LMFDB - Embedded newform 384.7.b.c.319.8

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:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 106x^{6} - 304x^{5} + 4359x^{4} - 8216x^{3} + 73366x^{2} - 69308x + 604693 $ x^8 - 4x^7 + 106x^6 - 304x^5 + 4359x^4 - 8216x^3 + 73366x^2 - 69308*x + 604693
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{32}\cdot 3^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			319.8
Root			$2.23205 + 6.41320i$ of defining polynomial
Character	$\chi$	$=$	384.319
Dual form			384.7.b.c.319.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+15.5885 q^{3} +195.235i q^{5} +277.510i q^{7} +243.000 q^{9} +O(q^{10})$	$q+15.5885 q^{3} +195.235i q^{5} +277.510i q^{7} +243.000 q^{9} -1753.29 q^{11} +1246.75i q^{13} +3043.41i q^{15} +6888.35 q^{17} +4401.67 q^{19} +4325.95i q^{21} +12728.7i q^{23} -22491.7 q^{25} +3788.00 q^{27} +8278.44i q^{29} +43940.1i q^{31} -27331.1 q^{33} -54179.6 q^{35} -12186.4i q^{37} +19434.9i q^{39} -54739.6 q^{41} +45453.4 q^{43} +47442.1i q^{45} +152336. i q^{47} +40637.3 q^{49} +107379. q^{51} -272550. i q^{53} -342303. i q^{55} +68615.3 q^{57} -213175. q^{59} +83964.8i q^{61} +67434.9i q^{63} -243409. q^{65} -373099. q^{67} +198420. i q^{69} -667812. i q^{71} -399457. q^{73} -350611. q^{75} -486555. i q^{77} -435376. i q^{79} +59049.0 q^{81} +246583. q^{83} +1.34485e6i q^{85} +129048. i q^{87} +80967.5 q^{89} -345985. q^{91} +684959. i q^{93} +859361. i q^{95} +877030. q^{97} -426049. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 1944 q^{9}+O(q^{10}) $ 8 * q + 1944 * q^9	$ 8 q + 1944 q^{9} - 6544 q^{17} - 56632 q^{25} - 33696 q^{33} - 499568 q^{41} - 414712 q^{49} - 375840 q^{57} - 36096 q^{65} - 1962640 q^{73} + 472392 q^{81} - 1694992 q^{89} + 7632752 q^{97}+O(q^{100}) $ 8 * q + 1944 * q^9 - 6544 * q^17 - 56632 * q^25 - 33696 * q^33 - 499568 * q^41 - 414712 * q^49 - 375840 * q^57 - 36096 * q^65 - 1962640 * q^73 + 472392 * q^81 - 1694992 * q^89 + 7632752 * 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
$5$	195.235i	1.56188i	0.624606	+	0.780940i	$0.285260\pi$
$5$	195.235i	1.56188i	−0.624606	+	0.780940i	$0.714740\pi$
$6$	0	0
$7$	277.510i	0.809067i	0.914523	+	0.404533i	$0.132566\pi$
$7$	277.510i	0.809067i	−0.914523	+	0.404533i	$0.867434\pi$
$8$	0	0
$9$	243.000	0.333333
$10$	0	0
$11$	−1753.29	−1.31727	−0.658636	−	0.752462i	$-0.728866\pi$
$11$	−1753.29	−1.31727	−0.658636	+	0.752462i	$0.728866\pi$
$12$	0	0
$13$	1246.75i	0.567478i	0.958902	+	0.283739i	$0.0915749\pi$
$13$	1246.75i	0.567478i	−0.958902	+	0.283739i	$0.908425\pi$
$14$	0	0
$15$	3043.41i	0.901752i
$16$	0	0
$17$	6888.35	1.40207	0.701033	−	0.713128i	$-0.252722\pi$
$17$	6888.35	1.40207	0.701033	+	0.713128i	$0.252722\pi$
$18$	0	0
$19$	4401.67	0.641737	0.320869	−	0.947124i	$-0.396025\pi$
$19$	4401.67	0.641737	0.320869	+	0.947124i	$0.396025\pi$
$20$	0	0
$21$	4325.95i	0.467115i
$22$	0	0
$23$	12728.7i	1.04616i	0.852283	+	0.523082i	$0.175218\pi$
$23$	12728.7i	1.04616i	−0.852283	+	0.523082i	$0.824782\pi$
$24$	0	0
$25$	−22491.7	−1.43947
$26$	0	0
$27$	3788.00	0.192450
$28$	0	0
$29$	8278.44i	0.339433i	0.985493	+	0.169717i	$0.0542852\pi$
$29$	8278.44i	0.339433i	−0.985493	+	0.169717i	$0.945715\pi$
$30$	0	0
$31$	43940.1i	1.47495i	0.675376	+	0.737474i	$0.263981\pi$
$31$	43940.1i	1.47495i	−0.675376	+	0.737474i	$0.736019\pi$
$32$	0	0
$33$	−27331.1	−0.760527
$34$	0	0
$35$	−54179.6	−1.26367
$36$	0	0
$37$	− 12186.4i	− 0.240585i	−0.992738	−	0.120293i	$-0.961617\pi$
$37$	− 12186.4i	− 0.240585i	0.992738	−	0.120293i	$-0.0383833\pi$
$38$	0	0
$39$	19434.9i	0.327633i
$40$	0	0
$41$	−54739.6	−0.794238	−0.397119	−	0.917767i	$-0.629990\pi$
$41$	−54739.6	−0.794238	−0.397119	+	0.917767i	$0.629990\pi$
$42$	0	0
$43$	45453.4	0.571691	0.285845	−	0.958276i	$-0.407726\pi$
$43$	45453.4	0.571691	0.285845	+	0.958276i	$0.407726\pi$
$44$	0	0
$45$	47442.1i	0.520627i
$46$	0	0
$47$	152336.i	1.46727i	0.679545	+	0.733634i	$0.262177\pi$
$47$	152336.i	1.46727i	−0.679545	+	0.733634i	$0.737823\pi$
$48$	0	0
$49$	40637.3	0.345411
$50$	0	0
$51$	107379.	0.809484
$52$	0	0
$53$	− 272550.i	− 1.83071i	−0.402649	−	0.915354i	$-0.631911\pi$
$53$	− 272550.i	− 1.83071i	0.402649	−	0.915354i	$-0.368089\pi$
$54$	0	0
$55$	− 342303.i	− 2.05742i
$56$	0	0
$57$	68615.3	0.370507
$58$	0	0
$59$	−213175.	−1.03796	−0.518978	−	0.854787i	$-0.673687\pi$
$59$	−213175.	−1.03796	−0.518978	+	0.854787i	$0.673687\pi$
$60$	0	0
$61$	83964.8i	0.369920i	0.982746	+	0.184960i	$0.0592156\pi$
$61$	83964.8i	0.369920i	−0.982746	+	0.184960i	$0.940784\pi$
$62$	0	0
$63$	67434.9i	0.269689i
$64$	0	0
$65$	−243409.	−0.886332
$66$	0	0
$67$	−373099.	−1.24051	−0.620254	−	0.784401i	$-0.712970\pi$
$67$	−373099.	−1.24051	−0.620254	+	0.784401i	$0.712970\pi$
$68$	0	0
$69$	198420.i	0.604003i
$70$	0	0
$71$	− 667812.i	− 1.86586i	−0.360057	−	0.932930i	$-0.617243\pi$
$71$	− 667812.i	− 1.86586i	0.360057	−	0.932930i	$-0.382757\pi$
$72$	0	0
