	−0.848108	−	0.529823i	$-0.822258\pi$
$19$		−	87.7589i		−	1.05965i	0.848108	−	0.529823i	$-0.177742\pi$
$20$		0			0
$21$	−31.4188	−	45.8242i	−0.326483	−	0.476174i
$22$		0			0
$23$		−	73.6175i		−	0.667405i	−0.942678	−	0.333703i	$-0.891702\pi$
$23$		−	73.6175i		−	0.667405i	0.942678	−	0.333703i	$-0.108298\pi$
$24$		0			0
$25$	−122.305			−0.978440
$26$		0			0
$27$			27.0000i			0.192450i
$28$		0			0
$29$			58.9537i			0.377498i	0.982025	+	0.188749i	$0.0604432\pi$
$29$			58.9537i			0.377498i	−0.982025	+	0.188749i	$0.939557\pi$
$30$		0			0
$31$	−124.909			−0.723686			−0.361843	−	0.932239i	$-0.617852\pi$
$31$	−124.909			−0.723686			−0.361843	+	0.932239i	$0.617852\pi$
$32$		0			0
$33$		−	61.0410i		−	0.321996i
$34$		0			0
$35$	−25.0759	+	17.1930i	−0.121103	+	0.0830327i
$36$		0			0
$37$		−	56.5972i		−	0.251474i	−0.992064	−	0.125737i	$-0.959871\pi$
$37$		−	56.5972i		−	0.251474i	0.992064	−	0.125737i	$-0.0401295\pi$
$38$		0			0
$39$			39.2998i			0.161359i
$40$		0			0
$41$			135.651i			0.516711i	0.966050	+	0.258356i	$0.0831806\pi$
$41$			135.651i			0.516711i	−0.966050	+	0.258356i	$0.916819\pi$
$42$		0			0
$43$	259.929			0.921833			0.460917	−	0.887443i	$-0.347521\pi$
$43$	259.929			0.921833			0.460917	+	0.887443i	$0.347521\pi$
$44$		0			0
$45$	14.7749			0.0489448
$46$		0			0
$47$	217.682			0.675580			0.337790	−	0.941222i	$-0.390321\pi$
$47$	217.682			0.675580			0.337790	+	0.941222i	$0.390321\pi$
$48$		0			0
$49$	123.635	−	319.943i	0.360452	−	0.932778i
$50$		0			0
$51$	71.8188			0.197189
$52$		0			0
$53$		−	529.342i		−	1.37190i	−0.727649	−	0.685950i	$-0.759387\pi$
$53$		−	529.342i		−	1.37190i	0.727649	−	0.685950i	$-0.240613\pi$
$54$		0			0
$55$	−33.4028			−0.0818916
$56$		0			0
$57$	−263.277			−0.611787
$58$		0			0
$59$		−	685.329i		−	1.51224i	−0.654433	−	0.756120i	$-0.727092\pi$
$59$		−	685.329i		−	1.51224i	0.654433	−	0.756120i	$-0.272908\pi$
$60$		0			0
$61$	−149.916			−0.314668			−0.157334	−	0.987545i	$-0.550290\pi$
$61$	−149.916			−0.314668			−0.157334	+	0.987545i	$0.550290\pi$
$62$		0			0
$63$	−137.473	+	94.2565i	−0.274919	+	0.188495i
$64$		0			0
$65$	21.5056			0.0410376
$66$		0			0
$67$	−409.156			−0.746066			−0.373033	−	0.927818i	$-0.621682\pi$
$67$	−409.156			−0.746066			−0.373033	+	0.927818i	$0.621682\pi$
$68$		0			0
$69$	−220.853			−0.385326
$70$		0			0
$71$			885.869i			1.48075i	0.672194	+	0.740375i	$0.265352\pi$
$71$			885.869i			1.48075i	−0.672194	+	0.740375i	$0.734648\pi$
$72$		0			0
$73$		−	269.426i		−	0.431971i	−0.976397	−	0.215986i	$-0.930704\pi$
$73$		−	269.426i		−	0.431971i	0.976397	−	0.215986i	$-0.0692964\pi$
$74$		0			0
$75$			366.915i			0.564902i
$76$		0			0
$77$	310.795	−	213.093i	0.459979	−	0.315379i
$78$		0			0
$79$			902.587i			1.28543i	0.766105	+	0.642715i	$0.222192\pi$
$79$			902.587i			1.28543i	−0.766105	+	0.642715i	$0.777808\pi$
$80$		0			0
$81$	81.0000			0.111111
$82$		0			0
$83$		−	623.963i		−	0.825166i	−0.910920	−	0.412583i	$-0.864627\pi$
$83$		−	623.963i		−	0.825166i	0.910920	−	0.412583i	$-0.135373\pi$
$84$		0			0
$85$		−	39.3006i		−	0.0501500i
$86$		0			0
$87$	176.861			0.217948
$88$		0			0
$89$		−	986.131i		−	1.17449i	−0.809409	−	0.587245i	$-0.800212\pi$
$89$		−	986.131i		−	1.17449i	0.809409	−	0.587245i	$-0.199788\pi$
$90$		0			0
$91$	−200.098	+	137.195i	−0.230505	+	0.158043i
$92$		0			0
$93$			374.726i			0.417821i
$94$		0			0
$95$			144.070i			0.155592i
$96$		0			0
$97$		−	179.437i		−	0.187825i	−0.995580	−	0.0939127i	$-0.970063\pi$
$97$		−	179.437i		−	0.187825i	0.995580	−	0.0939127i	$-0.0299375\pi$
$98$		0			0
$99$	−183.123			−0.185905
$100$		0			0
$101$	266.959			0.263004			0.131502	−	0.991316i	$-0.458020\pi$
$101$	266.959			0.263004			0.131502	+	0.991316i	$0.458020\pi$
$102$		0			0
$103$	−988.258			−0.945398			−0.472699	−	0.881224i	$-0.656720\pi$
$103$	−988.258			−0.945398			−0.472699	+	0.881224i	$0.656720\pi$
$104$		0			0
$105$	51.5789	+	75.2276i	0.0479389	+	0.0699187i
$106$		0			0
$107$	−1115.52			−1.00787			−0.503934	−	0.863742i	$-0.668114\pi$
$107$	−1115.52			−1.00787			−0.503934	+	0.863742i	$0.668114\pi$
$108$		0			0
$109$			286.196i			0.251492i	0.992062	+	0.125746i	$0.0401324\pi$
$109$			286.196i			0.251492i	−0.992062	+	0.125746i	$0.959868\pi$
$110$		0			0
$111$	−169.792			−0.145188
$112$		0			0
$113$	−728.322			−0.606326			−0.303163	−	0.952939i	$-0.598043\pi$
$113$	−728.322			−0.606326			−0.303163	+	0.952939i	$0.598043\pi$
$114$		0			0
$115$			120.855i			0.0979979i
$116$		0			0
$117$	117.899			0.0931607
$118$		0			0
$119$	250.718	+	365.671i	0.193137	+	0.281689i
$120$		0			0
$121$	−916.999			−0.688955
$122$		0			0
$123$	406.953			0.298323
$124$		0			0
$125$	405.990			0.290503
$126$		0			0
$127$		−	159.234i		−	0.111258i	−0.998452	−	0.0556288i	$-0.982284\pi$
$127$		−	159.234i		−	0.111258i	0.998452	−	0.0556288i	$-0.0177163\pi$
$128$		0			0
$129$		−	779.787i		−	0.532221i
$130$		0			0
$131$		−	748.323i		−	0.499094i	−0.968363	−	0.249547i	$-0.919718\pi$
$131$		−	748.323i		−	0.499094i	0.968363	−	0.249547i	$-0.0802817\pi$
$132$		0			0
$133$	−919.093	−	1340.49i	−0.599214	−	0.873951i
