LMFDB - Embedded newform 768.7.g.g.511.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,7,Mod(511,768)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(768, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("768.511");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 768 = 2^{8} \cdot 3 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	768.g (of order $2$, degree $1$, not 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:	$176.681536220$
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} - 2x^{7} - 53x^{6} - 2x^{5} + 2532x^{4} - 772x^{3} - 31349x^{2} - 33880x + 366025 $ x^8 - 2x^7 - 53x^6 - 2x^5 + 2532x^4 - 772x^3 - 31349x^2 - 33880*x + 366025
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{32}\cdot 3^{10} $
Twist minimal:	no (minimal twist has level 384)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			511.8
Root			$5.80399 - 2.77359i$ of defining polynomial
Character	$\chi$	$=$	768.511
Dual form			768.7.g.g.511.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+15.5885i q^{3} +195.235 q^{5} -277.510i q^{7} -243.000 q^{9} +O(q^{10})$	$q+15.5885i q^{3} +195.235 q^{5} -277.510i q^{7} -243.000 q^{9} +1753.29i q^{11} -1246.75 q^{13} +3043.41i q^{15} +6888.35 q^{17} +4401.67i q^{19} +4325.95 q^{21} -12728.7i q^{23} +22491.7 q^{25} -3788.00i q^{27} -8278.44 q^{29} +43940.1i q^{31} -27331.1 q^{33} -54179.6i q^{35} -12186.4 q^{37} -19434.9i q^{39} +54739.6 q^{41} -45453.4i q^{43} -47442.1 q^{45} +152336. i q^{47} +40637.3 q^{49} +107379. i q^{51} -272550. q^{53} +342303. i q^{55} -68615.3 q^{57} +213175. i q^{59} -83964.8 q^{61} +67434.9i q^{63} -243409. q^{65} -373099. i q^{67} +198420. q^{69} +667812. i q^{71} +399457. q^{73} +350611. i q^{75} +486555. q^{77} -435376. i q^{79} +59049.0 q^{81} +246583. i q^{83} +1.34485e6 q^{85} -129048. i q^{87} -80967.5 q^{89} +345985. i q^{91} -684959. q^{93} +859361. i q^{95} +877030. q^{97} -426049. i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/768\mathbb{Z}\right)^\times$.

$n$	$257$	$511$	$517$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	15.5885i	0.577350i
$3$	15.5885i	0.577350i
$4$	0	0
$5$	195.235	1.56188	0.780940	−	0.624606i	$-0.214740\pi$
$5$	195.235	1.56188	0.780940	+	0.624606i	$0.214740\pi$
$6$	0	0
$7$	− 277.510i	− 0.809067i	−0.914523	−	0.404533i	$-0.867434\pi$
$7$	− 277.510i	− 0.809067i	0.914523	−	0.404533i	$-0.132566\pi$
$8$	0	0
$9$	−243.000	−0.333333
$10$	0	0
$11$	1753.29i	1.31727i	0.752462	+	0.658636i	$0.228866\pi$
$11$	1753.29i	1.31727i	−0.752462	+	0.658636i	$0.771134\pi$
$12$	0	0
$13$	−1246.75	−0.567478	−0.283739	−	0.958902i	$-0.591575\pi$
$13$	−1246.75	−0.567478	−0.283739	+	0.958902i	$0.591575\pi$
$14$	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
$19$	4401.67i	0.641737i	0.947124	+	0.320869i	$0.103975\pi$
$19$	4401.67i	0.641737i	−0.947124	+	0.320869i	$0.896025\pi$
$20$	0	0
$21$	4325.95	0.467115
$22$	0	0
$23$	− 12728.7i	− 1.04616i	−0.852283	−	0.523082i	$-0.824782\pi$
$23$	− 12728.7i	− 1.04616i	0.852283	−	0.523082i	$-0.175218\pi$
$24$	0	0
$25$	22491.7	1.43947
$26$	0	0
$27$	− 3788.00i	− 0.192450i
$28$	0	0
$29$	−8278.44	−0.339433	−0.169717	−	0.985493i	$-0.554285\pi$
$29$	−8278.44	−0.339433	−0.169717	+	0.985493i	$0.554285\pi$
$30$	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
$35$	− 54179.6i	− 1.26367i
$36$	0	0
