LMFDB - Embedded newform 384.8.d.e.193.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,8,Mod(193,384)]

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([0, 1, 0]))

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("384.193");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	384.d (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:	$119.955849786$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 7628 x^{10} + 22070097 x^{8} - 30593373916 x^{6} + 21405948373596 x^{4} + \cdots + 90\!\cdots\!00 $ x^12 - 7628x^10 + 22070097x^8 - 30593373916x^6 + 21405948373596x^4 - 7160683139999288*x^2 + 907532972347562500
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{57}\cdot 3^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			193.7
Root			$-19.8448 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	384.193
Dual form			384.8.d.e.193.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+27.0000i q^{3} -467.267i q^{5} -31.2683 q^{7} -729.000 q^{9} +O(q^{10})$	$q+27.0000i q^{3} -467.267i q^{5} -31.2683 q^{7} -729.000 q^{9} +5276.60i q^{11} +806.828i q^{13} +12616.2 q^{15} +18.4070 q^{17} -13657.0i q^{19} -844.244i q^{21} +58696.4 q^{23} -140214. q^{25} -19683.0i q^{27} -119183. i q^{29} -101880. q^{31} -142468. q^{33} +14610.7i q^{35} +259208. i q^{37} -21784.4 q^{39} -809267. q^{41} +15343.1i q^{43} +340638. i q^{45} -183648. q^{47} -822565. q^{49} +496.988i q^{51} -907643. i q^{53} +2.46558e6 q^{55} +368739. q^{57} +1.43398e6i q^{59} +884055. i q^{61} +22794.6 q^{63} +377004. q^{65} -2.48901e6i q^{67} +1.58480e6i q^{69} +4.68753e6 q^{71} +3.46295e6 q^{73} -3.78577e6i q^{75} -164990. i q^{77} -4.34533e6 q^{79} +531441. q^{81} +8.38216e6i q^{83} -8600.98i q^{85} +3.21793e6 q^{87} +1.29208e7 q^{89} -25228.1i q^{91} -2.75075e6i q^{93} -6.38146e6 q^{95} +4.60705e6 q^{97} -3.84664e6i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 8748 q^{9}+O(q^{10}) $ 12 * q - 8748 * q^9	$ 12 q - 8748 q^{9} - 83096 q^{17} - 127812 q^{25} - 30672 q^{33} - 969032 q^{41} - 5087028 q^{49} - 69552 q^{57} - 240832 q^{65} + 18079656 q^{73} + 6377292 q^{81} + 43563144 q^{89} - 48458328 q^{97}+O(q^{100}) $ 12 * q - 8748 * q^9 - 83096 * q^17 - 127812 * q^25 - 30672 * q^33 - 969032 * q^41 - 5087028 * q^49 - 69552 * q^57 - 240832 * q^65 + 18079656 * q^73 + 6377292 * q^81 + 43563144 * q^89 - 48458328 * 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
$3$	27.0000i	0.577350i
$3$	27.0000i	0.577350i
$4$	0	0
$5$	− 467.267i	− 1.67175i	−0.548922	−	0.835873i	$-0.684962\pi$
$5$	− 467.267i	− 1.67175i	0.548922	−	0.835873i	$-0.315038\pi$
$6$	0	0
$7$	−31.2683	−0.0344557	−0.0172279	−	0.999852i	$-0.505484\pi$
$7$	−31.2683	−0.0344557	−0.0172279	+	0.999852i	$0.505484\pi$
$8$	0	0
$9$	−729.000	−0.333333
$10$	0	0
$11$	5276.60i	1.19531i	0.801754	+	0.597654i	$0.203900\pi$
$11$	5276.60i	1.19531i	−0.801754	+	0.597654i	$0.796100\pi$
$12$	0	0
$13$	806.828i	0.101854i	0.998702	+	0.0509271i	$0.0162176\pi$
$13$	806.828i	0.101854i	−0.998702	+	0.0509271i	$0.983782\pi$
$14$	0	0
$15$	12616.2	0.965183
$16$	0	0
$17$	18.4070	0.000908680 0	0.000454340	−	1.00000i	$-0.499855\pi$
$17$	18.4070	0.000908680 0	0.000454340		1.00000i	$0.499855\pi$
$18$	0	0
$19$	− 13657.0i	− 0.456791i	−0.973568	−	0.228395i	$-0.926652\pi$
$19$	− 13657.0i	− 0.456791i	0.973568	−	0.228395i	$-0.0733479\pi$
$20$	0	0
$21$	− 844.244i	− 0.0198930i
$22$	0	0
$23$	58696.4	1.00592	0.502961	−	0.864309i	$-0.332244\pi$
$23$	58696.4	1.00592	0.502961	+	0.864309i	$0.332244\pi$
$24$	0	0
$25$	−140214.	−1.79474
$26$	0	0
$27$	− 19683.0i	− 0.192450i
$28$	0	0
$29$	− 119183.i	− 0.907444i	−0.891143	−	0.453722i	$-0.850096\pi$
$29$	− 119183.i	− 0.907444i	0.891143	−	0.453722i	$-0.149904\pi$
$30$	0	0
$31$	−101880.	−0.614216	−0.307108	−	0.951675i	$-0.599361\pi$
$31$	−101880.	−0.614216	−0.307108	+	0.951675i	$0.599361\pi$
$32$	0	0
$33$	−142468.	−0.690111
$34$	0	0
$35$	14610.7i	0.0576012i
$36$	0	0
$37$	259208.i	0.841285i	0.907227	+	0.420642i	$0.138195\pi$
$37$	259208.i	0.841285i	−0.907227	+	0.420642i	$0.861805\pi$
$38$	0	0
$39$	−21784.4	−0.0588056
$40$	0	0
$41$	−809267.	−1.83378	−0.916892	−	0.399135i	$-0.869311\pi$
$41$	−809267.	−1.83378	−0.916892	+	0.399135i	$0.869311\pi$
$42$	0	0
$43$	15343.1i	0.0294289i	0.999892	+	0.0147145i	$0.00468392\pi$
$43$	15343.1i	0.0294289i	−0.999892	+	0.0147145i	$0.995316\pi$
$44$	0	0
$45$	340638.i	0.557249i
$46$	0	0
$47$	−183648.	−0.258015	−0.129007	−	0.991644i	$-0.541179\pi$
$47$	−183648.	−0.258015	−0.129007	+	0.991644i	$0.541179\pi$
$48$	0	0
$49$	−822565.	−0.998813
$50$	0	0
$51$	496.988i	0 0.000524627i
$52$	0	0
$53$	− 907643.i	− 0.837431i	−0.908117	−	0.418716i	$-0.862480\pi$
$53$	− 907643.i	− 0.837431i	0.908117	−	0.418716i	$-0.137520\pi$
$54$	0	0
$55$	2.46558e6	1.99825
$56$	0	0
$57$	368739.	0.263728
$58$	0	0
$59$	1.43398e6i	0.908993i	0.890748	+	0.454497i	$0.150181\pi$
$59$	1.43398e6i	0.908993i	−0.890748	+	0.454497i	$0.849819\pi$
$60$	0	0
$61$	884055.i	0.498683i	0.968416	+	0.249342i	$0.0802142\pi$
$61$	884055.i	0.498683i	−0.968416	+	0.249342i	$0.919786\pi$
