LMFDB - Embedded newform 1344.4.b.c.895.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1344,4,Mod(895,1344)]

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, 0, 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("1344.895");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1344 = 2^{6} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1344.b (of order $2$, degree $1$, not 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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-6}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 6 $ x^2 + 6
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 336)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			895.2
Root			$2.44949i$ of defining polynomial
Character	$\chi$	$=$	1344.895
Dual form			1344.4.b.c.895.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} +2.44949i q^{5} +(-17.0000 + 7.34847i) q^{7} +9.00000 q^{9} +O(q^{10})$	$q+3.00000 q^{3} +2.44949i q^{5} +(-17.0000 + 7.34847i) q^{7} +9.00000 q^{9} +56.3383i q^{11} -73.4847i q^{13} +7.34847i q^{15} -51.4393i q^{17} +10.0000 q^{19} +(-51.0000 + 22.0454i) q^{21} +115.126i q^{23} +119.000 q^{25} +27.0000 q^{27} -126.000 q^{29} -8.00000 q^{31} +169.015i q^{33} +(-18.0000 - 41.6413i) q^{35} -244.000 q^{37} -220.454i q^{39} +369.873i q^{41} +161.666i q^{43} +22.0454i q^{45} +180.000 q^{47} +(235.000 - 249.848i) q^{49} -154.318i q^{51} -594.000 q^{53} -138.000 q^{55} +30.0000 q^{57} -540.000 q^{59} -352.727i q^{61} +(-153.000 + 66.1362i) q^{63} +180.000 q^{65} -1058.18i q^{67} +345.378i q^{69} +178.813i q^{71} +661.362i q^{73} +357.000 q^{75} +(-414.000 - 957.750i) q^{77} -1028.79i q^{79} +81.0000 q^{81} -864.000 q^{83} +126.000 q^{85} -378.000 q^{87} -1290.88i q^{89} +(540.000 + 1249.24i) q^{91} -24.0000 q^{93} +24.4949i q^{95} -808.332i q^{97} +507.044i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{3} - 34 q^{7} + 18 q^{9}+O(q^{10}) $ 2 * q + 6 * q^3 - 34 * q^7 + 18 * q^9	$ 2 q + 6 q^{3} - 34 q^{7} + 18 q^{9} + 20 q^{19} - 102 q^{21} + 238 q^{25} + 54 q^{27} - 252 q^{29} - 16 q^{31} - 36 q^{35} - 488 q^{37} + 360 q^{47} + 470 q^{49} - 1188 q^{53} - 276 q^{55} + 60 q^{57} - 1080 q^{59} - 306 q^{63} + 360 q^{65} + 714 q^{75} - 828 q^{77} + 162 q^{81} - 1728 q^{83} + 252 q^{85} - 756 q^{87} + 1080 q^{91} - 48 q^{93}+O(q^{100}) $ 2 * q + 6 * q^3 - 34 * q^7 + 18 * q^9 + 20 * q^19 - 102 * q^21 + 238 * q^25 + 54 * q^27 - 252 * q^29 - 16 * q^31 - 36 * q^35 - 488 * q^37 + 360 * q^47 + 470 * q^49 - 1188 * q^53 - 276 * q^55 + 60 * q^57 - 1080 * q^59 - 306 * q^63 + 360 * q^65 + 714 * q^75 - 828 * q^77 + 162 * q^81 - 1728 * q^83 + 252 * q^85 - 756 * q^87 + 1080 * q^91 - 48 * q^93

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
$3$	3.00000			0.577350
$3$	3.00000			0.577350
$4$		0			0
$5$			2.44949i			0.219089i	0.993982	+	0.109545i	$0.0349392\pi$
$5$			2.44949i			0.219089i	−0.993982	+	0.109545i	$0.965061\pi$
$6$		0			0
$7$	−17.0000	+	7.34847i	−0.917914	+	0.396780i
$7$	−17.0000	+	7.34847i	−0.917914	+	0.396780i
$8$		0			0
$9$	9.00000			0.333333
$10$		0			0
$11$			56.3383i			1.54424i	0.635478	+	0.772119i	$0.280803\pi$
$11$			56.3383i			1.54424i	−0.635478	+	0.772119i	$0.719197\pi$
$12$		0			0
$13$		−	73.4847i		−	1.56777i	−0.620907	−	0.783884i	$-0.713236\pi$
$13$		−	73.4847i		−	1.56777i	0.620907	−	0.783884i	$-0.286764\pi$
$14$		0			0
$15$			7.34847i			0.126491i
$16$		0			0
$17$		−	51.4393i		−	0.733874i	−0.930246	−	0.366937i	$-0.880406\pi$
$17$		−	51.4393i		−	0.733874i	0.930246	−	0.366937i	$-0.119594\pi$
$18$		0			0
$19$	10.0000			0.120745			0.0603726	−	0.998176i	$-0.480771\pi$
$19$	10.0000			0.120745			0.0603726	+	0.998176i	$0.480771\pi$
$20$		0			0
$21$	−51.0000	+	22.0454i	−0.529958	+	0.229081i
$22$		0			0
$23$			115.126i			1.04371i	0.853033	+	0.521857i	$0.174761\pi$
$23$			115.126i			1.04371i	−0.853033	+	0.521857i	$0.825239\pi$
$24$		0			0
$25$	119.000			0.952000
$26$		0			0
$27$	27.0000			0.192450
$28$		0			0
$29$	−126.000			−0.806814			−0.403407	−	0.915021i	$-0.632174\pi$
$29$	−126.000			−0.806814			−0.403407	+	0.915021i	$0.632174\pi$
$30$		0			0
$31$	−8.00000			−0.0463498			−0.0231749	−	0.999731i	$-0.507377\pi$
$31$	−8.00000			−0.0463498			−0.0231749	+	0.999731i	$0.507377\pi$
$32$		0			0
$33$			169.015i			0.891567i
$34$		0			0
$35$	−18.0000	−	41.6413i	−0.0869302	−	0.201105i
$36$		0			0
$37$	−244.000			−1.08414			−0.542072	−	0.840332i	$-0.682360\pi$
$37$	−244.000			−1.08414			−0.542072	+	0.840332i	$0.682360\pi$
$38$		0			0
$39$		−	220.454i		−	0.905151i
$40$		0			0
$41$			369.873i			1.40889i	0.709759	+	0.704445i	$0.248804\pi$
$41$			369.873i			1.40889i	−0.709759	+	0.704445i	$0.751196\pi$
$42$		0			0
$43$			161.666i			0.573346i	0.958028	+	0.286673i	$0.0925493\pi$
$43$			161.666i			0.573346i	−0.958028	+	0.286673i	$0.907451\pi$
$44$		0			0
$45$			22.0454i			0.0730297i
$46$		0			0
$47$	180.000			0.558632			0.279316	−	0.960199i	$-0.409892\pi$
$47$	180.000			0.558632			0.279316	+	0.960199i	$0.409892\pi$
$48$		0			0
$49$	235.000	−	249.848i	0.685131	−	0.728420i
$50$		0			0
$51$		−	154.318i		−	0.423702i
$52$		0			0
$53$	−594.000			−1.53947			−0.769737	−	0.638361i	$-0.779613\pi$
$53$	−594.000			−1.53947			−0.769737	+	0.638361i	$0.779613\pi$
$54$		0			0
$55$	−138.000			−0.338326
$56$		0			0
$57$	30.0000			0.0697122
$58$		0			0
$59$	−540.000			−1.19156			−0.595780	−	0.803148i	$-0.703157\pi$
$59$	−540.000			−1.19156			−0.595780	+	0.803148i	$0.703157\pi$
$60$		0			0
$61$		−	352.727i		−	0.740361i	−0.928960	−	0.370180i	$-0.879296\pi$
$61$		−	352.727i		−	0.740361i	0.928960	−	0.370180i	$-0.120704\pi$
$62$		0			0
