LMFDB - Embedded newform 384.7.b.a.319.2

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:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			319.2
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	384.319
Dual form			384.7.b.a.319.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-15.5885 q^{3} +196.000i q^{5} +270.200i q^{7} +243.000 q^{9} +O(q^{10})$	$q-15.5885 q^{3} +196.000i q^{5} +270.200i q^{7} +243.000 q^{9} -1725.12 q^{11} +2880.00i q^{13} -3055.34i q^{15} -2898.00 q^{17} -9789.55 q^{19} -4212.00i q^{21} +1621.20i q^{23} -22791.0 q^{25} -3788.00 q^{27} +14596.0i q^{29} +6879.71i q^{31} +26892.0 q^{33} -52959.2 q^{35} -12168.0i q^{37} -44894.8i q^{39} +18738.0 q^{41} -135245. q^{43} +47628.0i q^{45} +151270. i q^{47} +44641.0 q^{49} +45175.3 q^{51} +144236. i q^{53} -338124. i q^{55} +152604. q^{57} +333655. q^{59} +333432. i q^{61} +65658.6i q^{63} -564480. q^{65} +319439. q^{67} -25272.0i q^{69} -420639. i q^{71} +628238. q^{73} +355277. q^{75} -466128. i q^{77} +291878. i q^{79} +59049.0 q^{81} +357017. q^{83} -568008. i q^{85} -227529. i q^{87} -1.37507e6 q^{89} -778176. q^{91} -107244. i q^{93} -1.91875e6i q^{95} +489710. q^{97} -419205. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 972 q^{9}+O(q^{10}) $ 4 * q + 972 * q^9	$ 4 q + 972 q^{9} - 11592 q^{17} - 91164 q^{25} + 107568 q^{33} + 74952 q^{41} + 178564 q^{49} + 610416 q^{57} - 2257920 q^{65} + 2512952 q^{73} + 236196 q^{81} - 5500296 q^{89} + 1958840 q^{97}+O(q^{100}) $ 4 * q + 972 * q^9 - 11592 * q^17 - 91164 * q^25 + 107568 * q^33 + 74952 * q^41 + 178564 * q^49 + 610416 * q^57 - 2257920 * q^65 + 2512952 * q^73 + 236196 * q^81 - 5500296 * q^89 + 1958840 * 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
$5$	196.000i	1.56800i	0.620761	+	0.784000i	$0.286824\pi$
$5$	196.000i	1.56800i	−0.620761	+	0.784000i	$0.713176\pi$
$6$	0	0
$7$	270.200i	0.787755i	0.919163	+	0.393877i	$0.128867\pi$
$7$	270.200i	0.787755i	−0.919163	+	0.393877i	$0.871133\pi$
$8$	0	0
$9$	243.000	0.333333
$10$	0	0
$11$	−1725.12	−1.29611	−0.648055	−	0.761593i	$-0.724417\pi$
$11$	−1725.12	−1.29611	−0.648055	+	0.761593i	$0.724417\pi$
$12$	0	0
$13$	2880.00i	1.31088i	0.755248	+	0.655439i	$0.227516\pi$
$13$	2880.00i	1.31088i	−0.755248	+	0.655439i	$0.772484\pi$
$14$	0	0
$15$	− 3055.34i	− 0.905285i
$16$	0	0
$17$	−2898.00	−0.589864	−0.294932	−	0.955518i	$-0.595297\pi$
$17$	−2898.00	−0.589864	−0.294932	+	0.955518i	$0.595297\pi$
$18$	0	0
$19$	−9789.55	−1.42726	−0.713628	−	0.700525i	$-0.752949\pi$
$19$	−9789.55	−1.42726	−0.713628	+	0.700525i	$0.752949\pi$
$20$	0	0
$21$	− 4212.00i	− 0.454810i
$22$	0	0
$23$	1621.20i	0.133246i	0.997778	+	0.0666228i	$0.0212224\pi$
$23$	1621.20i	0.133246i	−0.997778	+	0.0666228i	$0.978778\pi$
$24$	0	0
$25$	−22791.0	−1.45862
$26$	0	0
$27$	−3788.00	−0.192450
$28$	0	0
$29$	14596.0i	0.598467i	0.954180	+	0.299233i	$0.0967309\pi$
$29$	14596.0i	0.598467i	−0.954180	+	0.299233i	$0.903269\pi$
$30$	0	0
$31$	6879.71i	0.230932i	0.993311	+	0.115466i	$0.0368362\pi$
$31$	6879.71i	0.230932i	−0.993311	+	0.115466i	$0.963164\pi$
$32$	0	0
$33$	26892.0	0.748310
$34$	0	0
$35$	−52959.2	−1.23520
$36$	0	0
$37$	− 12168.0i	− 0.240223i	−0.992760	−	0.120111i	$-0.961675\pi$
$37$	− 12168.0i	− 0.240223i	0.992760	−	0.120111i	$-0.0383252\pi$
$38$	0	0
$39$	− 44894.8i	− 0.756836i
$40$	0	0
$41$	18738.0	0.271876	0.135938	−	0.990717i	$-0.456595\pi$
$41$	18738.0	0.271876	0.135938	+	0.990717i	$0.456595\pi$
$42$	0	0
$43$	−135245.	−1.70105	−0.850525	−	0.525934i	$-0.823716\pi$
$43$	−135245.	−1.70105	−0.850525	+	0.525934i	$0.823716\pi$
$44$	0	0
$45$	47628.0i	0.522667i
$46$	0	0
$47$	151270.i	1.45700i	0.685044	+	0.728501i	$0.259783\pi$
$47$	151270.i	1.45700i	−0.685044	+	0.728501i	$0.740217\pi$
$48$	0	0
$49$	44641.0	0.379442
$50$	0	0
$51$	45175.3	0.340558
$52$	0	0
$53$	144236.i	0.968827i	0.874839	+	0.484413i	$0.160967\pi$
$53$	144236.i	0.968827i	−0.874839	+	0.484413i	$0.839033\pi$
$54$	0	0
$55$	− 338124.i	− 2.03230i
$56$	0	0
$57$	152604.	0.824027
$58$	0	0
$59$	333655.	1.62458	0.812292	−	0.583251i	$-0.198220\pi$
$59$	333655.	1.62458	0.812292	+	0.583251i	$0.198220\pi$
$60$	0	0
$61$	333432.i	1.46899i	0.678616	+	0.734493i	$0.262580\pi$
$61$	333432.i	1.46899i	−0.678616	+	0.734493i	$0.737420\pi$
$62$	0	0
$63$	65658.6i	0.262585i
$64$	0	0
$65$	−564480.	−2.05546
$66$	0	0
$67$	319439.	1.06209	0.531047	−	0.847342i	$-0.321799\pi$
$67$	319439.	1.06209	0.531047	+	0.847342i	$0.321799\pi$
$68$	0	0
$69$	− 25272.0i	− 0.0769294i
$70$	0	0
$71$	− 420639.i	− 1.17526i	−0.809129	−	0.587631i	$-0.800061\pi$
$71$	− 420639.i	− 1.17526i	0.809129	−	0.587631i	$-0.199939\pi$
$72$	0	0
$73$	628238.	1.61494	0.807469	−	0.589911i	$-0.200837\pi$
$73$	628238.	1.61494	0.807469	+	0.589911i	$0.200837\pi$
$74$	0	0
$75$	355277.	0.842137
$76$	0	0
$77$	− 466128.i	− 1.02102i
$78$	0	0
$79$	291878.i	0.591998i	0.955188	+	0.295999i	$0.0956526\pi$