$134$		0			0
$135$		−	44.3247i		−	0.0282583i
$136$		0			0
$137$	−1189.48			−0.741780			−0.370890	−	0.928677i	$-0.620947\pi$
$137$	−1189.48			−0.741780			−0.370890	+	0.928677i	$0.620947\pi$
$138$		0			0
$139$		−	546.158i		−	0.333270i	−0.986019	−	0.166635i	$-0.946710\pi$
$139$		−	546.158i		−	0.333270i	0.986019	−	0.166635i	$-0.0532901\pi$
$140$		0			0
$141$		−	653.047i		−	0.390046i
$142$		0			0
$143$	−266.545			−0.155871
$144$		0			0
$145$		−	96.7818i		−	0.0554296i
$146$		0			0
$147$	−959.828	−	370.905i	−0.538539	−	0.208107i
$148$		0			0
$149$		−	2169.41i		−	1.19279i	−0.802693	−	0.596393i	$-0.796600\pi$
$149$		−	2169.41i		−	1.19279i	0.802693	−	0.596393i	$-0.203400\pi$
$150$		0			0
$151$			1873.28i			1.00957i	0.863244	+	0.504786i	$0.168429\pi$
$151$			1873.28i			1.00957i	−0.863244	+	0.504786i	$0.831571\pi$
$152$		0			0
$153$		−	215.457i		−	0.113847i
$154$		0			0
$155$	205.057			0.106262
$156$		0			0
$157$	2457.07			1.24902			0.624508	−	0.781018i	$-0.285299\pi$
$157$	2457.07			1.24902			0.624508	+	0.781018i	$0.285299\pi$
$158$		0			0
$159$	−1588.02			−0.792066
$160$		0			0
$161$	−770.992	−	1124.49i	−0.377408	−	0.550448i
$162$		0			0
$163$	−1184.76			−0.569308			−0.284654	−	0.958630i	$-0.591879\pi$
$163$	−1184.76			−0.569308			−0.284654	+	0.958630i	$0.591879\pi$
$164$		0			0
$165$			100.208i			0.0472801i
$166$		0			0
$167$	−2493.54			−1.15543			−0.577713	−	0.816240i	$-0.696055\pi$
$167$	−2493.54			−1.15543			−0.577713	+	0.816240i	$0.696055\pi$
$168$		0			0
$169$	−2025.39			−0.921890
$170$		0			0
$171$			789.830i			0.353215i
$172$		0			0
$173$	−336.961			−0.148085			−0.0740424	−	0.997255i	$-0.523590\pi$
$173$	−336.961			−0.148085			−0.0740424	+	0.997255i	$0.523590\pi$
$174$		0			0
$175$	−1868.18	+	1280.89i	−0.806976	+	0.553294i
$176$		0			0
$177$	−2055.99			−0.873092
$178$		0			0
$179$	403.536			0.168501			0.0842506	−	0.996445i	$-0.473150\pi$
$179$	403.536			0.168501			0.0842506	+	0.996445i	$0.473150\pi$
$180$		0			0
$181$	626.262			0.257180			0.128590	−	0.991698i	$-0.458955\pi$
$181$	626.262			0.257180			0.128590	+	0.991698i	$0.458955\pi$
$182$		0			0
$183$			449.747i			0.181674i
$184$		0			0
$185$			92.9132i			0.0369249i
$186$		0			0
$187$			487.100i			0.190483i
$188$		0			0
$189$	282.769	+	412.418i	0.108828	+	0.158725i
$190$		0			0
$191$			696.358i			0.263805i	0.991263	+	0.131902i	$0.0421085\pi$
$191$			696.358i			0.263805i	−0.991263	+	0.131902i	$0.957891\pi$
$192$		0			0
$193$	−3330.70			−1.24222			−0.621111	−	0.783723i	$-0.713318\pi$
$193$	−3330.70			−1.24222			−0.621111	+	0.783723i	$0.713318\pi$
$194$		0			0
$195$		−	64.5168i		−	0.0236930i
$196$		0			0
$197$		−	841.118i		−	0.304199i	−0.988365	−	0.152099i	$-0.951397\pi$
$197$		−	841.118i		−	0.304199i	0.988365	−	0.152099i	$-0.0486034\pi$
$198$		0			0
$199$	719.574			0.256328			0.128164	−	0.991753i	$-0.459092\pi$
$199$	719.574			0.256328			0.128164	+	0.991753i	$0.459092\pi$
$200$		0			0
$201$			1227.47i			0.430741i
$202$		0			0
$203$	617.419	+	900.503i	0.213470	+	0.311344i
$204$		0			0
$205$		−	222.693i		−	0.0758709i
$206$		0			0
$207$			662.558i			0.222468i
$208$		0			0
$209$		−	1785.63i		−	0.590979i
$210$		0			0
$211$	−4957.51			−1.61748			−0.808742	−	0.588163i	$-0.799851\pi$
$211$	−4957.51			−1.61748			−0.808742	+	0.588163i	$0.799851\pi$
$212$		0			0
$213$	2657.61			0.854912
$214$		0			0
$215$	−426.715			−0.135357
$216$		0			0
$217$	−1907.95	+	1308.16i	−0.596866	+	0.409234i
$218$		0			0
$219$	−808.277			−0.249399
$220$		0			0
$221$		−	313.607i		−	0.0954548i
$222$		0			0
$223$	−3947.42			−1.18538			−0.592688	−	0.805432i	$-0.701933\pi$
$223$	−3947.42			−1.18538			−0.592688	+	0.805432i	$0.701933\pi$
$224$		0			0
$225$	1100.74			0.326147
$226$		0			0
$227$		−	2044.37i		−	0.597751i	−0.954292	−	0.298876i	$-0.903388\pi$
$227$		−	2044.37i		−	0.597751i	0.954292	−	0.298876i	$-0.0966115\pi$
$228$		0			0
$229$	4391.90			1.26736			0.633679	−	0.773596i	$-0.281544\pi$
$229$	4391.90			1.26736			0.633679	+	0.773596i	$0.281544\pi$
$230$		0			0
$231$	−639.279	−	932.386i	−0.182084	−	0.265569i
$232$		0			0
$233$	−1970.16			−0.553946			−0.276973	−	0.960878i	$-0.589331\pi$
$233$	−1970.16			−0.553946			−0.276973	+	0.960878i	$0.589331\pi$
$234$		0			0
$235$	−357.360			−0.0991983
$236$		0			0
$237$	2707.76			0.742144
$238$		0			0
$239$			2964.89i			0.802437i	0.915982	+	0.401219i	$0.131413\pi$
$239$			2964.89i			0.802437i	−0.915982	+	0.401219i	$0.868587\pi$
$240$		0			0
$241$		−	3658.31i		−	0.977810i	−0.872337	−	0.488905i	$-0.837396\pi$
$241$		−	3658.31i		−	0.977810i	0.872337	−	0.488905i	$-0.162604\pi$
$242$		0			0
$243$		−	243.000i		−	0.0641500i
$244$		0			0
$245$	−202.966	+	525.236i	−0.0529267	+	0.136964i
$246$		0			0
$247$			1149.64i			0.296152i
$248$		0			0
$249$	−1871.89			−0.476410
$250$		0			0
$251$		−	2719.35i		−	0.683840i	−0.939729	−	0.341920i	$-0.888923\pi$
$251$		−	2719.35i		−	0.683840i	0.939729	−	0.341920i	$-0.111077\pi$
$252$		0			0
$253$		−	1497.90i		−	0.372221i
$254$		0			0
$255$	−117.902			−0.0289541
$256$		0			0
$257$		−	6798.11i		−	1.65002i	−0.565120	−	0.825009i	$-0.691170\pi$
