LMFDB - Embedded newform 1344.4.b.i.895.9

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			895.9
Character	$\chi$	$=$	1344.895
Dual form			1344.4.b.i.895.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} -5.77465i q^{5} +(3.77600 + 18.1312i) q^{7} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} -5.77465i q^{5} +(3.77600 + 18.1312i) q^{7} +9.00000 q^{9} +14.5391i q^{11} +75.4995i q^{13} +17.3240i q^{15} +81.5687i q^{17} -89.0300 q^{19} +(-11.3280 - 54.3937i) q^{21} -9.53823i q^{23} +91.6534 q^{25} -27.0000 q^{27} -259.288 q^{29} +13.5154 q^{31} -43.6174i q^{33} +(104.702 - 21.8051i) q^{35} +196.934 q^{37} -226.498i q^{39} +341.451i q^{41} -535.198i q^{43} -51.9719i q^{45} +26.7703 q^{47} +(-314.484 + 136.927i) q^{49} -244.706i q^{51} +671.696 q^{53} +83.9585 q^{55} +267.090 q^{57} -546.205 q^{59} -815.566i q^{61} +(33.9840 + 163.181i) q^{63} +435.983 q^{65} +605.151i q^{67} +28.6147i q^{69} +287.911i q^{71} -64.3771i q^{73} -274.960 q^{75} +(-263.612 + 54.8997i) q^{77} -131.742i q^{79} +81.0000 q^{81} -822.082 q^{83} +471.031 q^{85} +777.864 q^{87} +737.170i q^{89} +(-1368.90 + 285.086i) q^{91} -40.5461 q^{93} +514.117i q^{95} -348.816i q^{97} +130.852i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$		−	5.77465i		−	0.516501i	−0.966078	−	0.258250i	$-0.916854\pi$
$5$		−	5.77465i		−	0.516501i	0.966078	−	0.258250i	$-0.0831460\pi$
$6$		0			0
$7$	3.77600	+	18.1312i	0.203885	+	0.978995i
$7$	3.77600	+	18.1312i	0.203885	+	0.978995i
$8$		0			0
$9$	9.00000			0.333333
$10$		0			0
$11$			14.5391i			0.398519i	0.979947	+	0.199260i	$0.0638537\pi$
$11$			14.5391i			0.398519i	−0.979947	+	0.199260i	$0.936146\pi$
$12$		0			0
$13$			75.4995i			1.61075i	0.592764	+	0.805376i	$0.298037\pi$
$13$			75.4995i			1.61075i	−0.592764	+	0.805376i	$0.701963\pi$
$14$		0			0
$15$			17.3240i			0.298202i
$16$		0			0
$17$			81.5687i			1.16372i	0.813287	+	0.581862i	$0.197676\pi$
$17$			81.5687i			1.16372i	−0.813287	+	0.581862i	$0.802324\pi$
$18$		0			0
$19$	−89.0300			−1.07499			−0.537497	−	0.843266i	$-0.680630\pi$
$19$	−89.0300			−1.07499			−0.537497	+	0.843266i	$0.680630\pi$
$20$		0			0
$21$	−11.3280	−	54.3937i	−0.117713	−	0.565223i
$22$		0			0
$23$		−	9.53823i		−	0.0864721i	−0.999065	−	0.0432361i	$-0.986233\pi$
$23$		−	9.53823i		−	0.0864721i	0.999065	−	0.0432361i	$-0.0137668\pi$
$24$		0			0
$25$	91.6534			0.733227
$26$		0			0
$27$	−27.0000			−0.192450
$28$		0			0
$29$	−259.288			−1.66030			−0.830148	−	0.557543i	$-0.811744\pi$
$29$	−259.288			−1.66030			−0.830148	+	0.557543i	$0.811744\pi$
$30$		0			0
$31$	13.5154			0.0783042			0.0391521	−	0.999233i	$-0.487534\pi$
$31$	13.5154			0.0783042			0.0391521	+	0.999233i	$0.487534\pi$
$32$		0			0
$33$		−	43.6174i		−	0.230085i
$34$		0			0
$35$	104.702	−	21.8051i	0.505652	−	0.105307i
$36$		0			0
$37$	196.934			0.875020			0.437510	−	0.899214i	$-0.355860\pi$
$37$	196.934			0.875020			0.437510	+	0.899214i	$0.355860\pi$
$38$		0			0
$39$		−	226.498i		−	0.929968i
$40$		0			0
$41$			341.451i			1.30063i	0.759666	+	0.650313i	$0.225362\pi$
$41$			341.451i			1.30063i	−0.759666	+	0.650313i	$0.774638\pi$
$42$		0			0
$43$		−	535.198i		−	1.89807i	−0.315173	−	0.949034i	$-0.602062\pi$
$43$		−	535.198i		−	1.89807i	0.315173	−	0.949034i	$-0.397938\pi$
$44$		0			0
$45$		−	51.9719i		−	0.172167i
$46$		0			0
$47$	26.7703			0.0830818			0.0415409	−	0.999137i	$-0.486773\pi$
$47$	26.7703			0.0830818			0.0415409	+	0.999137i	$0.486773\pi$
$48$		0			0
$49$	−314.484	+	136.927i	−0.916862	+	0.399204i
$50$		0			0
$51$		−	244.706i		−	0.671877i
$52$		0			0
$53$	671.696			1.74084			0.870420	−	0.492310i	$-0.163847\pi$
$53$	671.696			1.74084			0.870420	+	0.492310i	$0.163847\pi$
$54$		0			0
$55$	83.9585			0.205836
$56$		0			0
$57$	267.090			0.620648
$58$		0			0
$59$	−546.205			−1.20525			−0.602626	−	0.798024i	$-0.705879\pi$
$59$	−546.205			−1.20525			−0.602626	+	0.798024i	$0.705879\pi$
$60$		0			0
$61$		−	815.566i		−	1.71184i	−0.517105	−	0.855922i	$-0.672990\pi$
$61$		−	815.566i		−	1.71184i	0.517105	−	0.855922i	$-0.327010\pi$
$62$		0			0
$63$	33.9840	+	163.181i	0.0679615	+	0.326332i
$64$		0			0
$65$	435.983			0.831955
$66$		0			0
$67$			605.151i			1.10345i	0.834027	+	0.551723i	$0.186030\pi$
$67$			605.151i			1.10345i	−0.834027	+	0.551723i	$0.813970\pi$
$68$		0			0
$69$			28.6147i			0.0499247i
$70$		0			0
$71$			287.911i			0.481250i	0.970618	+	0.240625i	$0.0773525\pi$
$71$			287.911i			0.481250i	−0.970618	+	0.240625i	$0.922648\pi$
$72$		0			0
$73$		−	64.3771i		−	0.103216i	−0.998667	−	0.0516080i	$-0.983565\pi$
$73$		−	64.3771i		−	0.103216i	0.998667	−	0.0516080i	$-0.0164346\pi$
$74$		0			0
$75$	−274.960			−0.423329
$76$		0			0
$77$	−263.612	+	54.8997i	−0.390148	+	0.0812519i
$78$		0			0
$79$		−	131.742i		−	0.187622i	−0.995590	−	0.0938111i	$-0.970095\pi$
$79$		−	131.742i		−	0.187622i	0.995590	−	0.0938111i	$-0.0299050\pi$
$80$		0			0
$81$	81.0000			0.111111
$82$		0			0
$83$	−822.082			−1.08717			−0.543586	−	0.839353i	$-0.682934\pi$
$83$	−822.082			−1.08717			−0.543586	+	0.839353i	$0.682934\pi$
$84$		0			0
$85$	471.031			0.601065
$86$		0			0
$87$	777.864			0.958572
$88$		0			0
$89$			737.170i			0.877976i	0.898493	+	0.438988i	$0.144663\pi$
$89$			737.170i			0.877976i	−0.898493	+	0.438988i	$0.855337\pi$
$90$		0			0
$91$	−1368.90	+	285.086i	−1.57692	+	0.328408i
