LMFDB - Embedded newform 1344.4.b.j.895.17

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:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 672)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			895.17
Character	$\chi$	$=$	1344.895
Dual form			1344.4.b.j.895.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} +8.63807i q^{5} +(-18.2735 - 3.01325i) q^{7} +9.00000 q^{9} +O(q^{10})$	$q+3.00000 q^{3} +8.63807i q^{5} +(-18.2735 - 3.01325i) q^{7} +9.00000 q^{9} +4.41160i q^{11} -47.2950i q^{13} +25.9142i q^{15} +130.535i q^{17} -44.8148 q^{19} +(-54.8205 - 9.03976i) q^{21} -104.571i q^{23} +50.3838 q^{25} +27.0000 q^{27} -168.152 q^{29} -27.5040 q^{31} +13.2348i q^{33} +(26.0287 - 157.848i) q^{35} +425.789 q^{37} -141.885i q^{39} -332.076i q^{41} +218.693i q^{43} +77.7426i q^{45} -390.577 q^{47} +(324.841 + 110.125i) q^{49} +391.606i q^{51} -736.988 q^{53} -38.1077 q^{55} -134.444 q^{57} -469.132 q^{59} -21.9630i q^{61} +(-164.461 - 27.1193i) q^{63} +408.537 q^{65} -365.638i q^{67} -313.714i q^{69} -620.001i q^{71} -938.039i q^{73} +151.151 q^{75} +(13.2933 - 80.6154i) q^{77} -883.933i q^{79} +81.0000 q^{81} +739.368 q^{83} -1127.57 q^{85} -504.457 q^{87} +66.4755i q^{89} +(-142.512 + 864.245i) q^{91} -82.5119 q^{93} -387.113i q^{95} -353.505i q^{97} +39.7044i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 72 q^{3} - 20 q^{7} + 216 q^{9}+O(q^{10}) $ 24 * q + 72 * q^3 - 20 * q^7 + 216 * q^9	$ 24 q + 72 q^{3} - 20 q^{7} + 216 q^{9} - 56 q^{19} - 60 q^{21} - 432 q^{25} + 648 q^{27} + 464 q^{31} + 568 q^{35} - 504 q^{37} + 560 q^{47} - 128 q^{49} + 784 q^{53} + 424 q^{55} - 168 q^{57} + 800 q^{59} - 180 q^{63} + 560 q^{65} - 1296 q^{75} + 1568 q^{77} + 1944 q^{81} + 1936 q^{83} - 3000 q^{85} - 496 q^{91} + 1392 q^{93}+O(q^{100}) $ 24 * q + 72 * q^3 - 20 * q^7 + 216 * q^9 - 56 * q^19 - 60 * q^21 - 432 * q^25 + 648 * q^27 + 464 * q^31 + 568 * q^35 - 504 * q^37 + 560 * q^47 - 128 * q^49 + 784 * q^53 + 424 * q^55 - 168 * q^57 + 800 * q^59 - 180 * q^63 + 560 * q^65 - 1296 * q^75 + 1568 * q^77 + 1944 * q^81 + 1936 * q^83 - 3000 * q^85 - 496 * q^91 + 1392 * 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
$5$			8.63807i			0.772612i	0.922371	+	0.386306i	$0.126249\pi$
$5$			8.63807i			0.772612i	−0.922371	+	0.386306i	$0.873751\pi$
$6$		0			0
$7$	−18.2735	−	3.01325i	−0.986676	−	0.162700i
$7$	−18.2735	−	3.01325i	−0.986676	−	0.162700i
$8$		0			0
$9$	9.00000			0.333333
$10$		0			0
$11$			4.41160i			0.120923i	0.998171	+	0.0604613i	$0.0192572\pi$
$11$			4.41160i			0.120923i	−0.998171	+	0.0604613i	$0.980743\pi$
$12$		0			0
$13$		−	47.2950i		−	1.00902i	−0.863406	−	0.504511i	$-0.831673\pi$
$13$		−	47.2950i		−	1.00902i	0.863406	−	0.504511i	$-0.168327\pi$
$14$		0			0
$15$			25.9142i			0.446068i
$16$		0			0
$17$			130.535i			1.86232i	0.364609	+	0.931161i	$0.381203\pi$
$17$			130.535i			1.86232i	−0.364609	+	0.931161i	$0.618797\pi$
$18$		0			0
$19$	−44.8148			−0.541117			−0.270558	−	0.962704i	$-0.587208\pi$
$19$	−44.8148			−0.541117			−0.270558	+	0.962704i	$0.587208\pi$
$20$		0			0
$21$	−54.8205	−	9.03976i	−0.569657	−	0.0939351i
$22$		0			0
$23$		−	104.571i		−	0.948028i	−0.880517	−	0.474014i	$-0.842805\pi$
$23$		−	104.571i		−	0.948028i	0.880517	−	0.474014i	$-0.157195\pi$
$24$		0			0
$25$	50.3838			0.403070
$26$		0			0
$27$	27.0000			0.192450
$28$		0			0
$29$	−168.152			−1.07673			−0.538364	−	0.842712i	$-0.680958\pi$
$29$	−168.152			−1.07673			−0.538364	+	0.842712i	$0.680958\pi$
$30$		0			0
$31$	−27.5040			−0.159350			−0.0796752	−	0.996821i	$-0.525388\pi$
$31$	−27.5040			−0.159350			−0.0796752	+	0.996821i	$0.525388\pi$
$32$		0			0
$33$			13.2348i			0.0698147i
$34$		0			0
$35$	26.0287	−	157.848i	0.125704	−	0.762318i
$36$		0			0
$37$	425.789			1.89187			0.945937	−	0.324350i	$-0.105146\pi$
$37$	425.789			1.89187			0.945937	+	0.324350i	$0.105146\pi$
$38$		0			0
$39$		−	141.885i		−	0.582559i
$40$		0			0
$41$		−	332.076i		−	1.26492i	−0.774595	−	0.632458i	$-0.782046\pi$
$41$		−	332.076i		−	1.26492i	0.774595	−	0.632458i	$-0.217954\pi$
$42$		0			0
$43$			218.693i			0.775590i	0.921746	+	0.387795i	$0.126763\pi$
$43$			218.693i			0.775590i	−0.921746	+	0.387795i	$0.873237\pi$
$44$		0			0
$45$			77.7426i			0.257537i
$46$		0			0
$47$	−390.577			−1.21216			−0.606080	−	0.795404i	$-0.707259\pi$
$47$	−390.577			−1.21216			−0.606080	+	0.795404i	$0.707259\pi$
$48$		0			0
$49$	324.841	+	110.125i	0.947057	+	0.321065i
$50$		0			0
$51$			391.606i			1.07521i
$52$		0			0
$53$	−736.988			−1.91006			−0.955029	−	0.296512i	$-0.904176\pi$
$53$	−736.988			−1.91006			−0.955029	+	0.296512i	$0.904176\pi$
$54$		0			0
$55$	−38.1077			−0.0934263
$56$		0			0
$57$	−134.444			−0.312414
$58$		0			0
$59$	−469.132			−1.03518			−0.517591	−	0.855628i	$-0.673171\pi$
$59$	−469.132			−1.03518			−0.517591	+	0.855628i	$0.673171\pi$
$60$		0			0
$61$		−	21.9630i		−	0.0460995i	−0.999734	−	0.0230498i	$-0.992662\pi$
$61$		−	21.9630i		−	0.0460995i	0.999734	−	0.0230498i	$-0.00733762\pi$
$62$		0			0
$63$	−164.461	−	27.1193i	−0.328892	−	0.0542335i
$64$		0			0
$65$	408.537			0.779582
$66$		0			0
$67$		−	365.638i		−	0.666714i	−0.942801	−	0.333357i	$-0.891819\pi$
$67$		−	365.638i		−	0.666714i	0.942801	−	0.333357i	$-0.108181\pi$
$68$		0			0
$69$		−	313.714i		−	0.547344i
$70$		0			0
$71$		−	620.001i		−	1.03635i	−0.855276	−	0.518173i	$-0.826612\pi$
$71$		−	620.001i		−	1.03635i	0.855276	−	0.518173i	$-0.173388\pi$
$72$		0			0
$73$		−	938.039i		−	1.50396i	−0.659185	−	0.751981i	$-0.729099\pi$