$73$	−399457.	−1.02684	−0.513419	−	0.858138i	$-0.671621\pi$
$73$	−399457.	−1.02684	−0.513419	+	0.858138i	$0.671621\pi$
$74$	0	0
$75$	−350611.	−0.831078
$76$	0	0
$77$	− 486555.i	− 1.06576i
$78$	0	0
$79$	− 435376.i	− 0.883046i	−0.897250	−	0.441523i	$-0.854438\pi$
$79$	− 435376.i	− 0.883046i	0.897250	−	0.441523i	$-0.145562\pi$
$80$	0	0
$81$	59049.0	0.111111
$82$	0	0
$83$	246583.	0.431250	0.215625	−	0.976476i	$-0.430821\pi$
$83$	246583.	0.431250	0.215625	+	0.976476i	$0.430821\pi$
$84$	0	0
$85$	1.34485e6i	2.18986i
$86$	0	0
$87$	129048.i	0.195972i
$88$	0	0
$89$	80967.5	0.114853	0.0574263	−	0.998350i	$-0.481711\pi$
$89$	80967.5	0.114853	0.0574263	+	0.998350i	$0.481711\pi$
$90$	0	0
$91$	−345985.	−0.459127
$92$	0	0
$93$	684959.i	0.851561i
$94$	0	0
$95$	859361.i	1.00232i
$96$	0	0
$97$	877030.	0.960947	0.480474	−	0.877009i	$-0.340465\pi$
$97$	877030.	0.960947	0.480474	+	0.877009i	$0.340465\pi$
$98$	0	0
$99$	−426049.	−0.439091
$100$	0	0
$101$	− 1.80491e6i	− 1.75183i	−0.482470	−	0.875913i	$-0.660260\pi$
$101$	− 1.80491e6i	− 1.75183i	0.482470	−	0.875913i	$-0.339740\pi$
$102$	0	0
$103$	826267.i	0.756151i	0.925775	+	0.378076i	$0.123414\pi$
$103$	826267.i	0.756151i	−0.925775	+	0.378076i	$0.876586\pi$
$104$	0	0
$105$	−844577.	−0.729577
$106$	0	0
$107$	−1.34587e6	−1.09863	−0.549314	−	0.835616i	$-0.685111\pi$
$107$	−1.34587e6	−1.09863	−0.549314	+	0.835616i	$0.685111\pi$
$108$	0	0
$109$	− 704622.i	− 0.544098i	−0.962283	−	0.272049i	$-0.912299\pi$
$109$	− 704622.i	− 0.544098i	0.962283	−	0.272049i	$-0.0877012\pi$
$110$	0	0
$111$	− 189967.i	− 0.138902i
$112$	0	0
$113$	2.55273e6	1.76917	0.884583	−	0.466382i	$-0.154443\pi$
$113$	2.55273e6	1.76917	0.884583	+	0.466382i	$0.154443\pi$
$114$	0	0
$115$	−2.48508e6	−1.63398
$116$	0	0
$117$	302960.i	0.189159i
$118$	0	0
$119$	1.91159e6i	1.13437i
$120$	0	0
$121$	1.30246e6	0.735205
$122$	0	0
$123$	−853307.	−0.458553
$124$	0	0
$125$	− 1.34062e6i	− 0.686399i
$126$	0	0
$127$	670190.i	0.327180i	0.986528	+	0.163590i	$0.0523074\pi$
$127$	670190.i	0.327180i	−0.986528	+	0.163590i	$0.947693\pi$
$128$	0	0
$129$	708549.	0.330066
$130$	0	0
$131$	−1.73132e6	−0.770128	−0.385064	−	0.922890i	$-0.625821\pi$
$131$	−1.73132e6	−0.770128	−0.385064	+	0.922890i	$0.625821\pi$
$132$	0	0
$133$	1.22151e6i	0.519208i
$134$	0	0
$135$	739549.i	0.300584i
$136$	0	0
$137$	−2.77355e6	−1.07863	−0.539317	−	0.842103i	$-0.681317\pi$
$137$	−2.77355e6	−1.07863	−0.539317	+	0.842103i	$0.681317\pi$
$138$	0	0
$139$	−4.31042e6	−1.60500	−0.802500	−	0.596652i	$-0.796497\pi$
$139$	−4.31042e6	−1.60500	−0.802500	+	0.596652i	$0.796497\pi$
$140$	0	0
$141$	2.37469e6i	0.847128i
$142$	0	0
$143$	− 2.18591e6i	− 0.747522i
$144$	0	0
$145$	−1.61624e6	−0.530154
$146$	0	0
$147$	633472.	0.199423
$148$	0	0
$149$	− 213236.i	− 0.0644616i	−0.999480	−	0.0322308i	$-0.989739\pi$
$149$	− 213236.i	− 0.0644616i	0.999480	−	0.0322308i	$-0.0102612\pi$
$150$	0	0
$151$	5.33710e6i	1.55015i	0.631868	+	0.775076i	$0.282289\pi$
$151$	5.33710e6i	1.55015i	−0.631868	+	0.775076i	$0.717711\pi$
$152$	0	0
$153$	1.67387e6	0.467356
$154$	0	0
$155$	−8.57866e6	−2.30369
$156$	0	0
$157$	− 4.08979e6i	− 1.05682i	−0.848988	−	0.528412i	$-0.822788\pi$
$157$	− 4.08979e6i	− 1.05682i	0.848988	−	0.528412i	$-0.177212\pi$
$158$	0	0
$159$	− 4.24864e6i	− 1.05696i
$160$	0	0
$161$	−3.53233e6	−0.846416
$162$	0	0
$163$	1.54126e6	0.355888	0.177944	−	0.984041i	$-0.443055\pi$
$163$	1.54126e6	0.355888	0.177944	+	0.984041i	$0.443055\pi$
$164$	0	0
$165$	− 5.33598e6i	− 1.18785i
$166$	0	0
$167$	− 7.66555e6i	− 1.64586i	−0.568140	−	0.822932i	$-0.692337\pi$
$167$	− 7.66555e6i	− 1.64586i	0.568140	−	0.822932i	$-0.307663\pi$
$168$	0	0
$169$	3.27243e6	0.677969
$170$	0	0
$171$	1.06961e6	0.213912
$172$	0	0
$173$	1.00380e7i	1.93870i	0.245693	+	0.969348i	$0.420984\pi$
$173$	1.00380e7i	1.93870i	−0.245693	+	0.969348i	$0.579016\pi$
$174$	0	0
$175$	− 6.24167e6i	− 1.16463i
$176$	0	0
$177$	−3.32306e6	−0.599265
$178$	0	0
$179$	−8.83296e6	−1.54009	−0.770047	−	0.637987i	$-0.779767\pi$
$179$	−8.83296e6	−1.54009	−0.770047	+	0.637987i	$0.779767\pi$
$180$	0	0
$181$	6.00091e6i	1.01200i	0.862533	+	0.506001i	$0.168877\pi$
$181$	6.00091e6i	1.01200i	−0.862533	+	0.506001i	$0.831123\pi$
$182$	0	0
$183$	1.30888e6i	0.213573i
$184$	0	0
$185$	2.37920e6	0.375765
$186$	0	0
$187$	−1.20773e7	−1.84690
$188$	0	0
$189$	1.05121e6i	0.155705i
$190$	0	0
$191$	603085.i	0.0865522i	0.999063	+	0.0432761i	$0.0137795\pi$
$191$	603085.i	0.0865522i	−0.999063	+	0.0432761i	$0.986220\pi$
$192$	0	0
$193$	7.75593e6	1.07885	0.539426	−	0.842033i	$-0.318641\pi$
$193$	7.75593e6	1.07885	0.539426	+	0.842033i	$0.318641\pi$
$194$	0	0
$195$	−3.79437e6	−0.511724
$196$	0	0
$197$	1.91770e6i	0.250832i	0.992104	+	0.125416i	$0.0400265\pi$
$197$	1.91770e6i	0.250832i	−0.992104	+	0.125416i	$0.959974\pi$
$198$	0	0
$199$	− 1.01444e7i	− 1.28726i	−0.765337	−	0.643629i	$-0.777428\pi$
$199$	− 1.01444e7i	− 1.28726i	0.765337	−	0.643629i	$-0.222572\pi$
$200$	0	0
$201$	−5.81603e6	−0.716207
