LMFDB - Embedded newform 1344.4.b.f.895.3

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:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 158x^{6} + 8461x^{4} + 180672x^{2} + 1232100 $ x^8 + 158x^6 + 8461x^4 + 180672*x^2 + 1232100
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{15}\cdot 3^{3} $
Twist minimal:	no (minimal twist has level 336)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			895.3
Root			$8.49618i$ of defining polynomial
Character	$\chi$	$=$	1344.895
Dual form			1344.4.b.f.895.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} -7.52281i q^{5} +(-4.86375 + 17.8702i) q^{7} +9.00000 q^{9} +O(q^{10})$	$q+3.00000 q^{3} -7.52281i q^{5} +(-4.86375 + 17.8702i) q^{7} +9.00000 q^{9} -29.4068i q^{11} +5.35686i q^{13} -22.5684i q^{15} -66.3364i q^{17} +22.3894 q^{19} +(-14.5913 + 53.6106i) q^{21} +33.9994i q^{23} +68.4073 q^{25} +27.0000 q^{27} +133.266 q^{29} -323.973 q^{31} -88.2204i q^{33} +(134.434 + 36.5891i) q^{35} -120.395 q^{37} +16.0706i q^{39} -140.290i q^{41} -19.0522i q^{43} -67.7053i q^{45} -376.658 q^{47} +(-295.688 - 173.832i) q^{49} -199.009i q^{51} +441.895 q^{53} -221.222 q^{55} +67.1681 q^{57} +241.815 q^{59} +130.894i q^{61} +(-43.7738 + 160.832i) q^{63} +40.2987 q^{65} -627.240i q^{67} +101.998i q^{69} -808.143i q^{71} -417.527i q^{73} +205.222 q^{75} +(525.505 + 143.027i) q^{77} +214.825i q^{79} +81.0000 q^{81} +639.640 q^{83} -499.037 q^{85} +399.798 q^{87} -686.995i q^{89} +(-95.7281 - 26.0545i) q^{91} -971.918 q^{93} -168.431i q^{95} -103.684i q^{97} -264.661i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 24 q^{3} - 4 q^{7} + 72 q^{9}+O(q^{10}) $ 8 * q + 24 * q^3 - 4 * q^7 + 72 * q^9	$ 8 q + 24 q^{3} - 4 q^{7} + 72 q^{9} + 56 q^{19} - 12 q^{21} - 656 q^{25} + 216 q^{27} - 240 q^{29} + 320 q^{31} - 600 q^{35} - 392 q^{37} - 816 q^{47} - 16 q^{49} - 288 q^{53} - 456 q^{55} + 168 q^{57} + 1824 q^{59} - 36 q^{63} - 816 q^{65} - 1968 q^{75} + 2064 q^{77} + 648 q^{81} - 1680 q^{83} - 2568 q^{85} - 720 q^{87} + 864 q^{91} + 960 q^{93}+O(q^{100}) $ 8 * q + 24 * q^3 - 4 * q^7 + 72 * q^9 + 56 * q^19 - 12 * q^21 - 656 * q^25 + 216 * q^27 - 240 * q^29 + 320 * q^31 - 600 * q^35 - 392 * q^37 - 816 * q^47 - 16 * q^49 - 288 * q^53 - 456 * q^55 + 168 * q^57 + 1824 * q^59 - 36 * q^63 - 816 * q^65 - 1968 * q^75 + 2064 * q^77 + 648 * q^81 - 1680 * q^83 - 2568 * q^85 - 720 * q^87 + 864 * q^91 + 960 * 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
$5$		−	7.52281i		−	0.672861i	−0.941708	−	0.336430i	$-0.890780\pi$
$5$		−	7.52281i		−	0.672861i	0.941708	−	0.336430i	$-0.109220\pi$
$6$		0			0
$7$	−4.86375	+	17.8702i	−0.262618	+	0.964900i
$7$	−4.86375	+	17.8702i	−0.262618	+	0.964900i
$8$		0			0
$9$	9.00000			0.333333
$10$		0			0
$11$		−	29.4068i		−	0.806044i	−0.915190	−	0.403022i	$-0.867960\pi$
$11$		−	29.4068i		−	0.806044i	0.915190	−	0.403022i	$-0.132040\pi$
$12$		0			0
$13$			5.35686i			0.114287i	0.998366	+	0.0571433i	$0.0181992\pi$
$13$			5.35686i			0.114287i	−0.998366	+	0.0571433i	$0.981801\pi$
$14$		0			0
$15$		−	22.5684i		−	0.388476i
$16$		0			0
$17$		−	66.3364i		−	0.946408i	−0.880953	−	0.473204i	$-0.843097\pi$
$17$		−	66.3364i		−	0.946408i	0.880953	−	0.473204i	$-0.156903\pi$
$18$		0			0
$19$	22.3894			0.270341			0.135170	−	0.990822i	$-0.456842\pi$
$19$	22.3894			0.270341			0.135170	+	0.990822i	$0.456842\pi$
$20$		0			0
$21$	−14.5913	+	53.6106i	−0.151623	+	0.557085i
$22$		0			0
$23$			33.9994i			0.308233i	0.988053	+	0.154117i	$0.0492532\pi$
$23$			33.9994i			0.308233i	−0.988053	+	0.154117i	$0.950747\pi$
$24$		0			0
$25$	68.4073			0.547258
$26$		0			0
$27$	27.0000			0.192450
$28$		0			0
$29$	133.266			0.853341			0.426671	−	0.904407i	$-0.359686\pi$
$29$	133.266			0.853341			0.426671	+	0.904407i	$0.359686\pi$
$30$		0			0
$31$	−323.973			−1.87701			−0.938504	−	0.345270i	$-0.887788\pi$
$31$	−323.973			−1.87701			−0.938504	+	0.345270i	$0.887788\pi$
$32$		0			0
$33$		−	88.2204i		−	0.465370i
$34$		0			0
$35$	134.434	+	36.5891i	0.649243	+	0.176705i
$36$		0			0
$37$	−120.395			−0.534940			−0.267470	−	0.963566i	$-0.586188\pi$
$37$	−120.395			−0.534940			−0.267470	+	0.963566i	$0.586188\pi$
$38$		0			0
$39$			16.0706i			0.0659834i
$40$		0			0
$41$		−	140.290i		−	0.534379i	−0.963644	−	0.267190i	$-0.913905\pi$
$41$		−	140.290i		−	0.534379i	0.963644	−	0.267190i	$-0.0860950\pi$
$42$		0			0
$43$		−	19.0522i		−	0.0675681i	−0.999429	−	0.0337841i	$-0.989244\pi$
$43$		−	19.0522i		−	0.0675681i	0.999429	−	0.0337841i	$-0.0107558\pi$
$44$		0			0
$45$		−	67.7053i		−	0.224287i
$46$		0			0
$47$	−376.658			−1.16896			−0.584481	−	0.811407i	$-0.698702\pi$
$47$	−376.658			−1.16896			−0.584481	+	0.811407i	$0.698702\pi$
$48$		0			0
$49$	−295.688	−	173.832i	−0.862064	−	0.506800i
$50$		0			0
$51$		−	199.009i		−	0.546409i
$52$		0			0
$53$	441.895			1.14526			0.572632	−	0.819813i	$-0.305922\pi$
$53$	441.895			1.14526			0.572632	+	0.819813i	$0.305922\pi$
$54$		0			0
$55$	−221.222			−0.542356
$56$		0			0
$57$	67.1681			0.156081
$58$		0			0
$59$	241.815			0.533586			0.266793	−	0.963754i	$-0.414036\pi$
$59$	241.815			0.533586			0.266793	+	0.963754i	$0.414036\pi$
$60$		0			0
$61$			130.894i			0.274743i	0.990520	+	0.137371i	$0.0438654\pi$
$61$			130.894i			0.274743i	−0.990520	+	0.137371i	$0.956135\pi$
$62$		0			0
$63$	−43.7738	+	160.832i	−0.0875394	+	0.321633i
$64$		0			0
$65$	40.2987			0.0768990
$66$		0			0
$67$		−	627.240i		−	1.14373i	−0.820349	−	0.571863i	$-0.806221\pi$
$67$		−	627.240i		−	1.14373i	0.820349	−	0.571863i	$-0.193779\pi$
$68$		0			0
$69$			101.998i			0.177959i
$70$		0			0