$79$	291878.i	0.591998i	−0.955188	+	0.295999i	$0.904347\pi$
$80$	0	0
$81$	59049.0	0.111111
$82$	0	0
$83$	357017.	0.624389	0.312194	−	0.950018i	$-0.398936\pi$
$83$	357017.	0.624389	0.312194	+	0.950018i	$0.398936\pi$
$84$	0	0
$85$	− 568008.i	− 0.924906i
$86$	0	0
$87$	− 227529.i	− 0.345525i
$88$	0	0
$89$	−1.37507e6	−1.95055	−0.975273	−	0.221005i	$-0.929066\pi$
$89$	−1.37507e6	−1.95055	−0.975273	+	0.221005i	$0.929066\pi$
$90$	0	0
$91$	−778176.	−1.03265
$92$	0	0
$93$	− 107244.i	− 0.133329i
$94$	0	0
$95$	− 1.91875e6i	− 2.23794i
$96$	0	0
$97$	489710.	0.536567	0.268283	−	0.963340i	$-0.413544\pi$
$97$	489710.	0.536567	0.268283	+	0.963340i	$0.413544\pi$
$98$	0	0
$99$	−419205.	−0.432037
$100$	0	0
$101$	1.13539e6i	1.10200i	0.834506	+	0.550998i	$0.185753\pi$
$101$	1.13539e6i	1.10200i	−0.834506	+	0.550998i	$0.814247\pi$
$102$	0	0
$103$	− 501262.i	− 0.458726i	−0.973341	−	0.229363i	$-0.926336\pi$
$103$	− 501262.i	− 0.458726i	0.973341	−	0.229363i	$-0.0736643\pi$
$104$	0	0
$105$	825552.	0.713143
$106$	0	0
$107$	−1.27948e6	−1.04444	−0.522218	−	0.852812i	$-0.674895\pi$
$107$	−1.27948e6	−1.04444	−0.522218	+	0.852812i	$0.674895\pi$
$108$	0	0
$109$	− 384336.i	− 0.296778i	−0.988929	−	0.148389i	$-0.952591\pi$
$109$	− 384336.i	− 0.296778i	0.988929	−	0.148389i	$-0.0474088\pi$
$110$	0	0
$111$	189680.i	0.138693i
$112$	0	0
$113$	−1.58125e6	−1.09588	−0.547941	−	0.836517i	$-0.684588\pi$
$113$	−1.58125e6	−1.09588	−0.547941	+	0.836517i	$0.684588\pi$
$114$	0	0
$115$	−317755.	−0.208929
$116$	0	0
$117$	699840.i	0.436959i
$118$	0	0
$119$	− 783039.i	− 0.464668i
$120$	0	0
$121$	1.20449e6	0.679902
$122$	0	0
$123$	−292097.	−0.156968
$124$	0	0
$125$	− 1.40454e6i	− 0.719122i
$126$	0	0
$127$	− 4.02180e6i	− 1.96340i	−0.190426	−	0.981701i	$-0.560987\pi$
$127$	− 4.02180e6i	− 1.96340i	0.190426	−	0.981701i	$-0.439013\pi$
$128$	0	0
$129$	2.10827e6	0.982102
$130$	0	0
$131$	−1.01277e6	−0.450503	−0.225251	−	0.974301i	$-0.572320\pi$
$131$	−1.01277e6	−0.450503	−0.225251	+	0.974301i	$0.572320\pi$
$132$	0	0
$133$	− 2.64514e6i	− 1.12433i
$134$	0	0
$135$	− 742447.i	− 0.301762i
$136$	0	0
$137$	2.29394e6	0.892113	0.446057	−	0.895005i	$-0.352828\pi$
$137$	2.29394e6	0.892113	0.446057	+	0.895005i	$0.352828\pi$
$138$	0	0
$139$	−15526.1	−0.00578120	−0.00289060	−	0.999996i	$-0.500920\pi$
$139$	−15526.1	−0.00578120	−0.00289060	+	0.999996i	$0.500920\pi$
$140$	0	0
$141$	− 2.35807e6i	− 0.841201i
$142$	0	0
$143$	− 4.96835e6i	− 1.69904i
$144$	0	0
$145$	−2.86082e6	−0.938396
$146$	0	0
$147$	−695884.	−0.219071
$148$	0	0
$149$	− 2.97326e6i	− 0.898823i	−0.893325	−	0.449411i	$-0.851634\pi$
$149$	− 2.97326e6i	− 0.898823i	0.893325	−	0.449411i	$-0.148366\pi$
$150$	0	0
$151$	− 5.62197e6i	− 1.63289i	−0.577422	−	0.816446i	$-0.695941\pi$
$151$	− 5.62197e6i	− 1.63289i	0.577422	−	0.816446i	$-0.304059\pi$
$152$	0	0
$153$	−704214.	−0.196621
$154$	0	0
$155$	−1.34842e6	−0.362102
$156$	0	0
$157$	5.27998e6i	1.36437i	0.731178	+	0.682186i	$0.238971\pi$
$157$	5.27998e6i	1.36437i	−0.731178	+	0.682186i	$0.761029\pi$
$158$	0	0
$159$	− 2.24842e6i	− 0.559352i
$160$	0	0
$161$	−438048.	−0.104965
$162$	0	0
$163$	6.90176e6	1.59366	0.796832	−	0.604201i	$-0.206507\pi$
$163$	6.90176e6	1.59366	0.796832	+	0.604201i	$0.206507\pi$
$164$	0	0
$165$	5.27083e6i	1.17335i
$166$	0	0
$167$	4.72567e6i	1.01465i	0.861756	+	0.507323i	$0.169365\pi$
$167$	4.72567e6i	1.01465i	−0.861756	+	0.507323i	$0.830635\pi$
$168$	0	0
$169$	−3.46759e6	−0.718402
$170$	0	0
$171$	−2.37886e6	−0.475752
$172$	0	0
$173$	5.49729e6i	1.06172i	0.847459	+	0.530861i	$0.178131\pi$
$173$	5.49729e6i	1.06172i	−0.847459	+	0.530861i	$0.821869\pi$
$174$	0	0
$175$	− 6.15813e6i	− 1.14904i
$176$	0	0
$177$	−5.20117e6	−0.937954
$178$	0	0
$179$	9.14334e6	1.59421	0.797105	−	0.603840i	$-0.206364\pi$
$179$	9.14334e6	1.59421	0.797105	+	0.603840i	$0.206364\pi$
$180$	0	0
$181$	− 8.71646e6i	− 1.46996i	−0.678090	−	0.734978i	$-0.737192\pi$
$181$	− 8.71646e6i	− 1.46996i	0.678090	−	0.734978i	$-0.262808\pi$
$182$	0	0
$183$	− 5.19769e6i	− 0.848120i
$184$	0	0
$185$	2.38493e6	0.376669
$186$	0	0
$187$	4.99941e6	0.764528
$188$	0	0
$189$	− 1.02352e6i	− 0.151603i
$190$	0	0
$191$	− 8.32374e6i	− 1.19459i	−0.802022	−	0.597294i	$-0.796242\pi$
$191$	− 8.32374e6i	− 1.19459i	0.802022	−	0.597294i	$-0.203758\pi$
$192$	0	0
$193$	4.03977e6	0.561934	0.280967	−	0.959717i	$-0.409345\pi$
$193$	4.03977e6	0.561934	0.280967	+	0.959717i	$0.409345\pi$
$194$	0	0
$195$	8.79937e6	1.18672
$196$	0	0
$197$	1.27599e7i	1.66897i	0.551027	+	0.834487i	$0.314236\pi$
$197$	1.27599e7i	1.66897i	−0.551027	+	0.834487i	$0.685764\pi$
$198$	0	0
$199$	963969.i	0.122322i	0.998128	+	0.0611609i	$0.0194803\pi$
$199$	963969.i	0.122322i	−0.998128	+	0.0611609i	$0.980520\pi$
$200$	0	0
$201$	−4.97956e6	−0.613200
$202$	0	0
$203$	−3.94384e6	−0.471445
$204$	0	0
$205$	3.67265e6i	0.426302i
$206$	0	0
$207$	393951.i	0.0444152i
$208$	0	0
$209$	1.68882e7	1.84988
$210$	0	0
$211$	−4.69955e6	−0.500275	−0.250137	−	0.968210i	$-0.580476\pi$