$202$	0	0
$203$	−2.29735e6	−0.274624
$204$	0	0
$205$	− 1.06871e7i	− 1.24050i
$206$	0	0
$207$	3.09307e6i	0.348721i
$208$	0	0
$209$	−7.71741e6	−0.845342
$210$	0	0
$211$	9.54322e6	1.01589	0.507946	−	0.861389i	$-0.330405\pi$
$211$	9.54322e6	1.01589	0.507946	+	0.861389i	$0.330405\pi$
$212$	0	0
$213$	− 1.04102e7i	− 1.07726i
$214$	0	0
$215$	8.87410e6i	0.892912i
$216$	0	0
$217$	−1.21938e7	−1.19333
$218$	0	0
$219$	−6.22692e6	−0.592845
$220$	0	0
$221$	8.58805e6i	0.795642i
$222$	0	0
$223$	1.58078e7i	1.42547i	0.701434	+	0.712735i	$0.252544\pi$
$223$	1.58078e7i	1.42547i	−0.701434	+	0.712735i	$0.747456\pi$
$224$	0	0
$225$	−5.46549e6	−0.479823
$226$	0	0
$227$	1.48574e7	1.27018	0.635092	−	0.772437i	$-0.280962\pi$
$227$	1.48574e7	1.27018	0.635092	+	0.772437i	$0.280962\pi$
$228$	0	0
$229$	− 1.28465e6i	− 0.106974i	−0.998569	−	0.0534872i	$-0.982966\pi$
$229$	− 1.28465e6i	− 0.106974i	0.998569	−	0.0534872i	$-0.0170336\pi$
$230$	0	0
$231$	− 7.58464e6i	− 0.615317i
$232$	0	0
$233$	262969.	0.0207892	0.0103946	−	0.999946i	$-0.496691\pi$
$233$	262969.	0.0207892	0.0103946	+	0.999946i	$0.496691\pi$
$234$	0	0
$235$	−2.97414e7	−2.29170
$236$	0	0
$237$	− 6.78684e6i	− 0.509827i
$238$	0	0
$239$	1.89260e7i	1.38633i	0.720781	+	0.693163i	$0.243783\pi$
$239$	1.89260e7i	1.38633i	−0.720781	+	0.693163i	$0.756217\pi$
$240$	0	0
$241$	−1.91538e6	−0.136837	−0.0684185	−	0.997657i	$-0.521795\pi$
$241$	−1.91538e6	−0.136837	−0.0684185	+	0.997657i	$0.521795\pi$
$242$	0	0
$243$	920483.	0.0641500
$244$	0	0
$245$	7.93382e6i	0.539491i
$246$	0	0
$247$	5.48778e6i	0.364172i
$248$	0	0
$249$	3.84385e6	0.248982
$250$	0	0
$251$	−1.10676e7	−0.699895	−0.349948	−	0.936769i	$-0.613801\pi$
$251$	−1.10676e7	−0.699895	−0.349948	+	0.936769i	$0.613801\pi$
$252$	0	0
$253$	− 2.23170e7i	− 1.37808i
$254$	0	0
$255$	2.09641e7i	1.26432i
$256$	0	0
$257$	678718.	0.0399843	0.0199922	−	0.999800i	$-0.493636\pi$
$257$	678718.	0.0399843	0.0199922	+	0.999800i	$0.493636\pi$
$258$	0	0
$259$	3.38184e6	0.194650
$260$	0	0
$261$	2.01166e6i	0.113144i
$262$	0	0
$263$	− 1.11128e7i	− 0.610882i	−0.952211	−	0.305441i	$-0.901196\pi$
$263$	− 1.11128e7i	− 0.610882i	0.952211	−	0.305441i	$-0.0988039\pi$
$264$	0	0
$265$	5.32114e7	2.85935
$266$	0	0
$267$	1.26216e6	0.0663101
$268$	0	0
$269$	1.00441e7i	0.516007i	0.966144	+	0.258004i	$0.0830647\pi$
$269$	1.00441e7i	0.516007i	−0.966144	+	0.258004i	$0.916935\pi$
$270$	0	0
$271$	− 6.62102e6i	− 0.332673i	−0.986069	−	0.166336i	$-0.946806\pi$
$271$	− 6.62102e6i	− 0.332673i	0.986069	−	0.166336i	$-0.0531938\pi$
$272$	0	0
$273$	−5.39337e6	−0.265077
$274$	0	0
$275$	3.94345e7	1.89617
$276$	0	0
$277$	2.44357e7i	1.14970i	0.818257	+	0.574852i	$0.194940\pi$
$277$	2.44357e7i	1.14970i	−0.818257	+	0.574852i	$0.805060\pi$
$278$	0	0
$279$	1.06775e7i	0.491649i
$280$	0	0
$281$	−1.39627e7	−0.629288	−0.314644	−	0.949210i	$-0.601885\pi$
$281$	−1.39627e7	−0.629288	−0.314644	+	0.949210i	$0.601885\pi$
$282$	0	0
$283$	4.28834e7	1.89204	0.946018	−	0.324113i	$-0.105066\pi$
$283$	4.28834e7	1.89204	0.946018	+	0.324113i	$0.105066\pi$
$284$	0	0
$285$	1.33961e7i	0.578688i
$286$	0	0
$287$	− 1.51908e7i	− 0.642591i
$288$	0	0
$289$	2.33119e7	0.965792
$290$	0	0
$291$	1.36716e7	0.554803
$292$	0	0
$293$	− 5.00254e6i	− 0.198878i	−0.995044	−	0.0994392i	$-0.968295\pi$
$293$	− 5.00254e6i	− 0.198878i	0.995044	−	0.0994392i	$-0.0317049\pi$
$294$	0	0
$295$	− 4.16191e7i	− 1.62116i
$296$	0	0
$297$	−6.64145e6	−0.253509
$298$	0	0
$299$	−1.58695e7	−0.593675
$300$	0	0
$301$	1.26138e7i	0.462536i
$302$	0	0
$303$	− 2.81357e7i	− 1.01142i
$304$	0	0
$305$	−1.63929e7	−0.577771
$306$	0	0
$307$	2.73510e7	0.945274	0.472637	−	0.881257i	$-0.343302\pi$
$307$	2.73510e7	0.945274	0.472637	+	0.881257i	$0.343302\pi$
$308$	0	0
$309$	1.28802e7i	0.436564i
$310$	0	0
$311$	3.37365e7i	1.12155i	0.827968	+	0.560775i	$0.189497\pi$
$311$	3.37365e7i	1.12155i	−0.827968	+	0.560775i	$0.810503\pi$
$312$	0	0
$313$	4.91891e7	1.60412	0.802058	−	0.597246i	$-0.203738\pi$
$313$	4.91891e7	1.60412	0.802058	+	0.597246i	$0.203738\pi$
$314$	0	0
$315$	−1.31657e7	−0.421222
$316$	0	0
$317$	− 1.97091e7i	− 0.618712i	−0.950946	−	0.309356i	$-0.899887\pi$
$317$	− 1.97091e7i	− 0.618712i	0.950946	−	0.309356i	$-0.100113\pi$
$318$	0	0
$319$	− 1.45145e7i	− 0.447126i
$320$	0	0
$321$	−2.09800e7	−0.634293
$322$	0	0
$323$	3.03203e7	0.899758
$324$	0	0
$325$	− 2.80415e7i	− 0.816867i
$326$	0	0
$327$	− 1.09840e7i	− 0.314135i
$328$	0	0
$329$	−4.22748e7	−1.18712
$330$	0	0
$331$	−1.59711e7	−0.440403	−0.220202	−	0.975454i	$-0.570672\pi$
$331$	−1.59711e7	−0.440403	−0.220202	+	0.975454i	$0.570672\pi$
$332$	0	0
$333$	− 2.96129e6i	− 0.0801951i
$334$	0	0
$335$	− 7.28419e7i	− 1.93752i
$336$	0	0
$337$	9.18400e6	0.239962	0.119981	−	0.992776i	$-0.461717\pi$
$337$	9.18400e6	0.239962	0.119981	+	0.992776i	$0.461717\pi$
$338$	0	0
$339$	3.97930e7	1.02143
$340$	0	0
$341$	− 7.70398e7i	− 1.94291i
$342$	0	0
$343$	4.39260e7i	1.08853i
$344$	0	0