$257$		−	6798.11i		−	1.65002i	0.565120	−	0.825009i	$-0.308830\pi$
$258$		0			0
$259$	−592.739	−	864.507i	−0.142205	−	0.207405i
$260$		0			0
$261$		−	530.584i		−	0.125833i
$262$		0			0
$263$			7965.94i			1.86768i	0.357686	+	0.933842i	$0.383566\pi$
$263$			7965.94i			1.86768i	−0.357686	+	0.933842i	$0.616434\pi$
$264$		0			0
$265$			868.997i			0.201442i
$266$		0			0
$267$	−2958.39			−0.678092
$268$		0			0
$269$	−5718.64			−1.29618			−0.648089	−	0.761565i	$-0.724431\pi$
$269$	−5718.64			−1.29618			−0.648089	+	0.761565i	$0.724431\pi$
$270$		0			0
$271$	1289.85			0.289124			0.144562	−	0.989496i	$-0.453823\pi$
$271$	1289.85			0.289124			0.144562	+	0.989496i	$0.453823\pi$
$272$		0			0
$273$	411.584	+	600.294i	0.0912462	+	0.133082i
$274$		0			0
$275$	−2488.54			−0.545690
$276$		0			0
$277$			4121.16i			0.893922i	0.894553	+	0.446961i	$0.147494\pi$
$277$			4121.16i			0.893922i	−0.894553	+	0.446961i	$0.852506\pi$
$278$		0			0
$279$	1124.18			0.241229
$280$		0			0
$281$	−2432.92			−0.516497			−0.258248	−	0.966079i	$-0.583145\pi$
$281$	−2432.92			−0.516497			−0.258248	+	0.966079i	$0.583145\pi$
$282$		0			0
$283$			0.386542i			8.11927e-5i	1.00000		4.05963e-5i	$1.29222e-5\pi$
$283$			0.386542i			8.11927e-5i	−1.00000		4.05963e-5i	$0.999987\pi$
$284$		0			0
$285$	432.210			0.0898312
$286$		0			0
$287$	1420.67	+	2072.04i	0.292193	+	0.426162i
$288$		0			0
$289$	4339.89			0.883349
$290$		0			0
$291$	−538.311			−0.108441
$292$		0			0
$293$	−3394.41			−0.676805			−0.338402	−	0.941002i	$-0.609886\pi$
$293$	−3394.41			−0.676805			−0.338402	+	0.941002i	$0.609886\pi$
$294$		0			0
$295$			1125.07i			0.222049i
$296$		0			0
$297$			549.369i			0.107332i
$298$		0			0
$299$			964.385i			0.186528i
$300$		0			0
$301$	3970.35	−	2722.22i	0.760289	−	0.521283i
$302$		0			0
$303$		−	800.877i		−	0.151845i
$304$		0			0
$305$	246.110			0.0462040
$306$		0			0
$307$			1720.47i			0.319844i	0.987130	+	0.159922i	$0.0511243\pi$
$307$			1720.47i			0.319844i	−0.987130	+	0.159922i	$0.948876\pi$
$308$		0			0
$309$			2964.77i			0.545826i
$310$		0			0
$311$	−4721.33			−0.860843			−0.430422	−	0.902628i	$-0.641635\pi$
$311$	−4721.33			−0.860843			−0.430422	+	0.902628i	$0.641635\pi$
$312$		0			0
$313$		−	483.679i		−	0.0873455i	−0.999046	−	0.0436728i	$-0.986094\pi$
$313$		−	483.679i		−	0.0873455i	0.999046	−	0.0436728i	$-0.0139059\pi$
$314$		0			0
$315$	225.683	−	154.737i	0.0403676	−	0.0276776i
$316$		0			0
$317$		−	6443.81i		−	1.14170i	−0.821053	−	0.570852i	$-0.806613\pi$
$317$		−	6443.81i		−	1.14170i	0.821053	−	0.570852i	$-0.193387\pi$
$318$		0			0
$319$			1199.53i			0.210536i
$320$		0			0
$321$			3346.57i			0.581893i
$322$		0			0
$323$	2100.91			0.361913
$324$		0			0
$325$	1602.19			0.273456
$326$		0			0
$327$	858.588			0.145199
$328$		0			0
$329$	3325.04	−	2279.78i	0.557190	−	0.382031i
$330$		0			0
$331$	−2784.85			−0.462444			−0.231222	−	0.972901i	$-0.574272\pi$
$331$	−2784.85			−0.462444			−0.231222	+	0.972901i	$0.574272\pi$
$332$		0			0
$333$			509.375i			0.0838245i
$334$		0			0
$335$	671.694			0.109548
$336$		0			0
$337$	−4181.48			−0.675904			−0.337952	−	0.941163i	$-0.609734\pi$
$337$	−4181.48			−0.675904			−0.337952	+	0.941163i	$0.609734\pi$
$338$		0			0
$339$			2184.97i			0.350062i
$340$		0			0
$341$	−2541.52			−0.403610
$342$		0			0
$343$	−1462.25	−	6181.86i	−0.230187	−	0.973146i
$344$		0			0
$345$	362.564			0.0565791
$346$		0			0
$347$	9524.25			1.47345			0.736727	−	0.676190i	$-0.236370\pi$
$347$	9524.25			1.47345			0.736727	+	0.676190i	$0.236370\pi$
$348$		0			0
$349$	−1248.33			−0.191465			−0.0957327	−	0.995407i	$-0.530519\pi$
$349$	−1248.33			−0.191465			−0.0957327	+	0.995407i	$0.530519\pi$
$350$		0			0
$351$		−	353.698i		−	0.0537864i
$352$		0			0
$353$		−	605.680i		−	0.0913233i	−0.998957	−	0.0456616i	$-0.985460\pi$
$353$		−	605.680i		−	0.0913233i	0.998957	−	0.0456616i	$-0.0145396\pi$
$354$		0			0
$355$		−	1454.29i		−	0.217425i
$356$		0			0
$357$	1097.01	−	752.154i	0.162633	−	0.111508i
$358$		0			0
$359$		−	83.1817i		−	0.0122289i	−0.999981	−	0.00611443i	$-0.998054\pi$
$359$		−	83.1817i		−	0.0122289i	0.999981	−	0.00611443i	$-0.00194630\pi$
$360$		0			0
$361$	−842.618			−0.122849
$362$		0			0
$363$			2751.00i			0.397768i
$364$		0			0
$365$			442.305i			0.0634282i
$366$		0			0
$367$	−4636.32			−0.659439			−0.329719	−	0.944079i	$-0.606954\pi$
$367$	−4636.32			−0.659439			−0.329719	+	0.944079i	$0.606954\pi$
$368$		0			0
$369$		−	1220.86i		−	0.172237i
$370$		0			0
$371$	−5543.76	−	8085.55i	−0.775789	−	1.13148i
$372$		0			0
$373$		−	2555.59i		−	0.354754i	−0.984143	−	0.177377i	$-0.943239\pi$
$373$		−	2555.59i		−	0.354754i	0.984143	−	0.177377i	$-0.0567612\pi$
$374$		0			0
$375$		−	1217.97i		−	0.167722i
$376$		0			0
$377$		−	772.290i		−	0.105504i
$378$		0			0
$379$	8981.28			1.21725			0.608624	−	0.793459i	$-0.291722\pi$
$379$	8981.28			1.21725			0.608624	+	0.793459i	$0.291722\pi$
$380$		0			0
$381$	−477.701			−0.0642346
$382$		0			0
$383$	12164.5			1.62291			0.811456	−	0.584413i	$-0.198675\pi$
$383$	12164.5			1.62291			0.811456	+	0.584413i	$0.198675\pi$