$211$	−4.69955e6	−0.500275	−0.250137	+	0.968210i	$0.580476\pi$
$212$	0	0
$213$	6.55711e6i	0.678537i
$214$	0	0
$215$	− 2.65081e7i	− 2.66725i
$216$	0	0
$217$	−1.85890e6	−0.181918
$218$	0	0
$219$	−9.79326e6	−0.932384
$220$	0	0
$221$	− 8.34624e6i	− 0.773240i
$222$	0	0
$223$	7.13047e6i	0.642989i	0.946911	+	0.321495i	$0.104185\pi$
$223$	7.13047e6i	0.642989i	−0.946911	+	0.321495i	$0.895815\pi$
$224$	0	0
$225$	−5.53821e6	−0.486208
$226$	0	0
$227$	−1.69883e6	−0.145235	−0.0726177	−	0.997360i	$-0.523135\pi$
$227$	−1.69883e6	−0.145235	−0.0726177	+	0.997360i	$0.523135\pi$
$228$	0	0
$229$	− 1.20413e7i	− 1.00269i	−0.865248	−	0.501344i	$-0.832839\pi$
$229$	− 1.20413e7i	− 1.00269i	0.865248	−	0.501344i	$-0.167161\pi$
$230$	0	0
$231$	7.26622e6i	0.589484i
$232$	0	0
$233$	2.57341e6	0.203442	0.101721	−	0.994813i	$-0.467565\pi$
$233$	2.57341e6	0.203442	0.101721	+	0.994813i	$0.467565\pi$
$234$	0	0
$235$	−2.96490e7	−2.28458
$236$	0	0
$237$	− 4.54993e6i	− 0.341790i
$238$	0	0
$239$	1.72461e7i	1.26327i	0.775266	+	0.631635i	$0.217616\pi$
$239$	1.72461e7i	1.26327i	−0.775266	+	0.631635i	$0.782384\pi$
$240$	0	0
$241$	2.02288e6	0.144517	0.0722584	−	0.997386i	$-0.476979\pi$
$241$	2.02288e6	0.144517	0.0722584	+	0.997386i	$0.476979\pi$
$242$	0	0
$243$	−920483.	−0.0641500
$244$	0	0
$245$	8.74964e6i	0.594965i
$246$	0	0
$247$	− 2.81939e7i	− 1.87096i
$248$	0	0
$249$	−5.56535e6	−0.360491
$250$	0	0
$251$	−2.60223e7	−1.64560	−0.822801	−	0.568330i	$-0.807590\pi$
$251$	−2.60223e7	−1.64560	−0.822801	+	0.568330i	$0.807590\pi$
$252$	0	0
$253$	− 2.79677e6i	− 0.172701i
$254$	0	0
$255$	8.85437e6i	0.533995i
$256$	0	0
$257$	1.19807e7	0.705800	0.352900	−	0.935661i	$-0.385196\pi$
$257$	1.19807e7	0.705800	0.352900	+	0.935661i	$0.385196\pi$
$258$	0	0
$259$	3.28779e6	0.189237
$260$	0	0
$261$	3.54683e6i	0.199489i
$262$	0	0
$263$	3.55118e6i	0.195211i	0.995225	+	0.0976056i	$0.0311184\pi$
$263$	3.55118e6i	0.195211i	−0.995225	+	0.0976056i	$0.968882\pi$
$264$	0	0
$265$	−2.82703e7	−1.51912
$266$	0	0
$267$	2.14353e7	1.12615
$268$	0	0
$269$	− 5.35329e6i	− 0.275020i	−0.990500	−	0.137510i	$-0.956090\pi$
$269$	− 5.35329e6i	− 0.275020i	0.990500	−	0.137510i	$-0.0439099\pi$
$270$	0	0
$271$	1.58492e7i	0.796342i	0.917311	+	0.398171i	$0.130355\pi$
$271$	1.58492e7i	0.796342i	−0.917311	+	0.398171i	$0.869645\pi$
$272$	0	0
$273$	1.21306e7	0.596201
$274$	0	0
$275$	3.93173e7	1.89054
$276$	0	0
$277$	− 2.29206e7i	− 1.07842i	−0.842172	−	0.539209i	$-0.818723\pi$
$277$	− 2.29206e7i	− 1.07842i	0.842172	−	0.539209i	$-0.181277\pi$
$278$	0	0
$279$	1.67177e6i	0.0769775i
$280$	0	0
$281$	2.02713e7	0.913614	0.456807	−	0.889566i	$-0.348993\pi$
$281$	2.02713e7	0.913614	0.456807	+	0.889566i	$0.348993\pi$
$282$	0	0
$283$	−1.08364e7	−0.478108	−0.239054	−	0.971006i	$-0.576837\pi$
$283$	−1.08364e7	−0.478108	−0.239054	+	0.971006i	$0.576837\pi$
$284$	0	0
$285$	2.99104e7i	1.29207i
$286$	0	0
$287$	5.06301e6i	0.214172i
$288$	0	0
$289$	−1.57392e7	−0.652061
$290$	0	0
$291$	−7.63382e6	−0.309787
$292$	0	0
$293$	1.99362e7i	0.792572i	0.918127	+	0.396286i	$0.129701\pi$
$293$	1.99362e7i	0.792572i	−0.918127	+	0.396286i	$0.870299\pi$
$294$	0	0
$295$	6.53964e7i	2.54735i
$296$	0	0
$297$	6.53476e6	0.249437
$298$	0	0
$299$	−4.66905e6	−0.174669
$300$	0	0
$301$	− 3.65433e7i	− 1.34001i
$302$	0	0
$303$	− 1.76989e7i	− 0.636238i
$304$	0	0
$305$	−6.53527e7	−2.30337
$306$	0	0
$307$	5.44670e7	1.88243	0.941214	−	0.337811i	$-0.109686\pi$
$307$	5.44670e7	1.88243	0.941214	+	0.337811i	$0.109686\pi$
$308$	0	0
$309$	7.81391e6i	0.264846i
$310$	0	0
$311$	2.77118e7i	0.921262i	0.887592	+	0.460631i	$0.152377\pi$
$311$	2.77118e7i	0.921262i	−0.887592	+	0.460631i	$0.847623\pi$
$312$	0	0
$313$	3.46046e7	1.12850	0.564250	−	0.825604i	$-0.309166\pi$
$313$	3.46046e7	1.12850	0.564250	+	0.825604i	$0.309166\pi$
$314$	0	0
$315$	−1.28691e7	−0.411733
$316$	0	0
$317$	− 7.74888e6i	− 0.243255i	−0.992576	−	0.121627i	$-0.961189\pi$
$317$	− 7.74888e6i	− 0.243255i	0.992576	−	0.121627i	$-0.0388113\pi$
$318$	0	0
$319$	− 2.51799e7i	− 0.775679i
$320$	0	0
$321$	1.99451e7	0.603006
$322$	0	0
$323$	2.83701e7	0.841887
$324$	0	0
$325$	− 6.56381e7i	− 1.91208i
$326$	0	0
$327$	5.99121e6i	0.171345i
$328$	0	0
$329$	−4.08732e7	−1.14776
$330$	0	0
$331$	−3.63895e7	−1.00344	−0.501721	−	0.865030i	$-0.667300\pi$
$331$	−3.63895e7	−1.00344	−0.501721	+	0.865030i	$0.667300\pi$
$332$	0	0
$333$	− 2.95682e6i	− 0.0800742i
$334$	0	0
$335$	6.26100e7i	1.66536i
$336$	0	0
$337$	2.72631e7	0.712338	0.356169	−	0.934422i	$-0.384083\pi$
$337$	2.72631e7	0.712338	0.356169	+	0.934422i	$0.384083\pi$
$338$	0	0
$339$	2.46492e7	0.632708
$340$	0	0
$341$	− 1.18683e7i	− 0.299314i
$342$	0	0
$343$	4.38507e7i	1.08666i
$344$	0	0
$345$	4.95331e6	0.120625
$346$	0	0
$347$	−7.13891e6	−0.170861	−0.0854306	−	0.996344i	$-0.527227\pi$
$347$	−7.13891e6	−0.170861	−0.0854306	+	0.996344i	$0.527227\pi$
$348$	0	0
$349$	− 4.72416e7i	− 1.11134i	−0.831402	−	0.555671i	$-0.812461\pi$
