LMFDB - Embedded newform 384.7.b.b.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:	$4$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{5} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			319.3
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	384.319
Dual form			384.7.b.b.319.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+15.5885 q^{3} -103.923i q^{5} -108.000i q^{7} +243.000 q^{9} +O(q^{10})$	$q+15.5885 q^{3} -103.923i q^{5} -108.000i q^{7} +243.000 q^{9} +394.908 q^{11} +2203.17i q^{13} -1620.00i q^{15} +974.000 q^{17} +976.877 q^{19} -1683.55i q^{21} +20952.0i q^{23} +4825.00 q^{25} +3788.00 q^{27} -9498.57i q^{29} +15660.0i q^{31} +6156.00 q^{33} -11223.7 q^{35} +51462.7i q^{37} +34344.0i q^{39} +33298.0 q^{41} +16357.5 q^{43} -25253.3i q^{45} -73224.0i q^{47} +105985. q^{49} +15183.2 q^{51} -164676. i q^{53} -41040.0i q^{55} +15228.0 q^{57} +75053.2 q^{59} +4822.03i q^{61} -26244.0i q^{63} +228960. q^{65} +261533. q^{67} +326609. i q^{69} +165672. i q^{71} -113618. q^{73} +75214.3 q^{75} -42650.0i q^{77} -658260. i q^{79} +59049.0 q^{81} +576461. q^{83} -101221. i q^{85} -148068. i q^{87} -464290. q^{89} +237942. q^{91} +244115. i q^{93} -101520. i q^{95} +51694.0 q^{97} +95962.5 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 972 q^{9}+O(q^{10}) $ 4 * q + 972 * q^9	$ 4 q + 972 q^{9} + 3896 q^{17} + 19300 q^{25} + 24624 q^{33} + 133192 q^{41} + 423940 q^{49} + 60912 q^{57} + 915840 q^{65} - 454472 q^{73} + 236196 q^{81} - 1857160 q^{89} + 206776 q^{97}+O(q^{100}) $ 4 * q + 972 * q^9 + 3896 * q^17 + 19300 * q^25 + 24624 * q^33 + 133192 * q^41 + 423940 * q^49 + 60912 * q^57 + 915840 * q^65 - 454472 * q^73 + 236196 * q^81 - 1857160 * q^89 + 206776 * 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
$5$	− 103.923i	− 0.831384i	−0.909505	−	0.415692i	$-0.863539\pi$
$5$	− 103.923i	− 0.831384i	0.909505	−	0.415692i	$-0.136461\pi$
$6$	0	0
$7$	− 108.000i	− 0.314869i	−0.987529	−	0.157434i	$-0.949678\pi$
$7$	− 108.000i	− 0.314869i	0.987529	−	0.157434i	$-0.0503223\pi$
$8$	0	0
$9$	243.000	0.333333
$10$	0	0
$11$	394.908	0.296700	0.148350	−	0.988935i	$-0.452604\pi$
$11$	394.908	0.296700	0.148350	+	0.988935i	$0.452604\pi$
$12$	0	0
$13$	2203.17i	1.00281i	0.865213	+	0.501404i	$0.167183\pi$
$13$	2203.17i	1.00281i	−0.865213	+	0.501404i	$0.832817\pi$
$14$	0	0
$15$	− 1620.00i	− 0.480000i
$16$	0	0
$17$	974.000	0.198250	0.0991248	−	0.995075i	$-0.468396\pi$
$17$	974.000	0.198250	0.0991248	+	0.995075i	$0.468396\pi$
$18$	0	0
$19$	976.877	0.142423	0.0712113	−	0.997461i	$-0.477314\pi$
$19$	976.877	0.142423	0.0712113	+	0.997461i	$0.477314\pi$
$20$	0	0
$21$	− 1683.55i	− 0.181790i
$22$	0	0
$23$	20952.0i	1.72204i	0.508575	+	0.861018i	$0.330172\pi$
$23$	20952.0i	1.72204i	−0.508575	+	0.861018i	$0.669828\pi$
$24$	0	0
$25$	4825.00	0.308800
$26$	0	0
$27$	3788.00	0.192450
$28$	0	0
$29$	− 9498.57i	− 0.389461i	−0.980857	−	0.194731i	$-0.937617\pi$
$29$	− 9498.57i	− 0.389461i	0.980857	−	0.194731i	$-0.0623832\pi$
$30$	0	0
$31$	15660.0i	0.525662i	0.964842	+	0.262831i	$0.0846562\pi$
$31$	15660.0i	0.525662i	−0.964842	+	0.262831i	$0.915344\pi$
$32$	0	0
$33$	6156.00	0.171300
$34$	0	0
$35$	−11223.7	−0.261777
$36$	0	0
$37$	51462.7i	1.01599i	0.861361	+	0.507993i	$0.169612\pi$
$37$	51462.7i	1.01599i	−0.861361	+	0.507993i	$0.830388\pi$
$38$	0	0
$39$	34344.0i	0.578971i
$40$	0	0
$41$	33298.0	0.483133	0.241566	−	0.970384i	$-0.422339\pi$
$41$	33298.0	0.483133	0.241566	+	0.970384i	$0.422339\pi$
$42$	0	0
$43$	16357.5	0.205736	0.102868	−	0.994695i	$-0.467198\pi$
$43$	16357.5	0.205736	0.102868	+	0.994695i	$0.467198\pi$
$44$	0	0
$45$	− 25253.3i	− 0.277128i
$46$	0	0
$47$	− 73224.0i	− 0.705277i	−0.935760	−	0.352639i	$-0.885284\pi$
$47$	− 73224.0i	− 0.705277i	0.935760	−	0.352639i	$-0.114716\pi$
$48$	0	0
$49$	105985.	0.900858
$50$	0	0
$51$	15183.2	0.114459
$52$	0	0
$53$	− 164676.i	− 1.10612i	−0.833140	−	0.553062i	$-0.813459\pi$
$53$	− 164676.i	− 1.10612i	0.833140	−	0.553062i	$-0.186541\pi$
$54$	0	0
$55$	− 41040.0i	− 0.246672i
$56$	0	0
$57$	15228.0	0.0822277
$58$	0	0
$59$	75053.2	0.365438	0.182719	−	0.983165i	$-0.441510\pi$
$59$	75053.2	0.365438	0.182719	+	0.983165i	$0.441510\pi$
$60$	0	0
$61$	4822.03i	0.0212442i	0.999944	+	0.0106221i	$0.00338118\pi$
$61$	4822.03i	0.0212442i	−0.999944	+	0.0106221i	$0.996619\pi$
$62$	0	0
$63$	− 26244.0i	− 0.104956i
$64$	0	0
$65$	228960.	0.833719
$66$	0	0
$67$	261533.	0.869564	0.434782	−	0.900536i	$-0.356825\pi$
$67$	261533.	0.869564	0.434782	+	0.900536i	$0.356825\pi$
$68$	0	0
$69$	326609.i	0.994217i
$70$	0	0
$71$	165672.i	0.462886i	0.972848	+	0.231443i	$0.0743447\pi$
$71$	165672.i	0.462886i	−0.972848	+	0.231443i	$0.925655\pi$
$72$	0	0
$73$	−113618.	−0.292064	−0.146032	−	0.989280i	$-0.546650\pi$
$73$	−113618.	−0.292064	−0.146032	+	0.989280i	$0.546650\pi$
$74$	0	0
$75$	75214.3	0.178286
$76$	0	0
$77$	− 42650.0i	− 0.0934215i
$78$	0	0