$384$		0			0
$385$	−510.219	+	349.826i	−0.0675407	+	0.0463085i
$386$		0			0
$387$	−2339.36			−0.307278
$388$		0			0
$389$		−	3569.55i		−	0.465252i	−0.972566	−	0.232626i	$-0.925268\pi$
$389$		−	3569.55i		−	0.465252i	0.972566	−	0.232626i	$-0.0747319\pi$
$390$		0			0
$391$	1762.38			0.227947
$392$		0			0
$393$	−2244.97			−0.288152
$394$		0			0
$395$		−	1481.74i		−	0.188745i
$396$		0			0
$397$	984.957			0.124518			0.0622589	−	0.998060i	$-0.480170\pi$
$397$	984.957			0.124518			0.0622589	+	0.998060i	$0.480170\pi$
$398$		0			0
$399$	−4021.48	+	2757.28i	−0.504576	+	0.345957i
$400$		0			0
$401$	1700.17			0.211727			0.105863	−	0.994381i	$-0.466239\pi$
$401$	1700.17			0.211727			0.105863	+	0.994381i	$0.466239\pi$
$402$		0			0
$403$	1636.30			0.202257
$404$		0			0
$405$	−132.974			−0.0163149
$406$		0			0
$407$		−	1151.58i		−	0.140250i
$408$		0			0
$409$			2739.61i			0.331210i	0.986192	+	0.165605i	$0.0529577\pi$
$409$			2739.61i			0.331210i	−0.986192	+	0.165605i	$0.947042\pi$
$410$		0			0
$411$			3568.43i			0.428267i
$412$		0			0
$413$	−7177.41	−	10468.2i	−0.855150	−	1.24723i
$414$		0			0
$415$			1024.33i			0.121163i
$416$		0			0
$417$	−1638.47			−0.192413
$418$		0			0
$419$		−	13461.7i		−	1.56957i	−0.619770	−	0.784783i	$-0.712774\pi$
$419$		−	13461.7i		−	1.56957i	0.619770	−	0.784783i	$-0.287226\pi$
$420$		0			0
$421$			5618.17i			0.650387i	0.945647	+	0.325194i	$0.105429\pi$
$421$			5618.17i			0.650387i	−0.945647	+	0.325194i	$0.894571\pi$
$422$		0			0
$423$	−1959.14			−0.225193
$424$		0			0
$425$		−	2927.93i		−	0.334178i
$426$		0			0
$427$	−2289.92	+	1570.06i	−0.259525	+	0.177940i
$428$		0			0
$429$			799.634i			0.0899923i
$430$		0			0
$431$		−	7137.27i		−	0.797657i	−0.917026	−	0.398828i	$-0.869417\pi$
$431$		−	7137.27i		−	0.797657i	0.917026	−	0.398828i	$-0.130583\pi$
$432$		0			0
$433$			1458.21i			0.161841i	0.996721	+	0.0809203i	$0.0257859\pi$
$433$			1458.21i			0.161841i	−0.996721	+	0.0809203i	$0.974214\pi$
$434$		0			0
$435$	−290.346			−0.0320023
$436$		0			0
$437$	−6460.59			−0.707213
$438$		0			0
$439$	3231.27			0.351299			0.175650	−	0.984453i	$-0.443797\pi$
$439$	3231.27			0.351299			0.175650	+	0.984453i	$0.443797\pi$
$440$		0			0
$441$	−1112.72	+	2879.48i	−0.120151	+	0.310926i
$442$		0			0
$443$	17504.9			1.87739			0.938694	−	0.344751i	$-0.112037\pi$
$443$	17504.9			1.87739			0.938694	+	0.344751i	$0.112037\pi$
$444$		0			0
$445$			1618.89i			0.172455i
$446$		0			0
$447$	−6508.23			−0.688655
$448$		0			0
$449$	9069.04			0.953217			0.476608	−	0.879116i	$-0.341866\pi$
$449$	9069.04			0.953217			0.476608	+	0.879116i	$0.341866\pi$
$450$		0			0
$451$			2760.10i			0.288177i
$452$		0			0
$453$	5619.84			0.582877
$454$		0			0
$455$	328.492	−	225.227i	0.0338461	−	0.0232062i
$456$		0			0
$457$	−1722.16			−0.176279			−0.0881394	−	0.996108i	$-0.528092\pi$
$457$	−1722.16			−0.176279			−0.0881394	+	0.996108i	$0.528092\pi$
$458$		0			0
$459$	−646.370			−0.0657297
$460$		0			0
$461$	−14064.0			−1.42088			−0.710442	−	0.703755i	$-0.751505\pi$
$461$	−14064.0			−1.42088			−0.710442	+	0.703755i	$0.751505\pi$
$462$		0			0
$463$		−	7092.58i		−	0.711923i	−0.934501	−	0.355961i	$-0.884153\pi$
$463$		−	7092.58i		−	0.711923i	0.934501	−	0.355961i	$-0.115847\pi$
$464$		0			0
$465$		−	615.172i		−	0.0613504i
$466$		0			0
$467$		−	8397.55i		−	0.832103i	−0.909341	−	0.416052i	$-0.863414\pi$
$467$		−	8397.55i		−	0.832103i	0.909341	−	0.416052i	$-0.136586\pi$
$468$		0			0
$469$	−6249.76	+	4285.07i	−0.615324	+	0.421889i
$470$		0			0
$471$		−	7371.21i		−	0.721120i
$472$		0			0
$473$	5288.78			0.514119
$474$		0			0
$475$			10733.3i			1.03680i
$476$		0			0
$477$			4764.07i			0.457300i
$478$		0			0
$479$	20229.8			1.92970			0.964849	−	0.262807i	$-0.0846482\pi$
$479$	20229.8			1.92970			0.964849	+	0.262807i	$0.0846482\pi$
$480$		0			0
$481$			741.419i			0.0702824i
$482$		0			0
$483$	−3373.46	+	2312.98i	−0.317801	+	0.217897i
$484$		0			0
$485$			294.574i			0.0275792i
$486$		0			0
$487$		−	1879.06i		−	0.174843i	−0.996171	−	0.0874213i	$-0.972137\pi$
$487$		−	1879.06i		−	0.174843i	0.996171	−	0.0874213i	$-0.0278626\pi$
$488$		0			0
$489$			3554.27i			0.328690i
$490$		0			0
$491$	9249.10			0.850114			0.425057	−	0.905167i	$-0.360254\pi$
$491$	9249.10			0.850114			0.425057	+	0.905167i	$0.360254\pi$
$492$		0			0
$493$	−1411.33			−0.128931
$494$		0			0
$495$	300.625			0.0272972
$496$		0			0
$497$	9277.65	+	13531.4i	0.837343	+	1.22126i
$498$		0			0
$499$	−18142.9			−1.62763			−0.813813	−	0.581126i	$-0.802612\pi$
$499$	−18142.9			−1.62763			−0.813813	+	0.581126i	$0.802612\pi$
$500$		0			0
$501$			7480.63i			0.667086i
$502$		0			0
$503$	7981.16			0.707480			0.353740	−	0.935344i	$-0.384910\pi$
$503$	7981.16			0.707480			0.353740	+	0.935344i	$0.384910\pi$
$504$		0			0
$505$	−438.255			−0.0386180
$506$		0			0
$507$			6076.18i			0.532253i
$508$		0			0
$509$	9093.08			0.791834			0.395917	−	0.918286i	$-0.370427\pi$
$509$	9093.08			0.791834			0.395917	+	0.918286i	$0.370427\pi$
$510$		0			0
$511$	−2821.68	−	4115.41i	−0.244274	−	0.356272i