$62$	0	0
$63$	22794.6	0.0114852
$64$	0	0
$65$	377004.	0.170274
$66$	0	0
$67$	− 2.48901e6i	− 1.01103i	−0.862817	−	0.505516i	$-0.831302\pi$
$67$	− 2.48901e6i	− 1.01103i	0.862817	−	0.505516i	$-0.168698\pi$
$68$	0	0
$69$	1.58480e6i	0.580769i
$70$	0	0
$71$	4.68753e6	1.55432	0.777158	−	0.629305i	$-0.216660\pi$
$71$	4.68753e6	1.55432	0.777158	+	0.629305i	$0.216660\pi$
$72$	0	0
$73$	3.46295e6	1.04188	0.520938	−	0.853594i	$-0.325582\pi$
$73$	3.46295e6	1.04188	0.520938	+	0.853594i	$0.325582\pi$
$74$	0	0
$75$	− 3.78577e6i	− 1.03619i
$76$	0	0
$77$	− 164990.i	− 0.0411852i
$78$	0	0
$79$	−4.34533e6	−0.991580	−0.495790	−	0.868442i	$-0.665121\pi$
$79$	−4.34533e6	−0.991580	−0.495790	+	0.868442i	$0.665121\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	0	0
$83$	8.38216e6i	1.60910i	0.593886	+	0.804549i	$0.297593\pi$
$83$	8.38216e6i	1.60910i	−0.593886	+	0.804549i	$0.702407\pi$
$84$	0	0
$85$	− 8600.98i	− 0.00151908i
$86$	0	0
$87$	3.21793e6	0.523913
$88$	0	0
$89$	1.29208e7	1.94278	0.971390	−	0.237489i	$-0.0763245\pi$
$89$	1.29208e7	1.94278	0.971390	+	0.237489i	$0.0763245\pi$
$90$	0	0
$91$	− 25228.1i	− 0.00350946i
$92$	0	0
$93$	− 2.75075e6i	− 0.354618i
$94$	0	0
$95$	−6.38146e6	−0.763638
$96$	0	0
$97$	4.60705e6	0.512533	0.256266	−	0.966606i	$-0.417508\pi$
$97$	4.60705e6	0.512533	0.256266	+	0.966606i	$0.417508\pi$
$98$	0	0
$99$	− 3.84664e6i	− 0.398436i
$100$	0	0
$101$	1.50932e7i	1.45766i	0.684696	+	0.728829i	$0.259935\pi$
$101$	1.50932e7i	1.45766i	−0.684696	+	0.728829i	$0.740065\pi$
$102$	0	0
$103$	6.45820e6	0.582346	0.291173	−	0.956670i	$-0.405954\pi$
$103$	6.45820e6	0.582346	0.291173	+	0.956670i	$0.405954\pi$
$104$	0	0
$105$	−394488.	−0.0332561
$106$	0	0
$107$	1.43047e7i	1.12885i	0.825484	+	0.564425i	$0.190902\pi$
$107$	1.43047e7i	1.12885i	−0.825484	+	0.564425i	$0.809098\pi$
$108$	0	0
$109$	1.46215e7i	1.08143i	0.841206	+	0.540715i	$0.181846\pi$
$109$	1.46215e7i	1.08143i	−0.841206	+	0.540715i	$0.818154\pi$
$110$	0	0
$111$	−6.99863e6	−0.485716
$112$	0	0
$113$	−1.44884e7	−0.944596	−0.472298	−	0.881439i	$-0.656575\pi$
$113$	−1.44884e7	−0.944596	−0.472298	+	0.881439i	$0.656575\pi$
$114$	0	0
$115$	− 2.74269e7i	− 1.68165i
$116$	0	0
$117$	− 588177.i	− 0.0339514i
$118$	0	0
$119$	−575.555	−3.13092e−5 0
$120$	0	0
$121$	−8.35532e6	−0.428760
$122$	0	0
$123$	− 2.18502e7i	− 1.05874i
$124$	0	0
$125$	2.90121e7i	1.32860i
$126$	0	0
$127$	1.06200e7	0.460055	0.230028	−	0.973184i	$-0.426118\pi$
$127$	1.06200e7	0.460055	0.230028	+	0.973184i	$0.426118\pi$
$128$	0	0
$129$	−414264.	−0.0169908
$130$	0	0
$131$	4.85018e7i	1.88499i	0.334227	+	0.942493i	$0.391525\pi$
$131$	4.85018e7i	1.88499i	−0.334227	+	0.942493i	$0.608475\pi$
$132$	0	0
$133$	427031.i	0.0157390i
$134$	0	0
$135$	−9.19722e6	−0.321728
$136$	0	0
$137$	−3.66799e7	−1.21873	−0.609363	−	0.792892i	$-0.708575\pi$
$137$	−3.66799e7	−1.21873	−0.609363	+	0.792892i	$0.708575\pi$
$138$	0	0
$139$	5.99325e6i	0.189282i	0.995511	+	0.0946412i	$0.0301704\pi$
$139$	5.99325e6i	0.189282i	−0.995511	+	0.0946412i	$0.969830\pi$
$140$	0	0
$141$	− 4.95850e6i	− 0.148965i
$142$	0	0
$143$	−4.25731e6	−0.121747
$144$	0	0
$145$	−5.56902e7	−1.51702
$146$	0	0
$147$	− 2.22093e7i	− 0.576665i
$148$	0	0
$149$	5.10670e7i	1.26470i	0.774682	+	0.632351i	$0.217910\pi$
$149$	5.10670e7i	1.26470i	−0.774682	+	0.632351i	$0.782090\pi$
$150$	0	0
$151$	−6.49428e7	−1.53501	−0.767506	−	0.641042i	$-0.778502\pi$
$151$	−6.49428e7	−1.53501	−0.767506	+	0.641042i	$0.778502\pi$
$152$	0	0
$153$	−13418.7	−0.000302893 0
$154$	0	0
$155$	4.76050e7i	1.02681i
$156$	0	0
$157$	8.27200e7i	1.70593i	0.521967	+	0.852966i	$0.325198\pi$
$157$	8.27200e7i	1.70593i	−0.521967	+	0.852966i	$0.674802\pi$
$158$	0	0
$159$	2.45063e7	0.483491
$160$	0	0
$161$	−1.83534e6	−0.0346598
$162$	0	0
$163$	− 4.55553e7i	− 0.823915i	−0.911203	−	0.411958i	$-0.864845\pi$
$163$	− 4.55553e7i	− 0.823915i	0.911203	−	0.411958i	$-0.135155\pi$
$164$	0	0
$165$	6.65707e7i	1.15369i
$166$	0	0
$167$	−8.80939e7	−1.46365	−0.731827	−	0.681491i	$-0.761332\pi$
$167$	−8.80939e7	−1.46365	−0.731827	+	0.681491i	$0.761332\pi$
$168$	0	0
$169$	6.20975e7	0.989626
$170$	0	0
$171$	9.95594e6i	0.152264i
$172$	0	0
$173$	− 2.34223e6i	− 0.0343929i	−0.999852	−	0.0171965i	$-0.994526\pi$
$173$	− 2.34223e6i	− 0.0343929i	0.999852	−	0.0171965i	$-0.00547407\pi$
$174$	0	0
$175$	4.38425e6	0.0618389
$176$	0	0
$177$	−3.87174e7	−0.524807
$178$	0	0
$179$	− 5.96459e7i	− 0.777312i	−0.921383	−	0.388656i	$-0.872939\pi$
$179$	− 5.96459e7i	− 0.777312i	0.921383	−	0.388656i	$-0.127061\pi$
$180$	0	0
$181$	4.32379e7i	0.541987i	0.962581	+	0.270994i	$0.0873522\pi$
$181$	4.32379e7i	0.541987i	−0.962581	+	0.270994i	$0.912648\pi$
$182$	0	0
$183$	−2.38695e7	−0.287915
$184$	0	0
$185$	1.21120e8	1.40642
$186$	0	0
$187$	97126.2i	0.00108615i
$188$	0	0
$189$	615454.i	0.00663100i
$190$	0	0
$191$	1.53658e8	1.59565	0.797826	−	0.602887i	$-0.205983\pi$
$191$	1.53658e8	1.59565	0.797826	+	0.602887i	$0.205983\pi$
$192$	0	0
$193$	7.22178e7	0.723093	0.361546	−	0.932354i	$-0.382249\pi$
