LMFDB - Embedded newform 384.7.b.d.319.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("384.319");

S:= CuspForms(chi, 7);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

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

Embedding invariants

Embedding label			319.4
Root			$2.21837 - 1.28078i$ of defining polynomial
Character	$\chi$	$=$	384.319
Dual form			384.7.b.d.319.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-15.5885 q^{3} +20.0000i q^{5} +529.850i q^{7} +243.000 q^{9} +O(q^{10})$	$q-15.5885 q^{3} +20.0000i q^{5} +529.850i q^{7} +243.000 q^{9} -435.847 q^{11} -341.182i q^{13} -311.769i q^{15} +7682.73 q^{17} +4300.52 q^{19} -8259.54i q^{21} -3175.32i q^{23} +15225.0 q^{25} -3788.00 q^{27} +19409.3i q^{29} +15297.3i q^{31} +6794.18 q^{33} -10597.0 q^{35} -61969.9i q^{37} +5318.50i q^{39} -33740.2 q^{41} +99312.6 q^{43} +4860.00i q^{45} -17726.4i q^{47} -163092. q^{49} -119762. q^{51} -224406. i q^{53} -8716.94i q^{55} -67038.5 q^{57} -199557. q^{59} +45671.7i q^{61} +128754. i q^{63} +6823.63 q^{65} +496811. q^{67} +49498.3i q^{69} +452032. i q^{71} +394349. q^{73} -237334. q^{75} -230933. i q^{77} +571742. i q^{79} +59049.0 q^{81} -324465. q^{83} +153655. i q^{85} -302561. i q^{87} +758607. q^{89} +180775. q^{91} -238462. i q^{93} +86010.5i q^{95} +25015.4 q^{97} -105911. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 1944 q^{9}+O(q^{10}) $ 8 * q + 1944 * q^9	$ 8 q + 1944 q^{9} + 4464 q^{17} + 121800 q^{25} - 116640 q^{33} - 98928 q^{41} - 278776 q^{49} - 23328 q^{57} - 230400 q^{65} + 76912 q^{73} + 472392 q^{81} + 6638832 q^{89} - 4929680 q^{97}+O(q^{100}) $ 8 * q + 1944 * q^9 + 4464 * q^17 + 121800 * q^25 - 116640 * q^33 - 98928 * q^41 - 278776 * q^49 - 23328 * q^57 - 230400 * q^65 + 76912 * q^73 + 472392 * q^81 + 6638832 * q^89 - 4929680 * q^97

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−15.5885	−0.577350
$3$	−15.5885	−0.577350
$4$	0	0
$5$	20.0000i	0.160000i	0.996795	+	0.0800000i	$0.0254920\pi$
$5$	20.0000i	0.160000i	−0.996795	+	0.0800000i	$0.974508\pi$
$6$	0	0
$7$	529.850i	1.54475i	0.635165	+	0.772376i	$0.280932\pi$
$7$	529.850i	1.54475i	−0.635165	+	0.772376i	$0.719068\pi$
$8$	0	0
$9$	243.000	0.333333
$10$	0	0
$11$	−435.847	−0.327458	−0.163729	−	0.986505i	$-0.552352\pi$
$11$	−435.847	−0.327458	−0.163729	+	0.986505i	$0.552352\pi$
$12$	0	0
$13$	− 341.182i	− 0.155294i	−0.996981	−	0.0776472i	$-0.975259\pi$
$13$	− 341.182i	− 0.155294i	0.996981	−	0.0776472i	$-0.0247408\pi$
$14$	0	0
$15$	− 311.769i	− 0.0923760i
$16$	0	0
$17$	7682.73	1.56375	0.781877	−	0.623432i	$-0.214262\pi$
$17$	7682.73	1.56375	0.781877	+	0.623432i	$0.214262\pi$
$18$	0	0
$19$	4300.52	0.626990	0.313495	−	0.949590i	$-0.398500\pi$
$19$	4300.52	0.626990	0.313495	+	0.949590i	$0.398500\pi$
$20$	0	0
$21$	− 8259.54i	− 0.891863i
$22$	0	0
$23$	− 3175.32i	− 0.260978i	−0.991450	−	0.130489i	$-0.958345\pi$
$23$	− 3175.32i	− 0.260978i	0.991450	−	0.130489i	$-0.0416547\pi$
$24$	0	0
$25$	15225.0	0.974400
$26$	0	0
$27$	−3788.00	−0.192450
$28$	0	0
$29$	19409.3i	0.795821i	0.917424	+	0.397910i	$0.130264\pi$
$29$	19409.3i	0.795821i	−0.917424	+	0.397910i	$0.869736\pi$
$30$	0	0
$31$	15297.3i	0.513488i	0.966479	+	0.256744i	$0.0826497\pi$
$31$	15297.3i	0.513488i	−0.966479	+	0.256744i	$0.917350\pi$
$32$	0	0
$33$	6794.18	0.189058
$34$	0	0
$35$	−10597.0	−0.247160
$36$	0	0
$37$	− 61969.9i	− 1.22342i	−0.791082	−	0.611710i	$-0.790482\pi$
$37$	− 61969.9i	− 1.22342i	0.791082	−	0.611710i	$-0.209518\pi$
$38$	0	0
$39$	5318.50i	0.0896592i
$40$	0	0
$41$	−33740.2	−0.489549	−0.244774	−	0.969580i	$-0.578714\pi$
$41$	−33740.2	−0.489549	−0.244774	+	0.969580i	$0.578714\pi$
$42$	0	0
$43$	99312.6	1.24911	0.624553	−	0.780983i	$-0.285281\pi$
$43$	99312.6	1.24911	0.624553	+	0.780983i	$0.285281\pi$
$44$	0	0
$45$	4860.00i	0.0533333i
$46$	0	0
$47$	− 17726.4i	− 0.170737i	−0.996349	−	0.0853685i	$-0.972793\pi$
$47$	− 17726.4i	− 0.170737i	0.996349	−	0.0853685i	$-0.0272068\pi$
$48$	0	0
$49$	−163092.	−1.38626
$50$	0	0
$51$	−119762.	−0.902834
$52$	0	0
$53$	− 224406.i	− 1.50733i	−0.657260	−	0.753664i	$-0.728285\pi$
$53$	− 224406.i	− 1.50733i	0.657260	−	0.753664i	$-0.271715\pi$
$54$	0	0
$55$	− 8716.94i	− 0.0523933i
$56$	0	0
$57$	−67038.5	−0.361993
$58$	0	0
$59$	−199557.	−0.971651	−0.485826	−	0.874056i	$-0.661481\pi$
$59$	−199557.	−0.971651	−0.485826	+	0.874056i	$0.661481\pi$
$60$	0	0
$61$	45671.7i	0.201214i	0.994926	+	0.100607i	$0.0320784\pi$
$61$	45671.7i	0.201214i	−0.994926	+	0.100607i	$0.967922\pi$
$62$	0	0
$63$	128754.i	0.514917i
$64$	0	0
$65$	6823.63	0.0248471
$66$	0	0
$67$	496811.	1.65183	0.825917	−	0.563791i	$-0.190658\pi$
$67$	496811.	1.65183	0.825917	+	0.563791i	$0.190658\pi$
$68$	0	0
$69$	49498.3i	0.150676i
$70$	0	0
$71$	452032.i	1.26297i	0.775387	+	0.631487i	$0.217555\pi$
$71$	452032.i	1.26297i	−0.775387	+	0.631487i	$0.782445\pi$
$72$	0	0
$73$	394349.	1.01371	0.506853	−	0.862032i	$-0.330809\pi$
