LMFDB - Embedded newform 384.9.b.b.319.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,9,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, 9, names="a")

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

chi := DirichletCharacter("384.319");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 9 $
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:	$156.433386263$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 4826x^{6} + 8748877x^{4} + 7060845096x^{2} + 2140819627716 $ x^8 + 4826x^6 + 8748877x^4 + 7060845096*x^2 + 2140819627716
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{32}\cdot 3^{11} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			319.8
Root			$-1.73205 + 33.5532i$ of defining polynomial
Character	$\chi$	$=$	384.319
Dual form			384.9.b.b.319.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+46.7654 q^{3} +1205.32i q^{5} -1133.44i q^{7} +2187.00 q^{9} +O(q^{10})$	$q+46.7654 q^{3} +1205.32i q^{5} -1133.44i q^{7} +2187.00 q^{9} +20623.5 q^{11} +25759.2i q^{13} +56367.2i q^{15} +2473.09 q^{17} -206001. q^{19} -53005.6i q^{21} +223202. i q^{23} -1.06217e6 q^{25} +102276. q^{27} -202690. i q^{29} +1.16928e6i q^{31} +964465. q^{33} +1.36615e6 q^{35} -2.62267e6i q^{37} +1.20464e6i q^{39} +1.46741e6 q^{41} -5.00580e6 q^{43} +2.63603e6i q^{45} +6.93618e6i q^{47} +4.48012e6 q^{49} +115655. q^{51} +1.16384e6i q^{53} +2.48579e7i q^{55} -9.63372e6 q^{57} +2.03352e7 q^{59} +3.06867e6i q^{61} -2.47883e6i q^{63} -3.10481e7 q^{65} -3.74270e6 q^{67} +1.04381e7i q^{69} +3.03025e7i q^{71} -1.81452e7 q^{73} -4.96728e7 q^{75} -2.33754e7i q^{77} -6.18230e7i q^{79} +4.78297e6 q^{81} -8.39301e6 q^{83} +2.98087e6i q^{85} -9.47888e6i q^{87} +1.73301e7 q^{89} +2.91964e7 q^{91} +5.46819e7i q^{93} -2.48297e8i q^{95} -1.34674e8 q^{97} +4.51035e7 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 17496 q^{9}+O(q^{10}) $ 8 * q + 17496 * q^9	$ 8 q + 17496 q^{9} + 276848 q^{17} - 2841976 q^{25} + 775008 q^{33} + 5826832 q^{41} - 16599928 q^{49} - 42366240 q^{57} - 132963072 q^{65} - 57759760 q^{73} + 38263752 q^{81} - 258778768 q^{89} - 771489424 q^{97}+O(q^{100}) $ 8 * q + 17496 * q^9 + 276848 * q^17 - 2841976 * q^25 + 775008 * q^33 + 5826832 * q^41 - 16599928 * q^49 - 42366240 * q^57 - 132963072 * q^65 - 57759760 * q^73 + 38263752 * q^81 - 258778768 * q^89 - 771489424 * 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
$3$	46.7654	0.577350
$3$	46.7654	0.577350
$4$	0	0
$5$	1205.32i	1.92851i	0.264973	+	0.964256i	$0.414637\pi$
$5$	1205.32i	1.92851i	−0.264973	+	0.964256i	$0.585363\pi$
$6$	0	0
$7$	− 1133.44i	− 0.472069i	−0.971745	−	0.236034i	$-0.924152\pi$
$7$	− 1133.44i	− 0.472069i	0.971745	−	0.236034i	$-0.0758478\pi$
$8$	0	0
$9$	2187.00	0.333333
$10$	0	0
$11$	20623.5	1.40861	0.704306	−	0.709897i	$-0.251259\pi$
$11$	20623.5	1.40861	0.704306	+	0.709897i	$0.251259\pi$
$12$	0	0
$13$	25759.2i	0.901901i	0.892549	+	0.450950i	$0.148915\pi$
$13$	25759.2i	0.901901i	−0.892549	+	0.450950i	$0.851085\pi$
$14$	0	0
$15$	56367.2i	1.11343i
$16$	0	0
$17$	2473.09	0.0296104	0.0148052	−	0.999890i	$-0.495287\pi$
$17$	2473.09	0.0296104	0.0148052	+	0.999890i	$0.495287\pi$
$18$	0	0
$19$	−206001.	−1.58072	−0.790361	−	0.612642i	$-0.790107\pi$
$19$	−206001.	−1.58072	−0.790361	+	0.612642i	$0.790107\pi$
$20$	0	0
$21$	− 53005.6i	− 0.272549i
$22$	0	0
$23$	223202.i	0.797603i	0.917037	+	0.398802i	$0.130574\pi$
$23$	223202.i	0.797603i	−0.917037	+	0.398802i	$0.869426\pi$
$24$	0	0
$25$	−1.06217e6	−2.71916
$26$	0	0
$27$	102276.	0.192450
$28$	0	0
$29$	− 202690.i	− 0.286577i	−0.989681	−	0.143288i	$-0.954232\pi$
$29$	− 202690.i	− 0.286577i	0.989681	−	0.143288i	$-0.0457676\pi$
$30$	0	0
$31$	1.16928e6i	1.26611i	0.774106	+	0.633056i	$0.218200\pi$
$31$	1.16928e6i	1.26611i	−0.774106	+	0.633056i	$0.781800\pi$
$32$	0	0
$33$	964465.	0.813262
$34$	0	0
$35$	1.36615e6	0.910390
$36$	0	0
$37$	− 2.62267e6i	− 1.39939i	−0.714444	−	0.699693i	$-0.753320\pi$
$37$	− 2.62267e6i	− 1.39939i	0.714444	−	0.699693i	$-0.246680\pi$
$38$	0	0
$39$	1.20464e6i	0.520713i
$40$	0	0
$41$	1.46741e6	0.519298	0.259649	−	0.965703i	$-0.416393\pi$
$41$	1.46741e6	0.519298	0.259649	+	0.965703i	$0.416393\pi$
$42$	0	0
$43$	−5.00580e6	−1.46420	−0.732099	−	0.681198i	$-0.761459\pi$
$43$	−5.00580e6	−1.46420	−0.732099	+	0.681198i	$0.761459\pi$
$44$	0	0
$45$	2.63603e6i	0.642837i
$46$	0	0
$47$	6.93618e6i	1.42144i	0.703474	+	0.710721i	$0.251631\pi$
$47$	6.93618e6i	1.42144i	−0.703474	+	0.710721i	$0.748369\pi$
$48$	0	0
$49$	4.48012e6	0.777151
$50$	0	0
$51$	115655.	0.0170956
$52$	0	0
$53$	1.16384e6i	0.147500i	0.997277	+	0.0737498i	$0.0234966\pi$
$53$	1.16384e6i	0.147500i	−0.997277	+	0.0737498i	$0.976503\pi$
$54$	0	0
$55$	2.48579e7i	2.71652i
$56$	0	0
$57$	−9.63372e6	−0.912630
$58$	0	0
$59$	2.03352e7	1.67819	0.839095	−	0.543986i	$-0.183085\pi$
$59$	2.03352e7	1.67819	0.839095	+	0.543986i	$0.183085\pi$
$60$	0	0
$61$	3.06867e6i	0.221631i	0.993841	+	0.110816i	$0.0353463\pi$
$61$	3.06867e6i	0.221631i	−0.993841	+	0.110816i	$0.964654\pi$
$62$	0	0
$63$	− 2.47883e6i	− 0.157356i