$71$		−	808.143i		−	1.35083i	−0.737438	−	0.675415i	$-0.763965\pi$
$71$		−	808.143i		−	1.35083i	0.737438	−	0.675415i	$-0.236035\pi$
$72$		0			0
$73$		−	417.527i		−	0.669423i	−0.942321	−	0.334711i	$-0.891361\pi$
$73$		−	417.527i		−	0.669423i	0.942321	−	0.334711i	$-0.108639\pi$
$74$		0			0
$75$	205.222			0.315960
$76$		0			0
$77$	525.505	+	143.027i	0.777752	+	0.211682i
$78$		0			0
$79$			214.825i			0.305946i	0.988230	+	0.152973i	$0.0488847\pi$
$79$			214.825i			0.305946i	−0.988230	+	0.152973i	$0.951115\pi$
$80$		0			0
$81$	81.0000			0.111111
$82$		0			0
$83$	639.640			0.845899			0.422949	−	0.906153i	$-0.360995\pi$
$83$	639.640			0.845899			0.422949	+	0.906153i	$0.360995\pi$
$84$		0			0
$85$	−499.037			−0.636801
$86$		0			0
$87$	399.798			0.492677
$88$		0			0
$89$		−	686.995i		−	0.818217i	−0.912486	−	0.409109i	$-0.865840\pi$
$89$		−	686.995i		−	0.818217i	0.912486	−	0.409109i	$-0.134160\pi$
$90$		0			0
$91$	−95.7281	−	26.0545i	−0.110275	−	0.0300137i
$92$		0			0
$93$	−971.918			−1.08369
$94$		0			0
$95$		−	168.431i		−	0.181902i
$96$		0			0
$97$		−	103.684i		−	0.108532i	−0.998527	−	0.0542658i	$-0.982718\pi$
$97$		−	103.684i		−	0.108532i	0.998527	−	0.0542658i	$-0.0172818\pi$
$98$		0			0
$99$		−	264.661i		−	0.268681i
$100$		0			0
$101$		−	1985.67i		−	1.95626i	−0.208000	−	0.978129i	$-0.566695\pi$
$101$		−	1985.67i		−	1.95626i	0.208000	−	0.978129i	$-0.433305\pi$
$102$		0			0
$103$	855.145			0.818058			0.409029	−	0.912521i	$-0.365867\pi$
$103$	855.145			0.818058			0.409029	+	0.912521i	$0.365867\pi$
$104$		0			0
$105$	403.303	+	109.767i	0.374841	+	0.102021i
$106$		0			0
$107$			436.104i			0.394016i	0.980402	+	0.197008i	$0.0631225\pi$
$107$			436.104i			0.394016i	−0.980402	+	0.197008i	$0.936877\pi$
$108$		0			0
$109$	−1424.63			−1.25188			−0.625938	−	0.779873i	$-0.715284\pi$
$109$	−1424.63			−1.25188			−0.625938	+	0.779873i	$0.715284\pi$
$110$		0			0
$111$	−361.184			−0.308848
$112$		0			0
$113$	−1217.43			−1.01351			−0.506753	−	0.862091i	$-0.669154\pi$
$113$	−1217.43			−1.01351			−0.506753	+	0.862091i	$0.669154\pi$
$114$		0			0
$115$	255.771			0.207398
$116$		0			0
$117$			48.2117i			0.0380955i
$118$		0			0
$119$	1185.44	+	322.644i	0.913189	+	0.248544i
$120$		0			0
$121$	466.240			0.350293
$122$		0			0
$123$		−	420.869i		−	0.308524i
$124$		0			0
$125$		−	1454.97i		−	1.04109i
$126$		0			0
$127$		−	1188.92i		−	0.830705i	−0.909660	−	0.415353i	$-0.863658\pi$
$127$		−	1188.92i		−	0.830705i	0.909660	−	0.415353i	$-0.136342\pi$
$128$		0			0
$129$		−	57.1565i		−	0.0390105i
$130$		0			0
$131$	−1652.03			−1.10182			−0.550911	−	0.834564i	$-0.685720\pi$
$131$	−1652.03			−1.10182			−0.550911	+	0.834564i	$0.685720\pi$
$132$		0			0
$133$	−108.896	+	400.102i	−0.0709963	+	0.260852i
$134$		0			0
$135$		−	203.116i		−	0.129492i
$136$		0			0
$137$	−2528.15			−1.57660			−0.788300	−	0.615291i	$-0.789039\pi$
$137$	−2528.15			−1.57660			−0.788300	+	0.615291i	$0.789039\pi$
$138$		0			0
$139$	−2522.86			−1.53947			−0.769734	−	0.638365i	$-0.779611\pi$
$139$	−2522.86			−1.53947			−0.769734	+	0.638365i	$0.779611\pi$
$140$		0			0
$141$	−1129.97			−0.674901
$142$		0			0
$143$	157.528			0.0921200
$144$		0			0
$145$		−	1002.54i		−	0.574180i
$146$		0			0
$147$	−887.063	−	521.497i	−0.497713	−	0.292601i
$148$		0			0
$149$	2456.90			1.35085			0.675425	−	0.737428i	$-0.263960\pi$
$149$	2456.90			1.35085			0.675425	+	0.737428i	$0.263960\pi$
$150$		0			0
$151$			1439.28i			0.775677i	0.921727	+	0.387838i	$0.126778\pi$
$151$			1439.28i			0.775677i	−0.921727	+	0.387838i	$0.873222\pi$
$152$		0			0
$153$		−	597.028i		−	0.315469i
$154$		0			0
$155$			2437.19i			1.26296i
$156$		0			0
$157$		−	3258.27i		−	1.65629i	−0.560512	−	0.828146i	$-0.689396\pi$
$157$		−	3258.27i		−	1.65629i	0.560512	−	0.828146i	$-0.310604\pi$
$158$		0			0
$159$	1325.69			0.661218
$160$		0			0
$161$	−607.576	−	165.365i	−0.297414	−	0.0809477i
$162$		0			0
$163$			1190.87i			0.572246i	0.958193	+	0.286123i	$0.0923666\pi$
$163$			1190.87i			0.572246i	−0.958193	+	0.286123i	$0.907633\pi$
$164$		0			0
$165$	−663.666			−0.313129
$166$		0			0
$167$	502.205			0.232706			0.116353	−	0.993208i	$-0.462880\pi$
$167$	502.205			0.232706			0.116353	+	0.993208i	$0.462880\pi$
$168$		0			0
$169$	2168.30			0.986939
$170$		0			0
$171$	201.504			0.0901135
$172$		0			0
$173$		−	1241.93i		−	0.545791i	−0.962044	−	0.272896i	$-0.912019\pi$
$173$		−	1241.93i		−	0.545791i	0.962044	−	0.272896i	$-0.0879813\pi$
$174$		0			0
$175$	−332.716	+	1222.45i	−0.143720	+	0.528049i
$176$		0			0
$177$	725.444			0.308066
$178$		0			0
$179$			2473.50i			1.03284i	0.856336	+	0.516419i	$0.172735\pi$
$179$			2473.50i			1.03284i	−0.856336	+	0.516419i	$0.827265\pi$
$180$		0			0
$181$		−	4292.14i		−	1.76261i	−0.472550	−	0.881304i	$-0.656666\pi$
$181$		−	4292.14i		−	1.76261i	0.472550	−	0.881304i	$-0.343334\pi$
$182$		0			0
$183$			392.683i			0.158623i
$184$		0			0
$185$			905.708i			0.359940i
$186$		0			0
$187$	−1950.74			−0.762847
$188$		0			0
$189$	−131.321	+	482.495i	−0.0505409	+	0.185695i
$190$		0			0
$191$		−	1252.21i		−	0.474383i	−0.971463	−	0.237191i	$-0.923773\pi$
$191$		−	1252.21i		−	0.474383i	0.971463	−	0.237191i	$-0.0762268\pi$
$192$		0			0
$193$	−901.321			−0.336158			−0.168079	−	0.985774i	$-0.553756\pi$
$193$	−901.321			−0.336158			−0.168079	+	0.985774i	$0.553756\pi$
$194$		0			0
$195$	120.896			0.0443976
$196$		0			0
$197$	−982.225			−0.355232			−0.177616	−	0.984100i	$-0.556838\pi$
$197$	−982.225			−0.355232			−0.177616	+	0.984100i	$0.556838\pi$
$198$		0			0
$199$	−1900.42			−0.676969			−0.338485	−	0.940972i	$-0.609914\pi$
