LMFDB - Embedded newform 384.7.b.d.319.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,7,Mod(319,384)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(384, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 0]))

N = Newforms(chi, 7, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("384.319");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	384.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$88.3407681100$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.1731891456.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 9x^{6} + 65x^{4} - 144x^{2} + 256 $ x^8 - 9x^6 + 65x^4 - 144*x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{32}\cdot 3^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			319.3
Root			$-1.35234 + 0.780776i$ of defining polynomial
Character	$\chi$	$=$	384.319
Dual form			384.7.b.d.319.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-15.5885 q^{3} +20.0000i q^{5} -155.727i q^{7} +243.000 q^{9} +O(q^{10})$	$q-15.5885 q^{3} +20.0000i q^{5} -155.727i q^{7} +243.000 q^{9} +2306.46 q^{11} +3221.18i q^{13} -311.769i q^{15} -6566.73 q^{17} -3926.40 q^{19} +2427.54i q^{21} +17392.0i q^{23} +15225.0 q^{25} -3788.00 q^{27} -44713.3i q^{29} -30636.4i q^{31} -35954.2 q^{33} +3114.54 q^{35} -44158.1i q^{37} -50213.3i q^{39} +9008.18 q^{41} +25270.3 q^{43} +4860.00i q^{45} +175606. i q^{47} +93398.1 q^{49} +102365. q^{51} +96206.3i q^{53} +46129.2i q^{55} +61206.5 q^{57} -35018.3 q^{59} -86135.7i q^{61} -37841.7i q^{63} -64423.6 q^{65} -424605. q^{67} -271114. i q^{69} +36572.4i q^{71} -375121. q^{73} -237334. q^{75} -359179. i q^{77} +520323. i q^{79} +59049.0 q^{81} +594208. q^{83} -131335. i q^{85} +697011. i q^{87} +901101. q^{89} +501625. q^{91} +477574. i q^{93} -78528.0i q^{95} -1.25744e6 q^{97} +560470. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 1944 q^{9}+O(q^{10}) $ 8 * q + 1944 * q^9	$ 8 q + 1944 q^{9} + 4464 q^{17} + 121800 q^{25} - 116640 q^{33} - 98928 q^{41} - 278776 q^{49} - 23328 q^{57} - 230400 q^{65} + 76912 q^{73} + 472392 q^{81} + 6638832 q^{89} - 4929680 q^{97}+O(q^{100}) $ 8 * q + 1944 * q^9 + 4464 * q^17 + 121800 * q^25 - 116640 * q^33 - 98928 * q^41 - 278776 * q^49 - 23328 * q^57 - 230400 * q^65 + 76912 * q^73 + 472392 * q^81 + 6638832 * q^89 - 4929680 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/384\mathbb{Z}\right)^\times$.

$n$	$127$	$133$	$257$
$\chi(n)$	$-1$	$-1$	$1$

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
$3$	−15.5885	−0.577350
$3$	−15.5885	−0.577350
$4$	0	0
$5$	20.0000i	0.160000i	0.996795	+	0.0800000i	$0.0254920\pi$
$5$	20.0000i	0.160000i	−0.996795	+	0.0800000i	$0.974508\pi$
$6$	0	0
$7$	− 155.727i	− 0.454015i	−0.973893	−	0.227007i	$-0.927106\pi$
$7$	− 155.727i	− 0.454015i	0.973893	−	0.227007i	$-0.0728942\pi$
$8$	0	0
$9$	243.000	0.333333
$10$	0	0
$11$	2306.46	1.73288	0.866439	−	0.499282i	$-0.166403\pi$
$11$	2306.46	1.73288	0.866439	+	0.499282i	$0.166403\pi$
$12$	0	0
$13$	3221.18i	1.46617i	0.680136	+	0.733086i	$0.261921\pi$
$13$	3221.18i	1.46617i	−0.680136	+	0.733086i	$0.738079\pi$
$14$	0	0
$15$	− 311.769i	− 0.0923760i
$16$	0	0
$17$	−6566.73	−1.33660	−0.668301	−	0.743891i	$-0.732978\pi$
$17$	−6566.73	−1.33660	−0.668301	+	0.743891i	$0.732978\pi$
$18$	0	0
$19$	−3926.40	−0.572445	−0.286223	−	0.958163i	$-0.592400\pi$
$19$	−3926.40	−0.572445	−0.286223	+	0.958163i	$0.592400\pi$
$20$	0	0
$21$	2427.54i	0.262126i
$22$	0	0
$23$	17392.0i	1.42944i	0.699411	+	0.714720i	$0.253446\pi$
$23$	17392.0i	1.42944i	−0.699411	+	0.714720i	$0.746554\pi$
$24$	0	0
$25$	15225.0	0.974400
$26$	0	0
$27$	−3788.00	−0.192450
$28$	0	0
$29$	− 44713.3i	− 1.83334i	−0.399648	−	0.916669i	$-0.630868\pi$
$29$	− 44713.3i	− 1.83334i	0.399648	−	0.916669i	$-0.369132\pi$
$30$	0	0
$31$	− 30636.4i	− 1.02838i	−0.857677	−	0.514188i	$-0.828093\pi$
$31$	− 30636.4i	− 1.02838i	0.857677	−	0.514188i	$-0.171907\pi$
$32$	0	0
$33$	−35954.2	−1.00048
$34$	0	0
$35$	3114.54	0.0726424
$36$	0	0
$37$	− 44158.1i	− 0.871776i	−0.900001	−	0.435888i	$-0.856434\pi$
$37$	− 44158.1i	− 0.871776i	0.900001	−	0.435888i	$-0.143566\pi$
$38$	0	0
$39$	− 50213.3i	− 0.846495i
$40$	0	0
$41$	9008.18	0.130703	0.0653515	−	0.997862i	$-0.479183\pi$
$41$	9008.18	0.130703	0.0653515	+	0.997862i	$0.479183\pi$
$42$	0	0
$43$	25270.3	0.317838	0.158919	−	0.987292i	$-0.449199\pi$
$43$	25270.3	0.317838	0.158919	+	0.987292i	$0.449199\pi$
$44$	0	0
$45$	4860.00i	0.0533333i
$46$	0	0
$47$	175606.i	1.69140i	0.533658	+	0.845700i	$0.320817\pi$
$47$	175606.i	1.69140i	−0.533658	+	0.845700i	$0.679183\pi$
$48$	0	0
$49$	93398.1	0.793871
$50$	0	0
$51$	102365.	0.771688
$52$	0	0
$53$	96206.3i	0.646214i	0.946363	+	0.323107i	$0.104727\pi$
$53$	96206.3i	0.646214i	−0.946363	+	0.323107i	$0.895273\pi$
$54$	0	0
$55$	46129.2i	0.277261i
$56$	0	0
$57$	61206.5	0.330501
$58$	0	0
$59$	−35018.3	−0.170506	−0.0852529	−	0.996359i	$-0.527170\pi$
$59$	−35018.3	−0.170506	−0.0852529	+	0.996359i	$0.527170\pi$
$60$	0	0
$61$	− 86135.7i	− 0.379484i	−0.981834	−	0.189742i	$-0.939235\pi$