$37$	−12186.4	−0.240585	−0.120293	−	0.992738i	$-0.538383\pi$
$37$	−12186.4	−0.240585	−0.120293	+	0.992738i	$0.538383\pi$
$38$	0	0
$39$	− 19434.9i	− 0.327633i
$40$	0	0
$41$	54739.6	0.794238	0.397119	−	0.917767i	$-0.370010\pi$
$41$	54739.6	0.794238	0.397119	+	0.917767i	$0.370010\pi$
$42$	0	0
$43$	− 45453.4i	− 0.571691i	−0.958276	−	0.285845i	$-0.907726\pi$
$43$	− 45453.4i	− 0.571691i	0.958276	−	0.285845i	$-0.0922744\pi$
$44$	0	0
$45$	−47442.1	−0.520627
$46$	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.i	0.809484i
$52$	0	0
$53$	−272550.	−1.83071	−0.915354	−	0.402649i	$-0.868089\pi$
$53$	−272550.	−1.83071	−0.915354	+	0.402649i	$0.868089\pi$
$54$	0	0
$55$	342303.i	2.05742i
$56$	0	0
$57$	−68615.3	−0.370507
$58$	0	0
$59$	213175.i	1.03796i	0.854787	+	0.518978i	$0.173687\pi$
$59$	213175.i	1.03796i	−0.854787	+	0.518978i	$0.826313\pi$
$60$	0	0
$61$	−83964.8	−0.369920	−0.184960	−	0.982746i	$-0.559216\pi$
$61$	−83964.8	−0.369920	−0.184960	+	0.982746i	$0.559216\pi$
$62$	0	0
$63$	67434.9i	0.269689i
$64$	0	0
$65$	−243409.	−0.886332
$66$	0	0
$67$	− 373099.i	− 1.24051i	−0.784401	−	0.620254i	$-0.787030\pi$
$67$	− 373099.i	− 1.24051i	0.784401	−	0.620254i	$-0.212970\pi$
$68$	0	0
$69$	198420.	0.604003
$70$	0	0
$71$	667812.i	1.86586i	0.360057	+	0.932930i	$0.382757\pi$
$71$	667812.i	1.86586i	−0.360057	+	0.932930i	$0.617243\pi$
$72$	0	0
$73$	399457.	1.02684	0.513419	−	0.858138i	$-0.328379\pi$
$73$	399457.	1.02684	0.513419	+	0.858138i	$0.328379\pi$
$74$	0	0
$75$	350611.i	0.831078i
$76$	0	0
$77$	486555.	1.06576
$78$	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.i	0.431250i	0.976476	+	0.215625i	$0.0691789\pi$
$83$	246583.i	0.431250i	−0.976476	+	0.215625i	$0.930821\pi$
$84$	0	0
$85$	1.34485e6	2.18986
$86$	0	0
$87$	− 129048.i	− 0.195972i
$88$	0	0
$89$	−80967.5	−0.114853	−0.0574263	−	0.998350i	$-0.518289\pi$
$89$	−80967.5	−0.114853	−0.0574263	+	0.998350i	$0.518289\pi$
$90$	0	0
$91$	345985.i	0.459127i
$92$	0	0
$93$	−684959.	−0.851561
$94$	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.i	− 0.439091i
$100$	0	0
$101$	−1.80491e6	−1.75183	−0.875913	−	0.482470i	$-0.839740\pi$
$101$	−1.80491e6	−1.75183	−0.875913	+	0.482470i	$0.839740\pi$
$102$	0	0
$103$	− 826267.i	− 0.756151i	−0.925775	−	0.378076i	$-0.876586\pi$
$103$	− 826267.i	− 0.756151i	0.925775	−	0.378076i	$-0.123414\pi$
$104$	0	0
$105$	844577.	0.729577
$106$	0	0
$107$	1.34587e6i	1.09863i	0.835616	+	0.549314i	$0.185111\pi$
$107$	1.34587e6i	1.09863i	−0.835616	+	0.549314i	$0.814889\pi$
$108$	0	0
$109$	704622.	0.544098	0.272049	−	0.962283i	$-0.412299\pi$
$109$	704622.	0.544098	0.272049	+	0.962283i	$0.412299\pi$
$110$	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
$115$	− 2.48508e6i	− 1.63398i
$116$	0	0
$117$	302960.	0.189159
$118$	0	0
$119$	− 1.91159e6i	− 1.13437i
$120$	0	0
$121$	−1.30246e6	−0.735205
$122$	0	0
$123$	853307.i	0.458553i
$124$	0	0
$125$	1.34062e6	0.686399
$126$	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
$131$	− 1.73132e6i	− 0.770128i	−0.922890	−	0.385064i	$-0.874179\pi$
$131$	− 1.73132e6i	− 0.770128i	0.922890	−	0.385064i	$-0.125821\pi$
$132$	0	0
$133$	1.22151e6	0.519208
$134$	0	0
$135$	− 739549.i	− 0.300584i
$136$	0	0
$137$	2.77355e6	1.07863	0.539317	−	0.842103i	$-0.318683\pi$
$137$	2.77355e6	1.07863	0.539317	+	0.842103i	$0.318683\pi$
$138$	0	0
$139$	4.31042e6i	1.60500i	0.596652	+	0.802500i	$0.296497\pi$