$345$	−3.87386e7	−0.943380
$346$	0	0
$347$	−3.95976e7	−0.947721	−0.473861	−	0.880600i	$-0.657140\pi$
$347$	−3.95976e7	−0.947721	−0.473861	+	0.880600i	$0.657140\pi$
$348$	0	0
$349$	7.85281e7i	1.84735i	0.383180	+	0.923674i	$0.374829\pi$
$349$	7.85281e7i	1.84735i	−0.383180	+	0.923674i	$0.625171\pi$
$350$	0	0
$351$	4.72268e6i	0.109211i
$352$	0	0
$353$	6.58915e7	1.49798	0.748989	−	0.662582i	$-0.230540\pi$
$353$	6.58915e7	1.49798	0.748989	+	0.662582i	$0.230540\pi$
$354$	0	0
$355$	1.30380e8	2.91425
$356$	0	0
$357$	2.97987e7i	0.654926i
$358$	0	0
$359$	9.24524e6i	0.199818i	0.994997	+	0.0999090i	$0.0318552\pi$
$359$	9.24524e6i	0.199818i	−0.994997	+	0.0999090i	$0.968145\pi$
$360$	0	0
$361$	−2.76711e7	−0.588173
$362$	0	0
$363$	2.03033e7	0.424471
$364$	0	0
$365$	− 7.79880e7i	− 1.60380i
$366$	0	0
$367$	1.05595e7i	0.213622i	0.994279	+	0.106811i	$0.0340640\pi$
$367$	1.05595e7i	0.213622i	−0.994279	+	0.106811i	$0.965936\pi$
$368$	0	0
$369$	−1.33017e7	−0.264746
$370$	0	0
$371$	7.56354e7	1.48117
$372$	0	0
$373$	− 1.92271e7i	− 0.370499i	−0.982691	−	0.185250i	$-0.940691\pi$
$373$	− 1.92271e7i	− 0.370499i	0.982691	−	0.185250i	$-0.0593094\pi$
$374$	0	0
$375$	− 2.08982e7i	− 0.396292i
$376$	0	0
$377$	−1.03211e7	−0.192621
$378$	0	0
$379$	1.53936e7	0.282764	0.141382	−	0.989955i	$-0.454845\pi$
$379$	1.53936e7	0.282764	0.141382	+	0.989955i	$0.454845\pi$
$380$	0	0
$381$	1.04472e7i	0.188897i
$382$	0	0
$383$	1.18517e7i	0.210953i	0.994422	+	0.105476i	$0.0336367\pi$
$383$	1.18517e7i	0.210953i	−0.994422	+	0.105476i	$0.966363\pi$
$384$	0	0
$385$	9.49926e7	1.66459
$386$	0	0
$387$	1.10452e7	0.190564
$388$	0	0
$389$	5.48941e7i	0.932561i	0.884637	+	0.466281i	$0.154406\pi$
$389$	5.48941e7i	0.932561i	−0.884637	+	0.466281i	$0.845594\pi$
$390$	0	0
$391$	8.76796e7i	1.46679i
$392$	0	0
$393$	−2.69886e7	−0.444634
$394$	0	0
$395$	8.50007e7	1.37921
$396$	0	0
$397$	5.07085e7i	0.810419i	0.914224	+	0.405209i	$0.132801\pi$
$397$	5.07085e7i	0.810419i	−0.914224	+	0.405209i	$0.867199\pi$
$398$	0	0
$399$	1.90414e7i	0.299765i
$400$	0	0
$401$	−1.11684e8	−1.73204	−0.866021	−	0.500007i	$-0.833331\pi$
$401$	−1.11684e8	−1.73204	−0.866021	+	0.500007i	$0.833331\pi$
$402$	0	0
$403$	−5.47823e7	−0.837000
$404$	0	0
$405$	1.15284e7i	0.173542i
$406$	0	0
$407$	2.13662e7i	0.316916i
$408$	0	0
$409$	8.03112e7	1.17383	0.586917	−	0.809647i	$-0.300342\pi$
$409$	8.03112e7	1.17383	0.586917	+	0.809647i	$0.300342\pi$
$410$	0	0
$411$	−4.32353e7	−0.622749
$412$	0	0
$413$	− 5.91580e7i	− 0.839776i
$414$	0	0
$415$	4.81417e7i	0.673561i
$416$	0	0
$417$	−6.71928e7	−0.926648
$418$	0	0
$419$	8.64533e7	1.17527	0.587637	−	0.809125i	$-0.300058\pi$
$419$	8.64533e7	1.17527	0.587637	+	0.809125i	$0.300058\pi$
$420$	0	0
$421$	− 1.94432e7i	− 0.260569i	−0.991477	−	0.130284i	$-0.958411\pi$
$421$	− 1.94432e7i	− 0.260569i	0.991477	−	0.130284i	$-0.0415890\pi$
$422$	0	0
$423$	3.70177e7i	0.489089i
$424$	0	0
$425$	−1.54931e8	−2.01823
$426$	0	0
$427$	−2.33011e7	−0.299290
$428$	0	0
$429$	− 3.40750e7i	− 0.431582i
$430$	0	0
$431$	1.05362e8i	1.31599i	0.753021	+	0.657996i	$0.228596\pi$
$431$	1.05362e8i	1.31599i	−0.753021	+	0.657996i	$0.771404\pi$
$432$	0	0
$433$	1.32564e8	1.63291	0.816457	−	0.577407i	$-0.195935\pi$
$433$	1.32564e8	1.63291	0.816457	+	0.577407i	$0.195935\pi$
$434$	0	0
$435$	−2.51947e7	−0.306085
$436$	0	0
$437$	5.60275e7i	0.671362i
$438$	0	0
$439$	− 5.71422e7i	− 0.675403i	−0.941253	−	0.337702i	$-0.890351\pi$
$439$	− 5.71422e7i	− 0.675403i	0.941253	−	0.337702i	$-0.109649\pi$
$440$	0	0
$441$	9.87485e6	0.115137
$442$	0	0
$443$	2.10479e7	0.242102	0.121051	−	0.992646i	$-0.461374\pi$
$443$	2.10479e7	0.242102	0.121051	+	0.992646i	$0.461374\pi$
$444$	0	0
$445$	1.58077e7i	0.179386i
$446$	0	0
$447$	− 3.32402e6i	− 0.0372169i
$448$	0	0
$449$	−6.46748e7	−0.714490	−0.357245	−	0.934011i	$-0.616284\pi$
$449$	−6.46748e7	−0.714490	−0.357245	+	0.934011i	$0.616284\pi$
$450$	0	0
$451$	9.59744e7	1.04623
$452$	0	0
$453$	8.31971e7i	0.894981i
$454$	0	0
$455$	− 6.75484e7i	− 0.717102i
$456$	0	0
$457$	5.37459e7	0.563114	0.281557	−	0.959544i	$-0.409149\pi$
$457$	5.37459e7	0.563114	0.281557	+	0.959544i	$0.409149\pi$
$458$	0	0
$459$	2.60931e7	0.269828
$460$	0	0
$461$	2.36640e7i	0.241538i	0.992681	+	0.120769i	$0.0385360\pi$
$461$	2.36640e7i	0.241538i	−0.992681	+	0.120769i	$0.961464\pi$
$462$	0	0
$463$	− 2.80452e7i	− 0.282563i	−0.989969	−	0.141281i	$-0.954878\pi$
$463$	− 2.80452e7i	− 0.282563i	0.989969	−	0.141281i	$-0.0451222\pi$
$464$	0	0
$465$	−1.33728e8	−1.33004
$466$	0	0
$467$	−3.61101e7	−0.354550	−0.177275	−	0.984161i	$-0.556728\pi$
$467$	−3.61101e7	−0.354550	−0.177275	+	0.984161i	$0.556728\pi$
$468$	0	0
$469$	− 1.03539e8i	− 1.00365i
$470$	0	0
$471$	− 6.37536e7i	− 0.610157i
$472$	0	0
$473$	−7.96930e7	−0.753072
$474$	0	0
$475$	−9.90012e7	−0.923761
$476$	0	0
$477$	− 6.62298e7i	− 0.610236i
$478$	0	0
$479$	− 4.70272e7i	− 0.427900i	−0.976845	−	0.213950i	$-0.931367\pi$
$479$	− 4.70272e7i	− 0.427900i	0.976845	−	0.213950i	$-0.0686330\pi$