$199$	−1900.42			−0.676969			−0.338485	+	0.940972i	$0.609914\pi$
$200$		0			0
$201$		−	1881.72i		−	0.660330i
$202$		0			0
$203$	−648.174	+	2381.49i	−0.224103	+	0.823389i
$204$		0			0
$205$	−1055.37			−0.359563
$206$		0			0
$207$			305.995i			0.102744i
$208$		0			0
$209$		−	658.399i		−	0.217906i
$210$		0			0
$211$		−	4281.82i		−	1.39703i	−0.715597	−	0.698513i	$-0.753845\pi$
$211$		−	4281.82i		−	1.39703i	0.715597	−	0.698513i	$-0.246155\pi$
$212$		0			0
$213$		−	2424.43i		−	0.779902i
$214$		0			0
$215$	−143.326			−0.0454640
$216$		0			0
$217$	1575.72	−	5789.46i	0.492936	−	1.81112i
$218$		0			0
$219$		−	1252.58i		−	0.386491i
$220$		0			0
$221$	355.355			0.108162
$222$		0			0
$223$	1677.62			0.503776			0.251888	−	0.967756i	$-0.418949\pi$
$223$	1677.62			0.503776			0.251888	+	0.967756i	$0.418949\pi$
$224$		0			0
$225$	615.665			0.182419
$226$		0			0
$227$	2852.74			0.834110			0.417055	−	0.908881i	$-0.363062\pi$
$227$	2852.74			0.834110			0.417055	+	0.908881i	$0.363062\pi$
$228$		0			0
$229$			3156.40i			0.910833i	0.890278	+	0.455417i	$0.150510\pi$
$229$			3156.40i			0.910833i	−0.890278	+	0.455417i	$0.849490\pi$
$230$		0			0
$231$	1576.52	+	429.082i	0.449035	+	0.122214i
$232$		0			0
$233$	−3827.73			−1.07624			−0.538118	−	0.842870i	$-0.680864\pi$
$233$	−3827.73			−1.07624			−0.538118	+	0.842870i	$0.680864\pi$
$234$		0			0
$235$			2833.53i			0.786549i
$236$		0			0
$237$			644.475i			0.176638i
$238$		0			0
$239$			515.516i			0.139523i	0.997564	+	0.0697614i	$0.0222238\pi$
$239$			515.516i			0.139523i	−0.997564	+	0.0697614i	$0.977776\pi$
$240$		0			0
$241$			3080.35i			0.823332i	0.911335	+	0.411666i	$0.135053\pi$
$241$			3080.35i			0.823332i	−0.911335	+	0.411666i	$0.864947\pi$
$242$		0			0
$243$	243.000			0.0641500
$244$		0			0
$245$	−1307.71	+	2224.40i	−0.341006	+	0.580049i
$246$		0			0
$247$			119.937i			0.0308963i
$248$		0			0
$249$	1918.92			0.488380
$250$		0			0
$251$	4782.52			1.20267			0.601335	−	0.798997i	$-0.294636\pi$
$251$	4782.52			1.20267			0.601335	+	0.798997i	$0.294636\pi$
$252$		0			0
$253$	999.814			0.248450
$254$		0			0
$255$	−1497.11			−0.367657
$256$		0			0
$257$		−	6031.52i		−	1.46395i	−0.681330	−	0.731976i	$-0.738598\pi$
$257$		−	6031.52i		−	1.46395i	0.681330	−	0.731976i	$-0.261402\pi$
$258$		0			0
$259$	585.571	−	2151.48i	0.140485	−	0.516164i
$260$		0			0
$261$	1199.39			0.284447
$262$		0			0
$263$			5984.65i			1.40315i	0.712594	+	0.701577i	$0.247520\pi$
$263$			5984.65i			1.40315i	−0.712594	+	0.701577i	$0.752480\pi$
$264$		0			0
$265$		−	3324.30i		−	0.770603i
$266$		0			0
$267$		−	2060.98i		−	0.472398i
$268$		0			0
$269$		−	8138.44i		−	1.84464i	−0.386422	−	0.922322i	$-0.626289\pi$
$269$		−	8138.44i		−	1.84464i	0.386422	−	0.922322i	$-0.373711\pi$
$270$		0			0
$271$	−1147.22			−0.257154			−0.128577	−	0.991700i	$-0.541041\pi$
$271$	−1147.22			−0.257154			−0.128577	+	0.991700i	$0.541041\pi$
$272$		0			0
$273$	−287.184	−	78.1634i	−0.0636674	−	0.0173284i
$274$		0			0
$275$		−	2011.64i		−	0.441114i
$276$		0			0
$277$	122.946			0.0266682			0.0133341	−	0.999911i	$-0.495755\pi$
$277$	122.946			0.0266682			0.0133341	+	0.999911i	$0.495755\pi$
$278$		0			0
$279$	−2915.75			−0.625669
$280$		0			0
$281$	−1308.64			−0.277817			−0.138909	−	0.990305i	$-0.544359\pi$
$281$	−1308.64			−0.277817			−0.138909	+	0.990305i	$0.544359\pi$
$282$		0			0
$283$	7443.18			1.56343			0.781715	−	0.623635i	$-0.214345\pi$
$283$	7443.18			1.56343			0.781715	+	0.623635i	$0.214345\pi$
$284$		0			0
$285$		−	505.293i		−	0.105021i
$286$		0			0
$287$	2507.00	+	682.334i	0.515623	+	0.140338i
$288$		0			0
$289$	512.480			0.104311
$290$		0			0
$291$		−	311.053i		−	0.0626607i
$292$		0			0
$293$		−	121.333i		−	0.0241924i	−0.999927	−	0.0120962i	$-0.996150\pi$
$293$		−	121.333i		−	0.0241924i	0.999927	−	0.0120962i	$-0.00385043\pi$
$294$		0			0
$295$		−	1819.13i		−	0.359029i
$296$		0			0
$297$		−	793.984i		−	0.155123i
$298$		0			0
$299$	−182.130			−0.0352269
$300$		0			0
$301$	340.466	+	92.6651i	0.0651965	+	0.0177446i
$302$		0			0
$303$		−	5957.02i		−	1.12945i
$304$		0			0
$305$	984.694			0.184864
$306$		0			0
$307$	5928.91			1.10222			0.551109	−	0.834433i	$-0.314205\pi$
$307$	5928.91			1.10222			0.551109	+	0.834433i	$0.314205\pi$
$308$		0			0
$309$	2565.44			0.472306
$310$		0			0
$311$	−6039.69			−1.10122			−0.550610	−	0.834763i	$-0.685605\pi$
$311$	−6039.69			−1.10122			−0.550610	+	0.834763i	$0.685605\pi$
$312$		0			0
$313$		−	6467.65i		−	1.16797i	−0.811766	−	0.583983i	$-0.801494\pi$
$313$		−	6467.65i		−	1.16797i	0.811766	−	0.583983i	$-0.198506\pi$
$314$		0			0
$315$	1209.91	+	329.302i	0.216414	+	0.0589018i
$316$		0			0
$317$	643.842			0.114075			0.0570375	−	0.998372i	$-0.481835\pi$
$317$	643.842			0.114075			0.0570375	+	0.998372i	$0.481835\pi$
$318$		0			0
$319$		−	3918.93i		−	0.687830i
$320$		0			0
$321$			1308.31i			0.227485i
$322$		0			0
$323$		−	1485.23i		−	0.255853i
$324$		0			0
$325$			366.448i			0.0625443i
$326$		0			0
$327$	−4273.88			−0.722771
$328$		0			0
$329$	1831.97	−	6730.95i	0.306991	−	1.12793i
$330$		0			0
$331$		−	2157.62i		−	0.358288i	−0.983823	−	0.179144i	$-0.942667\pi$
$331$		−	2157.62i		−	0.358288i	0.983823	−	0.179144i	$-0.0573328\pi$
$332$		0			0
$333$	−1083.55			−0.178313
$334$		0			0
$335$	−4718.61			−0.769568
$336$		0			0
$337$	−7841.90			−1.26758			−0.633791	−	0.773504i	$-0.718502\pi$
$337$	−7841.90			−1.26758			−0.633791	+	0.773504i	$0.718502\pi$
$338$		0			0
$339$	−3652.29			−0.585148
$340$		0			0
$341$			9527.00i			1.51295i
$342$		0			0
$343$	4544.57	−	4438.52i	0.715405	−	0.698710i
$344$		0			0