$64$	0	0
$65$	−3.10481e7	−1.73933
$66$	0	0
$67$	−3.74270e6	−0.185731	−0.0928657	−	0.995679i	$-0.529603\pi$
$67$	−3.74270e6	−0.185731	−0.0928657	+	0.995679i	$0.529603\pi$
$68$	0	0
$69$	1.04381e7i	0.460496i
$70$	0	0
$71$	3.03025e7i	1.19246i	0.802812	+	0.596232i	$0.203336\pi$
$71$	3.03025e7i	1.19246i	−0.802812	+	0.596232i	$0.796664\pi$
$72$	0	0
$73$	−1.81452e7	−0.638954	−0.319477	−	0.947594i	$-0.603507\pi$
$73$	−1.81452e7	−0.638954	−0.319477	+	0.947594i	$0.603507\pi$
$74$	0	0
$75$	−4.96728e7	−1.56991
$76$	0	0
$77$	− 2.33754e7i	− 0.664961i
$78$	0	0
$79$	− 6.18230e7i	− 1.58724i	−0.608416	−	0.793618i	$-0.708195\pi$
$79$	− 6.18230e7i	− 1.58724i	0.608416	−	0.793618i	$-0.291805\pi$
$80$	0	0
$81$	4.78297e6	0.111111
$82$	0	0
$83$	−8.39301e6	−0.176850	−0.0884250	−	0.996083i	$-0.528183\pi$
$83$	−8.39301e6	−0.176850	−0.0884250	+	0.996083i	$0.528183\pi$
$84$	0	0
$85$	2.98087e6i	0.0571040i
$86$	0	0
$87$	− 9.47888e6i	− 0.165455i
$88$	0	0
$89$	1.73301e7	0.276212	0.138106	−	0.990417i	$-0.455899\pi$
$89$	1.73301e7	0.276212	0.138106	+	0.990417i	$0.455899\pi$
$90$	0	0
$91$	2.91964e7	0.425759
$92$	0	0
$93$	5.46819e7i	0.730991i
$94$	0	0
$95$	− 2.48297e8i	− 3.04844i
$96$	0	0
$97$	−1.34674e8	−1.52124	−0.760620	−	0.649197i	$-0.775105\pi$
$97$	−1.34674e8	−1.52124	−0.760620	+	0.649197i	$0.775105\pi$
$98$	0	0
$99$	4.51035e7	0.469537
$100$	0	0
$101$	− 1.35048e8i	− 1.29778i	−0.760882	−	0.648890i	$-0.775233\pi$
$101$	− 1.35048e8i	− 1.29778i	0.760882	−	0.648890i	$-0.224767\pi$
$102$	0	0
$103$	6.71582e7i	0.596692i	0.954458	+	0.298346i	$0.0964349\pi$
$103$	6.71582e7i	0.596692i	−0.954458	+	0.298346i	$0.903565\pi$
$104$	0	0
$105$	6.38887e7	0.525614
$106$	0	0
$107$	−7.14494e7	−0.545084	−0.272542	−	0.962144i	$-0.587864\pi$
$107$	−7.14494e7	−0.545084	−0.272542	+	0.962144i	$0.587864\pi$
$108$	0	0
$109$	2.49238e8i	1.76566i	0.469690	+	0.882831i	$0.344366\pi$
$109$	2.49238e8i	1.76566i	−0.469690	+	0.882831i	$0.655634\pi$
$110$	0	0
$111$	− 1.22650e8i	− 0.807936i
$112$	0	0
$113$	2.58091e7	0.158292	0.0791460	−	0.996863i	$-0.474781\pi$
$113$	2.58091e7	0.158292	0.0791460	+	0.996863i	$0.474781\pi$
$114$	0	0
$115$	−2.69030e8	−1.53819
$116$	0	0
$117$	5.63353e7i	0.300634i
$118$	0	0
$119$	− 2.80309e6i	− 0.0139782i
$120$	0	0
$121$	2.10969e8	0.984185
$122$	0	0
$123$	6.86240e7	0.299817
$124$	0	0
$125$	− 8.09428e8i	− 3.31542i
$126$	0	0
$127$	− 1.05318e8i	− 0.404844i	−0.979298	−	0.202422i	$-0.935119\pi$
$127$	− 1.05318e8i	− 0.404844i	0.979298	−	0.202422i	$-0.0648812\pi$
$128$	0	0
$129$	−2.34098e8	−0.845356
$130$	0	0
$131$	−1.28724e8	−0.437093	−0.218547	−	0.975826i	$-0.570132\pi$
$131$	−1.28724e8	−0.437093	−0.218547	+	0.975826i	$0.570132\pi$
$132$	0	0
$133$	2.33489e8i	0.746209i
$134$	0	0
$135$	1.23275e8i	0.371142i
$136$	0	0
$137$	−3.20872e8	−0.910855	−0.455428	−	0.890273i	$-0.650514\pi$
$137$	−3.20872e8	−0.910855	−0.455428	+	0.890273i	$0.650514\pi$
$138$	0	0
$139$	1.98209e8	0.530964	0.265482	−	0.964116i	$-0.414469\pi$
$139$	1.98209e8	0.530964	0.265482	+	0.964116i	$0.414469\pi$
$140$	0	0
$141$	3.24373e8i	0.820670i
$142$	0	0
$143$	5.31244e8i	1.27043i
$144$	0	0
$145$	2.44306e8	0.552666
$146$	0	0
$147$	2.09515e8	0.448688
$148$	0	0
$149$	− 2.17324e8i	− 0.440922i	−0.975396	−	0.220461i	$-0.929244\pi$
$149$	− 2.17324e8i	− 0.440922i	0.975396	−	0.220461i	$-0.0707562\pi$
$150$	0	0
$151$	4.92507e8i	0.947338i	0.880703	+	0.473669i	$0.157071\pi$
$151$	4.92507e8i	0.947338i	−0.880703	+	0.473669i	$0.842929\pi$
$152$	0	0
$153$	5.40865e6	0.00987014
$154$	0	0
$155$	−1.40936e9	−2.44171
$156$	0	0
$157$	7.14935e8i	1.17671i	0.808604	+	0.588353i	$0.200223\pi$
$157$	7.14935e8i	1.17671i	−0.808604	+	0.588353i	$0.799777\pi$
$158$	0	0
$159$	5.44276e7i	0.0851590i
$160$	0	0
$161$	2.52985e8	0.376523
$162$	0	0
$163$	−8.75590e8	−1.24037	−0.620184	−	0.784457i	$-0.712942\pi$
$163$	−8.75590e8	−1.24037	−0.620184	+	0.784457i	$0.712942\pi$
$164$	0	0
$165$	1.16249e9i	1.56839i
$166$	0	0
$167$	3.17630e8i	0.408372i	0.978932	+	0.204186i	$0.0654548\pi$
$167$	3.17630e8i	0.408372i	−0.978932	+	0.204186i	$0.934545\pi$
$168$	0	0
$169$	1.52195e8	0.186575
$170$	0	0
$171$	−4.50525e8	−0.526907
$172$	0	0
$173$	− 1.05364e9i	− 1.17627i	−0.808762	−	0.588136i	$-0.799862\pi$
$173$	− 1.05364e9i	− 1.17627i	0.808762	−	0.588136i	$-0.200138\pi$
$174$	0	0
$175$	1.20390e9i	1.28363i
$176$	0	0
$177$	9.50984e8	0.968903
$178$	0	0
$179$	−5.57916e8	−0.543446	−0.271723	−	0.962375i	$-0.587594\pi$
$179$	−5.57916e8	−0.543446	−0.271723	+	0.962375i	$0.587594\pi$
$180$	0	0
$181$	− 6.12888e8i	− 0.571041i	−0.958373	−	0.285520i	$-0.907834\pi$
$181$	− 6.12888e8i	− 0.571041i	0.958373	−	0.285520i	$-0.0921664\pi$
$182$	0	0
$183$	1.43508e8i	0.127959i
$184$	0	0
$185$	3.16116e9	2.69873
$186$	0	0
$187$	5.10038e7	0.0417096
$188$	0	0
$189$	− 1.15923e8i	− 0.0908496i
$190$	0	0
$191$	1.36504e9i	1.02568i	0.858485	+	0.512839i	$0.171406\pi$
$191$	1.36504e9i	1.02568i	−0.858485	+	0.512839i	$0.828594\pi$
$192$	0	0
$193$	−7.89607e8	−0.569091	−0.284546	−	0.958663i	$-0.591843\pi$
$193$	−7.89607e8	−0.569091	−0.284546	+	0.958663i	$0.591843\pi$
$194$	0	0