$73$		−	938.039i		−	1.50396i	0.659185	−	0.751981i	$-0.270901\pi$
$74$		0			0
$75$	151.151			0.232713
$76$		0			0
$77$	13.2933	−	80.6154i	0.0196742	−	0.119311i
$78$		0			0
$79$		−	883.933i		−	1.25886i	−0.777055	−	0.629432i	$-0.783288\pi$
$79$		−	883.933i		−	1.25886i	0.777055	−	0.629432i	$-0.216712\pi$
$80$		0			0
$81$	81.0000			0.111111
$82$		0			0
$83$	739.368			0.977786			0.488893	−	0.872344i	$-0.337401\pi$
$83$	739.368			0.977786			0.488893	+	0.872344i	$0.337401\pi$
$84$		0			0
$85$	−1127.57			−1.43885
$86$		0			0
$87$	−504.457			−0.621650
$88$		0			0
$89$			66.4755i			0.0791729i	0.999216	+	0.0395864i	$0.0126040\pi$
$89$			66.4755i			0.0791729i	−0.999216	+	0.0395864i	$0.987396\pi$
$90$		0			0
$91$	−142.512	+	864.245i	−0.164168	+	0.995576i
$92$		0			0
$93$	−82.5119			−0.0920010
$94$		0			0
$95$		−	387.113i		−	0.418073i
$96$		0			0
$97$		−	353.505i		−	0.370031i	−0.982736	−	0.185015i	$-0.940766\pi$
$97$		−	353.505i		−	0.370031i	0.982736	−	0.185015i	$-0.0592335\pi$
$98$		0			0
$99$			39.7044i			0.0403075i
$100$		0			0
$101$		−	792.714i		−	0.780970i	−0.920609	−	0.390485i	$-0.872307\pi$
$101$		−	792.714i		−	0.780970i	0.920609	−	0.390485i	$-0.127693\pi$
$102$		0			0
$103$	1289.87			1.23393			0.616963	−	0.786992i	$-0.288363\pi$
$103$	1289.87			1.23393			0.616963	+	0.786992i	$0.288363\pi$
$104$		0			0
$105$	78.0861	−	473.543i	0.0725754	−	0.440124i
$106$		0			0
$107$		−	823.441i		−	0.743973i	−0.928238	−	0.371986i	$-0.878677\pi$
$107$		−	823.441i		−	0.743973i	0.928238	−	0.371986i	$-0.121323\pi$
$108$		0			0
$109$	−742.157			−0.652163			−0.326082	−	0.945342i	$-0.605728\pi$
$109$	−742.157			−0.652163			−0.326082	+	0.945342i	$0.605728\pi$
$110$		0			0
$111$	1277.37			1.09227
$112$		0			0
$113$	60.2761			0.0501796			0.0250898	−	0.999685i	$-0.492013\pi$
$113$	60.2761			0.0501796			0.0250898	+	0.999685i	$0.492013\pi$
$114$		0			0
$115$	903.295			0.732458
$116$		0			0
$117$		−	425.655i		−	0.336340i
$118$		0			0
$119$	393.336	−	2385.34i	0.303000	−	1.83751i
$120$		0			0
$121$	1311.54			0.985378
$122$		0			0
$123$		−	996.228i		−	0.730300i
$124$		0			0
$125$			1514.98i			1.08403i
$126$		0			0
$127$		−	2087.89i		−	1.45882i	−0.684076	−	0.729411i	$-0.739794\pi$
$127$		−	2087.89i		−	1.45882i	0.684076	−	0.729411i	$-0.260206\pi$
$128$		0			0
$129$			656.079i			0.447787i
$130$		0			0
$131$	−364.582			−0.243158			−0.121579	−	0.992582i	$-0.538796\pi$
$131$	−364.582			−0.243158			−0.121579	+	0.992582i	$0.538796\pi$
$132$		0			0
$133$	818.922	+	135.038i	0.533907	+	0.0880399i
$134$		0			0
$135$			233.228i			0.148689i
$136$		0			0
$137$	1459.31			0.910050			0.455025	−	0.890479i	$-0.349630\pi$
$137$	1459.31			0.910050			0.455025	+	0.890479i	$0.349630\pi$
$138$		0			0
$139$	−2780.03			−1.69640			−0.848198	−	0.529680i	$-0.822312\pi$
$139$	−2780.03			−1.69640			−0.848198	+	0.529680i	$0.822312\pi$
$140$		0			0
$141$	−1171.73			−0.699841
$142$		0			0
$143$	208.647			0.122013
$144$		0			0
$145$		−	1452.51i		−	0.831894i
$146$		0			0
$147$	974.522	+	330.376i	0.546784	+	0.185367i
$148$		0			0
$149$	577.447			0.317492			0.158746	−	0.987319i	$-0.449255\pi$
$149$	577.447			0.317492			0.158746	+	0.987319i	$0.449255\pi$
$150$		0			0
$151$		−	1675.18i		−	0.902811i	−0.892319	−	0.451405i	$-0.850923\pi$
$151$		−	1675.18i		−	0.902811i	0.892319	−	0.451405i	$-0.149077\pi$
$152$		0			0
$153$			1174.82i			0.620774i
$154$		0			0
$155$		−	237.581i		−	0.123116i
$156$		0			0
$157$			2470.29i			1.25574i	0.778320	+	0.627868i	$0.216072\pi$
$157$			2470.29i			1.25574i	−0.778320	+	0.627868i	$0.783928\pi$
$158$		0			0
$159$	−2210.96			−1.10277
$160$		0			0
$161$	−315.100	+	1910.88i	−0.154245	+	0.935396i
$162$		0			0
$163$		−	367.613i		−	0.176648i	−0.996092	−	0.0883241i	$-0.971849\pi$
$163$		−	367.613i		−	0.176648i	0.996092	−	0.0883241i	$-0.0281511\pi$
$164$		0			0
$165$	−114.323			−0.0539397
$166$		0			0
$167$	−2535.64			−1.17493			−0.587466	−	0.809249i	$-0.699874\pi$
$167$	−2535.64			−1.17493			−0.587466	+	0.809249i	$0.699874\pi$
$168$		0			0
$169$	−39.8175			−0.0181236
$170$		0			0
$171$	−403.333			−0.180372
$172$		0			0
$173$		−	85.1435i		−	0.0374182i	−0.999825	−	0.0187091i	$-0.994044\pi$
$173$		−	85.1435i		−	0.0374182i	0.999825	−	0.0187091i	$-0.00595563\pi$
$174$		0			0
$175$	−920.687	−	151.819i	−0.397699	−	0.0655797i
$176$		0			0
$177$	−1407.39			−0.597663
$178$		0			0
$179$		−	2428.33i		−	1.01398i	−0.861953	−	0.506989i	$-0.830759\pi$
$179$		−	2428.33i		−	1.01398i	0.861953	−	0.506989i	$-0.169241\pi$
$180$		0			0
$181$			610.613i			0.250754i	0.992109	+	0.125377i	$0.0400141\pi$
$181$			610.613i			0.250754i	−0.992109	+	0.125377i	$0.959986\pi$
$182$		0			0
$183$		−	65.8890i		−	0.0266156i
$184$		0			0
$185$			3678.00i			1.46169i
$186$		0			0
$187$	−575.870			−0.225197
$188$		0			0
$189$	−493.384	−	81.3578i	−0.189886	−	0.0313117i
$190$		0			0
$191$		−	1578.99i		−	0.598175i	−0.954226	−	0.299088i	$-0.903318\pi$
$191$		−	1578.99i		−	0.598175i	0.954226	−	0.299088i	$-0.0966823\pi$
$192$		0			0
$193$	1525.19			0.568836			0.284418	−	0.958700i	$-0.408200\pi$
$193$	1525.19			0.568836			0.284418	+	0.958700i	$0.408200\pi$
$194$		0			0
$195$	1225.61			0.450092
$196$		0			0
$197$	−3559.64			−1.28738			−0.643690	−	0.765287i	$-0.722597\pi$
$197$	−3559.64			−1.28738			−0.643690	+	0.765287i	$0.722597\pi$
$198$		0			0
$199$	−260.851			−0.0929208			−0.0464604	−	0.998920i	$-0.514794\pi$
$199$	−260.851			−0.0929208			−0.0464604	+	0.998920i	$0.514794\pi$
$200$		0			0
$201$		−	1096.91i		−	0.384927i
$202$		0			0