$345$	767.314			0.119741
$346$		0			0
$347$			4089.11i			0.632608i	0.948658	+	0.316304i	$0.102442\pi$
$347$			4089.11i			0.632608i	−0.948658	+	0.316304i	$0.897558\pi$
$348$		0			0
$349$		−	2627.48i		−	0.402997i	−0.979489	−	0.201499i	$-0.935419\pi$
$349$		−	2627.48i		−	0.402997i	0.979489	−	0.201499i	$-0.0645811\pi$
$350$		0			0
$351$			144.635i			0.0219945i
$352$		0			0
$353$			9452.79i			1.42527i	0.701534	+	0.712636i	$0.252499\pi$
$353$			9452.79i			1.42527i	−0.701534	+	0.712636i	$0.747501\pi$
$354$		0			0
$355$	−6079.51			−0.908921
$356$		0			0
$357$	3556.33	+	967.932i	0.527230	+	0.143497i
$358$		0			0
$359$			12522.8i			1.84103i	0.390710	+	0.920514i	$0.372230\pi$
$359$			12522.8i			1.84103i	−0.390710	+	0.920514i	$0.627770\pi$
$360$		0			0
$361$	−6357.72			−0.926916
$362$		0			0
$363$	1398.72			0.202242
$364$		0			0
$365$	−3140.98			−0.450428
$366$		0			0
$367$	−9111.30			−1.29593			−0.647965	−	0.761670i	$-0.724380\pi$
$367$	−9111.30			−1.29593			−0.647965	+	0.761670i	$0.724380\pi$
$368$		0			0
$369$		−	1262.61i		−	0.178126i
$370$		0			0
$371$	−2149.27	+	7896.75i	−0.300767	+	1.10506i
$372$		0			0
$373$	221.326			0.0307234			0.0153617	−	0.999882i	$-0.495110\pi$
$373$	221.326			0.0307234			0.0153617	+	0.999882i	$0.495110\pi$
$374$		0			0
$375$		−	4364.90i		−	0.601073i
$376$		0			0
$377$			713.888i			0.0975254i
$378$		0			0
$379$		−	7417.24i		−	1.00527i	−0.864498	−	0.502636i	$-0.832364\pi$
$379$		−	7417.24i		−	1.00527i	0.864498	−	0.502636i	$-0.167636\pi$
$380$		0			0
$381$		−	3566.76i		−	0.479608i
$382$		0			0
$383$	6689.28			0.892444			0.446222	−	0.894922i	$-0.352769\pi$
$383$	6689.28			0.892444			0.446222	+	0.894922i	$0.352769\pi$
$384$		0			0
$385$	1075.97	−	3953.28i	0.142432	−	0.523319i
$386$		0			0
$387$		−	171.470i		−	0.0225227i
$388$		0			0
$389$	−13138.4			−1.71246			−0.856228	−	0.516598i	$-0.827198\pi$
$389$	−13138.4			−1.71246			−0.856228	+	0.516598i	$0.827198\pi$
$390$		0			0
$391$	2255.40			0.291715
$392$		0			0
$393$	−4956.09			−0.636137
$394$		0			0
$395$	1616.09			0.205859
$396$		0			0
$397$		−	292.782i		−	0.0370134i	−0.999829	−	0.0185067i	$-0.994109\pi$
$397$		−	292.782i		−	0.0370134i	0.999829	−	0.0185067i	$-0.00589120\pi$
$398$		0			0
$399$	−326.689	+	1200.31i	−0.0409897	+	0.150603i
$400$		0			0
$401$	9071.46			1.12969			0.564847	−	0.825196i	$-0.308935\pi$
$401$	9071.46			1.12969			0.564847	+	0.825196i	$0.308935\pi$
$402$		0			0
$403$		−	1735.48i		−	0.214517i
$404$		0			0
$405$		−	609.348i		−	0.0747623i
$406$		0			0
$407$			3540.43i			0.431185i
$408$		0			0
$409$			16065.7i			1.94229i	0.238486	+	0.971146i	$0.423349\pi$
$409$			16065.7i			1.94229i	−0.238486	+	0.971146i	$0.576651\pi$
$410$		0			0
$411$	−7584.44			−0.910251
$412$		0			0
$413$	−1176.13	+	4321.27i	−0.140129	+	0.514857i
$414$		0			0
$415$		−	4811.89i		−	0.569172i
$416$		0			0
$417$	−7568.57			−0.888812
$418$		0			0
$419$	−8693.06			−1.01357			−0.506783	−	0.862074i	$-0.669165\pi$
$419$	−8693.06			−1.01357			−0.506783	+	0.862074i	$0.669165\pi$
$420$		0			0
$421$	999.478			0.115704			0.0578522	−	0.998325i	$-0.481575\pi$
$421$	999.478			0.115704			0.0578522	+	0.998325i	$0.481575\pi$
$422$		0			0
$423$	−3389.92			−0.389654
$424$		0			0
$425$		−	4537.89i		−	0.517930i
$426$		0			0
$427$	−2339.11	−	636.638i	−0.265099	−	0.0721524i
$428$		0			0
$429$	472.584			0.0531855
$430$		0			0
$431$			3194.94i			0.357065i	0.983934	+	0.178532i	$0.0571349\pi$
$431$			3194.94i			0.357065i	−0.983934	+	0.178532i	$0.942865\pi$
$432$		0			0
$433$			14150.0i			1.57045i	0.619209	+	0.785226i	$0.287453\pi$
$433$			14150.0i			1.57045i	−0.619209	+	0.785226i	$0.712547\pi$
$434$		0			0
$435$		−	3007.61i		−	0.331503i
$436$		0			0
$437$			761.225i			0.0833280i
$438$		0			0
$439$	4754.35			0.516886			0.258443	−	0.966027i	$-0.416791\pi$
$439$	4754.35			0.516886			0.258443	+	0.966027i	$0.416791\pi$
$440$		0			0
$441$	−2661.19	−	1564.49i	−0.287355	−	0.168933i
$442$		0			0
$443$		−	8322.04i		−	0.892533i	−0.894900	−	0.446266i	$-0.852753\pi$
$443$		−	8322.04i		−	0.892533i	0.894900	−	0.446266i	$-0.147247\pi$
$444$		0			0
$445$	−5168.14			−0.550546
$446$		0			0
$447$	7370.69			0.779914
$448$		0			0
$449$	15248.4			1.60270			0.801352	−	0.598193i	$-0.204114\pi$
$449$	15248.4			1.60270			0.801352	+	0.598193i	$0.204114\pi$
$450$		0			0
$451$	−4125.47			−0.430733
$452$		0			0
$453$			4317.85i			0.447837i
$454$		0			0
$455$	−196.003	+	720.145i	−0.0201951	+	0.0741998i
$456$		0			0
$457$	9050.07			0.926356			0.463178	−	0.886265i	$-0.346709\pi$
$457$	9050.07			0.926356			0.463178	+	0.886265i	$0.346709\pi$
$458$		0			0
$459$		−	1791.08i		−	0.182136i
$460$		0			0
$461$		−	1919.93i		−	0.193970i	−0.995286	−	0.0969851i	$-0.969080\pi$
$461$		−	1919.93i		−	0.193970i	0.995286	−	0.0969851i	$-0.0309199\pi$
$462$		0			0
$463$		−	7800.83i		−	0.783014i	−0.920175	−	0.391507i	$-0.871954\pi$
$463$		−	7800.83i		−	0.783014i	0.920175	−	0.391507i	$-0.128046\pi$
$464$		0			0
$465$			7311.56i			0.729173i
$466$		0			0
$467$	−4762.37			−0.471898			−0.235949	−	0.971765i	$-0.575820\pi$
$467$	−4762.37			−0.471898			−0.235949	+	0.971765i	$0.575820\pi$
$468$		0			0
$469$	11208.9	+	3050.74i	1.10358	+	0.300363i
$470$		0			0
$471$		−	9774.80i		−	0.956261i
$472$		0			0
$473$	−560.263			−0.0544629
$474$		0			0
$475$	1531.59			0.147946
$476$		0			0
$477$	3977.06			0.381755
$478$		0			0
$479$	8191.42			0.781369			0.390684	−	0.920525i	$-0.372238\pi$
$479$	8191.42			0.781369			0.390684	+	0.920525i	$0.372238\pi$
$480$		0			0
$481$		−	644.938i		−	0.0611365i
$482$		0			0
$483$	−1822.73	−	496.094i	−0.171712	−	0.0467352i