$512$		0			0
$513$	2369.49			0.203929
$514$		0			0
$515$	1622.38			0.138817
$516$		0			0
$517$	4429.19			0.376780
$518$		0			0
$519$			1010.88i			0.0854968i
$520$		0			0
$521$		−	10886.0i		−	0.915399i	−0.889107	−	0.457699i	$-0.848674\pi$
$521$		−	10886.0i		−	0.915399i	0.889107	−	0.457699i	$-0.151326\pi$
$522$		0			0
$523$			16709.0i			1.39701i	0.715607	+	0.698503i	$0.246150\pi$
$523$			16709.0i			1.39701i	−0.715607	+	0.698503i	$0.753850\pi$
$524$		0			0
$525$	3842.68	+	5604.53i	0.319444	+	0.465908i
$526$		0			0
$527$		−	2990.27i		−	0.247169i
$528$		0			0
$529$	6747.46			0.554570
$530$		0			0
$531$			6167.96i			0.504080i
$532$		0			0
$533$		−	1777.02i		−	0.144411i
$534$		0			0
$535$	1831.31			0.147990
$536$		0			0
$537$		−	1210.61i		−	0.0972842i
$538$		0			0
$539$	2515.60	−	6509.88i	0.201029	−	0.520223i
$540$		0			0
$541$			9846.26i			0.782484i	0.920288	+	0.391242i	$0.127954\pi$
$541$			9846.26i			0.782484i	−0.920288	+	0.391242i	$0.872046\pi$
$542$		0			0
$543$		−	1878.78i		−	0.148483i
$544$		0			0
$545$		−	469.836i		−	0.0369276i
$546$		0			0
$547$	−3956.47			−0.309262			−0.154631	−	0.987972i	$-0.549419\pi$
$547$	−3956.47			−0.309262			−0.154631	+	0.987972i	$0.549419\pi$
$548$		0			0
$549$	1349.24			0.104889
$550$		0			0
$551$	5173.71			0.400014
$552$		0			0
$553$	9452.74	+	13786.8i	0.726892	+	1.06017i
$554$		0			0
$555$	278.740			0.0213186
$556$		0			0
$557$			10754.5i			0.818106i	0.912511	+	0.409053i	$0.134141\pi$
$557$			10754.5i			0.818106i	−0.912511	+	0.409053i	$0.865859\pi$
$558$		0			0
$559$	−3405.05			−0.257636
$560$		0			0
$561$	1461.30			0.109975
$562$		0			0
$563$		−	9499.10i		−	0.711082i	−0.934661	−	0.355541i	$-0.884297\pi$
$563$		−	9499.10i		−	0.711082i	0.934661	−	0.355541i	$-0.115703\pi$
$564$		0			0
$565$	1195.66			0.0890294
$566$		0			0
$567$	1237.25	−	848.308i	0.0916398	−	0.0628317i
$568$		0			0
$569$	21157.1			1.55879			0.779395	−	0.626533i	$-0.215526\pi$
$569$	21157.1			1.55879			0.779395	+	0.626533i	$0.215526\pi$
$570$		0			0
$571$	21995.3			1.61204			0.806021	−	0.591887i	$-0.201617\pi$
$571$	21995.3			1.61204			0.806021	+	0.591887i	$0.201617\pi$
$572$		0			0
$573$	2089.07			0.152308
$574$		0			0
$575$			9003.79i			0.653016i
$576$		0			0
$577$			7582.38i			0.547068i	0.961862	+	0.273534i	$0.0881927\pi$
$577$			7582.38i			0.547068i	−0.961862	+	0.273534i	$0.911807\pi$
$578$		0			0
$579$			9992.09i			0.717197i
$580$		0			0
$581$	−6534.72	−	9530.86i	−0.466620	−	0.680563i
$582$		0			0
$583$		−	10770.5i		−	0.765128i
$584$		0			0
$585$	−193.550			−0.0136792
$586$		0			0
$587$		−	2177.57i		−	0.153114i	−0.997065	−	0.0765569i	$-0.975607\pi$
$587$		−	2177.57i		−	0.153114i	0.997065	−	0.0765569i	$-0.0243927\pi$
$588$		0			0
$589$			10961.9i			0.766851i
$590$		0			0
$591$	−2523.35			−0.175629
$592$		0			0
$593$			181.962i			0.0126008i	0.999980	+	0.00630042i	$0.00200550\pi$
$593$			181.962i			0.0126008i	−0.999980	+	0.00630042i	$0.997995\pi$
$594$		0			0
$595$	−411.593	−	600.307i	−0.0283591	−	0.0413616i
$596$		0			0
$597$		−	2158.72i		−	0.147991i
$598$		0			0
$599$		−	255.073i		−	0.0173990i	−0.999962	−	0.00869949i	$-0.997231\pi$
$599$		−	255.073i		−	0.0173990i	0.999962	−	0.00869949i	$-0.00276917\pi$
$600$		0			0
$601$			21122.3i			1.43361i	0.697275	+	0.716804i	$0.254396\pi$
$601$			21122.3i			1.43361i	−0.697275	+	0.716804i	$0.745604\pi$
$602$		0			0
$603$	3682.41			0.248689
$604$		0			0
$605$	1505.40			0.101162
$606$		0			0
$607$	20997.8			1.40408			0.702038	−	0.712139i	$-0.252273\pi$
$607$	20997.8			1.40408			0.702038	+	0.712139i	$0.252273\pi$
$608$		0			0
$609$	2701.51	−	1852.26i	0.179755	−	0.123247i
$610$		0			0
$611$	−2851.63			−0.188813
$612$		0			0
$613$			13781.4i			0.908032i	0.890993	+	0.454016i	$0.150009\pi$
$613$			13781.4i			0.908032i	−0.890993	+	0.454016i	$0.849991\pi$
$614$		0			0
$615$	−668.078			−0.0438041
$616$		0			0
$617$	15885.1			1.03648			0.518242	−	0.855234i	$-0.326586\pi$
$617$	15885.1			1.03648			0.518242	+	0.855234i	$0.326586\pi$
$618$		0			0
$619$			26764.3i			1.73788i	0.494917	+	0.868940i	$0.335198\pi$
$619$			26764.3i			1.73788i	−0.494917	+	0.868940i	$0.664802\pi$
$620$		0			0
$621$	1987.67			0.128442
$622$		0			0
$623$	−10327.7	−	15062.9i	−0.664157	−	0.968671i
$624$		0			0
$625$	14621.6			0.935784
$626$		0			0
$627$	−5356.89			−0.341202
$628$		0			0
$629$	1354.91			0.0858887
$630$		0			0
$631$		−	12538.1i		−	0.791022i	−0.918461	−	0.395511i	$-0.870568\pi$
$631$		−	12538.1i		−	0.791022i	0.918461	−	0.395511i	$-0.129432\pi$
$632$		0			0
$633$			14872.5i			0.933855i
$634$		0			0
$635$			261.407i			0.0163364i
$636$		0			0
$637$	−1619.61	+	4191.23i	−0.100740	+	0.260695i
$638$		0			0
$639$		−	7972.82i		−	0.493584i
$640$		0			0
$641$	−10139.1			−0.624761			−0.312380	−	0.949957i	$-0.601126\pi$
$641$	−10139.1			−0.624761			−0.312380	+	0.949957i	$0.601126\pi$
$642$		0			0
$643$		−	14901.0i		−	0.913902i	−0.889492	−	0.456951i	$-0.848941\pi$
$643$		−	14901.0i		−	0.913902i	0.889492	−	0.456951i	$-0.151059\pi$
$644$		0			0
$645$			1280.14i			0.0781482i
$646$		0			0