$195$	−1.45197e9	−1.00420
$196$	0	0
$197$	− 1.90708e9i	− 1.26620i	−0.774068	−	0.633102i	$-0.781781\pi$
$197$	− 1.90708e9i	− 1.26620i	0.774068	−	0.633102i	$-0.218219\pi$
$198$	0	0
$199$	− 2.00160e9i	− 1.27634i	−0.769896	−	0.638170i	$-0.779692\pi$
$199$	− 2.00160e9i	− 1.27634i	0.769896	−	0.638170i	$-0.220308\pi$
$200$	0	0
$201$	−1.75029e8	−0.107232
$202$	0	0
$203$	−2.29736e8	−0.135284
$204$	0	0
$205$	1.76870e9i	1.00147i
$206$	0	0
$207$	4.88143e8i	0.265868i
$208$	0	0
$209$	−4.24846e9	−2.22662
$210$	0	0
$211$	−3.30961e9	−1.66973	−0.834867	−	0.550451i	$-0.814456\pi$
$211$	−3.30961e9	−1.66973	−0.834867	+	0.550451i	$0.814456\pi$
$212$	0	0
$213$	1.41711e9i	0.688470i
$214$	0	0
$215$	− 6.03360e9i	− 2.82372i
$216$	0	0
$217$	1.32531e9	0.597692
$218$	0	0
$219$	−8.48565e8	−0.368900
$220$	0	0
$221$	6.37048e7i	0.0267057i
$222$	0	0
$223$	− 2.94041e9i	− 1.18902i	−0.804089	−	0.594510i	$-0.797346\pi$
$223$	− 2.94041e9i	− 1.18902i	0.804089	−	0.594510i	$-0.202654\pi$
$224$	0	0
$225$	−2.32297e9	−0.906386
$226$	0	0
$227$	−1.84361e9	−0.694331	−0.347165	−	0.937804i	$-0.612856\pi$
$227$	−1.84361e9	−0.694331	−0.347165	+	0.937804i	$0.612856\pi$
$228$	0	0
$229$	− 3.87979e9i	− 1.41080i	−0.708808	−	0.705402i	$-0.750767\pi$
$229$	− 3.87979e9i	− 1.41080i	0.708808	−	0.705402i	$-0.249233\pi$
$230$	0	0
$231$	− 1.09316e9i	− 0.383915i
$232$	0	0
$233$	−3.75956e9	−1.27560	−0.637799	−	0.770203i	$-0.720155\pi$
$233$	−3.75956e9	−1.27560	−0.637799	+	0.770203i	$0.720155\pi$
$234$	0	0
$235$	−8.36032e9	−2.74127
$236$	0	0
$237$	− 2.89118e9i	− 0.916392i
$238$	0	0
$239$	1.01650e9i	0.311541i	0.987793	+	0.155770i	$0.0497860\pi$
$239$	1.01650e9i	0.311541i	−0.987793	+	0.155770i	$0.950214\pi$
$240$	0	0
$241$	2.04339e9	0.605735	0.302868	−	0.953033i	$-0.402056\pi$
$241$	2.04339e9	0.605735	0.302868	+	0.953033i	$0.402056\pi$
$242$	0	0
$243$	2.23677e8	0.0641500
$244$	0	0
$245$	5.39998e9i	1.49875i
$246$	0	0
$247$	− 5.30642e9i	− 1.42565i
$248$	0	0
$249$	−3.92502e8	−0.102104
$250$	0	0
$251$	2.92562e9	0.737093	0.368547	−	0.929609i	$-0.379855\pi$
$251$	2.92562e9	0.737093	0.368547	+	0.929609i	$0.379855\pi$
$252$	0	0
$253$	4.60320e9i	1.12351i
$254$	0	0
$255$	1.39401e8i	0.0329690i
$256$	0	0
$257$	8.47836e9	1.94348	0.971739	−	0.236059i	$-0.0758559\pi$
$257$	8.47836e9	1.94348	0.971739	+	0.236059i	$0.0758559\pi$
$258$	0	0
$259$	−2.97264e9	−0.660606
$260$	0	0
$261$	− 4.43283e8i	− 0.0955255i
$262$	0	0
$263$	7.09637e9i	1.48325i	0.670817	+	0.741623i	$0.265944\pi$
$263$	7.09637e9i	1.48325i	−0.670817	+	0.741623i	$0.734056\pi$
$264$	0	0
$265$	−1.40280e9	−0.284455
$266$	0	0
$267$	8.10450e8	0.159471
$268$	0	0
$269$	7.19620e8i	0.137434i	0.997636	+	0.0687170i	$0.0218906\pi$
$269$	7.19620e8i	0.137434i	−0.997636	+	0.0687170i	$0.978109\pi$
$270$	0	0
$271$	2.70234e9i	0.501029i	0.968113	+	0.250514i	$0.0805997\pi$
$271$	2.70234e9i	0.501029i	−0.968113	+	0.250514i	$0.919400\pi$
$272$	0	0
$273$	1.36538e9	0.245812
$274$	0	0
$275$	−2.19057e10	−3.83024
$276$	0	0
$277$	− 6.00431e9i	− 1.01987i	−0.860213	−	0.509934i	$-0.829670\pi$
$277$	− 6.00431e9i	− 1.01987i	0.860213	−	0.509934i	$-0.170330\pi$
$278$	0	0
$279$	2.55722e9i	0.422038i
$280$	0	0
$281$	−8.33314e9	−1.33654	−0.668272	−	0.743917i	$-0.732966\pi$
$281$	−8.33314e9	−1.33654	−0.668272	+	0.743917i	$0.732966\pi$
$282$	0	0
$283$	−2.01898e9	−0.314765	−0.157382	−	0.987538i	$-0.550306\pi$
$283$	−2.01898e9	−0.314765	−0.157382	+	0.987538i	$0.550306\pi$
$284$	0	0
$285$	− 1.16117e10i	− 1.76002i
$286$	0	0
$287$	− 1.66322e9i	− 0.245144i
$288$	0	0
$289$	−6.96964e9	−0.999123
$290$	0	0
$291$	−6.29810e9	−0.878289
$292$	0	0
$293$	1.22776e10i	1.66588i	0.553362	+	0.832941i	$0.313345\pi$
$293$	1.22776e10i	1.66588i	−0.553362	+	0.832941i	$0.686655\pi$
$294$	0	0
$295$	2.45104e10i	3.23641i
$296$	0	0
$297$	2.10928e9	0.271087
$298$	0	0
$299$	−5.74950e9	−0.719359
$300$	0	0
$301$	5.67376e9i	0.691202i
$302$	0	0
$303$	− 6.31555e9i	− 0.749274i
$304$	0	0
$305$	−3.69873e9	−0.427418
$306$	0	0
$307$	4.15941e9	0.468251	0.234125	−	0.972206i	$-0.424777\pi$
$307$	4.15941e9	0.468251	0.234125	+	0.972206i	$0.424777\pi$
$308$	0	0
$309$	3.14068e9i	0.344500i
$310$	0	0
$311$	3.21765e9i	0.343951i	0.985101	+	0.171976i	$0.0550151\pi$
$311$	3.21765e9i	0.343951i	−0.985101	+	0.171976i	$0.944985\pi$
$312$	0	0
$313$	−5.39012e9	−0.561592	−0.280796	−	0.959767i	$-0.590598\pi$
$313$	−5.39012e9	−0.561592	−0.280796	+	0.959767i	$0.590598\pi$
$314$	0	0
$315$	2.98778e9	0.303463
$316$	0	0
$317$	1.15002e10i	1.13886i	0.822041	+	0.569428i	$0.192835\pi$
$317$	1.15002e10i	1.13886i	−0.822041	+	0.569428i	$0.807165\pi$
$318$	0	0
$319$	− 4.18018e9i	− 0.403675i
$320$	0	0
$321$	−3.34136e9	−0.314704
$322$	0	0
$323$	−5.09460e8	−0.0468058
$324$	0	0
$325$	− 2.73607e10i	− 2.45241i
$326$	0	0
$327$	1.16557e10i	1.01941i
$328$	0	0
$329$	7.86172e9	0.671018
$330$	0	0
$331$	6.39319e9	0.532606	0.266303	−	0.963889i	$-0.414198\pi$
$331$	6.39319e9	0.532606	0.266303	+	0.963889i	$0.414198\pi$
$332$	0	0
$333$	− 5.73579e9i	− 0.466462i
$334$	0	0
$335$	− 4.51115e9i	− 0.358185i
$336$	0	0
$337$	2.36857e10	1.83640	0.918198	−	0.396122i	$-0.129644\pi$