$484$		0			0
$485$	−779.999			−0.0730267
$486$		0			0
$487$		−	7326.15i		−	0.681683i	−0.940121	−	0.340841i	$-0.889288\pi$
$487$		−	7326.15i		−	0.681683i	0.940121	−	0.340841i	$-0.110712\pi$
$488$		0			0
$489$			3572.61i			0.330386i
$490$		0			0
$491$			16260.3i			1.49454i	0.664523	+	0.747268i	$0.268635\pi$
$491$			16260.3i			1.49454i	−0.664523	+	0.747268i	$0.731365\pi$
$492$		0			0
$493$		−	8840.39i		−	0.807609i
$494$		0			0
$495$	−1991.00			−0.180785
$496$		0			0
$497$	14441.7	+	3930.61i	1.30342	+	0.354752i
$498$		0			0
$499$			14395.5i			1.29144i	0.763573	+	0.645721i	$0.223443\pi$
$499$			14395.5i			1.29144i	−0.763573	+	0.645721i	$0.776557\pi$
$500$		0			0
$501$	1506.62			0.134353
$502$		0			0
$503$	16722.5			1.48235			0.741174	−	0.671313i	$-0.234270\pi$
$503$	16722.5			1.48235			0.741174	+	0.671313i	$0.234270\pi$
$504$		0			0
$505$	−14937.9			−1.31629
$506$		0			0
$507$	6504.91			0.569809
$508$		0			0
$509$		−	3391.00i		−	0.295292i	−0.989040	−	0.147646i	$-0.952830\pi$
$509$		−	3391.00i		−	0.295292i	0.989040	−	0.147646i	$-0.0471696\pi$
$510$		0			0
$511$	7461.29	+	2030.75i	0.645926	+	0.175803i
$512$		0			0
$513$	604.513			0.0520271
$514$		0			0
$515$		−	6433.10i		−	0.550439i
$516$		0			0
$517$			11076.3i			0.942235i
$518$		0			0
$519$		−	3725.78i		−	0.315113i
$520$		0			0
$521$			115.006i			0.00967084i	0.999988	+	0.00483542i	$0.00153917\pi$
$521$			115.006i			0.00967084i	−0.999988	+	0.00483542i	$0.998461\pi$
$522$		0			0
$523$	−9860.83			−0.824443			−0.412222	−	0.911084i	$-0.635247\pi$
$523$	−9860.83			−0.824443			−0.412222	+	0.911084i	$0.635247\pi$
$524$		0			0
$525$	−998.148	+	3667.35i	−0.0829767	+	0.304869i
$526$		0			0
$527$			21491.2i			1.77642i
$528$		0			0
$529$	11011.0			0.904992
$530$		0			0
$531$	2176.33			0.177862
$532$		0			0
$533$	751.512			0.0610724
$534$		0			0
$535$	3280.73			0.265118
$536$		0			0
$537$			7420.49i			0.596309i
$538$		0			0
$539$	−5111.86	+	8695.23i	−0.408503	+	0.694861i
$540$		0			0
$541$	15529.6			1.23414			0.617069	−	0.786909i	$-0.288320\pi$
$541$	15529.6			1.23414			0.617069	+	0.786909i	$0.288320\pi$
$542$		0			0
$543$		−	12876.4i		−	1.01764i
$544$		0			0
$545$			10717.2i			0.842338i
$546$		0			0
$547$		−	9476.01i		−	0.740704i	−0.928892	−	0.370352i	$-0.879237\pi$
$547$		−	9476.01i		−	0.740704i	0.928892	−	0.370352i	$-0.120763\pi$
$548$		0			0
$549$			1178.05i			0.0915809i
$550$		0			0
$551$	2983.74			0.230693
$552$		0			0
$553$	−3838.97	−	1044.86i	−0.295207	−	0.0803469i
$554$		0			0
$555$			2717.12i			0.207812i
$556$		0			0
$557$	4458.02			0.339125			0.169562	−	0.985519i	$-0.445765\pi$
$557$	4458.02			0.339125			0.169562	+	0.985519i	$0.445765\pi$
$558$		0			0
$559$	102.060			0.00772213
$560$		0			0
$561$	−5852.22			−0.440430
$562$		0			0
$563$	8596.00			0.643478			0.321739	−	0.946828i	$-0.395733\pi$
$563$	8596.00			0.643478			0.321739	+	0.946828i	$0.395733\pi$
$564$		0			0
$565$			9158.50i			0.681949i
$566$		0			0
$567$	−393.964	+	1447.49i	−0.0291798	+	0.107211i
$568$		0			0
$569$	−4098.66			−0.301976			−0.150988	−	0.988536i	$-0.548246\pi$
$569$	−4098.66			−0.301976			−0.150988	+	0.988536i	$0.548246\pi$
$570$		0			0
$571$			86.4091i			0.00633294i	0.999995	+	0.00316647i	$0.00100792\pi$
$571$			86.4091i			0.00633294i	−0.999995	+	0.00316647i	$0.998992\pi$
$572$		0			0
$573$		−	3756.64i		−	0.273885i
$574$		0			0
$575$			2325.81i			0.168683i
$576$		0			0
$577$			19490.5i			1.40624i	0.711071	+	0.703120i	$0.248210\pi$
$577$			19490.5i			1.40624i	−0.711071	+	0.703120i	$0.751790\pi$
$578$		0			0
$579$	−2703.96			−0.194081
$580$		0			0
$581$	−3111.05	+	11430.5i	−0.222148	+	0.816207i
$582$		0			0
$583$		−	12994.7i		−	0.923133i
$584$		0			0
$585$	362.688			0.0256330
$586$		0			0
$587$	21546.8			1.51504			0.757522	−	0.652810i	$-0.226410\pi$
$587$	21546.8			1.51504			0.757522	+	0.652810i	$0.226410\pi$
$588$		0			0
$589$	−7253.54			−0.507431
$590$		0			0
$591$	−2946.67			−0.205093
$592$		0			0
$593$			8845.46i			0.612546i	0.951944	+	0.306273i	$0.0990820\pi$
$593$			8845.46i			0.612546i	−0.951944	+	0.306273i	$0.900918\pi$
$594$		0			0
$595$	2427.19	−	8917.88i	0.167236	−	0.614450i
$596$		0			0
$597$	−5701.25			−0.390848
$598$		0			0
$599$			20160.5i			1.37518i	0.726098	+	0.687592i	$0.241332\pi$
$599$			20160.5i			1.37518i	−0.726098	+	0.687592i	$0.758668\pi$
$600$		0			0
$601$		−	6467.44i		−	0.438956i	−0.975618	−	0.219478i	$-0.929565\pi$
$601$		−	6467.44i		−	0.438956i	0.975618	−	0.219478i	$-0.0704353\pi$
$602$		0			0
$603$		−	5645.16i		−	0.381242i
$604$		0			0
$605$		−	3507.44i		−	0.235699i
$606$		0			0
$607$	−24606.0			−1.64535			−0.822673	−	0.568515i	$-0.807518\pi$
$607$	−24606.0			−1.64535			−0.822673	+	0.568515i	$0.807518\pi$
$608$		0			0
$609$	−1944.52	+	7144.47i	−0.129386	+	0.475384i
$610$		0			0
$611$		−	2017.70i		−	0.133597i
$612$		0			0
$613$	10251.2			0.675437			0.337719	−	0.941247i	$-0.390345\pi$
$613$	10251.2			0.675437			0.337719	+	0.941247i	$0.390345\pi$
$614$		0			0
$615$	−3166.12			−0.207594
$616$		0			0
$617$	19189.1			1.25206			0.626031	−	0.779798i	$-0.284678\pi$
$617$	19189.1			1.25206			0.626031	+	0.779798i	$0.284678\pi$
$618$		0			0
$619$	−17050.4			−1.10713			−0.553565	−	0.832806i	$-0.686733\pi$
$619$	−17050.4			−1.10713			−0.553565	+	0.832806i	$0.686733\pi$
$620$		0			0
$621$			917.984i			0.0593195i
$622$		0			0
$623$	12276.7	+	3341.37i	0.789498	+	0.214879i
$624$		0			0
$625$	−2394.54			−0.153250
$626$		0			0
$627$		−	1975.20i		−	0.125808i
$628$		0			0
$629$			7986.56i			0.506272i
$630$		0			0
$631$			1257.65i			0.0793442i	0.999213	+	0.0396721i	$0.0126313\pi$