$203$	3072.73	+	506.686i	1.06238	+	0.175184i
$204$		0			0
$205$	2868.49			0.977290
$206$		0			0
$207$		−	941.142i		−	0.316009i
$208$		0			0
$209$		−	197.705i		−	0.0654332i
$210$		0			0
$211$		−	5275.22i		−	1.72114i	−0.509329	−	0.860572i	$-0.670106\pi$
$211$		−	5275.22i		−	1.72114i	0.509329	−	0.860572i	$-0.329894\pi$
$212$		0			0
$213$		−	1860.00i		−	0.598334i
$214$		0			0
$215$	−1889.09			−0.599230
$216$		0			0
$217$	502.594	+	82.8765i	0.157227	+	0.0259264i
$218$		0			0
$219$		−	2814.12i		−	0.868312i
$220$		0			0
$221$	6173.67			1.87912
$222$		0			0
$223$	2251.47			0.676097			0.338048	−	0.941129i	$-0.390233\pi$
$223$	2251.47			0.676097			0.338048	+	0.941129i	$0.390233\pi$
$224$		0			0
$225$	453.454			0.134357
$226$		0			0
$227$	2469.05			0.721924			0.360962	−	0.932581i	$-0.382448\pi$
$227$	2469.05			0.721924			0.360962	+	0.932581i	$0.382448\pi$
$228$		0			0
$229$			1378.50i			0.397790i	0.980021	+	0.198895i	$0.0637353\pi$
$229$			1378.50i			0.397790i	−0.980021	+	0.198895i	$0.936265\pi$
$230$		0			0
$231$	39.8798	−	241.846i	0.0113589	−	0.0688844i
$232$		0			0
$233$	−5655.72			−1.59021			−0.795104	−	0.606473i	$-0.792584\pi$
$233$	−5655.72			−1.59021			−0.795104	+	0.606473i	$0.792584\pi$
$234$		0			0
$235$		−	3373.83i		−	0.936530i
$236$		0			0
$237$		−	2651.80i		−	0.726806i
$238$		0			0
$239$		−	1110.64i		−	0.300590i	−0.988641	−	0.150295i	$-0.951978\pi$
$239$		−	1110.64i		−	0.300590i	0.988641	−	0.150295i	$-0.0480224\pi$
$240$		0			0
$241$		−	475.766i		−	0.127165i	−0.997977	−	0.0635825i	$-0.979747\pi$
$241$		−	475.766i		−	0.127165i	0.997977	−	0.0635825i	$-0.0202526\pi$
$242$		0			0
$243$	243.000			0.0641500
$244$		0			0
$245$	−951.270	+	2806.00i	−0.248059	+	0.731708i
$246$		0			0
$247$			2119.52i			0.545998i
$248$		0			0
$249$	2218.10			0.564525
$250$		0			0
$251$	−2934.27			−0.737886			−0.368943	−	0.929452i	$-0.620280\pi$
$251$	−2934.27			−0.737886			−0.368943	+	0.929452i	$0.620280\pi$
$252$		0			0
$253$	461.327			0.114638
$254$		0			0
$255$	−3382.72			−0.830722
$256$		0			0
$257$			5352.36i			1.29911i	0.760315	+	0.649554i	$0.225044\pi$
$257$			5352.36i			1.29911i	−0.760315	+	0.649554i	$0.774956\pi$
$258$		0			0
$259$	−7780.66	−	1283.01i	−1.86667	−	0.307809i
$260$		0			0
$261$	−1513.37			−0.358910
$262$		0			0
$263$		−	526.211i		−	0.123375i	−0.998096	−	0.0616874i	$-0.980352\pi$
$263$		−	526.211i		−	0.123375i	0.998096	−	0.0616874i	$-0.0196482\pi$
$264$		0			0
$265$		−	6366.15i		−	1.47573i
$266$		0			0
$267$			199.426i			0.0457105i
$268$		0			0
$269$		−	3228.16i		−	0.731689i	−0.930676	−	0.365844i	$-0.880780\pi$
$269$		−	3228.16i		−	0.731689i	0.930676	−	0.365844i	$-0.119220\pi$
$270$		0			0
$271$	7864.89			1.76294			0.881472	−	0.472237i	$-0.156553\pi$
$271$	7864.89			1.76294			0.881472	+	0.472237i	$0.156553\pi$
$272$		0			0
$273$	−427.536	+	2592.73i	−0.0947825	+	0.574796i
$274$		0			0
$275$			222.273i			0.0487403i
$276$		0			0
$277$	3234.50			0.701597			0.350798	−	0.936451i	$-0.385910\pi$
$277$	3234.50			0.701597			0.350798	+	0.936451i	$0.385910\pi$
$278$		0			0
$279$	−247.536			−0.0531168
$280$		0			0
$281$	−367.370			−0.0779910			−0.0389955	−	0.999239i	$-0.512416\pi$
$281$	−367.370			−0.0779910			−0.0389955	+	0.999239i	$0.512416\pi$
$282$		0			0
$283$	−1482.82			−0.311465			−0.155733	−	0.987799i	$-0.549774\pi$
$283$	−1482.82			−0.311465			−0.155733	+	0.987799i	$0.549774\pi$
$284$		0			0
$285$		−	1161.34i		−	0.241375i
$286$		0			0
$287$	−1000.63	+	6068.19i	−0.205802	+	1.24806i
$288$		0			0
$289$	−12126.5			−2.46824
$290$		0			0
$291$		−	1060.51i		−	0.213637i
$292$		0			0
$293$			3146.63i			0.627400i	0.949522	+	0.313700i	$0.101569\pi$
$293$			3146.63i			0.627400i	−0.949522	+	0.313700i	$0.898431\pi$
$294$		0			0
$295$		−	4052.39i		−	0.799794i
$296$		0			0
$297$			119.113i			0.0232716i
$298$		0			0
$299$	−4945.70			−0.956580
$300$		0			0
$301$	658.978	−	3996.28i	0.126189	−	0.765256i
$302$		0			0
$303$		−	2378.14i		−	0.450893i
$304$		0			0
$305$	189.718			0.0356171
$306$		0			0
$307$	−6312.88			−1.17360			−0.586800	−	0.809732i	$-0.699612\pi$
$307$	−6312.88			−1.17360			−0.586800	+	0.809732i	$0.699612\pi$
$308$		0			0
$309$	3869.60			0.712408
$310$		0			0
$311$	−2777.97			−0.506509			−0.253255	−	0.967400i	$-0.581501\pi$
$311$	−2777.97			−0.506509			−0.253255	+	0.967400i	$0.581501\pi$
$312$		0			0
$313$		−	8369.07i		−	1.51133i	−0.654956	−	0.755667i	$-0.727313\pi$
$313$		−	8369.07i		−	1.51133i	0.654956	−	0.755667i	$-0.272687\pi$
$314$		0			0
$315$	234.258	−	1420.63i	0.0419014	−	0.254106i
$316$		0			0
$317$	−1783.66			−0.316026			−0.158013	−	0.987437i	$-0.550509\pi$
$317$	−1783.66			−0.316026			−0.158013	+	0.987437i	$0.550509\pi$
$318$		0			0
$319$		−	741.822i		−	0.130201i
$320$		0			0
$321$		−	2470.32i		−	0.429533i
$322$		0			0
$323$		−	5849.91i		−	1.00773i
$324$		0			0
$325$		−	2382.90i		−	0.406706i
$326$		0			0
$327$	−2226.47			−0.376527
$328$		0			0
$329$	7137.20	+	1176.91i	1.19601	+	0.197219i
$330$		0			0
$331$		−	8310.25i		−	1.37998i	−0.723820	−	0.689989i	$-0.757615\pi$
$331$		−	8310.25i		−	1.37998i	0.723820	−	0.689989i	$-0.242385\pi$
$332$		0			0
$333$	3832.10			0.630625
$334$		0			0
$335$	3158.41			0.515111
$336$		0			0
$337$	−8058.11			−1.30253			−0.651266	−	0.758850i	$-0.725762\pi$
$337$	−8058.11			−1.30253			−0.651266	+	0.758850i	$0.725762\pi$
$338$		0			0
$339$	180.828			0.0289712
$340$		0			0
$341$		−	121.337i		−	0.0192691i
$342$		0			0
$343$	−5604.13	−	2991.20i	−0.882201	−	0.470874i
$344$		0			0
$345$	2709.88			0.422885