$79$	− 658260.i	− 1.33511i	−0.744562	−	0.667554i	$-0.767341\pi$
$79$	− 658260.i	− 1.33511i	0.744562	−	0.667554i	$-0.232659\pi$
$80$	0	0
$81$	59049.0	0.111111
$82$	0	0
$83$	576461.	1.00817	0.504087	−	0.863653i	$-0.331829\pi$
$83$	576461.	1.00817	0.504087	+	0.863653i	$0.331829\pi$
$84$	0	0
$85$	− 101221.i	− 0.164822i
$86$	0	0
$87$	− 148068.i	− 0.224855i
$88$	0	0
$89$	−464290.	−0.658596	−0.329298	−	0.944226i	$-0.606812\pi$
$89$	−464290.	−0.658596	−0.329298	+	0.944226i	$0.606812\pi$
$90$	0	0
$91$	237942.	0.315753
$92$	0	0
$93$	244115.i	0.303491i
$94$	0	0
$95$	− 101520.i	− 0.118408i
$96$	0	0
$97$	51694.0	0.0566402	0.0283201	−	0.999599i	$-0.490984\pi$
$97$	51694.0	0.0566402	0.0283201	+	0.999599i	$0.490984\pi$
$98$	0	0
$99$	95962.5	0.0989000
$100$	0	0
$101$	− 802431.i	− 0.778832i	−0.921062	−	0.389416i	$-0.872677\pi$
$101$	− 802431.i	− 0.778832i	0.921062	−	0.389416i	$-0.127323\pi$
$102$	0	0
$103$	725004.i	0.663481i	0.943371	+	0.331741i	$0.107636\pi$
$103$	725004.i	0.663481i	−0.943371	+	0.331741i	$0.892364\pi$
$104$	0	0
$105$	−174960.	−0.151137
$106$	0	0
$107$	1.26950e6	1.03629	0.518146	−	0.855292i	$-0.326622\pi$
$107$	1.26950e6	1.03629	0.518146	+	0.855292i	$0.326622\pi$
$108$	0	0
$109$	1.00893e6i	0.779076i	0.921010	+	0.389538i	$0.127365\pi$
$109$	1.00893e6i	0.779076i	−0.921010	+	0.389538i	$0.872635\pi$
$110$	0	0
$111$	802224.i	0.586579i
$112$	0	0
$113$	−80318.0	−0.0556644	−0.0278322	−	0.999613i	$-0.508860\pi$
$113$	−80318.0	−0.0556644	−0.0278322	+	0.999613i	$0.508860\pi$
$114$	0	0
$115$	2.17740e6	1.43167
$116$	0	0
$117$	535370.i	0.334269i
$118$	0	0
$119$	− 105192.i	− 0.0624226i
$120$	0	0
$121$	−1.61561e6	−0.911969
$122$	0	0
$123$	519064.	0.278937
$124$	0	0
$125$	− 2.12523e6i	− 1.08812i
$126$	0	0
$127$	3.07228e6i	1.49985i	0.661520	+	0.749927i	$0.269911\pi$
$127$	3.07228e6i	1.49985i	−0.661520	+	0.749927i	$0.730089\pi$
$128$	0	0
$129$	254988.	0.118782
$130$	0	0
$131$	3.36792e6	1.49812	0.749062	−	0.662500i	$-0.230505\pi$
$131$	3.36792e6	1.49812	0.749062	+	0.662500i	$0.230505\pi$
$132$	0	0
$133$	− 105503.i	− 0.0448444i
$134$	0	0
$135$	− 393660.i	− 0.160000i
$136$	0	0
$137$	−1.10315e6	−0.429015	−0.214508	−	0.976722i	$-0.568815\pi$
$137$	−1.10315e6	−0.429015	−0.214508	+	0.976722i	$0.568815\pi$
$138$	0	0
$139$	3.05806e6	1.13868	0.569340	−	0.822102i	$-0.307199\pi$
$139$	3.05806e6	1.13868	0.569340	+	0.822102i	$0.307199\pi$
$140$	0	0
$141$	− 1.14145e6i	− 0.407192i
$142$	0	0
$143$	870048.i	0.297533i
$144$	0	0
$145$	−987120.	−0.323792
$146$	0	0
$147$	1.65214e6	0.520110
$148$	0	0
$149$	1.91449e6i	0.578755i	0.957215	+	0.289377i	$0.0934482\pi$
$149$	1.91449e6i	0.578755i	−0.957215	+	0.289377i	$0.906552\pi$
$150$	0	0
$151$	− 5.15257e6i	− 1.49656i	−0.663385	−	0.748278i	$-0.730881\pi$
$151$	− 5.15257e6i	− 1.49656i	0.663385	−	0.748278i	$-0.269119\pi$
$152$	0	0
$153$	236682.	0.0660832
$154$	0	0
$155$	1.62743e6	0.437027
$156$	0	0
$157$	3.99106e6i	1.03131i	0.856796	+	0.515655i	$0.172451\pi$
$157$	3.99106e6i	1.03131i	−0.856796	+	0.515655i	$0.827549\pi$
$158$	0	0
$159$	− 2.56705e6i	− 0.638621i
$160$	0	0
$161$	2.26282e6	0.542215
$162$	0	0
$163$	6.58912e6	1.52147	0.760737	−	0.649061i	$-0.224838\pi$
$163$	6.58912e6	1.52147	0.760737	+	0.649061i	$0.224838\pi$
$164$	0	0
$165$	− 639750.i	− 0.142416i
$166$	0	0
$167$	2160.00i	0 0.000463772i	1.00000		0.000231886i	$7.38116e-5\pi$
$167$	2160.00i	0 0.000463772i	−1.00000		0.000231886i	$0.999926\pi$
$168$	0	0
$169$	−27143.0	−0.00562338
$170$	0	0
$171$	237381.	0.0474742
$172$	0	0
$173$	− 841008.i	− 0.162428i	−0.996697	−	0.0812141i	$-0.974120\pi$
$173$	− 841008.i	− 0.162428i	0.996697	−	0.0812141i	$-0.0258798\pi$
$174$	0	0
$175$	− 521100.i	− 0.0972315i
$176$	0	0
$177$	1.16996e6	0.210986
$178$	0	0
$179$	8.57945e6	1.49589	0.747946	−	0.663759i	$-0.231040\pi$
$179$	8.57945e6	1.49589	0.747946	+	0.663759i	$0.231040\pi$
$180$	0	0
$181$	− 8.27173e6i	− 1.39496i	−0.716606	−	0.697479i	$-0.754305\pi$
$181$	− 8.27173e6i	− 1.39496i	0.716606	−	0.697479i	$-0.245695\pi$
$182$	0	0
$183$	75168.0i	0.0122653i
$184$	0	0
$185$	5.34816e6	0.844674
$186$	0	0
$187$	384640.	0.0588206
$188$	0	0
$189$	− 409103.i	− 0.0605965i
$190$	0	0
$191$	− 6.36811e6i	− 0.913925i	−0.889486	−	0.456963i	$-0.848937\pi$
$191$	− 6.36811e6i	− 0.913925i	0.889486	−	0.456963i	$-0.151063\pi$
$192$	0	0
$193$	−626690.	−0.0871728	−0.0435864	−	0.999050i	$-0.513878\pi$
$193$	−626690.	−0.0871728	−0.0435864	+	0.999050i	$0.513878\pi$
$194$	0	0
$195$	3.56913e6	0.481348
$196$	0	0
$197$	1.38376e7i	1.80993i	0.425487	+	0.904965i	$0.360103\pi$
$197$	1.38376e7i	1.80993i	−0.425487	+	0.904965i	$0.639897\pi$
$198$	0	0
$199$	8.83624e6i	1.12126i	0.828065	+	0.560632i	$0.189442\pi$
$199$	8.83624e6i	1.12126i	−0.828065	+	0.560632i	$0.810558\pi$
$200$	0	0
$201$	4.07689e6	0.502043
$202$	0	0
$203$	−1.02585e6	−0.122629
$204$	0	0
$205$	− 3.46043e6i	− 0.401669i
$206$	0	0
$207$	5.09134e6i	0.574012i
$208$	0	0
$209$	385776.	0.0422568
$210$	0	0