$647$	23608.6			1.43454			0.717271	−	0.696794i	$-0.245391\pi$
$647$	23608.6			1.43454			0.717271	+	0.696794i	$0.245391\pi$
$648$		0			0
$649$		−	13944.4i		−	0.843398i
$650$		0			0
$651$	3924.49	+	5723.85i	0.236272	+	0.344601i
$652$		0			0
$653$		−	1066.91i		−	0.0639380i	−0.999489	−	0.0319690i	$-0.989822\pi$
$653$		−	1066.91i		−	0.0639380i	0.999489	−	0.0319690i	$-0.0101778\pi$
$654$		0			0
$655$			1228.49i			0.0732841i
$656$		0			0
$657$			2424.83i			0.143990i
$658$		0			0
$659$	1256.55			0.0742766			0.0371383	−	0.999310i	$-0.488176\pi$
$659$	1256.55			0.0742766			0.0371383	+	0.999310i	$0.488176\pi$
$660$		0			0
$661$	11557.5			0.680080			0.340040	−	0.940411i	$-0.389559\pi$
$661$	11557.5			0.680080			0.340040	+	0.940411i	$0.389559\pi$
$662$		0			0
$663$	−940.822			−0.0551109
$664$		0			0
$665$	1508.84	+	2200.63i	0.0879852	+	0.128326i
$666$		0			0
$667$	4340.03			0.251944
$668$		0			0
$669$			11842.3i			0.684377i
$670$		0			0
$671$	−3050.34			−0.175495
$672$		0			0
$673$	4626.18			0.264972			0.132486	−	0.991185i	$-0.457704\pi$
$673$	4626.18			0.264972			0.132486	+	0.991185i	$0.457704\pi$
$674$		0			0
$675$		−	3302.23i		−	0.188301i
$676$		0			0
$677$	−33749.5			−1.91595			−0.957975	−	0.286850i	$-0.907392\pi$
$677$	−33749.5			−1.91595			−0.957975	+	0.286850i	$0.907392\pi$
$678$		0			0
$679$	−1879.23	−	2740.85i	−0.106213	−	0.154911i
$680$		0			0
$681$	−6133.11			−0.345112
$682$		0			0
$683$	22799.7			1.27732			0.638658	−	0.769491i	$-0.279490\pi$
$683$	22799.7			1.27732			0.638658	+	0.769491i	$0.279490\pi$
$684$		0			0
$685$	1952.71			0.108919
$686$		0			0
$687$		−	13175.7i		−	0.731709i
$688$		0			0
$689$			6934.34i			0.383421i
$690$		0			0
$691$			22334.8i			1.22960i	0.788682	+	0.614802i	$0.210764\pi$
$691$			22334.8i			1.22960i	−0.788682	+	0.614802i	$0.789236\pi$
$692$		0			0
$693$	−2797.16	+	1917.84i	−0.153326	+	0.105126i
$694$		0			0
$695$			896.603i			0.0489354i
$696$		0			0
$697$	−3247.44			−0.176478
$698$		0			0
$699$			5910.48i			0.319821i
$700$		0			0
$701$		−	8429.88i		−	0.454197i	−0.973872	−	0.227099i	$-0.927076\pi$
$701$		−	8429.88i		−	0.454197i	0.973872	−	0.227099i	$-0.0729240\pi$
$702$		0			0
$703$	−4966.91			−0.266473
$704$		0			0
$705$			1072.08i			0.0572722i
$706$		0			0
$707$	4077.73	−	2795.85i	0.216915	−	0.148725i
$708$		0			0
$709$			6186.00i			0.327673i	0.986488	+	0.163837i	$0.0523870\pi$
$709$			6186.00i			0.327673i	−0.986488	+	0.163837i	$0.947613\pi$
$710$		0			0
$711$		−	8123.29i		−	0.428477i
$712$		0			0
$713$			9195.48i			0.482992i
$714$		0			0
$715$	437.575			0.0228872
$716$		0			0
$717$	8894.66			0.463287
$718$		0			0
$719$	17866.0			0.926688			0.463344	−	0.886178i	$-0.346649\pi$
$719$	17866.0			0.926688			0.463344	+	0.886178i	$0.346649\pi$
$720$		0			0
$721$	−15095.4	+	10350.0i	−0.779725	+	0.534609i
$722$		0			0
$723$	−10974.9			−0.564539
$724$		0			0
$725$		−	7210.34i		−	0.369359i
$726$		0			0
$727$	−27651.7			−1.41065			−0.705326	−	0.708883i	$-0.749199\pi$
$727$	−27651.7			−1.41065			−0.705326	+	0.708883i	$0.749199\pi$
$728$		0			0
$729$	−729.000			−0.0370370
$730$		0			0
$731$			6222.60i			0.314844i
$732$		0			0
$733$	26895.7			1.35527			0.677636	−	0.735397i	$-0.263004\pi$
$733$	26895.7			1.35527			0.677636	+	0.735397i	$0.263004\pi$
$734$		0			0
$735$	1575.71	+	608.899i	0.0790761	+	0.0305573i
$736$		0			0
$737$	−8325.11			−0.416091
$738$		0			0
$739$	−22507.8			−1.12038			−0.560191	−	0.828364i	$-0.689272\pi$
$739$	−22507.8			−1.12038			−0.560191	+	0.828364i	$0.689272\pi$
$740$		0			0
$741$	3448.91			0.170983
$742$		0			0
$743$		−	19387.3i		−	0.957268i	−0.878014	−	0.478634i	$-0.841132\pi$
$743$		−	19387.3i		−	0.957268i	0.878014	−	0.478634i	$-0.158868\pi$
$744$		0			0
$745$			3561.43i			0.175142i
$746$		0			0
$747$			5615.66i			0.275055i
$748$		0			0
$749$	−17039.3	+	11682.8i	−0.831247	+	0.569935i
$750$		0			0
$751$		−	34865.6i		−	1.69409i	−0.531518	−	0.847047i	$-0.678378\pi$
$751$		−	34865.6i		−	1.69409i	0.531518	−	0.847047i	$-0.321622\pi$
$752$		0			0
$753$	−8158.04			−0.394815
$754$		0			0
$755$		−	3075.29i		−	0.148240i
$756$		0			0
$757$		−	19477.5i		−	0.935165i	−0.883949	−	0.467583i	$-0.845125\pi$
$757$		−	19477.5i		−	0.935165i	0.883949	−	0.467583i	$-0.154875\pi$
$758$		0			0
$759$	−4493.69			−0.214902
$760$		0			0
$761$		−	33587.3i		−	1.59992i	−0.600053	−	0.799960i	$-0.704854\pi$
$761$		−	33587.3i		−	1.59992i	0.600053	−	0.799960i	$-0.295146\pi$
$762$		0			0
$763$	2997.31	+	4371.57i	0.142215	+	0.207420i
$764$		0			0
$765$			353.706i			0.0167167i
$766$		0			0
$767$			8977.76i			0.422644i
$768$		0			0
$769$			30658.2i			1.43766i	0.695185	+	0.718831i	$0.255323\pi$
$769$			30658.2i			1.43766i	−0.695185	+	0.718831i	$0.744677\pi$
$770$		0			0
$771$	−20394.3			−0.952638
$772$		0			0
$773$	−12029.2			−0.559714			−0.279857	−	0.960042i	$-0.590287\pi$
$773$	−12029.2			−0.559714			−0.279857	+	0.960042i	$0.590287\pi$
$774$		0			0
$775$	15277.0			0.708084
$776$		0			0
$777$	−2593.52	+	1778.22i	−0.119745	+	0.0821019i
$778$		0			0
$779$	11904.6			0.547530
$780$		0			0
$781$			18024.8i			0.825836i
$782$		0			0