$139$	4.31042e6i	1.60500i	−0.596652	+	0.802500i	$0.703503\pi$
$140$	0	0
$141$	−2.37469e6	−0.847128
$142$	0	0
$143$	− 2.18591e6i	− 0.747522i
$144$	0	0
$145$	−1.61624e6	−0.530154
$146$	0	0
$147$	633472.i	0.199423i
$148$	0	0
$149$	−213236.	−0.0644616	−0.0322308	−	0.999480i	$-0.510261\pi$
$149$	−213236.	−0.0644616	−0.0322308	+	0.999480i	$0.510261\pi$
$150$	0	0
$151$	− 5.33710e6i	− 1.55015i	−0.631868	−	0.775076i	$-0.717711\pi$
$151$	− 5.33710e6i	− 1.55015i	0.631868	−	0.775076i	$-0.282289\pi$
$152$	0	0
$153$	−1.67387e6	−0.467356
$154$	0	0
$155$	8.57866e6i	2.30369i
$156$	0	0
$157$	4.08979e6	1.05682	0.528412	−	0.848988i	$-0.322788\pi$
$157$	4.08979e6	1.05682	0.528412	+	0.848988i	$0.322788\pi$
$158$	0	0
$159$	− 4.24864e6i	− 1.05696i
$160$	0	0
$161$	−3.53233e6	−0.846416
$162$	0	0
$163$	1.54126e6i	0.355888i	0.984041	+	0.177944i	$0.0569446\pi$
$163$	1.54126e6i	0.355888i	−0.984041	+	0.177944i	$0.943055\pi$
$164$	0	0
$165$	−5.33598e6	−1.18785
$166$	0	0
$167$	7.66555e6i	1.64586i	0.568140	+	0.822932i	$0.307663\pi$
$167$	7.66555e6i	1.64586i	−0.568140	+	0.822932i	$0.692337\pi$
$168$	0	0
$169$	−3.27243e6	−0.677969
$170$	0	0
$171$	− 1.06961e6i	− 0.213912i
$172$	0	0
$173$	−1.00380e7	−1.93870	−0.969348	−	0.245693i	$-0.920984\pi$
$173$	−1.00380e7	−1.93870	−0.969348	+	0.245693i	$0.920984\pi$
$174$	0	0
$175$	− 6.24167e6i	− 1.16463i
$176$	0	0
$177$	−3.32306e6	−0.599265
$178$	0	0
$179$	− 8.83296e6i	− 1.54009i	−0.637987	−	0.770047i	$-0.720233\pi$
$179$	− 8.83296e6i	− 1.54009i	0.637987	−	0.770047i	$-0.279767\pi$
$180$	0	0
$181$	6.00091e6	1.01200	0.506001	−	0.862533i	$-0.331123\pi$
$181$	6.00091e6	1.01200	0.506001	+	0.862533i	$0.331123\pi$
$182$	0	0
$183$	− 1.30888e6i	− 0.213573i
$184$	0	0
$185$	−2.37920e6	−0.375765
$186$	0	0
$187$	1.20773e7i	1.84690i
$188$	0	0
$189$	−1.05121e6	−0.155705
$190$	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
$195$	− 3.79437e6i	− 0.511724i
$196$	0	0
$197$	1.91770e6	0.250832	0.125416	−	0.992104i	$-0.459974\pi$
$197$	1.91770e6	0.250832	0.125416	+	0.992104i	$0.459974\pi$
$198$	0	0
$199$	1.01444e7i	1.28726i	0.765337	+	0.643629i	$0.222572\pi$
$199$	1.01444e7i	1.28726i	−0.765337	+	0.643629i	$0.777428\pi$
$200$	0	0
$201$	5.81603e6	0.716207
$202$	0	0
$203$	2.29735e6i	0.274624i
$204$	0	0
$205$	1.06871e7	1.24050
$206$	0	0
$207$	3.09307e6i	0.348721i
$208$	0	0
$209$	−7.71741e6	−0.845342
$210$	0	0
$211$	9.54322e6i	1.01589i	0.861389	+	0.507946i	$0.169595\pi$
$211$	9.54322e6i	1.01589i	−0.861389	+	0.507946i	$0.830405\pi$
$212$	0	0
$213$	−1.04102e7	−1.07726
$214$	0	0
$215$	− 8.87410e6i	− 0.892912i
$216$	0	0
$217$	1.21938e7	1.19333
$218$	0	0
$219$	6.22692e6i	0.592845i
$220$	0	0
$221$	−8.58805e6	−0.795642
$222$	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
$227$	1.48574e7i	1.27018i	0.772437	+	0.635092i	$0.219038\pi$
$227$	1.48574e7i	1.27018i	−0.772437	+	0.635092i	$0.780962\pi$
$228$	0	0
$229$	−1.28465e6	−0.106974	−0.0534872	−	0.998569i	$-0.517034\pi$
$229$	−1.28465e6	−0.106974	−0.0534872	+	0.998569i	$0.517034\pi$
$230$	0	0
$231$	7.58464e6i	0.615317i
$232$	0	0
$233$	−262969.	−0.0207892	−0.0103946	−	0.999946i	$-0.503309\pi$
$233$	−262969.	−0.0207892	−0.0103946	+	0.999946i	$0.503309\pi$
$234$	0	0
$235$	2.97414e7i	2.29170i
$236$	0	0
$237$	6.78684e6	0.509827
$238$	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.i	0.0641500i
$244$	0	0
$245$	7.93382e6	0.539491