$211$	1.21032e7	1.28841	0.644203	−	0.764855i	$-0.277189\pi$
$211$	1.21032e7	1.28841	0.644203	+	0.764855i	$0.277189\pi$
$212$	0	0
$213$	2.58257e6i	0.267247i
$214$	0	0
$215$	− 1.69992e6i	− 0.171046i
$216$	0	0
$217$	1.69128e6	0.165515
$218$	0	0
$219$	−1.77113e6	−0.168623
$220$	0	0
$221$	2.14589e6i	0.198806i
$222$	0	0
$223$	− 8.49539e6i	− 0.766070i	−0.923734	−	0.383035i	$-0.874879\pi$
$223$	− 8.49539e6i	− 0.766070i	0.923734	−	0.383035i	$-0.125121\pi$
$224$	0	0
$225$	1.17248e6	0.102933
$226$	0	0
$227$	1.54084e7	1.31728	0.658641	−	0.752457i	$-0.271132\pi$
$227$	1.54084e7	1.31728	0.658641	+	0.752457i	$0.271132\pi$
$228$	0	0
$229$	− 8.88821e6i	− 0.740129i	−0.929006	−	0.370065i	$-0.879336\pi$
$229$	− 8.88821e6i	− 0.740129i	0.929006	−	0.370065i	$-0.120664\pi$
$230$	0	0
$231$	− 664848.i	− 0.0539370i
$232$	0	0
$233$	5.40400e6	0.427216	0.213608	−	0.976919i	$-0.431478\pi$
$233$	5.40400e6	0.427216	0.213608	+	0.976919i	$0.431478\pi$
$234$	0	0
$235$	−7.60966e6	−0.586356
$236$	0	0
$237$	− 1.02613e7i	− 0.770825i
$238$	0	0
$239$	1.20675e7i	0.883941i	0.897030	+	0.441970i	$0.145720\pi$
$239$	1.20675e7i	0.883941i	−0.897030	+	0.441970i	$0.854280\pi$
$240$	0	0
$241$	−1.93628e7	−1.38330	−0.691652	−	0.722231i	$-0.743117\pi$
$241$	−1.93628e7	−1.38330	−0.691652	+	0.722231i	$0.743117\pi$
$242$	0	0
$243$	920483.	0.0641500
$244$	0	0
$245$	− 1.10143e7i	− 0.748959i
$246$	0	0
$247$	2.15222e6i	0.142822i
$248$	0	0
$249$	8.98614e6	0.582070
$250$	0	0
$251$	−1.30248e7	−0.823665	−0.411832	−	0.911260i	$-0.635111\pi$
$251$	−1.30248e7	−0.823665	−0.411832	+	0.911260i	$0.635111\pi$
$252$	0	0
$253$	8.27410e6i	0.510928i
$254$	0	0
$255$	− 1.57788e6i	− 0.0951598i
$256$	0	0
$257$	4.99482e6	0.294253	0.147126	−	0.989118i	$-0.452998\pi$
$257$	4.99482e6	0.294253	0.147126	+	0.989118i	$0.452998\pi$
$258$	0	0
$259$	5.55797e6	0.319902
$260$	0	0
$261$	− 2.30815e6i	− 0.129820i
$262$	0	0
$263$	1.99895e7i	1.09884i	0.835546	+	0.549420i	$0.185151\pi$
$263$	1.99895e7i	1.09884i	−0.835546	+	0.549420i	$0.814849\pi$
$264$	0	0
$265$	−1.71137e7	−0.919614
$266$	0	0
$267$	−7.23756e6	−0.380241
$268$	0	0
$269$	2.35776e7i	1.21128i	0.795740	+	0.605638i	$0.207082\pi$
$269$	2.35776e7i	1.21128i	−0.795740	+	0.605638i	$0.792918\pi$
$270$	0	0
$271$	551556.i	0.0277129i	0.999904	+	0.0138564i	$0.00441078\pi$
$271$	551556.i	0.0277129i	−0.999904	+	0.0138564i	$0.995589\pi$
$272$	0	0
$273$	3.70915e6	0.182300
$274$	0	0
$275$	1.90543e6	0.0916209
$276$	0	0
$277$	1.68319e7i	0.791944i	0.918263	+	0.395972i	$0.129592\pi$
$277$	1.68319e7i	0.791944i	−0.918263	+	0.395972i	$0.870408\pi$
$278$	0	0
$279$	3.80538e6i	0.175221i
$280$	0	0
$281$	4.57600e6	0.206237	0.103119	−	0.994669i	$-0.467118\pi$
$281$	4.57600e6	0.206237	0.103119	+	0.994669i	$0.467118\pi$
$282$	0	0
$283$	−3.36134e7	−1.48304	−0.741520	−	0.670930i	$-0.765895\pi$
$283$	−3.36134e7	−1.48304	−0.741520	+	0.670930i	$0.765895\pi$
$284$	0	0
$285$	− 1.58254e6i	− 0.0683629i
$286$	0	0
$287$	− 3.59618e6i	− 0.152123i
$288$	0	0
$289$	−2.31889e7	−0.960697
$290$	0	0
$291$	805830.	0.0327012
$292$	0	0
$293$	− 1.25131e7i	− 0.497463i	−0.968572	−	0.248731i	$-0.919986\pi$
$293$	− 1.25131e7i	− 0.497463i	0.968572	−	0.248731i	$-0.0800137\pi$
$294$	0	0
$295$	− 7.79976e6i	− 0.303819i
$296$	0	0
$297$	1.49591e6	0.0570999
$298$	0	0
$299$	−4.61608e7	−1.72687
$300$	0	0
$301$	− 1.76661e6i	− 0.0647800i
$302$	0	0
$303$	− 1.25087e7i	− 0.449659i
$304$	0	0
$305$	501120.	0.0176621
$306$	0	0
$307$	5.33514e6	0.184387	0.0921936	−	0.995741i	$-0.470612\pi$
$307$	5.33514e6	0.184387	0.0921936	+	0.995741i	$0.470612\pi$
$308$	0	0
$309$	1.13017e7i	0.383061i
$310$	0	0
$311$	− 4.35866e7i	− 1.44901i	−0.689268	−	0.724506i	$-0.742068\pi$
$311$	− 4.35866e7i	− 1.44901i	0.689268	−	0.724506i	$-0.257932\pi$
$312$	0	0
$313$	−2.88674e7	−0.941401	−0.470700	−	0.882293i	$-0.655999\pi$
$313$	−2.88674e7	−0.941401	−0.470700	+	0.882293i	$0.655999\pi$
$314$	0	0
$315$	−2.72736e6	−0.0872590
$316$	0	0
$317$	− 4.89688e7i	− 1.53724i	−0.639705	−	0.768620i	$-0.720944\pi$
$317$	− 4.89688e7i	− 1.53724i	0.639705	−	0.768620i	$-0.279056\pi$
$318$	0	0
$319$	− 3.75106e6i	− 0.115553i
$320$	0	0
$321$	1.97896e7	0.598304
$322$	0	0
$323$	951478.	0.0282352
$324$	0	0
$325$	1.06303e7i	0.309667i
$326$	0	0
$327$	1.57276e7i	0.449800i
$328$	0	0
$329$	−7.90819e6	−0.222070
$330$	0	0
$331$	−6.16606e7	−1.70029	−0.850147	−	0.526545i	$-0.823487\pi$
$331$	−6.16606e7	−1.70029	−0.850147	+	0.526545i	$0.823487\pi$
$332$	0	0
$333$	1.25054e7i	0.338662i
$334$	0	0
$335$	− 2.71793e7i	− 0.722942i
$336$	0	0
$337$	−5.25106e7	−1.37201	−0.686005	−	0.727597i	$-0.740637\pi$
$337$	−5.25106e7	−1.37201	−0.686005	+	0.727597i	$0.740637\pi$
$338$	0	0
$339$	−1.25203e6	−0.0321379
$340$	0	0
$341$	6.18425e6i	0.155964i
$342$	0	0
$343$	− 2.41525e7i	− 0.598521i
$344$	0	0
$345$	3.39422e7	0.826577
$346$	0	0
$347$	−6.14174e7	−1.46995	−0.734975	−	0.678094i	$-0.762806\pi$
$347$	−6.14174e7	−1.46995	−0.734975	+	0.678094i	$0.762806\pi$
$348$	0	0