$73$	394349.	1.01371	0.506853	+	0.862032i	$0.330809\pi$
$74$	0	0
$75$	−237334.	−0.562570
$76$	0	0
$77$	− 230933.i	− 0.505842i
$78$	0	0
$79$	571742.i	1.15963i	0.814749	+	0.579814i	$0.196875\pi$
$79$	571742.i	1.15963i	−0.814749	+	0.579814i	$0.803125\pi$
$80$	0	0
$81$	59049.0	0.111111
$82$	0	0
$83$	−324465.	−0.567458	−0.283729	−	0.958904i	$-0.591572\pi$
$83$	−324465.	−0.567458	−0.283729	+	0.958904i	$0.591572\pi$
$84$	0	0
$85$	153655.i	0.250201i
$86$	0	0
$87$	− 302561.i	− 0.459467i
$88$	0	0
$89$	758607.	1.07609	0.538043	−	0.842918i	$-0.319164\pi$
$89$	758607.	1.07609	0.538043	+	0.842918i	$0.319164\pi$
$90$	0	0
$91$	180775.	0.239891
$92$	0	0
$93$	− 238462.i	− 0.296462i
$94$	0	0
$95$	86010.5i	0.100318i
$96$	0	0
$97$	25015.4	0.0274089	0.0137045	−	0.999906i	$-0.495638\pi$
$97$	25015.4	0.0274089	0.0137045	+	0.999906i	$0.495638\pi$
$98$	0	0
$99$	−105911.	−0.109153
$100$	0	0
$101$	1.60937e6i	1.56204i	0.624508	+	0.781018i	$0.285300\pi$
$101$	1.60937e6i	1.56204i	−0.624508	+	0.781018i	$0.714700\pi$
$102$	0	0
$103$	647116.i	0.592203i	0.955156	+	0.296101i	$0.0956867\pi$
$103$	647116.i	0.592203i	−0.955156	+	0.296101i	$0.904313\pi$
$104$	0	0
$105$	165191.	0.142698
$106$	0	0
$107$	233041.	0.190231	0.0951156	−	0.995466i	$-0.469678\pi$
$107$	233041.	0.190231	0.0951156	+	0.995466i	$0.469678\pi$
$108$	0	0
$109$	− 525849.i	− 0.406052i	−0.979173	−	0.203026i	$-0.934922\pi$
$109$	− 525849.i	− 0.406052i	0.979173	−	0.203026i	$-0.0650776\pi$
$110$	0	0
$111$	966015.i	0.706342i
$112$	0	0
$113$	−1.21523e6	−0.842216	−0.421108	−	0.907011i	$-0.638359\pi$
$113$	−1.21523e6	−0.842216	−0.421108	+	0.907011i	$0.638359\pi$
$114$	0	0
$115$	63506.4	0.0417565
$116$	0	0
$117$	− 82907.1i	− 0.0517648i
$118$	0	0
$119$	4.07069e6i	2.41561i
$120$	0	0
$121$	−1.58160e6	−0.892771
$122$	0	0
$123$	525957.	0.282641
$124$	0	0
$125$	617000.i	0.315904i
$126$	0	0
$127$	3.23094e6i	1.57731i	0.614836	+	0.788655i	$0.289222\pi$
$127$	3.23094e6i	1.57731i	−0.614836	+	0.788655i	$0.710778\pi$
$128$	0	0
$129$	−1.54813e6	−0.721171
$130$	0	0
$131$	1.80119e6	0.801209	0.400604	−	0.916251i	$-0.368800\pi$
$131$	1.80119e6	0.801209	0.400604	+	0.916251i	$0.368800\pi$
$132$	0	0
$133$	2.27863e6i	0.968544i
$134$	0	0
$135$	− 75759.9i	− 0.0307920i
$136$	0	0
$137$	50362.3	0.0195859	0.00979295	−	0.999952i	$-0.496883\pi$
$137$	50362.3	0.0195859	0.00979295	+	0.999952i	$0.496883\pi$
$138$	0	0
$139$	−4.23758e6	−1.57788	−0.788940	−	0.614471i	$-0.789370\pi$
$139$	−4.23758e6	−1.57788	−0.788940	+	0.614471i	$0.789370\pi$
$140$	0	0
$141$	276328.i	0.0985751i
$142$	0	0
$143$	148703.i	0.0508524i
$144$	0	0
$145$	−388185.	−0.127331
$146$	0	0
$147$	2.54235e6	0.800357
$148$	0	0
$149$	− 4.40006e6i	− 1.33015i	−0.746778	−	0.665073i	$-0.768400\pi$
$149$	− 4.40006e6i	− 1.33015i	0.746778	−	0.665073i	$-0.231600\pi$
$150$	0	0
$151$	2.17763e6i	0.632488i	0.948678	+	0.316244i	$0.102422\pi$
$151$	2.17763e6i	0.632488i	−0.948678	+	0.316244i	$0.897578\pi$
$152$	0	0
$153$	1.86690e6	0.521252
$154$	0	0
$155$	−305946.	−0.0821580
$156$	0	0
$157$	5.73506e6i	1.48197i	0.671522	+	0.740985i	$0.265641\pi$
$157$	5.73506e6i	1.48197i	−0.671522	+	0.740985i	$0.734359\pi$
$158$	0	0
$159$	3.49815e6i	0.870256i
$160$	0	0
$161$	1.68244e6	0.403147
$162$	0	0
$163$	−2.99695e6	−0.692016	−0.346008	−	0.938232i	$-0.612463\pi$
$163$	−2.99695e6	−0.692016	−0.346008	+	0.938232i	$0.612463\pi$
$164$	0	0
$165$	135884.i	0.0302493i
$166$	0	0
$167$	8.03225e6i	1.72460i	0.506400	+	0.862298i	$0.330976\pi$
$167$	8.03225e6i	1.72460i	−0.506400	+	0.862298i	$0.669024\pi$
$168$	0	0
$169$	4.71040e6	0.975884
$170$	0	0
$171$	1.04503e6	0.208997
$172$	0	0
$173$	− 2.42976e6i	− 0.469272i	−0.972083	−	0.234636i	$-0.924610\pi$
$173$	− 2.42976e6i	− 0.469272i	0.972083	−	0.234636i	$-0.0753898\pi$
$174$	0	0
$175$	8.06697e6i	1.50521i
$176$	0	0
$177$	3.11078e6	0.560983
$178$	0	0
$179$	5.34862e6	0.932573	0.466286	−	0.884634i	$-0.345592\pi$
$179$	5.34862e6	0.932573	0.466286	+	0.884634i	$0.345592\pi$
$180$	0	0
$181$	− 9.11072e6i	− 1.53645i	−0.640183	−	0.768223i	$-0.721141\pi$
$181$	− 9.11072e6i	− 1.53645i	0.640183	−	0.768223i	$-0.278859\pi$
$182$	0	0
$183$	− 711952.i	− 0.116171i
$184$	0	0
$185$	1.23940e6	0.195747
$186$	0	0
$187$	−3.34849e6	−0.512064
$188$	0	0
$189$	− 2.00707e6i	− 0.297288i
$190$	0	0
$191$	1.19063e6i	0.170875i	0.996344	+	0.0854373i	$0.0272287\pi$
$191$	1.19063e6i	0.170875i	−0.996344	+	0.0854373i	$0.972771\pi$
$192$	0	0
$193$	2.07789e6	0.289034	0.144517	−	0.989502i	$-0.453837\pi$
$193$	2.07789e6	0.289034	0.144517	+	0.989502i	$0.453837\pi$
$194$	0	0
$195$	−106370.	−0.0143455
$196$	0	0
$197$	2.19700e6i	0.287364i	0.989624	+	0.143682i	$0.0458942\pi$
$197$	2.19700e6i	0.287364i	−0.989624	+	0.143682i	$0.954106\pi$
$198$	0	0
$199$	6.24588e6i	0.792564i	0.918129	+	0.396282i	$0.129700\pi$
$199$	6.24588e6i	0.792564i	−0.918129	+	0.396282i	$0.870300\pi$
$200$	0	0
$201$	−7.74451e6	−0.953687
$202$	0	0
$203$	−1.02840e7	−1.22935
$204$	0	0
$205$	− 674804.i	− 0.0783278i