$783$	−1591.75			−0.0726495
$784$		0			0
$785$	−4033.67			−0.183398
$786$		0			0
$787$		−	33352.6i		−	1.51066i	−0.655342	−	0.755332i	$-0.727476\pi$
$787$		−	33352.6i		−	1.51066i	0.655342	−	0.755332i	$-0.272524\pi$
$788$		0			0
$789$	23897.8			1.07831
$790$		0			0
$791$	−11124.9	+	7627.68i	−0.500072	+	0.342868i
$792$		0			0
$793$	1963.89			0.0879440
$794$		0			0
$795$	2606.99			0.116302
$796$		0			0
$797$	−44570.0			−1.98087			−0.990434	−	0.137989i	$-0.955936\pi$
$797$	−44570.0			−1.98087			−0.990434	+	0.137989i	$0.955936\pi$
$798$		0			0
$799$			5211.23i			0.230739i
$800$		0			0
$801$			8875.18i			0.391497i
$802$		0			0
$803$		−	5482.01i		−	0.240916i
$804$		0			0
$805$	1265.70	+	1846.02i	0.0554164	+	0.0808246i
$806$		0			0
$807$			17155.9i			0.748349i
$808$		0			0
$809$	33277.9			1.44621			0.723107	−	0.690736i	$-0.242713\pi$
$809$	33277.9			1.44621			0.723107	+	0.690736i	$0.242713\pi$
$810$		0			0
$811$		−	5936.77i		−	0.257051i	−0.991706	−	0.128525i	$-0.958976\pi$
$811$		−	5936.77i		−	0.257051i	0.991706	−	0.128525i	$-0.0410244\pi$
$812$		0			0
$813$		−	3869.54i		−	0.166926i
$814$		0			0
$815$	1944.96			0.0835939
$816$		0			0
$817$		−	22811.1i		−	0.976816i
$818$		0			0
$819$	1800.88	−	1234.75i	0.0768351	−	0.0526810i
$820$		0			0
$821$		−	12777.3i		−	0.543154i	−0.962417	−	0.271577i	$-0.912455\pi$
$821$		−	12777.3i		−	0.543154i	0.962417	−	0.271577i	$-0.0875452\pi$
$822$		0			0
$823$		−	10978.0i		−	0.464968i	−0.972600	−	0.232484i	$-0.925315\pi$
$823$		−	10978.0i		−	0.464968i	0.972600	−	0.232484i	$-0.0746854\pi$
$824$		0			0
$825$			7465.62i			0.315054i
$826$		0			0
$827$	40213.5			1.69088			0.845442	−	0.534067i	$-0.179337\pi$
$827$	40213.5			1.69088			0.845442	+	0.534067i	$0.179337\pi$
$828$		0			0
$829$	1332.10			0.0558090			0.0279045	−	0.999611i	$-0.491117\pi$
$829$	1332.10			0.0558090			0.0279045	+	0.999611i	$0.491117\pi$
$830$		0			0
$831$	12363.5			0.516106
$832$		0			0
$833$	7659.31	+	2959.78i	0.318583	+	0.123109i
$834$		0			0
$835$	4093.55			0.169656
$836$		0			0
$837$		−	3372.54i		−	0.139274i
$838$		0			0
$839$	−728.570			−0.0299798			−0.0149899	−	0.999888i	$-0.504772\pi$
$839$	−728.570			−0.0299798			−0.0149899	+	0.999888i	$0.504772\pi$
$840$		0			0
$841$	20913.5			0.857495
$842$		0			0
$843$			7298.75i			0.298200i
$844$		0			0
$845$	3325.00			0.135365
$846$		0			0
$847$	−14006.9	+	9603.68i	−0.568221	+	0.389594i
$848$		0			0
$849$	1.15963			4.68766e−5
$850$		0			0
$851$	−4166.55			−0.167835
$852$		0			0
$853$	−22239.8			−0.892703			−0.446352	−	0.894858i	$-0.647277\pi$
$853$	−22239.8			−0.892703			−0.446352	+	0.894858i	$0.647277\pi$
$854$		0			0
$855$		−	1296.63i		−	0.0518641i
$856$		0			0
$857$		−	40668.9i		−	1.62103i	−0.585716	−	0.810516i	$-0.699187\pi$
$857$		−	40668.9i		−	1.62103i	0.585716	−	0.810516i	$-0.300813\pi$
$858$		0			0
$859$			31416.1i			1.24785i	0.781485	+	0.623925i	$0.214463\pi$
$859$			31416.1i			1.24785i	−0.781485	+	0.623925i	$0.785537\pi$
$860$		0			0
$861$	6216.11	−	4262.00i	0.246045	−	0.168698i
$862$		0			0
$863$		−	7916.87i		−	0.312275i	−0.987735	−	0.156138i	$-0.950096\pi$
$863$		−	7916.87i		−	0.312275i	0.987735	−	0.156138i	$-0.0499043\pi$
$864$		0			0
$865$	553.175			0.0217439
$866$		0			0
$867$		−	13019.7i		−	0.510002i
$868$		0			0
$869$			18365.0i			0.716903i
$870$		0			0
$871$	5359.92			0.208512
$872$		0			0
$873$			1614.93i			0.0626085i
$874$		0			0
$875$	6201.39	−	4251.91i	0.239595	−	0.164275i
$876$		0			0
$877$		−	13698.3i		−	0.527432i	−0.964600	−	0.263716i	$-0.915052\pi$
$877$		−	13698.3i		−	0.527432i	0.964600	−	0.263716i	$-0.0849483\pi$
$878$		0			0
$879$			10183.2i			0.390753i
$880$		0			0
$881$			28771.4i			1.10026i	0.835078	+	0.550132i	$0.185423\pi$
$881$			28771.4i			1.10026i	−0.835078	+	0.550132i	$0.814577\pi$
$882$		0			0
$883$	−12122.7			−0.462019			−0.231009	−	0.972952i	$-0.574203\pi$
$883$	−12122.7			−0.462019			−0.231009	+	0.972952i	$0.574203\pi$
$884$		0			0
$885$	3375.22			0.128200
$886$		0			0
$887$	40858.2			1.54666			0.773328	−	0.634006i	$-0.218591\pi$
$887$	40858.2			1.54666			0.773328	+	0.634006i	$0.218591\pi$
$888$		0			0
$889$	−1667.65	−	2432.25i	−0.0629146	−	0.0917606i
$890$		0			0
$891$	1648.11			0.0619682
$892$		0			0
$893$		−	19103.6i		−	0.715875i
$894$		0			0
$895$	−662.468			−0.0247418
$896$		0			0
$897$	2893.15			0.107692
$898$		0			0
$899$		−	7363.84i		−	0.273190i
$900$		0			0
$901$	12672.2			0.468561
$902$		0			0
$903$	−8166.67	−	11911.0i	−0.300963	−	0.438953i
$904$		0			0
$905$	−1028.11			−0.0377629
$906$		0			0
$907$	12054.9			0.441318			0.220659	−	0.975351i	$-0.429179\pi$
$907$	12054.9			0.441318			0.220659	+	0.975351i	$0.429179\pi$
$908$		0			0
$909$	−2402.63			−0.0876680
$910$		0			0
$911$			41363.5i			1.50432i	0.658981	+	0.752160i	$0.270988\pi$
$911$			41363.5i			1.50432i	−0.658981	+	0.752160i	$0.729012\pi$
$912$		0			0
$913$		−	12695.8i		−	0.460207i
$914$		0			0
$915$		−	738.331i		−	0.0266759i
$916$		0			0
$917$	−7837.15	−	11430.4i	−0.282230	−	0.411632i
$918$		0			0
$919$			43591.9i			1.56470i	0.622836	+	0.782352i	$0.285980\pi$