$349$	− 547217.i	− 0.0128731i	−0.999979	−	0.00643655i	$-0.997951\pi$
$349$	− 547217.i	− 0.0128731i	0.999979	−	0.00643655i	$-0.00204883\pi$
$350$	0	0
$351$	8.34559e6i	0.192990i
$352$	0	0
$353$	6.98187e7	1.58726	0.793629	−	0.608401i	$-0.208189\pi$
$353$	6.98187e7	1.58726	0.793629	+	0.608401i	$0.208189\pi$
$354$	0	0
$355$	1.72171e7	0.384836
$356$	0	0
$357$	− 1.63978e6i	− 0.0360397i
$358$	0	0
$359$	3.93703e6i	0.0850914i	0.999095	+	0.0425457i	$0.0135468\pi$
$359$	3.93703e6i	0.0850914i	−0.999095	+	0.0425457i	$0.986453\pi$
$360$	0	0
$361$	−4.60916e7	−0.979716
$362$	0	0
$363$	−2.51849e7	−0.526526
$364$	0	0
$365$	1.18075e7i	0.242818i
$366$	0	0
$367$	6.27706e7i	1.26987i	0.772567	+	0.634933i	$0.218972\pi$
$367$	6.27706e7i	1.26987i	−0.772567	+	0.634933i	$0.781028\pi$
$368$	0	0
$369$	8.09141e6	0.161044
$370$	0	0
$371$	−1.77851e7	−0.348284
$372$	0	0
$373$	− 1.94384e7i	− 0.374572i	−0.982305	−	0.187286i	$-0.940031\pi$
$373$	− 1.94384e7i	− 0.374572i	0.982305	−	0.187286i	$-0.0599690\pi$
$374$	0	0
$375$	− 3.31290e7i	− 0.628224i
$376$	0	0
$377$	2.09269e7	0.390555
$378$	0	0
$379$	−8.72162e7	−1.60206	−0.801031	−	0.598623i	$-0.795715\pi$
$379$	−8.72162e7	−1.60206	−0.801031	+	0.598623i	$0.795715\pi$
$380$	0	0
$381$	4.78920e7i	0.865941i
$382$	0	0
$383$	1.03321e8i	1.83904i	0.393038	+	0.919522i	$0.371424\pi$
$383$	1.03321e8i	1.83904i	−0.393038	+	0.919522i	$0.628576\pi$
$384$	0	0
$385$	−4.43232e6	−0.0776692
$386$	0	0
$387$	3.97487e6	0.0685788
$388$	0	0
$389$	− 4.08320e7i	− 0.693668i	−0.937927	−	0.346834i	$-0.887257\pi$
$389$	− 4.08320e7i	− 0.693668i	0.937927	−	0.346834i	$-0.112743\pi$
$390$	0	0
$391$	2.04072e7i	0.341393i
$392$	0	0
$393$	5.25006e7	0.864942
$394$	0	0
$395$	−6.84084e7	−1.10999
$396$	0	0
$397$	− 5.81045e7i	− 0.928620i	−0.885673	−	0.464310i	$-0.846302\pi$
$397$	− 5.81045e7i	− 0.928620i	0.885673	−	0.464310i	$-0.153698\pi$
$398$	0	0
$399$	− 1.64462e6i	− 0.0258909i
$400$	0	0
$401$	6.80041e7	1.05463	0.527317	−	0.849668i	$-0.323198\pi$
$401$	6.80041e7	1.05463	0.527317	+	0.849668i	$0.323198\pi$
$402$	0	0
$403$	−3.45016e7	−0.527138
$404$	0	0
$405$	− 6.13655e6i	− 0.0923760i
$406$	0	0
$407$	2.03230e7i	0.301443i
$408$	0	0
$409$	−4.92576e7	−0.719952	−0.359976	−	0.932962i	$-0.617215\pi$
$409$	−4.92576e7	−0.719952	−0.359976	+	0.932962i	$0.617215\pi$
$410$	0	0
$411$	−1.71964e7	−0.247692
$412$	0	0
$413$	− 8.10575e6i	− 0.115065i
$414$	0	0
$415$	− 5.99076e7i	− 0.838181i
$416$	0	0
$417$	4.76704e7	0.657417
$418$	0	0
$419$	1.91861e7	0.260822	0.130411	−	0.991460i	$-0.458370\pi$
$419$	1.91861e7	0.260822	0.130411	+	0.991460i	$0.458370\pi$
$420$	0	0
$421$	1.03597e8i	1.38836i	0.719802	+	0.694179i	$0.244233\pi$
$421$	1.03597e8i	1.38836i	−0.719802	+	0.694179i	$0.755767\pi$
$422$	0	0
$423$	− 1.77934e7i	− 0.235092i
$424$	0	0
$425$	4.69955e6	0.0612195
$426$	0	0
$427$	520779.	0.00668914
$428$	0	0
$429$	1.35627e7i	0.171781i
$430$	0	0
$431$	− 6.34195e7i	− 0.792121i	−0.918225	−	0.396060i	$-0.870377\pi$
$431$	− 6.34195e7i	− 0.792121i	0.918225	−	0.396060i	$-0.129623\pi$
$432$	0	0
$433$	−1.05070e8	−1.29424	−0.647119	−	0.762389i	$-0.724026\pi$
$433$	−1.05070e8	−1.29424	−0.647119	+	0.762389i	$0.724026\pi$
$434$	0	0
$435$	−1.53877e7	−0.186941
$436$	0	0
$437$	2.04675e7i	0.245257i
$438$	0	0
$439$	9.91532e7i	1.17196i	0.810325	+	0.585980i	$0.199290\pi$
$439$	9.91532e7i	1.17196i	−0.810325	+	0.585980i	$0.800710\pi$
$440$	0	0
$441$	2.57544e7	0.300286
$442$	0	0
$443$	−1.17188e7	−0.134794	−0.0673972	−	0.997726i	$-0.521469\pi$
$443$	−1.17188e7	−0.134794	−0.0673972	+	0.997726i	$0.521469\pi$
$444$	0	0
$445$	4.82504e7i	0.547547i
$446$	0	0
$447$	2.98440e7i	0.334144i
$448$	0	0
$449$	1.29610e8	1.43186	0.715931	−	0.698171i	$-0.246003\pi$
$449$	1.29610e8	1.43186	0.715931	+	0.698171i	$0.246003\pi$
$450$	0	0
$451$	1.31496e7	0.143345
$452$	0	0
$453$	− 8.03206e7i	− 0.864038i
$454$	0	0
$455$	− 2.47277e7i	− 0.262512i
$456$	0	0
$457$	−1.16801e8	−1.22377	−0.611885	−	0.790947i	$-0.709588\pi$
$457$	−1.16801e8	−1.22377	−0.611885	+	0.790947i	$0.709588\pi$
$458$	0	0
$459$	3.68951e6	0.0381531
$460$	0	0
$461$	1.21659e6i	0.0124177i	0.999981	+	0.00620883i	$0.00197635\pi$
$461$	1.21659e6i	0.0124177i	−0.999981	+	0.00620883i	$0.998024\pi$
$462$	0	0
$463$	− 4.68881e7i	− 0.472411i	−0.971703	−	0.236205i	$-0.924096\pi$
$463$	− 4.68881e7i	− 0.472411i	0.971703	−	0.236205i	$-0.0759038\pi$
$464$	0	0
$465$	2.53692e7	0.252318
$466$	0	0
$467$	2.83632e7	0.278487	0.139244	−	0.990258i	$-0.455533\pi$
$467$	2.83632e7	0.278487	0.139244	+	0.990258i	$0.455533\pi$
$468$	0	0
$469$	− 2.82455e7i	− 0.273799i
$470$	0	0
$471$	6.22145e7i	0.595427i
$472$	0	0
$473$	6.45970e6	0.0610420
$474$	0	0
$475$	4.71343e6	0.0439801
$476$	0	0
$477$	− 4.00164e7i	− 0.368708i
$478$	0	0
$479$	− 1.22142e8i	− 1.11137i	−0.831394	−	0.555684i	$-0.812457\pi$
$479$	− 1.22142e8i	− 1.11137i	0.831394	−	0.555684i	$-0.187543\pi$
$480$	0	0
$481$	−1.13381e8	−1.01884
$482$	0	0
$483$	3.52738e7	0.313048
$484$	0	0