$206$	0	0
$207$	− 771603.i	− 0.0869927i
$208$	0	0
$209$	−1.87437e6	−0.205313
$210$	0	0
$211$	1.11658e7	1.18862	0.594311	−	0.804235i	$-0.297425\pi$
$211$	1.11658e7	1.18862	0.594311	+	0.804235i	$0.297425\pi$
$212$	0	0
$213$	− 7.04648e6i	− 0.729178i
$214$	0	0
$215$	1.98625e6i	0.199857i
$216$	0	0
$217$	−8.10528e6	−0.793211
$218$	0	0
$219$	−6.14730e6	−0.585264
$220$	0	0
$221$	− 2.62121e6i	− 0.242842i
$222$	0	0
$223$	658854.i	0.0594120i	0.999559	+	0.0297060i	$0.00945711\pi$
$223$	658854.i	0.0594120i	−0.999559	+	0.0297060i	$0.990543\pi$
$224$	0	0
$225$	3.69967e6	0.324800
$226$	0	0
$227$	1.04595e7	0.894198	0.447099	−	0.894484i	$-0.352457\pi$
$227$	1.04595e7	0.894198	0.447099	+	0.894484i	$0.352457\pi$
$228$	0	0
$229$	1.94802e7i	1.62214i	0.584951	+	0.811069i	$0.301114\pi$
$229$	1.94802e7i	1.62214i	−0.584951	+	0.811069i	$0.698886\pi$
$230$	0	0
$231$	3.59990e6i	0.292048i
$232$	0	0
$233$	−1.62947e7	−1.28819	−0.644095	−	0.764946i	$-0.722766\pi$
$233$	−1.62947e7	−1.28819	−0.644095	+	0.764946i	$0.722766\pi$
$234$	0	0
$235$	354529.	0.0273179
$236$	0	0
$237$	− 8.91257e6i	− 0.669511i
$238$	0	0
$239$	2.00798e7i	1.47084i	0.677609	+	0.735422i	$0.263016\pi$
$239$	2.00798e7i	1.47084i	−0.677609	+	0.735422i	$0.736984\pi$
$240$	0	0
$241$	−5.97568e6	−0.426910	−0.213455	−	0.976953i	$-0.568472\pi$
$241$	−5.97568e6	−0.426910	−0.213455	+	0.976953i	$0.568472\pi$
$242$	0	0
$243$	−920483.	−0.0641500
$244$	0	0
$245$	− 3.26184e6i	− 0.221802i
$246$	0	0
$247$	− 1.46726e6i	− 0.0973680i
$248$	0	0
$249$	5.05791e6	0.327622
$250$	0	0
$251$	−1.92078e7	−1.21466	−0.607331	−	0.794449i	$-0.707760\pi$
$251$	−1.92078e7	−1.21466	−0.607331	+	0.794449i	$0.707760\pi$
$252$	0	0
$253$	1.38395e6i	0.0854594i
$254$	0	0
$255$	− 2.39524e6i	− 0.144453i
$256$	0	0
$257$	5.96168e6	0.351212	0.175606	−	0.984461i	$-0.443811\pi$
$257$	5.96168e6	0.351212	0.175606	+	0.984461i	$0.443811\pi$
$258$	0	0
$259$	3.28348e7	1.88988
$260$	0	0
$261$	4.71645e6i	0.265274i
$262$	0	0
$263$	− 1.92686e7i	− 1.05921i	−0.848243	−	0.529607i	$-0.822339\pi$
$263$	− 1.92686e7i	− 1.05921i	0.848243	−	0.529607i	$-0.177661\pi$
$264$	0	0
$265$	4.48813e6	0.241172
$266$	0	0
$267$	−1.18255e7	−0.621278
$268$	0	0
$269$	− 1.74390e7i	− 0.895909i	−0.894056	−	0.447955i	$-0.852153\pi$
$269$	− 1.74390e7i	− 0.895909i	0.894056	−	0.447955i	$-0.147847\pi$
$270$	0	0
$271$	− 2.41844e7i	− 1.21514i	−0.794265	−	0.607571i	$-0.792144\pi$
$271$	− 2.41844e7i	− 1.21514i	0.794265	−	0.607571i	$-0.207856\pi$
$272$	0	0
$273$	−2.81800e6	−0.138501
$274$	0	0
$275$	−6.63577e6	−0.319075
$276$	0	0
$277$	3.19698e7i	1.50418i	0.659059	+	0.752091i	$0.270955\pi$
$277$	3.19698e7i	1.50418i	−0.659059	+	0.752091i	$0.729045\pi$
$278$	0	0
$279$	3.71725e6i	0.171163i
$280$	0	0
$281$	2.34865e7	1.05852	0.529259	−	0.848460i	$-0.322470\pi$
$281$	2.34865e7	1.05852	0.529259	+	0.848460i	$0.322470\pi$
$282$	0	0
$283$	−7.74978e6	−0.341924	−0.170962	−	0.985278i	$-0.554688\pi$
$283$	−7.74978e6	−0.341924	−0.170962	+	0.985278i	$0.554688\pi$
$284$	0	0
$285$	− 1.34077e6i	− 0.0579189i
$286$	0	0
$287$	− 1.78772e7i	− 0.756231i
$288$	0	0
$289$	3.48867e7	1.44533
$290$	0	0
$291$	−389951.	−0.0158246
$292$	0	0
$293$	− 1.14848e7i	− 0.456584i	−0.973593	−	0.228292i	$-0.926686\pi$
$293$	− 1.14848e7i	− 0.456584i	0.973593	−	0.228292i	$-0.0733140\pi$
$294$	0	0
$295$	− 3.99114e6i	− 0.155464i
$296$	0	0
$297$	1.65099e6	0.0630194
$298$	0	0
$299$	−1.08336e6	−0.0405284
$300$	0	0
$301$	5.26208e7i	1.92956i
$302$	0	0
$303$	− 2.50876e7i	− 0.901842i
$304$	0	0
$305$	−913434.	−0.0321942
$306$	0	0
$307$	2.30562e7	0.796841	0.398421	−	0.917203i	$-0.369558\pi$
$307$	2.30562e7	0.796841	0.398421	+	0.917203i	$0.369558\pi$
$308$	0	0
$309$	− 1.00875e7i	− 0.341909i
$310$	0	0
$311$	− 4.24296e7i	− 1.41055i	−0.708935	−	0.705274i	$-0.750824\pi$
$311$	− 4.24296e7i	− 1.41055i	0.708935	−	0.705274i	$-0.249176\pi$
$312$	0	0
$313$	−2.01928e7	−0.658513	−0.329257	−	0.944240i	$-0.606798\pi$
$313$	−2.01928e7	−0.658513	−0.329257	+	0.944240i	$0.606798\pi$
$314$	0	0
$315$	−2.57507e6	−0.0823868
$316$	0	0
$317$	5.83594e6i	0.183203i	0.995796	+	0.0916016i	$0.0291986\pi$
$317$	5.83594e6i	0.183203i	−0.995796	+	0.0916016i	$0.970801\pi$
$318$	0	0
$319$	− 8.45947e6i	− 0.260598i
$320$	0	0
$321$	−3.63276e6	−0.109830
$322$	0	0
$323$	3.30398e7	0.980458
$324$	0	0
$325$	− 5.19449e6i	− 0.151319i
$326$	0	0
$327$	8.19718e6i	0.234434i
$328$	0	0
$329$	9.39235e6	0.263746
$330$	0	0
$331$	−6.94601e7	−1.91536	−0.957682	−	0.287828i	$-0.907067\pi$
$331$	−6.94601e7	−1.91536	−0.957682	+	0.287828i	$0.907067\pi$
$332$	0	0
$333$	− 1.50587e7i	− 0.407807i
$334$	0	0
$335$	9.93621e6i	0.264294i
$336$	0	0
$337$	−2.65194e7	−0.692906	−0.346453	−	0.938067i	$-0.612614\pi$
$337$	−2.65194e7	−0.692906	−0.346453	+	0.938067i	$0.612614\pi$
$338$	0	0
$339$	1.89436e7	0.486254
$340$	0	0
$341$	− 6.66729e6i	− 0.168146i
$342$	0	0
$343$	− 2.40780e7i	− 0.596676i
$344$	0	0
$345$	−989967.	−0.0241081
$346$	0	0