$63$	−153.000	+	66.1362i	−0.305971	+	0.132260i
$64$		0			0
$65$	180.000			0.343481
$66$		0			0
$67$		−	1058.18i		−	1.92951i	−0.263149	−	0.964755i	$-0.584761\pi$
$67$		−	1058.18i		−	1.92951i	0.263149	−	0.964755i	$-0.415239\pi$
$68$		0			0
$69$			345.378i			0.602589i
$70$		0			0
$71$			178.813i			0.298890i	0.988770	+	0.149445i	$0.0477486\pi$
$71$			178.813i			0.298890i	−0.988770	+	0.149445i	$0.952251\pi$
$72$		0			0
$73$			661.362i			1.06036i	0.847884	+	0.530182i	$0.177876\pi$
$73$			661.362i			1.06036i	−0.847884	+	0.530182i	$0.822124\pi$
$74$		0			0
$75$	357.000			0.549637
$76$		0			0
$77$	−414.000	−	957.750i	−0.612723	−	1.41748i
$78$		0			0
$79$		−	1028.79i		−	1.46516i	−0.680682	−	0.732579i	$-0.738317\pi$
$79$		−	1028.79i		−	1.46516i	0.680682	−	0.732579i	$-0.261683\pi$
$80$		0			0
$81$	81.0000			0.111111
$82$		0			0
$83$	−864.000			−1.14261			−0.571303	−	0.820739i	$-0.693562\pi$
$83$	−864.000			−1.14261			−0.571303	+	0.820739i	$0.693562\pi$
$84$		0			0
$85$	126.000			0.160784
$86$		0			0
$87$	−378.000			−0.465814
$88$		0			0
$89$		−	1290.88i		−	1.53745i	−0.639579	−	0.768725i	$-0.720891\pi$
$89$		−	1290.88i		−	1.53745i	0.639579	−	0.768725i	$-0.279109\pi$
$90$		0			0
$91$	540.000	+	1249.24i	0.622059	+	1.43908i
$92$		0			0
$93$	−24.0000			−0.0267600
$94$		0			0
$95$			24.4949i			0.0264539i
$96$		0			0
$97$		−	808.332i		−	0.846120i	−0.906102	−	0.423060i	$-0.860956\pi$
$97$		−	808.332i		−	0.846120i	0.906102	−	0.423060i	$-0.139044\pi$
$98$		0			0
$99$			507.044i			0.514746i
$100$		0			0
$101$			36.7423i			0.0361980i	0.999836	+	0.0180990i	$0.00576141\pi$
$101$			36.7423i			0.0361980i	−0.999836	+	0.0180990i	$0.994239\pi$
$102$		0			0
$103$	−1376.00			−1.31632			−0.658162	−	0.752877i	$-0.728666\pi$
$103$	−1376.00			−1.31632			−0.658162	+	0.752877i	$0.728666\pi$
$104$		0			0
$105$	−54.0000	−	124.924i	−0.0501891	−	0.116108i
$106$		0			0
$107$		−	1183.10i		−	1.06892i	−0.845192	−	0.534462i	$-0.820514\pi$
$107$		−	1183.10i		−	1.06892i	0.845192	−	0.534462i	$-0.179486\pi$
$108$		0			0
$109$	−602.000			−0.529001			−0.264501	−	0.964386i	$-0.585207\pi$
$109$	−602.000			−0.529001			−0.264501	+	0.964386i	$0.585207\pi$
$110$		0			0
$111$	−732.000			−0.625931
$112$		0			0
$113$	−2106.00			−1.75324			−0.876619	−	0.481186i	$-0.840206\pi$
$113$	−2106.00			−1.75324			−0.876619	+	0.481186i	$0.840206\pi$
$114$		0			0
$115$	−282.000			−0.228666
$116$		0			0
$117$		−	661.362i		−	0.522589i
$118$		0			0
$119$	378.000	+	874.468i	0.291187	+	0.673633i
$120$		0			0
$121$	−1843.00			−1.38467
$122$		0			0
$123$			1109.62i			0.813422i
$124$		0			0
$125$			597.675i			0.427662i
$126$		0			0
$127$			1028.79i			0.718819i	0.933180	+	0.359409i	$0.117022\pi$
$127$			1028.79i			0.718819i	−0.933180	+	0.359409i	$0.882978\pi$
$128$		0			0
$129$			484.999i			0.331022i
$130$		0			0
$131$	−576.000			−0.384163			−0.192082	−	0.981379i	$-0.561524\pi$
$131$	−576.000			−0.384163			−0.192082	+	0.981379i	$0.561524\pi$
$132$		0			0
$133$	−170.000	+	73.4847i	−0.110834	+	0.0479093i
$134$		0			0
$135$			66.1362i			0.0421637i
$136$		0			0
$137$	−1674.00			−1.04394			−0.521969	−	0.852965i	$-0.674802\pi$
$137$	−1674.00			−1.04394			−0.521969	+	0.852965i	$0.674802\pi$
$138$		0			0
$139$	308.000			0.187944			0.0939720	−	0.995575i	$-0.470044\pi$
$139$	308.000			0.187944			0.0939720	+	0.995575i	$0.470044\pi$
$140$		0			0
$141$	540.000			0.322526
$142$		0			0
$143$	4140.00			2.42101
$144$		0			0
$145$		−	308.636i		−	0.176764i
$146$		0			0
$147$	705.000	−	749.544i	0.395561	−	0.420553i
$148$		0			0
$149$	522.000			0.287006			0.143503	−	0.989650i	$-0.454163\pi$
$149$	522.000			0.287006			0.143503	+	0.989650i	$0.454163\pi$
$150$		0			0
$151$		−	1366.82i		−	0.736622i	−0.929703	−	0.368311i	$-0.879936\pi$
$151$		−	1366.82i		−	0.736622i	0.929703	−	0.368311i	$-0.120064\pi$
$152$		0			0
$153$		−	462.954i		−	0.244625i
$154$		0			0
$155$		−	19.5959i		−	0.0101547i
$156$		0			0
$157$			2880.60i			1.46431i	0.681137	+	0.732156i	$0.261486\pi$
$157$			2880.60i			1.46431i	−0.681137	+	0.732156i	$0.738514\pi$
$158$		0			0
$159$	−1782.00			−0.888816
$160$		0			0
$161$	−846.000	−	1957.14i	−0.414125	−	0.958040i
$162$		0			0
$163$		−	293.939i		−	0.141246i	−0.997503	−	0.0706229i	$-0.977501\pi$
$163$		−	293.939i		−	0.141246i	0.997503	−	0.0706229i	$-0.0224987\pi$
$164$		0			0
$165$	−414.000			−0.195332
$166$		0			0
$167$	3564.00			1.65144			0.825720	−	0.564080i	$-0.190769\pi$
$167$	3564.00			1.65144			0.825720	+	0.564080i	$0.190769\pi$
$168$		0			0
$169$	−3203.00			−1.45790
$170$		0			0
$171$	90.0000			0.0402484
$172$		0			0
$173$		−	1383.96i		−	0.608212i	−0.952638	−	0.304106i	$-0.901642\pi$
$173$		−	1383.96i		−	0.608212i	0.952638	−	0.304106i	$-0.0983577\pi$
$174$		0			0
$175$	−2023.00	+	874.468i	−0.873854	+	0.377735i
$176$		0			0
$177$	−1620.00			−0.687947
$178$		0			0
$179$		−	139.621i		−	0.0583003i	−0.999575	−	0.0291502i	$-0.990720\pi$
$179$		−	139.621i		−	0.0583003i	0.999575	−	0.0291502i	$-0.00928010\pi$
$180$		0			0
$181$		−	3101.05i		−	1.27348i	−0.771079	−	0.636739i	$-0.780283\pi$
$181$		−	3101.05i		−	1.27348i	0.771079	−	0.636739i	$-0.219717\pi$
$182$		0			0
$183$		−	1058.18i		−	0.427447i
$184$		0			0
$185$		−	597.675i		−	0.237524i
$186$		0			0
$187$	2898.00			1.13328
$188$		0			0
$189$	−459.000	+	198.409i	−0.176653	+	0.0763604i
$190$		0			0
$191$			673.610i			0.255187i	0.991827	+	0.127593i	$0.0407253\pi$
$191$			673.610i			0.255187i	−0.991827	+	0.127593i	$0.959275\pi$
$192$		0			0
$193$	−2144.00			−0.799630			−0.399815	−	0.916596i	$-0.630926\pi$
$193$	−2144.00			−0.799630			−0.399815	+	0.916596i	$0.630926\pi$
$194$		0			0