$485$	− 5.37220e6i	− 0.0470898i
$486$	0	0
$487$	− 8.30931e7i	− 0.719413i	−0.933065	−	0.359707i	$-0.882877\pi$
$487$	− 8.30931e7i	− 0.719413i	0.933065	−	0.359707i	$-0.117123\pi$
$488$	0	0
$489$	1.02714e8	0.878423
$490$	0	0
$491$	7.46180e7	0.630375	0.315188	−	0.949029i	$-0.397933\pi$
$491$	7.46180e7	0.630375	0.315188	+	0.949029i	$0.397933\pi$
$492$	0	0
$493$	− 9.25160e6i	− 0.0772105i
$494$	0	0
$495$	− 9.97272e6i	− 0.0822239i
$496$	0	0
$497$	1.78926e7	0.145748
$498$	0	0
$499$	2.18859e7	0.176142	0.0880711	−	0.996114i	$-0.471930\pi$
$499$	2.18859e7	0.176142	0.0880711	+	0.996114i	$0.471930\pi$
$500$	0	0
$501$	33671.1i	0 0.000267759i
$502$	0	0
$503$	1.08518e8i	0.852705i	0.904557	+	0.426352i	$0.140202\pi$
$503$	1.08518e8i	0.852705i	−0.904557	+	0.426352i	$0.859798\pi$
$504$	0	0
$505$	−8.33911e7	−0.647509
$506$	0	0
$507$	−423117.	−0.00324666
$508$	0	0
$509$	1.17085e8i	0.887870i	0.896059	+	0.443935i	$0.146418\pi$
$509$	1.17085e8i	0.887870i	−0.896059	+	0.443935i	$0.853582\pi$
$510$	0	0
$511$	1.22707e7i	0.0919620i
$512$	0	0
$513$	3.70040e6	0.0274092
$514$	0	0
$515$	7.53446e7	0.551608
$516$	0	0
$517$	− 2.89167e7i	− 0.209256i
$518$	0	0
$519$	− 1.31100e7i	− 0.0937780i
$520$	0	0
$521$	−3.26446e7	−0.230833	−0.115417	−	0.993317i	$-0.536820\pi$
$521$	−3.26446e7	−0.230833	−0.115417	+	0.993317i	$0.536820\pi$
$522$	0	0
$523$	1.44252e8	1.00836	0.504180	−	0.863599i	$-0.331795\pi$
$523$	1.44252e8	1.00836	0.504180	+	0.863599i	$0.331795\pi$
$524$	0	0
$525$	− 8.12315e6i	− 0.0561366i
$526$	0	0
$527$	1.52528e7i	0.104212i
$528$	0	0
$529$	−2.90950e8	−1.96540
$530$	0	0
$531$	1.82379e7	0.121813
$532$	0	0
$533$	7.33611e7i	0.484489i
$534$	0	0
$535$	− 1.31931e8i	− 0.861558i
$536$	0	0
$537$	1.33740e8	0.863654
$538$	0	0
$539$	4.18543e7	0.267284
$540$	0	0
$541$	− 2.31669e8i	− 1.46311i	−0.681782	−	0.731555i	$-0.738795\pi$
$541$	− 2.31669e8i	− 1.46311i	0.681782	−	0.731555i	$-0.261205\pi$
$542$	0	0
$543$	− 1.28944e8i	− 0.805379i
$544$	0	0
$545$	1.04851e8	0.647712
$546$	0	0
$547$	1.23621e8	0.755320	0.377660	−	0.925944i	$-0.376729\pi$
$547$	1.23621e8	0.755320	0.377660	+	0.925944i	$0.376729\pi$
$548$	0	0
$549$	1.17175e6i	0.00708140i
$550$	0	0
$551$	− 9.27893e6i	− 0.0554681i
$552$	0	0
$553$	−7.10921e7	−0.420384
$554$	0	0
$555$	8.33696e7	0.487673
$556$	0	0
$557$	9.04852e7i	0.523615i	0.965120	+	0.261807i	$0.0843185\pi$
$557$	9.04852e7i	0.523615i	−0.965120	+	0.261807i	$0.915682\pi$
$558$	0	0
$559$	3.60383e7i	0.206314i
$560$	0	0
$561$	5.99594e6	0.0339601
$562$	0	0
$563$	−3.09901e7	−0.173659	−0.0868297	−	0.996223i	$-0.527674\pi$
$563$	−3.09901e7	−0.173659	−0.0868297	+	0.996223i	$0.527674\pi$
$564$	0	0
$565$	8.34689e6i	0.0462785i
$566$	0	0
$567$	− 6.37729e6i	− 0.0349854i
$568$	0	0
$569$	−2.12560e7	−0.115384	−0.0576918	−	0.998334i	$-0.518374\pi$
$569$	−2.12560e7	−0.115384	−0.0576918	+	0.998334i	$0.518374\pi$
$570$	0	0
$571$	1.86440e8	1.00145	0.500726	−	0.865606i	$-0.333067\pi$
$571$	1.86440e8	1.00145	0.500726	+	0.865606i	$0.333067\pi$
$572$	0	0
$573$	− 9.92690e7i	− 0.527655i
$574$	0	0
$575$	1.01093e8i	0.531764i
$576$	0	0
$577$	−2.49144e8	−1.29695	−0.648476	−	0.761236i	$-0.724593\pi$
$577$	−2.49144e8	−1.29695	−0.648476	+	0.761236i	$0.724593\pi$
$578$	0	0
$579$	−9.76913e6	−0.0503292
$580$	0	0
$581$	− 6.22578e7i	− 0.317443i
$582$	0	0
$583$	− 6.50320e7i	− 0.328187i
$584$	0	0
$585$	5.56373e7	0.277906
$586$	0	0
$587$	3.03562e8	1.50083	0.750417	−	0.660964i	$-0.229853\pi$
$587$	3.03562e8	1.50083	0.750417	+	0.660964i	$0.229853\pi$
$588$	0	0
$589$	1.52979e7i	0.0748662i
$590$	0	0
$591$	2.15707e8i	1.04496i
$592$	0	0
$593$	1.17518e8	0.563561	0.281780	−	0.959479i	$-0.409075\pi$
$593$	1.17518e8	0.563561	0.281780	+	0.959479i	$0.409075\pi$
$594$	0	0
$595$	−1.09319e7	−0.0518972
$596$	0	0
$597$	1.37743e8i	0.647362i
$598$	0	0
$599$	1.06061e8i	0.493486i	0.969081	+	0.246743i	$0.0793604\pi$
$599$	1.06061e8i	0.493486i	−0.969081	+	0.246743i	$0.920640\pi$
$600$	0	0
$601$	7.74177e7	0.356629	0.178315	−	0.983974i	$-0.442936\pi$
$601$	7.74177e7	0.356629	0.178315	+	0.983974i	$0.442936\pi$
$602$	0	0
$603$	6.35525e7	0.289855
$604$	0	0
$605$	1.67899e8i	0.758197i
$606$	0	0
$607$	2.47898e8i	1.10843i	0.832375	+	0.554213i	$0.186981\pi$
$607$	2.47898e8i	1.10843i	−0.832375	+	0.554213i	$0.813019\pi$
$608$	0	0
$609$	−1.59913e7	−0.0708000
$610$	0	0
$611$	1.61325e8	0.707257
$612$	0	0
$613$	4.31913e7i	0.187506i	0.995596	+	0.0937528i	$0.0298863\pi$
$613$	4.31913e7i	0.187506i	−0.995596	+	0.0937528i	$0.970114\pi$
$614$	0	0
$615$	− 5.39428e7i	− 0.231904i
$616$	0	0
$617$	−2.42180e7	−0.103106	−0.0515528	−	0.998670i	$-0.516417\pi$
$617$	−2.42180e7	−0.103106	−0.0515528	+	0.998670i	$0.516417\pi$
$618$	0	0
$619$	3.01450e8	1.27100	0.635498	−	0.772103i	$-0.280795\pi$
$619$	3.01450e8	1.27100	0.635498	+	0.772103i	$0.280795\pi$
$620$	0	0
$621$	7.93661e7i	0.331406i
$622$	0	0
$623$	5.01433e7i	0.207371i
$624$	0	0
$625$	−1.45469e8	−0.595843
$626$	0	0
$627$	6.01365e6	0.0243970