$347$	−5.53620e7	−1.32502	−0.662511	−	0.749052i	$-0.730509\pi$
$347$	−5.53620e7	−1.32502	−0.662511	+	0.749052i	$0.730509\pi$
$348$	0	0
$349$	4.23047e7i	0.995206i	0.867405	+	0.497603i	$0.165786\pi$
$349$	4.23047e7i	0.995206i	−0.867405	+	0.497603i	$0.834214\pi$
$350$	0	0
$351$	1.29239e6i	0.0298864i
$352$	0	0
$353$	−4.67810e7	−1.06352	−0.531760	−	0.846895i	$-0.678469\pi$
$353$	−4.67810e7	−1.06352	−0.531760	+	0.846895i	$0.678469\pi$
$354$	0	0
$355$	−9.04064e6	−0.202076
$356$	0	0
$357$	− 6.34558e7i	− 1.39466i
$358$	0	0
$359$	− 6.79965e7i	− 1.46961i	−0.678276	−	0.734807i	$-0.737273\pi$
$359$	− 6.79965e7i	− 1.46961i	0.678276	−	0.734807i	$-0.262727\pi$
$360$	0	0
$361$	−2.85514e7	−0.606884
$362$	0	0
$363$	2.46547e7	0.515442
$364$	0	0
$365$	7.88698e6i	0.162193i
$366$	0	0
$367$	− 8.52346e7i	− 1.72432i	−0.506637	−	0.862160i	$-0.669111\pi$
$367$	− 8.52346e7i	− 1.72432i	0.506637	−	0.862160i	$-0.330889\pi$
$368$	0	0
$369$	−8.19886e6	−0.163183
$370$	0	0
$371$	1.18902e8	2.32845
$372$	0	0
$373$	5.70643e7i	1.09961i	0.835293	+	0.549804i	$0.185298\pi$
$373$	5.70643e7i	1.09961i	−0.835293	+	0.549804i	$0.814702\pi$
$374$	0	0
$375$	− 9.61808e6i	− 0.182387i
$376$	0	0
$377$	6.62209e6	0.123586
$378$	0	0
$379$	−5.18779e7	−0.952939	−0.476469	−	0.879191i	$-0.658084\pi$
$379$	−5.18779e7	−0.952939	−0.476469	+	0.879191i	$0.658084\pi$
$380$	0	0
$381$	− 5.03653e7i	− 0.910661i
$382$	0	0
$383$	− 7.83098e7i	− 1.39386i	−0.717138	−	0.696931i	$-0.754548\pi$
$383$	− 7.83098e7i	− 1.39386i	0.717138	−	0.696931i	$-0.245452\pi$
$384$	0	0
$385$	4.61867e6	0.0809347
$386$	0	0
$387$	2.41330e7	0.416369
$388$	0	0
$389$	− 9.23410e7i	− 1.56872i	−0.620305	−	0.784361i	$-0.712991\pi$
$389$	− 9.23410e7i	− 1.56872i	0.620305	−	0.784361i	$-0.287009\pi$
$390$	0	0
$391$	− 2.43951e7i	− 0.408106i
$392$	0	0
$393$	−2.80778e7	−0.462578
$394$	0	0
$395$	−1.14348e7	−0.185540
$396$	0	0
$397$	3.75593e7i	0.600270i	0.953897	+	0.300135i	$0.0970317\pi$
$397$	3.75593e7i	0.600270i	−0.953897	+	0.300135i	$0.902968\pi$
$398$	0	0
$399$	− 3.55204e7i	− 0.559189i
$400$	0	0
$401$	−3.85071e7	−0.597183	−0.298592	−	0.954381i	$-0.596517\pi$
$401$	−3.85071e7	−0.597183	−0.298592	+	0.954381i	$0.596517\pi$
$402$	0	0
$403$	5.21916e6	0.0797417
$404$	0	0
$405$	1.18098e6i	0.0177778i
$406$	0	0
$407$	2.70094e7i	0.400619i
$408$	0	0
$409$	1.07783e8	1.57536	0.787681	−	0.616083i	$-0.211281\pi$
$409$	1.07783e8	1.57536	0.787681	+	0.616083i	$0.211281\pi$
$410$	0	0
$411$	−785070.	−0.0113079
$412$	0	0
$413$	− 1.05735e8i	− 1.50096i
$414$	0	0
$415$	− 6.48931e6i	− 0.0907933i
$416$	0	0
$417$	6.60574e7	0.910989
$418$	0	0
$419$	−7.92976e7	−1.07800	−0.538999	−	0.842306i	$-0.681197\pi$
$419$	−7.92976e7	−1.07800	−0.538999	+	0.842306i	$0.681197\pi$
$420$	0	0
$421$	6.67669e7i	0.894777i	0.894340	+	0.447389i	$0.147646\pi$
$421$	6.67669e7i	0.894777i	−0.894340	+	0.447389i	$0.852354\pi$
$422$	0	0
$423$	− 4.30752e6i	− 0.0569123i
$424$	0	0
$425$	1.16970e8	1.52372
$426$	0	0
$427$	−2.41992e7	−0.310826
$428$	0	0
$429$	− 2.31805e6i	− 0.0293596i
$430$	0	0
$431$	5.07275e7i	0.633595i	0.948493	+	0.316798i	$0.102608\pi$
$431$	5.07275e7i	0.633595i	−0.948493	+	0.316798i	$0.897392\pi$
$432$	0	0
$433$	−3.19683e7	−0.393782	−0.196891	−	0.980425i	$-0.563084\pi$
$433$	−3.19683e7	−0.393782	−0.196891	+	0.980425i	$0.563084\pi$
$434$	0	0
$435$	6.05121e6	0.0735148
$436$	0	0
$437$	− 1.36555e7i	− 0.163631i
$438$	0	0
$439$	− 7.14885e7i	− 0.844972i	−0.906369	−	0.422486i	$-0.861158\pi$
$439$	− 7.14885e7i	− 0.844972i	0.906369	−	0.422486i	$-0.138842\pi$
$440$	0	0
$441$	−3.96314e7	−0.462087
$442$	0	0
$443$	7.60820e7	0.875127	0.437563	−	0.899188i	$-0.355842\pi$
$443$	7.60820e7	0.875127	0.437563	+	0.899188i	$0.355842\pi$
$444$	0	0
$445$	1.51721e7i	0.172174i
$446$	0	0
$447$	6.85901e7i	0.767960i
$448$	0	0
$449$	9.00650e7	0.994986	0.497493	−	0.867468i	$-0.334254\pi$
$449$	9.00650e7	0.994986	0.497493	+	0.867468i	$0.334254\pi$
$450$	0	0
$451$	1.47056e7	0.160307
$452$	0	0
$453$	− 3.39458e7i	− 0.365167i
$454$	0	0
$455$	3.61550e6i	0.0383826i
$456$	0	0
$457$	−3.59392e7	−0.376548	−0.188274	−	0.982117i	$-0.560289\pi$
$457$	−3.59392e7	−0.376548	−0.188274	+	0.982117i	$0.560289\pi$
$458$	0	0
$459$	−2.91021e7	−0.300945
$460$	0	0
$461$	− 1.09802e8i	− 1.12075i	−0.828240	−	0.560373i	$-0.810658\pi$
$461$	− 1.09802e8i	− 1.12075i	0.828240	−	0.560373i	$-0.189342\pi$
$462$	0	0
$463$	5.53763e7i	0.557932i	0.960301	+	0.278966i	$0.0899917\pi$
$463$	5.53763e7i	0.557932i	−0.960301	+	0.278966i	$0.910008\pi$
$464$	0	0
$465$	4.76923e6	0.0474340
$466$	0	0
$467$	5.38248e7	0.528484	0.264242	−	0.964456i	$-0.414878\pi$
$467$	5.38248e7	0.528484	0.264242	+	0.964456i	$0.414878\pi$
$468$	0	0
$469$	2.63235e8i	2.55168i
$470$	0	0
$471$	− 8.94008e7i	− 0.855615i
$472$	0	0
$473$	−4.32851e7	−0.409030
$474$	0	0
$475$	6.54755e7	0.610939
$476$	0	0
$477$	− 5.45307e7i	− 0.502442i
$478$	0	0
$479$	− 1.71807e8i	− 1.56328i	−0.623733	−	0.781638i	$-0.714385\pi$