$92$		0			0
$93$	−40.5461			−0.0452090
$94$		0			0
$95$			514.117i			0.555235i
$96$		0			0
$97$		−	348.816i		−	0.365122i	−0.983194	−	0.182561i	$-0.941561\pi$
$97$		−	348.816i		−	0.365122i	0.983194	−	0.182561i	$-0.0584388\pi$
$98$		0			0
$99$			130.852i			0.132840i
$100$		0			0
$101$		−	947.473i		−	0.933436i	−0.884406	−	0.466718i	$-0.845436\pi$
$101$		−	947.473i		−	0.933436i	0.884406	−	0.466718i	$-0.154564\pi$
$102$		0			0
$103$	467.433			0.447161			0.223580	−	0.974686i	$-0.428226\pi$
$103$	467.433			0.447161			0.223580	+	0.974686i	$0.428226\pi$
$104$		0			0
$105$	−314.105	+	65.4152i	−0.291938	+	0.0607988i
$106$		0			0
$107$			277.654i			0.250859i	0.992103	+	0.125429i	$0.0400308\pi$
$107$			277.654i			0.250859i	−0.992103	+	0.125429i	$0.959969\pi$
$108$		0			0
$109$	1225.56			1.07695			0.538474	−	0.842642i	$-0.319001\pi$
$109$	1225.56			1.07695			0.538474	+	0.842642i	$0.319001\pi$
$110$		0			0
$111$	−590.802			−0.505193
$112$		0			0
$113$	−1993.16			−1.65930			−0.829648	−	0.558287i	$-0.811459\pi$
$113$	−1993.16			−1.65930			−0.829648	+	0.558287i	$0.811459\pi$
$114$		0			0
$115$	−55.0800			−0.0446629
$116$		0			0
$117$			679.495i			0.536918i
$118$		0			0
$119$	−1478.94	+	308.003i	−1.13928	+	0.237265i
$120$		0			0
$121$	1119.61			0.841182
$122$		0			0
$123$		−	1024.35i		−	0.750917i
$124$		0			0
$125$		−	1251.10i		−	0.895213i
$126$		0			0
$127$		−	1972.83i		−	1.37843i	−0.724558	−	0.689213i	$-0.757956\pi$
$127$		−	1972.83i		−	1.37843i	0.724558	−	0.689213i	$-0.242044\pi$
$128$		0			0
$129$			1605.59i			1.09585i
$130$		0			0
$131$	−2251.70			−1.50177			−0.750886	−	0.660432i	$-0.770373\pi$
$131$	−2251.70			−1.50177			−0.750886	+	0.660432i	$0.770373\pi$
$132$		0			0
$133$	−336.177	−	1614.22i	−0.219175	−	1.05241i
$134$		0			0
$135$			155.916i			0.0994006i
$136$		0			0
$137$	1185.62			0.739377			0.369689	−	0.929156i	$-0.379464\pi$
$137$	1185.62			0.739377			0.369689	+	0.929156i	$0.379464\pi$
$138$		0			0
$139$	−2597.62			−1.58509			−0.792544	−	0.609814i	$-0.791244\pi$
$139$	−2597.62			−1.58509			−0.792544	+	0.609814i	$0.791244\pi$
$140$		0			0
$141$	−80.3108			−0.0479673
$142$		0			0
$143$	−1097.70			−0.641916
$144$		0			0
$145$			1497.30i			0.857544i
$146$		0			0
$147$	943.451	−	410.781i	0.529351	−	0.230480i
$148$		0			0
$149$	−1445.20			−0.794601			−0.397300	−	0.917689i	$-0.630053\pi$
$149$	−1445.20			−0.794601			−0.397300	+	0.917689i	$0.630053\pi$
$150$		0			0
$151$		−	1876.14i		−	1.01111i	−0.862793	−	0.505557i	$-0.831287\pi$
$151$		−	1876.14i		−	1.01111i	0.862793	−	0.505557i	$-0.168713\pi$
$152$		0			0
$153$			734.118i			0.387908i
$154$		0			0
$155$		−	78.0466i		−	0.0404442i
$156$		0			0
$157$			371.136i			0.188661i	0.995541	+	0.0943307i	$0.0300711\pi$
$157$			371.136i			0.188661i	−0.995541	+	0.0943307i	$0.969929\pi$
$158$		0			0
$159$	−2015.09			−1.00507
$160$		0			0
$161$	172.940	−	36.0163i	0.0846558	−	0.0176303i
$162$		0			0
$163$			766.086i			0.368126i	0.982914	+	0.184063i	$0.0589250\pi$
$163$			766.086i			0.368126i	−0.982914	+	0.184063i	$0.941075\pi$
$164$		0			0
$165$	−251.875			−0.118839
$166$		0			0
$167$	2296.51			1.06413			0.532064	−	0.846704i	$-0.321417\pi$
$167$	2296.51			1.06413			0.532064	+	0.846704i	$0.321417\pi$
$168$		0			0
$169$	−3503.17			−1.59452
$170$		0			0
$171$	−801.270			−0.358331
$172$		0			0
$173$			1486.15i			0.653119i	0.945177	+	0.326560i	$0.105889\pi$
$173$			1486.15i			0.653119i	−0.945177	+	0.326560i	$0.894111\pi$
$174$		0			0
$175$	346.083	+	1661.79i	0.149494	+	0.717825i
$176$		0			0
$177$	1638.62			0.695853
$178$		0			0
$179$		−	1165.15i		−	0.486523i	−0.969961	−	0.243262i	$-0.921783\pi$
$179$		−	1165.15i		−	0.486523i	0.969961	−	0.243262i	$-0.0782173\pi$
$180$		0			0
$181$			1623.96i			0.666894i	0.942769	+	0.333447i	$0.108212\pi$
$181$			1623.96i			0.666894i	−0.942769	+	0.333447i	$0.891788\pi$
$182$		0			0
$183$			2446.70i			0.988333i
$184$		0			0
$185$		−	1137.23i		−	0.451949i
$186$		0			0
$187$	−1185.94			−0.463767
$188$		0			0
$189$	−101.952	−	489.543i	−0.0392376	−	0.188408i
$190$		0			0
$191$			1614.70i			0.611705i	0.952079	+	0.305852i	$0.0989414\pi$
$191$			1614.70i			0.611705i	−0.952079	+	0.305852i	$0.901059\pi$
$192$		0			0
$193$	−1997.58			−0.745021			−0.372510	−	0.928028i	$-0.621503\pi$
$193$	−1997.58			−0.745021			−0.372510	+	0.928028i	$0.621503\pi$
$194$		0			0
$195$	−1307.95			−0.480329
$196$		0			0
$197$	−1574.98			−0.569606			−0.284803	−	0.958586i	$-0.591928\pi$
$197$	−1574.98			−0.569606			−0.284803	+	0.958586i	$0.591928\pi$
$198$		0			0
$199$	−3994.83			−1.42305			−0.711523	−	0.702663i	$-0.751994\pi$
$199$	−3994.83			−1.42305			−0.711523	+	0.702663i	$0.751994\pi$
$200$		0			0
$201$		−	1815.45i		−	0.637075i
$202$		0			0
$203$	−979.070	−	4701.21i	−0.338509	−	1.62542i
$204$		0			0
$205$	1971.76			0.671775
$206$		0			0
$207$		−	85.8441i		−	0.0288240i
$208$		0			0
$209$		−	1294.42i		−	0.428406i
$210$		0			0
$211$			808.768i			0.263876i	0.991258	+	0.131938i	$0.0421200\pi$
$211$			808.768i			0.263876i	−0.991258	+	0.131938i	$0.957880\pi$
$212$		0			0
$213$		−	863.734i		−	0.277850i
$214$		0			0
$215$	−3090.58			−0.980354
$216$		0			0
$217$	51.0340	+	245.050i	0.0159650	+	0.0766595i
$218$		0			0
$219$			193.131i			0.0595918i
$220$		0			0
$221$	−6158.39			−1.87447