$195$	540.000			0.198309
$196$		0			0
$197$	−234.000			−0.0846285			−0.0423142	−	0.999104i	$-0.513473\pi$
$197$	−234.000			−0.0846285			−0.0423142	+	0.999104i	$0.513473\pi$
$198$		0			0
$199$	2654.00			0.945412			0.472706	−	0.881220i	$-0.343277\pi$
$199$	2654.00			0.945412			0.472706	+	0.881220i	$0.343277\pi$
$200$		0			0
$201$		−	3174.54i		−	1.11400i
$202$		0			0
$203$	2142.00	−	925.907i	0.740586	−	0.320128i
$204$		0			0
$205$	−906.000			−0.308672
$206$		0			0
$207$			1036.13i			0.347905i
$208$		0			0
$209$			563.383i			0.186459i
$210$		0			0
$211$			2307.42i			0.752840i	0.926449	+	0.376420i	$0.122845\pi$
$211$			2307.42i			0.752840i	−0.926449	+	0.376420i	$0.877155\pi$
$212$		0			0
$213$			536.438i			0.172564i
$214$		0			0
$215$	−396.000			−0.125614
$216$		0			0
$217$	136.000	−	58.7878i	0.0425451	−	0.0183907i
$218$		0			0
$219$			1984.09i			0.612202i
$220$		0			0
$221$	−3780.00			−1.15054
$222$		0			0
$223$	−838.000			−0.251644			−0.125822	−	0.992053i	$-0.540157\pi$
$223$	−838.000			−0.251644			−0.125822	+	0.992053i	$0.540157\pi$
$224$		0			0
$225$	1071.00			0.317333
$226$		0			0
$227$	−4860.00			−1.42101			−0.710506	−	0.703692i	$-0.751534\pi$
$227$	−4860.00			−1.42101			−0.710506	+	0.703692i	$0.751534\pi$
$228$		0			0
$229$			14.6969i			0.00424105i	0.999998	+	0.00212053i	$0.000674985\pi$
$229$			14.6969i			0.00424105i	−0.999998	+	0.00212053i	$0.999325\pi$
$230$		0			0
$231$	−1242.00	−	2873.25i	−0.353756	−	0.818381i
$232$		0			0
$233$	1458.00			0.409943			0.204972	−	0.978768i	$-0.434290\pi$
$233$	1458.00			0.409943			0.204972	+	0.978768i	$0.434290\pi$
$234$		0			0
$235$			440.908i			0.122390i
$236$		0			0
$237$		−	3086.36i		−	0.845909i
$238$		0			0
$239$			3211.28i			0.869123i	0.900642	+	0.434562i	$0.143097\pi$
$239$			3211.28i			0.869123i	−0.900642	+	0.434562i	$0.856903\pi$
$240$		0			0
$241$		−	808.332i		−	0.216055i	−0.994148	−	0.108027i	$-0.965547\pi$
$241$		−	808.332i		−	0.216055i	0.994148	−	0.108027i	$-0.0344534\pi$
$242$		0			0
$243$	243.000			0.0641500
$244$		0			0
$245$	612.000	+	575.630i	0.159589	+	0.150105i
$246$		0			0
$247$		−	734.847i		−	0.189300i
$248$		0			0
$249$	−2592.00			−0.659684
$250$		0			0
$251$	576.000			0.144848			0.0724239	−	0.997374i	$-0.476927\pi$
$251$	576.000			0.144848			0.0724239	+	0.997374i	$0.476927\pi$
$252$		0			0
$253$	−6486.00			−1.61174
$254$		0			0
$255$	378.000			0.0928285
$256$		0			0
$257$			4436.03i			1.07670i	0.842722	+	0.538350i	$0.180952\pi$
$257$			4436.03i			1.07670i	−0.842722	+	0.538350i	$0.819048\pi$
$258$		0			0
$259$	4148.00	−	1793.03i	0.995151	−	0.430167i
$260$		0			0
$261$	−1134.00			−0.268938
$262$		0			0
$263$			5420.72i			1.27094i	0.772128	+	0.635468i	$0.219193\pi$
$263$			5420.72i			1.27094i	−0.772128	+	0.635468i	$0.780807\pi$
$264$		0			0
$265$		−	1455.00i		−	0.337282i
$266$		0			0
$267$		−	3872.64i		−	0.887648i
$268$		0			0
$269$		−	6233.95i		−	1.41298i	−0.707725	−	0.706488i	$-0.750278\pi$
$269$		−	6233.95i		−	1.41298i	0.707725	−	0.706488i	$-0.249722\pi$
$270$		0			0
$271$	−5024.00			−1.12615			−0.563074	−	0.826406i	$-0.690382\pi$
$271$	−5024.00			−1.12615			−0.563074	+	0.826406i	$0.690382\pi$
$272$		0			0
$273$	1620.00	+	3747.72i	0.359146	+	0.830851i
$274$		0			0
$275$			6704.25i			1.47012i
$276$		0			0
$277$	−4024.00			−0.872847			−0.436424	−	0.899741i	$-0.643755\pi$
$277$	−4024.00			−0.872847			−0.436424	+	0.899741i	$0.643755\pi$
$278$		0			0
$279$	−72.0000			−0.0154499
$280$		0			0
$281$	630.000			0.133746			0.0668730	−	0.997761i	$-0.478698\pi$
$281$	630.000			0.133746			0.0668730	+	0.997761i	$0.478698\pi$
$282$		0			0
$283$	−9002.00			−1.89086			−0.945430	−	0.325825i	$-0.894358\pi$
$283$	−9002.00			−1.89086			−0.945430	+	0.325825i	$0.894358\pi$
$284$		0			0
$285$			73.4847i			0.0152732i
$286$		0			0
$287$	−2718.00	−	6287.84i	−0.559019	−	1.29324i
$288$		0			0
$289$	2267.00			0.461429
$290$		0			0
$291$		−	2424.99i		−	0.488508i
$292$		0			0
$293$			9148.84i			1.82417i	0.410004	+	0.912084i	$0.365527\pi$
$293$			9148.84i			1.82417i	−0.410004	+	0.912084i	$0.634473\pi$
$294$		0			0
$295$		−	1322.72i		−	0.261058i
$296$		0			0
$297$			1521.13i			0.297189i
$298$		0			0
$299$	8460.00			1.63630
$300$		0			0
$301$	−1188.00	−	2748.33i	−0.227492	−	0.526282i
$302$		0			0
$303$			110.227i			0.0208989i
$304$		0			0
$305$	864.000			0.162205
$306$		0			0
$307$	−5246.00			−0.975261			−0.487630	−	0.873050i	$-0.662139\pi$
$307$	−5246.00			−0.975261			−0.487630	+	0.873050i	$0.662139\pi$
$308$		0			0
$309$	−4128.00			−0.759980
$310$		0			0
$311$	8136.00			1.48344			0.741721	−	0.670709i	$-0.234010\pi$
$311$	8136.00			1.48344			0.741721	+	0.670709i	$0.234010\pi$
$312$		0			0
$313$		−	499.696i		−	0.0902380i	−0.998982	−	0.0451190i	$-0.985633\pi$
$313$		−	499.696i		−	0.0902380i	0.998982	−	0.0451190i	$-0.0143667\pi$
$314$		0			0
$315$	−162.000	−	374.772i	−0.0289767	−	0.0670349i
$316$		0			0
$317$	−8838.00			−1.56590			−0.782952	−	0.622082i	$-0.786287\pi$
$317$	−8838.00			−1.56590			−0.782952	+	0.622082i	$0.786287\pi$
$318$		0			0
$319$		−	7098.62i		−	1.24591i
$320$		0			0
$321$		−	3549.31i		−	0.617144i
$322$		0			0
$323$		−	514.393i		−	0.0886117i
$324$		0			0
$325$		−	8744.68i		−	1.49252i
$326$		0			0
$327$	−1806.00			−0.305419
$328$		0			0
$329$	−3060.00	+	1322.72i	−0.512776	+	0.221654i
$330$		0			0
$331$		−	9068.01i		−	1.50581i	−0.658129	−	0.752905i	$-0.728652\pi$
$331$		−	9068.01i		−	1.50581i	0.658129	−	0.752905i	$-0.271348\pi$
$332$		0			0
$333$	−2196.00			−0.361382
$334$		0			0
$335$	2592.00			0.422735
$336$		0			0
$337$	7982.00			1.29023			0.645115	−	0.764086i	$-0.276810\pi$