$193$	7.22178e7	0.723093	0.361546	+	0.932354i	$0.382249\pi$
$194$	0	0
$195$	1.01791e7i	0.0983080i
$196$	0	0
$197$	1.48312e8i	1.38212i	0.722799	+	0.691058i	$0.242855\pi$
$197$	1.48312e8i	1.38212i	−0.722799	+	0.691058i	$0.757145\pi$
$198$	0	0
$199$	1.21013e8	1.08855	0.544274	−	0.838907i	$-0.316805\pi$
$199$	1.21013e8	1.08855	0.544274	+	0.838907i	$0.316805\pi$
$200$	0	0
$201$	6.72033e7	0.583720
$202$	0	0
$203$	3.72664e6i	0.0312666i
$204$	0	0
$205$	3.78144e8i	3.06562i
$206$	0	0
$207$	−4.27897e7	−0.335307
$208$	0	0
$209$	7.20624e7	0.546005
$210$	0	0
$211$	1.37134e8i	1.00498i	0.864584	+	0.502489i	$0.167582\pi$
$211$	1.37134e8i	1.00498i	−0.864584	+	0.502489i	$0.832418\pi$
$212$	0	0
$213$	1.26563e8i	0.897385i
$214$	0	0
$215$	7.16934e6	0.0491977
$216$	0	0
$217$	3.18560e6	0.0211632
$218$	0	0
$219$	9.34996e7i	0.601528i
$220$	0	0
$221$	14851.3i	0 9.25529e-5i
$222$	0	0
$223$	2.14479e8	1.29514	0.647572	−	0.762004i	$-0.275785\pi$
$223$	2.14479e8	1.29514	0.647572	+	0.762004i	$0.275785\pi$
$224$	0	0
$225$	1.02216e8	0.598246
$226$	0	0
$227$	− 2.91532e8i	− 1.65423i	−0.562035	−	0.827113i	$-0.689981\pi$
$227$	− 2.91532e8i	− 1.65423i	0.562035	−	0.827113i	$-0.310019\pi$
$228$	0	0
$229$	− 1.31691e8i	− 0.724657i	−0.932050	−	0.362329i	$-0.881982\pi$
$229$	− 1.31691e8i	− 0.724657i	0.932050	−	0.362329i	$-0.118018\pi$
$230$	0	0
$231$	4.45474e6	0.0237783
$232$	0	0
$233$	−1.40022e8	−0.725189	−0.362595	−	0.931947i	$-0.618109\pi$
$233$	−1.40022e8	−0.725189	−0.362595	+	0.931947i	$0.618109\pi$
$234$	0	0
$235$	8.58129e7i	0.431335i
$236$	0	0
$237$	− 1.17324e8i	− 0.572489i
$238$	0	0
$239$	−3.32407e8	−1.57499	−0.787495	−	0.616321i	$-0.788622\pi$
$239$	−3.32407e8	−1.57499	−0.787495	+	0.616321i	$0.788622\pi$
$240$	0	0
$241$	1.15264e8	0.530436	0.265218	−	0.964188i	$-0.414556\pi$
$241$	1.15264e8	0.530436	0.265218	+	0.964188i	$0.414556\pi$
$242$	0	0
$243$	1.43489e7i	0.0641500i
$244$	0	0
$245$	3.84358e8i	1.66976i
$246$	0	0
$247$	1.10188e7	0.0465261
$248$	0	0
$249$	−2.26318e8	−0.929014
$250$	0	0
$251$	− 6.42315e7i	− 0.256384i	−0.991749	−	0.128192i	$-0.959083\pi$
$251$	− 6.42315e7i	− 0.256384i	0.991749	−	0.128192i	$-0.0409173\pi$
$252$	0	0
$253$	3.09718e8i	1.20239i
$254$	0	0
$255$	232226.	0.000877043 0
$256$	0	0
$257$	−1.87951e8	−0.690682	−0.345341	−	0.938477i	$-0.612237\pi$
$257$	−1.87951e8	−0.690682	−0.345341	+	0.938477i	$0.612237\pi$
$258$	0	0
$259$	− 8.10501e6i	− 0.0289871i
$260$	0	0
$261$	8.68841e7i	0.302481i
$262$	0	0
$263$	4.53334e8	1.53664	0.768322	−	0.640064i	$-0.221092\pi$
$263$	4.53334e8	1.53664	0.768322	+	0.640064i	$0.221092\pi$
$264$	0	0
$265$	−4.24112e8	−1.39997
$266$	0	0
$267$	3.48861e8i	1.12166i
$268$	0	0
$269$	3.11428e8i	0.975495i	0.872985	+	0.487747i	$0.162181\pi$
$269$	3.11428e8i	0.975495i	−0.872985	+	0.487747i	$0.837819\pi$
$270$	0	0
$271$	−3.59086e8	−1.09599	−0.547994	−	0.836483i	$-0.684608\pi$
$271$	−3.59086e8	−1.09599	−0.547994	+	0.836483i	$0.684608\pi$
$272$	0	0
$273$	681160.	0.00202619
$274$	0	0
$275$	− 7.39852e8i	− 2.14526i
$276$	0	0
$277$	− 2.12866e8i	− 0.601767i	−0.953661	−	0.300883i	$-0.902719\pi$
$277$	− 2.12866e8i	− 0.601767i	0.953661	−	0.300883i	$-0.0972815\pi$
$278$	0	0
$279$	7.42702e7	0.204739
$280$	0	0
$281$	−2.34768e8	−0.631200	−0.315600	−	0.948892i	$-0.602206\pi$
$281$	−2.34768e8	−0.631200	−0.315600	+	0.948892i	$0.602206\pi$
$282$	0	0
$283$	4.61638e8i	1.21073i	0.795947	+	0.605367i	$0.206974\pi$
$283$	4.61638e8i	1.21073i	−0.795947	+	0.605367i	$0.793026\pi$
$284$	0	0
$285$	− 1.72299e8i	− 0.440887i
$286$	0	0
$287$	2.53044e7	0.0631843
$288$	0	0
$289$	−4.10338e8	−0.999999
$290$	0	0
$291$	1.24390e8i	0.295911i
$292$	0	0
$293$	2.81937e8i	0.654811i	0.944884	+	0.327406i	$0.106174\pi$
$293$	2.81937e8i	0.654811i	−0.944884	+	0.327406i	$0.893826\pi$
$294$	0	0
$295$	6.70051e8	1.51961
$296$	0	0
$297$	1.03859e8	0.230037
$298$	0	0
$299$	4.73579e7i	0.102457i
$300$	0	0
$301$	− 479753.i	− 0.00101399i
$302$	0	0
$303$	−4.07515e8	−0.841579
$304$	0	0
$305$	4.13090e8	0.833672
$306$	0	0
$307$	− 2.31592e8i	− 0.456813i	−0.973566	−	0.228407i	$-0.926648\pi$
$307$	− 2.31592e8i	− 0.456813i	0.973566	−	0.228407i	$-0.0733515\pi$
$308$	0	0
$309$	1.74371e8i	0.336218i
$310$	0	0
$311$	−6.15818e8	−1.16089	−0.580445	−	0.814299i	$-0.697121\pi$
$311$	−6.15818e8	−1.16089	−0.580445	+	0.814299i	$0.697121\pi$
$312$	0	0
$313$	2.86280e8	0.527699	0.263849	−	0.964564i	$-0.415008\pi$
$313$	2.86280e8	0.527699	0.263849	+	0.964564i	$0.415008\pi$
$314$	0	0
$315$	− 1.06512e7i	− 0.0192004i
$316$	0	0
$317$	− 3.20478e8i	− 0.565055i	−0.959259	−	0.282527i	$-0.908827\pi$
$317$	− 3.20478e8i	− 0.565055i	0.959259	−	0.282527i	$-0.0911728\pi$
$318$	0	0
$319$	6.28879e8	1.08467
$320$	0	0
$321$	−3.86228e8	−0.651742
$322$	0	0
$323$	− 251384.i	0 0.000415077i
$324$	0	0
$325$	− 1.13128e8i	− 0.182802i
$326$	0	0
$327$	−3.94780e8	−0.624364
$328$	0	0
$329$	5.74237e6	0.00889008
$330$	0	0
$331$	6.95072e8i	1.05349i	0.850022	+	0.526746i	$0.176588\pi$
$331$	6.95072e8i	1.05349i	−0.850022	+	0.526746i	$0.823412\pi$
$332$	0	0
$333$	− 1.88963e8i	− 0.280428i
$334$	0	0