$246$	0	0
$247$	− 5.48778e6i	− 0.364172i
$248$	0	0
$249$	−3.84385e6	−0.248982
$250$	0	0
$251$	1.10676e7i	0.699895i	0.936769	+	0.349948i	$0.113801\pi$
$251$	1.10676e7i	0.699895i	−0.936769	+	0.349948i	$0.886199\pi$
$252$	0	0
$253$	2.23170e7	1.37808
$254$	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
$259$	3.38184e6i	0.194650i
$260$	0	0
$261$	2.01166e6	0.113144
$262$	0	0
$263$	1.11128e7i	0.610882i	0.952211	+	0.305441i	$0.0988039\pi$
$263$	1.11128e7i	0.610882i	−0.952211	+	0.305441i	$0.901196\pi$
$264$	0	0
$265$	−5.32114e7	−2.85935
$266$	0	0
$267$	− 1.26216e6i	− 0.0663101i
$268$	0	0
$269$	−1.00441e7	−0.516007	−0.258004	−	0.966144i	$-0.583065\pi$
$269$	−1.00441e7	−0.516007	−0.258004	+	0.966144i	$0.583065\pi$
$270$	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
$275$	3.94345e7i	1.89617i
$276$	0	0
$277$	2.44357e7	1.14970	0.574852	−	0.818257i	$-0.305060\pi$
$277$	2.44357e7	1.14970	0.574852	+	0.818257i	$0.305060\pi$
$278$	0	0
$279$	− 1.06775e7i	− 0.491649i
$280$	0	0
$281$	1.39627e7	0.629288	0.314644	−	0.949210i	$-0.398115\pi$
$281$	1.39627e7	0.629288	0.314644	+	0.949210i	$0.398115\pi$
$282$	0	0
$283$	− 4.28834e7i	− 1.89204i	−0.324113	−	0.946018i	$-0.605066\pi$
$283$	− 4.28834e7i	− 1.89204i	0.324113	−	0.946018i	$-0.394934\pi$
$284$	0	0
$285$	−1.33961e7	−0.578688
$286$	0	0
$287$	− 1.51908e7i	− 0.642591i
$288$	0	0
$289$	2.33119e7	0.965792
$290$	0	0
$291$	1.36716e7i	0.554803i
$292$	0	0
$293$	−5.00254e6	−0.198878	−0.0994392	−	0.995044i	$-0.531705\pi$
$293$	−5.00254e6	−0.198878	−0.0994392	+	0.995044i	$0.531705\pi$
$294$	0	0
$295$	4.16191e7i	1.62116i
$296$	0	0
$297$	6.64145e6	0.253509
$298$	0	0
$299$	1.58695e7i	0.593675i
$300$	0	0
$301$	−1.26138e7	−0.462536
$302$	0	0
$303$	− 2.81357e7i	− 1.01142i
$304$	0	0
$305$	−1.63929e7	−0.577771
$306$	0	0
$307$	2.73510e7i	0.945274i	0.881257	+	0.472637i	$0.156698\pi$
$307$	2.73510e7i	0.945274i	−0.881257	+	0.472637i	$0.843302\pi$
$308$	0	0
$309$	1.28802e7	0.436564
$310$	0	0
$311$	− 3.37365e7i	− 1.12155i	−0.827968	−	0.560775i	$-0.810503\pi$
$311$	− 3.37365e7i	− 1.12155i	0.827968	−	0.560775i	$-0.189497\pi$
$312$	0	0
$313$	−4.91891e7	−1.60412	−0.802058	−	0.597246i	$-0.796262\pi$
$313$	−4.91891e7	−1.60412	−0.802058	+	0.597246i	$0.796262\pi$
$314$	0	0
$315$	1.31657e7i	0.421222i
$316$	0	0
$317$	1.97091e7	0.618712	0.309356	−	0.950946i	$-0.399887\pi$
$317$	1.97091e7	0.618712	0.309356	+	0.950946i	$0.399887\pi$
$318$	0	0
$319$	− 1.45145e7i	− 0.447126i
$320$	0	0
$321$	−2.09800e7	−0.634293
$322$	0	0
$323$	3.03203e7i	0.899758i
$324$	0	0
$325$	−2.80415e7	−0.816867
$326$	0	0
$327$	1.09840e7i	0.314135i
$328$	0	0
$329$	4.22748e7	1.18712
$330$	0	0
$331$	1.59711e7i	0.440403i	0.975454	+	0.220202i	$0.0706715\pi$
$331$	1.59711e7i	0.440403i	−0.975454	+	0.220202i	$0.929328\pi$
$332$	0	0
$333$	2.96129e6	0.0801951
$334$	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
$339$	3.97930e7i	1.02143i
$340$	0	0
$341$	−7.70398e7	−1.94291
$342$	0	0
$343$	− 4.39260e7i	− 1.08853i
$344$	0	0
$345$	3.87386e7	0.943380
$346$	0	0
$347$	3.95976e7i	0.947721i	0.880600	+	0.473861i	$0.157140\pi$
$347$	3.95976e7i	0.947721i	−0.880600	+	0.473861i	$0.842860\pi$
$348$	0	0
$349$	−7.85281e7	−1.84735	−0.923674	−	0.383180i	$-0.874829\pi$
$349$	−7.85281e7	−1.84735	−0.923674	+	0.383180i	$0.874829\pi$
$350$	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