$479$	− 1.71807e8i	− 1.56328i	0.623733	−	0.781638i	$-0.285615\pi$
$480$	0	0
$481$	−2.11430e7	−0.189990
$482$	0	0
$483$	−2.62267e7	−0.232757
$484$	0	0
$485$	500308.i	0.00438543i
$486$	0	0
$487$	5.72463e7i	0.495634i	0.968807	+	0.247817i	$0.0797131\pi$
$487$	5.72463e7i	0.495634i	−0.968807	+	0.247817i	$0.920287\pi$
$488$	0	0
$489$	4.67178e7	0.399536
$490$	0	0
$491$	−2.13055e8	−1.79989	−0.899946	−	0.436001i	$-0.856395\pi$
$491$	−2.13055e8	−1.79989	−0.899946	+	0.436001i	$0.856395\pi$
$492$	0	0
$493$	1.49116e8i	1.24447i
$494$	0	0
$495$	− 2.11822e6i	− 0.0174644i
$496$	0	0
$497$	−2.39509e8	−1.95098
$498$	0	0
$499$	9.04068e7	0.727611	0.363806	−	0.931475i	$-0.381477\pi$
$499$	9.04068e7	0.727611	0.363806	+	0.931475i	$0.381477\pi$
$500$	0	0
$501$	− 1.25210e8i	− 0.995697i
$502$	0	0
$503$	2.29271e8i	1.80155i	0.434290	+	0.900773i	$0.356999\pi$
$503$	2.29271e8i	1.80155i	−0.434290	+	0.900773i	$0.643001\pi$
$504$	0	0
$505$	−3.21874e7	−0.249926
$506$	0	0
$507$	−7.34279e7	−0.563427
$508$	0	0
$509$	− 1.29225e7i	− 0.0979930i	−0.998799	−	0.0489965i	$-0.984398\pi$
$509$	− 1.29225e7i	− 0.0979930i	0.998799	−	0.0489965i	$-0.0156023\pi$
$510$	0	0
$511$	2.08946e8i	1.56593i
$512$	0	0
$513$	−1.62904e7	−0.120664
$514$	0	0
$515$	−1.29423e7	−0.0947525
$516$	0	0
$517$	7.72601e6i	0.0559092i
$518$	0	0
$519$	3.78762e7i	0.270934i
$520$	0	0
$521$	−7.99955e7	−0.565656	−0.282828	−	0.959171i	$-0.591273\pi$
$521$	−7.99955e7	−0.565656	−0.282828	+	0.959171i	$0.591273\pi$
$522$	0	0
$523$	1.28458e8	0.897957	0.448979	−	0.893543i	$-0.351788\pi$
$523$	1.28458e8	0.897957	0.448979	+	0.893543i	$0.351788\pi$
$524$	0	0
$525$	− 1.25752e8i	− 0.869031i
$526$	0	0
$527$	1.17525e8i	0.802969i
$528$	0	0
$529$	1.37953e8	0.931890
$530$	0	0
$531$	−4.84923e7	−0.323884
$532$	0	0
$533$	1.15115e7i	0.0760241i
$534$	0	0
$535$	4.66083e6i	0.0304370i
$536$	0	0
$537$	−8.33768e7	−0.538421
$538$	0	0
$539$	7.10832e7	0.453942
$540$	0	0
$541$	2.90048e8i	1.83180i	0.401410	+	0.915899i	$0.368520\pi$
$541$	2.90048e8i	1.83180i	−0.401410	+	0.915899i	$0.631480\pi$
$542$	0	0
$543$	1.42022e8i	0.887067i
$544$	0	0
$545$	1.05170e7	0.0649684
$546$	0	0
$547$	2.91198e7	0.177921	0.0889604	−	0.996035i	$-0.471646\pi$
$547$	2.91198e7	0.177921	0.0889604	+	0.996035i	$0.471646\pi$
$548$	0	0
$549$	1.10982e7i	0.0670713i
$550$	0	0
$551$	8.34700e7i	0.498972i
$552$	0	0
$553$	−3.02937e8	−1.79134
$554$	0	0
$555$	−1.93203e7	−0.113015
$556$	0	0
$557$	1.95014e8i	1.12849i	0.825606	+	0.564247i	$0.190834\pi$
$557$	1.95014e8i	1.12849i	−0.825606	+	0.564247i	$0.809166\pi$
$558$	0	0
$559$	− 3.38836e7i	− 0.193979i
$560$	0	0
$561$	5.21978e7	0.295640
$562$	0	0
$563$	−3.06065e8	−1.71510	−0.857548	−	0.514404i	$-0.828013\pi$
$563$	−3.06065e8	−1.71510	−0.857548	+	0.514404i	$0.828013\pi$
$564$	0	0
$565$	− 2.43046e7i	− 0.134755i
$566$	0	0
$567$	3.12871e7i	0.171639i
$568$	0	0
$569$	3.27388e8	1.77716	0.888578	−	0.458725i	$-0.151694\pi$
$569$	3.27388e8	1.77716	0.888578	+	0.458725i	$0.151694\pi$
$570$	0	0
$571$	1.37293e8	0.737461	0.368731	−	0.929536i	$-0.379792\pi$
$571$	1.37293e8	0.737461	0.368731	+	0.929536i	$0.379792\pi$
$572$	0	0
$573$	− 1.85601e7i	− 0.0986544i
$574$	0	0
$575$	− 4.83443e7i	− 0.254297i
$576$	0	0
$577$	2.84639e8	1.48172	0.740861	−	0.671658i	$-0.234417\pi$
$577$	2.84639e8	1.48172	0.740861	+	0.671658i	$0.234417\pi$
$578$	0	0
$579$	−3.23910e7	−0.166874
$580$	0	0
$581$	− 1.71918e8i	− 0.876583i
$582$	0	0
$583$	9.78068e7i	0.493587i
$584$	0	0
$585$	1.65814e6	0.00828236
$586$	0	0
$587$	3.11468e8	1.53992	0.769962	−	0.638090i	$-0.220275\pi$
$587$	3.11468e8	1.53992	0.769962	+	0.638090i	$0.220275\pi$
$588$	0	0
$589$	6.57865e7i	0.321952i
$590$	0	0
$591$	− 3.42479e7i	− 0.165909i
$592$	0	0
$593$	−7.54519e7	−0.361831	−0.180916	−	0.983499i	$-0.557906\pi$
$593$	−7.54519e7	−0.361831	−0.180916	+	0.983499i	$0.557906\pi$
$594$	0	0
$595$	−8.14139e7	−0.386498
$596$	0	0
$597$	− 9.73636e7i	− 0.457587i
$598$	0	0
$599$	2.08861e8i	0.971800i	0.874014	+	0.485900i	$0.161508\pi$
$599$	2.08861e8i	0.971800i	−0.874014	+	0.485900i	$0.838492\pi$
$600$	0	0
$601$	−2.88006e7	−0.132672	−0.0663359	−	0.997797i	$-0.521131\pi$
$601$	−2.88006e7	−0.132672	−0.0663359	+	0.997797i	$0.521131\pi$
$602$	0	0
$603$	1.20725e8	0.550611
$604$	0	0
$605$	− 3.16320e7i	− 0.142843i
$606$	0	0
$607$	2.89154e8i	1.29290i	0.762958	+	0.646448i	$0.223746\pi$
$607$	2.89154e8i	1.29290i	−0.762958	+	0.646448i	$0.776254\pi$
$608$	0	0
$609$	1.60312e8	0.709763
$610$	0	0
$611$	−6.04793e6	−0.0265145
$612$	0	0
$613$	3.53775e8i	1.53584i	0.640548	+	0.767919i	$0.278707\pi$
$613$	3.53775e8i	1.53584i	−0.640548	+	0.767919i	$0.721293\pi$
$614$	0	0
$615$	1.05191e7i	0.0452226i
$616$	0	0
$617$	1.88929e8	0.804348	0.402174	−	0.915563i	$-0.368255\pi$
$617$	1.88929e8	0.804348	0.402174	+	0.915563i	$0.368255\pi$
$618$	0	0
$619$	−8.45244e7	−0.356377	−0.178189	−	0.983996i	$-0.557024\pi$
$619$	−8.45244e7	−0.356377	−0.178189	+	0.983996i	$0.557024\pi$
$620$	0	0
$621$	1.20281e7i	0.0502253i
$622$	0	0