$349$	− 4.72416e7i	− 1.11134i	0.831402	−	0.555671i	$-0.187539\pi$
$350$	0	0
$351$	− 1.09094e7i	− 0.252279i
$352$	0	0
$353$	1.46933e7	0.334039	0.167019	−	0.985954i	$-0.446586\pi$
$353$	1.46933e7	0.334039	0.167019	+	0.985954i	$0.446586\pi$
$354$	0	0
$355$	8.24452e7	1.84281
$356$	0	0
$357$	1.22064e7i	0.268276i
$358$	0	0
$359$	7.67789e7i	1.65943i	0.558189	+	0.829714i	$0.311497\pi$
$359$	7.67789e7i	1.65943i	−0.558189	+	0.829714i	$0.688503\pi$
$360$	0	0
$361$	4.87894e7	1.03706
$362$	0	0
$363$	−1.87761e7	−0.392541
$364$	0	0
$365$	1.23135e8i	2.53222i
$366$	0	0
$367$	3.02395e7i	0.611754i	0.952071	+	0.305877i	$0.0989496\pi$
$367$	3.02395e7i	0.611754i	−0.952071	+	0.305877i	$0.901050\pi$
$368$	0	0
$369$	4.55333e6	0.0906255
$370$	0	0
$371$	−3.89726e7	−0.763198
$372$	0	0
$373$	7.98541e7i	1.53876i	0.638791	+	0.769380i	$0.279435\pi$
$373$	7.98541e7i	1.53876i	−0.638791	+	0.769380i	$0.720565\pi$
$374$	0	0
$375$	2.18945e7i	0.415186i
$376$	0	0
$377$	−4.20365e7	−0.784517
$378$	0	0
$379$	−2.61074e7	−0.479563	−0.239781	−	0.970827i	$-0.577076\pi$
$379$	−2.61074e7	−0.479563	−0.239781	+	0.970827i	$0.577076\pi$
$380$	0	0
$381$	6.26937e7i	1.13357i
$382$	0	0
$383$	− 1.22550e7i	− 0.218131i	−0.994035	−	0.109066i	$-0.965214\pi$
$383$	− 1.22550e7i	− 0.218131i	0.994035	−	0.109066i	$-0.0347859\pi$
$384$	0	0
$385$	9.13611e7	1.60095
$386$	0	0
$387$	−3.28646e7	−0.567017
$388$	0	0
$389$	2.25279e6i	0.0382712i	0.999817	+	0.0191356i	$0.00609141\pi$
$389$	2.25279e6i	0.0382712i	−0.999817	+	0.0191356i	$0.993909\pi$
$390$	0	0
$391$	− 4.69824e6i	− 0.0785967i
$392$	0	0
$393$	1.57875e7	0.260098
$394$	0	0
$395$	−5.72081e7	−0.928253
$396$	0	0
$397$	− 1.03690e8i	− 1.65716i	−0.559873	−	0.828578i	$-0.689150\pi$
$397$	− 1.03690e8i	− 1.65716i	0.559873	−	0.828578i	$-0.310850\pi$
$398$	0	0
$399$	4.12336e7i	0.649131i
$400$	0	0
$401$	−1.15940e8	−1.79805	−0.899023	−	0.437901i	$-0.855722\pi$
$401$	−1.15940e8	−1.79805	−0.899023	+	0.437901i	$0.855722\pi$
$402$	0	0
$403$	−1.98136e7	−0.302724
$404$	0	0
$405$	1.15736e7i	0.174222i
$406$	0	0
$407$	2.09913e7i	0.311355i
$408$	0	0
$409$	−3.81538e7	−0.557658	−0.278829	−	0.960341i	$-0.589946\pi$
$409$	−3.81538e7	−0.557658	−0.278829	+	0.960341i	$0.589946\pi$
$410$	0	0
$411$	−3.57590e7	−0.515062
$412$	0	0
$413$	9.01536e7i	1.27977i
$414$	0	0
$415$	6.99754e7i	0.979041i
$416$	0	0
$417$	242028.	0.00333778
$418$	0	0
$419$	5.37135e6	0.0730199	0.0365099	−	0.999333i	$-0.488376\pi$
$419$	5.37135e6	0.0730199	0.0365099	+	0.999333i	$0.488376\pi$
$420$	0	0
$421$	− 7.40412e7i	− 0.992264i	−0.868247	−	0.496132i	$-0.834753\pi$
$421$	− 7.40412e7i	− 0.992264i	0.868247	−	0.496132i	$-0.165247\pi$
$422$	0	0
$423$	3.67587e7i	0.485668i
$424$	0	0
$425$	6.60483e7	0.860389
$426$	0	0
$427$	−9.00933e7	−1.15720
$428$	0	0
$429$	7.74490e7i	0.980943i
$430$	0	0
$431$	1.81463e7i	0.226651i	0.993558	+	0.113325i	$0.0361502\pi$
$431$	1.81463e7i	0.226651i	−0.993558	+	0.113325i	$0.963850\pi$
$432$	0	0
$433$	−1.56953e8	−1.93333	−0.966665	−	0.256046i	$-0.917580\pi$
$433$	−1.56953e8	−1.93333	−0.966665	+	0.256046i	$0.917580\pi$
$434$	0	0
$435$	4.45957e7	0.541783
$436$	0	0
$437$	− 1.58708e7i	− 0.190176i
$438$	0	0
$439$	6.59653e7i	0.779691i	0.920880	+	0.389845i	$0.127472\pi$
$439$	6.59653e7i	0.779691i	−0.920880	+	0.389845i	$0.872528\pi$
$440$	0	0
$441$	1.08478e7	0.126481
$442$	0	0
$443$	−1.07019e8	−1.23098	−0.615489	−	0.788145i	$-0.711042\pi$
$443$	−1.07019e8	−1.23098	−0.615489	+	0.788145i	$0.711042\pi$
$444$	0	0
$445$	− 2.69515e8i	− 3.05846i
$446$	0	0
$447$	4.63485e7i	0.518936i
$448$	0	0
$449$	−4.53200e7	−0.500669	−0.250335	−	0.968159i	$-0.580541\pi$
$449$	−4.53200e7	−0.500669	−0.250335	+	0.968159i	$0.580541\pi$
$450$	0	0
$451$	−3.23253e7	−0.352382
$452$	0	0
$453$	8.76378e7i	0.942751i
$454$	0	0
$455$	− 1.52522e8i	− 1.61920i
$456$	0	0
$457$	−2.38273e6	−0.0249647	−0.0124824	−	0.999922i	$-0.503973\pi$
$457$	−2.38273e6	−0.0249647	−0.0124824	+	0.999922i	$0.503973\pi$
$458$	0	0
$459$	1.09776e7	0.113519
$460$	0	0
$461$	1.20106e7i	0.122591i	0.998120	+	0.0612957i	$0.0195233\pi$
$461$	1.20106e7i	0.122591i	−0.998120	+	0.0612957i	$0.980477\pi$
$462$	0	0
$463$	− 9.28538e6i	− 0.0935528i	−0.998905	−	0.0467764i	$-0.985105\pi$
$463$	− 9.28538e6i	− 0.0935528i	0.998905	−	0.0467764i	$-0.0148948\pi$
$464$	0	0
$465$	2.10198e7	0.209060
$466$	0	0
$467$	1.33286e8	1.30868	0.654341	−	0.756199i	$-0.272946\pi$
$467$	1.33286e8	1.30868	0.654341	+	0.756199i	$0.272946\pi$
$468$	0	0
$469$	8.63123e7i	0.836670i
$470$	0	0
$471$	− 8.23067e7i	− 0.787721i
$472$	0	0
$473$	2.33315e8	2.20475
$474$	0	0
$475$	2.23114e8	2.08183
$476$	0	0
$477$	3.50493e7i	0.322942i
$478$	0	0
$479$	− 6.64790e7i	− 0.604892i	−0.953166	−	0.302446i	$-0.902197\pi$
$479$	− 6.64790e7i	− 0.604892i	0.953166	−	0.302446i	$-0.0978033\pi$
$480$	0	0
$481$	3.50438e7	0.314903
$482$	0	0
$483$	6.82849e6	0.0606015
$484$	0	0
$485$	9.59832e7i	0.841337i
$486$	0	0