$355$	1.30380e8i	2.91425i
$356$	0	0
$357$	2.97987e7	0.654926
$358$	0	0
$359$	− 9.24524e6i	− 0.199818i	−0.994997	−	0.0999090i	$-0.968145\pi$
$359$	− 9.24524e6i	− 0.199818i	0.994997	−	0.0999090i	$-0.0318552\pi$
$360$	0	0
$361$	2.76711e7	0.588173
$362$	0	0
$363$	− 2.03033e7i	− 0.424471i
$364$	0	0
$365$	7.79880e7	1.60380
$366$	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
$371$	7.56354e7i	1.48117i
$372$	0	0
$373$	−1.92271e7	−0.370499	−0.185250	−	0.982691i	$-0.559309\pi$
$373$	−1.92271e7	−0.370499	−0.185250	+	0.982691i	$0.559309\pi$
$374$	0	0
$375$	2.08982e7i	0.396292i
$376$	0	0
$377$	1.03211e7	0.192621
$378$	0	0
$379$	− 1.53936e7i	− 0.282764i	−0.989955	−	0.141382i	$-0.954845\pi$
$379$	− 1.53936e7i	− 0.282764i	0.989955	−	0.141382i	$-0.0451545\pi$
$380$	0	0
$381$	−1.04472e7	−0.188897
$382$	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
$387$	1.10452e7i	0.190564i
$388$	0	0
$389$	5.48941e7	0.932561	0.466281	−	0.884637i	$-0.345594\pi$
$389$	5.48941e7	0.932561	0.466281	+	0.884637i	$0.345594\pi$
$390$	0	0
$391$	− 8.76796e7i	− 1.46679i
$392$	0	0
$393$	2.69886e7	0.444634
$394$	0	0
$395$	− 8.50007e7i	− 1.37921i
$396$	0	0
$397$	−5.07085e7	−0.810419	−0.405209	−	0.914224i	$-0.632801\pi$
$397$	−5.07085e7	−0.810419	−0.405209	+	0.914224i	$0.632801\pi$
$398$	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
$403$	− 5.47823e7i	− 0.837000i
$404$	0	0
$405$	1.15284e7	0.173542
$406$	0	0
$407$	− 2.13662e7i	− 0.316916i
$408$	0	0
$409$	−8.03112e7	−1.17383	−0.586917	−	0.809647i	$-0.699658\pi$
$409$	−8.03112e7	−1.17383	−0.586917	+	0.809647i	$0.699658\pi$
$410$	0	0
$411$	4.32353e7i	0.622749i
$412$	0	0
$413$	5.91580e7	0.839776
$414$	0	0
$415$	4.81417e7i	0.673561i
$416$	0	0
$417$	−6.71928e7	−0.926648
$418$	0	0
$419$	8.64533e7i	1.17527i	0.809125	+	0.587637i	$0.199942\pi$
$419$	8.64533e7i	1.17527i	−0.809125	+	0.587637i	$0.800058\pi$
$420$	0	0
$421$	−1.94432e7	−0.260569	−0.130284	−	0.991477i	$-0.541589\pi$
$421$	−1.94432e7	−0.260569	−0.130284	+	0.991477i	$0.541589\pi$
$422$	0	0
$423$	− 3.70177e7i	− 0.489089i
$424$	0	0
$425$	1.54931e8	2.01823
$426$	0	0
$427$	2.33011e7i	0.299290i
$428$	0	0
$429$	3.40750e7	0.431582
$430$	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
$435$	− 2.51947e7i	− 0.306085i
$436$	0	0
$437$	5.60275e7	0.671362
$438$	0	0
$439$	5.71422e7i	0.675403i	0.941253	+	0.337702i	$0.109649\pi$
$439$	5.71422e7i	0.675403i	−0.941253	+	0.337702i	$0.890351\pi$
$440$	0	0
$441$	−9.87485e6	−0.115137
$442$	0	0
$443$	− 2.10479e7i	− 0.242102i	−0.992646	−	0.121051i	$-0.961374\pi$
$443$	− 2.10479e7i	− 0.242102i	0.992646	−	0.121051i	$-0.0386265\pi$
$444$	0	0
$445$	−1.58077e7	−0.179386
$446$	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
$451$	9.59744e7i	1.04623i
$452$	0	0
$453$	8.31971e7	0.894981
$454$	0	0
$455$	6.75484e7i	0.717102i
$456$	0	0
$457$	−5.37459e7	−0.563114	−0.281557	−	0.959544i	$-0.590851\pi$
$457$	−5.37459e7	−0.563114	−0.281557	+	0.959544i	$0.590851\pi$
$458$	0	0
$459$	− 2.60931e7i	− 0.269828i
$460$	0	0
$461$	−2.36640e7	−0.241538	−0.120769	−	0.992681i	$-0.538536\pi$
$461$	−2.36640e7	−0.241538	−0.120769	+	0.992681i	$0.538536\pi$
$462$	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
$467$	− 3.61101e7i	− 0.354550i	−0.984161	−	0.177275i	$-0.943272\pi$
$467$	− 3.61101e7i	− 0.354550i	0.984161	−	0.177275i	$-0.0567283\pi$
$468$	0	0
$469$	−1.03539e8	−1.00365