$61$	− 86135.7i	− 0.379484i	0.981834	−	0.189742i	$-0.0607652\pi$
$62$	0	0
$63$	− 37841.7i	− 0.151338i
$64$	0	0
$65$	−64423.6	−0.234588
$66$	0	0
$67$	−424605.	−1.41176	−0.705880	−	0.708332i	$-0.749448\pi$
$67$	−424605.	−1.41176	−0.705880	+	0.708332i	$0.749448\pi$
$68$	0	0
$69$	− 271114.i	− 0.825287i
$70$	0	0
$71$	36572.4i	0.102183i	0.998694	+	0.0510915i	$0.0162700\pi$
$71$	36572.4i	0.102183i	−0.998694	+	0.0510915i	$0.983730\pi$
$72$	0	0
$73$	−375121.	−0.964280	−0.482140	−	0.876094i	$-0.660140\pi$
$73$	−375121.	−0.964280	−0.482140	+	0.876094i	$0.660140\pi$
$74$	0	0
$75$	−237334.	−0.562570
$76$	0	0
$77$	− 359179.i	− 0.786753i
$78$	0	0
$79$	520323.i	1.05534i	0.849450	+	0.527670i	$0.176934\pi$
$79$	520323.i	1.05534i	−0.849450	+	0.527670i	$0.823066\pi$
$80$	0	0
$81$	59049.0	0.111111
$82$	0	0
$83$	594208.	1.03921	0.519606	−	0.854406i	$-0.326079\pi$
$83$	594208.	1.03921	0.519606	+	0.854406i	$0.326079\pi$
$84$	0	0
$85$	− 131335.i	− 0.213856i
$86$	0	0
$87$	697011.i	1.05848i
$88$	0	0
$89$	901101.	1.27821	0.639107	−	0.769118i	$-0.279304\pi$
$89$	901101.	1.27821	0.639107	+	0.769118i	$0.279304\pi$
$90$	0	0
$91$	501625.	0.665664
$92$	0	0
$93$	477574.i	0.593733i
$94$	0	0
$95$	− 78528.0i	− 0.0915912i
$96$	0	0
$97$	−1.25744e6	−1.37775	−0.688875	−	0.724880i	$-0.741895\pi$
$97$	−1.25744e6	−1.37775	−0.688875	+	0.724880i	$0.741895\pi$
$98$	0	0
$99$	560470.	0.577626
$100$	0	0
$101$	391040.i	0.379539i	0.981829	+	0.189770i	$0.0607741\pi$
$101$	391040.i	0.379539i	−0.981829	+	0.189770i	$0.939226\pi$
$102$	0	0
$103$	1.36080e6i	1.24533i	0.782490	+	0.622663i	$0.213949\pi$
$103$	1.36080e6i	1.24533i	−0.782490	+	0.622663i	$0.786051\pi$
$104$	0	0
$105$	−48550.9	−0.0419401
$106$	0	0
$107$	−1.28620e6	−1.04992	−0.524960	−	0.851127i	$-0.675920\pi$
$107$	−1.28620e6	−1.04992	−0.524960	+	0.851127i	$0.675920\pi$
$108$	0	0
$109$	1.75763e6i	1.35721i	0.734504	+	0.678605i	$0.237415\pi$
$109$	1.75763e6i	1.35721i	−0.734504	+	0.678605i	$0.762585\pi$
$110$	0	0
$111$	688357.i	0.503320i
$112$	0	0
$113$	−730749.	−0.506446	−0.253223	−	0.967408i	$-0.581491\pi$
$113$	−730749.	−0.506446	−0.253223	+	0.967408i	$0.581491\pi$
$114$	0	0
$115$	−347840.	−0.228710
$116$	0	0
$117$	782747.i	0.488724i
$118$	0	0
$119$	1.02262e6i	0.606837i
$120$	0	0
$121$	3.54820e6	2.00287
$122$	0	0
$123$	−140424.	−0.0754614
$124$	0	0
$125$	617000.i	0.315904i
$126$	0	0
$127$	2.59678e6i	1.26772i	0.773448	+	0.633860i	$0.218531\pi$
$127$	2.59678e6i	1.26772i	−0.773448	+	0.633860i	$0.781469\pi$
$128$	0	0
$129$	−393925.	−0.183504
$130$	0	0
$131$	605543.	0.269359	0.134679	−	0.990889i	$-0.457000\pi$
$131$	605543.	0.269359	0.134679	+	0.990889i	$0.457000\pi$
$132$	0	0
$133$	611447.i	0.259899i
$134$	0	0
$135$	− 75759.9i	− 0.0307920i
$136$	0	0
$137$	−1.75932e6	−0.684199	−0.342100	−	0.939664i	$-0.611138\pi$
$137$	−1.75932e6	−0.684199	−0.342100	+	0.939664i	$0.611138\pi$
$138$	0	0
$139$	303680.	0.113076	0.0565382	−	0.998400i	$-0.481994\pi$
$139$	303680.	0.113076	0.0565382	+	0.998400i	$0.481994\pi$
$140$	0	0
$141$	− 2.73743e6i	− 0.976531i
$142$	0	0
$143$	7.42953e6i	2.54070i
$144$	0	0
$145$	894265.	0.293334
$146$	0	0
$147$	−1.45593e6	−0.458341
$148$	0	0
$149$	− 1.32218e6i	− 0.399696i	−0.979827	−	0.199848i	$-0.935955\pi$
$149$	− 1.32218e6i	− 0.399696i	0.979827	−	0.199848i	$-0.0640449\pi$
$150$	0	0
$151$	4.67518e6i	1.35790i	0.734184	+	0.678950i	$0.237565\pi$
$151$	4.67518e6i	1.35790i	−0.734184	+	0.678950i	$0.762435\pi$
$152$	0	0
$153$	−1.59571e6	−0.445534
$154$	0	0
$155$	612727.	0.164540
$156$	0	0
$157$	− 2.44769e6i	− 0.632495i	−0.948677	−	0.316247i	$-0.897577\pi$
$157$	− 2.44769e6i	− 0.632495i	0.948677	−	0.316247i	$-0.102423\pi$
$158$	0	0
$159$	− 1.49971e6i	− 0.373092i
$160$	0	0
$161$	2.70840e6	0.648987
$162$	0	0
$163$	6.62033e6	1.52868	0.764340	−	0.644813i	$-0.223065\pi$
$163$	6.62033e6	1.52868	0.764340	+	0.644813i	$0.223065\pi$
$164$	0	0
$165$	− 719084.i	− 0.160076i
$166$	0	0
$167$	7.27537e6i	1.56209i	0.624476	+	0.781044i	$0.285313\pi$
$167$	7.27537e6i	1.56209i	−0.624476	+	0.781044i	$0.714687\pi$
$168$	0	0
$169$	−5.54920e6	−1.14966
$170$	0	0
$171$	−954116.	−0.190815
$172$	0	0
$173$	3.72601e6i	0.719623i	0.933025	+	0.359812i	$0.117159\pi$
$173$	3.72601e6i	0.719623i	−0.933025	+	0.359812i	$0.882841\pi$
$174$	0	0
$175$	− 2.37094e6i	− 0.442392i
$176$	0	0
$177$	545881.	0.0984415
$178$	0	0
$179$	−201811.	−0.0351873	−0.0175937	−	0.999845i	$-0.505601\pi$
$179$	−201811.	−0.0351873	−0.0175937	+	0.999845i	$0.505601\pi$
$180$	0	0
$181$	5.43440e6i	0.916466i	0.888832	+	0.458233i	$0.151517\pi$
$181$	5.43440e6i	0.916466i	−0.888832	+	0.458233i	$0.848483\pi$
$182$	0	0
$183$	1.34272e6i	0.219095i
$184$	0	0
$185$	883162.	0.139484
$186$	0	0
$187$	−1.51459e7	−2.31617
$188$	0	0
$189$	589893.i	0.0873752i
$190$	0	0
$191$	6.82608e6i	0.979650i	0.871821	+	0.489825i	$0.162939\pi$
$191$	6.82608e6i	0.979650i	−0.871821	+	0.489825i	$0.837061\pi$
$192$	0	0
$193$	1.30841e6	0.182001	0.0910004	−	0.995851i	$-0.470994\pi$
$193$	1.30841e6	0.182001	0.0910004	+	0.995851i	$0.470994\pi$
$194$	0	0