$335$	−1.16303e9	−1.69019
$336$	0	0
$337$	7.90458e8	1.12506	0.562528	−	0.826778i	$-0.309829\pi$
$337$	7.90458e8	1.12506	0.562528	+	0.826778i	$0.309829\pi$
$338$	0	0
$339$	− 3.91187e8i	− 0.545363i
$340$	0	0
$341$	− 5.37577e8i	− 0.734177i
$342$	0	0
$343$	5.14710e7	0.0688705
$344$	0	0
$345$	7.40527e8	0.970899
$346$	0	0
$347$	8.49503e8i	1.09147i	0.837958	+	0.545735i	$0.183749\pi$
$347$	8.49503e8i	1.09147i	−0.837958	+	0.545735i	$0.816251\pi$
$348$	0	0
$349$	3.96115e8i	0.498806i	0.968400	+	0.249403i	$0.0802345\pi$
$349$	3.96115e8i	0.498806i	−0.968400	+	0.249403i	$0.919766\pi$
$350$	0	0
$351$	1.58808e7	0.0196019
$352$	0	0
$353$	6.57428e8	0.795494	0.397747	−	0.917495i	$-0.369792\pi$
$353$	6.57428e8	0.795494	0.397747	+	0.917495i	$0.369792\pi$
$354$	0	0
$355$	− 2.19033e9i	− 2.59842i
$356$	0	0
$357$	− 15540.0i	0 1.80764e-5i
$358$	0	0
$359$	−5.97068e8	−0.681072	−0.340536	−	0.940231i	$-0.610609\pi$
$359$	−5.97068e8	−0.681072	−0.340536	+	0.940231i	$0.610609\pi$
$360$	0	0
$361$	7.07359e8	0.791342
$362$	0	0
$363$	− 2.25594e8i	− 0.247545i
$364$	0	0
$365$	− 1.61812e9i	− 1.74175i
$366$	0	0
$367$	−2.31319e8	−0.244275	−0.122138	−	0.992513i	$-0.538975\pi$
$367$	−2.31319e8	−0.244275	−0.122138	+	0.992513i	$0.538975\pi$
$368$	0	0
$369$	5.89956e8	0.611262
$370$	0	0
$371$	2.83804e7i	0.0288543i
$372$	0	0
$373$	− 1.47548e9i	− 1.47215i	−0.676897	−	0.736077i	$-0.736676\pi$
$373$	− 1.47548e9i	− 1.47215i	0.676897	−	0.736077i	$-0.263324\pi$
$374$	0	0
$375$	−7.83326e8	−0.767067
$376$	0	0
$377$	9.61599e7	0.0924270
$378$	0	0
$379$	9.58315e8i	0.904214i	0.891964	+	0.452107i	$0.149327\pi$
$379$	9.58315e8i	0.904214i	−0.891964	+	0.452107i	$0.850673\pi$
$380$	0	0
$381$	2.86739e8i	0.265613i
$382$	0	0
$383$	1.90346e9	1.73121	0.865603	−	0.500731i	$-0.166935\pi$
$383$	1.90346e9	1.73121	0.865603	+	0.500731i	$0.166935\pi$
$384$	0	0
$385$	−7.70946e7	−0.0688512
$386$	0	0
$387$	− 1.11851e7i	− 0.00980963i
$388$	0	0
$389$	− 1.68380e9i	− 1.45033i	−0.688575	−	0.725165i	$-0.741764\pi$
$389$	− 1.68380e9i	− 1.45033i	0.688575	−	0.725165i	$-0.258236\pi$
$390$	0	0
$391$	1.08042e6	0.000914061 0
$392$	0	0
$393$	−1.30955e9	−1.08830
$394$	0	0
$395$	2.03043e9i	1.65767i
$396$	0	0
$397$	− 9.50529e8i	− 0.762428i	−0.924487	−	0.381214i	$-0.875506\pi$
$397$	− 9.50529e8i	− 0.762428i	0.924487	−	0.381214i	$-0.124494\pi$
$398$	0	0
$399$	−1.15298e7	−0.00908694
$400$	0	0
$401$	−5.52085e8	−0.427564	−0.213782	−	0.976881i	$-0.568578\pi$
$401$	−5.52085e8	−0.427564	−0.213782	+	0.976881i	$0.568578\pi$
$402$	0	0
$403$	− 8.21992e7i	− 0.0625605i
$404$	0	0
$405$	− 2.48325e8i	− 0.185750i
$406$	0	0
$407$	−1.36774e9	−1.00559
$408$	0	0
$409$	−4.84172e8	−0.349919	−0.174960	−	0.984576i	$-0.555980\pi$
$409$	−4.84172e8	−0.349919	−0.174960	+	0.984576i	$0.555980\pi$
$410$	0	0
$411$	− 9.90357e8i	− 0.703631i
$412$	0	0
$413$	− 4.48381e7i	− 0.0313200i
$414$	0	0
$415$	3.91671e9	2.69001
$416$	0	0
$417$	−1.61818e8	−0.109282
$418$	0	0
$419$	4.02693e8i	0.267439i	0.991019	+	0.133719i	$0.0426921\pi$
$419$	4.02693e8i	0.267439i	−0.991019	+	0.133719i	$0.957308\pi$
$420$	0	0
$421$	1.16230e9i	0.759159i	0.925159	+	0.379579i	$0.123931\pi$
$421$	1.16230e9i	0.759159i	−0.925159	+	0.379579i	$0.876069\pi$
$422$	0	0
$423$	1.33880e8	0.0860049
$424$	0	0
$425$	−2.58091e6	−0.00163084
$426$	0	0
$427$	− 2.76429e7i	− 0.0171825i
$428$	0	0
$429$	− 1.14947e8i	− 0.0702907i
$430$	0	0
$431$	−8.65230e8	−0.520548	−0.260274	−	0.965535i	$-0.583813\pi$
$431$	−8.65230e8	−0.520548	−0.260274	+	0.965535i	$0.583813\pi$
$432$	0	0
$433$	−1.88729e9	−1.11720	−0.558599	−	0.829438i	$-0.688661\pi$
$433$	−1.88729e9	−1.11720	−0.558599	+	0.829438i	$0.688661\pi$
$434$	0	0
$435$	− 1.50363e9i	− 0.875850i
$436$	0	0
$437$	− 8.01616e8i	− 0.459496i
$438$	0	0
$439$	1.78047e9	1.00440	0.502202	−	0.864751i	$-0.332524\pi$
$439$	1.78047e9	1.00440	0.502202	+	0.864751i	$0.332524\pi$
$440$	0	0
$441$	5.99650e8	0.332938
$442$	0	0
$443$	2.21282e9i	1.20930i	0.796492	+	0.604649i	$0.206686\pi$
$443$	2.21282e9i	1.20930i	−0.796492	+	0.604649i	$0.793314\pi$
$444$	0	0
$445$	− 6.03746e9i	− 3.24784i
$446$	0	0
$447$	−1.37881e9	−0.730176
$448$	0	0
$449$	−1.81035e9	−0.943844	−0.471922	−	0.881640i	$-0.656440\pi$
$449$	−1.81035e9	−0.943844	−0.471922	+	0.881640i	$0.656440\pi$
$450$	0	0
$451$	− 4.27018e9i	− 2.19194i
$452$	0	0
$453$	− 1.75346e9i	− 0.886239i
$454$	0	0
$455$	−1.17883e7	−0.00586693
$456$	0	0
$457$	−2.26015e9	−1.10772	−0.553861	−	0.832609i	$-0.686846\pi$
$457$	−2.26015e9	−1.10772	−0.553861	+	0.832609i	$0.686846\pi$
$458$	0	0
$459$	− 362304.i	0 0.000174876i
$460$	0	0
$461$	2.29944e8i	0.109312i	0.998505	+	0.0546561i	$0.0174062\pi$
$461$	2.29944e8i	0.109312i	−0.998505	+	0.0546561i	$0.982594\pi$
$462$	0	0
$463$	2.87071e9	1.34418	0.672088	−	0.740472i	$-0.265398\pi$
$463$	2.87071e9	1.34418	0.672088	+	0.740472i	$0.265398\pi$
$464$	0	0
$465$	−1.28533e9	−0.592831
$466$	0	0
$467$	− 8.25672e8i	− 0.375145i	−0.982251	−	0.187572i	$-0.939938\pi$
$467$	− 8.25672e8i	− 0.375145i	0.982251	−	0.187572i	$-0.0600619\pi$
$468$	0	0
$469$	7.78272e7i	0.0348359i