$628$	0	0
$629$	5.01247e7i	0.201419i
$630$	0	0
$631$	3.91172e8i	1.55697i	0.627664	+	0.778484i	$0.284011\pi$
$631$	3.91172e8i	1.55697i	−0.627664	+	0.778484i	$0.715989\pi$
$632$	0	0
$633$	1.88670e8	0.743861
$634$	0	0
$635$	3.19280e8	1.24696
$636$	0	0
$637$	2.33503e8i	0.903387i
$638$	0	0
$639$	4.02583e7i	0.154295i
$640$	0	0
$641$	3.56478e8	1.35350	0.676750	−	0.736213i	$-0.263388\pi$
$641$	3.56478e8	1.35350	0.676750	+	0.736213i	$0.263388\pi$
$642$	0	0
$643$	7.13886e7	0.268532	0.134266	−	0.990945i	$-0.457132\pi$
$643$	7.13886e7	0.268532	0.134266	+	0.990945i	$0.457132\pi$
$644$	0	0
$645$	− 2.64991e7i	− 0.0987535i
$646$	0	0
$647$	− 3.11785e8i	− 1.15118i	−0.817740	−	0.575588i	$-0.804773\pi$
$647$	− 3.11785e8i	− 1.15118i	0.817740	−	0.575588i	$-0.195227\pi$
$648$	0	0
$649$	2.96391e7	0.108425
$650$	0	0
$651$	2.63644e7	0.0955599
$652$	0	0
$653$	− 3.67358e8i	− 1.31932i	−0.751565	−	0.659659i	$-0.770701\pi$
$653$	− 3.67358e8i	− 1.31932i	0.751565	−	0.659659i	$-0.229299\pi$
$654$	0	0
$655$	− 3.50004e8i	− 1.24552i
$656$	0	0
$657$	−2.76092e7	−0.0973548
$658$	0	0
$659$	−2.60909e8	−0.911659	−0.455830	−	0.890067i	$-0.650657\pi$
$659$	−2.60909e8	−0.911659	−0.455830	+	0.890067i	$0.650657\pi$
$660$	0	0
$661$	− 4.51263e8i	− 1.56252i	−0.624207	−	0.781259i	$-0.714578\pi$
$661$	− 4.51263e8i	− 1.56252i	0.624207	−	0.781259i	$-0.285422\pi$
$662$	0	0
$663$	3.34511e7i	0.114781i
$664$	0	0
$665$	−1.09642e7	−0.0372830
$666$	0	0
$667$	1.99014e8	0.670666
$668$	0	0
$669$	− 1.32430e8i	− 0.442291i
$670$	0	0
$671$	1.90426e6i	0.00630315i
$672$	0	0
$673$	−1.15721e8	−0.379635	−0.189818	−	0.981819i	$-0.560790\pi$
$673$	−1.15721e8	−0.379635	−0.189818	+	0.981819i	$0.560790\pi$
$674$	0	0
$675$	1.82771e7	0.0594286
$676$	0	0
$677$	4.95719e7i	0.159760i	0.996804	+	0.0798802i	$0.0254538\pi$
$677$	4.95719e7i	0.159760i	−0.996804	+	0.0798802i	$0.974546\pi$
$678$	0	0
$679$	− 5.58295e6i	− 0.0178342i
$680$	0	0
$681$	2.40193e8	0.760533
$682$	0	0
$683$	3.50124e8	1.09890	0.549452	−	0.835526i	$-0.314837\pi$
$683$	3.50124e8	1.09890	0.549452	+	0.835526i	$0.314837\pi$
$684$	0	0
$685$	1.14643e8i	0.356677i
$686$	0	0
$687$	− 1.38553e8i	− 0.427314i
$688$	0	0
$689$	3.62810e8	1.10923
$690$	0	0
$691$	−3.82306e8	−1.15872	−0.579358	−	0.815074i	$-0.696696\pi$
$691$	−3.82306e8	−1.15872	−0.579358	+	0.815074i	$0.696696\pi$
$692$	0	0
$693$	− 1.03640e7i	− 0.0311405i
$694$	0	0
$695$	− 3.17803e8i	− 0.946681i
$696$	0	0
$697$	3.24323e7	0.0957809
$698$	0	0
$699$	8.42400e7	0.246653
$700$	0	0
$701$	− 4.72251e8i	− 1.37094i	−0.728101	−	0.685470i	$-0.759597\pi$
$701$	− 4.72251e8i	− 1.37094i	0.728101	−	0.685470i	$-0.240403\pi$
$702$	0	0
$703$	5.02727e7i	0.144699i
$704$	0	0
$705$	−1.18623e8	−0.338533
$706$	0	0
$707$	−8.66626e7	−0.245230
$708$	0	0
$709$	6.80249e8i	1.90866i	0.298751	+	0.954331i	$0.403430\pi$
$709$	6.80249e8i	1.90866i	−0.298751	+	0.954331i	$0.596570\pi$
$710$	0	0
$711$	− 1.59957e8i	− 0.445036i
$712$	0	0
$713$	−3.28108e8	−0.905209
$714$	0	0
$715$	9.04180e7	0.247364
$716$	0	0
$717$	1.88114e8i	0.510343i
$718$	0	0
$719$	2.86915e7i	0.0771910i	0.999255	+	0.0385955i	$0.0122884\pi$
$719$	2.86915e7i	0.0771910i	−0.999255	+	0.0385955i	$0.987712\pi$
$720$	0	0
$721$	7.83004e7	0.208910
$722$	0	0
$723$	−3.01836e8	−0.798651
$724$	0	0
$725$	− 4.58306e7i	− 0.120266i
$726$	0	0
$727$	− 4.47238e8i	− 1.16395i	−0.813205	−	0.581977i	$-0.802279\pi$
$727$	− 4.47238e8i	− 1.16395i	0.813205	−	0.581977i	$-0.197721\pi$
$728$	0	0
$729$	1.43489e7	0.0370370
$730$	0	0
$731$	1.59322e7	0.0407872
$732$	0	0
$733$	6.48484e8i	1.64660i	0.567608	+	0.823299i	$0.307869\pi$
$733$	6.48484e8i	1.64660i	−0.567608	+	0.823299i	$0.692131\pi$
$734$	0	0
$735$	− 1.71696e8i	− 0.432412i
$736$	0	0
$737$	1.03281e8	0.258000
$738$	0	0
$739$	1.59084e7	0.0394180	0.0197090	−	0.999806i	$-0.493726\pi$
$739$	1.59084e7	0.0394180	0.0197090	+	0.999806i	$0.493726\pi$
$740$	0	0
$741$	3.35499e7i	0.0824586i
$742$	0	0
$743$	4.84815e8i	1.18198i	0.806679	+	0.590990i	$0.201263\pi$
$743$	4.84815e8i	1.18198i	−0.806679	+	0.590990i	$0.798737\pi$
$744$	0	0
$745$	1.98960e8	0.481168
$746$	0	0
$747$	1.40080e8	0.336058
$748$	0	0
$749$	− 1.37106e8i	− 0.326296i
$750$	0	0
$751$	1.36659e8i	0.322641i	0.986902	+	0.161321i	$0.0515753\pi$
$751$	1.36659e8i	0.322641i	−0.986902	+	0.161321i	$0.948425\pi$
$752$	0	0
$753$	−2.03037e8	−0.475543
$754$	0	0
$755$	−5.35471e8	−1.24421
$756$	0	0
$757$	− 8.19641e8i	− 1.88945i	−0.327858	−	0.944727i	$-0.606327\pi$
$757$	− 8.19641e8i	− 1.88945i	0.327858	−	0.944727i	$-0.393673\pi$
$758$	0	0
$759$	1.28981e8i	0.294984i
$760$	0	0
$761$	−4.58635e8	−1.04067	−0.520335	−	0.853962i	$-0.674193\pi$
$761$	−4.58635e8	−1.04067	−0.520335	+	0.853962i	$0.674193\pi$
$762$	0	0
$763$	1.08964e8	0.245307
$764$	0	0
$765$	− 2.45967e7i	− 0.0549405i
$766$	0	0
$767$	1.65355e8i	0.366464i
$768$	0	0
$769$	−1.29060e8	−0.283801	−0.141900	−	0.989881i	$-0.545321\pi$
$769$	−1.29060e8	−0.283801	−0.141900	+	0.989881i	$0.545321\pi$
$770$	0	0