$222$		0			0
$223$	−5103.24			−1.53246			−0.766229	−	0.642567i	$-0.777869\pi$
$223$	−5103.24			−1.53246			−0.766229	+	0.642567i	$0.777869\pi$
$224$		0			0
$225$	824.880			0.244409
$226$		0			0
$227$	−2534.63			−0.741099			−0.370549	−	0.928813i	$-0.620831\pi$
$227$	−2534.63			−0.741099			−0.370549	+	0.928813i	$0.620831\pi$
$228$		0			0
$229$			2323.15i			0.670386i	0.942150	+	0.335193i	$0.108802\pi$
$229$			2323.15i			0.670386i	−0.942150	+	0.335193i	$0.891198\pi$
$230$		0			0
$231$	790.837	−	164.699i	0.225252	−	0.0469108i
$232$		0			0
$233$	−1272.46			−0.357776			−0.178888	−	0.983869i	$-0.557250\pi$
$233$	−1272.46			−0.357776			−0.178888	+	0.983869i	$0.557250\pi$
$234$		0			0
$235$		−	154.589i		−	0.0429118i
$236$		0			0
$237$			395.226i			0.108324i
$238$		0			0
$239$		−	312.303i		−	0.0845239i	−0.999107	−	0.0422620i	$-0.986544\pi$
$239$		−	312.303i		−	0.0845239i	0.999107	−	0.0422620i	$-0.0134564\pi$
$240$		0			0
$241$		−	1090.73i		−	0.291536i	−0.989319	−	0.145768i	$-0.953435\pi$
$241$		−	1090.73i		−	0.291536i	0.989319	−	0.145768i	$-0.0465653\pi$
$242$		0			0
$243$	−243.000			−0.0641500
$244$		0			0
$245$	790.706	+	1816.03i	0.206189	+	0.473560i
$246$		0			0
$247$		−	6721.72i		−	1.73155i
$248$		0			0
$249$	2466.25			0.627679
$250$		0			0
$251$	4320.91			1.08659			0.543294	−	0.839543i	$-0.317177\pi$
$251$	4320.91			1.08659			0.543294	+	0.839543i	$0.317177\pi$
$252$		0			0
$253$	138.678			0.0344608
$254$		0			0
$255$	−1413.09			−0.347025
$256$		0			0
$257$			138.980i			0.0337327i	0.999858	+	0.0168664i	$0.00536899\pi$
$257$			138.980i			0.0337327i	−0.999858	+	0.0168664i	$0.994631\pi$
$258$		0			0
$259$	743.622	+	3570.66i	0.178403	+	0.856640i
$260$		0			0
$261$	−2333.59			−0.553432
$262$		0			0
$263$		−	6782.11i		−	1.59012i	−0.606528	−	0.795062i	$-0.707438\pi$
$263$		−	6782.11i		−	1.59012i	0.606528	−	0.795062i	$-0.292562\pi$
$264$		0			0
$265$		−	3878.81i		−	0.899145i
$266$		0			0
$267$		−	2211.51i		−	0.506899i
$268$		0			0
$269$		−	5605.82i		−	1.27061i	−0.772263	−	0.635303i	$-0.780875\pi$
$269$		−	5605.82i		−	1.27061i	0.772263	−	0.635303i	$-0.219125\pi$
$270$		0			0
$271$	747.876			0.167639			0.0838196	−	0.996481i	$-0.473288\pi$
$271$	747.876			0.167639			0.0838196	+	0.996481i	$0.473288\pi$
$272$		0			0
$273$	4106.70	−	855.257i	0.910434	−	0.189606i
$274$		0			0
$275$			1332.56i			0.292205i
$276$		0			0
$277$	5603.44			1.21544			0.607722	−	0.794150i	$-0.292083\pi$
$277$	5603.44			1.21544			0.607722	+	0.794150i	$0.292083\pi$
$278$		0			0
$279$	121.638			0.0261014
$280$		0			0
$281$	1954.56			0.414945			0.207472	−	0.978241i	$-0.433476\pi$
$281$	1954.56			0.414945			0.207472	+	0.978241i	$0.433476\pi$
$282$		0			0
$283$	5011.56			1.05267			0.526336	−	0.850277i	$-0.323565\pi$
$283$	5011.56			1.05267			0.526336	+	0.850277i	$0.323565\pi$
$284$		0			0
$285$		−	1542.35i		−	0.320565i
$286$		0			0
$287$	−6190.93	+	1289.32i	−1.27331	+	0.265178i
$288$		0			0
$289$	−1740.45			−0.354254
$290$		0			0
$291$			1046.45i			0.210804i
$292$		0			0
$293$			8253.06i			1.64556i	0.568361	+	0.822779i	$0.307578\pi$
$293$			8253.06i			1.64556i	−0.568361	+	0.822779i	$0.692422\pi$
$294$		0			0
$295$			3154.15i			0.622514i
$296$		0			0
$297$		−	392.557i		−	0.0766951i
$298$		0			0
$299$	720.131			0.139285
$300$		0			0
$301$	9703.81	−	2020.91i	1.85820	−	0.386987i
$302$		0			0
$303$			2842.42i			0.538920i
$304$		0			0
$305$	−4709.61			−0.884169
$306$		0			0
$307$	−10554.1			−1.96207			−0.981034	−	0.193837i	$-0.937907\pi$
$307$	−10554.1			−1.96207			−0.981034	+	0.193837i	$0.937907\pi$
$308$		0			0
$309$	−1402.30			−0.258168
$310$		0			0
$311$	−9756.96			−1.77899			−0.889496	−	0.456943i	$-0.848944\pi$
$311$	−9756.96			−1.77899			−0.889496	+	0.456943i	$0.848944\pi$
$312$		0			0
$313$		−	8663.75i		−	1.56455i	−0.622934	−	0.782275i	$-0.714059\pi$
$313$		−	8663.75i		−	1.56455i	0.622934	−	0.782275i	$-0.285941\pi$
$314$		0			0
$315$	942.315	−	196.246i	0.168551	−	0.0351022i
$316$		0			0
$317$	−9685.17			−1.71600			−0.858002	−	0.513646i	$-0.828294\pi$
$317$	−9685.17			−1.71600			−0.858002	+	0.513646i	$0.828294\pi$
$318$		0			0
$319$		−	3769.82i		−	0.661660i
$320$		0			0
$321$		−	832.963i		−	0.144833i
$322$		0			0
$323$		−	7262.06i		−	1.25100i
$324$		0			0
$325$			6919.78i			1.18105i
$326$		0			0
$327$	−3676.68			−0.621776
$328$		0			0
$329$	101.084	+	485.378i	0.0169391	+	0.0813366i
$330$		0			0
$331$			10134.4i			1.68288i	0.540347	+	0.841442i	$0.318293\pi$
$331$			10134.4i			1.68288i	−0.540347	+	0.841442i	$0.681707\pi$
$332$		0			0
$333$	1772.41			0.291673
$334$		0			0
$335$	3494.54			0.569931
$336$		0			0
$337$	700.301			0.113198			0.0565992	−	0.998397i	$-0.481974\pi$
$337$	700.301			0.113198			0.0565992	+	0.998397i	$0.481974\pi$
$338$		0			0
$339$	5979.47			0.957995
$340$		0			0
$341$			196.502i			0.0312058i
$342$		0			0
$343$	−3670.14	−	5184.94i	−0.577753	−	0.816212i
$344$		0			0
$345$	165.240			0.0257861
$346$		0			0
$347$		−	6590.30i		−	1.01955i	−0.860306	−	0.509777i	$-0.829728\pi$
$347$		−	6590.30i		−	1.01955i	0.860306	−	0.509777i	$-0.170272\pi$
$348$		0			0
$349$			2636.31i			0.404351i	0.979349	+	0.202175i	$0.0648011\pi$
$349$			2636.31i			0.404351i	−0.979349	+	0.202175i	$0.935199\pi$
$350$		0			0
$351$		−	2038.49i		−	0.309989i
$352$		0			0