$487$	− 1.25653e8i	− 1.08789i	−0.839121	−	0.543945i	$-0.816930\pi$
$487$	− 1.25653e8i	− 1.08789i	0.839121	−	0.543945i	$-0.183070\pi$
$488$	0	0
$489$	−1.07588e8	−0.920103
$490$	0	0
$491$	1.89229e8	1.59861	0.799304	−	0.600927i	$-0.205202\pi$
$491$	1.89229e8	1.59861	0.799304	+	0.600927i	$0.205202\pi$
$492$	0	0
$493$	− 4.22992e7i	− 0.353014i
$494$	0	0
$495$	− 8.21641e7i	− 0.677434i
$496$	0	0
$497$	1.13657e8	0.925818
$498$	0	0
$499$	−1.09389e7	−0.0880385	−0.0440193	−	0.999031i	$-0.514016\pi$
$499$	−1.09389e7	−0.0880385	−0.0440193	+	0.999031i	$0.514016\pi$
$500$	0	0
$501$	− 7.36659e7i	− 0.585806i
$502$	0	0
$503$	1.12787e8i	0.886247i	0.896460	+	0.443124i	$0.146130\pi$
$503$	1.12787e8i	0.886247i	−0.896460	+	0.443124i	$0.853870\pi$
$504$	0	0
$505$	−2.22536e8	−1.72793
$506$	0	0
$507$	5.40544e7	0.414770
$508$	0	0
$509$	4.40530e7i	0.334058i	0.985952	+	0.167029i	$0.0534173\pi$
$509$	4.40530e7i	0.334058i	−0.985952	+	0.167029i	$0.946583\pi$
$510$	0	0
$511$	1.69750e8i	1.27217i
$512$	0	0
$513$	3.70828e7	0.274676
$514$	0	0
$515$	9.82474e7	0.719283
$516$	0	0
$517$	− 2.60960e8i	− 1.88844i
$518$	0	0
$519$	− 8.56943e7i	− 0.612985i
$520$	0	0
$521$	1.54449e8	1.09213	0.546063	−	0.837744i	$-0.316126\pi$
$521$	1.54449e8	1.09213	0.546063	+	0.837744i	$0.316126\pi$
$522$	0	0
$523$	−9.91307e6	−0.0692952	−0.0346476	−	0.999400i	$-0.511031\pi$
$523$	−9.91307e6	−0.0692952	−0.0346476	+	0.999400i	$0.511031\pi$
$524$	0	0
$525$	9.59957e7i	0.663398i
$526$	0	0
$527$	− 1.99374e7i	− 0.136219i
$528$	0	0
$529$	1.45408e8	0.982246
$530$	0	0
$531$	8.10782e7	0.541528
$532$	0	0
$533$	5.39654e7i	0.356397i
$534$	0	0
$535$	− 2.50778e8i	− 1.63768i
$536$	0	0
$537$	−1.42531e8	−0.920418
$538$	0	0
$539$	−7.70112e7	−0.491799
$540$	0	0
$541$	− 1.41127e8i	− 0.891287i	−0.895211	−	0.445644i	$-0.852975\pi$
$541$	− 1.41127e8i	− 0.891287i	0.895211	−	0.445644i	$-0.147025\pi$
$542$	0	0
$543$	1.35876e8i	0.848680i
$544$	0	0
$545$	7.53299e7	0.465348
$546$	0	0
$547$	−2.92266e8	−1.78573	−0.892865	−	0.450324i	$-0.851308\pi$
$547$	−2.92266e8	−1.78573	−0.892865	+	0.450324i	$0.851308\pi$
$548$	0	0
$549$	8.10240e7i	0.489662i
$550$	0	0
$551$	− 1.42888e8i	− 0.854165i
$552$	0	0
$553$	−7.88655e7	−0.466350
$554$	0	0
$555$	−3.71773e7	−0.217470
$556$	0	0
$557$	− 6.89434e7i	− 0.398958i	−0.979902	−	0.199479i	$-0.936075\pi$
$557$	− 6.89434e7i	− 0.398958i	0.979902	−	0.199479i	$-0.0639249\pi$
$558$	0	0
$559$	− 3.89507e8i	− 2.22987i
$560$	0	0
$561$	−7.79330e7	−0.441401
$562$	0	0
$563$	−7.26589e7	−0.407159	−0.203579	−	0.979058i	$-0.565257\pi$
$563$	−7.26589e7	−0.407159	−0.203579	+	0.979058i	$0.565257\pi$
$564$	0	0
$565$	− 3.09924e8i	− 1.71834i
$566$	0	0
$567$	1.59550e7i	0.0875283i
$568$	0	0
$569$	−1.16272e8	−0.631161	−0.315581	−	0.948899i	$-0.602199\pi$
$569$	−1.16272e8	−0.631161	−0.315581	+	0.948899i	$0.602199\pi$
$570$	0	0
$571$	−7.62675e7	−0.409667	−0.204834	−	0.978797i	$-0.565665\pi$
$571$	−7.62675e7	−0.409667	−0.204834	+	0.978797i	$0.565665\pi$
$572$	0	0
$573$	1.29754e8i	0.689696i
$574$	0	0
$575$	− 3.69488e7i	− 0.194355i
$576$	0	0
$577$	−3.84696e6	−0.0200258	−0.0100129	−	0.999950i	$-0.503187\pi$
$577$	−3.84696e6	−0.0200258	−0.0100129	+	0.999950i	$0.503187\pi$
$578$	0	0
$579$	−6.29738e7	−0.324433
$580$	0	0
$581$	9.64660e7i	0.491865i
$582$	0	0
$583$	− 2.48825e8i	− 1.25571i
$584$	0	0
$585$	−1.37169e8	−0.685152
$586$	0	0
$587$	−1.80286e8	−0.891348	−0.445674	−	0.895195i	$-0.647036\pi$
$587$	−1.80286e8	−0.891348	−0.445674	+	0.895195i	$0.647036\pi$
$588$	0	0
$589$	− 6.73492e7i	− 0.329600i
$590$	0	0
$591$	− 1.98908e8i	− 0.963583i
$592$	0	0
$593$	2.58548e7	0.123987	0.0619936	−	0.998077i	$-0.480254\pi$
$593$	2.58548e7	0.123987	0.0619936	+	0.998077i	$0.480254\pi$
$594$	0	0
$595$	1.53476e8	0.728599
$596$	0	0
$597$	− 1.50268e7i	− 0.0706226i
$598$	0	0
$599$	1.37569e8i	0.640088i	0.947403	+	0.320044i	$0.103698\pi$
$599$	1.37569e8i	0.640088i	−0.947403	+	0.320044i	$0.896302\pi$
$600$	0	0
$601$	3.09688e8	1.42659	0.713297	−	0.700862i	$-0.247201\pi$
$601$	3.09688e8	1.42659	0.713297	+	0.700862i	$0.247201\pi$
$602$	0	0
$603$	7.76236e7	0.354031
$604$	0	0
$605$	2.36079e8i	1.06609i
$606$	0	0
$607$	− 2.21384e8i	− 0.989875i	−0.868929	−	0.494937i	$-0.835191\pi$
$607$	− 2.21384e8i	− 0.989875i	0.868929	−	0.494937i	$-0.164809\pi$
$608$	0	0
$609$	6.14784e7	0.272189
$610$	0	0
$611$	−4.35659e8	−1.90995
$612$	0	0
$613$	− 3.15371e8i	− 1.36912i	−0.728957	−	0.684559i	$-0.759995\pi$
$613$	− 3.15371e8i	− 1.36912i	0.728957	−	0.684559i	$-0.240005\pi$
$614$	0	0
$615$	− 5.72509e7i	− 0.246126i
$616$	0	0
$617$	−2.80709e7	−0.119509	−0.0597546	−	0.998213i	$-0.519032\pi$
$617$	−2.80709e7	−0.119509	−0.0597546	+	0.998213i	$0.519032\pi$
$618$	0	0
$619$	−1.78560e7	−0.0752855	−0.0376427	−	0.999291i	$-0.511985\pi$
$619$	−1.78560e7	−0.0752855	−0.0376427	+	0.999291i	$0.511985\pi$
$620$	0	0
$621$	− 6.14110e6i	− 0.0256431i
$622$	0	0
$623$	− 3.71545e8i	− 1.53655i
$624$	0	0
$625$	−8.08203e7	−0.331040
$626$	0	0