$195$	1.00427e6	0.135439
$196$	0	0
$197$	− 7.61375e6i	− 0.995863i	−0.867216	−	0.497932i	$-0.834093\pi$
$197$	− 7.61375e6i	− 0.995863i	0.867216	−	0.497932i	$-0.165907\pi$
$198$	0	0
$199$	6.47692e6i	0.821882i	0.911662	+	0.410941i	$0.134800\pi$
$199$	6.47692e6i	0.821882i	−0.911662	+	0.410941i	$0.865200\pi$
$200$	0	0
$201$	6.61894e6	0.815080
$202$	0	0
$203$	−6.96307e6	−0.832362
$204$	0	0
$205$	180164.i	0.0209125i
$206$	0	0
$207$	4.22625e6i	0.476480i
$208$	0	0
$209$	−9.05609e6	−0.991978
$210$	0	0
$211$	1.58058e7	1.68256	0.841278	−	0.540603i	$-0.181804\pi$
$211$	1.58058e7	1.68256	0.841278	+	0.540603i	$0.181804\pi$
$212$	0	0
$213$	− 570108.i	− 0.0589954i
$214$	0	0
$215$	505406.i	0.0508540i
$216$	0	0
$217$	−4.77091e6	−0.466898
$218$	0	0
$219$	5.84756e6	0.556727
$220$	0	0
$221$	− 2.11526e7i	− 1.95969i
$222$	0	0
$223$	2.07099e7i	1.86751i	0.357907	+	0.933757i	$0.383491\pi$
$223$	2.07099e7i	1.86751i	−0.357907	+	0.933757i	$0.616509\pi$
$224$	0	0
$225$	3.69967e6	0.324800
$226$	0	0
$227$	7.06179e6	0.603722	0.301861	−	0.953352i	$-0.402392\pi$
$227$	7.06179e6	0.603722	0.301861	+	0.953352i	$0.402392\pi$
$228$	0	0
$229$	− 4.64766e6i	− 0.387015i	−0.981099	−	0.193507i	$-0.938014\pi$
$229$	− 4.64766e6i	− 0.387015i	0.981099	−	0.193507i	$-0.0619863\pi$
$230$	0	0
$231$	5.59904e6i	0.454232i
$232$	0	0
$233$	1.13777e7	0.899469	0.449735	−	0.893162i	$-0.351519\pi$
$233$	1.13777e7	0.899469	0.449735	+	0.893162i	$0.351519\pi$
$234$	0	0
$235$	−3.51213e6	−0.270624
$236$	0	0
$237$	− 8.11104e6i	− 0.609300i
$238$	0	0
$239$	− 1.93765e7i	− 1.41932i	−0.704542	−	0.709662i	$-0.748848\pi$
$239$	− 1.93765e7i	− 1.41932i	0.704542	−	0.709662i	$-0.251152\pi$
$240$	0	0
$241$	6.84883e6	0.489289	0.244644	−	0.969613i	$-0.421329\pi$
$241$	6.84883e6	0.489289	0.244644	+	0.969613i	$0.421329\pi$
$242$	0	0
$243$	−920483.	−0.0641500
$244$	0	0
$245$	1.86796e6i	0.127019i
$246$	0	0
$247$	− 1.26477e7i	− 0.839304i
$248$	0	0
$249$	−9.26279e6	−0.599989
$250$	0	0
$251$	−129527.	−0.00819104	−0.00409552	−	0.999992i	$-0.501304\pi$
$251$	−129527.	−0.00819104	−0.00409552	+	0.999992i	$0.501304\pi$
$252$	0	0
$253$	4.01140e7i	2.47705i
$254$	0	0
$255$	2.04730e6i	0.123470i
$256$	0	0
$257$	1.18895e7	0.700427	0.350213	−	0.936670i	$-0.386109\pi$
$257$	1.18895e7	0.700427	0.350213	+	0.936670i	$0.386109\pi$
$258$	0	0
$259$	−6.87661e6	−0.395799
$260$	0	0
$261$	− 1.08653e7i	− 0.611113i
$262$	0	0
$263$	− 1.35674e7i	− 0.745811i	−0.927869	−	0.372906i	$-0.878362\pi$
$263$	− 1.35674e7i	− 0.745811i	0.927869	−	0.372906i	$-0.121638\pi$
$264$	0	0
$265$	−1.92413e6	−0.103394
$266$	0	0
$267$	−1.40468e7	−0.737977
$268$	0	0
$269$	1.71231e7i	0.879680i	0.898076	+	0.439840i	$0.144965\pi$
$269$	1.71231e7i	0.879680i	−0.898076	+	0.439840i	$0.855035\pi$
$270$	0	0
$271$	320201.i	0.0160885i	0.999968	+	0.00804423i	$0.00256058\pi$
$271$	320201.i	0.0160885i	−0.999968	+	0.00804423i	$0.997439\pi$
$272$	0	0
$273$	−7.81956e6	−0.384321
$274$	0	0
$275$	3.51159e7	1.68852
$276$	0	0
$277$	1.50949e7i	0.710216i	0.934825	+	0.355108i	$0.115556\pi$
$277$	1.50949e7i	0.710216i	−0.934825	+	0.355108i	$0.884444\pi$
$278$	0	0
$279$	− 7.44463e6i	− 0.342792i
$280$	0	0
$281$	−4.32580e7	−1.94961	−0.974804	−	0.223063i	$-0.928394\pi$
$281$	−4.32580e7	−1.94961	−0.974804	+	0.223063i	$0.928394\pi$
$282$	0	0
$283$	−3.22002e7	−1.42069	−0.710345	−	0.703854i	$-0.751461\pi$
$283$	−3.22002e7	−1.42069	−0.710345	+	0.703854i	$0.751461\pi$
$284$	0	0
$285$	1.22413e6i	0.0528802i
$286$	0	0
$287$	− 1.40282e6i	− 0.0593411i
$288$	0	0
$289$	1.89843e7	0.786505
$290$	0	0
$291$	1.96015e7	0.795444
$292$	0	0
$293$	3.01307e7i	1.19786i	0.800801	+	0.598931i	$0.204408\pi$
$293$	3.01307e7i	1.19786i	−0.800801	+	0.598931i	$0.795592\pi$
$294$	0	0
$295$	− 700366.i	− 0.0272809i
$296$	0	0
$297$	−8.73687e6	−0.333493
$298$	0	0
$299$	−5.60228e7	−2.09581
$300$	0	0
$301$	− 3.93527e6i	− 0.144303i
$302$	0	0
$303$	− 6.09571e6i	− 0.219127i
$304$	0	0
$305$	1.72271e6	0.0607175
$306$	0	0
$307$	2.09501e7	0.724053	0.362026	−	0.932168i	$-0.382085\pi$
$307$	2.09501e7	0.724053	0.362026	+	0.932168i	$0.382085\pi$
$308$	0	0
$309$	− 2.12128e7i	− 0.718990i
$310$	0	0
$311$	4.94101e6i	0.164261i	0.996622	+	0.0821305i	$0.0261724\pi$
$311$	4.94101e6i	0.164261i	−0.996622	+	0.0821305i	$0.973828\pi$
$312$	0	0
$313$	2.67449e7	0.872182	0.436091	−	0.899903i	$-0.356363\pi$
$313$	2.67449e7	0.872182	0.436091	+	0.899903i	$0.356363\pi$
$314$	0	0
$315$	756834.	0.0242141
$316$	0	0
$317$	− 4.74499e7i	− 1.48956i	−0.667311	−	0.744779i	$-0.732555\pi$
$317$	− 4.74499e7i	− 1.48956i	0.667311	−	0.744779i	$-0.267445\pi$
$318$	0	0
$319$	− 1.03129e8i	− 3.17695i
$320$	0	0
$321$	2.00498e7	0.606172
$322$	0	0
$323$	2.57836e7	0.765131
$324$	0	0
$325$	4.90425e7i	1.42864i
$326$	0	0
$327$	− 2.73987e7i	− 0.783585i
$328$	0	0
$329$	2.73467e7	0.767921
$330$	0	0
$331$	4.84318e7	1.33551	0.667754	−	0.744382i	$-0.267256\pi$
$331$	4.84318e7	1.33551	0.667754	+	0.744382i	$0.267256\pi$
$332$	0	0
$333$	− 1.07304e7i	− 0.290592i
$334$	0	0
$335$	− 8.49210e6i	− 0.225881i
$336$	0	0