$919$			43591.9i			1.56470i	−0.622836	+	0.782352i	$0.714020\pi$
$920$		0			0
$921$	5161.40			0.184662
$922$		0			0
$923$		−	11604.8i		−	0.413843i
$924$		0			0
$925$			6922.12i			0.246052i
$926$		0			0
$927$	8894.32			0.315133
$928$		0			0
$929$		−	23421.9i		−	0.827179i	−0.910464	−	0.413589i	$-0.864275\pi$
$929$		−	23421.9i		−	0.827179i	0.910464	−	0.413589i	$-0.135725\pi$
$930$		0			0
$931$	−28077.8	−	10850.1i	−0.988414	−	0.381951i
$932$		0			0
$933$			14164.0i			0.497008i
$934$		0			0
$935$		−	799.651i		−	0.0279694i
$936$		0			0
$937$		−	12461.9i		−	0.434486i	−0.976118	−	0.217243i	$-0.930294\pi$
$937$		−	12461.9i		−	0.434486i	0.976118	−	0.217243i	$-0.0697064\pi$
$938$		0			0
$939$	−1451.04			−0.0504290
$940$		0			0
$941$	4981.43			0.172572			0.0862858	−	0.996270i	$-0.472500\pi$
$941$	4981.43			0.172572			0.0862858	+	0.996270i	$0.472500\pi$
$942$		0			0
$943$	9986.30			0.344856
$944$		0			0
$945$	−464.210	−	677.049i	−0.0159796	−	0.0233062i
$946$		0			0
$947$	−9015.91			−0.309374			−0.154687	−	0.987963i	$-0.549437\pi$
$947$	−9015.91			−0.309374			−0.154687	+	0.987963i	$0.549437\pi$
$948$		0			0
$949$			3529.46i			0.120728i
$950$		0			0
$951$	−19331.4			−0.659163
$952$		0			0
$953$	−25785.1			−0.876455			−0.438227	−	0.898864i	$-0.644394\pi$
$953$	−25785.1			−0.876455			−0.438227	+	0.898864i	$0.644394\pi$
$954$		0			0
$955$		−	1143.18i		−	0.0387356i
$956$		0			0
$957$	3598.60			0.121553
$958$		0			0
$959$	−18168.9	+	12457.3i	−0.611789	+	0.419466i
$960$		0			0
$961$	−14188.8			−0.476278
$962$		0			0
$963$	10039.7			0.335956
$964$		0			0
$965$	5467.86			0.182401
$966$		0			0
$967$			11328.2i			0.376721i	0.982100	+	0.188361i	$0.0603174\pi$
$967$			11328.2i			0.376721i	−0.982100	+	0.188361i	$0.939683\pi$
$968$		0			0
$969$		−	6302.74i		−	0.208951i
$970$		0			0
$971$			21613.7i			0.714331i	0.934041	+	0.357166i	$0.116257\pi$
$971$			21613.7i			0.714331i	−0.934041	+	0.357166i	$0.883743\pi$
$972$		0			0
$973$	−5719.88	−	8342.41i	−0.188459	−	0.274867i
$974$		0			0
$975$		−	4806.56i		−	0.157880i
$976$		0			0
$977$	−289.064			−0.00946570			−0.00473285	−	0.999989i	$-0.501507\pi$
$977$	−289.064			−0.00946570			−0.00473285	+	0.999989i	$0.501507\pi$
$978$		0			0
$979$		−	20064.8i		−	0.655030i
$980$		0			0
$981$		−	2575.76i		−	0.0838306i
$982$		0			0
$983$	6611.03			0.214506			0.107253	−	0.994232i	$-0.465795\pi$
$983$	6611.03			0.214506			0.107253	+	0.994232i	$0.465795\pi$
$984$		0			0
$985$			1380.83i			0.0446668i
$986$		0			0
$987$	−6839.33	−	9975.13i	−0.220566	−	0.321694i
$988$		0			0
$989$		−	19135.3i		−	0.615236i
$990$		0			0
$991$			6406.80i			0.205367i	0.994714	+	0.102684i	$0.0327429\pi$
$991$			6406.80i			0.205367i	−0.994714	+	0.102684i	$0.967257\pi$
$992$		0			0
$993$			8354.54i			0.266992i
$994$		0			0
$995$	−1181.29			−0.0376377
$996$		0			0
$997$	21635.3			0.687257			0.343629	−	0.939106i	$-0.388344\pi$
$997$	21635.3			0.687257			0.343629	+	0.939106i	$0.388344\pi$
$998$		0			0
$999$	1528.12			0.0483961

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.4.p.d.223.17	yes	32
4.3	odd	2	inner	1344.4.p.d.223.20	yes	32
7.6	odd	2		1344.4.p.c.223.24	yes	32
8.3	odd	2		1344.4.p.c.223.23	yes	32
8.5	even	2		1344.4.p.c.223.16	yes	32
28.27	even	2		1344.4.p.c.223.15	✓	32
56.13	odd	2	inner	1344.4.p.d.223.19	yes	32
56.27	even	2	inner	1344.4.p.d.223.18	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1344.4.p.c.223.15	✓	32	28.27	even	2
1344.4.p.c.223.16	yes	32	8.5	even	2
1344.4.p.c.223.23	yes	32	8.3	odd	2
1344.4.p.c.223.24	yes	32	7.6	odd	2
1344.4.p.d.223.17	yes	32	1.1	even	1	trivial
1344.4.p.d.223.18	yes	32	56.27	even	2	inner
1344.4.p.d.223.19	yes	32	56.13	odd	2	inner
1344.4.p.d.223.20	yes	32	4.3	odd	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:	\( N \)	\(=\)	\( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1344.p (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.00000i q^{3} -1.64166 q^{5} +(15.2747 - 10.4729i) q^{7} -9.00000 q^{9} +O(q^{10})\)	\(q-3.00000i q^{3} -1.64166 q^{5} +(15.2747 - 10.4729i) q^{7} -9.00000 q^{9} +20.3470 q^{11} -13.0999 q^{13} +4.92497i q^{15} +23.9396i q^{17} -87.7589i q^{19} +(-31.4188 - 45.8242i) q^{21} -73.6175i q^{23} -122.305 q^{25} +27.0000i q^{27} +58.9537i q^{29} -124.909 q^{31} -61.0410i q^{33} +(-25.0759 + 17.1930i) q^{35} -56.5972i q^{37} +39.2998i q^{39} +135.651i q^{41} +259.929 q^{43} +14.7749 q^{45} +217.682 q^{47} +(123.635 - 319.943i) q^{49} +71.8188 q^{51} -529.342i q^{53} -33.4028 q^{55} -263.277 q^{57} -685.329i q^{59} -149.916 q^{61} +(-137.473 + 94.2565i) q^{63} +21.5056 q^{65} -409.156 q^{67} -220.853 q^{69} +885.869i q^{71} -269.426i q^{73} +366.915i q^{75} +(310.795 - 213.093i) q^{77} +902.587i q^{79} +81.0000 q^{81} -623.963i q^{83} -39.3006i q^{85} +176.861 q^{87} -986.131i q^{89} +(-200.098 + 137.195i) q^{91} +374.726i q^{93} +144.070i q^{95} -179.437i q^{97} -183.123 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 288 q^{9}+O(q^{10}) \) 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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