$353$			5950.82i			0.897252i	0.893720	+	0.448626i	$0.148086\pi$
$353$			5950.82i			0.897252i	−0.893720	+	0.448626i	$0.851914\pi$
$354$		0			0
$355$	1662.59			0.248566
$356$		0			0
$357$	4436.82	−	924.009i	0.657764	−	0.136985i
$358$		0			0
$359$			921.748i			0.135510i	0.997702	+	0.0677549i	$0.0215836\pi$
$359$			921.748i			0.135510i	−0.997702	+	0.0677549i	$0.978416\pi$
$360$		0			0
$361$	1067.34			0.155611
$362$		0			0
$363$	−3358.84			−0.485657
$364$		0			0
$365$	−371.756			−0.0533112
$366$		0			0
$367$	2627.64			0.373738			0.186869	−	0.982385i	$-0.440166\pi$
$367$	2627.64			0.373738			0.186869	+	0.982385i	$0.440166\pi$
$368$		0			0
$369$			3073.06i			0.433542i
$370$		0			0
$371$	2536.32	+	12178.7i	0.354930	+	1.70427i
$372$		0			0
$373$	−2183.43			−0.303093			−0.151546	−	0.988450i	$-0.548425\pi$
$373$	−2183.43			−0.303093			−0.151546	+	0.988450i	$0.548425\pi$
$374$		0			0
$375$			3753.29i			0.516852i
$376$		0			0
$377$		−	19576.1i		−	2.67433i
$378$		0			0
$379$			5540.25i			0.750880i	0.926847	+	0.375440i	$0.122508\pi$
$379$			5540.25i			0.750880i	−0.926847	+	0.375440i	$0.877492\pi$
$380$		0			0
$381$			5918.48i			0.795835i
$382$		0			0
$383$	4048.48			0.540124			0.270062	−	0.962843i	$-0.412956\pi$
$383$	4048.48			0.540124			0.270062	+	0.962843i	$0.412956\pi$
$384$		0			0
$385$	317.027	+	1522.27i	0.0419667	+	0.201512i
$386$		0			0
$387$		−	4816.78i		−	0.632690i
$388$		0			0
$389$	−3380.49			−0.440611			−0.220305	−	0.975431i	$-0.570705\pi$
$389$	−3380.49			−0.440611			−0.220305	+	0.975431i	$0.570705\pi$
$390$		0			0
$391$	778.021			0.100630
$392$		0			0
$393$	6755.10			0.867048
$394$		0			0
$395$	−760.765			−0.0969070
$396$		0			0
$397$			11865.9i			1.50008i	0.661392	+	0.750040i	$0.269966\pi$
$397$			11865.9i			1.50008i	−0.661392	+	0.750040i	$0.730034\pi$
$398$		0			0
$399$	1008.53	+	4842.67i	0.126541	+	0.607611i
$400$		0			0
$401$	−4336.29			−0.540011			−0.270005	−	0.962859i	$-0.587025\pi$
$401$	−4336.29			−0.540011			−0.270005	+	0.962859i	$0.587025\pi$
$402$		0			0
$403$			1020.40i			0.126129i
$404$		0			0
$405$		−	467.747i		−	0.0573890i
$406$		0			0
$407$			2863.25i			0.348712i
$408$		0			0
$409$			11019.1i			1.33217i	0.745874	+	0.666087i	$0.232032\pi$
$409$			11019.1i			1.33217i	−0.745874	+	0.666087i	$0.767968\pi$
$410$		0			0
$411$	−3556.87			−0.426880
$412$		0			0
$413$	−2062.47	−	9903.38i	−0.245732	−	1.17994i
$414$		0			0
$415$			4747.24i			0.561525i
$416$		0			0
$417$	7792.86			0.915151
$418$		0			0
$419$	9791.38			1.14162			0.570812	−	0.821081i	$-0.306629\pi$
$419$	9791.38			1.14162			0.570812	+	0.821081i	$0.306629\pi$
$420$		0			0
$421$	10701.6			1.23887			0.619437	−	0.785046i	$-0.287361\pi$
$421$	10701.6			1.23887			0.619437	+	0.785046i	$0.287361\pi$
$422$		0			0
$423$	240.932			0.0276939
$424$		0			0
$425$			7476.04i			0.853274i
$426$		0			0
$427$	14787.2	−	3079.57i	1.67589	−	0.349019i
$428$		0			0
$429$	3293.09			0.370610
$430$		0			0
$431$			11506.1i			1.28592i	0.765901	+	0.642959i	$0.222293\pi$
$431$			11506.1i			1.28592i	−0.765901	+	0.642959i	$0.777707\pi$
$432$		0			0
$433$			1134.62i			0.125926i	0.998016	+	0.0629632i	$0.0200551\pi$
$433$			1134.62i			0.125926i	−0.998016	+	0.0629632i	$0.979945\pi$
$434$		0			0
$435$		−	4491.90i		−	0.495103i
$436$		0			0
$437$			849.188i			0.0929570i
$438$		0			0
$439$	4908.19			0.533611			0.266806	−	0.963750i	$-0.414032\pi$
$439$	4908.19			0.533611			0.266806	+	0.963750i	$0.414032\pi$
$440$		0			0
$441$	−2830.35	+	1232.34i	−0.305621	+	0.133068i
$442$		0			0
$443$		−	127.763i		−	0.0137025i	−0.999977	−	0.00685124i	$-0.997819\pi$
$443$		−	127.763i		−	0.0137025i	0.999977	−	0.00685124i	$-0.00218083\pi$
$444$		0			0
$445$	4256.90			0.453475
$446$		0			0
$447$	4335.60			0.458763
$448$		0			0
$449$	−6741.08			−0.708533			−0.354267	−	0.935144i	$-0.615269\pi$
$449$	−6741.08			−0.708533			−0.354267	+	0.935144i	$0.615269\pi$
$450$		0			0
$451$	−4964.40			−0.518325
$452$		0			0
$453$			5628.42i			0.583767i
$454$		0			0
$455$	1646.27	+	7904.92i	0.169623	+	0.814480i
$456$		0			0
$457$	2976.15			0.304635			0.152318	−	0.988332i	$-0.451326\pi$
$457$	2976.15			0.304635			0.152318	+	0.988332i	$0.451326\pi$
$458$		0			0
$459$		−	2202.35i		−	0.223959i
$460$		0			0
$461$		−	6326.93i		−	0.639207i	−0.947551	−	0.319604i	$-0.896450\pi$
$461$		−	6326.93i		−	0.639207i	0.947551	−	0.319604i	$-0.103550\pi$
$462$		0			0
$463$		−	13109.7i		−	1.31589i	−0.753066	−	0.657945i	$-0.771426\pi$
$463$		−	13109.7i		−	1.31589i	0.753066	−	0.657945i	$-0.228574\pi$
$464$		0			0
$465$			234.140i			0.0233505i
$466$		0			0
$467$	−2556.42			−0.253313			−0.126656	−	0.991947i	$-0.540425\pi$
$467$	−2556.42			−0.253313			−0.126656	+	0.991947i	$0.540425\pi$
$468$		0			0
$469$	−10972.1	+	2285.05i	−1.08027	+	0.224976i
$470$		0			0
$471$		−	1113.41i		−	0.108924i
$472$		0			0
$473$	7781.32			0.756417
$474$		0			0
$475$	−8159.90			−0.788214
$476$		0			0
$477$	6045.26			0.580280
$478$		0			0
$479$	−4199.36			−0.400571			−0.200286	−	0.979738i	$-0.564187\pi$
$479$	−4199.36			−0.400571			−0.200286	+	0.979738i	$0.564187\pi$
$480$		0			0
$481$			14868.4i			1.40944i
$482$		0			0
$483$	−518.820	+	108.049i	−0.0488760	+	0.0101789i
$484$		0			0
$485$	−2014.29			−0.188586
$486$		0			0
$487$		−	4659.54i		−	0.433561i	−0.976220	−	0.216780i	$-0.930444\pi$