$337$	−5.96066e7	−1.55742	−0.778709	−	0.627386i	$-0.784125\pi$
$337$	−5.96066e7	−1.55742	−0.778709	+	0.627386i	$0.784125\pi$
$338$	0	0
$339$	1.13913e7	0.292397
$340$	0	0
$341$	− 7.06616e7i	− 1.78205i
$342$	0	0
$343$	− 3.28657e7i	− 0.814444i
$344$	0	0
$345$	5.42229e6	0.132046
$346$	0	0
$347$	−2.95157e7	−0.706423	−0.353211	−	0.935544i	$-0.614910\pi$
$347$	−2.95157e7	−0.706423	−0.353211	+	0.935544i	$0.614910\pi$
$348$	0	0
$349$	5.55247e7i	1.30620i	0.757272	+	0.653100i	$0.226532\pi$
$349$	5.55247e7i	1.30620i	−0.757272	+	0.653100i	$0.773468\pi$
$350$	0	0
$351$	− 1.22018e7i	− 0.282165i
$352$	0	0
$353$	−1.81396e7	−0.412386	−0.206193	−	0.978511i	$-0.566107\pi$
$353$	−1.81396e7	−0.412386	−0.206193	+	0.978511i	$0.566107\pi$
$354$	0	0
$355$	−731449.	−0.0163493
$356$	0	0
$357$	− 1.59410e7i	− 0.350358i
$358$	0	0
$359$	− 8.83261e6i	− 0.190900i	−0.995434	−	0.0954499i	$-0.969571\pi$
$359$	− 8.83261e6i	− 0.190900i	0.995434	−	0.0954499i	$-0.0304290\pi$
$360$	0	0
$361$	−3.16293e7	−0.672307
$362$	0	0
$363$	−5.53110e7	−1.15636
$364$	0	0
$365$	− 7.50242e6i	− 0.154285i
$366$	0	0
$367$	4.83247e7i	0.977623i	0.872389	+	0.488811i	$0.162569\pi$
$367$	4.83247e7i	0.977623i	−0.872389	+	0.488811i	$0.837431\pi$
$368$	0	0
$369$	2.18899e6	0.0435677
$370$	0	0
$371$	1.49819e7	0.293391
$372$	0	0
$373$	8.98844e7i	1.73204i	0.500010	+	0.866020i	$0.333330\pi$
$373$	8.98844e7i	1.73204i	−0.500010	+	0.866020i	$0.666670\pi$
$374$	0	0
$375$	− 9.61808e6i	− 0.182387i
$376$	0	0
$377$	1.44030e8	2.68799
$378$	0	0
$379$	−8.60279e7	−1.58024	−0.790118	−	0.612955i	$-0.789981\pi$
$379$	−8.60279e7	−1.58024	−0.790118	+	0.612955i	$0.789981\pi$
$380$	0	0
$381$	− 4.04798e7i	− 0.731919i
$382$	0	0
$383$	− 6.65042e7i	− 1.18373i	−0.806037	−	0.591865i	$-0.798392\pi$
$383$	− 6.65042e7i	− 1.18373i	0.806037	−	0.591865i	$-0.201608\pi$
$384$	0	0
$385$	7.18357e6	0.125880
$386$	0	0
$387$	6.14069e6	0.105946
$388$	0	0
$389$	− 5.66889e7i	− 0.963050i	−0.876432	−	0.481525i	$-0.840083\pi$
$389$	− 5.66889e7i	− 0.963050i	0.876432	−	0.481525i	$-0.159917\pi$
$390$	0	0
$391$	− 1.14208e8i	− 1.91059i
$392$	0	0
$393$	−9.43949e6	−0.155514
$394$	0	0
$395$	−1.04065e7	−0.168854
$396$	0	0
$397$	3.21267e7i	0.513446i	0.966485	+	0.256723i	$0.0826429\pi$
$397$	3.21267e7i	0.513446i	−0.966485	+	0.256723i	$0.917357\pi$
$398$	0	0
$399$	− 9.53152e6i	− 0.150053i
$400$	0	0
$401$	−7.28625e7	−1.12998	−0.564991	−	0.825097i	$-0.691120\pi$
$401$	−7.28625e7	−1.12998	−0.564991	+	0.825097i	$0.691120\pi$
$402$	0	0
$403$	9.86853e7	1.50778
$404$	0	0
$405$	1.18098e6i	0.0177778i
$406$	0	0
$407$	− 1.01849e8i	− 1.51068i
$408$	0	0
$409$	−2.07185e7	−0.302823	−0.151412	−	0.988471i	$-0.548382\pi$
$409$	−2.07185e7	−0.302823	−0.151412	+	0.988471i	$0.548382\pi$
$410$	0	0
$411$	2.74251e7	0.395023
$412$	0	0
$413$	5.45330e6i	0.0774121i
$414$	0	0
$415$	1.18842e7i	0.166274i
$416$	0	0
$417$	−4.73390e6	−0.0652846
$418$	0	0
$419$	6.79561e7	0.923818	0.461909	−	0.886927i	$-0.347165\pi$
$419$	6.79561e7	0.923818	0.461909	+	0.886927i	$0.347165\pi$
$420$	0	0
$421$	− 522580.i	− 0.00700336i	−0.999994	−	0.00350168i	$-0.998885\pi$
$421$	− 522580.i	− 0.00700336i	0.999994	−	0.00350168i	$-0.00111462\pi$
$422$	0	0
$423$	4.26723e7i	0.563800i
$424$	0	0
$425$	−9.99784e7	−1.30239
$426$	0	0
$427$	−1.34137e7	−0.172291
$428$	0	0
$429$	− 1.15815e8i	− 1.46687i
$430$	0	0
$431$	− 1.10533e8i	− 1.38057i	−0.723538	−	0.690285i	$-0.757485\pi$
$431$	− 1.10533e8i	− 1.38057i	0.723538	−	0.690285i	$-0.242515\pi$
$432$	0	0
$433$	6.65239e7	0.819435	0.409717	−	0.912213i	$-0.365627\pi$
$433$	6.65239e7	0.819435	0.409717	+	0.912213i	$0.365627\pi$
$434$	0	0
$435$	−1.39402e7	−0.169356
$436$	0	0
$437$	− 6.82879e7i	− 0.818276i
$438$	0	0
$439$	1.06995e8i	1.26465i	0.774702	+	0.632327i	$0.217900\pi$
$439$	1.06995e8i	1.26465i	−0.774702	+	0.632327i	$0.782100\pi$
$440$	0	0
$441$	2.26957e7	0.264624
$442$	0	0
$443$	1.01597e7	0.116861	0.0584304	−	0.998291i	$-0.481390\pi$
$443$	1.01597e7	0.116861	0.0584304	+	0.998291i	$0.481390\pi$
$444$	0	0
$445$	1.80220e7i	0.204514i
$446$	0	0
$447$	2.06107e7i	0.230765i
$448$	0	0
$449$	−4.39511e7	−0.485546	−0.242773	−	0.970083i	$-0.578057\pi$
$449$	−4.39511e7	−0.485546	−0.242773	+	0.970083i	$0.578057\pi$
$450$	0	0
$451$	2.07770e7	0.226492
$452$	0	0
$453$	− 7.28789e7i	− 0.783984i
$454$	0	0
$455$	1.00325e7i	0.106506i
$456$	0	0
$457$	−5.92987e6	−0.0621293	−0.0310646	−	0.999517i	$-0.509890\pi$
$457$	−5.92987e6	−0.0621293	−0.0310646	+	0.999517i	$0.509890\pi$
$458$	0	0
$459$	2.48747e7	0.257229
$460$	0	0
$461$	− 4.31145e7i	− 0.440069i	−0.975492	−	0.220035i	$-0.929383\pi$
$461$	− 4.31145e7i	− 0.440069i	0.975492	−	0.220035i	$-0.0706170\pi$
$462$	0	0
$463$	− 8.09555e7i	− 0.815649i	−0.913060	−	0.407825i	$-0.866288\pi$
$463$	− 8.09555e7i	− 0.815649i	0.913060	−	0.407825i	$-0.133712\pi$
$464$	0	0
$465$	−9.55147e6	−0.0949973
$466$	0	0
$467$	−1.20309e8	−1.18127	−0.590633	−	0.806940i	$-0.701122\pi$
$467$	−1.20309e8	−1.18127	−0.590633	+	0.806940i	$0.701122\pi$
$468$	0	0
$469$	6.61225e7i	0.640960i
$470$	0	0
$471$	3.81557e7i	0.365171i