$470$	0	0
$471$	−2.23344e9	−0.984920
$472$	0	0
$473$	−8.09595e7	−0.0351766
$474$	0	0
$475$	1.91490e9i	0.819819i
$476$	0	0
$477$	6.61671e8i	0.279144i
$478$	0	0
$479$	−3.09397e9	−1.28630	−0.643150	−	0.765740i	$-0.722373\pi$
$479$	−3.09397e9	−1.28630	−0.643150	+	0.765740i	$0.722373\pi$
$480$	0	0
$481$	−2.09137e8	−0.0856884
$482$	0	0
$483$	− 4.95541e7i	− 0.0200108i
$484$	0	0
$485$	− 2.15272e9i	− 0.856825i
$486$	0	0
$487$	2.88844e9	1.13321	0.566607	−	0.823988i	$-0.308256\pi$
$487$	2.88844e9	1.13321	0.566607	+	0.823988i	$0.308256\pi$
$488$	0	0
$489$	1.22999e9	0.475688
$490$	0	0
$491$	3.69018e9i	1.40690i	0.710746	+	0.703449i	$0.248357\pi$
$491$	3.69018e9i	1.40690i	−0.710746	+	0.703449i	$0.751643\pi$
$492$	0	0
$493$	− 2.19379e6i	0 0.000824577i
$494$	0	0
$495$	−1.79741e9	−0.666084
$496$	0	0
$497$	−1.46571e8	−0.0535551
$498$	0	0
$499$	1.03673e9i	0.373519i	0.982406	+	0.186759i	$0.0597985\pi$
$499$	1.03673e9i	0.373519i	−0.982406	+	0.186759i	$0.940202\pi$
$500$	0	0
$501$	− 2.37854e9i	− 0.845040i
$502$	0	0
$503$	6.16186e8	0.215886	0.107943	−	0.994157i	$-0.465574\pi$
$503$	6.16186e8	0.215886	0.107943	+	0.994157i	$0.465574\pi$
$504$	0	0
$505$	7.05254e9	2.43683
$506$	0	0
$507$	1.67663e9i	0.571361i
$508$	0	0
$509$	1.70431e9i	0.572844i	0.958103	+	0.286422i	$0.0924660\pi$
$509$	1.70431e9i	0.572844i	−0.958103	+	0.286422i	$0.907534\pi$
$510$	0	0
$511$	−1.08281e8	−0.0358986
$512$	0	0
$513$	−2.68810e8	−0.0879094
$514$	0	0
$515$	− 3.01771e9i	− 0.973535i
$516$	0	0
$517$	− 9.69038e8i	− 0.308407i
$518$	0	0
$519$	6.32403e7	0.0198568
$520$	0	0
$521$	−1.12737e9	−0.349249	−0.174624	−	0.984635i	$-0.555871\pi$
$521$	−1.12737e9	−0.349249	−0.174624	+	0.984635i	$0.555871\pi$
$522$	0	0
$523$	− 7.00952e8i	− 0.214256i	−0.994245	−	0.107128i	$-0.965835\pi$
$523$	− 7.00952e8i	− 0.214256i	0.994245	−	0.107128i	$-0.0341654\pi$
$524$	0	0
$525$	1.18375e8i	0.0357027i
$526$	0	0
$527$	−1.87529e6	−0.000558126 0
$528$	0	0
$529$	4.04465e7	0.0118792
$530$	0	0
$531$	− 1.04537e9i	− 0.302998i
$532$	0	0
$533$	− 6.52939e8i	− 0.186779i
$534$	0	0
$535$	6.68413e9	1.88715
$536$	0	0
$537$	1.61044e9	0.448781
$538$	0	0
$539$	− 4.34035e9i	− 1.19389i
$540$	0	0
$541$	− 2.93472e9i	− 0.796851i	−0.917201	−	0.398425i	$-0.869557\pi$
$541$	− 2.93472e9i	− 0.796851i	0.917201	−	0.398425i	$-0.130443\pi$
$542$	0	0
$543$	−1.16742e9	−0.312916
$544$	0	0
$545$	6.83214e9	1.80788
$546$	0	0
$547$	3.77357e9i	0.985819i	0.870080	+	0.492910i	$0.164067\pi$
$547$	3.77357e9i	0.985819i	−0.870080	+	0.492910i	$0.835933\pi$
$548$	0	0
$549$	− 6.44476e8i	− 0.166228i
$550$	0	0
$551$	−1.62767e9	−0.414512
$552$	0	0
$553$	1.35871e8	0.0341656
$554$	0	0
$555$	3.27023e9i	0.811994i
$556$	0	0
$557$	− 1.41429e9i	− 0.346774i	−0.984854	−	0.173387i	$-0.944529\pi$
$557$	− 1.41429e9i	− 0.346774i	0.984854	−	0.173387i	$-0.0554711\pi$
$558$	0	0
$559$	−1.23793e7	−0.00299746
$560$	0	0
$561$	−2.62241e6	−0.000627090 0
$562$	0	0
$563$	− 6.13087e9i	− 1.44791i	−0.689845	−	0.723957i	$-0.742321\pi$
$563$	− 6.13087e9i	− 1.44791i	0.689845	−	0.723957i	$-0.257679\pi$
$564$	0	0
$565$	6.76996e9i	1.57913i
$566$	0	0
$567$	−1.66173e7	−0.00382841
$568$	0	0
$569$	1.93798e9	0.441017	0.220509	−	0.975385i	$-0.429228\pi$
$569$	1.93798e9	0.441017	0.220509	+	0.975385i	$0.429228\pi$
$570$	0	0
$571$	8.05170e8i	0.180993i	0.995897	+	0.0904964i	$0.0288453\pi$
$571$	8.05170e8i	0.180993i	−0.995897	+	0.0904964i	$0.971155\pi$
$572$	0	0
$573$	4.14877e9i	0.921250i
$574$	0	0
$575$	−8.23005e9	−1.80537
$576$	0	0
$577$	−1.00273e9	−0.217305	−0.108653	−	0.994080i	$-0.534654\pi$
$577$	−1.00273e9	−0.217305	−0.108653	+	0.994080i	$0.534654\pi$
$578$	0	0
$579$	1.94988e9i	0.417478i
$580$	0	0
$581$	− 2.62096e8i	− 0.0554426i
$582$	0	0
$583$	4.78927e9	1.00099
$584$	0	0
$585$	−2.74836e8	−0.0567582
$586$	0	0
$587$	− 1.87262e9i	− 0.382135i	−0.981577	−	0.191067i	$-0.938805\pi$
$587$	− 1.87262e9i	− 0.382135i	0.981577	−	0.191067i	$-0.0611949\pi$
$588$	0	0
$589$	1.39137e9i	0.280568i
$590$	0	0
$591$	−4.00442e9	−0.797965
$592$	0	0
$593$	5.41357e9	1.06609	0.533043	−	0.846088i	$-0.321048\pi$
$593$	5.41357e9	1.06609	0.533043	+	0.846088i	$0.321048\pi$
$594$	0	0
$595$	268938.i	0 5.23411e-5i
$596$	0	0
$597$	3.26736e9i	0.628474i
$598$	0	0
$599$	−4.71914e9	−0.897159	−0.448579	−	0.893743i	$-0.648070\pi$
$599$	−4.71914e9	−0.897159	−0.448579	+	0.893743i	$0.648070\pi$
$600$	0	0
$601$	−2.69993e9	−0.507331	−0.253666	−	0.967292i	$-0.581636\pi$
$601$	−2.69993e9	−0.507331	−0.253666	+	0.967292i	$0.581636\pi$
$602$	0	0
$603$	1.81449e9i	0.337011i
$604$	0	0
$605$	3.90417e9i	0.716778i
$606$	0	0
$607$	3.86996e8	0.0702337	0.0351169	−	0.999383i	$-0.488820\pi$
$607$	3.86996e8	0.0702337	0.0351169	+	0.999383i	$0.488820\pi$
$608$	0	0
$609$	−1.00619e8	−0.0180518
$610$	0	0
$611$	− 1.48173e8i	− 0.0262799i
$612$	0	0
$613$	7.20217e9i	1.26285i	0.775437	+	0.631425i	$0.217530\pi$
$613$	7.20217e9i	1.26285i	−0.775437	+	0.631425i	$0.782470\pi$
$614$	0	0
$615$	−1.02099e10	−1.76994
$616$	0	0
$617$	−1.11331e10	−1.90817	−0.954087	−	0.299531i	$-0.903170\pi$