$346$		0			0
$347$		−	9675.12i		−	1.49679i	−0.663251	−	0.748397i	$-0.730824\pi$
$347$		−	9675.12i		−	1.49679i	0.663251	−	0.748397i	$-0.269176\pi$
$348$		0			0
$349$			12518.5i			1.92006i	0.279902	+	0.960028i	$0.409698\pi$
$349$			12518.5i			1.92006i	−0.279902	+	0.960028i	$0.590302\pi$
$350$		0			0
$351$		−	1276.97i		−	0.194186i
$352$		0			0
$353$			1508.86i			0.227503i	0.993509	+	0.113751i	$0.0362867\pi$
$353$			1508.86i			0.227503i	−0.993509	+	0.113751i	$0.963713\pi$
$354$		0			0
$355$	5355.61			0.800693
$356$		0			0
$357$	1180.01	−	7156.01i	0.174937	−	1.06089i
$358$		0			0
$359$			12334.0i			1.81327i	0.421912	+	0.906637i	$0.361359\pi$
$359$			12334.0i			1.81327i	−0.421912	+	0.906637i	$0.638641\pi$
$360$		0			0
$361$	−4850.63			−0.707193
$362$		0			0
$363$	3934.61			0.568908
$364$		0			0
$365$	8102.85			1.16198
$366$		0			0
$367$	4703.48			0.668991			0.334495	−	0.942397i	$-0.391434\pi$
$367$	4703.48			0.668991			0.334495	+	0.942397i	$0.391434\pi$
$368$		0			0
$369$		−	2988.68i		−	0.421639i
$370$		0			0
$371$	13467.3	+	2220.73i	1.88461	+	0.310767i
$372$		0			0
$373$	−6002.47			−0.833234			−0.416617	−	0.909082i	$-0.636784\pi$
$373$	−6002.47			−0.833234			−0.416617	+	0.909082i	$0.636784\pi$
$374$		0			0
$375$			4544.93i			0.625865i
$376$		0			0
$377$			7952.77i			1.08644i
$378$		0			0
$379$			13785.1i			1.86832i	0.356852	+	0.934161i	$0.383850\pi$
$379$			13785.1i			1.86832i	−0.356852	+	0.934161i	$0.616150\pi$
$380$		0			0
$381$		−	6263.67i		−	0.842251i
$382$		0			0
$383$	6107.74			0.814858			0.407429	−	0.913237i	$-0.366425\pi$
$383$	6107.74			0.814858			0.407429	+	0.913237i	$0.366425\pi$
$384$		0			0
$385$	696.361	+	114.828i	0.0921814	+	0.0152005i
$386$		0			0
$387$			1968.24i			0.258530i
$388$		0			0
$389$	−1732.22			−0.225776			−0.112888	−	0.993608i	$-0.536010\pi$
$389$	−1732.22			−0.225776			−0.112888	+	0.993608i	$0.536010\pi$
$390$		0			0
$391$	13650.3			1.76553
$392$		0			0
$393$	−1093.74			−0.140387
$394$		0			0
$395$	7635.48			0.972614
$396$		0			0
$397$		−	1736.09i		−	0.219476i	−0.993961	−	0.109738i	$-0.964999\pi$
$397$		−	1736.09i		−	0.219476i	0.993961	−	0.109738i	$-0.0350011\pi$
$398$		0			0
$399$	2456.77	+	405.115i	0.308251	+	0.0508299i
$400$		0			0
$401$	5595.25			0.696792			0.348396	−	0.937348i	$-0.386727\pi$
$401$	5595.25			0.696792			0.348396	+	0.937348i	$0.386727\pi$
$402$		0			0
$403$			1300.80i			0.160788i
$404$		0			0
$405$			699.684i			0.0858458i
$406$		0			0
$407$			1878.41i			0.228770i
$408$		0			0
$409$		−	2612.18i		−	0.315804i	−0.987455	−	0.157902i	$-0.949527\pi$
$409$		−	2612.18i		−	0.315804i	0.987455	−	0.157902i	$-0.0504730\pi$
$410$		0			0
$411$	4377.92			0.525418
$412$		0			0
$413$	8572.67	+	1413.61i	1.02139	+	0.168425i
$414$		0			0
$415$			6386.71i			0.755449i
$416$		0			0
$417$	−8340.09			−0.979414
$418$		0			0
$419$	−5893.63			−0.687166			−0.343583	−	0.939122i	$-0.611641\pi$
$419$	−5893.63			−0.687166			−0.343583	+	0.939122i	$0.611641\pi$
$420$		0			0
$421$	−7228.41			−0.836797			−0.418398	−	0.908264i	$-0.637408\pi$
$421$	−7228.41			−0.836797			−0.418398	+	0.908264i	$0.637408\pi$
$422$		0			0
$423$	−3515.19			−0.404053
$424$		0			0
$425$			6576.86i			0.750646i
$426$		0			0
$427$	−66.1801	+	401.340i	−0.00750042	+	0.0454853i
$428$		0			0
$429$	625.940			0.0704445
$430$		0			0
$431$		−	13447.1i		−	1.50283i	−0.659827	−	0.751417i	$-0.729371\pi$
$431$		−	13447.1i		−	1.50283i	0.659827	−	0.751417i	$-0.270629\pi$
$432$		0			0
$433$		−	3605.02i		−	0.400106i	−0.979785	−	0.200053i	$-0.935889\pi$
$433$		−	3605.02i		−	0.400106i	0.979785	−	0.200053i	$-0.0641115\pi$
$434$		0			0
$435$		−	4357.54i		−	0.480294i
$436$		0			0
$437$			4686.34i			0.512994i
$438$		0			0
$439$	−7561.25			−0.822047			−0.411023	−	0.911625i	$-0.634829\pi$
$439$	−7561.25			−0.822047			−0.411023	+	0.911625i	$0.634829\pi$
$440$		0			0
$441$	2923.57	+	991.128i	0.315686	+	0.107022i
$442$		0			0
$443$			371.134i			0.0398038i	0.999802	+	0.0199019i	$0.00633540\pi$
$443$			371.134i			0.0398038i	−0.999802	+	0.0199019i	$0.993665\pi$
$444$		0			0
$445$	−574.220			−0.0611699
$446$		0			0
$447$	1732.34			0.183304
$448$		0			0
$449$	−2453.53			−0.257882			−0.128941	−	0.991652i	$-0.541158\pi$
$449$	−2453.53			−0.257882			−0.128941	+	0.991652i	$0.541158\pi$
$450$		0			0
$451$	1464.99			0.152957
$452$		0			0
$453$		−	5025.55i		−	0.521238i
$454$		0			0
$455$	−7465.40	−	1231.03i	−0.769195	−	0.126838i
$456$		0			0
$457$	14931.7			1.52839			0.764196	−	0.644985i	$-0.223136\pi$
$457$	14931.7			1.52839			0.764196	+	0.644985i	$0.223136\pi$
$458$		0			0
$459$			3524.45i			0.358404i
$460$		0			0
$461$			14153.6i			1.42993i	0.699161	+	0.714965i	$0.253557\pi$
$461$			14153.6i			1.42993i	−0.699161	+	0.714965i	$0.746443\pi$
$462$		0			0
$463$			10658.3i			1.06983i	0.844906	+	0.534915i	$0.179656\pi$
$463$			10658.3i			1.06983i	−0.844906	+	0.534915i	$0.820344\pi$
$464$		0			0
$465$		−	712.744i		−	0.0710811i
$466$		0			0
$467$	−8719.82			−0.864036			−0.432018	−	0.901865i	$-0.642198\pi$
$467$	−8719.82			−0.864036			−0.432018	+	0.901865i	$0.642198\pi$
$468$		0			0
$469$	−1101.76	+	6681.48i	−0.108475	+	0.657830i
$470$		0			0
$471$			7410.87i			0.724999i
$472$		0			0
$473$	−964.787			−0.0937864
$474$		0			0
$475$	−2257.94			−0.218108
$476$		0			0
$477$	−6632.89			−0.636686
$478$		0			0
$479$	−12627.0			−1.20447			−0.602236	−	0.798318i	$-0.705723\pi$
$479$	−12627.0			−1.20447			−0.602236	+	0.798318i	$0.705723\pi$
$480$		0			0
$481$		−	20137.7i		−	1.90894i
$482$		0			0