$487$		−	4659.54i		−	0.433561i	0.976220	−	0.216780i	$-0.0695555\pi$
$488$		0			0
$489$		−	2298.26i		−	0.212538i
$490$		0			0
$491$			6311.43i			0.580104i	0.957011	+	0.290052i	$0.0936726\pi$
$491$			6311.43i			0.580104i	−0.957011	+	0.290052i	$0.906327\pi$
$492$		0			0
$493$		−	21149.8i		−	1.93213i
$494$		0			0
$495$	755.626			0.0686119
$496$		0			0
$497$	−5220.19	+	1087.15i	−0.471142	+	0.0981195i
$498$		0			0
$499$			3080.68i			0.276373i	0.990406	+	0.138187i	$0.0441273\pi$
$499$			3080.68i			0.276373i	−0.990406	+	0.138187i	$0.955873\pi$
$500$		0			0
$501$	−6889.53			−0.614374
$502$		0			0
$503$	14985.9			1.32840			0.664201	−	0.747554i	$-0.268772\pi$
$503$	14985.9			1.32840			0.664201	+	0.747554i	$0.268772\pi$
$504$		0			0
$505$	−5471.33			−0.482121
$506$		0			0
$507$	10509.5			0.920599
$508$		0			0
$509$		−	5360.33i		−	0.466783i	−0.972383	−	0.233392i	$-0.925018\pi$
$509$		−	5360.33i		−	0.466783i	0.972383	−	0.233392i	$-0.0749824\pi$
$510$		0			0
$511$	1167.24	−	243.088i	0.101048	−	0.0210442i
$512$		0			0
$513$	2403.81			0.206883
$514$		0			0
$515$		−	2699.26i		−	0.230959i
$516$		0			0
$517$			389.216i			0.0331097i
$518$		0			0
$519$		−	4458.44i		−	0.377079i
$520$		0			0
$521$		−	4858.45i		−	0.408546i	−0.978914	−	0.204273i	$-0.934517\pi$
$521$		−	4858.45i		−	0.408546i	0.978914	−	0.204273i	$-0.0654831\pi$
$522$		0			0
$523$	11679.7			0.976517			0.488259	−	0.872699i	$-0.337632\pi$
$523$	11679.7			0.976517			0.488259	+	0.872699i	$0.337632\pi$
$524$		0			0
$525$	−1038.25	−	4985.37i	−0.0863102	−	0.414437i
$526$		0			0
$527$			1102.43i			0.0911246i
$528$		0			0
$529$	12076.0			0.992523
$530$		0			0
$531$	−4915.85			−0.401751
$532$		0			0
$533$	−25779.4			−2.09499
$534$		0			0
$535$	1603.36			0.129569
$536$		0			0
$537$			3495.46i			0.280894i
$538$		0			0
$539$	−1990.80	−	4572.32i	−0.159090	−	0.365387i
$540$		0			0
$541$	22704.0			1.80429			0.902147	−	0.431429i	$-0.141991\pi$
$541$	22704.0			1.80429			0.902147	+	0.431429i	$0.141991\pi$
$542$		0			0
$543$		−	4871.87i		−	0.385031i
$544$		0			0
$545$		−	7077.18i		−	0.556245i
$546$		0			0
$547$		−	9236.63i		−	0.721992i	−0.932567	−	0.360996i	$-0.882437\pi$
$547$		−	9236.63i		−	0.721992i	0.932567	−	0.360996i	$-0.117563\pi$
$548$		0			0
$549$		−	7340.09i		−	0.570615i
$550$		0			0
$551$	23084.4			1.78481
$552$		0			0
$553$	2388.65	−	497.458i	0.183681	−	0.0382533i
$554$		0			0
$555$			3411.68i			0.260933i
$556$		0			0
$557$	−2287.70			−0.174027			−0.0870136	−	0.996207i	$-0.527732\pi$
$557$	−2287.70			−0.174027			−0.0870136	+	0.996207i	$0.527732\pi$
$558$		0			0
$559$	40407.2			3.05732
$560$		0			0
$561$	3557.81			0.267756
$562$		0			0
$563$	15794.1			1.18231			0.591157	−	0.806556i	$-0.298671\pi$
$563$	15794.1			1.18231			0.591157	+	0.806556i	$0.298671\pi$
$564$		0			0
$565$			11509.8i			0.857027i
$566$		0			0
$567$	305.856	+	1468.63i	0.0226538	+	0.108777i
$568$		0			0
$569$	−19977.7			−1.47189			−0.735947	−	0.677039i	$-0.763263\pi$
$569$	−19977.7			−1.47189			−0.735947	+	0.677039i	$0.763263\pi$
$570$		0			0
$571$		−	20104.0i		−	1.47342i	−0.676207	−	0.736711i	$-0.736378\pi$
$571$		−	20104.0i		−	1.47342i	0.676207	−	0.736711i	$-0.263622\pi$
$572$		0			0
$573$		−	4844.10i		−	0.353168i
$574$		0			0
$575$		−	874.211i		−	0.0634037i
$576$		0			0
$577$			14026.0i			1.01198i	0.862540	+	0.505990i	$0.168873\pi$
$577$			14026.0i			1.01198i	−0.862540	+	0.505990i	$0.831127\pi$
$578$		0			0
$579$	5992.74			0.430138
$580$		0			0
$581$	−3104.18	−	14905.4i	−0.221658	−	1.06434i
$582$		0			0
$583$			9765.88i			0.693759i
$584$		0			0
$585$	3923.85			0.277318
$586$		0			0
$587$	−27143.4			−1.90857			−0.954283	−	0.298904i	$-0.903379\pi$
$587$	−27143.4			−1.90857			−0.954283	+	0.298904i	$0.903379\pi$
$588$		0			0
$589$	−1203.27			−0.0841766
$590$		0			0
$591$	4724.93			0.328862
$592$		0			0
$593$			6180.97i			0.428030i	0.976830	+	0.214015i	$0.0686542\pi$
$593$			6180.97i			0.428030i	−0.976830	+	0.214015i	$0.931346\pi$
$594$		0			0
$595$	1778.61	+	8540.38i	0.122548	+	0.588439i
$596$		0			0
$597$	11984.5			0.821596
$598$		0			0
$599$		−	11988.1i		−	0.817730i	−0.912595	−	0.408865i	$-0.865925\pi$
$599$		−	11988.1i		−	0.817730i	0.912595	−	0.408865i	$-0.134075\pi$
$600$		0			0
$601$		−	27089.7i		−	1.83862i	−0.393534	−	0.919310i	$-0.628748\pi$
$601$		−	27089.7i		−	1.83862i	0.393534	−	0.919310i	$-0.371252\pi$
$602$		0			0
$603$			5446.36i			0.367815i
$604$		0			0
$605$		−	6465.38i		−	0.434471i
$606$		0			0
$607$	1352.56			0.0904425			0.0452212	−	0.998977i	$-0.485601\pi$
$607$	1352.56			0.0904425			0.0452212	+	0.998977i	$0.485601\pi$
$608$		0			0
$609$	2937.21	+	14103.6i	0.195438	+	0.938437i
$610$		0			0
$611$			2021.14i			0.133824i
$612$		0			0
$613$	13838.7			0.911808			0.455904	−	0.890029i	$-0.349316\pi$
$613$	13838.7			0.911808			0.455904	+	0.890029i	$0.349316\pi$
$614$		0			0
$615$	−5915.28			−0.387849
$616$		0			0
$617$	−14746.5			−0.962191			−0.481096	−	0.876668i	$-0.659761\pi$
$617$	−14746.5			−0.962191			−0.481096	+	0.876668i	$0.659761\pi$
$618$		0			0
$619$	1773.86			0.115182			0.0575909	−	0.998340i	$-0.481658\pi$
$619$	1773.86			0.115182			0.0575909	+	0.998340i	$0.481658\pi$
$620$		0			0
$621$			257.532i			0.0166416i
$622$		0			0
$623$	−13365.8	+	2783.55i	−0.859534	+	0.179006i
$624$		0			0
$625$	4232.01			0.270849
$626$		0			0