$337$	2.36857e10	1.83640	0.918198	+	0.396122i	$0.129644\pi$
$338$	0	0
$339$	1.20697e9	0.0913899
$340$	0	0
$341$	2.41147e10i	1.78346i
$342$	0	0
$343$	− 1.16120e10i	− 0.838937i
$344$	0	0
$345$	−1.25813e10	−0.888073
$346$	0	0
$347$	−2.00975e10	−1.38619	−0.693096	−	0.720845i	$-0.743754\pi$
$347$	−2.00975e10	−1.38619	−0.693096	+	0.720845i	$0.743754\pi$
$348$	0	0
$349$	− 1.88548e9i	− 0.127093i	−0.997979	−	0.0635464i	$-0.979759\pi$
$349$	− 1.88548e9i	− 0.127093i	0.997979	−	0.0635464i	$-0.0202411\pi$
$350$	0	0
$351$	2.63454e9i	0.173571i
$352$	0	0
$353$	−1.45724e10	−0.938496	−0.469248	−	0.883066i	$-0.655475\pi$
$353$	−1.45724e10	−0.938496	−0.469248	+	0.883066i	$0.655475\pi$
$354$	0	0
$355$	−3.65242e10	−2.29968
$356$	0	0
$357$	− 1.31088e8i	− 0.00807029i
$358$	0	0
$359$	1.57991e10i	0.951163i	0.879672	+	0.475581i	$0.157762\pi$
$359$	1.57991e10i	0.951163i	−0.879672	+	0.475581i	$0.842238\pi$
$360$	0	0
$361$	2.54529e10	1.49868
$362$	0	0
$363$	9.86604e9	0.568220
$364$	0	0
$365$	− 2.18707e10i	− 1.23223i
$366$	0	0
$367$	− 3.24057e9i	− 0.178631i	−0.996003	−	0.0893156i	$-0.971532\pi$
$367$	− 3.24057e9i	− 0.178631i	0.996003	−	0.0893156i	$-0.0284680\pi$
$368$	0	0
$369$	3.20923e9	0.173099
$370$	0	0
$371$	1.31914e9	0.0696300
$372$	0	0
$373$	− 2.27941e10i	− 1.17757i	−0.808288	−	0.588787i	$-0.799606\pi$
$373$	− 2.27941e10i	− 1.17757i	0.808288	−	0.588787i	$-0.200394\pi$
$374$	0	0
$375$	− 3.78532e10i	− 1.91416i
$376$	0	0
$377$	5.22113e9	0.258464
$378$	0	0
$379$	2.83101e10	1.37210	0.686049	−	0.727556i	$-0.259344\pi$
$379$	2.83101e10	1.37210	0.686049	+	0.727556i	$0.259344\pi$
$380$	0	0
$381$	− 4.92523e9i	− 0.233737i
$382$	0	0
$383$	− 3.54736e10i	− 1.64858i	−0.566169	−	0.824289i	$-0.691575\pi$
$383$	− 3.54736e10i	− 1.64858i	0.566169	−	0.824289i	$-0.308425\pi$
$384$	0	0
$385$	2.81748e10	1.28239
$386$	0	0
$387$	−1.09477e10	−0.488066
$388$	0	0
$389$	1.40513e10i	0.613646i	0.951767	+	0.306823i	$0.0992659\pi$
$389$	1.40513e10i	0.613646i	−0.951767	+	0.306823i	$0.900734\pi$
$390$	0	0
$391$	5.51999e8i	0.0236174i
$392$	0	0
$393$	−6.01982e9	−0.252356
$394$	0	0
$395$	7.45165e10	3.06100
$396$	0	0
$397$	3.82104e10i	1.53822i	0.639115	+	0.769111i	$0.279301\pi$
$397$	3.82104e10i	1.53822i	−0.639115	+	0.769111i	$0.720699\pi$
$398$	0	0
$399$	1.09192e10i	0.430824i
$400$	0	0
$401$	2.13716e10	0.826530	0.413265	−	0.910611i	$-0.364388\pi$
$401$	2.13716e10	0.826530	0.413265	+	0.910611i	$0.364388\pi$
$402$	0	0
$403$	−3.01197e10	−1.14191
$404$	0	0
$405$	5.76501e9i	0.214279i
$406$	0	0
$407$	− 5.40887e10i	− 1.97119i
$408$	0	0
$409$	−1.90485e10	−0.680717	−0.340359	−	0.940296i	$-0.610548\pi$
$409$	−1.90485e10	−0.680717	−0.340359	+	0.940296i	$0.610548\pi$
$410$	0	0
$411$	−1.50057e10	−0.525882
$412$	0	0
$413$	− 2.30487e10i	− 0.792220i
$414$	0	0
$415$	− 1.01163e10i	− 0.341057i
$416$	0	0
$417$	9.26933e9	0.306552
$418$	0	0
$419$	3.04260e9	0.0987163	0.0493582	−	0.998781i	$-0.484282\pi$
$419$	3.04260e9	0.0987163	0.0493582	+	0.998781i	$0.484282\pi$
$420$	0	0
$421$	3.68586e10i	1.17330i	0.809839	+	0.586652i	$0.199554\pi$
$421$	3.68586e10i	1.17330i	−0.809839	+	0.586652i	$0.800446\pi$
$422$	0	0
$423$	1.51694e10i	0.473814i
$424$	0	0
$425$	−2.62685e9	−0.0805154
$426$	0	0
$427$	3.47814e9	0.104625
$428$	0	0
$429$	2.48438e10i	0.733482i
$430$	0	0
$431$	5.30543e10i	1.53749i	0.639557	+	0.768744i	$0.279118\pi$
$431$	5.30543e10i	1.53749i	−0.639557	+	0.768744i	$0.720882\pi$
$432$	0	0
$433$	−2.82756e10	−0.804379	−0.402190	−	0.915556i	$-0.631751\pi$
$433$	−2.82756e10	−0.804379	−0.402190	+	0.915556i	$0.631751\pi$
$434$	0	0
$435$	1.14251e10	0.319082
$436$	0	0
$437$	− 4.59799e10i	− 1.26079i
$438$	0	0
$439$	− 6.30601e10i	− 1.69784i	−0.528522	−	0.848920i	$-0.677254\pi$
$439$	− 6.30601e10i	− 1.69784i	0.528522	−	0.848920i	$-0.322746\pi$
$440$	0	0
$441$	9.79803e9	0.259050
$442$	0	0
$443$	3.71295e10	0.964060	0.482030	−	0.876155i	$-0.339900\pi$
$443$	3.71295e10	0.964060	0.482030	+	0.876155i	$0.339900\pi$
$444$	0	0
$445$	2.08883e10i	0.532677i
$446$	0	0
$447$	− 1.01632e10i	− 0.254566i
$448$	0	0
$449$	−3.04464e10	−0.749118	−0.374559	−	0.927203i	$-0.622206\pi$
$449$	−3.04464e10	−0.749118	−0.374559	+	0.927203i	$0.622206\pi$
$450$	0	0
$451$	3.02631e10	0.731488
$452$	0	0
$453$	2.30323e10i	0.546946i
$454$	0	0
$455$	3.51910e10i	0.821081i
$456$	0	0
$457$	2.91731e10	0.668832	0.334416	−	0.942425i	$-0.391461\pi$
$457$	2.91731e10	0.668832	0.334416	+	0.942425i	$0.391461\pi$
$458$	0	0
$459$	2.52938e8	0.00569853
$460$	0	0
$461$	− 2.33681e10i	− 0.517392i	−0.965959	−	0.258696i	$-0.916707\pi$
$461$	− 2.33681e10i	− 0.517392i	0.965959	−	0.258696i	$-0.0832929\pi$
$462$	0	0
$463$	8.43080e10i	1.83461i	0.398181	+	0.917307i	$0.369642\pi$
$463$	8.43080e10i	1.83461i	−0.398181	+	0.917307i	$0.630358\pi$
$464$	0	0
$465$	−6.59092e10	−1.40972
$466$	0	0
$467$	−7.98604e9	−0.167905	−0.0839526	−	0.996470i	$-0.526754\pi$
$467$	−7.98604e9	−0.167905	−0.0839526	+	0.996470i	$0.526754\pi$
$468$	0	0
$469$	4.24211e9i	0.0876780i
$470$	0	0
$471$	3.34342e10i	0.679371i
$472$	0	0
$473$	−1.03237e11	−2.06249