$483$	−945.300	+	5732.65i	−0.0890531	+	0.540051i
$484$		0			0
$485$	3053.60			0.285890
$486$		0			0
$487$		−	14172.2i		−	1.31869i	−0.751841	−	0.659345i	$-0.770834\pi$
$487$		−	14172.2i		−	1.31869i	0.751841	−	0.659345i	$-0.229166\pi$
$488$		0			0
$489$		−	1102.84i		−	0.101988i
$490$		0			0
$491$			10366.0i			0.952776i	0.879235	+	0.476388i	$0.158054\pi$
$491$			10366.0i			0.952776i	−0.879235	+	0.476388i	$0.841946\pi$
$492$		0			0
$493$		−	21949.8i		−	2.00521i
$494$		0			0
$495$	−342.970			−0.0311421
$496$		0			0
$497$	−1868.22	+	11329.6i	−0.168614	+	1.02254i
$498$		0			0
$499$			18379.9i			1.64890i	0.565938	+	0.824448i	$0.308514\pi$
$499$			18379.9i			1.64890i	−0.565938	+	0.824448i	$0.691486\pi$
$500$		0			0
$501$	−7606.92			−0.678347
$502$		0			0
$503$	59.4388			0.00526888			0.00263444	−	0.999997i	$-0.499161\pi$
$503$	59.4388			0.00526888			0.00263444	+	0.999997i	$0.499161\pi$
$504$		0			0
$505$	6847.51			0.603387
$506$		0			0
$507$	−119.452			−0.0104636
$508$		0			0
$509$		−	10957.8i		−	0.954215i	−0.878845	−	0.477108i	$-0.841685\pi$
$509$		−	10957.8i		−	0.954215i	0.878845	−	0.477108i	$-0.158315\pi$
$510$		0			0
$511$	−2826.55	+	17141.2i	−0.244695	+	1.48392i
$512$		0			0
$513$	−1210.00			−0.104138
$514$		0			0
$515$			11142.0i			0.953347i
$516$		0			0
$517$		−	1723.07i		−	0.146578i
$518$		0			0
$519$		−	255.431i		−	0.0216034i
$520$		0			0
$521$			14559.6i			1.22431i	0.790738	+	0.612155i	$0.209697\pi$
$521$			14559.6i			1.22431i	−0.790738	+	0.612155i	$0.790303\pi$
$522$		0			0
$523$	−11752.0			−0.982557			−0.491279	−	0.871003i	$-0.663470\pi$
$523$	−11752.0			−0.982557			−0.491279	+	0.871003i	$0.663470\pi$
$524$		0			0
$525$	−2762.06	−	455.457i	−0.229612	−	0.0378624i
$526$		0			0
$527$		−	3590.24i		−	0.296762i
$528$		0			0
$529$	1231.83			0.101243
$530$		0			0
$531$	−4222.18			−0.345061
$532$		0			0
$533$	−15705.5			−1.27633
$534$		0			0
$535$	7112.94			0.574803
$536$		0			0
$537$		−	7284.99i		−	0.585420i
$538$		0			0
$539$	−485.829	+	1433.07i	−0.0388240	+	0.114521i
$540$		0			0
$541$	6568.82			0.522025			0.261013	−	0.965335i	$-0.415944\pi$
$541$	6568.82			0.522025			0.261013	+	0.965335i	$0.415944\pi$
$542$		0			0
$543$			1831.84i			0.144773i
$544$		0			0
$545$		−	6410.81i		−	0.503869i
$546$		0			0
$547$		−	6138.75i		−	0.479843i	−0.970792	−	0.239921i	$-0.922878\pi$
$547$		−	6138.75i		−	0.479843i	0.970792	−	0.239921i	$-0.0771217\pi$
$548$		0			0
$549$		−	197.667i		−	0.0153665i
$550$		0			0
$551$	7535.72			0.582636
$552$		0			0
$553$	−2663.52	+	16152.5i	−0.204818	+	1.24209i
$554$		0			0
$555$			11034.0i			0.843904i
$556$		0			0
$557$	−945.128			−0.0718965			−0.0359483	−	0.999354i	$-0.511445\pi$
$557$	−945.128			−0.0718965			−0.0359483	+	0.999354i	$0.511445\pi$
$558$		0			0
$559$	10343.1			0.782587
$560$		0			0
$561$	−1727.61			−0.130017
$562$		0			0
$563$	9503.47			0.711410			0.355705	−	0.934598i	$-0.384241\pi$
$563$	9503.47			0.711410			0.355705	+	0.934598i	$0.384241\pi$
$564$		0			0
$565$			520.669i			0.0387694i
$566$		0			0
$567$	−1480.15	−	244.074i	−0.109631	−	0.0180778i
$568$		0			0
$569$	19196.7			1.41435			0.707177	−	0.707037i	$-0.249968\pi$
$569$	19196.7			1.41435			0.707177	+	0.707037i	$0.249968\pi$
$570$		0			0
$571$			18073.8i			1.32463i	0.749225	+	0.662316i	$0.230426\pi$
$571$			18073.8i			1.32463i	−0.749225	+	0.662316i	$0.769574\pi$
$572$		0			0
$573$		−	4736.96i		−	0.345357i
$574$		0			0
$575$		−	5268.70i		−	0.382122i
$576$		0			0
$577$			996.513i			0.0718984i	0.999354	+	0.0359492i	$0.0114455\pi$
$577$			996.513i			0.0718984i	−0.999354	+	0.0359492i	$0.988555\pi$
$578$		0			0
$579$	4575.56			0.328418
$580$		0			0
$581$	−13510.8	−	2227.90i	−0.964757	−	0.159086i
$582$		0			0
$583$		−	3251.30i		−	0.230969i
$584$		0			0
$585$	3676.84			0.259861
$586$		0			0
$587$	−14089.9			−0.990722			−0.495361	−	0.868687i	$-0.664964\pi$
$587$	−14089.9			−0.990722			−0.495361	+	0.868687i	$0.664964\pi$
$588$		0			0
$589$	1232.59			0.0862271
$590$		0			0
$591$	−10678.9			−0.743269
$592$		0			0
$593$		−	296.425i		−	0.0205274i	−0.999947	−	0.0102637i	$-0.996733\pi$
$593$		−	296.425i		−	0.0205274i	0.999947	−	0.0102637i	$-0.00326709\pi$
$594$		0			0
$595$	20604.7	+	3397.66i	1.41968	+	0.234102i
$596$		0			0
$597$	−782.553			−0.0536478
$598$		0			0
$599$			15062.7i			1.02745i	0.857954	+	0.513726i	$0.171735\pi$
$599$			15062.7i			1.02745i	−0.857954	+	0.513726i	$0.828265\pi$
$600$		0			0
$601$		−	14616.3i		−	0.992035i	−0.868312	−	0.496018i	$-0.834795\pi$
$601$		−	14616.3i		−	0.992035i	0.868312	−	0.496018i	$-0.165205\pi$
$602$		0			0
$603$		−	3290.74i		−	0.222238i
$604$		0			0
$605$			11329.2i			0.761315i
$606$		0			0
$607$	−20947.6			−1.40072			−0.700359	−	0.713790i	$-0.746977\pi$
$607$	−20947.6			−1.40072			−0.700359	+	0.713790i	$0.746977\pi$
$608$		0			0
$609$	9218.20	+	1520.06i	0.613366	+	0.101143i
$610$		0			0
$611$			18472.3i			1.22310i
$612$		0			0
$613$	−13224.6			−0.871346			−0.435673	−	0.900105i	$-0.643490\pi$
$613$	−13224.6			−0.871346			−0.435673	+	0.900105i	$0.643490\pi$
$614$		0			0
$615$	8605.48			0.564238
$616$		0			0
$617$	9029.28			0.589149			0.294575	−	0.955628i	$-0.404822\pi$
$617$	9029.28			0.589149			0.294575	+	0.955628i	$0.404822\pi$
$618$		0			0
$619$	22946.3			1.48997			0.744984	−	0.667082i	$-0.232457\pi$
$619$	22946.3			1.48997			0.744984	+	0.667082i	$0.232457\pi$
$620$		0			0
$621$		−	2823.43i		−	0.182448i
$622$		0			0
$623$	200.307	−	1214.74i	0.0128815	−	0.0781179i
$624$		0			0
$625$	−6788.50			−0.434464
$626$		0			0