$337$	7982.00			1.29023			0.645115	+	0.764086i	$0.276810\pi$
$338$		0			0
$339$	−6318.00			−1.01223
$340$		0			0
$341$		−	450.706i		−	0.0715751i
$342$		0			0
$343$	−2159.00	+	5974.31i	−0.339869	+	0.940473i
$344$		0			0
$345$	−846.000			−0.132021
$346$		0			0
$347$			4308.65i			0.666572i	0.942826	+	0.333286i	$0.108158\pi$
$347$			4308.65i			0.666572i	−0.942826	+	0.333286i	$0.891842\pi$
$348$		0			0
$349$		−	4291.51i		−	0.658221i	−0.944291	−	0.329110i	$-0.893251\pi$
$349$		−	4291.51i		−	0.658221i	0.944291	−	0.329110i	$-0.106749\pi$
$350$		0			0
$351$		−	1984.09i		−	0.301717i
$352$		0			0
$353$		−	6258.45i		−	0.943636i	−0.881696	−	0.471818i	$-0.843598\pi$
$353$		−	6258.45i		−	0.943636i	0.881696	−	0.471818i	$-0.156402\pi$
$354$		0			0
$355$	−438.000			−0.0654835
$356$		0			0
$357$	1134.00	+	2623.40i	0.168117	+	0.388922i
$358$		0			0
$359$			8605.06i			1.26506i	0.774535	+	0.632531i	$0.217984\pi$
$359$			8605.06i			1.26506i	−0.774535	+	0.632531i	$0.782016\pi$
$360$		0			0
$361$	−6759.00			−0.985421
$362$		0			0
$363$	−5529.00			−0.799441
$364$		0			0
$365$	−1620.00			−0.232314
$366$		0			0
$367$	5570.00			0.792239			0.396119	−	0.918199i	$-0.370357\pi$
$367$	5570.00			0.792239			0.396119	+	0.918199i	$0.370357\pi$
$368$		0			0
$369$			3328.86i			0.469630i
$370$		0			0
$371$	10098.0	−	4364.99i	1.41311	−	0.610833i
$372$		0			0
$373$	7282.00			1.01085			0.505426	−	0.862870i	$-0.331335\pi$
$373$	7282.00			1.01085			0.505426	+	0.862870i	$0.331335\pi$
$374$		0			0
$375$			1793.03i			0.246911i
$376$		0			0
$377$			9259.07i			1.26490i
$378$		0			0
$379$			984.695i			0.133457i	0.997771	+	0.0667287i	$0.0212562\pi$
$379$			984.695i			0.133457i	−0.997771	+	0.0667287i	$0.978744\pi$
$380$		0			0
$381$			3086.36i			0.415010i
$382$		0			0
$383$	−11628.0			−1.55134			−0.775670	−	0.631139i	$-0.782588\pi$
$383$	−11628.0			−1.55134			−0.775670	+	0.631139i	$0.782588\pi$
$384$		0			0
$385$	2346.00	−	1014.09i	0.310554	−	0.134241i
$386$		0			0
$387$			1455.00i			0.191115i
$388$		0			0
$389$	4518.00			0.588873			0.294437	−	0.955671i	$-0.404868\pi$
$389$	4518.00			0.588873			0.294437	+	0.955671i	$0.404868\pi$
$390$		0			0
$391$	5922.00			0.765955
$392$		0			0
$393$	−1728.00			−0.221797
$394$		0			0
$395$	2520.00			0.321000
$396$		0			0
$397$			7260.29i			0.917842i	0.888477	+	0.458921i	$0.151764\pi$
$397$			7260.29i			0.917842i	−0.888477	+	0.458921i	$0.848236\pi$
$398$		0			0
$399$	−510.000	+	220.454i	−0.0639898	+	0.0276604i
$400$		0			0
$401$	10062.0			1.25305			0.626524	−	0.779402i	$-0.284477\pi$
$401$	10062.0			1.25305			0.626524	+	0.779402i	$0.284477\pi$
$402$		0			0
$403$			587.878i			0.0726657i
$404$		0			0
$405$			198.409i			0.0243432i
$406$		0			0
$407$		−	13746.5i		−	1.67418i
$408$		0			0
$409$		−	6804.68i		−	0.822665i	−0.911485	−	0.411332i	$-0.865064\pi$
$409$		−	6804.68i		−	0.822665i	0.911485	−	0.411332i	$-0.134936\pi$
$410$		0			0
$411$	−5022.00			−0.602718
$412$		0			0
$413$	9180.00	−	3968.17i	1.09375	−	0.472787i
$414$		0			0
$415$		−	2116.36i		−	0.250332i
$416$		0			0
$417$	924.000			0.108510
$418$		0			0
$419$	12564.0			1.46490			0.732448	−	0.680823i	$-0.238378\pi$
$419$	12564.0			1.46490			0.732448	+	0.680823i	$0.238378\pi$
$420$		0			0
$421$	3752.00			0.434350			0.217175	−	0.976133i	$-0.430316\pi$
$421$	3752.00			0.434350			0.217175	+	0.976133i	$0.430316\pi$
$422$		0			0
$423$	1620.00			0.186211
$424$		0			0
$425$		−	6121.27i		−	0.698648i
$426$		0			0
$427$	2592.00	+	5996.35i	0.293760	+	0.679587i
$428$		0			0
$429$	12420.0			1.39777
$430$		0			0
$431$		−	6086.98i		−	0.680278i	−0.940375	−	0.340139i	$-0.889526\pi$
$431$		−	6086.98i		−	0.680278i	0.940375	−	0.340139i	$-0.110474\pi$
$432$		0			0
$433$		−	3321.51i		−	0.368641i	−0.982866	−	0.184320i	$-0.940992\pi$
$433$		−	3321.51i		−	0.368641i	0.982866	−	0.184320i	$-0.0590084\pi$
$434$		0			0
$435$		−	925.907i		−	0.102055i
$436$		0			0
$437$			1151.26i			0.126023i
$438$		0			0
$439$	−14506.0			−1.57707			−0.788535	−	0.614990i	$-0.789160\pi$
$439$	−14506.0			−1.57707			−0.788535	+	0.614990i	$0.789160\pi$
$440$		0			0
$441$	2115.00	−	2248.63i	0.228377	−	0.242807i
$442$		0			0
$443$			11858.0i			1.27176i	0.771788	+	0.635880i	$0.219363\pi$
$443$			11858.0i			1.27176i	−0.771788	+	0.635880i	$0.780637\pi$
$444$		0			0
$445$	3162.00			0.336839
$446$		0			0
$447$	1566.00			0.165703
$448$		0			0
$449$	−7866.00			−0.826769			−0.413385	−	0.910556i	$-0.635654\pi$
$449$	−7866.00			−0.826769			−0.413385	+	0.910556i	$0.635654\pi$
$450$		0			0
$451$	−20838.0			−2.17566
$452$		0			0
$453$		−	4100.45i		−	0.425289i
$454$		0			0
$455$	−3060.00	+	1322.72i	−0.315286	+	0.136286i
$456$		0			0
$457$	−6058.00			−0.620090			−0.310045	−	0.950722i	$-0.600344\pi$
$457$	−6058.00			−0.620090			−0.310045	+	0.950722i	$0.600344\pi$
$458$		0			0
$459$		−	1388.86i		−	0.141234i
$460$		0			0
$461$		−	10481.4i		−	1.05893i	−0.848332	−	0.529464i	$-0.822393\pi$
$461$		−	10481.4i		−	1.05893i	0.848332	−	0.529464i	$-0.177607\pi$
$462$		0			0
$463$			13726.9i			1.37785i	0.724832	+	0.688926i	$0.241917\pi$
$463$			13726.9i			1.37785i	−0.724832	+	0.688926i	$0.758083\pi$
$464$		0			0
$465$		−	58.7878i		−	0.00586283i
$466$		0			0
$467$	14472.0			1.43401			0.717007	−	0.697066i	$-0.245512\pi$
$467$	14472.0			1.43401			0.717007	+	0.697066i	$0.245512\pi$
$468$		0			0
$469$	7776.00	+	17989.1i	0.765591	+	1.77112i
$470$		0			0
$471$			8641.80i			0.845421i
$472$		0			0
$473$	−9108.00			−0.885383
$474$		0			0
$475$	1190.00			0.114949
$476$		0			0
$477$	−5346.00			−0.513158
$478$		0			0
$479$	3816.00			0.364003			0.182002	−	0.983298i	$-0.441742\pi$