$480$	0	0
$481$	1.51933e7	0.136527
$482$	0	0
$483$	−5.50636e7	−0.488679
$484$	0	0
$485$	1.71227e8i	1.50088i
$486$	0	0
$487$	− 9.56947e7i	− 0.828516i	−0.910160	−	0.414258i	$-0.864041\pi$
$487$	− 9.56947e7i	− 0.828516i	0.910160	−	0.414258i	$-0.135959\pi$
$488$	0	0
$489$	2.40259e7	0.205472
$490$	0	0
$491$	−9.69668e7	−0.819178	−0.409589	−	0.912270i	$-0.634328\pi$
$491$	−9.69668e7	−0.819178	−0.409589	+	0.912270i	$0.634328\pi$
$492$	0	0
$493$	5.70248e7i	0.475908i
$494$	0	0
$495$	− 8.31797e7i	− 0.685807i
$496$	0	0
$497$	1.85324e8	1.50961
$498$	0	0
$499$	5.16768e7	0.415905	0.207952	−	0.978139i	$-0.433320\pi$
$499$	5.16768e7	0.415905	0.207952	+	0.978139i	$0.433320\pi$
$500$	0	0
$501$	− 1.19494e8i	− 0.950240i
$502$	0	0
$503$	− 2.28417e8i	− 1.79483i	−0.441185	−	0.897416i	$-0.645442\pi$
$503$	− 2.28417e8i	− 1.79483i	0.441185	−	0.897416i	$-0.354558\pi$
$504$	0	0
$505$	3.52381e8	2.73614
$506$	0	0
$507$	5.10121e7	0.391426
$508$	0	0
$509$	1.31906e8i	1.00025i	0.865952	+	0.500127i	$0.166713\pi$
$509$	1.31906e8i	1.00025i	−0.865952	+	0.500127i	$0.833287\pi$
$510$	0	0
$511$	− 1.10853e8i	− 0.830780i
$512$	0	0
$513$	1.66735e7	0.123502
$514$	0	0
$515$	−1.61316e8	−1.18102
$516$	0	0
$517$	− 2.67089e8i	− 1.93279i
$518$	0	0
$519$	1.56477e8i	1.11931i
$520$	0	0
$521$	1.21687e7	0.0860462	0.0430231	−	0.999074i	$-0.486301\pi$
$521$	1.21687e7	0.0860462	0.0430231	+	0.999074i	$0.486301\pi$
$522$	0	0
$523$	−8.96786e7	−0.626879	−0.313440	−	0.949608i	$-0.601481\pi$
$523$	−8.96786e7	−0.626879	−0.313440	+	0.949608i	$0.601481\pi$
$524$	0	0
$525$	− 9.72980e7i	− 0.672398i
$526$	0	0
$527$	3.02675e8i	2.06797i
$528$	0	0
$529$	−1.39832e7	−0.0944579
$530$	0	0
$531$	−5.18014e7	−0.345986
$532$	0	0
$533$	− 6.82466e7i	− 0.450712i
$534$	0	0
$535$	− 2.62760e8i	− 1.71592i
$536$	0	0
$537$	−1.37692e8	−0.889174
$538$	0	0
$539$	−7.12488e7	−0.455000
$540$	0	0
$541$	− 2.72097e8i	− 1.71843i	−0.511613	−	0.859216i	$-0.670952\pi$
$541$	− 2.72097e8i	− 1.71843i	0.511613	−	0.859216i	$-0.329048\pi$
$542$	0	0
$543$	9.35449e7i	0.584279i
$544$	0	0
$545$	1.37567e8	0.849815
$546$	0	0
$547$	−1.84749e8	−1.12881	−0.564404	−	0.825499i	$-0.690894\pi$
$547$	−1.84749e8	−1.12881	−0.564404	+	0.825499i	$0.690894\pi$
$548$	0	0
$549$	2.04035e7i	0.123307i
$550$	0	0
$551$	3.64390e7i	0.217827i
$552$	0	0
$553$	1.20821e8	0.714443
$554$	0	0
$555$	3.70881e7	0.216948
$556$	0	0
$557$	9.60239e7i	0.555666i	0.960629	+	0.277833i	$0.0896162\pi$
$557$	9.60239e7i	0.555666i	−0.960629	+	0.277833i	$0.910384\pi$
$558$	0	0
$559$	5.66690e7i	0.324422i
$560$	0	0
$561$	−1.88266e8	−1.06631
$562$	0	0
$563$	−2.37375e8	−1.33018	−0.665089	−	0.746764i	$-0.731607\pi$
$563$	−2.37375e8	−1.33018	−0.665089	+	0.746764i	$0.731607\pi$
$564$	0	0
$565$	4.98381e8i	2.76323i
$566$	0	0
$567$	1.63867e7i	0.0898963i
$568$	0	0
$569$	−3.26674e8	−1.77328	−0.886641	−	0.462458i	$-0.846968\pi$
$569$	−3.26674e8	−1.77328	−0.886641	+	0.462458i	$0.846968\pi$
$570$	0	0
$571$	−971103.	−0.00521623	−0.00260812	−	0.999997i	$-0.500830\pi$
$571$	−971103.	−0.00521623	−0.00260812	+	0.999997i	$0.500830\pi$
$572$	0	0
$573$	9.40116e6i	0.0499709i
$574$	0	0
$575$	− 2.86290e8i	− 1.50592i
$576$	0	0
$577$	1.87315e8	0.975092	0.487546	−	0.873097i	$-0.337892\pi$
$577$	1.87315e8	0.975092	0.487546	+	0.873097i	$0.337892\pi$
$578$	0	0
$579$	1.20903e8	0.622876
$580$	0	0
$581$	6.84293e7i	0.348910i
$582$	0	0
$583$	4.77860e8i	2.41154i
$584$	0	0
$585$	−5.91484e7	−0.295444
$586$	0	0
$587$	−8.47232e7	−0.418878	−0.209439	−	0.977822i	$-0.567164\pi$
$587$	−8.47232e7	−0.418878	−0.209439	+	0.977822i	$0.567164\pi$
$588$	0	0
$589$	1.93410e8i	0.946528i
$590$	0	0
$591$	2.98940e7i	0.144818i
$592$	0	0
$593$	−2.30616e8	−1.10592	−0.552962	−	0.833206i	$-0.686503\pi$
$593$	−2.30616e8	−1.10592	−0.552962	+	0.833206i	$0.686503\pi$
$594$	0	0
$595$	−3.73209e8	−1.77174
$596$	0	0
$597$	− 1.58135e8i	− 0.743199i
$598$	0	0
$599$	3.65098e8i	1.69875i	0.527791	+	0.849374i	$0.323020\pi$
$599$	3.65098e8i	1.69875i	−0.527791	+	0.849374i	$0.676980\pi$
$600$	0	0
$601$	2.63307e8	1.21294	0.606469	−	0.795107i	$-0.292585\pi$
$601$	2.63307e8	1.21294	0.606469	+	0.795107i	$0.292585\pi$
$602$	0	0
$603$	−9.06630e7	−0.413502
$604$	0	0
$605$	2.54286e8i	1.14830i
$606$	0	0
$607$	1.38978e8i	0.621413i	0.950506	+	0.310707i	$0.100566\pi$
$607$	1.38978e8i	0.621413i	−0.950506	+	0.310707i	$0.899434\pi$
$608$	0	0
$609$	−3.58121e7	−0.158554
$610$	0	0
$611$	−1.89925e8	−0.832642
$612$	0	0
$613$	− 4.20201e8i	− 1.82421i	−0.409953	−	0.912106i	$-0.634455\pi$
$613$	− 4.20201e8i	− 1.82421i	0.409953	−	0.912106i	$-0.365545\pi$
$614$	0	0
$615$	− 1.66595e8i	− 0.716205i
$616$	0	0
$617$	1.25856e7	0.0535818	0.0267909	−	0.999641i	$-0.491471\pi$
$617$	1.25856e7	0.0535818	0.0267909	+	0.999641i	$0.491471\pi$
$618$	0	0
$619$	2.43243e8	1.02558	0.512789	−	0.858515i	$-0.328612\pi$
$619$	2.43243e8	1.02558	0.512789	+	0.858515i	$0.328612\pi$
$620$	0	0
$621$	4.82161e7i	0.201334i
$622$	0	0
$623$	2.24693e7i	0.0929234i