$474$	0	0
$475$	2.18808e11	4.29823
$476$	0	0
$477$	2.54533e9i	0.0491666i
$478$	0	0
$479$	5.14200e10i	0.976765i	0.872630	+	0.488382i	$0.162413\pi$
$479$	5.14200e10i	0.976765i	−0.872630	+	0.488382i	$0.837587\pi$
$480$	0	0
$481$	6.75580e10	1.26211
$482$	0	0
$483$	1.18310e10	0.217386
$484$	0	0
$485$	− 1.62326e11i	− 2.93373i
$486$	0	0
$487$	6.06516e10i	1.07827i	0.842220	+	0.539134i	$0.181248\pi$
$487$	6.06516e10i	1.07827i	−0.842220	+	0.539134i	$0.818752\pi$
$488$	0	0
$489$	−4.09473e10	−0.716126
$490$	0	0
$491$	7.27344e10	1.25145	0.625726	−	0.780043i	$-0.284803\pi$
$491$	7.27344e10	1.25145	0.625726	+	0.780043i	$0.284803\pi$
$492$	0	0
$493$	− 5.01271e8i	− 0.00848565i
$494$	0	0
$495$	5.43642e10i	0.905508i
$496$	0	0
$497$	3.43460e10	0.562925
$498$	0	0
$499$	−5.93007e10	−0.956439	−0.478220	−	0.878240i	$-0.658718\pi$
$499$	−5.93007e10	−0.956439	−0.478220	+	0.878240i	$0.658718\pi$
$500$	0	0
$501$	1.48541e10i	0.235774i
$502$	0	0
$503$	1.07478e10i	0.167899i	0.996470	+	0.0839495i	$0.0267535\pi$
$503$	1.07478e10i	0.167899i	−0.996470	+	0.0839495i	$0.973247\pi$
$504$	0	0
$505$	1.62776e11	2.50279
$506$	0	0
$507$	7.11746e9	0.107719
$508$	0	0
$509$	1.30363e9i	0.0194215i	0.999953	+	0.00971074i	$0.00309107\pi$
$509$	1.30363e9i	0.0194215i	−0.999953	+	0.00971074i	$0.996909\pi$
$510$	0	0
$511$	2.05664e10i	0.301630i
$512$	0	0
$513$	−2.10690e10	−0.304210
$514$	0	0
$515$	−8.09471e10	−1.15073
$516$	0	0
$517$	1.43048e11i	2.00226i
$518$	0	0
$519$	− 4.92739e10i	− 0.679121i
$520$	0	0
$521$	3.82717e10	0.519429	0.259715	−	0.965685i	$-0.416371\pi$
$521$	3.82717e10	0.519429	0.259715	+	0.965685i	$0.416371\pi$
$522$	0	0
$523$	1.70009e10	0.227229	0.113615	−	0.993525i	$-0.463757\pi$
$523$	1.70009e10	0.227229	0.113615	+	0.993525i	$0.463757\pi$
$524$	0	0
$525$	5.63010e10i	0.741104i
$526$	0	0
$527$	2.89174e9i	0.0374901i
$528$	0	0
$529$	2.84918e10	0.363829
$530$	0	0
$531$	4.44731e10	0.559396
$532$	0	0
$533$	3.77993e10i	0.468355i
$534$	0	0
$535$	− 8.61193e10i	− 1.05120i
$536$	0	0
$537$	−2.60911e10	−0.313759
$538$	0	0
$539$	9.23957e10	1.09470
$540$	0	0
$541$	− 1.28391e11i	− 1.49881i	−0.662111	−	0.749406i	$-0.730339\pi$
$541$	− 1.28391e11i	− 1.49881i	0.662111	−	0.749406i	$-0.269661\pi$
$542$	0	0
$543$	− 2.86619e10i	− 0.329690i
$544$	0	0
$545$	−3.00411e11	−3.40510
$546$	0	0
$547$	8.72133e10	0.974167	0.487083	−	0.873356i	$-0.338061\pi$
$547$	8.72133e10	0.974167	0.487083	+	0.873356i	$0.338061\pi$
$548$	0	0
$549$	6.71118e9i	0.0738771i
$550$	0	0
$551$	4.17544e10i	0.452998i
$552$	0	0
$553$	−7.00725e10	−0.749285
$554$	0	0
$555$	1.47833e11	1.55811
$556$	0	0
$557$	− 3.76332e10i	− 0.390977i	−0.980706	−	0.195488i	$-0.937371\pi$
$557$	− 3.76332e10i	− 0.390977i	0.980706	−	0.195488i	$-0.0626292\pi$
$558$	0	0
$559$	− 1.28945e11i	− 1.32056i
$560$	0	0
$561$	2.38521e9	0.0240810
$562$	0	0
$563$	7.92375e10	0.788674	0.394337	−	0.918966i	$-0.370974\pi$
$563$	7.92375e10	0.788674	0.394337	+	0.918966i	$0.370974\pi$
$564$	0	0
$565$	3.11082e10i	0.305268i
$566$	0	0
$567$	− 5.42119e9i	− 0.0524521i
$568$	0	0
$569$	1.50563e11	1.43638	0.718191	−	0.695846i	$-0.244970\pi$
$569$	1.50563e11	1.43638	0.718191	+	0.695846i	$0.244970\pi$
$570$	0	0
$571$	−3.89532e10	−0.366437	−0.183218	−	0.983072i	$-0.558652\pi$
$571$	−3.89532e10	−0.366437	−0.183218	+	0.983072i	$0.558652\pi$
$572$	0	0
$573$	6.38365e10i	0.592176i
$574$	0	0
$575$	− 2.37079e11i	− 2.16881i
$576$	0	0
$577$	−7.76797e10	−0.700817	−0.350408	−	0.936597i	$-0.613957\pi$
$577$	−7.76797e10	−0.700817	−0.350408	+	0.936597i	$0.613957\pi$
$578$	0	0
$579$	−3.69263e10	−0.328565
$580$	0	0
$581$	9.51294e9i	0.0834854i
$582$	0	0
$583$	2.40025e10i	0.207770i
$584$	0	0
$585$	−6.79021e10	−0.579775
$586$	0	0
$587$	1.18050e11	0.994293	0.497146	−	0.867667i	$-0.334381\pi$
$587$	1.18050e11	0.994293	0.497146	+	0.867667i	$0.334381\pi$
$588$	0	0
$589$	− 2.40873e11i	− 2.00137i
$590$	0	0
$591$	− 8.91853e10i	− 0.731044i
$592$	0	0
$593$	−1.28434e10	−0.103863	−0.0519317	−	0.998651i	$-0.516538\pi$
$593$	−1.28434e10	−0.103863	−0.0519317	+	0.998651i	$0.516538\pi$
$594$	0	0
$595$	3.37862e9	0.0269570
$596$	0	0
$597$	− 9.36058e10i	− 0.736895i
$598$	0	0
$599$	− 4.09653e10i	− 0.318206i	−0.987262	−	0.159103i	$-0.949140\pi$
$599$	− 4.09653e10i	− 0.318206i	0.987262	−	0.159103i	$-0.0508602\pi$
$600$	0	0
$601$	8.68532e10	0.665714	0.332857	−	0.942977i	$-0.391987\pi$
$601$	8.68532e10	0.665714	0.332857	+	0.942977i	$0.391987\pi$
$602$	0	0
$603$	−8.18527e9	−0.0619105
$604$	0	0
$605$	2.54285e11i	1.89801i
$606$	0	0
$607$	− 1.20813e11i	− 0.889934i	−0.895547	−	0.444967i	$-0.853215\pi$
$607$	− 1.20813e11i	− 0.889934i	0.895547	−	0.444967i	$-0.146785\pi$
$608$	0	0
$609$	−1.07437e10	−0.0781061
$610$	0	0
$611$	−1.78670e11	−1.28200
$612$	0	0
$613$	1.16441e11i	0.824636i	0.911040	+	0.412318i	$0.135281\pi$
$613$	1.16441e11i	0.824636i	−0.911040	+	0.412318i	$0.864719\pi$
$614$	0	0
$615$	8.27139e10i	0.578200i
$616$	0	0
$617$	−4.81037e10	−0.331923	−0.165962	−	0.986132i	$-0.553073\pi$
$617$	−4.81037e10	−0.331923	−0.165962	+	0.986132i	$0.553073\pi$
$618$	0	0