$631$			1257.65i			0.0793442i	−0.999213	+	0.0396721i	$0.987369\pi$
$632$		0			0
$633$		−	12845.5i		−	0.806573i
$634$		0			0
$635$	−8944.02			−0.558949
$636$		0			0
$637$	931.196	−	1583.96i	0.0579205	−	0.0985223i
$638$		0			0
$639$		−	7273.29i		−	0.450277i
$640$		0			0
$641$	−9903.08			−0.610215			−0.305108	−	0.952318i	$-0.598692\pi$
$641$	−9903.08			−0.610215			−0.305108	+	0.952318i	$0.598692\pi$
$642$		0			0
$643$	20327.6			1.24672			0.623361	−	0.781934i	$-0.285767\pi$
$643$	20327.6			1.24672			0.623361	+	0.781934i	$0.285767\pi$
$644$		0			0
$645$	−429.978			−0.0262486
$646$		0			0
$647$	−21089.7			−1.28149			−0.640743	−	0.767755i	$-0.721374\pi$
$647$	−21089.7			−1.28149			−0.640743	+	0.767755i	$0.721374\pi$
$648$		0			0
$649$		−	7110.99i		−	0.430094i
$650$		0			0
$651$	4727.17	−	17368.4i	0.284597	−	1.04565i
$652$		0			0
$653$	20130.9			1.20641			0.603204	−	0.797587i	$-0.293890\pi$
$653$	20130.9			1.20641			0.603204	+	0.797587i	$0.293890\pi$
$654$		0			0
$655$			12427.9i			0.741373i
$656$		0			0
$657$		−	3757.74i		−	0.223141i
$658$		0			0
$659$			31889.2i			1.88502i	0.334181	+	0.942509i	$0.391540\pi$
$659$			31889.2i			1.88502i	−0.334181	+	0.942509i	$0.608460\pi$
$660$		0			0
$661$			25390.7i			1.49408i	0.664780	+	0.747039i	$0.268525\pi$
$661$			25390.7i			1.49408i	−0.664780	+	0.747039i	$0.731475\pi$
$662$		0			0
$663$	1066.06			0.0624472
$664$		0			0
$665$	3009.90	+	819.207i	0.175517	+	0.0477707i
$666$		0			0
$667$			4530.97i			0.263028i
$668$		0			0
$669$	5032.87			0.290855
$670$		0			0
$671$	3849.18			0.221455
$672$		0			0
$673$	28068.6			1.60767			0.803836	−	0.594851i	$-0.202789\pi$
$673$	28068.6			1.60767			0.803836	+	0.594851i	$0.202789\pi$
$674$		0			0
$675$	1847.00			0.105320
$676$		0			0
$677$			31036.6i			1.76194i	0.473171	+	0.880971i	$0.343109\pi$
$677$			31036.6i			1.76194i	−0.473171	+	0.880971i	$0.656891\pi$
$678$		0			0
$679$	1852.86	+	504.296i	0.104722	+	0.0285024i
$680$		0			0
$681$	8558.22			0.481574
$682$		0			0
$683$			11387.9i			0.637986i	0.947757	+	0.318993i	$0.103345\pi$
$683$			11387.9i			0.637986i	−0.947757	+	0.318993i	$0.896655\pi$
$684$		0			0
$685$			19018.8i			1.06083i
$686$		0			0
$687$			9469.20i			0.525870i
$688$		0			0
$689$			2367.17i			0.130888i
$690$		0			0
$691$	3707.61			0.204116			0.102058	−	0.994778i	$-0.467457\pi$
$691$	3707.61			0.204116			0.102058	+	0.994778i	$0.467457\pi$
$692$		0			0
$693$	4729.55	+	1287.25i	0.259251	+	0.0705606i
$694$		0			0
$695$			18979.0i			1.03585i
$696$		0			0
$697$	−9306.31			−0.505741
$698$		0			0
$699$	−11483.2			−0.621365
$700$		0			0
$701$	−16000.1			−0.862079			−0.431039	−	0.902333i	$-0.641853\pi$
$701$	−16000.1			−0.862079			−0.431039	+	0.902333i	$0.641853\pi$
$702$		0			0
$703$	−2695.56			−0.144616
$704$		0			0
$705$			8500.59i			0.454114i
$706$		0			0
$707$	35484.4	+	9657.84i	1.88759	+	0.513749i
$708$		0			0
$709$	−19839.3			−1.05089			−0.525444	−	0.850828i	$-0.676101\pi$
$709$	−19839.3			−1.05089			−0.525444	+	0.850828i	$0.676101\pi$
$710$		0			0
$711$			1933.43i			0.101982i
$712$		0			0
$713$		−	11014.9i		−	0.578556i
$714$		0			0
$715$		−	1185.05i		−	0.0619840i
$716$		0			0
$717$			1546.55i			0.0805536i
$718$		0			0
$719$	−31744.9			−1.64657			−0.823286	−	0.567626i	$-0.807862\pi$
$719$	−31744.9			−1.64657			−0.823286	+	0.567626i	$0.807862\pi$
$720$		0			0
$721$	−4159.22	+	15281.6i	−0.214837	+	0.789344i
$722$		0			0
$723$			9241.06i			0.475351i
$724$		0			0
$725$	9116.37			0.466998
$726$		0			0
$727$	−13264.7			−0.676697			−0.338349	−	0.941021i	$-0.609868\pi$
$727$	−13264.7			−0.676697			−0.338349	+	0.941021i	$0.609868\pi$
$728$		0			0
$729$	729.000			0.0370370
$730$		0			0
$731$	−1263.85			−0.0639470
$732$		0			0
$733$		−	15237.1i		−	0.767795i	−0.923376	−	0.383897i	$-0.874582\pi$
$733$		−	15237.1i		−	0.767795i	0.923376	−	0.383897i	$-0.125418\pi$
$734$		0			0
$735$	−3923.13	+	6673.21i	−0.196880	+	0.334891i
$736$		0			0
$737$	−18445.1			−0.921893
$738$		0			0
$739$		−	9609.13i		−	0.478319i	−0.970980	−	0.239159i	$-0.923128\pi$
$739$		−	9609.13i		−	0.478319i	0.970980	−	0.239159i	$-0.0768719\pi$
$740$		0			0
$741$			359.810i			0.0178380i
$742$		0			0
$743$		−	2760.49i		−	0.136302i	−0.997675	−	0.0681510i	$-0.978290\pi$
$743$		−	2760.49i		−	0.136302i	0.997675	−	0.0681510i	$-0.0217100\pi$
$744$		0			0
$745$		−	18482.8i		−	0.908935i
$746$		0			0
$747$	5756.76			0.281966
$748$		0			0
$749$	−7793.26	−	2121.10i	−0.380186	−	0.103476i
$750$		0			0
$751$		−	3559.77i		−	0.172966i	−0.996253	−	0.0864832i	$-0.972437\pi$
$751$		−	3559.77i		−	0.172966i	0.996253	−	0.0864832i	$-0.0275629\pi$
$752$		0			0
$753$	14347.6			0.694362
$754$		0			0
$755$	10827.5			0.521922
$756$		0			0
$757$	18894.6			0.907180			0.453590	−	0.891210i	$-0.350143\pi$
$757$	18894.6			0.907180			0.453590	+	0.891210i	$0.350143\pi$
$758$		0			0
$759$	2999.44			0.143442
$760$		0			0
$761$			22833.2i			1.08765i	0.839199	+	0.543825i	$0.183024\pi$
$761$			22833.2i			1.08765i	−0.839199	+	0.543825i	$0.816976\pi$
$762$		0			0
$763$	6929.03	−	25458.4i	0.328765	−	1.20793i
$764$		0			0
$765$	−4491.33			−0.212267
$766$		0			0
$767$			1295.37i			0.0609817i
$768$		0			0
$769$			25450.6i			1.19346i	0.802441	+	0.596731i	$0.203534\pi$
$769$			25450.6i			1.19346i	−0.802441	+	0.596731i	$0.796466\pi$
$770$		0			0
$771$		−	18094.6i		−	0.845213i
$772$		0			0
$773$			19338.8i			0.899831i	0.893071	+	0.449915i	$0.148546\pi$
$773$			19338.8i			0.899831i	−0.893071	+	0.449915i	$0.851454\pi$
$774$		0			0
$775$	−22162.1			−1.02721
$776$		0			0