$624$	0	0
$625$	−8.96966e7	−0.367397
$626$	0	0
$627$	−1.20302e8	−0.488059
$628$	0	0
$629$	− 8.39440e7i	− 0.337317i
$630$	0	0
$631$	5.45064e7i	0.216950i	0.994099	+	0.108475i	$0.0345967\pi$
$631$	5.45064e7i	0.216950i	−0.994099	+	0.108475i	$0.965403\pi$
$632$	0	0
$633$	1.48764e8	0.586526
$634$	0	0
$635$	−1.30844e8	−0.511016
$636$	0	0
$637$	5.06644e7i	0.196013i
$638$	0	0
$639$	− 1.62278e8i	− 0.621954i
$640$	0	0
$641$	−1.69420e8	−0.643268	−0.321634	−	0.946864i	$-0.604232\pi$
$641$	−1.69420e8	−0.643268	−0.321634	+	0.946864i	$0.604232\pi$
$642$	0	0
$643$	−4.20644e8	−1.58227	−0.791137	−	0.611639i	$-0.790510\pi$
$643$	−4.20644e8	−1.58227	−0.791137	+	0.611639i	$0.790510\pi$
$644$	0	0
$645$	1.38333e8i	0.515523i
$646$	0	0
$647$	3.19909e8i	1.18117i	0.806974	+	0.590587i	$0.201104\pi$
$647$	3.19909e8i	1.18117i	−0.806974	+	0.590587i	$0.798896\pi$
$648$	0	0
$649$	3.73756e8	1.36727
$650$	0	0
$651$	−1.90083e8	−0.688970
$652$	0	0
$653$	− 2.98645e8i	− 1.07254i	−0.844045	−	0.536272i	$-0.819832\pi$
$653$	− 2.98645e8i	− 1.07254i	0.844045	−	0.536272i	$-0.180168\pi$
$654$	0	0
$655$	− 3.38014e8i	− 1.20285i
$656$	0	0
$657$	−9.70681e7	−0.342279
$658$	0	0
$659$	−3.27357e8	−1.14384	−0.571920	−	0.820309i	$-0.693801\pi$
$659$	−3.27357e8	−1.14384	−0.571920	+	0.820309i	$0.693801\pi$
$660$	0	0
$661$	4.03255e8i	1.39629i	0.715956	+	0.698145i	$0.245991\pi$
$661$	4.03255e8i	1.39629i	−0.715956	+	0.698145i	$0.754009\pi$
$662$	0	0
$663$	1.33874e8i	0.459364i
$664$	0	0
$665$	−2.38481e8	−0.810941
$666$	0	0
$667$	−1.05373e8	−0.355103
$668$	0	0
$669$	2.46420e8i	0.822995i
$670$	0	0
$671$	− 1.47215e8i	− 0.487285i
$672$	0	0
$673$	−1.48356e8	−0.486697	−0.243349	−	0.969939i	$-0.578246\pi$
$673$	−1.48356e8	−0.486697	−0.243349	+	0.969939i	$0.578246\pi$
$674$	0	0
$675$	−8.51985e7	−0.277026
$676$	0	0
$677$	− 4.50729e8i	− 1.45261i	−0.687371	−	0.726306i	$-0.741235\pi$
$677$	− 4.50729e8i	− 1.45261i	0.687371	−	0.726306i	$-0.258765\pi$
$678$	0	0
$679$	2.43385e8i	0.777470i
$680$	0	0
$681$	2.31605e8	0.733341
$682$	0	0
$683$	5.93405e8	1.86247	0.931235	−	0.364418i	$-0.118732\pi$
$683$	5.93405e8	1.86247	0.931235	+	0.364418i	$0.118732\pi$
$684$	0	0
$685$	− 5.41493e8i	− 1.68470i
$686$	0	0
$687$	− 2.00258e7i	− 0.0617617i
$688$	0	0
$689$	3.39802e8	1.03889
$690$	0	0
$691$	−2.84794e8	−0.863170	−0.431585	−	0.902072i	$-0.642045\pi$
$691$	−2.84794e8	−0.863170	−0.431585	+	0.902072i	$0.642045\pi$
$692$	0	0
$693$	− 1.18233e8i	− 0.355254i
$694$	0	0
$695$	− 8.41545e8i	− 2.50682i
$696$	0	0
$697$	−3.77066e8	−1.11357
$698$	0	0
$699$	4.09928e6	0.0120026
$700$	0	0
$701$	− 1.51280e8i	− 0.439165i	−0.975594	−	0.219583i	$-0.929530\pi$
$701$	− 1.51280e8i	− 0.439165i	0.975594	−	0.219583i	$-0.0704695\pi$
$702$	0	0
$703$	− 5.36404e7i	− 0.154392i
$704$	0	0
$705$	−4.63622e8	−1.32311
$706$	0	0
$707$	5.00880e8	1.41734
$708$	0	0
$709$	6.04474e8i	1.69605i	0.529956	+	0.848025i	$0.322208\pi$
$709$	6.04474e8i	1.69605i	−0.529956	+	0.848025i	$0.677792\pi$
$710$	0	0
$711$	− 1.05796e8i	− 0.294349i
$712$	0	0
$713$	−5.59300e8	−1.54304
$714$	0	0
$715$	4.26766e8	1.16754
$716$	0	0
$717$	2.95027e8i	0.800396i
$718$	0	0
$719$	− 1.92148e8i	− 0.516951i	−0.966018	−	0.258476i	$-0.916780\pi$
$719$	− 1.92148e8i	− 0.516951i	0.966018	−	0.258476i	$-0.0832201\pi$
$720$	0	0
$721$	−2.29297e8	−0.611777
$722$	0	0
$723$	−2.98578e7	−0.0790029
$724$	0	0
$725$	− 1.86196e8i	− 0.488604i
$726$	0	0
$727$	− 1.33711e8i	− 0.347989i	−0.984747	−	0.173994i	$-0.944333\pi$
$727$	− 1.33711e8i	− 0.347989i	0.984747	−	0.173994i	$-0.0556675\pi$
$728$	0	0
$729$	1.43489e7	0.0370370
$730$	0	0
$731$	3.13099e8	0.801549
$732$	0	0
$733$	− 5.99936e8i	− 1.52333i	−0.647973	−	0.761663i	$-0.724383\pi$
$733$	− 5.99936e8i	− 1.52333i	0.647973	−	0.761663i	$-0.275617\pi$
$734$	0	0
$735$	1.23676e8i	0.311475i
$736$	0	0
$737$	6.54150e8	1.63409
$738$	0	0
$739$	4.83646e7	0.119838	0.0599190	−	0.998203i	$-0.480916\pi$
$739$	4.83646e7	0.119838	0.0599190	+	0.998203i	$0.480916\pi$
$740$	0	0
$741$	8.55461e7i	0.210255i
$742$	0	0
$743$	3.69904e7i	0.0901826i	0.998983	+	0.0450913i	$0.0143579\pi$
$743$	3.69904e7i	0.0901826i	−0.998983	+	0.0450913i	$0.985642\pi$
$744$	0	0
$745$	4.16311e7	0.100681
$746$	0	0
$747$	5.99197e7	0.143750
$748$	0	0
$749$	− 3.73491e8i	− 0.888863i
$750$	0	0
$751$	1.99551e8i	0.471123i	0.971859	+	0.235562i	$0.0756929\pi$
$751$	1.99551e8i	0.471123i	−0.971859	+	0.235562i	$0.924307\pi$
$752$	0	0
$753$	−1.72527e8	−0.404085
$754$	0	0
$755$	−1.04199e9	−2.42115
$756$	0	0
$757$	4.79819e8i	1.10609i	0.833152	+	0.553045i	$0.186534\pi$
$757$	4.79819e8i	1.10609i	−0.833152	+	0.553045i	$0.813466\pi$
$758$	0	0
$759$	− 3.47888e8i	− 0.795636i
$760$	0	0
$761$	−2.43069e8	−0.551537	−0.275769	−	0.961224i	$-0.588932\pi$
$761$	−2.43069e8	−0.551537	−0.275769	+	0.961224i	$0.588932\pi$
$762$	0	0
$763$	1.95540e8	0.440211
$764$	0	0
$765$	3.26798e8i	0.729953i
$766$	0	0
$767$	− 2.65775e8i	− 0.589017i
$768$	0	0
$769$	4.11830e8	0.905605	0.452802	−	0.891611i	$-0.350424\pi$