$617$	−1.11331e10	−1.90817	−0.954087	+	0.299531i	$0.903170\pi$
$618$	0	0
$619$	5.50625e9i	0.933123i	0.884489	+	0.466561i	$0.154507\pi$
$619$	5.50625e9i	0.933123i	−0.884489	+	0.466561i	$0.845493\pi$
$620$	0	0
$621$	− 1.15532e9i	− 0.193590i
$622$	0	0
$623$	−4.04011e8	−0.0669399
$624$	0	0
$625$	2.60219e9	0.426343
$626$	0	0
$627$	1.94569e9i	0.315236i
$628$	0	0
$629$	4.77124e6i	0 0.000764459i
$630$	0	0
$631$	4.99750e9	0.791863	0.395931	−	0.918280i	$-0.370422\pi$
$631$	4.99750e9	0.791863	0.395931	+	0.918280i	$0.370422\pi$
$632$	0	0
$633$	−3.70262e9	−0.580224
$634$	0	0
$635$	− 4.96236e9i	− 0.769096i
$636$	0	0
$637$	− 6.63669e8i	− 0.101733i
$638$	0	0
$639$	−3.41721e9	−0.518106
$640$	0	0
$641$	−8.92704e9	−1.33877	−0.669383	−	0.742918i	$-0.733441\pi$
$641$	−8.92704e9	−1.33877	−0.669383	+	0.742918i	$0.733441\pi$
$642$	0	0
$643$	1.87437e9i	0.278046i	0.990289	+	0.139023i	$0.0443963\pi$
$643$	1.87437e9i	0.278046i	−0.990289	+	0.139023i	$0.955604\pi$
$644$	0	0
$645$	1.93572e8i	0.0284043i
$646$	0	0
$647$	−6.12131e9	−0.888545	−0.444273	−	0.895892i	$-0.646538\pi$
$647$	−6.12131e9	−0.888545	−0.444273	+	0.895892i	$0.646538\pi$
$648$	0	0
$649$	−7.56653e9	−1.08653
$650$	0	0
$651$	8.60112e7i	0.0122186i
$652$	0	0
$653$	3.55775e9i	0.500010i	0.968245	+	0.250005i	$0.0804323\pi$
$653$	3.55775e9i	0.500010i	−0.968245	+	0.250005i	$0.919568\pi$
$654$	0	0
$655$	2.26633e10	3.15122
$656$	0	0
$657$	−2.52449e9	−0.347292
$658$	0	0
$659$	− 2.14258e9i	− 0.291634i	−0.989312	−	0.145817i	$-0.953419\pi$
$659$	− 2.14258e9i	− 0.291634i	0.989312	−	0.145817i	$-0.0465811\pi$
$660$	0	0
$661$	− 1.02631e9i	− 0.138221i	−0.997609	−	0.0691105i	$-0.977984\pi$
$661$	− 1.02631e9i	− 0.138221i	0.997609	−	0.0691105i	$-0.0220161\pi$
$662$	0	0
$663$	−400984.	−5.34355e−5 0
$664$	0	0
$665$	1.99537e8	0.0263117
$666$	0	0
$667$	− 6.99560e9i	− 0.912818i
$668$	0	0
$669$	5.79094e9i	0.747752i
$670$	0	0
$671$	−4.66480e9	−0.596080
$672$	0	0
$673$	−4.26856e9	−0.539794	−0.269897	−	0.962889i	$-0.586990\pi$
$673$	−4.26856e9	−0.539794	−0.269897	+	0.962889i	$0.586990\pi$
$674$	0	0
$675$	2.75983e9i	0.345397i
$676$	0	0
$677$	5.61660e9i	0.695686i	0.937553	+	0.347843i	$0.113086\pi$
$677$	5.61660e9i	0.695686i	−0.937553	+	0.347843i	$0.886914\pi$
$678$	0	0
$679$	−1.44055e8	−0.0176597
$680$	0	0
$681$	7.87135e9	0.955068
$682$	0	0
$683$	− 5.51352e9i	− 0.662150i	−0.943604	−	0.331075i	$-0.892589\pi$
$683$	− 5.51352e9i	− 0.662150i	0.943604	−	0.331075i	$-0.107411\pi$
$684$	0	0
$685$	1.71393e10i	2.03740i
$686$	0	0
$687$	3.55566e9	0.418381
$688$	0	0
$689$	7.32311e8	0.0852959
$690$	0	0
$691$	1.03033e10i	1.18796i	0.804480	+	0.593980i	$0.202444\pi$
$691$	1.03033e10i	1.18796i	−0.804480	+	0.593980i	$0.797556\pi$
$692$	0	0
$693$	1.20278e8i	0.0137284i
$694$	0	0
$695$	2.80045e9	0.316432
$696$	0	0
$697$	−1.48962e7	−0.00166632
$698$	0	0
$699$	− 3.78060e9i	− 0.418688i
$700$	0	0
$701$	1.05192e10i	1.15337i	0.816966	+	0.576686i	$0.195655\pi$
$701$	1.05192e10i	1.15337i	−0.816966	+	0.576686i	$0.804345\pi$
$702$	0	0
$703$	3.54000e9	0.384291
$704$	0	0
$705$	−2.31695e9	−0.249031
$706$	0	0
$707$	− 4.71938e8i	− 0.0502246i
$708$	0	0
$709$	7.67229e9i	0.808468i	0.914656	+	0.404234i	$0.132462\pi$
$709$	7.67229e9i	0.808468i	−0.914656	+	0.404234i	$0.867538\pi$
$710$	0	0
$711$	3.16774e9	0.330527
$712$	0	0
$713$	−5.97996e9	−0.617853
$714$	0	0
$715$	1.98930e9i	0.203530i
$716$	0	0
$717$	− 8.97500e9i	− 0.909321i
$718$	0	0
$719$	1.54882e10	1.55400	0.776999	−	0.629502i	$-0.216741\pi$
$719$	1.54882e10	1.55400	0.776999	+	0.629502i	$0.216741\pi$
$720$	0	0
$721$	−2.01937e8	−0.0200652
$722$	0	0
$723$	3.11212e9i	0.306247i
$724$	0	0
$725$	1.67110e10i	1.62862i
$726$	0	0
$727$	−1.36057e9	−0.131326	−0.0656631	−	0.997842i	$-0.520916\pi$
$727$	−1.36057e9	−0.131326	−0.0656631	+	0.997842i	$0.520916\pi$
$728$	0	0
$729$	−3.87420e8	−0.0370370
$730$	0	0
$731$	282420.i	0 2.67415e-5i
$732$	0	0
$733$	− 9.50290e8i	− 0.0891235i	−0.999007	−	0.0445617i	$-0.985811\pi$
$733$	− 9.50290e8i	− 0.0891235i	0.999007	−	0.0445617i	$-0.0141891\pi$
$734$	0	0
$735$	−1.03777e10	−0.964037
$736$	0	0
$737$	1.31335e10	1.20850
$738$	0	0
$739$	− 1.61058e10i	− 1.46800i	−0.679150	−	0.733999i	$-0.737651\pi$
$739$	− 1.61058e10i	− 1.46800i	0.679150	−	0.733999i	$-0.262349\pi$
$740$	0	0
$741$	2.97508e8i	0.0268618i
$742$	0	0
$743$	−1.62026e10	−1.44918	−0.724592	−	0.689178i	$-0.757972\pi$
$743$	−1.62026e10	−1.44918	−0.724592	+	0.689178i	$0.757972\pi$
$744$	0	0
$745$	2.38619e10	2.11426
$746$	0	0
$747$	− 6.11060e9i	− 0.536366i
$748$	0	0
$749$	− 4.47284e8i	− 0.0388953i
$750$	0	0
$751$	1.60043e10	1.37879	0.689393	−	0.724388i	$-0.257878\pi$
$751$	1.60043e10	1.37879	0.689393	+	0.724388i	$0.257878\pi$
$752$	0	0
$753$	1.73425e9	0.148023
$754$	0	0
$755$	3.03457e10i	2.56615i
$756$	0	0
$757$	− 1.44950e10i	− 1.21446i	−0.794527	−	0.607229i	$-0.792281\pi$
$757$	− 1.44950e10i	− 1.21446i	0.794527	−	0.607229i	$-0.207719\pi$
$758$	0	0
$759$	−8.36237e9	−0.694198
$760$	0	0
$761$	3.48731e8	0.0286843	0.0143421	−	0.999897i	$-0.495435\pi$
$761$	3.48731e8	0.0286843	0.0143421	+	0.999897i	$0.495435\pi$