$472$	0	0
$473$	5.82850e7	0.550774
$474$	0	0
$475$	−5.97795e7	−0.557791
$476$	0	0
$477$	2.33781e7i	0.215405i
$478$	0	0
$479$	1.54238e8i	1.40341i	0.712468	+	0.701705i	$0.247578\pi$
$479$	1.54238e8i	1.40341i	−0.712468	+	0.701705i	$0.752422\pi$
$480$	0	0
$481$	1.42241e8	1.27817
$482$	0	0
$483$	−4.22198e7	−0.374693
$484$	0	0
$485$	− 2.51487e7i	− 0.220440i
$486$	0	0
$487$	6.53601e7i	0.565882i	0.959137	+	0.282941i	$0.0913101\pi$
$487$	6.53601e7i	0.565882i	−0.959137	+	0.282941i	$0.908690\pi$
$488$	0	0
$489$	−1.03201e8	−0.882584
$490$	0	0
$491$	−4.51212e7	−0.381185	−0.190593	−	0.981669i	$-0.561041\pi$
$491$	−4.51212e7	−0.381185	−0.190593	+	0.981669i	$0.561041\pi$
$492$	0	0
$493$	2.93620e8i	2.45044i
$494$	0	0
$495$	1.12094e7i	0.0924202i
$496$	0	0
$497$	5.69532e6	0.0463926
$498$	0	0
$499$	5.03581e7	0.405292	0.202646	−	0.979252i	$-0.435046\pi$
$499$	5.03581e7	0.405292	0.202646	+	0.979252i	$0.435046\pi$
$500$	0	0
$501$	− 1.13412e8i	− 0.901872i
$502$	0	0
$503$	− 4.41054e6i	− 0.0346567i	−0.999850	−	0.0173284i	$-0.994484\pi$
$503$	− 4.41054e6i	− 0.0346567i	0.999850	−	0.0173284i	$-0.00551607\pi$
$504$	0	0
$505$	−7.82080e6	−0.0607263
$506$	0	0
$507$	8.65035e7	0.663758
$508$	0	0
$509$	− 2.44005e7i	− 0.185031i	−0.995711	−	0.0925156i	$-0.970509\pi$
$509$	− 2.44005e7i	− 0.185031i	0.995711	−	0.0925156i	$-0.0294908\pi$
$510$	0	0
$511$	5.84165e7i	0.437797i
$512$	0	0
$513$	1.48732e7	0.110167
$514$	0	0
$515$	−2.72160e7	−0.199252
$516$	0	0
$517$	4.05029e8i	2.93099i
$518$	0	0
$519$	− 5.80827e7i	− 0.415475i
$520$	0	0
$521$	6.53917e7	0.462391	0.231195	−	0.972907i	$-0.425736\pi$
$521$	6.53917e7	0.462391	0.231195	+	0.972907i	$0.425736\pi$
$522$	0	0
$523$	−1.19428e8	−0.834833	−0.417417	−	0.908715i	$-0.637064\pi$
$523$	−1.19428e8	−0.834833	−0.417417	+	0.908715i	$0.637064\pi$
$524$	0	0
$525$	3.69594e7i	0.255415i
$526$	0	0
$527$	2.01181e8i	1.37453i
$528$	0	0
$529$	−1.54446e8	−1.04330
$530$	0	0
$531$	−8.50945e6	−0.0568352
$532$	0	0
$533$	2.90170e7i	0.191633i
$534$	0	0
$535$	− 2.57240e7i	− 0.167987i
$536$	0	0
$537$	3.14593e6	0.0203154
$538$	0	0
$539$	2.15419e8	1.37568
$540$	0	0
$541$	3.06540e7i	0.193596i	0.995304	+	0.0967979i	$0.0308600\pi$
$541$	3.06540e7i	0.193596i	−0.995304	+	0.0967979i	$0.969140\pi$
$542$	0	0
$543$	− 8.47140e7i	− 0.529122i
$544$	0	0
$545$	−3.51525e7	−0.217153
$546$	0	0
$547$	−1.60733e8	−0.982071	−0.491035	−	0.871140i	$-0.663381\pi$
$547$	−1.60733e8	−0.982071	−0.491035	+	0.871140i	$0.663381\pi$
$548$	0	0
$549$	− 2.09310e7i	− 0.126495i
$550$	0	0
$551$	1.75562e8i	1.04949i
$552$	0	0
$553$	8.10284e7	0.479140
$554$	0	0
$555$	−1.37671e7	−0.0805313
$556$	0	0
$557$	− 3.11683e8i	− 1.80363i	−0.432125	−	0.901814i	$-0.642236\pi$
$557$	− 3.11683e8i	− 1.80363i	0.432125	−	0.901814i	$-0.357764\pi$
$558$	0	0
$559$	8.14003e7i	0.466005i
$560$	0	0
$561$	2.36101e8	1.33724
$562$	0	0
$563$	2.81653e8	1.57830	0.789149	−	0.614201i	$-0.210522\pi$
$563$	2.81653e8	1.57830	0.789149	+	0.614201i	$0.210522\pi$
$564$	0	0
$565$	− 1.46150e7i	− 0.0810313i
$566$	0	0
$567$	− 9.19553e6i	− 0.0504461i
$568$	0	0
$569$	8.82677e7	0.479143	0.239571	−	0.970879i	$-0.422993\pi$
$569$	8.82677e7	0.479143	0.239571	+	0.970879i	$0.422993\pi$
$570$	0	0
$571$	1.66103e8	0.892217	0.446108	−	0.894979i	$-0.352810\pi$
$571$	1.66103e8	0.892217	0.446108	+	0.894979i	$0.352810\pi$
$572$	0	0
$573$	− 1.06408e8i	− 0.565601i
$574$	0	0
$575$	2.64793e8i	1.39285i
$576$	0	0
$577$	−5.26455e7	−0.274053	−0.137026	−	0.990567i	$-0.543754\pi$
$577$	−5.26455e7	−0.274053	−0.137026	+	0.990567i	$0.543754\pi$
$578$	0	0
$579$	−2.03962e7	−0.105078
$580$	0	0
$581$	− 9.25343e7i	− 0.471818i
$582$	0	0
$583$	2.21896e8i	1.11981i
$584$	0	0
$585$	−1.56549e7	−0.0781959
$586$	0	0
$587$	−3.88069e7	−0.191865	−0.0959323	−	0.995388i	$-0.530583\pi$
$587$	−3.88069e7	−0.191865	−0.0959323	+	0.995388i	$0.530583\pi$
$588$	0	0
$589$	1.20291e8i	0.588689i
$590$	0	0
$591$	1.18687e8i	0.574962i
$592$	0	0
$593$	−3.26100e8	−1.56382	−0.781909	−	0.623392i	$-0.785754\pi$
$593$	−3.26100e8	−1.56382	−0.781909	+	0.623392i	$0.785754\pi$
$594$	0	0
$595$	−2.04523e7	−0.0970939
$596$	0	0
$597$	− 1.00965e8i	− 0.474514i
$598$	0	0
$599$	3.03039e8i	1.41000i	0.709210	+	0.704998i	$0.249052\pi$
$599$	3.03039e8i	1.41000i	−0.709210	+	0.704998i	$0.750948\pi$
$600$	0	0
$601$	−3.98916e8	−1.83763	−0.918815	−	0.394689i	$-0.870852\pi$
$601$	−3.98916e8	−1.83763	−0.918815	+	0.394689i	$0.870852\pi$
$602$	0	0
$603$	−1.03179e8	−0.470586
$604$	0	0
$605$	7.09641e7i	0.320459i
$606$	0	0
$607$	2.46327e8i	1.10140i	0.834703	+	0.550701i	$0.185639\pi$
$607$	2.46327e8i	1.10140i	−0.834703	+	0.550701i	$0.814361\pi$
$608$	0	0
$609$	1.08543e8	0.480565
$610$	0	0
$611$	−5.65660e8	−2.47989
$612$	0	0
$613$	− 2.16670e8i	− 0.940628i	−0.882499	−	0.470314i	$-0.844141\pi$
$613$	− 2.16670e8i	− 0.940628i	0.882499	−	0.470314i	$-0.155859\pi$
$614$	0	0
$615$	− 2.80847e6i	− 0.0120738i
$616$	0	0
$617$	1.27942e8	0.544699	0.272350	−	0.962198i	$-0.412199\pi$
$617$	1.27942e8	0.544699	0.272350	+	0.962198i	$0.412199\pi$
$618$	0	0