$769$	4.11830e8	0.905605	0.452802	+	0.891611i	$0.350424\pi$
$770$	0	0
$771$	1.05802e7	0.0230850
$772$	0	0
$773$	9.17077e7i	0.198549i	0.995060	+	0.0992744i	$0.0316522\pi$
$773$	9.17077e7i	0.198549i	−0.995060	+	0.0992744i	$0.968348\pi$
$774$	0	0
$775$	− 9.88289e8i	− 2.12314i
$776$	0	0
$777$	5.27176e7	0.112381
$778$	0	0
$779$	−2.40946e8	−0.509692
$780$	0	0
$781$	1.17087e9i	2.45785i
$782$	0	0
$783$	3.13587e7i	0.0653240i
$784$	0	0
$785$	7.98471e8	1.65063
$786$	0	0
$787$	−5.23454e8	−1.07388	−0.536938	−	0.843622i	$-0.680419\pi$
$787$	−5.23454e8	−1.07388	−0.536938	+	0.843622i	$0.680419\pi$
$788$	0	0
$789$	− 1.73232e8i	− 0.352693i
$790$	0	0
$791$	7.08406e8i	1.43137i
$792$	0	0
$793$	−1.04683e8	−0.209921
$794$	0	0
$795$	8.29483e8	1.65085
$796$	0	0
$797$	− 1.58874e7i	− 0.0313818i	−0.999877	−	0.0156909i	$-0.995005\pi$
$797$	− 1.58874e7i	− 0.0313818i	0.999877	−	0.0156909i	$-0.00499477\pi$
$798$	0	0
$799$	1.04935e9i	2.05721i
$800$	0	0
$801$	1.96751e7	0.0382842
$802$	0	0
$803$	7.00364e8	1.35262
$804$	0	0
$805$	− 6.89635e8i	− 1.32200i
$806$	0	0
$807$	1.56573e8i	0.297917i
$808$	0	0
$809$	−5.17580e8	−0.977534	−0.488767	−	0.872414i	$-0.662553\pi$
$809$	−5.17580e8	−0.977534	−0.488767	+	0.872414i	$0.662553\pi$
$810$	0	0
$811$	9.43642e8	1.76907	0.884534	−	0.466476i	$-0.154476\pi$
$811$	9.43642e8	1.76907	0.884534	+	0.466476i	$0.154476\pi$
$812$	0	0
$813$	− 1.03212e8i	− 0.192069i
$814$	0	0
$815$	3.00908e8i	0.555855i
$816$	0	0
$817$	2.00071e8	0.366875
$818$	0	0
$819$	−8.40744e7	−0.153042
$820$	0	0
$821$	− 6.51239e8i	− 1.17682i	−0.808562	−	0.588411i	$-0.799754\pi$
$821$	− 6.51239e8i	− 1.17682i	0.808562	−	0.588411i	$-0.200246\pi$
$822$	0	0
$823$	1.12175e8i	0.201231i	0.994925	+	0.100615i	$0.0320812\pi$
$823$	1.12175e8i	0.201231i	−0.994925	+	0.100615i	$0.967919\pi$
$824$	0	0
$825$	6.14722e8	1.09476
$826$	0	0
$827$	6.24440e8	1.10401	0.552007	−	0.833840i	$-0.313862\pi$
$827$	6.24440e8	1.10401	0.552007	+	0.833840i	$0.313862\pi$
$828$	0	0
$829$	− 5.96016e8i	− 1.04615i	−0.852286	−	0.523076i	$-0.824784\pi$
$829$	− 5.96016e8i	− 1.04615i	0.852286	−	0.523076i	$-0.175216\pi$
$830$	0	0
$831$	3.80915e8i	0.663782i
$832$	0	0
$833$	2.79924e8	0.484289
$834$	0	0
$835$	1.49658e9	2.57064
$836$	0	0
$837$	1.66445e8i	0.283854i
$838$	0	0
$839$	4.46780e7i	0.0756497i	0.999284	+	0.0378249i	$0.0120429\pi$
$839$	4.46780e7i	0.0756497i	−0.999284	+	0.0378249i	$0.987957\pi$
$840$	0	0
$841$	5.26291e8	0.884785
$842$	0	0
$843$	−2.17656e8	−0.363320
$844$	0	0
$845$	6.38892e8i	1.05891i
$846$	0	0
$847$	3.61446e8i	0.594830i
$848$	0	0
$849$	6.68485e8	1.09237
$850$	0	0
$851$	1.55116e8	0.251691
$852$	0	0
$853$	1.12761e9i	1.81683i	0.418072	+	0.908414i	$0.362706\pi$
$853$	1.12761e9i	1.81683i	−0.418072	+	0.908414i	$0.637294\pi$
$854$	0	0
$855$	2.08825e8i	0.334105i
$856$	0	0
$857$	5.47949e8	0.870557	0.435279	−	0.900296i	$-0.356650\pi$
$857$	5.47949e8	0.870557	0.435279	+	0.900296i	$0.356650\pi$
$858$	0	0
$859$	−6.34544e8	−1.00111	−0.500556	−	0.865704i	$-0.666871\pi$
$859$	−6.34544e8	−1.00111	−0.500556	+	0.865704i	$0.666871\pi$
$860$	0	0
$861$	− 2.36801e8i	− 0.371000i
$862$	0	0
$863$	1.20080e9i	1.86826i	0.356928	+	0.934132i	$0.383824\pi$
$863$	1.20080e9i	1.86826i	−0.356928	+	0.934132i	$0.616176\pi$
$864$	0	0
$865$	−1.95977e9	−3.02801
$866$	0	0
$867$	3.63396e8	0.557600
$868$	0	0
$869$	7.63340e8i	1.16321i
$870$	0	0
$871$	− 4.65160e8i	− 0.703960i
$872$	0	0
$873$	2.13118e8	0.320316
$874$	0	0
$875$	3.72036e8	0.555342
$876$	0	0
$877$	6.11977e6i	0.00907270i	0.999990	+	0.00453635i	$0.00144397\pi$
$877$	6.11977e6i	0.00907270i	−0.999990	+	0.00453635i	$0.998556\pi$
$878$	0	0
$879$	− 7.79819e7i	− 0.114823i
$880$	0	0
$881$	5.39118e8	0.788417	0.394209	−	0.919021i	$-0.371019\pi$
$881$	5.39118e8	0.788417	0.394209	+	0.919021i	$0.371019\pi$
$882$	0	0
$883$	4.91469e8	0.713861	0.356931	−	0.934131i	$-0.383823\pi$
$883$	4.91469e8	0.713861	0.356931	+	0.934131i	$0.383823\pi$
$884$	0	0
$885$	− 6.48778e8i	− 0.935980i
$886$	0	0
$887$	6.95698e8i	0.996897i	0.866919	+	0.498448i	$0.166097\pi$
$887$	6.95698e8i	0.996897i	−0.866919	+	0.498448i	$0.833903\pi$
$888$	0	0
$889$	−1.85984e8	−0.264710
$890$	0	0
$891$	−1.03530e8	−0.146364
$892$	0	0
$893$	6.70534e8i	0.941600i
$894$	0	0
$895$	− 1.72450e9i	− 2.40544i
$896$	0	0
$897$	−2.47380e8	−0.342758
$898$	0	0
$899$	−3.63756e8	−0.500646
$900$	0	0
$901$	− 1.87742e9i	− 2.56678i
$902$	0	0
$903$	1.96629e8i	0.267045i
$904$	0	0
$905$	−1.17159e9	−1.58063
$906$	0	0
$907$	1.10192e9	1.47682	0.738411	−	0.674351i	$-0.235577\pi$
$907$	1.10192e9	1.47682	0.738411	+	0.674351i	$0.235577\pi$
$908$	0	0
$909$	− 4.38592e8i	− 0.583942i
$910$	0	0
$911$	− 3.89543e8i	− 0.515229i	−0.966248	−	0.257615i	$-0.917063\pi$
$911$	− 3.89543e8i	− 0.515229i	0.966248	−	0.257615i	$-0.0829365\pi$
$912$	0	0
$913$	−4.32331e8	−0.568073
$914$	0	0
$915$	−2.55540e8	−0.333576
$916$	0	0
$917$	− 4.80458e8i	− 0.623085i
$918$	0	0
$919$	1.13853e9i	1.46690i	0.679746	+	0.733448i	$0.262090\pi$