$627$			3883.26i			0.247340i
$628$		0			0
$629$			16063.6i			1.01828i
$630$		0			0
$631$			16563.6i			1.04499i	0.852643	+	0.522494i	$0.174998\pi$
$631$			16563.6i			1.04499i	−0.852643	+	0.522494i	$0.825002\pi$
$632$		0			0
$633$		−	2426.30i		−	0.152349i
$634$		0			0
$635$	−11392.4			−0.711959
$636$		0			0
$637$	−10337.9	−	23743.4i	−0.643019	−	1.47684i
$638$		0			0
$639$			2591.20i			0.160417i
$640$		0			0
$641$	−22907.9			−1.41156			−0.705779	−	0.708432i	$-0.749403\pi$
$641$	−22907.9			−1.41156			−0.705779	+	0.708432i	$0.749403\pi$
$642$		0			0
$643$	2919.92			0.179083			0.0895414	−	0.995983i	$-0.471460\pi$
$643$	2919.92			0.179083			0.0895414	+	0.995983i	$0.471460\pi$
$644$		0			0
$645$	9271.75			0.566008
$646$		0			0
$647$	23638.1			1.43634			0.718169	−	0.695868i	$-0.244980\pi$
$647$	23638.1			1.43634			0.718169	+	0.695868i	$0.244980\pi$
$648$		0			0
$649$		−	7941.35i		−	0.480316i
$650$		0			0
$651$	−153.102	−	735.151i	−0.00921741	−	0.0442594i
$652$		0			0
$653$	−3983.07			−0.238698			−0.119349	−	0.992852i	$-0.538081\pi$
$653$	−3983.07			−0.238698			−0.119349	+	0.992852i	$0.538081\pi$
$654$		0			0
$655$			13002.8i			0.775666i
$656$		0			0
$657$		−	579.394i		−	0.0344053i
$658$		0			0
$659$			13889.1i			0.821004i	0.911860	+	0.410502i	$0.134646\pi$
$659$			13889.1i			0.821004i	−0.911860	+	0.410502i	$0.865354\pi$
$660$		0			0
$661$		−	11415.7i		−	0.671736i	−0.941909	−	0.335868i	$-0.890970\pi$
$661$		−	11415.7i		−	0.671736i	0.941909	−	0.335868i	$-0.109030\pi$
$662$		0			0
$663$	18475.2			1.08223
$664$		0			0
$665$	−9321.58	+	1941.30i	−0.543572	+	0.113204i
$666$		0			0
$667$			2473.15i			0.143569i
$668$		0			0
$669$	15309.7			0.884765
$670$		0			0
$671$	11857.6			0.682203
$672$		0			0
$673$	10799.6			0.618567			0.309284	−	0.950970i	$-0.399911\pi$
$673$	10799.6			0.618567			0.309284	+	0.950970i	$0.399911\pi$
$674$		0			0
$675$	−2474.64			−0.141110
$676$		0			0
$677$			11603.5i			0.658730i	0.944203	+	0.329365i	$0.106835\pi$
$677$			11603.5i			0.658730i	−0.944203	+	0.329365i	$0.893165\pi$
$678$		0			0
$679$	6324.46	−	1317.13i	0.357453	−	0.0744428i
$680$		0			0
$681$	7603.90			0.427874
$682$		0			0
$683$			18698.8i			1.04757i	0.851851	+	0.523784i	$0.175480\pi$
$683$			18698.8i			1.04757i	−0.851851	+	0.523784i	$0.824520\pi$
$684$		0			0
$685$		−	6846.57i		−	0.381889i
$686$		0			0
$687$		−	6969.46i		−	0.387047i
$688$		0			0
$689$			50712.7i			2.80406i
$690$		0			0
$691$	7549.67			0.415634			0.207817	−	0.978168i	$-0.433364\pi$
$691$	7549.67			0.415634			0.207817	+	0.978168i	$0.433364\pi$
$692$		0			0
$693$	−2372.51	+	494.097i	−0.130049	+	0.0270840i
$694$		0			0
$695$			15000.4i			0.818699i
$696$		0			0
$697$	−27851.7			−1.51357
$698$		0			0
$699$	3817.38			0.206562
$700$		0			0
$701$	−6127.34			−0.330138			−0.165069	−	0.986282i	$-0.552785\pi$
$701$	−6127.34			−0.330138			−0.165069	+	0.986282i	$0.552785\pi$
$702$		0			0
$703$	−17533.0			−0.940641
$704$		0			0
$705$			463.767i			0.0247751i
$706$		0			0
$707$	17178.9	−	3577.65i	0.913829	−	0.190313i
$708$		0			0
$709$	−3209.29			−0.169996			−0.0849981	−	0.996381i	$-0.527088\pi$
$709$	−3209.29			−0.169996			−0.0849981	+	0.996381i	$0.527088\pi$
$710$		0			0
$711$		−	1185.68i		−	0.0625407i
$712$		0			0
$713$		−	128.913i		−	0.00677113i
$714$		0			0
$715$			6338.82i			0.331550i
$716$		0			0
$717$			936.910i			0.0487999i
$718$		0			0
$719$	5415.53			0.280897			0.140449	−	0.990088i	$-0.455146\pi$
$719$	5415.53			0.280897			0.140449	+	0.990088i	$0.455146\pi$
$720$		0			0
$721$	1765.02	+	8475.14i	0.0911691	+	0.437768i
$722$		0			0
$723$			3272.20i			0.168319i
$724$		0			0
$725$	−23764.6			−1.21737
$726$		0			0
$727$	12165.9			0.620643			0.310322	−	0.950632i	$-0.399563\pi$
$727$	12165.9			0.620643			0.310322	+	0.950632i	$0.399563\pi$
$728$		0			0
$729$	729.000			0.0370370
$730$		0			0
$731$	43655.4			2.20883
$732$		0			0
$733$		−	7503.91i		−	0.378122i	−0.981965	−	0.189061i	$-0.939456\pi$
$733$		−	7503.91i		−	0.378122i	0.981965	−	0.189061i	$-0.0605444\pi$
$734$		0			0
$735$	−2372.12	−	5448.10i	−0.119043	−	0.273410i
$736$		0			0
$737$	−8798.36			−0.439745
$738$		0			0
$739$		−	25591.6i		−	1.27389i	−0.770910	−	0.636944i	$-0.780198\pi$
$739$		−	25591.6i		−	1.27389i	0.770910	−	0.636944i	$-0.219802\pi$
$740$		0			0
$741$			20165.1i			0.999710i
$742$		0			0
$743$			13474.7i			0.665328i	0.943045	+	0.332664i	$0.107947\pi$
$743$			13474.7i			0.665328i	−0.943045	+	0.332664i	$0.892053\pi$
$744$		0			0
$745$			8345.54i			0.410412i
$746$		0			0
$747$	−7398.74			−0.362391
$748$		0			0
$749$	−5034.22	+	1048.42i	−0.245589	+	0.0511462i
$750$		0			0
$751$		−	6265.52i		−	0.304437i	−0.988347	−	0.152219i	$-0.951358\pi$
$751$		−	6265.52i		−	0.304437i	0.988347	−	0.152219i	$-0.0486418\pi$
$752$		0			0
$753$	−12962.7			−0.627342
$754$		0			0
$755$	−10834.1			−0.522241
$756$		0			0
$757$	−12600.0			−0.604960			−0.302480	−	0.953156i	$-0.597815\pi$
$757$	−12600.0			−0.604960			−0.302480	+	0.953156i	$0.597815\pi$
$758$		0			0
$759$	−416.033			−0.0198960
$760$		0			0
$761$		−	14110.8i		−	0.672163i	−0.941833	−	0.336082i	$-0.890898\pi$
$761$		−	14110.8i		−	0.672163i	0.941833	−	0.336082i	$-0.109102\pi$
$762$		0			0
$763$	4627.71	+	22220.9i	0.219573	+	1.05433i
$764$		0			0
$765$	4239.28			0.200355
$766$		0			0