$623$	4.01948e8i	1.66229i
$624$	0	0
$625$	2.25551e8	0.923855
$626$	0	0
$627$	2.92185e7	0.118538
$628$	0	0
$629$	− 4.76098e8i	− 1.91313i
$630$	0	0
$631$	3.49408e8i	1.39074i	0.718653	+	0.695369i	$0.244759\pi$
$631$	3.49408e8i	1.39074i	−0.718653	+	0.695369i	$0.755241\pi$
$632$	0	0
$633$	−1.74058e8	−0.686251
$634$	0	0
$635$	−6.46187e7	−0.252370
$636$	0	0
$637$	5.56440e7i	0.215278i
$638$	0	0
$639$	1.09844e8i	0.420991i
$640$	0	0
$641$	−2.70070e8	−1.02542	−0.512710	−	0.858562i	$-0.671359\pi$
$641$	−2.70070e8	−1.02542	−0.512710	+	0.858562i	$0.671359\pi$
$642$	0	0
$643$	−4.94695e8	−1.86082	−0.930411	−	0.366518i	$-0.880550\pi$
$643$	−4.94695e8	−1.86082	−0.930411	+	0.366518i	$0.880550\pi$
$644$	0	0
$645$	− 3.09626e7i	− 0.115387i
$646$	0	0
$647$	1.19819e7i	0.0442396i	0.999755	+	0.0221198i	$0.00704153\pi$
$647$	1.19819e7i	0.0442396i	−0.999755	+	0.0221198i	$0.992958\pi$
$648$	0	0
$649$	8.69762e7	0.318175
$650$	0	0
$651$	1.26349e8	0.457961
$652$	0	0
$653$	− 5.21595e8i	− 1.87324i	−0.350345	−	0.936621i	$-0.613936\pi$
$653$	− 5.21595e8i	− 1.87324i	0.350345	−	0.936621i	$-0.386064\pi$
$654$	0	0
$655$	3.60238e7i	0.128193i
$656$	0	0
$657$	9.58269e7	0.337902
$658$	0	0
$659$	4.10833e8	1.43552	0.717759	−	0.696292i	$-0.245168\pi$
$659$	4.10833e8	1.43552	0.717759	+	0.696292i	$0.245168\pi$
$660$	0	0
$661$	− 8.48526e7i	− 0.293806i	−0.989151	−	0.146903i	$-0.953069\pi$
$661$	− 8.48526e7i	− 0.293806i	0.989151	−	0.146903i	$-0.0469305\pi$
$662$	0	0
$663$	4.08605e7i	0.140205i
$664$	0	0
$665$	−4.55727e7	−0.154967
$666$	0	0
$667$	6.16306e7	0.207692
$668$	0	0
$669$	− 1.02705e7i	− 0.0343016i
$670$	0	0
$671$	− 1.99059e7i	− 0.0658891i
$672$	0	0
$673$	−3.39619e8	−1.11416	−0.557079	−	0.830460i	$-0.688078\pi$
$673$	−3.39619e8	−1.11416	−0.557079	+	0.830460i	$0.688078\pi$
$674$	0	0
$675$	−5.76722e7	−0.187523
$676$	0	0
$677$	4.04369e8i	1.30320i	0.758561	+	0.651602i	$0.225903\pi$
$677$	4.04369e8i	1.30320i	−0.758561	+	0.651602i	$0.774097\pi$
$678$	0	0
$679$	1.32544e7i	0.0423400i
$680$	0	0
$681$	−1.63048e8	−0.516266
$682$	0	0
$683$	2.05590e8	0.645268	0.322634	−	0.946524i	$-0.395432\pi$
$683$	2.05590e8	0.645268	0.322634	+	0.946524i	$0.395432\pi$
$684$	0	0
$685$	1.00725e6i	0.00313374i
$686$	0	0
$687$	− 3.03667e8i	− 0.936542i
$688$	0	0
$689$	−7.65633e7	−0.234079
$690$	0	0
$691$	−1.25719e8	−0.381037	−0.190518	−	0.981684i	$-0.561017\pi$
$691$	−1.25719e8	−0.381037	−0.190518	+	0.981684i	$0.561017\pi$
$692$	0	0
$693$	− 5.61168e7i	− 0.168614i
$694$	0	0
$695$	− 8.47517e7i	− 0.252461i
$696$	0	0
$697$	−2.59217e8	−0.765534
$698$	0	0
$699$	2.54010e8	0.743737
$700$	0	0
$701$	− 1.36521e8i	− 0.396320i	−0.980170	−	0.198160i	$-0.936503\pi$
$701$	− 1.36521e8i	− 0.396320i	0.980170	−	0.198160i	$-0.0634966\pi$
$702$	0	0
$703$	− 2.66503e8i	− 0.767072i
$704$	0	0
$705$	−5.52655e6	−0.0157720
$706$	0	0
$707$	−8.52724e8	−2.41296
$708$	0	0
$709$	1.19437e8i	0.335119i	0.985862	+	0.167559i	$0.0535886\pi$
$709$	1.19437e8i	0.335119i	−0.985862	+	0.167559i	$0.946411\pi$
$710$	0	0
$711$	1.38933e8i	0.386543i
$712$	0	0
$713$	4.85739e7	0.134009
$714$	0	0
$715$	−2.97406e6	−0.00813638
$716$	0	0
$717$	− 3.13014e8i	− 0.849192i
$718$	0	0
$719$	2.90874e8i	0.782561i	0.920272	+	0.391280i	$0.127968\pi$
$719$	2.90874e8i	0.782561i	−0.920272	+	0.391280i	$0.872032\pi$
$720$	0	0
$721$	−3.42875e8	−0.914807
$722$	0	0
$723$	9.31516e7	0.246477
$724$	0	0
$725$	2.95506e8i	0.775448i
$726$	0	0
$727$	− 8.94298e7i	− 0.232744i	−0.993206	−	0.116372i	$-0.962873\pi$
$727$	− 8.94298e7i	− 0.232744i	0.993206	−	0.116372i	$-0.0371265\pi$
$728$	0	0
$729$	1.43489e7	0.0370370
$730$	0	0
$731$	7.62992e8	1.95329
$732$	0	0
$733$	− 1.28445e8i	− 0.326141i	−0.986614	−	0.163071i	$-0.947860\pi$
$733$	− 1.28445e8i	− 0.326141i	0.986614	−	0.163071i	$-0.0521398\pi$
$734$	0	0
$735$	5.08471e7i	0.128057i
$736$	0	0
$737$	−2.16533e8	−0.540907
$738$	0	0
$739$	−7.15451e8	−1.77275	−0.886373	−	0.462972i	$-0.846783\pi$
$739$	−7.15451e8	−1.77275	−0.886373	+	0.462972i	$0.846783\pi$
$740$	0	0
$741$	2.28723e7i	0.0562154i
$742$	0	0
$743$	− 3.45246e8i	− 0.841708i	−0.907128	−	0.420854i	$-0.861730\pi$
$743$	− 3.45246e8i	− 0.841708i	0.907128	−	0.420854i	$-0.138270\pi$
$744$	0	0
$745$	8.80011e7	0.212823
$746$	0	0
$747$	−7.88451e7	−0.189153
$748$	0	0
$749$	1.23477e8i	0.293860i
$750$	0	0
$751$	− 1.75422e8i	− 0.414157i	−0.978324	−	0.207079i	$-0.933604\pi$
$751$	− 1.75422e8i	− 0.414157i	0.978324	−	0.207079i	$-0.0663956\pi$
$752$	0	0
$753$	2.99419e8	0.701286
$754$	0	0
$755$	−4.35525e7	−0.101198
$756$	0	0
$757$	− 4.03668e8i	− 0.930544i	−0.885168	−	0.465272i	$-0.845956\pi$
$757$	− 4.03668e8i	− 0.930544i	0.885168	−	0.465272i	$-0.154044\pi$
$758$	0	0
$759$	− 2.15737e7i	− 0.0493400i
$760$	0	0
$761$	5.80482e8	1.31715	0.658574	−	0.752516i	$-0.271160\pi$
$761$	5.80482e8	1.31715	0.658574	+	0.752516i	$0.271160\pi$
$762$	0	0
$763$	2.78621e8	0.627250
$764$	0	0
$765$	3.73381e7i	0.0834002i
$766$	0	0
$767$	6.80851e7i	0.150892i
$768$	0	0