$479$	3816.00			0.364003			0.182002	+	0.983298i	$0.441742\pi$
$480$		0			0
$481$			17930.3i			1.69969i
$482$		0			0
$483$	−2538.00	−	5871.43i	−0.239095	−	0.553125i
$484$		0			0
$485$	1980.00			0.185376
$486$		0			0
$487$			14799.8i			1.37709i	0.725193	+	0.688546i	$0.241751\pi$
$487$			14799.8i			1.37709i	−0.725193	+	0.688546i	$0.758249\pi$
$488$		0			0
$489$		−	881.816i		−	0.0815483i
$490$		0			0
$491$		−	1349.67i		−	0.124052i	−0.998075	−	0.0620262i	$-0.980244\pi$
$491$		−	1349.67i		−	0.124052i	0.998075	−	0.0620262i	$-0.0197562\pi$
$492$		0			0
$493$			6481.35i			0.592100i
$494$		0			0
$495$	−1242.00			−0.112775
$496$		0			0
$497$	−1314.00	−	3039.82i	−0.118593	−	0.274355i
$498$		0			0
$499$		−	2689.54i		−	0.241283i	−0.992696	−	0.120642i	$-0.961505\pi$
$499$		−	2689.54i		−	0.241283i	0.992696	−	0.120642i	$-0.0384952\pi$
$500$		0			0
$501$	10692.0			0.953460
$502$		0			0
$503$	7596.00			0.673338			0.336669	−	0.941623i	$-0.390700\pi$
$503$	7596.00			0.673338			0.336669	+	0.941623i	$0.390700\pi$
$504$		0			0
$505$	−90.0000			−0.00793059
$506$		0			0
$507$	−9609.00			−0.841717
$508$		0			0
$509$			19671.9i			1.71304i	0.516110	+	0.856522i	$0.327379\pi$
$509$			19671.9i			1.71304i	−0.516110	+	0.856522i	$0.672621\pi$
$510$		0			0
$511$	−4860.00	−	11243.2i	−0.420731	−	0.973323i
$512$		0			0
$513$	270.000			0.0232374
$514$		0			0
$515$		−	3370.50i		−	0.288392i
$516$		0			0
$517$			10140.9i			0.862661i
$518$		0			0
$519$		−	4151.89i		−	0.351151i
$520$		0			0
$521$		−	10393.2i		−	0.873961i	−0.899471	−	0.436981i	$-0.856048\pi$
$521$		−	10393.2i		−	0.873961i	0.899471	−	0.436981i	$-0.143952\pi$
$522$		0			0
$523$	−17956.0			−1.50126			−0.750632	−	0.660721i	$-0.770251\pi$
$523$	−17956.0			−1.50126			−0.750632	+	0.660721i	$0.770251\pi$
$524$		0			0
$525$	−6069.00	+	2623.40i	−0.504520	+	0.218085i
$526$		0			0
$527$			411.514i			0.0340149i
$528$		0			0
$529$	−1087.00			−0.0893400
$530$		0			0
$531$	−4860.00			−0.397187
$532$		0			0
$533$	27180.0			2.20881
$534$		0			0
$535$	2898.00			0.234190
$536$		0			0
$537$		−	418.863i		−	0.0336597i
$538$		0			0
$539$	14076.0	+	13239.5i	1.12485	+	1.05801i
$540$		0			0
$541$	14192.0			1.12784			0.563920	−	0.825829i	$-0.309293\pi$
$541$	14192.0			1.12784			0.563920	+	0.825829i	$0.309293\pi$
$542$		0			0
$543$		−	9303.16i		−	0.735243i
$544$		0			0
$545$		−	1474.59i		−	0.115898i
$546$		0			0
$547$			2733.63i			0.213678i	0.994276	+	0.106839i	$0.0340729\pi$
$547$			2733.63i			0.213678i	−0.994276	+	0.106839i	$0.965927\pi$
$548$		0			0
$549$		−	3174.54i		−	0.246787i
$550$		0			0
$551$	−1260.00			−0.0974189
$552$		0			0
$553$	7560.00	+	17489.4i	0.581345	+	1.34489i
$554$		0			0
$555$		−	1793.03i		−	0.137135i
$556$		0			0
$557$	−16254.0			−1.23645			−0.618226	−	0.786000i	$-0.712148\pi$
$557$	−16254.0			−1.23645			−0.618226	+	0.786000i	$0.712148\pi$
$558$		0			0
$559$	11880.0			0.898874
$560$		0			0
$561$	8694.00			0.654298
$562$		0			0
$563$	3672.00			0.274878			0.137439	−	0.990510i	$-0.456113\pi$
$563$	3672.00			0.274878			0.137439	+	0.990510i	$0.456113\pi$
$564$		0			0
$565$		−	5158.63i		−	0.384115i
$566$		0			0
$567$	−1377.00	+	595.226i	−0.101990	+	0.0440867i
$568$		0			0
$569$	11466.0			0.844780			0.422390	−	0.906414i	$-0.361191\pi$
$569$	11466.0			0.844780			0.422390	+	0.906414i	$0.361191\pi$
$570$		0			0
$571$		−	20869.7i		−	1.52954i	−0.644303	−	0.764770i	$-0.722852\pi$
$571$		−	20869.7i		−	1.52954i	0.644303	−	0.764770i	$-0.277148\pi$
$572$		0			0
$573$			2020.83i			0.147332i
$574$		0			0
$575$			13700.0i			0.993616i
$576$		0			0
$577$			19752.7i			1.42516i	0.701593	+	0.712578i	$0.252473\pi$
$577$			19752.7i			1.42516i	−0.701593	+	0.712578i	$0.747527\pi$
$578$		0			0
$579$	−6432.00			−0.461666
$580$		0			0
$581$	14688.0	−	6349.08i	1.04881	−	0.453363i
$582$		0			0
$583$		−	33464.9i		−	2.37732i
$584$		0			0
$585$	1620.00			0.114494
$586$		0			0
$587$	18720.0			1.31628			0.658141	−	0.752895i	$-0.271343\pi$
$587$	18720.0			1.31628			0.658141	+	0.752895i	$0.271343\pi$
$588$		0			0
$589$	−80.0000			−0.00559651
$590$		0			0
$591$	−702.000			−0.0488603
$592$		0			0
$593$			1643.61i			0.113819i	0.998379	+	0.0569097i	$0.0181247\pi$
$593$			1643.61i			0.113819i	−0.998379	+	0.0569097i	$0.981875\pi$
$594$		0			0
$595$	−2142.00	+	925.907i	−0.147586	+	0.0637958i
$596$		0			0
$597$	7962.00			0.545834
$598$		0			0
$599$		−	10099.2i		−	0.688888i	−0.938807	−	0.344444i	$-0.888067\pi$
$599$		−	10099.2i		−	0.688888i	0.938807	−	0.344444i	$-0.111933\pi$
$600$		0			0
$601$		−	7466.04i		−	0.506733i	−0.967370	−	0.253366i	$-0.918462\pi$
$601$		−	7466.04i		−	0.506733i	0.967370	−	0.253366i	$-0.0815378\pi$
$602$		0			0
$603$		−	9523.62i		−	0.643170i
$604$		0			0
$605$		−	4514.41i		−	0.303367i
$606$		0			0
$607$	−7198.00			−0.481314			−0.240657	−	0.970610i	$-0.577363\pi$
$607$	−7198.00			−0.481314			−0.240657	+	0.970610i	$0.577363\pi$
$608$		0			0
$609$	6426.00	−	2777.72i	0.427577	−	0.184826i
$610$		0			0
$611$		−	13227.2i		−	0.875805i
$612$		0			0
$613$	5662.00			0.373060			0.186530	−	0.982449i	$-0.440276\pi$
$613$	5662.00			0.373060			0.186530	+	0.982449i	$0.440276\pi$
$614$		0			0
$615$	−2718.00			−0.178212
$616$		0			0
$617$	−3546.00			−0.231372			−0.115686	−	0.993286i	$-0.536907\pi$
$617$	−3546.00			−0.231372			−0.115686	+	0.993286i	$0.536907\pi$
$618$		0			0
$619$	8180.00			0.531150			0.265575	−	0.964090i	$-0.414438\pi$
$619$	8180.00			0.531150			0.265575	+	0.964090i	$0.414438\pi$
$620$		0			0
$621$			3108.40i			0.200863i
$622$		0			0
$623$	9486.00	+	21945.0i	0.610030	+	1.41125i
$624$		0			0
$625$	13411.0			0.858304
$626$		0			0