$777$	1756.71	−	6454.44i	0.0811090	−	0.298007i
$778$		0			0
$779$		−	3140.99i		−	0.144464i
$780$		0			0
$781$	−23764.9			−1.08883
$782$		0			0
$783$	3598.18			0.164226
$784$		0			0
$785$	−24511.3			−1.11445
$786$		0			0
$787$	4298.43			0.194692			0.0973459	−	0.995251i	$-0.468965\pi$
$787$	4298.43			0.194692			0.0973459	+	0.995251i	$0.468965\pi$
$788$		0			0
$789$			17954.0i			0.810111i
$790$		0			0
$791$	5921.28	−	21755.7i	0.266165	−	0.977932i
$792$		0			0
$793$	−701.183			−0.0313994
$794$		0			0
$795$		−	9972.89i		−	0.444908i
$796$		0			0
$797$		−	10181.8i		−	0.452519i	−0.974067	−	0.226260i	$-0.927350\pi$
$797$		−	10181.8i		−	0.452519i	0.974067	−	0.226260i	$-0.0726498\pi$
$798$		0			0
$799$			24986.1i			1.10632i
$800$		0			0
$801$		−	6182.95i		−	0.272739i
$802$		0			0
$803$	−12278.1			−0.539584
$804$		0			0
$805$	−1244.01	+	4570.68i	−0.0544665	+	0.200119i
$806$		0			0
$807$		−	24415.3i		−	1.06501i
$808$		0			0
$809$	−29969.2			−1.30242			−0.651211	−	0.758896i	$-0.725739\pi$
$809$	−29969.2			−1.30242			−0.651211	+	0.758896i	$0.725739\pi$
$810$		0			0
$811$	−3426.41			−0.148357			−0.0741785	−	0.997245i	$-0.523633\pi$
$811$	−3426.41			−0.148357			−0.0741785	+	0.997245i	$0.523633\pi$
$812$		0			0
$813$	−3441.66			−0.148468
$814$		0			0
$815$	8958.69			0.385042
$816$		0			0
$817$		−	426.566i		−	0.0182664i
$818$		0			0
$819$	−861.553	−	234.490i	−0.0367584	−	0.0100046i
$820$		0			0
$821$	44007.6			1.87074			0.935368	−	0.353676i	$-0.115068\pi$
$821$	44007.6			1.87074			0.935368	+	0.353676i	$0.115068\pi$
$822$		0			0
$823$		−	19248.3i		−	0.815255i	−0.913148	−	0.407627i	$-0.866356\pi$
$823$		−	19248.3i		−	0.815255i	0.913148	−	0.407627i	$-0.133644\pi$
$824$		0			0
$825$		−	6034.92i		−	0.254677i
$826$		0			0
$827$			39040.4i			1.64156i	0.571245	+	0.820779i	$0.306460\pi$
$827$			39040.4i			1.64156i	−0.571245	+	0.820779i	$0.693540\pi$
$828$		0			0
$829$		−	882.320i		−	0.0369653i	−0.999829	−	0.0184827i	$-0.994116\pi$
$829$		−	882.320i		−	0.0369653i	0.999829	−	0.0184827i	$-0.00588355\pi$
$830$		0			0
$831$	368.838			0.0153969
$832$		0			0
$833$	−11531.4	+	19614.9i	−0.479640	+	0.815864i
$834$		0			0
$835$		−	3778.00i		−	0.156578i
$836$		0			0
$837$	−8747.26			−0.361230
$838$		0			0
$839$	35277.8			1.45164			0.725819	−	0.687886i	$-0.241461\pi$
$839$	35277.8			1.45164			0.725819	+	0.687886i	$0.241461\pi$
$840$		0			0
$841$	−6629.15			−0.271809
$842$		0			0
$843$	−3925.91			−0.160398
$844$		0			0
$845$		−	16311.7i		−	0.664072i
$846$		0			0
$847$	−2267.68	+	8331.80i	−0.0919933	+	0.337998i
$848$		0			0
$849$	22329.5			0.902647
$850$		0			0
$851$		−	4093.35i		−	0.164886i
$852$		0			0
$853$			11073.8i			0.444501i	0.974990	+	0.222250i	$0.0713403\pi$
$853$			11073.8i			0.444501i	−0.974990	+	0.222250i	$0.928660\pi$
$854$		0			0
$855$		−	1515.88i		−	0.0606339i
$856$		0			0
$857$		−	39961.2i		−	1.59282i	−0.604757	−	0.796410i	$-0.706730\pi$
$857$		−	39961.2i		−	1.59282i	0.604757	−	0.796410i	$-0.293270\pi$
$858$		0			0
$859$	−36708.5			−1.45807			−0.729033	−	0.684479i	$-0.760030\pi$
$859$	−36708.5			−1.45807			−0.729033	+	0.684479i	$0.760030\pi$
$860$		0			0
$861$	7521.01	+	2047.00i	0.297695	+	0.0810240i
$862$		0			0
$863$			35497.2i			1.40016i	0.714064	+	0.700081i	$0.246853\pi$
$863$			35497.2i			1.40016i	−0.714064	+	0.700081i	$0.753147\pi$
$864$		0			0
$865$	−9342.78			−0.367242
$866$		0			0
$867$	1537.44			0.0602240
$868$		0			0
$869$	6317.32			0.246606
$870$		0			0
$871$	3360.04			0.130712
$872$		0			0
$873$		−	933.160i		−	0.0361772i
$874$		0			0
$875$	26000.5	+	7076.60i	1.00455	+	0.273409i
$876$		0			0
$877$	−26876.6			−1.03485			−0.517423	−	0.855730i	$-0.673109\pi$
$877$	−26876.6			−1.03485			−0.517423	+	0.855730i	$0.673109\pi$
$878$		0			0
$879$		−	364.000i		−	0.0139675i
$880$		0			0
$881$			9934.42i			0.379908i	0.981793	+	0.189954i	$0.0608339\pi$
$881$			9934.42i			0.379908i	−0.981793	+	0.189954i	$0.939166\pi$
$882$		0			0
$883$			20298.1i			0.773596i	0.922164	+	0.386798i	$0.126419\pi$
$883$			20298.1i			0.773596i	−0.922164	+	0.386798i	$0.873581\pi$
$884$		0			0
$885$		−	5457.38i		−	0.207286i
$886$		0			0
$887$	−22739.8			−0.860799			−0.430399	−	0.902639i	$-0.641627\pi$
$887$	−22739.8			−0.860799			−0.430399	+	0.902639i	$0.641627\pi$
$888$		0			0
$889$	21246.2	+	5782.61i	0.801548	+	0.218158i
$890$		0			0
$891$		−	2381.95i		−	0.0895604i
$892$		0			0
$893$	−8433.13			−0.316018
$894$		0			0
$895$	18607.7			0.694956
$896$		0			0
$897$	−546.390			−0.0203383
$898$		0			0
$899$	−43174.6			−1.60173
$900$		0			0
$901$		−	29313.7i		−	1.08389i
$902$		0			0
$903$	1021.40	+	277.995i	0.0376412	+	0.0102449i
$904$		0			0
$905$	−32288.9			−1.18599
$906$		0			0
$907$		−	11598.9i		−	0.424624i	−0.977202	−	0.212312i	$-0.931901\pi$
$907$		−	11598.9i		−	0.424624i	0.977202	−	0.212312i	$-0.0680993\pi$
$908$		0			0
$909$		−	17871.1i		−	0.652086i
$910$		0			0
$911$		−	44200.3i		−	1.60749i	−0.594975	−	0.803744i	$-0.702838\pi$
$911$		−	44200.3i		−	1.60749i	0.594975	−	0.803744i	$-0.297162\pi$
$912$		0			0
$913$		−	18809.8i		−	0.681831i
$914$		0			0
$915$	2954.08			0.106731
$916$		0			0
$917$	8035.07	−	29522.1i	0.289358	−	1.06315i
$918$		0			0
$919$		−	48975.3i		−	1.75794i	−0.476878	−	0.878969i	$-0.658232\pi$
$919$		−	48975.3i		−	1.75794i	0.476878	−	0.878969i	$-0.341768\pi$
$920$		0			0
$921$	17786.7			0.636366
$922$		0			0
$923$	4329.11			0.154382
$924$		0			0
$925$	−8235.88			−0.292750
$926$		0			0
$927$	7696.31			0.272686
$928$		0			0
$929$			55123.2i			1.94675i	0.229212	+	0.973377i	$0.426385\pi$
$929$			55123.2i			1.94675i	−0.229212	+	0.973377i	$0.573615\pi$