$619$	1.96738e8	0.829499	0.414749	−	0.909936i	$-0.363869\pi$
$619$	1.96738e8	0.829499	0.414749	+	0.909936i	$0.363869\pi$
$620$	0	0
$621$	− 6.58808e7i	− 0.275096i
$622$	0	0
$623$	− 1.40326e8i	− 0.580328i
$624$	0	0
$625$	2.25551e8	0.923855
$626$	0	0
$627$	1.41171e8	0.572719
$628$	0	0
$629$	2.89974e8i	1.16522i
$630$	0	0
$631$	9.81163e7i	0.390529i	0.980751	+	0.195264i	$0.0625565\pi$
$631$	9.81163e7i	0.390529i	−0.980751	+	0.195264i	$0.937443\pi$
$632$	0	0
$633$	−2.46388e8	−0.971424
$634$	0	0
$635$	−5.19355e7	−0.202835
$636$	0	0
$637$	3.00852e8i	1.16395i
$638$	0	0
$639$	8.88710e6i	0.0340610i
$640$	0	0
$641$	2.85929e8	1.08564	0.542819	−	0.839850i	$-0.317357\pi$
$641$	2.85929e8	1.08564	0.542819	+	0.839850i	$0.317357\pi$
$642$	0	0
$643$	−2.98146e8	−1.12149	−0.560745	−	0.827988i	$-0.689485\pi$
$643$	−2.98146e8	−1.12149	−0.560745	+	0.827988i	$0.689485\pi$
$644$	0	0
$645$	− 7.87850e6i	− 0.0293606i
$646$	0	0
$647$	− 3.84615e7i	− 0.142008i	−0.997476	−	0.0710042i	$-0.977380\pi$
$647$	− 3.84615e7i	− 0.142008i	0.997476	−	0.0710042i	$-0.0226204\pi$
$648$	0	0
$649$	−8.07684e7	−0.295466
$650$	0	0
$651$	7.43711e7	0.269564
$652$	0	0
$653$	− 9.67190e7i	− 0.347354i	−0.984803	−	0.173677i	$-0.944435\pi$
$653$	− 9.67190e7i	− 0.347354i	0.984803	−	0.173677i	$-0.0555648\pi$
$654$	0	0
$655$	1.21109e7i	0.0430974i
$656$	0	0
$657$	−9.11545e7	−0.321427
$658$	0	0
$659$	−6.49525e7	−0.226955	−0.113477	−	0.993541i	$-0.536199\pi$
$659$	−6.49525e7	−0.226955	−0.113477	+	0.993541i	$0.536199\pi$
$660$	0	0
$661$	3.29739e8i	1.14174i	0.821042	+	0.570868i	$0.193393\pi$
$661$	3.29739e8i	1.14174i	−0.821042	+	0.570868i	$0.806607\pi$
$662$	0	0
$663$	3.29737e8i	1.13143i
$664$	0	0
$665$	−1.22289e7	−0.0415838
$666$	0	0
$667$	7.77653e8	2.62065
$668$	0	0
$669$	− 3.22836e8i	− 1.07821i
$670$	0	0
$671$	− 1.98669e8i	− 0.657600i
$672$	0	0
$673$	5.17571e8	1.69795	0.848975	−	0.528433i	$-0.177220\pi$
$673$	5.17571e8	1.69795	0.848975	+	0.528433i	$0.177220\pi$
$674$	0	0
$675$	−5.76722e7	−0.187523
$676$	0	0
$677$	− 3.25473e8i	− 1.04894i	−0.851430	−	0.524469i	$-0.824264\pi$
$677$	− 3.25473e8i	− 1.04894i	0.851430	−	0.524469i	$-0.175736\pi$
$678$	0	0
$679$	1.95817e8i	0.625519i
$680$	0	0
$681$	−1.10082e8	−0.348559
$682$	0	0
$683$	5.37538e8	1.68713	0.843563	−	0.537031i	$-0.180454\pi$
$683$	5.37538e8	1.68713	0.843563	+	0.537031i	$0.180454\pi$
$684$	0	0
$685$	− 3.51864e7i	− 0.109472i
$686$	0	0
$687$	7.24498e7i	0.223443i
$688$	0	0
$689$	−3.09898e8	−0.947461
$690$	0	0
$691$	−4.64528e8	−1.40792	−0.703960	−	0.710239i	$-0.748587\pi$
$691$	−4.64528e8	−1.40792	−0.703960	+	0.710239i	$0.748587\pi$
$692$	0	0
$693$	− 8.72804e7i	− 0.262251i
$694$	0	0
$695$	6.07360e6i	0.0180922i
$696$	0	0
$697$	−5.91543e7	−0.174698
$698$	0	0
$699$	−1.77361e8	−0.519309
$700$	0	0
$701$	− 3.59130e7i	− 0.104255i	−0.998640	−	0.0521275i	$-0.983400\pi$
$701$	− 3.59130e7i	− 0.104255i	0.998640	−	0.0521275i	$-0.0166002\pi$
$702$	0	0
$703$	1.73382e8i	0.499044i
$704$	0	0
$705$	5.47486e7	0.156245
$706$	0	0
$707$	6.08955e7	0.172317
$708$	0	0
$709$	− 1.86788e8i	− 0.524094i	−0.965055	−	0.262047i	$-0.915602\pi$
$709$	− 1.86788e8i	− 0.524094i	0.965055	−	0.262047i	$-0.0843976\pi$
$710$	0	0
$711$	1.26439e8i	0.351780i
$712$	0	0
$713$	5.32827e8	1.47000
$714$	0	0
$715$	−1.48591e8	−0.406512
$716$	0	0
$717$	3.02050e8i	0.819447i
$718$	0	0
$719$	3.76368e7i	0.101257i	0.998718	+	0.0506286i	$0.0161225\pi$
$719$	3.76368e7i	0.101257i	−0.998718	+	0.0506286i	$0.983878\pi$
$720$	0	0
$721$	2.11914e8	0.565397
$722$	0	0
$723$	−1.06763e8	−0.282491
$724$	0	0
$725$	− 6.80760e8i	− 1.78640i
$726$	0	0
$727$	− 2.58872e8i	− 0.673724i	−0.941554	−	0.336862i	$-0.890634\pi$
$727$	− 2.58872e8i	− 0.673724i	0.941554	−	0.336862i	$-0.109366\pi$
$728$	0	0
$729$	1.43489e7	0.0370370
$730$	0	0
$731$	−1.65943e8	−0.424822
$732$	0	0
$733$	− 5.31872e8i	− 1.35050i	−0.737588	−	0.675251i	$-0.764035\pi$
$733$	− 5.31872e8i	− 1.35050i	0.737588	−	0.675251i	$-0.235965\pi$
$734$	0	0
$735$	− 2.91186e7i	− 0.0733346i
$736$	0	0
$737$	−9.79335e8	−2.44641
$738$	0	0
$739$	−1.70943e8	−0.423564	−0.211782	−	0.977317i	$-0.567927\pi$
$739$	−1.70943e8	−0.423564	−0.211782	+	0.977317i	$0.567927\pi$
$740$	0	0
$741$	1.97157e8i	0.484572i
$742$	0	0
$743$	− 4.23377e8i	− 1.03219i	−0.856531	−	0.516096i	$-0.827385\pi$
$743$	− 4.23377e8i	− 1.03219i	0.856531	−	0.516096i	$-0.172615\pi$
$744$	0	0
$745$	2.64435e7	0.0639514
$746$	0	0
$747$	1.44393e8	0.346404
$748$	0	0
$749$	2.00296e8i	0.476679i
$750$	0	0
$751$	3.04170e8i	0.718118i	0.933315	+	0.359059i	$0.116902\pi$
$751$	3.04170e8i	0.718118i	−0.933315	+	0.359059i	$0.883098\pi$
$752$	0	0
$753$	2.01912e6	0.00472910
$754$	0	0
$755$	−9.35037e7	−0.217264
$756$	0	0
$757$	3.06435e8i	0.706401i	0.935548	+	0.353201i	$0.114907\pi$
$757$	3.06435e8i	0.706401i	−0.935548	+	0.353201i	$0.885093\pi$
$758$	0	0
$759$	− 6.25315e8i	− 1.43012i
$760$	0	0
$761$	4.23923e8	0.961907	0.480954	−	0.876746i	$-0.340291\pi$
$761$	4.23923e8	0.961907	0.480954	+	0.876746i	$0.340291\pi$
$762$	0	0
$763$	2.73710e8	0.616193
$764$	0	0
$765$	− 3.19143e7i	− 0.0712854i