$627$		−	593.115i		−	0.0377779i
$628$		0			0
$629$			55580.5i			3.52328i
$630$		0			0
$631$			8205.29i			0.517666i	0.965922	+	0.258833i	$0.0833380\pi$
$631$			8205.29i			0.517666i	−0.965922	+	0.258833i	$0.916662\pi$
$632$		0			0
$633$		−	15825.7i		−	0.993703i
$634$		0			0
$635$	18035.3			1.12710
$636$		0			0
$637$	5208.38	−	15363.3i	0.323961	−	0.955601i
$638$		0			0
$639$		−	5580.01i		−	0.345449i
$640$		0			0
$641$	−15372.8			−0.947254			−0.473627	−	0.880725i	$-0.657056\pi$
$641$	−15372.8			−0.947254			−0.473627	+	0.880725i	$0.657056\pi$
$642$		0			0
$643$	−15543.5			−0.953309			−0.476654	−	0.879091i	$-0.658151\pi$
$643$	−15543.5			−0.953309			−0.476654	+	0.879091i	$0.658151\pi$
$644$		0			0
$645$	−5667.26			−0.345966
$646$		0			0
$647$	−653.815			−0.0397282			−0.0198641	−	0.999803i	$-0.506323\pi$
$647$	−653.815			−0.0397282			−0.0198641	+	0.999803i	$0.506323\pi$
$648$		0			0
$649$		−	2069.62i		−	0.125177i
$650$		0			0
$651$	1507.78	+	248.629i	0.0907751	+	0.0149686i
$652$		0			0
$653$	−4675.33			−0.280183			−0.140092	−	0.990139i	$-0.544740\pi$
$653$	−4675.33			−0.280183			−0.140092	+	0.990139i	$0.544740\pi$
$654$		0			0
$655$		−	3149.28i		−	0.187867i
$656$		0			0
$657$		−	8442.35i		−	0.501320i
$658$		0			0
$659$			13350.9i			0.789190i	0.918855	+	0.394595i	$0.129115\pi$
$659$			13350.9i			0.789190i	−0.918855	+	0.394595i	$0.870885\pi$
$660$		0			0
$661$		−	3008.54i		−	0.177033i	−0.996075	−	0.0885164i	$-0.971787\pi$
$661$		−	3008.54i		−	0.177033i	0.996075	−	0.0885164i	$-0.0282126\pi$
$662$		0			0
$663$	18521.0			1.08491
$664$		0			0
$665$	−1166.47	+	7073.91i	−0.0680207	+	0.412503i
$666$		0			0
$667$			17583.9i			1.02077i
$668$		0			0
$669$	6754.41			0.390345
$670$		0			0
$671$	96.8920			0.00557448
$672$		0			0
$673$	8488.74			0.486207			0.243103	−	0.970000i	$-0.421835\pi$
$673$	8488.74			0.486207			0.243103	+	0.970000i	$0.421835\pi$
$674$		0			0
$675$	1360.36			0.0775709
$676$		0			0
$677$			16663.1i			0.945959i	0.881073	+	0.472979i	$0.156821\pi$
$677$			16663.1i			0.945959i	−0.881073	+	0.472979i	$0.843179\pi$
$678$		0			0
$679$	−1065.20	+	6459.77i	−0.0602042	+	0.365100i
$680$		0			0
$681$	7407.16			0.416803
$682$		0			0
$683$			9688.30i			0.542771i	0.962471	+	0.271385i	$0.0874818\pi$
$683$			9688.30i			0.542771i	−0.962471	+	0.271385i	$0.912518\pi$
$684$		0			0
$685$			12605.6i			0.703116i
$686$		0			0
$687$			4135.51i			0.229664i
$688$		0			0
$689$			34855.8i			1.92729i
$690$		0			0
$691$	−28730.2			−1.58169			−0.790846	−	0.612016i	$-0.790359\pi$
$691$	−28730.2			−1.58169			−0.790846	+	0.612016i	$0.790359\pi$
$692$		0			0
$693$	119.640	−	725.538i	0.00655805	−	0.0397705i
$694$		0			0
$695$		−	24014.1i		−	1.31066i
$696$		0			0
$697$	43347.6			2.35568
$698$		0			0
$699$	−16967.2			−0.918107
$700$		0			0
$701$	31392.0			1.69139			0.845693	−	0.533670i	$-0.179188\pi$
$701$	31392.0			1.69139			0.845693	+	0.533670i	$0.179188\pi$
$702$		0			0
$703$	−19081.7			−1.02372
$704$		0			0
$705$		−	10121.5i		−	0.540706i
$706$		0			0
$707$	−2388.65	+	14485.6i	−0.127064	+	0.770564i
$708$		0			0
$709$	−8916.26			−0.472295			−0.236148	−	0.971717i	$-0.575885\pi$
$709$	−8916.26			−0.472295			−0.236148	+	0.971717i	$0.575885\pi$
$710$		0			0
$711$		−	7955.40i		−	0.419621i
$712$		0			0
$713$			2876.13i			0.151069i
$714$		0			0
$715$			1802.31i			0.0942691i
$716$		0			0
$717$		−	3331.91i		−	0.173546i
$718$		0			0
$719$	22265.8			1.15490			0.577451	−	0.816425i	$-0.304047\pi$
$719$	22265.8			1.15490			0.577451	+	0.816425i	$0.304047\pi$
$720$		0			0
$721$	−23570.4	−	3886.70i	−1.21748	−	0.200760i
$722$		0			0
$723$		−	1427.30i		−	0.0734187i
$724$		0			0
$725$	−8472.16			−0.433997
$726$		0			0
$727$	14442.9			0.736804			0.368402	−	0.929667i	$-0.379905\pi$
$727$	14442.9			0.736804			0.368402	+	0.929667i	$0.379905\pi$
$728$		0			0
$729$	729.000			0.0370370
$730$		0			0
$731$	−28547.2			−1.44440
$732$		0			0
$733$		−	4384.61i		−	0.220940i	−0.993879	−	0.110470i	$-0.964764\pi$
$733$		−	4384.61i		−	0.220940i	0.993879	−	0.110470i	$-0.0352356\pi$
$734$		0			0
$735$	−2853.81	+	8417.99i	−0.143217	+	0.422452i
$736$		0			0
$737$	1613.05			0.0806207
$738$		0			0
$739$		−	12361.2i		−	0.615308i	−0.951498	−	0.307654i	$-0.900456\pi$
$739$		−	12361.2i		−	0.615308i	0.951498	−	0.307654i	$-0.0995440\pi$
$740$		0			0
$741$			6358.55i			0.315232i
$742$		0			0
$743$			31174.3i			1.53926i	0.638487	+	0.769632i	$0.279560\pi$
$743$			31174.3i			1.53926i	−0.638487	+	0.769632i	$0.720440\pi$
$744$		0			0
$745$			4988.03i			0.245298i
$746$		0			0
$747$	6654.31			0.325929
$748$		0			0
$749$	−2481.24	+	15047.1i	−0.121045	+	0.734060i
$750$		0			0
$751$			37794.0i			1.83638i	0.396139	+	0.918190i	$0.370350\pi$
$751$			37794.0i			1.83638i	−0.396139	+	0.918190i	$0.629650\pi$
$752$		0			0
$753$	−8802.80			−0.426019
$754$		0			0
$755$	14470.3			0.697523
$756$		0			0
$757$	10155.7			0.487604			0.243802	−	0.969825i	$-0.421605\pi$
$757$	10155.7			0.487604			0.243802	+	0.969825i	$0.421605\pi$
$758$		0			0
$759$	1383.98			0.0661863
$760$		0			0
$761$		−	23997.8i		−	1.14313i	−0.820558	−	0.571564i	$-0.806337\pi$
$761$		−	23997.8i		−	1.14313i	0.820558	−	0.571564i	$-0.193663\pi$
$762$		0			0
$763$	13561.8	+	2236.31i	0.643473	+	0.106107i
$764$		0			0
$765$	−10148.2			−0.479617
$766$		0			0
$767$			22187.6i			1.04452i
$768$		0			0
$769$			35632.1i			1.67090i	0.549563	+	0.835452i	$0.314794\pi$
$769$			35632.1i			1.67090i	−0.549563	+	0.835452i	$0.685206\pi$
$770$		0			0
$771$			16057.1i			0.750040i