$930$		0			0
$931$	−6620.26	−	3892.00i	−0.233051	−	0.137009i
$932$		0			0
$933$	−18119.1			−0.635789
$934$		0			0
$935$			14675.1i			0.513290i
$936$		0			0
$937$		−	41003.8i		−	1.42960i	−0.699329	−	0.714800i	$-0.746518\pi$
$937$		−	41003.8i		−	1.42960i	0.699329	−	0.714800i	$-0.253482\pi$
$938$		0			0
$939$		−	19402.9i		−	0.674325i
$940$		0			0
$941$		−	6567.48i		−	0.227517i	−0.993508	−	0.113759i	$-0.963711\pi$
$941$		−	6567.48i		−	0.227517i	0.993508	−	0.113759i	$-0.0362890\pi$
$942$		0			0
$943$	4769.77			0.164714
$944$		0			0
$945$	3629.72	+	987.906i	0.124947	+	0.0340070i
$946$		0			0
$947$		−	27665.3i		−	0.949316i	−0.880170	−	0.474658i	$-0.842572\pi$
$947$		−	27665.3i		−	0.949316i	0.880170	−	0.474658i	$-0.157428\pi$
$948$		0			0
$949$	2236.63			0.0765060
$950$		0			0
$951$	1931.53			0.0658612
$952$		0			0
$953$	20375.2			0.692568			0.346284	−	0.938130i	$-0.387443\pi$
$953$	20375.2			0.692568			0.346284	+	0.938130i	$0.387443\pi$
$954$		0			0
$955$	−9420.18			−0.319193
$956$		0			0
$957$		−	11756.8i		−	0.397119i
$958$		0			0
$959$	12296.3	−	45178.5i	0.414044	−	1.52126i
$960$		0			0
$961$	75167.3			2.52316
$962$		0			0
$963$			3924.93i			0.131339i
$964$		0			0
$965$			6780.47i			0.226188i
$966$		0			0
$967$			32218.8i			1.07144i	0.844394	+	0.535722i	$0.179961\pi$
$967$			32218.8i			1.07144i	−0.844394	+	0.535722i	$0.820039\pi$
$968$		0			0
$969$		−	4455.69i		−	0.147717i
$970$		0			0
$971$	−6664.98			−0.220277			−0.110139	−	0.993916i	$-0.535129\pi$
$971$	−6664.98			−0.220277			−0.110139	+	0.993916i	$0.535129\pi$
$972$		0			0
$973$	12270.6	−	45084.0i	0.404292	−	1.48543i
$974$		0			0
$975$			1099.34i			0.0361099i
$976$		0			0
$977$	−1037.07			−0.0339598			−0.0169799	−	0.999856i	$-0.505405\pi$
$977$	−1037.07			−0.0339598			−0.0169799	+	0.999856i	$0.505405\pi$
$978$		0			0
$979$	−20202.3			−0.659519
$980$		0			0
$981$	−12821.6			−0.417292
$982$		0			0
$983$	31143.8			1.01051			0.505256	−	0.862969i	$-0.331398\pi$
$983$	31143.8			1.01051			0.505256	+	0.862969i	$0.331398\pi$
$984$		0			0
$985$			7389.09i			0.239021i
$986$		0			0
$987$	5495.92	−	20192.9i	0.177241	−	0.651212i
$988$		0			0
$989$	647.763			0.0208268
$990$		0			0
$991$		−	39874.3i		−	1.27815i	−0.769144	−	0.639076i	$-0.779317\pi$
$991$		−	39874.3i		−	1.27815i	0.769144	−	0.639076i	$-0.220683\pi$
$992$		0			0
$993$		−	6472.85i		−	0.206858i
$994$		0			0
$995$			14296.5i			0.455506i
$996$		0			0
$997$			55124.7i			1.75107i	0.483154	+	0.875535i	$0.339491\pi$
$997$			55124.7i			1.75107i	−0.483154	+	0.875535i	$0.660509\pi$
$998$		0			0
$999$	−3250.66			−0.102949

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.4.b.f.895.3		8
4.3	odd	2		1344.4.b.e.895.3		8
7.6	odd	2		1344.4.b.e.895.6		8
8.3	odd	2		336.4.b.f.223.6	yes	8
8.5	even	2		336.4.b.e.223.6	yes	8
24.5	odd	2		1008.4.b.i.559.3		8
24.11	even	2		1008.4.b.k.559.3		8
28.27	even	2	inner	1344.4.b.f.895.6		8
56.13	odd	2		336.4.b.f.223.3	yes	8
56.27	even	2		336.4.b.e.223.3	✓	8
168.83	odd	2		1008.4.b.i.559.6		8
168.125	even	2		1008.4.b.k.559.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.4.b.e.223.3	✓	8	56.27	even	2
336.4.b.e.223.6	yes	8	8.5	even	2
336.4.b.f.223.3	yes	8	56.13	odd	2
336.4.b.f.223.6	yes	8	8.3	odd	2
1008.4.b.i.559.3		8	24.5	odd	2
1008.4.b.i.559.6		8	168.83	odd	2
1008.4.b.k.559.3		8	24.11	even	2
1008.4.b.k.559.6		8	168.125	even	2
1344.4.b.e.895.3		8	4.3	odd	2
1344.4.b.e.895.6		8	7.6	odd	2
1344.4.b.f.895.3		8	1.1	even	1	trivial
1344.4.b.f.895.6		8	28.27	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1344.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} -7.52281i q^{5} +(-4.86375 + 17.8702i) q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q+3.00000 q^{3} -7.52281i q^{5} +(-4.86375 + 17.8702i) q^{7} +9.00000 q^{9} -29.4068i q^{11} +5.35686i q^{13} -22.5684i q^{15} -66.3364i q^{17} +22.3894 q^{19} +(-14.5913 + 53.6106i) q^{21} +33.9994i q^{23} +68.4073 q^{25} +27.0000 q^{27} +133.266 q^{29} -323.973 q^{31} -88.2204i q^{33} +(134.434 + 36.5891i) q^{35} -120.395 q^{37} +16.0706i q^{39} -140.290i q^{41} -19.0522i q^{43} -67.7053i q^{45} -376.658 q^{47} +(-295.688 - 173.832i) q^{49} -199.009i q^{51} +441.895 q^{53} -221.222 q^{55} +67.1681 q^{57} +241.815 q^{59} +130.894i q^{61} +(-43.7738 + 160.832i) q^{63} +40.2987 q^{65} -627.240i q^{67} +101.998i q^{69} -808.143i q^{71} -417.527i q^{73} +205.222 q^{75} +(525.505 + 143.027i) q^{77} +214.825i q^{79} +81.0000 q^{81} +639.640 q^{83} -499.037 q^{85} +399.798 q^{87} -686.995i q^{89} +(-95.7281 - 26.0545i) q^{91} -971.918 q^{93} -168.431i q^{95} -103.684i q^{97} -264.661i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 24 q^{3} - 4 q^{7} + 72 q^{9}+O(q^{10}) \) 8 * q + 24 * q^3 - 4 * q^7 + 72 * q^9	\( 8 q + 24 q^{3} - 4 q^{7} + 72 q^{9} + 56 q^{19} - 12 q^{21} - 656 q^{25} + 216 q^{27} - 240 q^{29} + 320 q^{31} - 600 q^{35} - 392 q^{37} - 816 q^{47} - 16 q^{49} - 288 q^{53} - 456 q^{55} + 168 q^{57} + 1824 q^{59} - 36 q^{63} - 816 q^{65} - 1968 q^{75} + 2064 q^{77} + 648 q^{81} - 1680 q^{83} - 2568 q^{85} - 720 q^{87} + 864 q^{91} + 960 q^{93}+O(q^{100}) \) 8 * q + 24 * q^3 - 4 * q^7 + 72 * q^9 + 56 * q^19 - 12 * q^21 - 656 * q^25 + 216 * q^27 - 240 * q^29 + 320 * q^31 - 600 * q^35 - 392 * q^37 - 816 * q^47 - 16 * q^49 - 288 * q^53 - 456 * q^55 + 168 * q^57 + 1824 * q^59 - 36 * q^63 - 816 * q^65 - 1968 * q^75 + 2064 * q^77 + 648 * q^81 - 1680 * q^83 - 2568 * q^85 - 720 * q^87 + 864 * q^91 + 960 * q^93

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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