$762$	0	0
$763$	− 4.57189e8i	− 0.0372615i
$764$	0	0
$765$	6.27011e6i	0 0.000506361i
$766$	0	0
$767$	−1.15697e9	−0.0925848
$768$	0	0
$769$	−4.31946e9	−0.342521	−0.171260	−	0.985226i	$-0.554784\pi$
$769$	−4.31946e9	−0.342521	−0.171260	+	0.985226i	$0.554784\pi$
$770$	0	0
$771$	− 5.07467e9i	− 0.398765i
$772$	0	0
$773$	1.47386e10i	1.14770i	0.818961	+	0.573849i	$0.194551\pi$
$773$	1.47386e10i	1.14770i	−0.818961	+	0.573849i	$0.805449\pi$
$774$	0	0
$775$	1.42849e10	1.10236
$776$	0	0
$777$	2.18835e8	0.0167357
$778$	0	0
$779$	1.10521e10i	0.837655i
$780$	0	0
$781$	2.47342e10i	1.85789i
$782$	0	0
$783$	−2.34587e9	−0.174638
$784$	0	0
$785$	3.86523e10	2.85189
$786$	0	0
$787$	− 8.74976e9i	− 0.639859i	−0.947441	−	0.319930i	$-0.896341\pi$
$787$	− 8.74976e9i	− 0.639859i	0.947441	−	0.319930i	$-0.103659\pi$
$788$	0	0
$789$	1.22400e10i	0.887182i
$790$	0	0
$791$	4.53028e8	0.0325467
$792$	0	0
$793$	−7.13280e8	−0.0507930
$794$	0	0
$795$	− 1.14510e10i	− 0.808275i
$796$	0	0
$797$	1.95855e9i	0.137035i	0.997650	+	0.0685173i	$0.0218268\pi$
$797$	1.95855e9i	0.137035i	−0.997650	+	0.0685173i	$0.978173\pi$
$798$	0	0
$799$	−3.38041e6	−0.000234453 0
$800$	0	0
$801$	−9.41925e9	−0.647593
$802$	0	0
$803$	1.82726e10i	1.24536i
$804$	0	0
$805$	8.57594e8i	0.0579423i
$806$	0	0
$807$	−8.40856e9	−0.563202
$808$	0	0
$809$	−1.64242e10	−1.09060	−0.545299	−	0.838242i	$-0.683584\pi$
$809$	−1.64242e10	−1.09060	−0.545299	+	0.838242i	$0.683584\pi$
$810$	0	0
$811$	− 2.27404e10i	− 1.49701i	−0.663127	−	0.748507i	$-0.730771\pi$
$811$	− 2.27404e10i	− 1.49701i	0.663127	−	0.748507i	$-0.269229\pi$
$812$	0	0
$813$	− 9.69531e9i	− 0.632768i
$814$	0	0
$815$	−2.12865e10	−1.37738
$816$	0	0
$817$	2.09541e8	0.0134428
$818$	0	0
$819$	1.83913e7i	0.00116982i
$820$	0	0
$821$	2.18898e9i	0.138052i	0.997615	+	0.0690258i	$0.0219891\pi$
$821$	2.18898e9i	0.138052i	−0.997615	+	0.0690258i	$0.978011\pi$
$822$	0	0
$823$	2.09467e10	1.30983	0.654916	−	0.755702i	$-0.272704\pi$
$823$	2.09467e10	1.30983	0.654916	+	0.755702i	$0.272704\pi$
$824$	0	0
$825$	1.99760e10	1.23857
$826$	0	0
$827$	− 1.60221e10i	− 0.985033i	−0.870303	−	0.492517i	$-0.836077\pi$
$827$	− 1.60221e10i	− 0.985033i	0.870303	−	0.492517i	$-0.163923\pi$
$828$	0	0
$829$	− 8.26352e9i	− 0.503761i	−0.967758	−	0.251880i	$-0.918951\pi$
$829$	− 8.26352e9i	− 0.503761i	0.967758	−	0.251880i	$-0.0810490\pi$
$830$	0	0
$831$	5.74739e9	0.347430
$832$	0	0
$833$	−1.51409e7	−0.000907601 0
$834$	0	0
$835$	4.11634e10i	2.44686i
$836$	0	0
$837$	2.00529e9i	0.118206i
$838$	0	0
$839$	9.38387e9	0.548549	0.274274	−	0.961651i	$-0.411562\pi$
$839$	9.38387e9	0.548549	0.274274	+	0.961651i	$0.411562\pi$
$840$	0	0
$841$	3.04538e9	0.176545
$842$	0	0
$843$	− 6.33874e9i	− 0.364424i
$844$	0	0
$845$	− 2.90162e10i	− 1.65440i
$846$	0	0
$847$	2.61257e8	0.0147732
$848$	0	0
$849$	−1.24642e10	−0.699017
$850$	0	0
$851$	1.52146e10i	0.846267i
$852$	0	0
$853$	− 3.49929e10i	− 1.93045i	−0.261424	−	0.965224i	$-0.584192\pi$
$853$	− 3.49929e10i	− 1.93045i	0.261424	−	0.965224i	$-0.415808\pi$
$854$	0	0
$855$	4.65209e9	0.254546
$856$	0	0
$857$	−2.73150e10	−1.48241	−0.741205	−	0.671279i	$-0.765745\pi$
$857$	−2.73150e10	−1.48241	−0.741205	+	0.671279i	$0.765745\pi$
$858$	0	0
$859$	− 3.65662e10i	− 1.96836i	−0.177181	−	0.984178i	$-0.556698\pi$
$859$	− 3.65662e10i	− 1.96836i	0.177181	−	0.984178i	$-0.443302\pi$
$860$	0	0
$861$	6.83219e8i	0.0364795i
$862$	0	0
$863$	−1.34559e10	−0.712646	−0.356323	−	0.934363i	$-0.615970\pi$
$863$	−1.34559e10	−0.712646	−0.356323	+	0.934363i	$0.615970\pi$
$864$	0	0
$865$	−1.09445e9	−0.0574962
$866$	0	0
$867$	− 1.10791e10i	− 0.577350i
$868$	0	0
$869$	− 2.29286e10i	− 1.18524i
$870$	0	0
$871$	2.00820e9	0.102978
$872$	0	0
$873$	−3.35854e9	−0.170844
$874$	0	0
$875$	− 9.07158e8i	− 0.0457778i
$876$	0	0
$877$	3.73698e10i	1.87078i	0.353624	+	0.935388i	$0.384949\pi$
$877$	3.73698e10i	1.87078i	−0.353624	+	0.935388i	$0.615051\pi$
$878$	0	0
$879$	−7.61231e9	−0.378055
$880$	0	0
$881$	−3.50627e10	−1.72754	−0.863772	−	0.503883i	$-0.831904\pi$
$881$	−3.50627e10	−1.72754	−0.863772	+	0.503883i	$0.831904\pi$
$882$	0	0
$883$	− 1.59555e10i	− 0.779918i	−0.920832	−	0.389959i	$-0.872489\pi$
$883$	− 1.59555e10i	− 0.779918i	0.920832	−	0.389959i	$-0.127511\pi$
$884$	0	0
$885$	1.80914e10i	0.877345i
$886$	0	0
$887$	1.32602e10	0.637993	0.318996	−	0.947756i	$-0.396654\pi$
$887$	1.32602e10	0.637993	0.318996	+	0.947756i	$0.396654\pi$
$888$	0	0
$889$	−3.32068e8	−0.0158515
$890$	0	0
$891$	2.80420e9i	0.132812i
$892$	0	0
$893$	2.50808e9i	0.117859i
$894$	0	0
$895$	−2.78706e10	−1.29947
$896$	0	0
$897$	−1.27866e9	−0.0591538
$898$	0	0
$899$	1.21423e10i	0.557366i
$900$	0	0
$901$	− 1.67069e7i	0 0.000760957i
$902$	0	0
$903$	1.29533e7	0.000585429 0
$904$	0	0
$905$	2.02036e10	0.906065
$906$	0	0
$907$	4.01584e10i	1.78711i	0.448953	+	0.893555i	$0.351797\pi$
$907$	4.01584e10i	1.78711i	−0.448953	+	0.893555i	$0.648203\pi$
$908$	0	0
$909$	− 1.10029e10i	− 0.485886i
$910$	0	0
$911$	−9.17715e8	−0.0402155	−0.0201078	−	0.999798i	$-0.506401\pi$