$772$		0			0
$773$		−	34767.5i		−	1.61772i	−0.587998	−	0.808862i	$-0.700084\pi$
$773$		−	34767.5i		−	1.61772i	0.587998	−	0.808862i	$-0.299916\pi$
$774$		0			0
$775$	−1385.75			−0.0642294
$776$		0			0
$777$	−23342.0	−	3849.03i	−1.07772	−	0.177713i
$778$		0			0
$779$			14881.9i			0.684467i
$780$		0			0
$781$	2735.20			0.125318
$782$		0			0
$783$	−4540.12			−0.207217
$784$		0			0
$785$	−21338.5			−0.970197
$786$		0			0
$787$	42534.2			1.92653			0.963267	−	0.268546i	$-0.0865430\pi$
$787$	42534.2			1.92653			0.963267	+	0.268546i	$0.0865430\pi$
$788$		0			0
$789$		−	1578.63i		−	0.0712305i
$790$		0			0
$791$	−1101.45	−	181.627i	−0.0495110	−	0.00816425i
$792$		0			0
$793$	−1038.74			−0.0465154
$794$		0			0
$795$		−	19098.5i		−	0.852016i
$796$		0			0
$797$			23504.9i			1.04465i	0.852747	+	0.522324i	$0.174935\pi$
$797$			23504.9i			1.04465i	−0.852747	+	0.522324i	$0.825065\pi$
$798$		0			0
$799$		−	50984.1i		−	2.25743i
$800$		0			0
$801$			598.279i			0.0263910i
$802$		0			0
$803$	4138.26			0.181863
$804$		0			0
$805$	−16506.3	−	2721.86i	−0.722698	−	0.119171i
$806$		0			0
$807$		−	9684.47i		−	0.422441i
$808$		0			0
$809$	31194.0			1.35565			0.677827	−	0.735222i	$-0.262922\pi$
$809$	31194.0			1.35565			0.677827	+	0.735222i	$0.262922\pi$
$810$		0			0
$811$	25148.9			1.08890			0.544450	−	0.838793i	$-0.316738\pi$
$811$	25148.9			1.08890			0.544450	+	0.838793i	$0.316738\pi$
$812$		0			0
$813$	23594.7			1.01784
$814$		0			0
$815$	3175.47			0.136481
$816$		0			0
$817$		−	9800.68i		−	0.419685i
$818$		0			0
$819$	−1282.61	+	7778.20i	−0.0547227	+	0.331859i
$820$		0			0
$821$	−13930.5			−0.592177			−0.296088	−	0.955161i	$-0.595682\pi$
$821$	−13930.5			−0.592177			−0.296088	+	0.955161i	$0.595682\pi$
$822$		0			0
$823$			17913.7i			0.758725i	0.925248	+	0.379363i	$0.123857\pi$
$823$			17913.7i			0.758725i	−0.925248	+	0.379363i	$0.876143\pi$
$824$		0			0
$825$			666.820i			0.0281402i
$826$		0			0
$827$		−	9779.57i		−	0.411208i	−0.978635	−	0.205604i	$-0.934084\pi$
$827$		−	9779.57i		−	0.411208i	0.978635	−	0.205604i	$-0.0659159\pi$
$828$		0			0
$829$		−	39697.2i		−	1.66314i	−0.555421	−	0.831570i	$-0.687443\pi$
$829$		−	39697.2i		−	1.66314i	0.555421	−	0.831570i	$-0.312557\pi$
$830$		0			0
$831$	9703.50			0.405067
$832$		0			0
$833$	−14375.2	+	42403.2i	−0.597926	+	1.76372i
$834$		0			0
$835$		−	21903.0i		−	0.907767i
$836$		0			0
$837$	−742.607			−0.0306670
$838$		0			0
$839$	36208.2			1.48992			0.744962	−	0.667107i	$-0.232468\pi$
$839$	36208.2			1.48992			0.744962	+	0.667107i	$0.232468\pi$
$840$		0			0
$841$	3886.26			0.159345
$842$		0			0
$843$	−1102.11			−0.0450281
$844$		0			0
$845$		−	343.946i		−	0.0140025i
$846$		0			0
$847$	−23966.4	−	3952.00i	−0.972248	−	0.160321i
$848$		0			0
$849$	−4448.47			−0.179824
$850$		0			0
$851$		−	44525.4i		−	1.79355i
$852$		0			0
$853$		−	6875.81i		−	0.275994i	−0.990433	−	0.137997i	$-0.955934\pi$
$853$		−	6875.81i		−	0.275994i	0.990433	−	0.137997i	$-0.0440665\pi$
$854$		0			0
$855$		−	3484.02i		−	0.139358i
$856$		0			0
$857$		−	6319.67i		−	0.251897i	−0.992037	−	0.125949i	$-0.959803\pi$
$857$		−	6319.67i		−	0.251897i	0.992037	−	0.125949i	$-0.0401974\pi$
$858$		0			0
$859$	16033.2			0.636841			0.318421	−	0.947949i	$-0.396848\pi$
$859$	16033.2			0.636841			0.318421	+	0.947949i	$0.396848\pi$
$860$		0			0
$861$	−3001.89	+	18204.6i	−0.118820	+	0.720569i
$862$		0			0
$863$			20690.6i			0.816125i	0.912954	+	0.408063i	$0.133795\pi$
$863$			20690.6i			0.816125i	−0.912954	+	0.408063i	$0.866205\pi$
$864$		0			0
$865$	735.475			0.0289097
$866$		0			0
$867$	−36379.4			−1.42504
$868$		0			0
$869$	3899.56			0.152225
$870$		0			0
$871$	−17292.9			−0.672728
$872$		0			0
$873$		−	3181.54i		−	0.123344i
$874$		0			0
$875$	4565.01	−	27683.9i	0.176372	−	1.06959i
$876$		0			0
$877$	32212.1			1.24028			0.620140	−	0.784491i	$-0.287076\pi$
$877$	32212.1			1.24028			0.620140	+	0.784491i	$0.287076\pi$
$878$		0			0
$879$			9439.90i			0.362230i
$880$		0			0
$881$		−	14473.8i		−	0.553503i	−0.960942	−	0.276751i	$-0.910742\pi$
$881$		−	14473.8i		−	0.553503i	0.960942	−	0.276751i	$-0.0892578\pi$
$882$		0			0
$883$		−	14958.3i		−	0.570087i	−0.958515	−	0.285044i	$-0.907992\pi$
$883$		−	14958.3i		−	0.570087i	0.958515	−	0.285044i	$-0.0920081\pi$
$884$		0			0
$885$		−	12157.2i		−	0.461761i
$886$		0			0
$887$	281.770			0.0106662			0.00533310	−	0.999986i	$-0.498302\pi$
$887$	281.770			0.0106662			0.00533310	+	0.999986i	$0.498302\pi$
$888$		0			0
$889$	−6291.34	+	38153.0i	−0.237351	+	1.43938i
$890$		0			0
$891$			357.340i			0.0134358i
$892$		0			0
$893$	17503.6			0.655920
$894$		0			0
$895$	20976.1			0.783411
$896$		0			0
$897$	−14837.1			−0.552282
$898$		0			0
$899$	4624.86			0.171577
$900$		0			0
$901$		−	96202.9i		−	3.55714i
$902$		0			0
$903$	1976.93	−	11988.9i	0.0728552	−	0.441821i
$904$		0			0
$905$	−5274.52			−0.193736
$906$		0			0
$907$			31098.1i			1.13847i	0.822174	+	0.569236i	$0.192761\pi$
$907$			31098.1i			1.13847i	−0.822174	+	0.569236i	$0.807239\pi$
$908$		0			0
$909$		−	7134.42i		−	0.260323i
$910$		0			0
$911$			32095.2i			1.16725i	0.812025	+	0.583623i	$0.198365\pi$
$911$			32095.2i			1.16725i	−0.812025	+	0.583623i	$0.801635\pi$
$912$		0			0
$913$			3261.80i			0.118236i
$914$		0			0
$915$	569.153			0.0205635
$916$		0			0
$917$	6662.18	+	1098.58i	0.239918	+	0.0395618i
$918$		0			0
$919$		−	30290.4i		−	1.08725i	−0.839327	−	0.543627i	$-0.817051\pi$
$919$		−	30290.4i		−	1.08725i	0.839327	−	0.543627i	$-0.182949\pi$
$920$		0			0
$921$	−18938.6			−0.677578