$767$		−	41238.2i		−	1.94136i
$768$		0			0
$769$		−	18732.3i		−	0.878419i	−0.898385	−	0.439210i	$-0.855259\pi$
$769$		−	18732.3i		−	0.878419i	0.898385	−	0.439210i	$-0.144741\pi$
$770$		0			0
$771$		−	416.939i		−	0.0194756i
$772$		0			0
$773$			17750.9i			0.825943i	0.910744	+	0.412972i	$0.135509\pi$
$773$			17750.9i			0.825943i	−0.910744	+	0.412972i	$0.864491\pi$
$774$		0			0
$775$	1238.73			0.0574148
$776$		0			0
$777$	−2230.86	−	10712.0i	−0.103001	−	0.494581i
$778$		0			0
$779$		−	30399.4i		−	1.39817i
$780$		0			0
$781$	−4185.98			−0.191788
$782$		0			0
$783$	7000.78			0.319524
$784$		0			0
$785$	2143.18			0.0974438
$786$		0			0
$787$	12685.4			0.574567			0.287283	−	0.957846i	$-0.407248\pi$
$787$	12685.4			0.574567			0.287283	+	0.957846i	$0.407248\pi$
$788$		0			0
$789$			20346.3i			0.918059i
$790$		0			0
$791$	−7526.15	−	36138.4i	−0.338305	−	1.62444i
$792$		0			0
$793$	61574.8			2.75736
$794$		0			0
$795$			11636.4i			0.519122i
$796$		0			0
$797$		−	21163.9i		−	0.940608i	−0.882505	−	0.470304i	$-0.844144\pi$
$797$		−	21163.9i		−	0.940608i	0.882505	−	0.470304i	$-0.155856\pi$
$798$		0			0
$799$			2183.62i			0.0966843i
$800$		0			0
$801$			6634.53i			0.292659i
$802$		0			0
$803$	935.987			0.0411336
$804$		0			0
$805$	−207.982	−	998.668i	−0.00910608	−	0.0437248i
$806$		0			0
$807$			16817.5i			0.733585i
$808$		0			0
$809$	−18524.3			−0.805042			−0.402521	−	0.915411i	$-0.631866\pi$
$809$	−18524.3			−0.805042			−0.402521	+	0.915411i	$0.631866\pi$
$810$		0			0
$811$	9847.07			0.426359			0.213180	−	0.977013i	$-0.431618\pi$
$811$	9847.07			0.426359			0.213180	+	0.977013i	$0.431618\pi$
$812$		0			0
$813$	−2243.63			−0.0967865
$814$		0			0
$815$	4423.88			0.190137
$816$		0			0
$817$			47648.7i			2.04041i
$818$		0			0
$819$	−12320.1	+	2565.77i	−0.525640	+	0.109469i
$820$		0			0
$821$	29068.7			1.23570			0.617848	−	0.786298i	$-0.288005\pi$
$821$	29068.7			1.23570			0.617848	+	0.786298i	$0.288005\pi$
$822$		0			0
$823$			39075.4i			1.65502i	0.561450	+	0.827511i	$0.310244\pi$
$823$			39075.4i			1.65502i	−0.561450	+	0.827511i	$0.689756\pi$
$824$		0			0
$825$		−	3997.68i		−	0.168705i
$826$		0			0
$827$			2846.21i			0.119677i	0.998208	+	0.0598383i	$0.0190585\pi$
$827$			2846.21i			0.119677i	−0.998208	+	0.0598383i	$0.980941\pi$
$828$		0			0
$829$			1936.00i			0.0811099i	0.999177	+	0.0405550i	$0.0129126\pi$
$829$			1936.00i			0.0811099i	−0.999177	+	0.0405550i	$0.987087\pi$
$830$		0			0
$831$	−16810.3			−0.701737
$832$		0			0
$833$	−11169.0	−	25652.0i	−0.464563	−	1.06697i
$834$		0			0
$835$		−	13261.6i		−	0.549623i
$836$		0			0
$837$	−364.915			−0.0150697
$838$		0			0
$839$	17052.5			0.701690			0.350845	−	0.936434i	$-0.385894\pi$
$839$	17052.5			0.701690			0.350845	+	0.936434i	$0.385894\pi$
$840$		0			0
$841$	42841.3			1.75658
$842$		0			0
$843$	−5863.69			−0.239568
$844$		0			0
$845$			20229.6i			0.823573i
$846$		0			0
$847$	4227.66	+	20300.0i	0.171504	+	0.823513i
$848$		0			0
$849$	−15034.7			−0.607761
$850$		0			0
$851$		−	1878.40i		−	0.0756648i
$852$		0			0
$853$			24807.3i			0.995764i	0.867245	+	0.497882i	$0.165889\pi$
$853$			24807.3i			0.995764i	−0.867245	+	0.497882i	$0.834111\pi$
$854$		0			0
$855$			4627.06i			0.185078i
$856$		0			0
$857$		−	35069.3i		−	1.39784i	−0.715202	−	0.698918i	$-0.753665\pi$
$857$		−	35069.3i		−	1.39784i	0.715202	−	0.698918i	$-0.246335\pi$
$858$		0			0
$859$	−42164.9			−1.67479			−0.837397	−	0.546596i	$-0.815923\pi$
$859$	−42164.9			−1.67479			−0.837397	+	0.546596i	$0.815923\pi$
$860$		0			0
$861$	18572.8	−	3867.95i	0.735144	−	0.153100i
$862$		0			0
$863$			25133.2i			0.991360i	0.868505	+	0.495680i	$0.165081\pi$
$863$			25133.2i			0.991360i	−0.868505	+	0.495680i	$0.834919\pi$
$864$		0			0
$865$	8581.98			0.337337
$866$		0			0
$867$	5221.35			0.204529
$868$		0			0
$869$	1915.42			0.0747710
$870$		0			0
$871$	−45688.5			−1.77738
$872$		0			0
$873$		−	3139.34i		−	0.121707i
$874$		0			0
$875$	22684.0	−	4724.14i	0.876409	−	0.182520i
$876$		0			0
$877$	−47942.9			−1.84597			−0.922985	−	0.384835i	$-0.874258\pi$
$877$	−47942.9			−1.84597			−0.922985	+	0.384835i	$0.874258\pi$
$878$		0			0
$879$		−	24759.2i		−	0.950064i
$880$		0			0
$881$		−	40982.4i		−	1.56723i	−0.621244	−	0.783617i	$-0.713373\pi$
$881$		−	40982.4i		−	1.56723i	0.621244	−	0.783617i	$-0.286627\pi$
$882$		0			0
$883$		−	23832.4i		−	0.908295i	−0.890927	−	0.454147i	$-0.849944\pi$
$883$		−	23832.4i		−	0.908295i	0.890927	−	0.454147i	$-0.150056\pi$
$884$		0			0
$885$		−	9462.44i		−	0.359408i
$886$		0			0
$887$	−33825.6			−1.28044			−0.640220	−	0.768191i	$-0.721157\pi$
$887$	−33825.6			−1.28044			−0.640220	+	0.768191i	$0.721157\pi$
$888$		0			0
$889$	35769.8	−	7449.39i	1.34947	−	0.281040i
$890$		0			0
$891$			1177.67i			0.0442799i
$892$		0			0
$893$	−2383.36			−0.0893124
$894$		0			0
$895$	−6728.36			−0.251290
$896$		0			0
$897$	−2160.39			−0.0804163
$898$		0			0
$899$	−3504.37			−0.130008
$900$		0			0
$901$			54789.4i			2.02586i
$902$		0			0
$903$	−29111.4	+	6062.72i	−1.07283	+	0.223427i
$904$		0			0
$905$	9377.79			0.344451
$906$		0			0
$907$			43691.4i			1.59950i	0.600330	+	0.799752i	$0.295036\pi$
$907$			43691.4i			1.59950i	−0.600330	+	0.799752i	$0.704964\pi$
$908$		0			0
$909$		−	8527.26i		−	0.311145i
$910$		0			0
$911$		−	16717.9i		−	0.608001i	−0.952672	−	0.304000i	$-0.901678\pi$