$619$	2.05490e11	1.39968	0.699838	−	0.714301i	$-0.253255\pi$
$619$	2.05490e11	1.39968	0.699838	+	0.714301i	$0.253255\pi$
$620$	0	0
$621$	2.28282e10i	0.153499i
$622$	0	0
$623$	− 1.96426e10i	− 0.130391i
$624$	0	0
$625$	5.60709e11	3.67466
$626$	0	0
$627$	−1.98681e11	−1.28554
$628$	0	0
$629$	− 6.48612e9i	− 0.0414364i
$630$	0	0
$631$	2.45214e11i	1.54678i	0.633932	+	0.773389i	$0.281440\pi$
$631$	2.45214e11i	1.54678i	−0.633932	+	0.773389i	$0.718560\pi$
$632$	0	0
$633$	−1.54775e11	−0.964022
$634$	0	0
$635$	1.26942e11	0.780746
$636$	0	0
$637$	1.15404e11i	0.700913i
$638$	0	0
$639$	6.62716e10i	0.397488i
$640$	0	0
$641$	6.07017e10	0.359558	0.179779	−	0.983707i	$-0.442462\pi$
$641$	6.07017e10	0.359558	0.179779	+	0.983707i	$0.442462\pi$
$642$	0	0
$643$	−7.52262e10	−0.440074	−0.220037	−	0.975492i	$-0.570618\pi$
$643$	−7.52262e10	−0.440074	−0.220037	+	0.975492i	$0.570618\pi$
$644$	0	0
$645$	− 2.82163e11i	− 1.63028i
$646$	0	0
$647$	− 3.65466e10i	− 0.208559i	−0.994548	−	0.104280i	$-0.966746\pi$
$647$	− 3.65466e10i	− 0.208559i	0.994548	−	0.104280i	$-0.0332537\pi$
$648$	0	0
$649$	4.19383e11	2.36392
$650$	0	0
$651$	6.19785e10	0.345078
$652$	0	0
$653$	2.44063e11i	1.34230i	0.741322	+	0.671150i	$0.234199\pi$
$653$	2.44063e11i	1.34230i	−0.741322	+	0.671150i	$0.765801\pi$
$654$	0	0
$655$	− 1.55154e11i	− 0.842940i
$656$	0	0
$657$	−3.96835e10	−0.212985
$658$	0	0
$659$	1.68123e11	0.891427	0.445714	−	0.895176i	$-0.352950\pi$
$659$	1.68123e11	0.891427	0.445714	+	0.895176i	$0.352950\pi$
$660$	0	0
$661$	− 5.28700e10i	− 0.276951i	−0.990366	−	0.138476i	$-0.955780\pi$
$661$	− 5.28700e10i	− 0.276951i	0.990366	−	0.138476i	$-0.0442203\pi$
$662$	0	0
$663$	2.97918e9i	0.0154185i
$664$	0	0
$665$	−2.81429e11	−1.43907
$666$	0	0
$667$	4.52409e10	0.228574
$668$	0	0
$669$	− 1.37509e11i	− 0.686480i
$670$	0	0
$671$	6.32866e10i	0.312192i
$672$	0	0
$673$	2.17550e11	1.06047	0.530237	−	0.847850i	$-0.322103\pi$
$673$	2.17550e11	1.06047	0.530237	+	0.847850i	$0.322103\pi$
$674$	0	0
$675$	−1.08634e11	−0.523302
$676$	0	0
$677$	1.10635e11i	0.526668i	0.964705	+	0.263334i	$0.0848222\pi$
$677$	1.10635e11i	0.526668i	−0.964705	+	0.263334i	$0.915178\pi$
$678$	0	0
$679$	1.52645e11i	0.718130i
$680$	0	0
$681$	−8.62172e10	−0.400872
$682$	0	0
$683$	−9.80246e10	−0.450456	−0.225228	−	0.974306i	$-0.572313\pi$
$683$	−9.80246e10	−0.450456	−0.225228	+	0.974306i	$0.572313\pi$
$684$	0	0
$685$	− 3.86753e11i	− 1.75659i
$686$	0	0
$687$	− 1.81440e11i	− 0.814528i
$688$	0	0
$689$	−2.99797e10	−0.133030
$690$	0	0
$691$	−2.42053e11	−1.06169	−0.530846	−	0.847468i	$-0.678126\pi$
$691$	−2.42053e11	−1.06169	−0.530846	+	0.847468i	$0.678126\pi$
$692$	0	0
$693$	− 5.11220e10i	− 0.221654i
$694$	0	0
$695$	2.38906e11i	1.02397i
$696$	0	0
$697$	3.62904e9	0.0153766
$698$	0	0
$699$	−1.75817e11	−0.736467
$700$	0	0
$701$	6.51496e10i	0.269798i	0.990859	+	0.134899i	$0.0430711\pi$
$701$	6.51496e10i	0.269798i	−0.990859	+	0.134899i	$0.956929\pi$
$702$	0	0
$703$	5.40274e11i	2.21204i
$704$	0	0
$705$	−3.90973e11	−1.58267
$706$	0	0
$707$	−1.53068e11	−0.612641
$708$	0	0
$709$	1.77744e11i	0.703412i	0.936111	+	0.351706i	$0.114398\pi$
$709$	1.77744e11i	0.703412i	−0.936111	+	0.351706i	$0.885602\pi$
$710$	0	0
$711$	− 1.35207e11i	− 0.529079i
$712$	0	0
$713$	−2.60986e11	−1.00986
$714$	0	0
$715$	−6.40319e11	−2.45003
$716$	0	0
$717$	4.75369e10i	0.179868i
$718$	0	0
$719$	4.48867e11i	1.67958i	0.542908	+	0.839792i	$0.317323\pi$
$719$	4.48867e11i	1.67958i	−0.542908	+	0.839792i	$0.682677\pi$
$720$	0	0
$721$	7.61196e10	0.281680
$722$	0	0
$723$	9.55598e10	0.349721
$724$	0	0
$725$	2.15292e11i	0.779247i
$726$	0	0
$727$	1.46455e11i	0.524284i	0.965029	+	0.262142i	$0.0844289\pi$
$727$	1.46455e11i	0.524284i	−0.965029	+	0.262142i	$0.915571\pi$
$728$	0	0
$729$	1.04604e10	0.0370370
$730$	0	0
$731$	−1.23798e10	−0.0433555
$732$	0	0
$733$	− 1.76979e11i	− 0.613063i	−0.951861	−	0.306531i	$-0.900832\pi$
$733$	− 1.76979e11i	− 0.613063i	0.951861	−	0.306531i	$-0.0991684\pi$
$734$	0	0
$735$	2.52532e11i	0.865301i
$736$	0	0
$737$	−7.71874e10	−0.261623
$738$	0	0
$739$	−3.79399e10	−0.127209	−0.0636046	−	0.997975i	$-0.520260\pi$
$739$	−3.79399e10	−0.127209	−0.0636046	+	0.997975i	$0.520260\pi$
$740$	0	0
$741$	− 2.48157e11i	− 0.823102i
$742$	0	0
$743$	− 4.78465e11i	− 1.56998i	−0.619507	−	0.784991i	$-0.712667\pi$
$743$	− 4.78465e11i	− 1.56998i	0.619507	−	0.784991i	$-0.287333\pi$
$744$	0	0
$745$	2.61944e11	0.850323
$746$	0	0
$747$	−1.83555e10	−0.0589500
$748$	0	0
$749$	8.09833e10i	0.257317i
$750$	0	0
$751$	5.44597e10i	0.171205i	0.996329	+	0.0856023i	$0.0272814\pi$
$751$	5.44597e10i	0.171205i	−0.996329	+	0.0856023i	$0.972719\pi$
$752$	0	0
$753$	1.36818e11	0.425561
$754$	0	0
$755$	−5.93629e11	−1.82695
$756$	0	0
$757$	− 4.87219e11i	− 1.48368i	−0.670575	−	0.741841i	$-0.733953\pi$
$757$	− 4.87219e11i	− 1.48368i	0.670575	−	0.741841i	$-0.266047\pi$
$758$	0	0
$759$	2.15270e11i	0.648660i
$760$	0	0
$761$	2.82246e11	0.841567	0.420783	−	0.907161i	$-0.361755\pi$
$761$	2.82246e11	0.841567	0.420783	+	0.907161i	$0.361755\pi$
$762$	0	0
$763$	2.82495e11	0.833514
$764$	0	0
$765$	6.51916e9i	0.0190347i