$627$	−2.63261e8	−1.06803
$628$	0	0
$629$	3.52629e7i	0.141699i
$630$	0	0
$631$	4.60375e8i	1.83241i	0.400706	+	0.916207i	$0.368765\pi$
$631$	4.60375e8i	1.83241i	−0.400706	+	0.916207i	$0.631235\pi$
$632$	0	0
$633$	7.32587e7	0.288834
$634$	0	0
$635$	7.88273e8	3.07862
$636$	0	0
$637$	1.28566e8i	0.497403i
$638$	0	0
$639$	− 1.02215e8i	− 0.391754i
$640$	0	0
$641$	−4.39176e8	−1.66750	−0.833748	−	0.552145i	$-0.813810\pi$
$641$	−4.39176e8	−1.66750	−0.833748	+	0.552145i	$0.813810\pi$
$642$	0	0
$643$	−1.13354e8	−0.426387	−0.213194	−	0.977010i	$-0.568387\pi$
$643$	−1.13354e8	−0.426387	−0.213194	+	0.977010i	$0.568387\pi$
$644$	0	0
$645$	4.13221e8i	1.53994i
$646$	0	0
$647$	− 3.52127e8i	− 1.30013i	−0.759879	−	0.650064i	$-0.774742\pi$
$647$	− 3.52127e8i	− 1.30013i	0.759879	−	0.650064i	$-0.225258\pi$
$648$	0	0
$649$	−5.75596e8	−2.10564
$650$	0	0
$651$	2.89773e7	0.105030
$652$	0	0
$653$	− 3.37871e8i	− 1.21342i	−0.794923	−	0.606711i	$-0.792489\pi$
$653$	− 3.37871e8i	− 1.21342i	0.794923	−	0.606711i	$-0.207511\pi$
$654$	0	0
$655$	− 1.98503e8i	− 0.706389i
$656$	0	0
$657$	1.52662e8	0.538312
$658$	0	0
$659$	−4.27619e8	−1.49417	−0.747086	−	0.664727i	$-0.768548\pi$
$659$	−4.27619e8	−1.49417	−0.747086	+	0.664727i	$0.768548\pi$
$660$	0	0
$661$	4.35404e8i	1.50761i	0.657100	+	0.753803i	$0.271783\pi$
$661$	4.35404e8i	1.50761i	−0.657100	+	0.753803i	$0.728217\pi$
$662$	0	0
$663$	1.30105e8i	0.446430i
$664$	0	0
$665$	5.18447e8	1.76295
$666$	0	0
$667$	−2.36630e7	−0.0797430
$668$	0	0
$669$	− 1.11153e8i	− 0.371230i
$670$	0	0
$671$	− 5.75211e8i	− 1.90397i
$672$	0	0
$673$	5.20237e8	1.70669	0.853347	−	0.521343i	$-0.174569\pi$
$673$	5.20237e8	1.70669	0.853347	+	0.521343i	$0.174569\pi$
$674$	0	0
$675$	8.63322e7	0.280712
$676$	0	0
$677$	5.23676e8i	1.68771i	0.536574	+	0.843853i	$0.319718\pi$
$677$	5.23676e8i	1.68771i	−0.536574	+	0.843853i	$0.680282\pi$
$678$	0	0
$679$	1.32320e8i	0.422683i
$680$	0	0
$681$	2.64821e7	0.0838517
$682$	0	0
$683$	−2.29151e7	−0.0719216	−0.0359608	−	0.999353i	$-0.511449\pi$
$683$	−2.29151e7	−0.0719216	−0.0359608	+	0.999353i	$0.511449\pi$
$684$	0	0
$685$	4.49612e8i	1.39883i
$686$	0	0
$687$	1.87705e8i	0.578903i
$688$	0	0
$689$	−4.15400e8	−1.27001
$690$	0	0
$691$	3.31509e8	1.00476	0.502379	−	0.864647i	$-0.332458\pi$
$691$	3.31509e8	1.00476	0.502379	+	0.864647i	$0.332458\pi$
$692$	0	0
$693$	− 1.13269e8i	− 0.340339i
$694$	0	0
$695$	− 3.04312e6i	− 0.00906492i
$696$	0	0
$697$	−5.43027e7	−0.160370
$698$	0	0
$699$	−4.01154e7	−0.117457
$700$	0	0
$701$	3.52665e8i	1.02378i	0.859050	+	0.511891i	$0.171055\pi$
$701$	3.52665e8i	1.02378i	−0.859050	+	0.511891i	$0.828945\pi$
$702$	0	0
$703$	1.19119e8i	0.342859i
$704$	0	0
$705$	4.62182e8	1.31900
$706$	0	0
$707$	−3.06782e8	−0.868103
$708$	0	0
$709$	− 6.26217e8i	− 1.75706i	−0.477688	−	0.878530i	$-0.658525\pi$
$709$	− 6.26217e8i	− 1.75706i	0.477688	−	0.878530i	$-0.341475\pi$
$710$	0	0
$711$	7.09264e7i	0.197333i
$712$	0	0
$713$	−1.11534e7	−0.0307707
$714$	0	0
$715$	9.73797e8	2.66410
$716$	0	0
$717$	− 2.68840e8i	− 0.729350i
$718$	0	0
$719$	3.63123e8i	0.976937i	0.872581	+	0.488469i	$0.162444\pi$
$719$	3.63123e8i	0.976937i	−0.872581	+	0.488469i	$0.837556\pi$
$720$	0	0
$721$	1.35441e8	0.361364
$722$	0	0
$723$	−3.15335e7	−0.0834369
$724$	0	0
$725$	− 3.32657e8i	− 0.872938i
$726$	0	0
$727$	− 3.63836e8i	− 0.946897i	−0.880821	−	0.473449i	$-0.843009\pi$
$727$	− 3.63836e8i	− 0.946897i	0.880821	−	0.473449i	$-0.156991\pi$
$728$	0	0
$729$	1.43489e7	0.0370370
$730$	0	0
$731$	3.91941e8	1.00339
$732$	0	0
$733$	− 5.50829e8i	− 1.39864i	−0.714811	−	0.699318i	$-0.753487\pi$
$733$	− 5.50829e8i	− 1.39864i	0.714811	−	0.699318i	$-0.246513\pi$
$734$	0	0
$735$	− 1.36393e8i	− 0.343503i
$736$	0	0
$737$	−5.51071e8	−1.37659
$738$	0	0
$739$	−2.89298e8	−0.716823	−0.358412	−	0.933564i	$-0.616682\pi$
$739$	−2.89298e8	−0.716823	−0.358412	+	0.933564i	$0.616682\pi$
$740$	0	0
$741$	4.39500e8i	1.08020i
$742$	0	0
$743$	− 9.00629e7i	− 0.219573i	−0.993955	−	0.109787i	$-0.964983\pi$
$743$	− 9.00629e7i	− 0.219573i	0.993955	−	0.109787i	$-0.0350167\pi$
$744$	0	0
$745$	5.82759e8	1.40935
$746$	0	0
$747$	8.67552e7	0.208130
$748$	0	0
$749$	− 3.45715e8i	− 0.822760i
$750$	0	0
$751$	5.56903e8i	1.31480i	0.753542	+	0.657400i	$0.228344\pi$
$751$	5.56903e8i	1.31480i	−0.753542	+	0.657400i	$0.771656\pi$
$752$	0	0
$753$	4.05648e8	0.950089
$754$	0	0
$755$	1.10191e9	2.56037
$756$	0	0
$757$	− 1.67647e8i	− 0.386463i	−0.981153	−	0.193232i	$-0.938103\pi$
$757$	− 1.67647e8i	− 0.386463i	0.981153	−	0.193232i	$-0.0618969\pi$
$758$	0	0
$759$	4.35973e7i	0.0997090i
$760$	0	0
$761$	−7.51246e6	−0.0170462	−0.00852311	−	0.999964i	$-0.502713\pi$
$761$	−7.51246e6	−0.0170462	−0.00852311	+	0.999964i	$0.502713\pi$
$762$	0	0
$763$	1.03848e8	0.233788
$764$	0	0
$765$	− 1.38026e8i	− 0.308302i
$766$	0	0
$767$	9.60927e8i	2.12963i
$768$	0	0
$769$	−1.93147e8	−0.424727	−0.212364	−	0.977191i	$-0.568116\pi$
$769$	−1.93147e8	−0.424727	−0.212364	+	0.977191i	$0.568116\pi$