$766$	0	0
$767$	− 1.12800e8i	− 0.249991i
$768$	0	0
$769$	−5.19668e8	−1.14274	−0.571369	−	0.820693i	$-0.693588\pi$
$769$	−5.19668e8	−1.14274	−0.571369	+	0.820693i	$0.693588\pi$
$770$	0	0
$771$	−1.85338e8	−0.404391
$772$	0	0
$773$	1.89126e7i	0.0409462i	0.999790	+	0.0204731i	$0.00651724\pi$
$773$	1.89126e7i	0.0409462i	−0.999790	+	0.0204731i	$0.993483\pi$
$774$	0	0
$775$	− 4.66439e8i	− 1.00205i
$776$	0	0
$777$	1.07196e8	0.228515
$778$	0	0
$779$	−3.53697e7	−0.0748203
$780$	0	0
$781$	8.43529e7i	0.177071i
$782$	0	0
$783$	1.69374e8i	0.352826i
$784$	0	0
$785$	4.89537e7	0.101199
$786$	0	0
$787$	2.65036e8	0.543726	0.271863	−	0.962336i	$-0.412360\pi$
$787$	2.65036e8	0.543726	0.271863	+	0.962336i	$0.412360\pi$
$788$	0	0
$789$	2.11495e8i	0.430594i
$790$	0	0
$791$	1.13797e8i	0.229934i
$792$	0	0
$793$	2.77459e8	0.556390
$794$	0	0
$795$	2.99942e7	0.0596947
$796$	0	0
$797$	6.82061e8i	1.34725i	0.739073	+	0.673626i	$0.235264\pi$
$797$	6.82061e8i	1.34725i	−0.739073	+	0.673626i	$0.764736\pi$
$798$	0	0
$799$	− 1.15316e9i	− 2.26073i
$800$	0	0
$801$	2.18968e8	0.426071
$802$	0	0
$803$	−8.65203e8	−1.67098
$804$	0	0
$805$	5.41681e7i	0.103838i
$806$	0	0
$807$	− 2.66922e8i	− 0.507884i
$808$	0	0
$809$	5.50511e8	1.03973	0.519865	−	0.854248i	$-0.325982\pi$
$809$	5.50511e8	1.03973	0.519865	+	0.854248i	$0.325982\pi$
$810$	0	0
$811$	−3.11549e8	−0.584068	−0.292034	−	0.956408i	$-0.594332\pi$
$811$	−3.11549e8	−0.584068	−0.292034	+	0.956408i	$0.594332\pi$
$812$	0	0
$813$	− 4.99143e6i	− 0.00928867i
$814$	0	0
$815$	1.32407e8i	0.244589i
$816$	0	0
$817$	−9.92214e7	−0.181945
$818$	0	0
$819$	1.21895e8	0.221888
$820$	0	0
$821$	7.17579e8i	1.29670i	0.761342	+	0.648351i	$0.224541\pi$
$821$	7.17579e8i	1.29670i	−0.761342	+	0.648351i	$0.775459\pi$
$822$	0	0
$823$	5.41195e8i	0.970854i	0.874277	+	0.485427i	$0.161336\pi$
$823$	5.41195e8i	0.970854i	−0.874277	+	0.485427i	$0.838664\pi$
$824$	0	0
$825$	−5.47402e8	−0.974866
$826$	0	0
$827$	8.79442e8	1.55486	0.777429	−	0.628971i	$-0.216524\pi$
$827$	8.79442e8	1.55486	0.777429	+	0.628971i	$0.216524\pi$
$828$	0	0
$829$	4.29123e8i	0.753213i	0.926373	+	0.376607i	$0.122909\pi$
$829$	4.29123e8i	0.753213i	−0.926373	+	0.376607i	$0.877091\pi$
$830$	0	0
$831$	− 2.35306e8i	− 0.410043i
$832$	0	0
$833$	−6.13320e8	−1.06109
$834$	0	0
$835$	−1.45507e8	−0.249934
$836$	0	0
$837$	1.16050e8i	0.197911i
$838$	0	0
$839$	5.86636e8i	0.993305i	0.867949	+	0.496653i	$0.165438\pi$
$839$	5.86636e8i	0.993305i	−0.867949	+	0.496653i	$0.834562\pi$
$840$	0	0
$841$	−1.40445e9	−2.36113
$842$	0	0
$843$	6.74325e8	1.12561
$844$	0	0
$845$	− 1.10984e8i	− 0.183946i
$846$	0	0
$847$	− 5.52552e8i	− 0.909332i
$848$	0	0
$849$	5.01951e8	0.820236
$850$	0	0
$851$	7.67997e8	1.24615
$852$	0	0
$853$	− 7.00029e8i	− 1.12790i	−0.825810	−	0.563948i	$-0.809282\pi$
$853$	− 7.00029e8i	− 1.12790i	0.825810	−	0.563948i	$-0.190718\pi$
$854$	0	0
$855$	− 1.90823e7i	− 0.0305304i
$856$	0	0
$857$	9.52114e8	1.51268	0.756339	−	0.654180i	$-0.226986\pi$
$857$	9.52114e8	1.51268	0.756339	+	0.654180i	$0.226986\pi$
$858$	0	0
$859$	4.16968e8	0.657844	0.328922	−	0.944357i	$-0.393315\pi$
$859$	4.16968e8	0.657844	0.328922	+	0.944357i	$0.393315\pi$
$860$	0	0
$861$	2.18678e7i	0.0342606i
$862$	0	0
$863$	− 4.62686e8i	− 0.719869i	−0.932978	−	0.359935i	$-0.882799\pi$
$863$	− 4.62686e8i	− 0.719869i	0.932978	−	0.359935i	$-0.117201\pi$
$864$	0	0
$865$	−7.45201e7	−0.115140
$866$	0	0
$867$	−2.95936e8	−0.454089
$868$	0	0
$869$	1.20011e9i	1.82877i
$870$	0	0
$871$	− 1.36773e9i	− 2.06988i
$872$	0	0
$873$	−3.05557e8	−0.459250
$874$	0	0
$875$	9.60836e7	0.143425
$876$	0	0
$877$	− 6.73329e8i	− 0.998225i	−0.866537	−	0.499113i	$-0.833659\pi$
$877$	− 6.73329e8i	− 0.998225i	0.866537	−	0.499113i	$-0.166341\pi$
$878$	0	0
$879$	− 4.69692e8i	− 0.691586i
$880$	0	0
$881$	1.77155e8	0.259074	0.129537	−	0.991575i	$-0.458651\pi$
$881$	1.77155e8	0.259074	0.129537	+	0.991575i	$0.458651\pi$
$882$	0	0
$883$	1.35707e8	0.197115	0.0985574	−	0.995131i	$-0.468577\pi$
$883$	1.35707e8	0.197115	0.0985574	+	0.995131i	$0.468577\pi$
$884$	0	0
$885$	1.09176e7i	0.0157506i
$886$	0	0
$887$	− 1.09553e9i	− 1.56984i	−0.619598	−	0.784919i	$-0.712704\pi$
$887$	− 1.09553e9i	− 1.56984i	0.619598	−	0.784919i	$-0.287296\pi$
$888$	0	0
$889$	4.04389e8	0.575564
$890$	0	0
$891$	1.36194e8	0.192542
$892$	0	0
$893$	− 6.89501e8i	− 0.968234i
$894$	0	0
$895$	− 4.03623e6i	− 0.00562997i
$896$	0	0
$897$	8.73309e8	1.21001
$898$	0	0
$899$	−1.36985e9	−1.88536
$900$	0	0
$901$	− 6.31761e8i	− 0.863731i
$902$	0	0
$903$	6.13448e7i	0.0833133i
$904$	0	0
$905$	−1.08688e8	−0.146635
$906$	0	0
$907$	−1.34159e9	−1.79803	−0.899015	−	0.437917i	$-0.855716\pi$
$907$	−1.34159e9	−1.79803	−0.899015	+	0.437917i	$0.855716\pi$
$908$	0	0
$909$	9.50227e7i	0.126513i
$910$	0	0
$911$	− 1.16107e9i	− 1.53569i	−0.640638	−	0.767843i	$-0.721330\pi$
$911$	− 1.16107e9i	− 1.53569i	0.640638	−	0.767843i	$-0.278670\pi$
$912$	0	0
$913$	1.37052e9	1.80083
$914$	0	0
$915$	−2.68545e7	−0.0350553
$916$	0	0
$917$	− 9.42995e7i	− 0.122293i
$918$	0	0