$766$	0	0
$767$	5.23819e11i	1.51356i
$768$	0	0
$769$	−1.56182e10	−0.0446607	−0.0223303	−	0.999751i	$-0.507109\pi$
$769$	−1.56182e10	−0.0446607	−0.0223303	+	0.999751i	$0.507109\pi$
$770$	0	0
$771$	3.96494e11	1.12207
$772$	0	0
$773$	2.70943e11i	0.758856i	0.925221	+	0.379428i	$0.123879\pi$
$773$	2.70943e11i	0.758856i	−0.925221	+	0.379428i	$0.876121\pi$
$774$	0	0
$775$	− 1.24198e12i	− 3.44276i
$776$	0	0
$777$	−1.39016e11	−0.381401
$778$	0	0
$779$	−3.02288e11	−0.820865
$780$	0	0
$781$	6.24943e11i	1.67972i
$782$	0	0
$783$	− 2.07303e10i	− 0.0551517i
$784$	0	0
$785$	−8.61725e11	−2.26929
$786$	0	0
$787$	−3.02095e11	−0.787488	−0.393744	−	0.919220i	$-0.628820\pi$
$787$	−3.02095e11	−0.787488	−0.393744	+	0.919220i	$0.628820\pi$
$788$	0	0
$789$	3.31864e11i	0.856353i
$790$	0	0
$791$	− 2.92530e10i	− 0.0747246i
$792$	0	0
$793$	−7.90465e10	−0.199889
$794$	0	0
$795$	−6.56026e10	−0.164230
$796$	0	0
$797$	− 3.70077e11i	− 0.917189i	−0.888646	−	0.458595i	$-0.848353\pi$
$797$	− 3.70077e11i	− 0.917189i	0.888646	−	0.458595i	$-0.151647\pi$
$798$	0	0
$799$	1.71538e10i	0.0420895i
$800$	0	0
$801$	3.79010e10	0.0920705
$802$	0	0
$803$	−3.74216e11	−0.900037
$804$	0	0
$805$	3.04928e11i	0.726130i
$806$	0	0
$807$	3.36533e10i	0.0793476i
$808$	0	0
$809$	−8.12768e9	−0.0189746	−0.00948730	−	0.999955i	$-0.503020\pi$
$809$	−8.12768e9	−0.0189746	−0.00948730	+	0.999955i	$0.503020\pi$
$810$	0	0
$811$	−4.82470e11	−1.11529	−0.557643	−	0.830081i	$-0.688294\pi$
$811$	−4.82470e11	−1.11529	−0.557643	+	0.830081i	$0.688294\pi$
$812$	0	0
$813$	1.26376e11i	0.289269i
$814$	0	0
$815$	− 1.05537e12i	− 2.39206i
$816$	0	0
$817$	1.03120e12	2.31449
$818$	0	0
$819$	6.38525e10	0.141920
$820$	0	0
$821$	− 3.20753e10i	− 0.0705989i	−0.999377	−	0.0352995i	$-0.988762\pi$
$821$	− 3.20753e10i	− 0.0705989i	0.999377	−	0.0352995i	$-0.0112385\pi$
$822$	0	0
$823$	9.03986e10i	0.197044i	0.995135	+	0.0985218i	$0.0314114\pi$
$823$	9.03986e10i	0.197044i	−0.995135	+	0.0985218i	$0.968589\pi$
$824$	0	0
$825$	−1.02443e12	−2.21139
$826$	0	0
$827$	−8.40230e11	−1.79629	−0.898144	−	0.439701i	$-0.855084\pi$
$827$	−8.40230e11	−1.79629	−0.898144	+	0.439701i	$0.855084\pi$
$828$	0	0
$829$	− 1.29286e11i	− 0.273737i	−0.990589	−	0.136868i	$-0.956296\pi$
$829$	− 1.29286e11i	− 0.273737i	0.990589	−	0.136868i	$-0.0437038\pi$
$830$	0	0
$831$	− 2.80794e11i	− 0.588821i
$832$	0	0
$833$	1.10798e10	0.0230118
$834$	0	0
$835$	−3.82846e11	−0.787551
$836$	0	0
$837$	1.19589e11i	0.243664i
$838$	0	0
$839$	− 6.65798e11i	− 1.34368i	−0.740698	−	0.671838i	$-0.765505\pi$
$839$	− 6.65798e11i	− 1.34368i	0.740698	−	0.671838i	$-0.234495\pi$
$840$	0	0
$841$	4.59163e11	0.917874
$842$	0	0
$843$	−3.89702e11	−0.771654
$844$	0	0
$845$	1.83444e11i	0.359812i
$846$	0	0
$847$	− 2.39120e11i	− 0.464603i
$848$	0	0
$849$	−9.44183e10	−0.181730
$850$	0	0
$851$	5.85386e11	1.11615
$852$	0	0
$853$	− 2.81152e11i	− 0.531062i	−0.964102	−	0.265531i	$-0.914453\pi$
$853$	− 2.81152e11i	− 0.531062i	0.964102	−	0.265531i	$-0.0855473\pi$
$854$	0	0
$855$	− 5.43026e11i	− 1.01615i
$856$	0	0
$857$	−7.53826e11	−1.39749	−0.698744	−	0.715372i	$-0.746257\pi$
$857$	−7.53826e11	−1.39749	−0.698744	+	0.715372i	$0.746257\pi$
$858$	0	0
$859$	−1.75148e11	−0.321687	−0.160843	−	0.986980i	$-0.551421\pi$
$859$	−1.75148e11	−0.321687	−0.160843	+	0.986980i	$0.551421\pi$
$860$	0	0
$861$	− 7.77810e10i	− 0.141534i
$862$	0	0
$863$	− 2.00425e11i	− 0.361334i	−0.983544	−	0.180667i	$-0.942174\pi$
$863$	− 2.00425e11i	− 0.361334i	0.983544	−	0.180667i	$-0.0578257\pi$
$864$	0	0
$865$	1.26997e12	2.26845
$866$	0	0
$867$	−3.25938e11	−0.576844
$868$	0	0
$869$	− 1.27501e12i	− 2.23580i
$870$	0	0
$871$	− 9.64088e10i	− 0.167511i
$872$	0	0
$873$	−2.94533e11	−0.507080
$874$	0	0
$875$	−9.17435e11	−1.56510
$876$	0	0
$877$	− 3.29390e11i	− 0.556816i	−0.960463	−	0.278408i	$-0.910193\pi$
$877$	− 3.29390e11i	− 0.556816i	0.960463	−	0.278408i	$-0.0898068\pi$
$878$	0	0
$879$	5.74168e11i	0.961797i
$880$	0	0
$881$	−8.87218e11	−1.47274	−0.736371	−	0.676578i	$-0.763462\pi$
$881$	−8.87218e11	−1.47274	−0.736371	+	0.676578i	$0.763462\pi$
$882$	0	0
$883$	8.69085e11	1.42962	0.714808	−	0.699321i	$-0.246514\pi$
$883$	8.69085e11	1.42962	0.714808	+	0.699321i	$0.246514\pi$
$884$	0	0
$885$	1.14624e12i	1.86854i
$886$	0	0
$887$	1.19260e12i	1.92663i	0.268364	+	0.963317i	$0.413517\pi$
$887$	1.19260e12i	1.92663i	−0.268364	+	0.963317i	$0.586483\pi$
$888$	0	0
$889$	−1.19371e11	−0.191114
$890$	0	0
$891$	9.86414e10	0.156512
$892$	0	0
$893$	− 1.42886e12i	− 2.24690i
$894$	0	0
$895$	− 6.72467e11i	− 1.04804i
$896$	0	0
$897$	−2.68878e11	−0.415322
$898$	0	0
$899$	2.37002e11	0.362838
$900$	0	0
$901$	2.87829e9i	0.00436753i
$902$	0	0
$903$	2.65336e11i	0.399066i
$904$	0	0
$905$	7.38726e11	1.10126
$906$	0	0
$907$	5.06869e11	0.748974	0.374487	−	0.927232i	$-0.377819\pi$
$907$	5.06869e11	0.748974	0.374487	+	0.927232i	$0.377819\pi$
$908$	0	0
$909$	− 2.95349e11i	− 0.432594i
$910$	0	0
$911$	6.36648e11i	0.924327i	0.886795	+	0.462164i	$0.152927\pi$
$911$	6.36648e11i	0.924327i	−0.886795	+	0.462164i	$0.847073\pi$
$912$	0	0