$627$			1690.15i			0.107652i
$628$		0			0
$629$			12551.2i			0.795626i
$630$		0			0
$631$		−	2410.30i		−	0.152064i	−0.997105	−	0.0760320i	$-0.975775\pi$
$631$		−	2410.30i		−	0.152064i	0.997105	−	0.0760320i	$-0.0242251\pi$
$632$		0			0
$633$			6922.26i			0.434653i
$634$		0			0
$635$	−2520.00			−0.157485
$636$		0			0
$637$	−18360.0	−	17268.9i	−1.14199	−	1.07413i
$638$		0			0
$639$			1609.31i			0.0996299i
$640$		0			0
$641$	26010.0			1.60270			0.801352	−	0.598193i	$-0.204114\pi$
$641$	26010.0			1.60270			0.801352	+	0.598193i	$0.204114\pi$
$642$		0			0
$643$	4582.00			0.281021			0.140510	−	0.990079i	$-0.455126\pi$
$643$	4582.00			0.281021			0.140510	+	0.990079i	$0.455126\pi$
$644$		0			0
$645$	−1188.00			−0.0725232
$646$		0			0
$647$	5832.00			0.354373			0.177187	−	0.984177i	$-0.443300\pi$
$647$	5832.00			0.354373			0.177187	+	0.984177i	$0.443300\pi$
$648$		0			0
$649$		−	30422.7i		−	1.84005i
$650$		0			0
$651$	408.000	−	176.363i	0.0245634	−	0.0106179i
$652$		0			0
$653$	−21474.0			−1.28690			−0.643448	−	0.765490i	$-0.722497\pi$
$653$	−21474.0			−1.28690			−0.643448	+	0.765490i	$0.722497\pi$
$654$		0			0
$655$		−	1410.91i		−	0.0841659i
$656$		0			0
$657$			5952.26i			0.353455i
$658$		0			0
$659$			27936.4i			1.65136i	0.564136	+	0.825682i	$0.309209\pi$
$659$			27936.4i			1.65136i	−0.564136	+	0.825682i	$0.690791\pi$
$660$		0			0
$661$			18430.0i			1.08448i	0.840223	+	0.542241i	$0.182424\pi$
$661$			18430.0i			1.08448i	−0.840223	+	0.542241i	$0.817576\pi$
$662$		0			0
$663$	−11340.0			−0.664267
$664$		0			0
$665$	−180.000	−	416.413i	−0.0104964	−	0.0242824i
$666$		0			0
$667$		−	14505.9i		−	0.842084i
$668$		0			0
$669$	−2514.00			−0.145287
$670$		0			0
$671$	19872.0			1.14329
$672$		0			0
$673$	14344.0			0.821576			0.410788	−	0.911731i	$-0.365254\pi$
$673$	14344.0			0.821576			0.410788	+	0.911731i	$0.365254\pi$
$674$		0			0
$675$	3213.00			0.183212
$676$		0			0
$677$		−	26599.0i		−	1.51002i	−0.655714	−	0.755010i	$-0.727632\pi$
$677$		−	26599.0i		−	1.51002i	0.655714	−	0.755010i	$-0.272368\pi$
$678$		0			0
$679$	5940.00	+	13741.6i	0.335724	+	0.776665i
$680$		0			0
$681$	−14580.0			−0.820421
$682$		0			0
$683$			27407.3i			1.53545i	0.640779	+	0.767725i	$0.278611\pi$
$683$			27407.3i			1.53545i	−0.640779	+	0.767725i	$0.721389\pi$
$684$		0			0
$685$		−	4100.45i		−	0.228715i
$686$		0			0
$687$			44.0908i			0.00244857i
$688$		0			0
$689$			43649.9i			2.41354i
$690$		0			0
$691$	6716.00			0.369738			0.184869	−	0.982763i	$-0.440814\pi$
$691$	6716.00			0.369738			0.184869	+	0.982763i	$0.440814\pi$
$692$		0			0
$693$	−3726.00	−	8619.75i	−0.204241	−	0.472493i
$694$		0			0
$695$			754.443i			0.0411765i
$696$		0			0
$697$	19026.0			1.03395
$698$		0			0
$699$	4374.00			0.236681
$700$		0			0
$701$	12546.0			0.675971			0.337986	−	0.941151i	$-0.390254\pi$
$701$	12546.0			0.675971			0.337986	+	0.941151i	$0.390254\pi$
$702$		0			0
$703$	−2440.00			−0.130905
$704$		0			0
$705$			1322.72i			0.0706620i
$706$		0			0
$707$	−270.000	−	624.620i	−0.0143627	−	0.0332267i
$708$		0			0
$709$	24562.0			1.30105			0.650526	−	0.759484i	$-0.274549\pi$
$709$	24562.0			1.30105			0.650526	+	0.759484i	$0.274549\pi$
$710$		0			0
$711$		−	9259.07i		−	0.488386i
$712$		0			0
$713$		−	921.008i		−	0.0483759i
$714$		0			0
$715$			10140.9i			0.530416i
$716$		0			0
$717$			9633.84i			0.501789i
$718$		0			0
$719$	7704.00			0.399598			0.199799	−	0.979837i	$-0.435971\pi$
$719$	7704.00			0.399598			0.199799	+	0.979837i	$0.435971\pi$
$720$		0			0
$721$	23392.0	−	10111.5i	1.20827	−	0.522291i
$722$		0			0
$723$		−	2424.99i		−	0.124739i
$724$		0			0
$725$	−14994.0			−0.768087
$726$		0			0
$727$	−22160.0			−1.13049			−0.565247	−	0.824922i	$-0.691219\pi$
$727$	−22160.0			−1.13049			−0.565247	+	0.824922i	$0.691219\pi$
$728$		0			0
$729$	729.000			0.0370370
$730$		0			0
$731$	8316.00			0.420764
$732$		0			0
$733$		−	17004.4i		−	0.856849i	−0.903578	−	0.428424i	$-0.859069\pi$
$733$		−	17004.4i		−	0.856849i	0.903578	−	0.428424i	$-0.140931\pi$
$734$		0			0
$735$	1836.00	+	1726.89i	0.0921386	+	0.0866630i
$736$		0			0
$737$	59616.0			2.97962
$738$		0			0
$739$			18371.2i			0.914472i	0.889345	+	0.457236i	$0.151160\pi$
$739$			18371.2i			0.914472i	−0.889345	+	0.457236i	$0.848840\pi$
$740$		0			0
$741$		−	2204.54i		−	0.109293i
$742$		0			0
$743$			23532.2i			1.16193i	0.813928	+	0.580965i	$0.197325\pi$
$743$			23532.2i			1.16193i	−0.813928	+	0.580965i	$0.802675\pi$
$744$		0			0
$745$			1278.63i			0.0628799i
$746$		0			0
$747$	−7776.00			−0.380869
$748$		0			0
$749$	8694.00	+	20112.8i	0.424128	+	0.981181i
$750$		0			0
$751$			10831.6i			0.526301i	0.964755	+	0.263151i	$0.0847616\pi$
$751$			10831.6i			0.526301i	−0.964755	+	0.263151i	$0.915238\pi$
$752$		0			0
$753$	1728.00			0.0836279
$754$		0			0
$755$	3348.00			0.161386
$756$		0			0
$757$	−12206.0			−0.586043			−0.293022	−	0.956106i	$-0.594661\pi$
$757$	−12206.0			−0.586043			−0.293022	+	0.956106i	$0.594661\pi$
$758$		0			0
$759$	−19458.0			−0.930541
$760$		0			0
$761$			23640.0i			1.12608i	0.826428	+	0.563042i	$0.190369\pi$
$761$			23640.0i			1.12608i	−0.826428	+	0.563042i	$0.809631\pi$
$762$		0			0
$763$	10234.0	−	4423.78i	0.485578	−	0.209897i
$764$		0			0
$765$	1134.00			0.0535946
$766$		0			0
$767$			39681.7i			1.86809i
$768$		0			0
$769$			27718.4i			1.29981i	0.760017	+	0.649904i	$0.225191\pi$
$769$			27718.4i			1.29981i	−0.760017	+	0.649904i	$0.774809\pi$
$770$		0			0
$771$			13308.1i			0.621633i
$772$		0			0
$773$			40272.1i			1.87385i	0.349532	+	0.936925i	$0.386341\pi$
$773$			40272.1i			1.87385i	−0.349532	+	0.936925i	$0.613659\pi$
$774$		0			0