$470$	0	0
$471$	6.37536e7i	0.610157i
$472$	0	0
$473$	7.96930e7	0.753072
$474$	0	0
$475$	9.90012e7i	0.923761i
$476$	0	0
$477$	6.62298e7	0.610236
$478$	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
$483$	− 5.50636e7i	− 0.488679i
$484$	0	0
$485$	1.71227e8	1.50088
$486$	0	0
$487$	9.56947e7i	0.828516i	0.910160	+	0.414258i	$0.135959\pi$
$487$	9.56947e7i	0.828516i	−0.910160	+	0.414258i	$0.864041\pi$
$488$	0	0
$489$	−2.40259e7	−0.205472
$490$	0	0
$491$	9.69668e7i	0.819178i	0.912270	+	0.409589i	$0.134328\pi$
$491$	9.69668e7i	0.819178i	−0.912270	+	0.409589i	$0.865672\pi$
$492$	0	0
$493$	−5.70248e7	−0.475908
$494$	0	0
$495$	− 8.31797e7i	− 0.685807i
$496$	0	0
$497$	1.85324e8	1.50961
$498$	0	0
$499$	5.16768e7i	0.415905i	0.978139	+	0.207952i	$0.0666799\pi$
$499$	5.16768e7i	0.415905i	−0.978139	+	0.207952i	$0.933320\pi$
$500$	0	0
$501$	−1.19494e8	−0.950240
$502$	0	0
$503$	2.28417e8i	1.79483i	0.441185	+	0.897416i	$0.354558\pi$
$503$	2.28417e8i	1.79483i	−0.441185	+	0.897416i	$0.645442\pi$
$504$	0	0
$505$	−3.52381e8	−2.73614
$506$	0	0
$507$	− 5.10121e7i	− 0.391426i
$508$	0	0
$509$	−1.31906e8	−1.00025	−0.500127	−	0.865952i	$-0.666713\pi$
$509$	−1.31906e8	−1.00025	−0.500127	+	0.865952i	$0.666713\pi$
$510$	0	0
$511$	− 1.10853e8i	− 0.830780i
$512$	0	0
$513$	1.66735e7	0.123502
$514$	0	0
$515$	− 1.61316e8i	− 1.18102i
$516$	0	0
$517$	−2.67089e8	−1.93279
$518$	0	0
$519$	− 1.56477e8i	− 1.11931i
$520$	0	0
$521$	−1.21687e7	−0.0860462	−0.0430231	−	0.999074i	$-0.513699\pi$
$521$	−1.21687e7	−0.0860462	−0.0430231	+	0.999074i	$0.513699\pi$
$522$	0	0
$523$	8.96786e7i	0.626879i	0.949608	+	0.313440i	$0.101481\pi$
$523$	8.96786e7i	0.626879i	−0.949608	+	0.313440i	$0.898519\pi$
$524$	0	0
$525$	9.72980e7	0.672398
$526$	0	0
$527$	3.02675e8i	2.06797i
$528$	0	0
$529$	−1.39832e7	−0.0944579
$530$	0	0
$531$	− 5.18014e7i	− 0.345986i
$532$	0	0
$533$	−6.82466e7	−0.450712
$534$	0	0
$535$	2.62760e8i	1.71592i
$536$	0	0
$537$	1.37692e8	0.889174
$538$	0	0
$539$	7.12488e7i	0.455000i
$540$	0	0
$541$	2.72097e8	1.71843	0.859216	−	0.511613i	$-0.170952\pi$
$541$	2.72097e8	1.71843	0.859216	+	0.511613i	$0.170952\pi$
$542$	0	0
$543$	9.35449e7i	0.584279i
$544$	0	0
$545$	1.37567e8	0.849815
$546$	0	0
$547$	− 1.84749e8i	− 1.12881i	−0.825499	−	0.564404i	$-0.809106\pi$
$547$	− 1.84749e8i	− 1.12881i	0.825499	−	0.564404i	$-0.190894\pi$
$548$	0	0
$549$	2.04035e7	0.123307
$550$	0	0
$551$	− 3.64390e7i	− 0.217827i
$552$	0	0
$553$	−1.20821e8	−0.714443
$554$	0	0
$555$	− 3.70881e7i	− 0.216948i
$556$	0	0
$557$	−9.60239e7	−0.555666	−0.277833	−	0.960629i	$-0.589616\pi$
$557$	−9.60239e7	−0.555666	−0.277833	+	0.960629i	$0.589616\pi$
$558$	0	0
$559$	5.66690e7i	0.324422i
$560$	0	0
$561$	−1.88266e8	−1.06631
$562$	0	0
$563$	− 2.37375e8i	− 1.33018i	−0.746764	−	0.665089i	$-0.768393\pi$
$563$	− 2.37375e8i	− 1.33018i	0.746764	−	0.665089i	$-0.231607\pi$
$564$	0	0
$565$	4.98381e8	2.76323
$566$	0	0
$567$	− 1.63867e7i	− 0.0898963i
$568$	0	0
$569$	3.26674e8	1.77328	0.886641	−	0.462458i	$-0.153032\pi$
$569$	3.26674e8	1.77328	0.886641	+	0.462458i	$0.153032\pi$
$570$	0	0
$571$	971103.i	0.00521623i	0.999997	+	0.00260812i	$0.000830190\pi$
$571$	971103.i	0.00521623i	−0.999997	+	0.00260812i	$0.999170\pi$
$572$	0	0
$573$	−9.40116e6	−0.0499709
$574$	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
$579$	1.20903e8i	0.622876i
$580$	0	0
$581$	6.84293e7	0.348910