$919$	− 1.03538e9i	− 1.33399i	−0.745061	−	0.666996i	$-0.767580\pi$
$919$	− 1.03538e9i	− 1.33399i	0.745061	−	0.666996i	$-0.232420\pi$
$920$	0	0
$921$	−3.26579e8	−0.418032
$922$	0	0
$923$	−1.17806e8	−0.149818
$924$	0	0
$925$	− 6.72307e8i	− 0.849459i
$926$	0	0
$927$	3.30675e8i	0.415109i
$928$	0	0
$929$	−1.47698e9	−1.84217	−0.921083	−	0.389367i	$-0.872694\pi$
$929$	−1.47698e9	−1.84217	−0.921083	+	0.389367i	$0.872694\pi$
$930$	0	0
$931$	−3.66718e8	−0.454447
$932$	0	0
$933$	− 7.70227e7i	− 0.0948361i
$934$	0	0
$935$	− 3.02918e8i	− 0.370587i
$936$	0	0
$937$	4.90868e8	0.596686	0.298343	−	0.954459i	$-0.403566\pi$
$937$	4.90868e8	0.596686	0.298343	+	0.954459i	$0.403566\pi$
$938$	0	0
$939$	−4.16911e8	−0.503555
$940$	0	0
$941$	1.54536e8i	0.185464i	0.995691	+	0.0927321i	$0.0295600\pi$
$941$	1.54536e8i	0.185464i	−0.995691	+	0.0927321i	$0.970440\pi$
$942$	0	0
$943$	1.56670e8i	0.186832i
$944$	0	0
$945$	−1.17979e7	−0.0139800
$946$	0	0
$947$	1.24667e9	1.46792	0.733961	−	0.679192i	$-0.237669\pi$
$947$	1.24667e9	1.46792	0.733961	+	0.679192i	$0.237669\pi$
$948$	0	0
$949$	− 1.20833e9i	− 1.41380i
$950$	0	0
$951$	7.39671e8i	0.859997i
$952$	0	0
$953$	−4.21385e8	−0.486856	−0.243428	−	0.969919i	$-0.578272\pi$
$953$	−4.21385e8	−0.486856	−0.243428	+	0.969919i	$0.578272\pi$
$954$	0	0
$955$	−1.36522e8	−0.156744
$956$	0	0
$957$	1.60763e9i	1.83421i
$958$	0	0
$959$	2.73973e8i	0.310637i
$960$	0	0
$961$	−5.10826e7	−0.0575576
$962$	0	0
$963$	−3.12546e8	−0.349973
$964$	0	0
$965$	2.61683e7i	0.0291201i
$966$	0	0
$967$	1.35010e9i	1.49309i	0.665334	+	0.746546i	$0.268289\pi$
$967$	1.35010e9i	1.49309i	−0.665334	+	0.746546i	$0.731711\pi$
$968$	0	0
$969$	−4.01927e8	−0.441749
$970$	0	0
$971$	2.75696e8	0.301143	0.150571	−	0.988599i	$-0.451889\pi$
$971$	2.75696e8	0.301143	0.150571	+	0.988599i	$0.451889\pi$
$972$	0	0
$973$	− 4.72912e7i	− 0.0513383i
$974$	0	0
$975$	− 7.64497e8i	− 0.824825i
$976$	0	0
$977$	−4.86639e8	−0.521823	−0.260911	−	0.965363i	$-0.584023\pi$
$977$	−4.86639e8	−0.521823	−0.260911	+	0.965363i	$0.584023\pi$
$978$	0	0
$979$	2.07836e9	2.21499
$980$	0	0
$981$	4.27103e8i	0.452403i
$982$	0	0
$983$	− 1.34935e9i	− 1.42058i	−0.703909	−	0.710290i	$-0.748564\pi$
$983$	− 1.34935e9i	− 1.42058i	0.703909	−	0.710290i	$-0.251436\pi$
$984$	0	0
$985$	1.52275e8	0.159338
$986$	0	0
$987$	−4.26292e8	−0.443359
$988$	0	0
$989$	4.39501e8i	0.454330i
$990$	0	0
$991$	5.76213e8i	0.592056i	0.955179	+	0.296028i	$0.0956621\pi$
$991$	5.76213e8i	0.592056i	−0.955179	+	0.296028i	$0.904338\pi$
$992$	0	0
$993$	−7.54976e8	−0.771055
$994$	0	0
$995$	−1.29538e8	−0.131501
$996$	0	0
$997$	− 8.84682e7i	− 0.0892692i	−0.999003	−	0.0446346i	$-0.985788\pi$
$997$	− 8.84682e7i	− 0.0892692i	0.999003	−	0.0446346i	$-0.0142124\pi$
$998$	0	0
$999$	1.67271e8i	0.167773i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.7.b.d.319.3	yes	8
4.3	odd	2	inner	384.7.b.d.319.8	yes	8
8.3	odd	2	inner	384.7.b.d.319.2	✓	8
8.5	even	2	inner	384.7.b.d.319.5	yes	8
16.3	odd	4		768.7.g.e.511.3		4
16.5	even	4		768.7.g.c.511.4		4
16.11	odd	4		768.7.g.c.511.1		4
16.13	even	4		768.7.g.e.511.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.7.b.d.319.2	✓	8	8.3	odd	2	inner
384.7.b.d.319.3	yes	8	1.1	even	1	trivial
384.7.b.d.319.5	yes	8	8.5	even	2	inner
384.7.b.d.319.8	yes	8	4.3	odd	2	inner
768.7.g.c.511.1		4	16.11	odd	4
768.7.g.c.511.4		4	16.5	even	4
768.7.g.e.511.2		4	16.13	even	4
768.7.g.e.511.3		4	16.3	odd	4

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:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	384.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-15.5885 q^{3} +20.0000i q^{5} -155.727i q^{7} +243.000 q^{9} +O(q^{10})\)	\(q-15.5885 q^{3} +20.0000i q^{5} -155.727i q^{7} +243.000 q^{9} +2306.46 q^{11} +3221.18i q^{13} -311.769i q^{15} -6566.73 q^{17} -3926.40 q^{19} +2427.54i q^{21} +17392.0i q^{23} +15225.0 q^{25} -3788.00 q^{27} -44713.3i q^{29} -30636.4i q^{31} -35954.2 q^{33} +3114.54 q^{35} -44158.1i q^{37} -50213.3i q^{39} +9008.18 q^{41} +25270.3 q^{43} +4860.00i q^{45} +175606. i q^{47} +93398.1 q^{49} +102365. q^{51} +96206.3i q^{53} +46129.2i q^{55} +61206.5 q^{57} -35018.3 q^{59} -86135.7i q^{61} -37841.7i q^{63} -64423.6 q^{65} -424605. q^{67} -271114. i q^{69} +36572.4i q^{71} -375121. q^{73} -237334. q^{75} -359179. i q^{77} +520323. i q^{79} +59049.0 q^{81} +594208. q^{83} -131335. i q^{85} +697011. i q^{87} +901101. q^{89} +501625. q^{91} +477574. i q^{93} -78528.0i q^{95} -1.25744e6 q^{97} +560470. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 1944 q^{9}+O(q^{10}) \) 8 * q + 1944 * q^9	\( 8 q + 1944 q^{9} + 4464 q^{17} + 121800 q^{25} - 116640 q^{33} - 98928 q^{41} - 278776 q^{49} - 23328 q^{57} - 230400 q^{65} + 76912 q^{73} + 472392 q^{81} + 6638832 q^{89} - 4929680 q^{97}+O(q^{100}) \) 8 * q + 1944 * q^9 + 4464 * q^17 + 121800 * q^25 - 116640 * q^33 - 98928 * q^41 - 278776 * q^49 - 23328 * q^57 - 230400 * q^65 + 76912 * q^73 + 472392 * q^81 + 6638832 * q^89 - 4929680 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more