$775$	−952.000			−0.0441250
$776$		0			0
$777$	12444.0	−	5379.08i	0.574551	−	0.248357i
$778$		0			0
$779$			3698.73i			0.170116i
$780$		0			0
$781$	−10074.0			−0.461557
$782$		0			0
$783$	−3402.00			−0.155271
$784$		0			0
$785$	−7056.00			−0.320815
$786$		0			0
$787$	12188.0			0.552040			0.276020	−	0.961152i	$-0.410984\pi$
$787$	12188.0			0.552040			0.276020	+	0.961152i	$0.410984\pi$
$788$		0			0
$789$			16262.2i			0.733775i
$790$		0			0
$791$	35802.0	−	15475.9i	1.60932	−	0.695650i
$792$		0			0
$793$	−25920.0			−1.16071
$794$		0			0
$795$		−	4364.99i		−	0.194730i
$796$		0			0
$797$		−	9109.65i		−	0.404869i	−0.979296	−	0.202434i	$-0.935115\pi$
$797$		−	9109.65i		−	0.404869i	0.979296	−	0.202434i	$-0.0648853\pi$
$798$		0			0
$799$		−	9259.07i		−	0.409965i
$800$		0			0
$801$		−	11617.9i		−	0.512484i
$802$		0			0
$803$	−37260.0			−1.63746
$804$		0			0
$805$	4794.00	−	2072.27i	0.209896	−	0.0907303i
$806$		0			0
$807$		−	18701.9i		−	0.815782i
$808$		0			0
$809$	−19710.0			−0.856572			−0.428286	−	0.903643i	$-0.640882\pi$
$809$	−19710.0			−0.856572			−0.428286	+	0.903643i	$0.640882\pi$
$810$		0			0
$811$	−8164.00			−0.353486			−0.176743	−	0.984257i	$-0.556556\pi$
$811$	−8164.00			−0.353486			−0.176743	+	0.984257i	$0.556556\pi$
$812$		0			0
$813$	−15072.0			−0.650182
$814$		0			0
$815$	720.000			0.0309454
$816$		0			0
$817$			1616.66i			0.0692287i
$818$		0			0
$819$	4860.00	+	11243.2i	0.207353	+	0.479692i
$820$		0			0
$821$	−17226.0			−0.732267			−0.366134	−	0.930562i	$-0.619319\pi$
$821$	−17226.0			−0.732267			−0.366134	+	0.930562i	$0.619319\pi$
$822$		0			0
$823$		−	16636.9i		−	0.704650i	−0.935878	−	0.352325i	$-0.885391\pi$
$823$		−	16636.9i		−	0.704650i	0.935878	−	0.352325i	$-0.114609\pi$
$824$		0			0
$825$			20112.8i			0.848771i
$826$		0			0
$827$		−	39517.6i		−	1.66162i	−0.556554	−	0.830811i	$-0.687877\pi$
$827$		−	39517.6i		−	1.66162i	0.556554	−	0.830811i	$-0.312123\pi$
$828$		0			0
$829$		−	38873.4i		−	1.62862i	−0.580428	−	0.814312i	$-0.697115\pi$
$829$		−	38873.4i		−	1.62862i	0.580428	−	0.814312i	$-0.302885\pi$
$830$		0			0
$831$	−12072.0			−0.503939
$832$		0			0
$833$	−12852.0	−	12088.2i	−0.534568	−	0.502800i
$834$		0			0
$835$			8729.98i			0.361813i
$836$		0			0
$837$	−216.000			−0.00892001
$838$		0			0
$839$	33804.0			1.39099			0.695497	−	0.718529i	$-0.255184\pi$
$839$	33804.0			1.39099			0.695497	+	0.718529i	$0.255184\pi$
$840$		0			0
$841$	−8513.00			−0.349051
$842$		0			0
$843$	1890.00			0.0772183
$844$		0			0
$845$		−	7845.72i		−	0.319409i
$846$		0			0
$847$	31331.0	−	13543.2i	1.27101	−	0.549411i
$848$		0			0
$849$	−27006.0			−1.09169
$850$		0			0
$851$		−	28090.7i		−	1.13154i
$852$		0			0
$853$			6260.90i			0.251312i	0.992074	+	0.125656i	$0.0401035\pi$
$853$			6260.90i			0.251312i	−0.992074	+	0.125656i	$0.959896\pi$
$854$		0			0
$855$			220.454i			0.00881798i
$856$		0			0
$857$		−	12240.1i		−	0.487881i	−0.969790	−	0.243940i	$-0.921560\pi$
$857$		−	12240.1i		−	0.487881i	0.969790	−	0.243940i	$-0.0784401\pi$
$858$		0			0
$859$	−40382.0			−1.60398			−0.801988	−	0.597340i	$-0.796224\pi$
$859$	−40382.0			−1.60398			−0.801988	+	0.597340i	$0.796224\pi$
$860$		0			0
$861$	−8154.00	−	18863.5i	−0.322750	−	0.746652i
$862$		0			0
$863$		−	9369.30i		−	0.369565i	−0.982779	−	0.184783i	$-0.940842\pi$
$863$		−	9369.30i		−	0.369565i	0.982779	−	0.184783i	$-0.0591581\pi$
$864$		0			0
$865$	3390.00			0.133253
$866$		0			0
$867$	6801.00			0.266406
$868$		0			0
$869$	57960.0			2.26255
$870$		0			0
$871$	−77760.0			−3.02503
$872$		0			0
$873$		−	7274.98i		−	0.282040i
$874$		0			0
$875$	−4392.00	−	10160.5i	−0.169688	−	0.392557i
$876$		0			0
$877$	−578.000			−0.0222550			−0.0111275	−	0.999938i	$-0.503542\pi$
$877$	−578.000			−0.0222550			−0.0111275	+	0.999938i	$0.503542\pi$
$878$		0			0
$879$			27446.5i			1.05318i
$880$		0			0
$881$		−	11902.1i		−	0.455154i	−0.973760	−	0.227577i	$-0.926920\pi$
$881$		−	11902.1i		−	0.455154i	0.973760	−	0.227577i	$-0.0730804\pi$
$882$		0			0
$883$		−	46633.4i		−	1.77728i	−0.458605	−	0.888640i	$-0.651651\pi$
$883$		−	46633.4i		−	1.77728i	0.458605	−	0.888640i	$-0.348349\pi$
$884$		0			0
$885$		−	3968.17i		−	0.150722i
$886$		0			0
$887$	23724.0			0.898054			0.449027	−	0.893518i	$-0.351771\pi$
$887$	23724.0			0.898054			0.449027	+	0.893518i	$0.351771\pi$
$888$		0			0
$889$	−7560.00	−	17489.4i	−0.285213	−	0.659813i
$890$		0			0
$891$			4563.40i			0.171582i
$892$		0			0
$893$	1800.00			0.0674521
$894$		0			0
$895$	342.000			0.0127730
$896$		0			0
$897$	25380.0			0.944720
$898$		0			0
$899$	1008.00			0.0373956
$900$		0			0
$901$			30554.9i			1.12978i
$902$		0			0
$903$	−3564.00	−	8244.98i	−0.131343	−	0.303849i
$904$		0			0
$905$	7596.00			0.279005
$906$		0			0
$907$			25998.9i			0.951796i	0.879501	+	0.475898i	$0.157877\pi$
$907$			25998.9i			0.951796i	−0.879501	+	0.475898i	$0.842123\pi$
$908$		0			0
$909$			330.681i			0.0120660i
$910$		0			0
$911$		−	45126.9i		−	1.64119i	−0.571511	−	0.820594i	$-0.693643\pi$
$911$		−	45126.9i		−	1.64119i	0.571511	−	0.820594i	$-0.306357\pi$
$912$		0			0
$913$		−	48676.3i		−	1.76446i
$914$		0			0
$915$	2592.00			0.0936490
$916$		0			0
$917$	9792.00	−	4232.72i	0.352628	−	0.152428i
$918$		0			0
$919$			35316.7i			1.26767i	0.773467	+	0.633837i	$0.218521\pi$
$919$			35316.7i			1.26767i	−0.773467	+	0.633837i	$0.781479\pi$
$920$		0			0
$921$	−15738.0			−0.563067
$922$		0			0
$923$	13140.0			0.468590
$924$		0			0
$925$	−29036.0			−1.03211
$926$		0			0
$927$	−12384.0			−0.438774
$928$		0			0
$929$		−	27064.4i		−	0.955818i	−0.878409	−	0.477909i	$-0.841395\pi$
$929$		−	27064.4i		−	0.955818i	0.878409	−	0.477909i	$-0.158605\pi$