$771$	7.78615e7	0.169887
$772$	0	0
$773$	− 4.15620e8i	− 0.899826i	−0.893072	−	0.449913i	$-0.851455\pi$
$773$	− 4.15620e8i	− 0.899826i	0.893072	−	0.449913i	$-0.148545\pi$
$774$	0	0
$775$	7.55595e7i	0.162324i
$776$	0	0
$777$	8.66402e7	0.184696
$778$	0	0
$779$	3.25280e7	0.0688090
$780$	0	0
$781$	6.54251e7i	0.137338i
$782$	0	0
$783$	− 3.59805e7i	− 0.0749518i
$784$	0	0
$785$	4.14763e8	0.857415
$786$	0	0
$787$	−8.33479e8	−1.70990	−0.854949	−	0.518712i	$-0.826412\pi$
$787$	−8.33479e8	−1.70990	−0.854949	+	0.518712i	$0.826412\pi$
$788$	0	0
$789$	3.11606e8i	0.634416i
$790$	0	0
$791$	8.67434e6i	0.0175270i
$792$	0	0
$793$	−1.06237e7	−0.0213038
$794$	0	0
$795$	−2.66776e8	−0.530940
$796$	0	0
$797$	2.29848e8i	0.454011i	0.973894	+	0.227005i	$0.0728935\pi$
$797$	2.29848e8i	0.454011i	−0.973894	+	0.227005i	$0.927106\pi$
$798$	0	0
$799$	− 7.13202e7i	− 0.139821i
$800$	0	0
$801$	−1.12822e8	−0.219532
$802$	0	0
$803$	−4.48686e7	−0.0866555
$804$	0	0
$805$	− 2.35159e8i	− 0.450789i
$806$	0	0
$807$	3.67539e8i	0.699331i
$808$	0	0
$809$	−6.60006e8	−1.24653	−0.623265	−	0.782011i	$-0.714194\pi$
$809$	−6.60006e8	−1.24653	−0.623265	+	0.782011i	$0.714194\pi$
$810$	0	0
$811$	1.11459e7	0.0208954	0.0104477	−	0.999945i	$-0.496674\pi$
$811$	1.11459e7	0.0208954	0.0104477	+	0.999945i	$0.496674\pi$
$812$	0	0
$813$	8.59791e6i	0.0160000i
$814$	0	0
$815$	− 6.84761e8i	− 1.26493i
$816$	0	0
$817$	1.59792e7	0.0293015
$818$	0	0
$819$	5.78200e7	0.105251
$820$	0	0
$821$	9.53052e8i	1.72221i	0.508424	+	0.861107i	$0.330228\pi$
$821$	9.53052e8i	1.72221i	−0.508424	+	0.861107i	$0.669772\pi$
$822$	0	0
$823$	2.97112e8i	0.532992i	0.963836	+	0.266496i	$0.0858660\pi$
$823$	2.97112e8i	0.532992i	−0.963836	+	0.266496i	$0.914134\pi$
$824$	0	0
$825$	2.97027e7	0.0528974
$826$	0	0
$827$	−5.58756e8	−0.987883	−0.493942	−	0.869495i	$-0.664444\pi$
$827$	−5.58756e8	−0.987883	−0.493942	+	0.869495i	$0.664444\pi$
$828$	0	0
$829$	− 5.44984e8i	− 0.956577i	−0.878203	−	0.478289i	$-0.841257\pi$
$829$	− 5.44984e8i	− 0.956577i	0.878203	−	0.478289i	$-0.158743\pi$
$830$	0	0
$831$	2.62384e8i	0.457229i
$832$	0	0
$833$	1.03229e8	0.178595
$834$	0	0
$835$	224474.	0.000385573 0
$836$	0	0
$837$	5.93200e7i	0.101164i
$838$	0	0
$839$	5.63883e7i	0.0954780i	0.998860	+	0.0477390i	$0.0152016\pi$
$839$	5.63883e7i	0.0954780i	−0.998860	+	0.0477390i	$0.984798\pi$
$840$	0	0
$841$	5.04601e8	0.848320
$842$	0	0
$843$	7.13327e7	0.119071
$844$	0	0
$845$	2.82078e6i	0.00467519i
$846$	0	0
$847$	1.74486e8i	0.287151i
$848$	0	0
$849$	−5.23981e8	−0.856234
$850$	0	0
$851$	−1.07825e9	−1.74956
$852$	0	0
$853$	7.29363e8i	1.17516i	0.809166	+	0.587579i	$0.199919\pi$
$853$	7.29363e8i	1.17516i	−0.809166	+	0.587579i	$0.800081\pi$
$854$	0	0
$855$	− 2.46694e7i	− 0.0394693i
$856$	0	0
$857$	−1.05038e9	−1.66880	−0.834402	−	0.551156i	$-0.814187\pi$
$857$	−1.05038e9	−1.66880	−0.834402	+	0.551156i	$0.814187\pi$
$858$	0	0
$859$	−4.30853e8	−0.679751	−0.339875	−	0.940470i	$-0.610385\pi$
$859$	−4.30853e8	−0.679751	−0.339875	+	0.940470i	$0.610385\pi$
$860$	0	0
$861$	− 5.60590e7i	− 0.0878285i
$862$	0	0
$863$	− 2.19495e7i	− 0.0341501i	−0.999854	−	0.0170751i	$-0.994565\pi$
$863$	− 2.19495e7i	− 0.0341501i	0.999854	−	0.0170751i	$-0.00543542\pi$
$864$	0	0
$865$	−8.74001e7	−0.135040
$866$	0	0
$867$	−3.61479e8	−0.554659
$868$	0	0
$869$	− 2.59952e8i	− 0.396126i
$870$	0	0
$871$	5.76201e8i	0.872006i
$872$	0	0
$873$	1.25616e7	0.0188801
$874$	0	0
$875$	−2.29524e8	−0.342614
$876$	0	0
$877$	− 5.14486e8i	− 0.762736i	−0.924423	−	0.381368i	$-0.875453\pi$
$877$	− 5.14486e8i	− 0.762736i	0.924423	−	0.381368i	$-0.124547\pi$
$878$	0	0
$879$	− 1.95059e8i	− 0.287210i
$880$	0	0
$881$	1.00296e9	1.46675	0.733377	−	0.679822i	$-0.237943\pi$
$881$	1.00296e9	1.46675	0.733377	+	0.679822i	$0.237943\pi$
$882$	0	0
$883$	−1.19134e9	−1.73043	−0.865217	−	0.501397i	$-0.832820\pi$
$883$	−1.19134e9	−1.73043	−0.865217	+	0.501397i	$0.832820\pi$
$884$	0	0
$885$	− 1.21586e8i	− 0.175410i
$886$	0	0
$887$	4.09000e8i	0.586074i	0.956101	+	0.293037i	$0.0946660\pi$
$887$	4.09000e8i	0.586074i	−0.956101	+	0.293037i	$0.905334\pi$
$888$	0	0
$889$	3.31806e8	0.472257
$890$	0	0
$891$	2.33189e7	0.0329667
$892$	0	0
$893$	− 7.15308e7i	− 0.100447i
$894$	0	0
$895$	− 8.91603e8i	− 1.24366i
$896$	0	0
$897$	−7.19575e8	−0.997009
$898$	0	0
$899$	1.48748e8	0.204725
$900$	0	0
$901$	− 1.60395e8i	− 0.219289i
$902$	0	0
$903$	− 2.75387e7i	− 0.0374007i
$904$	0	0
$905$	−8.59624e8	−1.15975
$906$	0	0
$907$	−2.37225e8	−0.317935	−0.158967	−	0.987284i	$-0.550816\pi$
$907$	−2.37225e8	−0.317935	−0.158967	+	0.987284i	$0.550816\pi$
$908$	0	0
$909$	− 1.94991e8i	− 0.259611i
$910$	0	0
$911$	9.75538e8i	1.29030i	0.764058	+	0.645148i	$0.223204\pi$
$911$	9.75538e8i	1.29030i	−0.764058	+	0.645148i	$0.776796\pi$
$912$	0	0
$913$	2.27649e8	0.299125
$914$	0	0
$915$	7.81169e6	0.0101972
$916$	0	0
$917$	− 3.63735e8i	− 0.471712i
$918$	0	0
$919$	− 5.29410e8i	− 0.682096i	−0.940046	−	0.341048i	$-0.889218\pi$