$770$	0	0
$771$	−1.86760e8	−0.407494
$772$	0	0
$773$	1.25486e8i	0.271679i	0.990731	+	0.135839i	$0.0433731\pi$
$773$	1.25486e8i	0.271679i	−0.990731	+	0.135839i	$0.956627\pi$
$774$	0	0
$775$	− 1.56795e8i	− 0.336843i
$776$	0	0
$777$	−5.12516e7	−0.109256
$778$	0	0
$779$	−1.83437e8	−0.388037
$780$	0	0
$781$	7.25654e8i	1.52327i
$782$	0	0
$783$	− 5.52896e7i	− 0.115175i
$784$	0	0
$785$	−1.03488e9	−2.13934
$786$	0	0
$787$	−5.71718e8	−1.17289	−0.586445	−	0.809989i	$-0.699473\pi$
$787$	−5.71718e8	−1.17289	−0.586445	+	0.809989i	$0.699473\pi$
$788$	0	0
$789$	− 5.53573e7i	− 0.112705i
$790$	0	0
$791$	− 4.27253e8i	− 0.863287i
$792$	0	0
$793$	−9.60284e8	−1.92566
$794$	0	0
$795$	4.40690e8	0.877064
$796$	0	0
$797$	4.13512e8i	0.816796i	0.912804	+	0.408398i	$0.133912\pi$
$797$	4.13512e8i	0.816796i	−0.912804	+	0.408398i	$0.866088\pi$
$798$	0	0
$799$	− 4.38382e8i	− 0.859433i
$800$	0	0
$801$	−3.34143e8	−0.650182
$802$	0	0
$803$	−1.08379e9	−2.09314
$804$	0	0
$805$	− 8.58574e7i	− 0.164585i
$806$	0	0
$807$	8.34496e7i	0.158783i
$808$	0	0
$809$	4.21028e8	0.795180	0.397590	−	0.917563i	$-0.369847\pi$
$809$	4.21028e8	0.795180	0.397590	+	0.917563i	$0.369847\pi$
$810$	0	0
$811$	−5.42556e7	−0.101714	−0.0508572	−	0.998706i	$-0.516195\pi$
$811$	−5.42556e7	−0.101714	−0.0508572	+	0.998706i	$0.516195\pi$
$812$	0	0
$813$	− 2.47065e8i	− 0.459768i
$814$	0	0
$815$	1.35274e9i	2.49887i
$816$	0	0
$817$	1.32399e9	2.42784
$818$	0	0
$819$	−1.89097e8	−0.344217
$820$	0	0
$821$	− 2.90848e8i	− 0.525578i	−0.964853	−	0.262789i	$-0.915358\pi$
$821$	− 2.90848e8i	− 0.525578i	0.964853	−	0.262789i	$-0.0846423\pi$
$822$	0	0
$823$	3.58287e8i	0.642734i	0.946955	+	0.321367i	$0.104142\pi$
$823$	3.58287e8i	0.642734i	−0.946955	+	0.321367i	$0.895858\pi$
$824$	0	0
$825$	−6.12896e8	−1.09150
$826$	0	0
$827$	−5.49604e8	−0.971702	−0.485851	−	0.874042i	$-0.661490\pi$
$827$	−5.49604e8	−0.971702	−0.485851	+	0.874042i	$0.661490\pi$
$828$	0	0
$829$	1.91879e8i	0.336794i	0.985719	+	0.168397i	$0.0538591\pi$
$829$	1.91879e8i	0.336794i	−0.985719	+	0.168397i	$0.946141\pi$
$830$	0	0
$831$	3.57297e8i	0.622625i
$832$	0	0
$833$	−1.29370e8	−0.223819
$834$	0	0
$835$	−9.26232e8	−1.59096
$836$	0	0
$837$	− 2.60603e7i	− 0.0444430i
$838$	0	0
$839$	8.59129e8i	1.45470i	0.686269	+	0.727348i	$0.259247\pi$
$839$	8.59129e8i	1.45470i	−0.686269	+	0.727348i	$0.740753\pi$
$840$	0	0
$841$	3.81780e8	0.641838
$842$	0	0
$843$	−3.15998e8	−0.527475
$844$	0	0
$845$	− 6.79648e8i	− 1.12645i
$846$	0	0
$847$	3.25452e8i	0.535596i
$848$	0	0
$849$	1.68923e8	0.276036
$850$	0	0
$851$	1.97268e7	0.0320086
$852$	0	0
$853$	7.92564e8i	1.27699i	0.769626	+	0.638495i	$0.220442\pi$
$853$	7.92564e8i	1.27699i	−0.769626	+	0.638495i	$0.779558\pi$
$854$	0	0
$855$	− 4.66257e8i	− 0.745979i
$856$	0	0
$857$	2.82625e8	0.449022	0.224511	−	0.974472i	$-0.427922\pi$
$857$	2.82625e8	0.449022	0.224511	+	0.974472i	$0.427922\pi$
$858$	0	0
$859$	−4.51038e8	−0.711596	−0.355798	−	0.934563i	$-0.615791\pi$
$859$	−4.51038e8	−0.711596	−0.355798	+	0.934563i	$0.615791\pi$
$860$	0	0
$861$	− 7.89245e7i	− 0.123652i
$862$	0	0
$863$	− 5.56438e8i	− 0.865734i	−0.901458	−	0.432867i	$-0.857502\pi$
$863$	− 5.56438e8i	− 0.865734i	0.901458	−	0.432867i	$-0.142498\pi$
$864$	0	0
$865$	−1.07747e9	−1.66478
$866$	0	0
$867$	2.45349e8	0.376468
$868$	0	0
$869$	− 5.03526e8i	− 0.767295i
$870$	0	0
$871$	9.19983e8i	1.39228i
$872$	0	0
$873$	1.19000e8	0.178856
$874$	0	0
$875$	3.79506e8	0.566492
$876$	0	0
$877$	4.73043e8i	0.701296i	0.936507	+	0.350648i	$0.114039\pi$
$877$	4.73043e8i	0.701296i	−0.936507	+	0.350648i	$0.885961\pi$
$878$	0	0
$879$	− 3.10774e8i	− 0.457592i
$880$	0	0
$881$	2.82594e8	0.413271	0.206636	−	0.978418i	$-0.433749\pi$
$881$	2.82594e8	0.413271	0.206636	+	0.978418i	$0.433749\pi$
$882$	0	0
$883$	−1.66922e8	−0.242454	−0.121227	−	0.992625i	$-0.538683\pi$
$883$	−1.66922e8	−0.242454	−0.121227	+	0.992625i	$0.538683\pi$
$884$	0	0
$885$	− 1.01943e9i	− 1.47071i
$886$	0	0
$887$	5.01664e8i	0.718857i	0.933173	+	0.359428i	$0.117028\pi$
$887$	5.01664e8i	0.718857i	−0.933173	+	0.359428i	$0.882972\pi$
$888$	0	0
$889$	1.08669e9	1.54668
$890$	0	0
$891$	−1.01867e8	−0.144012
$892$	0	0
$893$	− 1.48087e9i	− 2.07952i
$894$	0	0
$895$	1.79209e9i	2.49972i
$896$	0	0
$897$	7.27834e7	0.100845
$898$	0	0
$899$	−1.00416e8	−0.138205
$900$	0	0
$901$	− 4.17996e8i	− 0.571476i
$902$	0	0
$903$	5.69654e8i	0.773656i
$904$	0	0
$905$	1.70843e9	2.30489
$906$	0	0
$907$	11909.6	1.59615e−5 0	7.98077e−6	−	1.00000i	$-0.499997\pi$
$907$	11909.6	1.59615e−5 0	7.98077e−6		1.00000i	$0.499997\pi$
$908$	0	0
$909$	2.75899e8i	0.367332i
$910$	0	0
$911$	− 3.13710e8i	− 0.414929i	−0.978243	−	0.207464i	$-0.933479\pi$
$911$	− 3.13710e8i	− 0.414929i	0.978243	−	0.207464i	$-0.0665211\pi$
$912$	0	0
$913$	−6.15899e8	−0.809276
$914$	0	0
$915$	1.01875e9	1.32985
$916$	0	0
$917$	− 2.73651e8i	− 0.354886i
$918$	0	0
$919$	− 1.77960e8i	− 0.229285i	−0.993407	−	0.114642i	$-0.963428\pi$