$769$	5.90678e8	1.29889	0.649444	−	0.760409i	$-0.275002\pi$
$769$	5.90678e8	1.29889	0.649444	+	0.760409i	$0.275002\pi$
$770$	0	0
$771$	−9.29335e7	−0.202772
$772$	0	0
$773$	3.10093e8i	0.671357i	0.941977	+	0.335679i	$0.108966\pi$
$773$	3.10093e8i	0.671357i	−0.941977	+	0.335679i	$0.891034\pi$
$774$	0	0
$775$	2.32902e8i	0.500342i
$776$	0	0
$777$	−5.11843e8	−1.09112
$778$	0	0
$779$	−1.45100e8	−0.306942
$780$	0	0
$781$	− 1.97017e8i	− 0.413571i
$782$	0	0
$783$	− 7.35222e7i	− 0.153156i
$784$	0	0
$785$	−1.14701e8	−0.237115
$786$	0	0
$787$	5.73439e8	1.17642	0.588210	−	0.808708i	$-0.299833\pi$
$787$	5.73439e8	1.17642	0.588210	+	0.808708i	$0.299833\pi$
$788$	0	0
$789$	3.00368e8i	0.611538i
$790$	0	0
$791$	− 6.43890e8i	− 1.30101i
$792$	0	0
$793$	1.55824e7	0.0312474
$794$	0	0
$795$	−6.99630e7	−0.139241
$796$	0	0
$797$	6.24736e8i	1.23402i	0.786956	+	0.617009i	$0.211656\pi$
$797$	6.24736e8i	1.23402i	−0.786956	+	0.617009i	$0.788344\pi$
$798$	0	0
$799$	− 1.36187e8i	− 0.266991i
$800$	0	0
$801$	1.84341e8	0.358695
$802$	0	0
$803$	−1.71876e8	−0.331947
$804$	0	0
$805$	3.36489e7i	0.0645034i
$806$	0	0
$807$	2.71847e8i	0.517253i
$808$	0	0
$809$	9.88696e8	1.86731	0.933657	−	0.358169i	$-0.116599\pi$
$809$	9.88696e8	1.86731	0.933657	+	0.358169i	$0.116599\pi$
$810$	0	0
$811$	4.02063e8	0.753757	0.376879	−	0.926263i	$-0.376997\pi$
$811$	4.02063e8	0.753757	0.376879	+	0.926263i	$0.376997\pi$
$812$	0	0
$813$	3.76997e8i	0.701563i
$814$	0	0
$815$	− 5.99389e7i	− 0.110723i
$816$	0	0
$817$	4.27096e8	0.783177
$818$	0	0
$819$	4.39284e7	0.0799638
$820$	0	0
$821$	− 6.26365e8i	− 1.13187i	−0.824448	−	0.565937i	$-0.808514\pi$
$821$	− 6.26365e8i	− 1.13187i	0.824448	−	0.565937i	$-0.191486\pi$
$822$	0	0
$823$	− 8.23953e8i	− 1.47810i	−0.673652	−	0.739049i	$-0.735275\pi$
$823$	− 8.23953e8i	− 1.47810i	0.673652	−	0.739049i	$-0.264725\pi$
$824$	0	0
$825$	1.03441e8	0.184218
$826$	0	0
$827$	−3.02093e7	−0.0534102	−0.0267051	−	0.999643i	$-0.508502\pi$
$827$	−3.02093e7	−0.0534102	−0.0267051	+	0.999643i	$0.508502\pi$
$828$	0	0
$829$	2.17842e7i	0.0382366i	0.999817	+	0.0191183i	$0.00608591\pi$
$829$	2.17842e7i	0.0382366i	−0.999817	+	0.0191183i	$0.993914\pi$
$830$	0	0
$831$	− 4.98360e8i	− 0.868440i
$832$	0	0
$833$	−1.25299e9	−2.16777
$834$	0	0
$835$	−1.60645e8	−0.275936
$836$	0	0
$837$	− 5.79461e7i	− 0.0988208i
$838$	0	0
$839$	− 5.51909e8i	− 0.934505i	−0.884124	−	0.467252i	$-0.845244\pi$
$839$	− 5.51909e8i	− 0.934505i	0.884124	−	0.467252i	$-0.154756\pi$
$840$	0	0
$841$	2.18104e8	0.366670
$842$	0	0
$843$	−3.66118e8	−0.611136
$844$	0	0
$845$	9.42081e7i	0.156141i
$846$	0	0
$847$	− 8.38010e8i	− 1.37911i
$848$	0	0
$849$	1.20807e8	0.197410
$850$	0	0
$851$	−1.96774e8	−0.319286
$852$	0	0
$853$	4.15243e8i	0.669045i	0.942388	+	0.334523i	$0.108575\pi$
$853$	4.15243e8i	0.669045i	−0.942388	+	0.334523i	$0.891425\pi$
$854$	0	0
$855$	2.09005e7i	0.0334395i
$856$	0	0
$857$	4.44449e8	0.706121	0.353061	−	0.935600i	$-0.385141\pi$
$857$	4.44449e8	0.706121	0.353061	+	0.935600i	$0.385141\pi$
$858$	0	0
$859$	4.67604e8	0.737733	0.368866	−	0.929482i	$-0.379746\pi$
$859$	4.67604e8	0.737733	0.368866	+	0.929482i	$0.379746\pi$
$860$	0	0
$861$	2.78679e8i	0.436610i
$862$	0	0
$863$	− 5.95476e8i	− 0.926472i	−0.886235	−	0.463236i	$-0.846688\pi$
$863$	− 5.95476e8i	− 0.926472i	0.886235	−	0.463236i	$-0.153312\pi$
$864$	0	0
$865$	4.85952e7	0.0750835
$866$	0	0
$867$	−5.43830e8	−0.834461
$868$	0	0
$869$	− 2.49192e8i	− 0.379730i
$870$	0	0
$871$	− 1.69503e8i	− 0.256521i
$872$	0	0
$873$	6.07874e6	0.00913631
$874$	0	0
$875$	−3.26917e8	−0.487993
$876$	0	0
$877$	7.21425e8i	1.06953i	0.845001	+	0.534765i	$0.179600\pi$
$877$	7.21425e8i	1.06953i	−0.845001	+	0.534765i	$0.820400\pi$
$878$	0	0
$879$	1.79030e8i	0.263609i
$880$	0	0
$881$	−6.77300e8	−0.990497	−0.495248	−	0.868751i	$-0.664923\pi$
$881$	−6.77300e8	−0.990497	−0.495248	+	0.868751i	$0.664923\pi$
$882$	0	0
$883$	8.41700e8	1.22257	0.611287	−	0.791409i	$-0.290652\pi$
$883$	8.41700e8	1.22257	0.611287	+	0.791409i	$0.290652\pi$
$884$	0	0
$885$	6.22157e7i	0.0897573i
$886$	0	0
$887$	− 2.23932e8i	− 0.320882i	−0.987045	−	0.160441i	$-0.948708\pi$
$887$	− 2.23932e8i	− 0.320882i	0.987045	−	0.160441i	$-0.0512917\pi$
$888$	0	0
$889$	−1.71191e9	−2.43655
$890$	0	0
$891$	−2.57363e7	−0.0363842
$892$	0	0
$893$	− 7.62329e7i	− 0.107050i
$894$	0	0
$895$	1.06972e8i	0.149212i
$896$	0	0
$897$	1.68879e7	0.0233991
$898$	0	0
$899$	−2.96910e8	−0.408644
$900$	0	0
$901$	− 1.72405e9i	− 2.35709i
$902$	0	0
$903$	− 8.20277e8i	− 1.11403i
$904$	0	0
$905$	1.82214e8	0.245831
$906$	0	0
$907$	6.79727e8	0.910988	0.455494	−	0.890239i	$-0.349463\pi$
$907$	6.79727e8	0.910988	0.455494	+	0.890239i	$0.349463\pi$
$908$	0	0
$909$	3.91076e8i	0.520679i
$910$	0	0
$911$	− 9.24099e8i	− 1.22226i	−0.791530	−	0.611130i	$-0.790715\pi$
$911$	− 9.24099e8i	− 1.22226i	0.791530	−	0.611130i	$-0.209285\pi$
$912$	0	0
$913$	1.41417e8	0.185819
$914$	0	0
$915$	1.42390e7	0.0185873
$916$	0	0
$917$	9.54361e8i	1.23767i