$919$	− 5.29410e8i	− 0.682096i	0.940046	−	0.341048i	$-0.110782\pi$
$920$	0	0
$921$	8.31666e7	0.106456
$922$	0	0
$923$	−3.65003e8	−0.464186
$924$	0	0
$925$	2.48307e8i	0.313736i
$926$	0	0
$927$	1.76176e8i	0.221160i
$928$	0	0
$929$	−1.15445e8	−0.143989	−0.0719944	−	0.997405i	$-0.522936\pi$
$929$	−1.15445e8	−0.143989	−0.0719944	+	0.997405i	$0.522936\pi$
$930$	0	0
$931$	1.03534e8	0.128302
$932$	0	0
$933$	− 6.79448e8i	− 0.836588i
$934$	0	0
$935$	− 3.99730e7i	− 0.0489025i
$936$	0	0
$937$	1.31295e9	1.59598	0.797991	−	0.602669i	$-0.205896\pi$
$937$	1.31295e9	1.59598	0.797991	+	0.602669i	$0.205896\pi$
$938$	0	0
$939$	−4.49998e8	−0.543518
$940$	0	0
$941$	1.31295e9i	1.57572i	0.615855	+	0.787860i	$0.288811\pi$
$941$	1.31295e9i	1.57572i	−0.615855	+	0.787860i	$0.711189\pi$
$942$	0	0
$943$	6.97660e8i	0.831972i
$944$	0	0
$945$	−4.25153e7	−0.0503790
$946$	0	0
$947$	−7.58646e8	−0.893283	−0.446642	−	0.894713i	$-0.647380\pi$
$947$	−7.58646e8	−0.893283	−0.446642	+	0.894713i	$0.647380\pi$
$948$	0	0
$949$	− 2.50320e8i	− 0.292884i
$950$	0	0
$951$	− 7.63348e8i	− 0.887526i
$952$	0	0
$953$	−5.81061e8	−0.671341	−0.335670	−	0.941980i	$-0.608963\pi$
$953$	−5.81061e8	−0.671341	−0.335670	+	0.941980i	$0.608963\pi$
$954$	0	0
$955$	−6.61794e8	−0.759823
$956$	0	0
$957$	− 5.84732e7i	− 0.0667146i
$958$	0	0
$959$	1.19140e8i	0.135084i
$960$	0	0
$961$	6.42268e8	0.723679
$962$	0	0
$963$	3.08489e8	0.345431
$964$	0	0
$965$	6.51275e7i	0.0724741i
$966$	0	0
$967$	− 7.56646e8i	− 0.836784i	−0.908267	−	0.418392i	$-0.862594\pi$
$967$	− 7.56646e8i	− 0.836784i	0.908267	−	0.418392i	$-0.137406\pi$
$968$	0	0
$969$	1.48321e7	0.0163016
$970$	0	0
$971$	1.26034e8	0.137667	0.0688337	−	0.997628i	$-0.478072\pi$
$971$	1.26034e8	0.137667	0.0688337	+	0.997628i	$0.478072\pi$
$972$	0	0
$973$	− 3.30271e8i	− 0.358535i
$974$	0	0
$975$	1.65710e8i	0.178786i
$976$	0	0
$977$	3.93556e7	0.0422010	0.0211005	−	0.999777i	$-0.493283\pi$
$977$	3.93556e7	0.0422010	0.0211005	+	0.999777i	$0.493283\pi$
$978$	0	0
$979$	−1.83352e8	−0.195405
$980$	0	0
$981$	2.45169e8i	0.259692i
$982$	0	0
$983$	− 1.58331e9i	− 1.66689i	−0.552603	−	0.833444i	$-0.686366\pi$
$983$	− 1.58331e9i	− 1.66689i	0.552603	−	0.833444i	$-0.313634\pi$
$984$	0	0
$985$	1.43804e9	1.50475
$986$	0	0
$987$	−1.23277e8	−0.128212
$988$	0	0
$989$	3.42722e8i	0.354285i
$990$	0	0
$991$	− 1.05716e9i	− 1.08623i	−0.839659	−	0.543114i	$-0.817245\pi$
$991$	− 1.05716e9i	− 1.08623i	0.839659	−	0.543114i	$-0.182755\pi$
$992$	0	0
$993$	−9.61194e8	−0.981665
$994$	0	0
$995$	9.18289e8	0.932202
$996$	0	0
$997$	1.08192e9i	1.09172i	0.837877	+	0.545859i	$0.183796\pi$
$997$	1.08192e9i	1.09172i	−0.837877	+	0.545859i	$0.816204\pi$
$998$	0	0
$999$	1.94940e8i	0.195526i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.7.b.b.319.3	yes	4
4.3	odd	2	inner	384.7.b.b.319.1	✓	4
8.3	odd	2	inner	384.7.b.b.319.4	yes	4
8.5	even	2	inner	384.7.b.b.319.2	yes	4
16.3	odd	4		768.7.g.d.511.1		4
16.5	even	4		768.7.g.d.511.2		4
16.11	odd	4		768.7.g.d.511.4		4
16.13	even	4		768.7.g.d.511.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.7.b.b.319.1	✓	4	4.3	odd	2	inner
384.7.b.b.319.2	yes	4	8.5	even	2	inner
384.7.b.b.319.3	yes	4	1.1	even	1	trivial
384.7.b.b.319.4	yes	4	8.3	odd	2	inner
768.7.g.d.511.1		4	16.3	odd	4
768.7.g.d.511.2		4	16.5	even	4
768.7.g.d.511.3		4	16.13	even	4
768.7.g.d.511.4		4	16.11	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)

\(f(q)\)	\(=\)	\(q+15.5885 q^{3} -103.923i q^{5} -108.000i q^{7} +243.000 q^{9} +O(q^{10})\)	\(q+15.5885 q^{3} -103.923i q^{5} -108.000i q^{7} +243.000 q^{9} +394.908 q^{11} +2203.17i q^{13} -1620.00i q^{15} +974.000 q^{17} +976.877 q^{19} -1683.55i q^{21} +20952.0i q^{23} +4825.00 q^{25} +3788.00 q^{27} -9498.57i q^{29} +15660.0i q^{31} +6156.00 q^{33} -11223.7 q^{35} +51462.7i q^{37} +34344.0i q^{39} +33298.0 q^{41} +16357.5 q^{43} -25253.3i q^{45} -73224.0i q^{47} +105985. q^{49} +15183.2 q^{51} -164676. i q^{53} -41040.0i q^{55} +15228.0 q^{57} +75053.2 q^{59} +4822.03i q^{61} -26244.0i q^{63} +228960. q^{65} +261533. q^{67} +326609. i q^{69} +165672. i q^{71} -113618. q^{73} +75214.3 q^{75} -42650.0i q^{77} -658260. i q^{79} +59049.0 q^{81} +576461. q^{83} -101221. i q^{85} -148068. i q^{87} -464290. q^{89} +237942. q^{91} +244115. i q^{93} -101520. i q^{95} +51694.0 q^{97} +95962.5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 972 q^{9}+O(q^{10}) \) 4 * q + 972 * q^9	\( 4 q + 972 q^{9} + 3896 q^{17} + 19300 q^{25} + 24624 q^{33} + 133192 q^{41} + 423940 q^{49} + 60912 q^{57} + 915840 q^{65} - 454472 q^{73} + 236196 q^{81} - 1857160 q^{89} + 206776 q^{97}+O(q^{100}) \) 4 * q + 972 * q^9 + 3896 * q^17 + 19300 * q^25 + 24624 * q^33 + 133192 * q^41 + 423940 * q^49 + 60912 * q^57 + 915840 * q^65 - 454472 * q^73 + 236196 * q^81 - 1857160 * q^89 + 206776 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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