$582$	0	0
$583$	− 4.77860e8i	− 2.41154i
$584$	0	0
$585$	5.91484e7	0.295444
$586$	0	0
$587$	8.47232e7i	0.418878i	0.977822	+	0.209439i	$0.0671638\pi$
$587$	8.47232e7i	0.418878i	−0.977822	+	0.209439i	$0.932836\pi$
$588$	0	0
$589$	−1.93410e8	−0.946528
$590$	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
$595$	− 3.73209e8i	− 1.77174i
$596$	0	0
$597$	−1.58135e8	−0.743199
$598$	0	0
$599$	− 3.65098e8i	− 1.69875i	−0.527791	−	0.849374i	$-0.676980\pi$
$599$	− 3.65098e8i	− 1.69875i	0.527791	−	0.849374i	$-0.323020\pi$
$600$	0	0
$601$	−2.63307e8	−1.21294	−0.606469	−	0.795107i	$-0.707415\pi$
$601$	−2.63307e8	−1.21294	−0.606469	+	0.795107i	$0.707415\pi$
$602$	0	0
$603$	9.06630e7i	0.413502i
$604$	0	0
$605$	−2.54286e8	−1.14830
$606$	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
$611$	− 1.89925e8i	− 0.832642i
$612$	0	0
$613$	−4.20201e8	−1.82421	−0.912106	−	0.409953i	$-0.865545\pi$
$613$	−4.20201e8	−1.82421	−0.912106	+	0.409953i	$0.865545\pi$
$614$	0	0
$615$	1.66595e8i	0.716205i
$616$	0	0
$617$	−1.25856e7	−0.0535818	−0.0267909	−	0.999641i	$-0.508529\pi$
$617$	−1.25856e7	−0.0535818	−0.0267909	+	0.999641i	$0.508529\pi$
$618$	0	0
$619$	− 2.43243e8i	− 1.02558i	−0.858515	−	0.512789i	$-0.828612\pi$
$619$	− 2.43243e8i	− 1.02558i	0.858515	−	0.512789i	$-0.171388\pi$
$620$	0	0
$621$	−4.82161e7	−0.201334
$622$	0	0
$623$	2.24693e7i	0.0929234i
$624$	0	0
$625$	−8.96966e7	−0.367397
$626$	0	0
$627$	− 1.20302e8i	− 0.488059i
$628$	0	0
$629$	−8.39440e7	−0.337317
$630$	0	0
$631$	− 5.45064e7i	− 0.216950i	−0.994099	−	0.108475i	$-0.965403\pi$
$631$	− 5.45064e7i	− 0.216950i	0.994099	−	0.108475i	$-0.0345967\pi$
$632$	0	0
$633$	−1.48764e8	−0.586526
$634$	0	0
$635$	1.30844e8i	0.511016i
$636$	0	0
$637$	−5.06644e7	−0.196013
$638$	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
$643$	− 4.20644e8i	− 1.58227i	−0.611639	−	0.791137i	$-0.709490\pi$
$643$	− 4.20644e8i	− 1.58227i	0.611639	−	0.791137i	$-0.290510\pi$
$644$	0	0
$645$	1.38333e8	0.515523
$646$	0	0
$647$	− 3.19909e8i	− 1.18117i	−0.806974	−	0.590587i	$-0.798896\pi$
$647$	− 3.19909e8i	− 1.18117i	0.806974	−	0.590587i	$-0.201104\pi$
$648$	0	0
$649$	−3.73756e8	−1.36727
$650$	0	0
$651$	1.90083e8i	0.688970i
$652$	0	0
$653$	2.98645e8	1.07254	0.536272	−	0.844045i	$-0.319832\pi$
$653$	2.98645e8	1.07254	0.536272	+	0.844045i	$0.319832\pi$
$654$	0	0
$655$	− 3.38014e8i	− 1.20285i
$656$	0	0
$657$	−9.70681e7	−0.342279
$658$	0	0
$659$	− 3.27357e8i	− 1.14384i	−0.820309	−	0.571920i	$-0.806199\pi$
$659$	− 3.27357e8i	− 1.14384i	0.820309	−	0.571920i	$-0.193801\pi$
$660$	0	0
$661$	4.03255e8	1.39629	0.698145	−	0.715956i	$-0.254009\pi$
$661$	4.03255e8	1.39629	0.698145	+	0.715956i	$0.254009\pi$
$662$	0	0
$663$	− 1.33874e8i	− 0.459364i
$664$	0	0
$665$	2.38481e8	0.810941
$666$	0	0
$667$	1.05373e8i	0.355103i
$668$	0	0
$669$	−2.46420e8	−0.822995
$670$	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
$675$	− 8.51985e7i	− 0.277026i
$676$	0	0
$677$	−4.50729e8	−1.45261	−0.726306	−	0.687371i	$-0.758765\pi$
$677$	−4.50729e8	−1.45261	−0.726306	+	0.687371i	$0.758765\pi$
$678$	0	0
$679$	− 2.43385e8i	− 0.777470i
$680$	0	0
$681$	−2.31605e8	−0.733341
$682$	0	0
$683$	− 5.93405e8i	− 1.86247i	−0.364418	−	0.931235i	$-0.618732\pi$
$683$	− 5.93405e8i	− 1.86247i	0.364418	−	0.931235i	$-0.381268\pi$
$684$	0	0
$685$	5.41493e8	1.68470
$686$	0	0
$687$	− 2.00258e7i	− 0.0617617i