$918$	0	0
$919$	3.27998e8i	0.422595i	0.977422	+	0.211297i	$0.0677689\pi$
$919$	3.27998e8i	0.422595i	−0.977422	+	0.211297i	$0.932231\pi$
$920$	0	0
$921$	−3.59410e8	−0.460056
$922$	0	0
$923$	1.54225e8	0.196133
$924$	0	0
$925$	− 9.43492e8i	− 1.19210i
$926$	0	0
$927$	1.57249e8i	0.197401i
$928$	0	0
$929$	−8.75671e8	−1.09218	−0.546090	−	0.837727i	$-0.683884\pi$
$929$	−8.75671e8	−1.09218	−0.546090	+	0.837727i	$0.683884\pi$
$930$	0	0
$931$	−7.01381e8	−0.869171
$932$	0	0
$933$	6.61412e8i	0.814381i
$934$	0	0
$935$	− 6.69698e7i	− 0.0819303i
$936$	0	0
$937$	−1.46253e8	−0.177782	−0.0888908	−	0.996041i	$-0.528332\pi$
$937$	−1.46253e8	−0.177782	−0.0888908	+	0.996041i	$0.528332\pi$
$938$	0	0
$939$	3.14775e8	0.380193
$940$	0	0
$941$	9.23237e8i	1.10801i	0.832513	+	0.554006i	$0.186901\pi$
$941$	9.23237e8i	1.10801i	−0.832513	+	0.554006i	$0.813099\pi$
$942$	0	0
$943$	1.07136e8i	0.127761i
$944$	0	0
$945$	4.01414e7	0.0475660
$946$	0	0
$947$	4.29334e8	0.505529	0.252764	−	0.967528i	$-0.418660\pi$
$947$	4.29334e8	0.505529	0.252764	+	0.967528i	$0.418660\pi$
$948$	0	0
$949$	− 1.34545e8i	− 0.157423i
$950$	0	0
$951$	− 9.09733e7i	− 0.105772i
$952$	0	0
$953$	9.16653e8	1.05907	0.529537	−	0.848287i	$-0.322366\pi$
$953$	9.16653e8	1.05907	0.529537	+	0.848287i	$0.322366\pi$
$954$	0	0
$955$	−2.38126e7	−0.0273399
$956$	0	0
$957$	1.31870e8i	0.150456i
$958$	0	0
$959$	2.66845e7i	0.0302554i
$960$	0	0
$961$	6.53496e8	0.736330
$962$	0	0
$963$	5.66291e7	0.0634104
$964$	0	0
$965$	4.15577e7i	0.0462455i
$966$	0	0
$967$	− 1.74117e9i	− 1.92558i	−0.270247	−	0.962791i	$-0.587105\pi$
$967$	− 1.74117e9i	− 1.92558i	0.270247	−	0.962791i	$-0.412895\pi$
$968$	0	0
$969$	−5.15039e8	−0.566068
$970$	0	0
$971$	−5.75119e8	−0.628203	−0.314102	−	0.949389i	$-0.601703\pi$
$971$	−5.75119e8	−0.628203	−0.314102	+	0.949389i	$0.601703\pi$
$972$	0	0
$973$	− 2.24528e9i	− 2.43743i
$974$	0	0
$975$	8.09741e7i	0.0873639i
$976$	0	0
$977$	1.49329e8	0.160125	0.0800625	−	0.996790i	$-0.474488\pi$
$977$	1.49329e8	0.160125	0.0800625	+	0.996790i	$0.474488\pi$
$978$	0	0
$979$	−3.30636e8	−0.352373
$980$	0	0
$981$	− 1.27781e8i	− 0.135351i
$982$	0	0
$983$	6.57661e8i	0.692376i	0.938165	+	0.346188i	$0.112524\pi$
$983$	6.57661e8i	0.692376i	−0.938165	+	0.346188i	$0.887476\pi$
$984$	0	0
$985$	−4.39400e7	−0.0459782
$986$	0	0
$987$	−1.46412e8	−0.152274
$988$	0	0
$989$	− 3.15349e8i	− 0.325989i
$990$	0	0
$991$	− 1.37005e9i	− 1.40771i	−0.710341	−	0.703857i	$-0.751459\pi$
$991$	− 1.37005e9i	− 1.40771i	0.710341	−	0.703857i	$-0.248541\pi$
$992$	0	0
$993$	1.08278e9	1.10584
$994$	0	0
$995$	−1.24918e8	−0.126810
$996$	0	0
$997$	− 1.05005e9i	− 1.05956i	−0.848135	−	0.529780i	$-0.822274\pi$
$997$	− 1.05005e9i	− 1.05956i	0.848135	−	0.529780i	$-0.177726\pi$
$998$	0	0
$999$	2.34742e8i	0.235447i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.7.b.d.319.4	yes	8
4.3	odd	2	inner	384.7.b.d.319.7	yes	8
8.3	odd	2	inner	384.7.b.d.319.1	✓	8
8.5	even	2	inner	384.7.b.d.319.6	yes	8
16.3	odd	4		768.7.g.e.511.4		4
16.5	even	4		768.7.g.c.511.3		4
16.11	odd	4		768.7.g.c.511.2		4
16.13	even	4		768.7.g.e.511.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.7.b.d.319.1	✓	8	8.3	odd	2	inner
384.7.b.d.319.4	yes	8	1.1	even	1	trivial
384.7.b.d.319.6	yes	8	8.5	even	2	inner
384.7.b.d.319.7	yes	8	4.3	odd	2	inner
768.7.g.c.511.2		4	16.11	odd	4
768.7.g.c.511.3		4	16.5	even	4
768.7.g.e.511.1		4	16.13	even	4
768.7.g.e.511.4		4	16.3	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-15.5885 q^{3} +20.0000i q^{5} +529.850i q^{7} +243.000 q^{9} +O(q^{10})\)	\(q-15.5885 q^{3} +20.0000i q^{5} +529.850i q^{7} +243.000 q^{9} -435.847 q^{11} -341.182i q^{13} -311.769i q^{15} +7682.73 q^{17} +4300.52 q^{19} -8259.54i q^{21} -3175.32i q^{23} +15225.0 q^{25} -3788.00 q^{27} +19409.3i q^{29} +15297.3i q^{31} +6794.18 q^{33} -10597.0 q^{35} -61969.9i q^{37} +5318.50i q^{39} -33740.2 q^{41} +99312.6 q^{43} +4860.00i q^{45} -17726.4i q^{47} -163092. q^{49} -119762. q^{51} -224406. i q^{53} -8716.94i q^{55} -67038.5 q^{57} -199557. q^{59} +45671.7i q^{61} +128754. i q^{63} +6823.63 q^{65} +496811. q^{67} +49498.3i q^{69} +452032. i q^{71} +394349. q^{73} -237334. q^{75} -230933. i q^{77} +571742. i q^{79} +59049.0 q^{81} -324465. q^{83} +153655. i q^{85} -302561. i q^{87} +758607. q^{89} +180775. q^{91} -238462. i q^{93} +86010.5i q^{95} +25015.4 q^{97} -105911. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 1944 q^{9}+O(q^{10}) \) 8 * q + 1944 * q^9	\( 8 q + 1944 q^{9} + 4464 q^{17} + 121800 q^{25} - 116640 q^{33} - 98928 q^{41} - 278776 q^{49} - 23328 q^{57} - 230400 q^{65} + 76912 q^{73} + 472392 q^{81} + 6638832 q^{89} - 4929680 q^{97}+O(q^{100}) \) 8 * q + 1944 * q^9 + 4464 * q^17 + 121800 * q^25 - 116640 * q^33 - 98928 * q^41 - 278776 * q^49 - 23328 * q^57 - 230400 * q^65 + 76912 * q^73 + 472392 * q^81 + 6638832 * q^89 - 4929680 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more