$919$	− 1.77960e8i	− 0.229285i	0.993407	−	0.114642i	$-0.0365722\pi$
$920$	0	0
$921$	−8.49057e8	−1.08682
$922$	0	0
$923$	1.21144e9	1.54062
$924$	0	0
$925$	2.77321e8i	0.350395i
$926$	0	0
$927$	− 1.21807e8i	− 0.152909i
$928$	0	0
$929$	−1.99728e8	−0.249110	−0.124555	−	0.992213i	$-0.539750\pi$
$929$	−1.99728e8	−0.249110	−0.124555	+	0.992213i	$0.539750\pi$
$930$	0	0
$931$	−4.37015e8	−0.541561
$932$	0	0
$933$	− 4.31984e8i	− 0.531891i
$934$	0	0
$935$	9.79883e8i	1.19878i
$936$	0	0
$937$	−1.50949e9	−1.83490	−0.917450	−	0.397852i	$-0.869756\pi$
$937$	−1.50949e9	−1.83490	−0.917450	+	0.397852i	$0.869756\pi$
$938$	0	0
$939$	−5.39433e8	−0.651539
$940$	0	0
$941$	1.62445e9i	1.94957i	0.223150	+	0.974784i	$0.428366\pi$
$941$	1.62445e9i	1.94957i	−0.223150	+	0.974784i	$0.571634\pi$
$942$	0	0
$943$	3.03780e7i	0.0362264i
$944$	0	0
$945$	2.00609e8	0.237714
$946$	0	0
$947$	7.64857e8	0.900597	0.450298	−	0.892878i	$-0.351318\pi$
$947$	7.64857e8	0.900597	0.450298	+	0.892878i	$0.351318\pi$
$948$	0	0
$949$	1.80933e9i	2.11699i
$950$	0	0
$951$	1.20793e8i	0.140443i
$952$	0	0
$953$	1.12339e9	1.29794	0.648968	−	0.760815i	$-0.275201\pi$
$953$	1.12339e9	1.29794	0.648968	+	0.760815i	$0.275201\pi$
$954$	0	0
$955$	1.63145e9	1.87311
$956$	0	0
$957$	3.92516e8i	0.447838i
$958$	0	0
$959$	6.19822e8i	0.702767i
$960$	0	0
$961$	8.40173e8	0.946670
$962$	0	0
$963$	−3.10914e8	−0.348146
$964$	0	0
$965$	7.91796e8i	0.881112i
$966$	0	0
$967$	9.82431e8i	1.08648i	0.839577	+	0.543241i	$0.182803\pi$
$967$	9.82431e8i	1.08648i	−0.839577	+	0.543241i	$0.817197\pi$
$968$	0	0
$969$	−4.42246e8	−0.486063
$970$	0	0
$971$	−1.00598e9	−1.09883	−0.549414	−	0.835550i	$-0.685149\pi$
$971$	−1.00598e9	−1.09883	−0.549414	+	0.835550i	$0.685149\pi$
$972$	0	0
$973$	− 4.19515e6i	− 0.00455417i
$974$	0	0
$975$	1.02320e9i	1.10394i
$976$	0	0
$977$	1.50555e7	0.0161440	0.00807199	−	0.999967i	$-0.497431\pi$
$977$	1.50555e7	0.0161440	0.00807199	+	0.999967i	$0.497431\pi$
$978$	0	0
$979$	2.37217e9	2.52812
$980$	0	0
$981$	− 9.33936e7i	− 0.0989260i
$982$	0	0
$983$	− 1.01319e9i	− 1.06667i	−0.845905	−	0.533333i	$-0.820939\pi$
$983$	− 1.01319e9i	− 1.06667i	0.845905	−	0.533333i	$-0.179061\pi$
$984$	0	0
$985$	−2.50095e9	−2.61695
$986$	0	0
$987$	6.37151e8	0.662660
$988$	0	0
$989$	− 2.19260e8i	− 0.226658i
$990$	0	0
$991$	− 1.84753e9i	− 1.89832i	−0.314791	−	0.949161i	$-0.601934\pi$
$991$	− 1.84753e9i	− 1.89832i	0.314791	−	0.949161i	$-0.398066\pi$
$992$	0	0
$993$	5.67256e8	0.579337
$994$	0	0
$995$	−1.88938e8	−0.191801
$996$	0	0
$997$	5.12629e8i	0.517270i	0.965975	+	0.258635i	$0.0832727\pi$
$997$	5.12629e8i	0.517270i	−0.965975	+	0.258635i	$0.916727\pi$
$998$	0	0
$999$	4.60923e7i	0.0462309i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.7.b.a.319.2	yes	4
4.3	odd	2	inner	384.7.b.a.319.4	yes	4
8.3	odd	2	inner	384.7.b.a.319.1	✓	4
8.5	even	2	inner	384.7.b.a.319.3	yes	4
16.3	odd	4		768.7.g.b.511.2		2
16.5	even	4		768.7.g.a.511.2		2
16.11	odd	4		768.7.g.a.511.1		2
16.13	even	4		768.7.g.b.511.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.7.b.a.319.1	✓	4	8.3	odd	2	inner
384.7.b.a.319.2	yes	4	1.1	even	1	trivial
384.7.b.a.319.3	yes	4	8.5	even	2	inner
384.7.b.a.319.4	yes	4	4.3	odd	2	inner
768.7.g.a.511.1		2	16.11	odd	4
768.7.g.a.511.2		2	16.5	even	4
768.7.g.b.511.1		2	16.13	even	4
768.7.g.b.511.2		2	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)

\(f(q)\)	\(=\)	\(q-15.5885 q^{3} +196.000i q^{5} +270.200i q^{7} +243.000 q^{9} +O(q^{10})\)	\(q-15.5885 q^{3} +196.000i q^{5} +270.200i q^{7} +243.000 q^{9} -1725.12 q^{11} +2880.00i q^{13} -3055.34i q^{15} -2898.00 q^{17} -9789.55 q^{19} -4212.00i q^{21} +1621.20i q^{23} -22791.0 q^{25} -3788.00 q^{27} +14596.0i q^{29} +6879.71i q^{31} +26892.0 q^{33} -52959.2 q^{35} -12168.0i q^{37} -44894.8i q^{39} +18738.0 q^{41} -135245. q^{43} +47628.0i q^{45} +151270. i q^{47} +44641.0 q^{49} +45175.3 q^{51} +144236. i q^{53} -338124. i q^{55} +152604. q^{57} +333655. q^{59} +333432. i q^{61} +65658.6i q^{63} -564480. q^{65} +319439. q^{67} -25272.0i q^{69} -420639. i q^{71} +628238. q^{73} +355277. q^{75} -466128. i q^{77} +291878. i q^{79} +59049.0 q^{81} +357017. q^{83} -568008. i q^{85} -227529. i q^{87} -1.37507e6 q^{89} -778176. q^{91} -107244. i q^{93} -1.91875e6i q^{95} +489710. q^{97} -419205. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 972 q^{9}+O(q^{10}) \) 4 * q + 972 * q^9	\( 4 q + 972 q^{9} - 11592 q^{17} - 91164 q^{25} + 107568 q^{33} + 74952 q^{41} + 178564 q^{49} + 610416 q^{57} - 2257920 q^{65} + 2512952 q^{73} + 236196 q^{81} - 5500296 q^{89} + 1958840 q^{97}+O(q^{100}) \) 4 * q + 972 * q^9 - 11592 * q^17 - 91164 * q^25 + 107568 * q^33 + 74952 * q^41 + 178564 * q^49 + 610416 * q^57 - 2257920 * q^65 + 2512952 * q^73 + 236196 * q^81 - 5500296 * q^89 + 1958840 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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