$919$	1.13853e9i	1.46690i	−0.679746	+	0.733448i	$0.737910\pi$
$920$	0	0
$921$	4.26360e8	0.545754
$922$	0	0
$923$	8.32594e8	1.05883
$924$	0	0
$925$	2.74092e8i	0.346315i
$926$	0	0
$927$	2.00783e8i	0.252050i
$928$	0	0
$929$	5.92474e8	0.738962	0.369481	−	0.929238i	$-0.379535\pi$
$929$	5.92474e8	0.738962	0.369481	+	0.929238i	$0.379535\pi$
$930$	0	0
$931$	1.78872e8	0.221663
$932$	0	0
$933$	5.25900e8i	0.647528i
$934$	0	0
$935$	− 2.35791e9i	− 2.88464i
$936$	0	0
$937$	−6.16456e7	−0.0749348	−0.0374674	−	0.999298i	$-0.511929\pi$
$937$	−6.16456e7	−0.0749348	−0.0374674	+	0.999298i	$0.511929\pi$
$938$	0	0
$939$	7.66782e8	0.926137
$940$	0	0
$941$	− 3.10725e8i	− 0.372913i	−0.982463	−	0.186457i	$-0.940300\pi$
$941$	− 3.10725e8i	− 0.372913i	0.982463	−	0.186457i	$-0.0597004\pi$
$942$	0	0
$943$	− 6.96763e8i	− 0.830902i
$944$	0	0
$945$	−2.05232e8	−0.243192
$946$	0	0
$947$	1.13722e8	0.133904	0.0669522	−	0.997756i	$-0.478672\pi$
$947$	1.13722e8	0.133904	0.0669522	+	0.997756i	$0.478672\pi$
$948$	0	0
$949$	− 4.98023e8i	− 0.582707i
$950$	0	0
$951$	− 3.07234e8i	− 0.357214i
$952$	0	0
$953$	−5.25518e8	−0.607168	−0.303584	−	0.952805i	$-0.598183\pi$
$953$	−5.25518e8	−0.607168	−0.303584	+	0.952805i	$0.598183\pi$
$954$	0	0
$955$	−1.17743e8	−0.135184
$956$	0	0
$957$	− 2.26258e8i	− 0.258148i
$958$	0	0
$959$	− 7.69687e8i	− 0.872686i
$960$	0	0
$961$	−1.04323e9	−1.17547
$962$	0	0
$963$	−3.27045e8	−0.366209
$964$	0	0
$965$	1.51423e9i	1.68504i
$966$	0	0
$967$	5.50448e8i	0.608747i	0.952553	+	0.304373i	$0.0984470\pi$
$967$	5.50448e8i	0.608747i	−0.952553	+	0.304373i	$0.901553\pi$
$968$	0	0
$969$	4.72647e8	0.519476
$970$	0	0
$971$	1.54851e9	1.69144	0.845722	−	0.533624i	$-0.179170\pi$
$971$	1.54851e9	1.69144	0.845722	+	0.533624i	$0.179170\pi$
$972$	0	0
$973$	− 1.19618e9i	− 1.29855i
$974$	0	0
$975$	− 4.37124e8i	− 0.471618i
$976$	0	0
$977$	1.73673e9	1.86229	0.931146	−	0.364645i	$-0.118810\pi$
$977$	1.73673e9	1.86229	0.931146	+	0.364645i	$0.118810\pi$
$978$	0	0
$979$	−1.41959e8	−0.151292
$980$	0	0
$981$	− 1.71223e8i	− 0.181366i
$982$	0	0
$983$	7.96734e8i	0.838789i	0.907804	+	0.419395i	$0.137758\pi$
$983$	7.96734e8i	0.838789i	−0.907804	+	0.419395i	$0.862242\pi$
$984$	0	0
$985$	−3.74402e8	−0.391769
$986$	0	0
$987$	−6.58999e8	−0.685383
$988$	0	0
$989$	5.78562e8i	0.598082i
$990$	0	0
$991$	3.33810e8i	0.342988i	0.985185	+	0.171494i	$0.0548594\pi$
$991$	3.33810e8i	0.342988i	−0.985185	+	0.171494i	$0.945141\pi$
$992$	0	0
$993$	−2.48965e8	−0.254267
$994$	0	0
$995$	1.98054e9	2.01054
$996$	0	0
$997$	4.81174e8i	0.485531i	0.970085	+	0.242765i	$0.0780545\pi$
$997$	4.81174e8i	0.485531i	−0.970085	+	0.242765i	$0.921945\pi$
$998$	0	0
$999$	− 4.61619e7i	− 0.0463007i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.7.b.c.319.8	yes	8
4.3	odd	2	inner	384.7.b.c.319.4	yes	8
8.3	odd	2	inner	384.7.b.c.319.5	yes	8
8.5	even	2	inner	384.7.b.c.319.1	✓	8
16.3	odd	4		768.7.g.g.511.4		8
16.5	even	4		768.7.g.g.511.1		8
16.11	odd	4		768.7.g.g.511.5		8
16.13	even	4		768.7.g.g.511.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.7.b.c.319.1	✓	8	8.5	even	2	inner
384.7.b.c.319.4	yes	8	4.3	odd	2	inner
384.7.b.c.319.5	yes	8	8.3	odd	2	inner
384.7.b.c.319.8	yes	8	1.1	even	1	trivial
768.7.g.g.511.1		8	16.5	even	4
768.7.g.g.511.4		8	16.3	odd	4
768.7.g.g.511.5		8	16.11	odd	4
768.7.g.g.511.8		8	16.13	even	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)

\(f(q)\)	\(=\)	\(q+15.5885 q^{3} +195.235i q^{5} +277.510i q^{7} +243.000 q^{9} +O(q^{10})\)	\(q+15.5885 q^{3} +195.235i q^{5} +277.510i q^{7} +243.000 q^{9} -1753.29 q^{11} +1246.75i q^{13} +3043.41i q^{15} +6888.35 q^{17} +4401.67 q^{19} +4325.95i q^{21} +12728.7i q^{23} -22491.7 q^{25} +3788.00 q^{27} +8278.44i q^{29} +43940.1i q^{31} -27331.1 q^{33} -54179.6 q^{35} -12186.4i q^{37} +19434.9i q^{39} -54739.6 q^{41} +45453.4 q^{43} +47442.1i q^{45} +152336. i q^{47} +40637.3 q^{49} +107379. q^{51} -272550. i q^{53} -342303. i q^{55} +68615.3 q^{57} -213175. q^{59} +83964.8i q^{61} +67434.9i q^{63} -243409. q^{65} -373099. q^{67} +198420. i q^{69} -667812. i q^{71} -399457. q^{73} -350611. q^{75} -486555. i q^{77} -435376. i q^{79} +59049.0 q^{81} +246583. q^{83} +1.34485e6i q^{85} +129048. i q^{87} +80967.5 q^{89} -345985. q^{91} +684959. i q^{93} +859361. i q^{95} +877030. q^{97} -426049. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 1944 q^{9}+O(q^{10}) \) 8 * q + 1944 * q^9	\( 8 q + 1944 q^{9} - 6544 q^{17} - 56632 q^{25} - 33696 q^{33} - 499568 q^{41} - 414712 q^{49} - 375840 q^{57} - 36096 q^{65} - 1962640 q^{73} + 472392 q^{81} - 1694992 q^{89} + 7632752 q^{97}+O(q^{100}) \) 8 * q + 1944 * q^9 - 6544 * q^17 - 56632 * q^25 - 33696 * q^33 - 499568 * q^41 - 414712 * q^49 - 375840 * q^57 - 36096 * q^65 - 1962640 * q^73 + 472392 * q^81 - 1694992 * q^89 + 7632752 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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