$913$	−1.73093e11	−0.249113
$914$	0	0
$915$	−1.72972e11	−0.246770
$916$	0	0
$917$	1.45900e11i	0.206338i
$918$	0	0
$919$	1.02943e12i	1.44322i	0.692299	+	0.721611i	$0.256598\pi$
$919$	1.02943e12i	1.44322i	−0.692299	+	0.721611i	$0.743402\pi$
$920$	0	0
$921$	1.94516e11	0.270345
$922$	0	0
$923$	−7.80568e11	−1.07548
$924$	0	0
$925$	2.78573e12i	3.80515i
$926$	0	0
$927$	1.46875e11i	0.198897i
$928$	0	0
$929$	−9.00103e11	−1.20845	−0.604226	−	0.796813i	$-0.706518\pi$
$929$	−9.00103e11	−1.20845	−0.604226	+	0.796813i	$0.706518\pi$
$930$	0	0
$931$	−9.22911e11	−1.22846
$932$	0	0
$933$	1.50475e11i	0.198580i
$934$	0	0
$935$	6.14758e10i	0.0804374i
$936$	0	0
$937$	−1.42921e11	−0.185412	−0.0927060	−	0.995694i	$-0.529552\pi$
$937$	−1.42921e11	−0.185412	−0.0927060	+	0.995694i	$0.529552\pi$
$938$	0	0
$939$	−2.52071e11	−0.324235
$940$	0	0
$941$	− 2.35339e11i	− 0.300148i	−0.988675	−	0.150074i	$-0.952049\pi$
$941$	− 2.35339e11i	− 0.300148i	0.988675	−	0.150074i	$-0.0479511\pi$
$942$	0	0
$943$	3.27529e11i	0.414193i
$944$	0	0
$945$	1.39725e11	0.175205
$946$	0	0
$947$	7.13889e11	0.887628	0.443814	−	0.896119i	$-0.353625\pi$
$947$	7.13889e11	0.887628	0.443814	+	0.896119i	$0.353625\pi$
$948$	0	0
$949$	− 4.67405e11i	− 0.576273i
$950$	0	0
$951$	5.37811e11i	0.657518i
$952$	0	0
$953$	9.16680e11	1.11134	0.555669	−	0.831403i	$-0.312462\pi$
$953$	9.16680e11	1.11134	0.555669	+	0.831403i	$0.312462\pi$
$954$	0	0
$955$	−1.64531e12	−1.97803
$956$	0	0
$957$	− 1.95487e11i	− 0.233062i
$958$	0	0
$959$	3.63688e11i	0.429986i
$960$	0	0
$961$	−5.14329e11	−0.603042
$962$	0	0
$963$	−1.56260e11	−0.181695
$964$	0	0
$965$	− 9.51729e11i	− 1.09750i
$966$	0	0
$967$	1.25148e12i	1.43126i	0.698480	+	0.715630i	$0.253860\pi$
$967$	1.25148e12i	1.43126i	−0.698480	+	0.715630i	$0.746140\pi$
$968$	0	0
$969$	−2.38251e10	−0.0270234
$970$	0	0
$971$	1.49155e12	1.67788	0.838938	−	0.544227i	$-0.183177\pi$
$971$	1.49155e12	1.67788	0.838938	+	0.544227i	$0.183177\pi$
$972$	0	0
$973$	− 2.24658e11i	− 0.250651i
$974$	0	0
$975$	− 1.27953e12i	− 1.41590i
$976$	0	0
$977$	1.16356e12	1.27705	0.638527	−	0.769600i	$-0.279544\pi$
$977$	1.16356e12	1.27705	0.638527	+	0.769600i	$0.279544\pi$
$978$	0	0
$979$	3.57408e11	0.389075
$980$	0	0
$981$	5.45083e11i	0.588554i
$982$	0	0
$983$	1.21391e12i	1.30008i	0.759898	+	0.650042i	$0.225249\pi$
$983$	1.21391e12i	1.30008i	−0.759898	+	0.650042i	$0.774751\pi$
$984$	0	0
$985$	2.29864e12	2.44189
$986$	0	0
$987$	3.67656e11	0.387412
$988$	0	0
$989$	− 1.11731e12i	− 1.16785i
$990$	0	0
$991$	2.61795e10i	0.0271436i	0.999908	+	0.0135718i	$0.00432017\pi$
$991$	2.61795e10i	0.0271436i	−0.999908	+	0.0135718i	$0.995680\pi$
$992$	0	0
$993$	2.98980e11	0.307500
$994$	0	0
$995$	2.41257e12	2.46143
$996$	0	0
$997$	1.49208e12i	1.51012i	0.655653	+	0.755062i	$0.272393\pi$
$997$	1.49208e12i	1.51012i	−0.655653	+	0.755062i	$0.727607\pi$
$998$	0	0
$999$	− 2.68236e11i	− 0.269312i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.9.b.b.319.8	yes	8
4.3	odd	2	inner	384.9.b.b.319.4	yes	8
8.3	odd	2	inner	384.9.b.b.319.5	yes	8
8.5	even	2	inner	384.9.b.b.319.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.9.b.b.319.1	✓	8	8.5	even	2	inner
384.9.b.b.319.4	yes	8	4.3	odd	2	inner
384.9.b.b.319.5	yes	8	8.3	odd	2	inner
384.9.b.b.319.8	yes	8	1.1	even	1	trivial

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:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	384.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+46.7654 q^{3} +1205.32i q^{5} -1133.44i q^{7} +2187.00 q^{9} +O(q^{10})\)	\(q+46.7654 q^{3} +1205.32i q^{5} -1133.44i q^{7} +2187.00 q^{9} +20623.5 q^{11} +25759.2i q^{13} +56367.2i q^{15} +2473.09 q^{17} -206001. q^{19} -53005.6i q^{21} +223202. i q^{23} -1.06217e6 q^{25} +102276. q^{27} -202690. i q^{29} +1.16928e6i q^{31} +964465. q^{33} +1.36615e6 q^{35} -2.62267e6i q^{37} +1.20464e6i q^{39} +1.46741e6 q^{41} -5.00580e6 q^{43} +2.63603e6i q^{45} +6.93618e6i q^{47} +4.48012e6 q^{49} +115655. q^{51} +1.16384e6i q^{53} +2.48579e7i q^{55} -9.63372e6 q^{57} +2.03352e7 q^{59} +3.06867e6i q^{61} -2.47883e6i q^{63} -3.10481e7 q^{65} -3.74270e6 q^{67} +1.04381e7i q^{69} +3.03025e7i q^{71} -1.81452e7 q^{73} -4.96728e7 q^{75} -2.33754e7i q^{77} -6.18230e7i q^{79} +4.78297e6 q^{81} -8.39301e6 q^{83} +2.98087e6i q^{85} -9.47888e6i q^{87} +1.73301e7 q^{89} +2.91964e7 q^{91} +5.46819e7i q^{93} -2.48297e8i q^{95} -1.34674e8 q^{97} +4.51035e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 17496 q^{9}+O(q^{10}) \) 8 * q + 17496 * q^9	\( 8 q + 17496 q^{9} + 276848 q^{17} - 2841976 q^{25} + 775008 q^{33} + 5826832 q^{41} - 16599928 q^{49} - 42366240 q^{57} - 132963072 q^{65} - 57759760 q^{73} + 38263752 q^{81} - 258778768 q^{89} - 771489424 q^{97}+O(q^{100}) \) 8 * q + 17496 * q^9 + 276848 * q^17 - 2841976 * q^25 + 775008 * q^33 + 5826832 * q^41 - 16599928 * q^49 - 42366240 * q^57 - 132963072 * q^65 - 57759760 * q^73 + 38263752 * q^81 - 258778768 * q^89 - 771489424 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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