$911$		−	16717.9i		−	0.608001i	0.952672	−	0.304000i	$-0.0983224\pi$
$912$		0			0
$913$		−	11952.4i		−	0.433259i
$914$		0			0
$915$	14128.8			0.510475
$916$		0			0
$917$	−8502.41	−	40826.1i	−0.306188	−	1.47023i
$918$		0			0
$919$			7962.36i			0.285804i	0.989737	+	0.142902i	$0.0456434\pi$
$919$			7962.36i			0.285804i	−0.989737	+	0.142902i	$0.954357\pi$
$920$		0			0
$921$	31662.3			1.13280
$922$		0			0
$923$	−21737.1			−0.775175
$924$		0			0
$925$	18049.7			0.641588
$926$		0			0
$927$	4206.90			0.149054
$928$		0			0
$929$			24057.0i			0.849606i	0.905286	+	0.424803i	$0.139657\pi$
$929$			24057.0i			0.849606i	−0.905286	+	0.424803i	$0.860343\pi$
$930$		0			0
$931$	27998.5	−	12190.6i	0.985621	−	0.429142i
$932$		0			0
$933$	29270.9			1.02710
$934$		0			0
$935$			6848.38i			0.239536i
$936$		0			0
$937$		−	42305.5i		−	1.47498i	−0.675356	−	0.737492i	$-0.736010\pi$
$937$		−	42305.5i		−	1.47498i	0.675356	−	0.737492i	$-0.263990\pi$
$938$		0			0
$939$			25991.2i			0.903293i
$940$		0			0
$941$		−	43425.7i		−	1.50440i	−0.658936	−	0.752199i	$-0.728993\pi$
$941$		−	43425.7i		−	1.50440i	0.658936	−	0.752199i	$-0.271007\pi$
$942$		0			0
$943$	3256.84			0.112468
$944$		0			0
$945$	−2826.94	+	588.737i	−0.0973127	+	0.0202663i
$946$		0			0
$947$			40700.2i			1.39660i	0.715806	+	0.698299i	$0.246059\pi$
$947$			40700.2i			1.39660i	−0.715806	+	0.698299i	$0.753941\pi$
$948$		0			0
$949$	4860.44			0.166255
$950$		0			0
$951$	29055.5			0.990735
$952$		0			0
$953$	335.224			0.0113945			0.00569726	−	0.999984i	$-0.498186\pi$
$953$	335.224			0.0113945			0.00569726	+	0.999984i	$0.498186\pi$
$954$		0			0
$955$	9324.33			0.315946
$956$		0			0
$957$			11309.5i			0.382010i
$958$		0			0
$959$	4476.91	+	21496.8i	0.150748	+	0.723847i
$960$		0			0
$961$	−29608.3			−0.993868
$962$		0			0
$963$			2498.89i			0.0836195i
$964$		0			0
$965$			11535.3i			0.384804i
$966$		0			0
$967$		−	23773.5i		−	0.790595i	−0.918553	−	0.395298i	$-0.870641\pi$
$967$		−	23773.5i		−	0.790595i	0.918553	−	0.395298i	$-0.129359\pi$
$968$		0			0
$969$			21786.2i			0.722263i
$970$		0			0
$971$	30129.6			0.995781			0.497891	−	0.867240i	$-0.334108\pi$
$971$	30129.6			0.995781			0.497891	+	0.867240i	$0.334108\pi$
$972$		0			0
$973$	−9808.60	−	47098.1i	−0.323175	−	1.55179i
$974$		0			0
$975$		−	20759.3i		−	0.681878i
$976$		0			0
$977$	−2562.41			−0.0839086			−0.0419543	−	0.999120i	$-0.513358\pi$
$977$	−2562.41			−0.0839086			−0.0419543	+	0.999120i	$0.513358\pi$
$978$		0			0
$979$	−10717.8			−0.349890
$980$		0			0
$981$	11030.0			0.358983
$982$		0			0
$983$	−50346.8			−1.63358			−0.816792	−	0.576932i	$-0.804250\pi$
$983$	−50346.8			−1.63358			−0.816792	+	0.576932i	$0.804250\pi$
$984$		0			0
$985$			9094.94i			0.294202i
$986$		0			0
$987$	−303.253	−	1456.13i	−0.00977979	−	0.0469597i
$988$		0			0
$989$	−5104.84			−0.164130
$990$		0			0
$991$			6553.24i			0.210061i	0.994469	+	0.105031i	$0.0334941\pi$
$991$			6553.24i			0.210061i	−0.994469	+	0.105031i	$0.966506\pi$
$992$		0			0
$993$		−	30403.1i		−	0.971614i
$994$		0			0
$995$			23068.8i			0.735004i
$996$		0			0
$997$			34220.3i			1.08703i	0.839400	+	0.543514i	$0.182907\pi$
$997$			34220.3i			1.08703i	−0.839400	+	0.543514i	$0.817093\pi$
$998$		0			0
$999$	−5317.22			−0.168398

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.4.b.i.895.9		24
4.3	odd	2		1344.4.b.j.895.9		24
7.6	odd	2		1344.4.b.j.895.16		24
8.3	odd	2		672.4.b.a.223.16	yes	24
8.5	even	2		672.4.b.b.223.16	yes	24
28.27	even	2	inner	1344.4.b.i.895.16		24
56.13	odd	2		672.4.b.a.223.9	✓	24
56.27	even	2		672.4.b.b.223.9	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.4.b.a.223.9	✓	24	56.13	odd	2
672.4.b.a.223.16	yes	24	8.3	odd	2
672.4.b.b.223.9	yes	24	56.27	even	2
672.4.b.b.223.16	yes	24	8.5	even	2
1344.4.b.i.895.9		24	1.1	even	1	trivial
1344.4.b.i.895.16		24	28.27	even	2	inner
1344.4.b.j.895.9		24	4.3	odd	2
1344.4.b.j.895.16		24	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} -5.77465i q^{5} +(3.77600 + 18.1312i) q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} -5.77465i q^{5} +(3.77600 + 18.1312i) q^{7} +9.00000 q^{9} +14.5391i q^{11} +75.4995i q^{13} +17.3240i q^{15} +81.5687i q^{17} -89.0300 q^{19} +(-11.3280 - 54.3937i) q^{21} -9.53823i q^{23} +91.6534 q^{25} -27.0000 q^{27} -259.288 q^{29} +13.5154 q^{31} -43.6174i q^{33} +(104.702 - 21.8051i) q^{35} +196.934 q^{37} -226.498i q^{39} +341.451i q^{41} -535.198i q^{43} -51.9719i q^{45} +26.7703 q^{47} +(-314.484 + 136.927i) q^{49} -244.706i q^{51} +671.696 q^{53} +83.9585 q^{55} +267.090 q^{57} -546.205 q^{59} -815.566i q^{61} +(33.9840 + 163.181i) q^{63} +435.983 q^{65} +605.151i q^{67} +28.6147i q^{69} +287.911i q^{71} -64.3771i q^{73} -274.960 q^{75} +(-263.612 + 54.8997i) q^{77} -131.742i q^{79} +81.0000 q^{81} -822.082 q^{83} +471.031 q^{85} +777.864 q^{87} +737.170i q^{89} +(-1368.90 + 285.086i) q^{91} -40.5461 q^{93} +514.117i q^{95} -348.816i q^{97} +130.852i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 72 q^{3} + 20 q^{7} + 216 q^{9}+O(q^{10}) \) 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

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Representations

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Database

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