$930$		0			0
$931$	2350.00	−	2498.48i	0.0827263	−	0.0879531i
$932$		0			0
$933$	24408.0			0.856465
$934$		0			0
$935$			7098.62i			0.248288i
$936$		0			0
$937$			13286.0i			0.463219i	0.972809	+	0.231609i	$0.0743991\pi$
$937$			13286.0i			0.463219i	−0.972809	+	0.231609i	$0.925601\pi$
$938$		0			0
$939$		−	1499.09i		−	0.0520989i
$940$		0			0
$941$			8850.01i			0.306591i	0.988180	+	0.153295i	$0.0489886\pi$
$941$			8850.01i			0.306591i	−0.988180	+	0.153295i	$0.951011\pi$
$942$		0			0
$943$	−42582.0			−1.47048
$944$		0			0
$945$	−486.000	−	1124.32i	−0.0167297	−	0.0387026i
$946$		0			0
$947$			3397.44i			0.116581i	0.998300	+	0.0582904i	$0.0185649\pi$
$947$			3397.44i			0.116581i	−0.998300	+	0.0582904i	$0.981435\pi$
$948$		0			0
$949$	48600.0			1.66241
$950$		0			0
$951$	−26514.0			−0.904075
$952$		0			0
$953$	−41346.0			−1.40538			−0.702691	−	0.711496i	$-0.748018\pi$
$953$	−41346.0			−1.40538			−0.702691	+	0.711496i	$0.748018\pi$
$954$		0			0
$955$	−1650.00			−0.0559086
$956$		0			0
$957$		−	21295.9i		−	0.719329i
$958$		0			0
$959$	28458.0	−	12301.3i	0.958245	−	0.414214i
$960$		0			0
$961$	−29727.0			−0.997852
$962$		0			0
$963$		−	10647.9i		−	0.356308i
$964$		0			0
$965$		−	5251.71i		−	0.175190i
$966$		0			0
$967$		−	43561.7i		−	1.44866i	−0.689455	−	0.724328i	$-0.742150\pi$
$967$		−	43561.7i		−	1.44866i	0.689455	−	0.724328i	$-0.257850\pi$
$968$		0			0
$969$		−	1543.18i		−	0.0511600i
$970$		0			0
$971$	−54252.0			−1.79303			−0.896514	−	0.443016i	$-0.853908\pi$
$971$	−54252.0			−1.79303			−0.896514	+	0.443016i	$0.853908\pi$
$972$		0			0
$973$	−5236.00	+	2263.33i	−0.172516	+	0.0745724i
$974$		0			0
$975$		−	26234.0i		−	0.861704i
$976$		0			0
$977$	13734.0			0.449733			0.224867	−	0.974390i	$-0.427805\pi$
$977$	13734.0			0.449733			0.224867	+	0.974390i	$0.427805\pi$
$978$		0			0
$979$	72726.0			2.37419
$980$		0			0
$981$	−5418.00			−0.176334
$982$		0			0
$983$	−17172.0			−0.557174			−0.278587	−	0.960411i	$-0.589866\pi$
$983$	−17172.0			−0.557174			−0.278587	+	0.960411i	$0.589866\pi$
$984$		0			0
$985$		−	573.181i		−	0.0185412i
$986$		0			0
$987$	−9180.00	+	3968.17i	−0.296051	+	0.127972i
$988$		0			0
$989$	−18612.0			−0.598410
$990$		0			0
$991$		−	13712.2i		−	0.439540i	−0.975552	−	0.219770i	$-0.929469\pi$
$991$		−	13712.2i		−	0.439540i	0.975552	−	0.219770i	$-0.0705306\pi$
$992$		0			0
$993$		−	27204.0i		−	0.869380i
$994$		0			0
$995$			6500.95i			0.207129i
$996$		0			0
$997$		−	14461.8i		−	0.459388i	−0.973263	−	0.229694i	$-0.926228\pi$
$997$		−	14461.8i		−	0.459388i	0.973263	−	0.229694i	$-0.0737725\pi$
$998$		0			0
$999$	−6588.00			−0.208644

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.4.b.c.895.2		2
4.3	odd	2		1344.4.b.b.895.2		2
7.6	odd	2		1344.4.b.b.895.1		2
8.3	odd	2		336.4.b.d.223.1	yes	2
8.5	even	2		336.4.b.a.223.1	✓	2
24.5	odd	2		1008.4.b.a.559.2		2
24.11	even	2		1008.4.b.f.559.2		2
28.27	even	2	inner	1344.4.b.c.895.1		2
56.13	odd	2		336.4.b.d.223.2	yes	2
56.27	even	2		336.4.b.a.223.2	yes	2
168.83	odd	2		1008.4.b.a.559.1		2
168.125	even	2		1008.4.b.f.559.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.4.b.a.223.1	✓	2	8.5	even	2
336.4.b.a.223.2	yes	2	56.27	even	2
336.4.b.d.223.1	yes	2	8.3	odd	2
336.4.b.d.223.2	yes	2	56.13	odd	2
1008.4.b.a.559.1		2	168.83	odd	2
1008.4.b.a.559.2		2	24.5	odd	2
1008.4.b.f.559.1		2	168.125	even	2
1008.4.b.f.559.2		2	24.11	even	2
1344.4.b.b.895.1		2	7.6	odd	2
1344.4.b.b.895.2		2	4.3	odd	2
1344.4.b.c.895.1		2	28.27	even	2	inner
1344.4.b.c.895.2		2	1.1	even	1	trivial

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

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} +2.44949i q^{5} +(-17.0000 + 7.34847i) q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q+3.00000 q^{3} +2.44949i q^{5} +(-17.0000 + 7.34847i) q^{7} +9.00000 q^{9} +56.3383i q^{11} -73.4847i q^{13} +7.34847i q^{15} -51.4393i q^{17} +10.0000 q^{19} +(-51.0000 + 22.0454i) q^{21} +115.126i q^{23} +119.000 q^{25} +27.0000 q^{27} -126.000 q^{29} -8.00000 q^{31} +169.015i q^{33} +(-18.0000 - 41.6413i) q^{35} -244.000 q^{37} -220.454i q^{39} +369.873i q^{41} +161.666i q^{43} +22.0454i q^{45} +180.000 q^{47} +(235.000 - 249.848i) q^{49} -154.318i q^{51} -594.000 q^{53} -138.000 q^{55} +30.0000 q^{57} -540.000 q^{59} -352.727i q^{61} +(-153.000 + 66.1362i) q^{63} +180.000 q^{65} -1058.18i q^{67} +345.378i q^{69} +178.813i q^{71} +661.362i q^{73} +357.000 q^{75} +(-414.000 - 957.750i) q^{77} -1028.79i q^{79} +81.0000 q^{81} -864.000 q^{83} +126.000 q^{85} -378.000 q^{87} -1290.88i q^{89} +(540.000 + 1249.24i) q^{91} -24.0000 q^{93} +24.4949i q^{95} -808.332i q^{97} +507.044i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{3} - 34 q^{7} + 18 q^{9}+O(q^{10}) \) 2 * q + 6 * q^3 - 34 * q^7 + 18 * q^9	\( 2 q + 6 q^{3} - 34 q^{7} + 18 q^{9} + 20 q^{19} - 102 q^{21} + 238 q^{25} + 54 q^{27} - 252 q^{29} - 16 q^{31} - 36 q^{35} - 488 q^{37} + 360 q^{47} + 470 q^{49} - 1188 q^{53} - 276 q^{55} + 60 q^{57} - 1080 q^{59} - 306 q^{63} + 360 q^{65} + 714 q^{75} - 828 q^{77} + 162 q^{81} - 1728 q^{83} + 252 q^{85} - 756 q^{87} + 1080 q^{91} - 48 q^{93}+O(q^{100}) \) 2 * q + 6 * q^3 - 34 * q^7 + 18 * q^9 + 20 * q^19 - 102 * q^21 + 238 * q^25 + 54 * q^27 - 252 * q^29 - 16 * q^31 - 36 * q^35 - 488 * q^37 + 360 * q^47 + 470 * q^49 - 1188 * q^53 - 276 * q^55 + 60 * q^57 - 1080 * q^59 - 306 * q^63 + 360 * q^65 + 714 * q^75 - 828 * q^77 + 162 * q^81 - 1728 * q^83 + 252 * q^85 - 756 * q^87 + 1080 * q^91 - 48 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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