$911$	−9.17715e8	−0.0402155	−0.0201078	+	0.999798i	$0.506401\pi$
$912$	0	0
$913$	−4.42293e10	−1.92337
$914$	0	0
$915$	1.11534e10i	0.481321i
$916$	0	0
$917$	− 1.51657e9i	− 0.0649485i
$918$	0	0
$919$	−1.08713e10	−0.462039	−0.231019	−	0.972949i	$-0.574206\pi$
$919$	−1.08713e10	−0.462039	−0.231019	+	0.972949i	$0.574206\pi$
$920$	0	0
$921$	6.25297e9	0.263741
$922$	0	0
$923$	3.78203e9i	0.158314i
$924$	0	0
$925$	− 3.63446e10i	− 1.50988i
$926$	0	0
$927$	−4.70803e9	−0.194115
$928$	0	0
$929$	5.19109e9	0.212424	0.106212	−	0.994344i	$-0.466128\pi$
$929$	5.19109e9	0.212424	0.106212	+	0.994344i	$0.466128\pi$
$930$	0	0
$931$	1.12338e10i	0.456248i
$932$	0	0
$933$	− 1.66271e10i	− 0.670240i
$934$	0	0
$935$	4.53839e7	0.00181577
$936$	0	0
$937$	−1.42896e10	−0.567453	−0.283727	−	0.958905i	$-0.591571\pi$
$937$	−1.42896e10	−0.567453	−0.283727	+	0.958905i	$0.591571\pi$
$938$	0	0
$939$	7.72957e9i	0.304667i
$940$	0	0
$941$	− 4.37976e10i	− 1.71351i	−0.515722	−	0.856756i	$-0.672476\pi$
$941$	− 4.37976e10i	− 1.71351i	0.515722	−	0.856756i	$-0.327524\pi$
$942$	0	0
$943$	−4.75011e10	−1.84464
$944$	0	0
$945$	2.87582e8	0.0110854
$946$	0	0
$947$	1.19780e10i	0.458309i	0.973390	+	0.229154i	$0.0735961\pi$
$947$	1.19780e10i	0.458309i	−0.973390	+	0.229154i	$0.926404\pi$
$948$	0	0
$949$	2.79400e9i	0.106120i
$950$	0	0
$951$	8.65290e9	0.326234
$952$	0	0
$953$	−1.67815e10	−0.628068	−0.314034	−	0.949412i	$-0.601681\pi$
$953$	−1.67815e10	−0.628068	−0.314034	+	0.949412i	$0.601681\pi$
$954$	0	0
$955$	− 7.17994e10i	− 2.66753i
$956$	0	0
$957$	1.69797e10i	0.626237i
$958$	0	0
$959$	1.14692e9	0.0419920
$960$	0	0
$961$	−1.71332e10	−0.622739
$962$	0	0
$963$	− 1.04281e10i	− 0.376283i
$964$	0	0
$965$	− 3.37450e10i	− 1.20883i
$966$	0	0
$967$	−7.39750e9	−0.263083	−0.131541	−	0.991311i	$-0.541993\pi$
$967$	−7.39750e9	−0.263083	−0.131541	+	0.991311i	$0.541993\pi$
$968$	0	0
$969$	6.78736e6	0.000239645 0
$970$	0	0
$971$	− 3.13191e10i	− 1.09785i	−0.835872	−	0.548924i	$-0.815038\pi$
$971$	− 3.13191e10i	− 1.09785i	0.835872	−	0.548924i	$-0.184962\pi$
$972$	0	0
$973$	− 1.87399e8i	− 0.00652186i
$974$	0	0
$975$	3.05447e9	0.105541
$976$	0	0
$977$	1.00101e10	0.343405	0.171703	−	0.985149i	$-0.445073\pi$
$977$	1.00101e10	0.343405	0.171703	+	0.985149i	$0.445073\pi$
$978$	0	0
$979$	6.81778e10i	2.32222i
$980$	0	0
$981$	− 1.06591e10i	− 0.360477i
$982$	0	0
$983$	−4.04299e10	−1.35758	−0.678789	−	0.734333i	$-0.737495\pi$
$983$	−4.04299e10	−1.35758	−0.678789	+	0.734333i	$0.737495\pi$
$984$	0	0
$985$	6.93014e10	2.31055
$986$	0	0
$987$	1.55044e8i	0.00513269i
$988$	0	0
$989$	9.00586e8i	0.0296032i
$990$	0	0
$991$	6.02049e10	1.96505	0.982526	−	0.186125i	$-0.0595930\pi$
$991$	6.02049e10	1.96505	0.982526	+	0.186125i	$0.0595930\pi$
$992$	0	0
$993$	−1.87669e10	−0.608234
$994$	0	0
$995$	− 5.65456e10i	− 1.81978i
$996$	0	0
$997$	− 4.18147e10i	− 1.33628i	−0.744037	−	0.668138i	$-0.767091\pi$
$997$	− 4.18147e10i	− 1.33628i	0.744037	−	0.668138i	$-0.232909\pi$
$998$	0	0
$999$	5.10200e9	0.161905

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.8.d.e.193.7	yes	12
4.3	odd	2	inner	384.8.d.e.193.1	✓	12
8.3	odd	2	inner	384.8.d.e.193.12	yes	12
8.5	even	2	inner	384.8.d.e.193.6	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.8.d.e.193.1	✓	12	4.3	odd	2	inner
384.8.d.e.193.6	yes	12	8.5	even	2	inner
384.8.d.e.193.7	yes	12	1.1	even	1	trivial
384.8.d.e.193.12	yes	12	8.3	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	384.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+27.0000i q^{3} -467.267i q^{5} -31.2683 q^{7} -729.000 q^{9} +O(q^{10})\)	\(q+27.0000i q^{3} -467.267i q^{5} -31.2683 q^{7} -729.000 q^{9} +5276.60i q^{11} +806.828i q^{13} +12616.2 q^{15} +18.4070 q^{17} -13657.0i q^{19} -844.244i q^{21} +58696.4 q^{23} -140214. q^{25} -19683.0i q^{27} -119183. i q^{29} -101880. q^{31} -142468. q^{33} +14610.7i q^{35} +259208. i q^{37} -21784.4 q^{39} -809267. q^{41} +15343.1i q^{43} +340638. i q^{45} -183648. q^{47} -822565. q^{49} +496.988i q^{51} -907643. i q^{53} +2.46558e6 q^{55} +368739. q^{57} +1.43398e6i q^{59} +884055. i q^{61} +22794.6 q^{63} +377004. q^{65} -2.48901e6i q^{67} +1.58480e6i q^{69} +4.68753e6 q^{71} +3.46295e6 q^{73} -3.78577e6i q^{75} -164990. i q^{77} -4.34533e6 q^{79} +531441. q^{81} +8.38216e6i q^{83} -8600.98i q^{85} +3.21793e6 q^{87} +1.29208e7 q^{89} -25228.1i q^{91} -2.75075e6i q^{93} -6.38146e6 q^{95} +4.60705e6 q^{97} -3.84664e6i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 8748 q^{9}+O(q^{10}) \) 12 * q - 8748 * q^9	\( 12 q - 8748 q^{9} - 83096 q^{17} - 127812 q^{25} - 30672 q^{33} - 969032 q^{41} - 5087028 q^{49} - 69552 q^{57} - 240832 q^{65} + 18079656 q^{73} + 6377292 q^{81} + 43563144 q^{89} - 48458328 q^{97}+O(q^{100}) \) 12 * q - 8748 * q^9 - 83096 * q^17 - 127812 * q^25 - 30672 * q^33 - 969032 * q^41 - 5087028 * q^49 - 69552 * q^57 - 240832 * q^65 + 18079656 * q^73 + 6377292 * q^81 + 43563144 * q^89 - 48458328 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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