$922$		0			0
$923$	−29322.9			−1.04569
$924$		0			0
$925$	21452.9			0.762558
$926$		0			0
$927$	11608.8			0.411309
$928$		0			0
$929$		−	26721.7i		−	0.943715i	−0.881675	−	0.471858i	$-0.843584\pi$
$929$		−	26721.7i		−	0.943715i	0.881675	−	0.471858i	$-0.156416\pi$
$930$		0			0
$931$	−14557.7	−	4935.24i	−0.512468	−	0.173734i
$932$		0			0
$933$	−8333.92			−0.292433
$934$		0			0
$935$		−	4974.40i		−	0.173990i
$936$		0			0
$937$		−	48521.7i		−	1.69171i	−0.533411	−	0.845856i	$-0.679090\pi$
$937$		−	48521.7i		−	1.69171i	0.533411	−	0.845856i	$-0.320910\pi$
$938$		0			0
$939$		−	25107.2i		−	0.872570i
$940$		0			0
$941$			21786.5i			0.754749i	0.926061	+	0.377374i	$0.123173\pi$
$941$			21786.5i			0.754749i	−0.926061	+	0.377374i	$0.876827\pi$
$942$		0			0
$943$	−34725.6			−1.19918
$944$		0			0
$945$	702.775	−	4261.89i	0.0241918	−	0.146708i
$946$		0			0
$947$		−	483.014i		−	0.0165743i	−0.999966	−	0.00828715i	$-0.997362\pi$
$947$		−	483.014i		−	0.0165743i	0.999966	−	0.00828715i	$-0.00263791\pi$
$948$		0			0
$949$	−44364.6			−1.51753
$950$		0			0
$951$	−5350.97			−0.182458
$952$		0			0
$953$	38751.5			1.31719			0.658596	−	0.752496i	$-0.271151\pi$
$953$	38751.5			1.31719			0.658596	+	0.752496i	$0.271151\pi$
$954$		0			0
$955$	13639.4			0.462158
$956$		0			0
$957$		−	2225.47i		−	0.0751715i
$958$		0			0
$959$	−26666.6	−	4397.26i	−0.897924	−	0.148066i
$960$		0			0
$961$	−29034.5			−0.974607
$962$		0			0
$963$		−	7410.97i		−	0.247991i
$964$		0			0
$965$			13174.7i			0.439490i
$966$		0			0
$967$			37370.5i			1.24276i	0.783508	+	0.621382i	$0.213429\pi$
$967$			37370.5i			1.24276i	−0.783508	+	0.621382i	$0.786571\pi$
$968$		0			0
$969$		−	17549.7i		−	0.581815i
$970$		0			0
$971$	−13582.6			−0.448903			−0.224452	−	0.974485i	$-0.572059\pi$
$971$	−13582.6			−0.448903			−0.224452	+	0.974485i	$0.572059\pi$
$972$		0			0
$973$	50800.8	+	8376.93i	1.67379	+	0.276004i
$974$		0			0
$975$		−	7148.70i		−	0.234812i
$976$		0			0
$977$	−42262.6			−1.38393			−0.691965	−	0.721931i	$-0.743255\pi$
$977$	−42262.6			−1.38393			−0.691965	+	0.721931i	$0.743255\pi$
$978$		0			0
$979$	−293.263			−0.00957379
$980$		0			0
$981$	−6679.42			−0.217388
$982$		0			0
$983$	−45606.3			−1.47977			−0.739886	−	0.672733i	$-0.765120\pi$
$983$	−45606.3			−1.47977			−0.739886	+	0.672733i	$0.765120\pi$
$984$		0			0
$985$		−	30748.4i		−	0.994645i
$986$		0			0
$987$	21411.6	+	3530.72i	0.690516	+	0.113864i
$988$		0			0
$989$	22869.0			0.735281
$990$		0			0
$991$		−	19857.9i		−	0.636535i	−0.948001	−	0.318267i	$-0.896899\pi$
$991$		−	19857.9i		−	0.636535i	0.948001	−	0.318267i	$-0.103101\pi$
$992$		0			0
$993$		−	24930.7i		−	0.796730i
$994$		0			0
$995$		−	2253.25i		−	0.0717917i
$996$		0			0
$997$			18857.1i			0.599008i	0.954095	+	0.299504i	$0.0968212\pi$
$997$			18857.1i			0.599008i	−0.954095	+	0.299504i	$0.903179\pi$
$998$		0			0
$999$	11496.3			0.364091

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.4.b.j.895.17		24
4.3	odd	2		1344.4.b.i.895.17		24
7.6	odd	2		1344.4.b.i.895.8		24
8.3	odd	2		672.4.b.b.223.8	yes	24
8.5	even	2		672.4.b.a.223.8	✓	24
28.27	even	2	inner	1344.4.b.j.895.8		24
56.13	odd	2		672.4.b.b.223.17	yes	24
56.27	even	2		672.4.b.a.223.17	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.4.b.a.223.8	✓	24	8.5	even	2
672.4.b.a.223.17	yes	24	56.27	even	2
672.4.b.b.223.8	yes	24	8.3	odd	2
672.4.b.b.223.17	yes	24	56.13	odd	2
1344.4.b.i.895.8		24	7.6	odd	2
1344.4.b.i.895.17		24	4.3	odd	2
1344.4.b.j.895.8		24	28.27	even	2	inner
1344.4.b.j.895.17		24	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)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} +8.63807i q^{5} +(-18.2735 - 3.01325i) q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q+3.00000 q^{3} +8.63807i q^{5} +(-18.2735 - 3.01325i) q^{7} +9.00000 q^{9} +4.41160i q^{11} -47.2950i q^{13} +25.9142i q^{15} +130.535i q^{17} -44.8148 q^{19} +(-54.8205 - 9.03976i) q^{21} -104.571i q^{23} +50.3838 q^{25} +27.0000 q^{27} -168.152 q^{29} -27.5040 q^{31} +13.2348i q^{33} +(26.0287 - 157.848i) q^{35} +425.789 q^{37} -141.885i q^{39} -332.076i q^{41} +218.693i q^{43} +77.7426i q^{45} -390.577 q^{47} +(324.841 + 110.125i) q^{49} +391.606i q^{51} -736.988 q^{53} -38.1077 q^{55} -134.444 q^{57} -469.132 q^{59} -21.9630i q^{61} +(-164.461 - 27.1193i) q^{63} +408.537 q^{65} -365.638i q^{67} -313.714i q^{69} -620.001i q^{71} -938.039i q^{73} +151.151 q^{75} +(13.2933 - 80.6154i) q^{77} -883.933i q^{79} +81.0000 q^{81} +739.368 q^{83} -1127.57 q^{85} -504.457 q^{87} +66.4755i q^{89} +(-142.512 + 864.245i) q^{91} -82.5119 q^{93} -387.113i q^{95} -353.505i q^{97} +39.7044i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 72 q^{3} - 20 q^{7} + 216 q^{9}+O(q^{10}) \) 24 * q + 72 * q^3 - 20 * q^7 + 216 * q^9	\( 24 q + 72 q^{3} - 20 q^{7} + 216 q^{9} - 56 q^{19} - 60 q^{21} - 432 q^{25} + 648 q^{27} + 464 q^{31} + 568 q^{35} - 504 q^{37} + 560 q^{47} - 128 q^{49} + 784 q^{53} + 424 q^{55} - 168 q^{57} + 800 q^{59} - 180 q^{63} + 560 q^{65} - 1296 q^{75} + 1568 q^{77} + 1944 q^{81} + 1936 q^{83} - 3000 q^{85} - 496 q^{91} + 1392 q^{93}+O(q^{100}) \) 24 * q + 72 * q^3 - 20 * q^7 + 216 * q^9 - 56 * q^19 - 60 * q^21 - 432 * q^25 + 648 * q^27 + 464 * q^31 + 568 * q^35 - 504 * q^37 + 560 * q^47 - 128 * q^49 + 784 * q^53 + 424 * q^55 - 168 * q^57 + 800 * q^59 - 180 * q^63 + 560 * q^65 - 1296 * q^75 + 1568 * q^77 + 1944 * q^81 + 1936 * q^83 - 3000 * q^85 - 496 * q^91 + 1392 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more