$688$	0	0
$689$	3.39802e8	1.03889
$690$	0	0
$691$	− 2.84794e8i	− 0.863170i	−0.902072	−	0.431585i	$-0.857955\pi$
$691$	− 2.84794e8i	− 0.863170i	0.902072	−	0.431585i	$-0.142045\pi$
$692$	0	0
$693$	−1.18233e8	−0.355254
$694$	0	0
$695$	8.41545e8i	2.50682i
$696$	0	0
$697$	3.77066e8	1.11357
$698$	0	0
$699$	− 4.09928e6i	− 0.0120026i
$700$	0	0
$701$	1.51280e8	0.439165	0.219583	−	0.975594i	$-0.429530\pi$
$701$	1.51280e8	0.439165	0.219583	+	0.975594i	$0.429530\pi$
$702$	0	0
$703$	− 5.36404e7i	− 0.154392i
$704$	0	0
$705$	−4.63622e8	−1.32311
$706$	0	0
$707$	5.00880e8i	1.41734i
$708$	0	0
$709$	6.04474e8	1.69605	0.848025	−	0.529956i	$-0.177792\pi$
$709$	6.04474e8	1.69605	0.848025	+	0.529956i	$0.177792\pi$
$710$	0	0
$711$	1.05796e8i	0.294349i
$712$	0	0
$713$	5.59300e8	1.54304
$714$	0	0
$715$	− 4.26766e8i	− 1.16754i
$716$	0	0
$717$	−2.95027e8	−0.800396
$718$	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
$723$	− 2.98578e7i	− 0.0790029i
$724$	0	0
$725$	−1.86196e8	−0.488604
$726$	0	0
$727$	1.33711e8i	0.347989i	0.984747	+	0.173994i	$0.0556675\pi$
$727$	1.33711e8i	0.347989i	−0.984747	+	0.173994i	$0.944333\pi$
$728$	0	0
$729$	−1.43489e7	−0.0370370
$730$	0	0
$731$	− 3.13099e8i	− 0.801549i
$732$	0	0
$733$	5.99936e8	1.52333	0.761663	−	0.647973i	$-0.224383\pi$
$733$	5.99936e8	1.52333	0.761663	+	0.647973i	$0.224383\pi$
$734$	0	0
$735$	1.23676e8i	0.311475i
$736$	0	0
$737$	6.54150e8	1.63409
$738$	0	0
$739$	4.83646e7i	0.119838i	0.998203	+	0.0599190i	$0.0190842\pi$
$739$	4.83646e7i	0.119838i	−0.998203	+	0.0599190i	$0.980916\pi$
$740$	0	0
$741$	8.55461e7	0.210255
$742$	0	0
$743$	− 3.69904e7i	− 0.0901826i	−0.998983	−	0.0450913i	$-0.985642\pi$
$743$	− 3.69904e7i	− 0.0901826i	0.998983	−	0.0450913i	$-0.0143579\pi$
$744$	0	0
$745$	−4.16311e7	−0.100681
$746$	0	0
$747$	− 5.99197e7i	− 0.143750i
$748$	0	0
$749$	3.73491e8	0.888863
$750$	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
$755$	− 1.04199e9i	− 2.42115i
$756$	0	0
$757$	4.79819e8	1.10609	0.553045	−	0.833152i	$-0.313466\pi$
$757$	4.79819e8	1.10609	0.553045	+	0.833152i	$0.313466\pi$
$758$	0	0
$759$	3.47888e8i	0.795636i
$760$	0	0
$761$	2.43069e8	0.551537	0.275769	−	0.961224i	$-0.411068\pi$
$761$	2.43069e8	0.551537	0.275769	+	0.961224i	$0.411068\pi$
$762$	0	0
$763$	− 1.95540e8i	− 0.440211i
$764$	0	0
$765$	−3.26798e8	−0.729953
$766$	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
$771$	1.05802e7i	0.0230850i
$772$	0	0
$773$	9.17077e7	0.198549	0.0992744	−	0.995060i	$-0.468348\pi$
$773$	9.17077e7	0.198549	0.0992744	+	0.995060i	$0.468348\pi$
$774$	0	0
$775$	9.88289e8i	2.12314i
$776$	0	0
$777$	−5.27176e7	−0.112381
$778$	0	0
$779$	2.40946e8i	0.509692i
$780$	0	0
$781$	−1.17087e9	−2.45785
$782$	0	0
$783$	3.13587e7i	0.0653240i
$784$	0	0
$785$	7.98471e8	1.65063
$786$	0	0
$787$	− 5.23454e8i	− 1.07388i	−0.843622	−	0.536938i	$-0.819581\pi$
$787$	− 5.23454e8i	− 1.07388i	0.843622	−	0.536938i	$-0.180419\pi$
$788$	0	0
$789$	−1.73232e8	−0.352693
$790$	0	0
$791$	− 7.08406e8i	− 1.43137i
$792$	0	0
$793$	1.04683e8	0.209921
$794$	0	0
$795$	− 8.29483e8i	− 1.65085i
$796$	0	0
$797$	1.58874e7	0.0313818	0.0156909	−	0.999877i	$-0.495005\pi$
$797$	1.58874e7	0.0313818	0.0156909	+	0.999877i	$0.495005\pi$
$798$	0	0
$799$	1.04935e9i	2.05721i
$800$	0	0
$801$	1.96751e7	0.0382842
$802$	0	0
$803$	7.00364e8i	1.35262i
$804$	0	0
$805$	−6.89635e8	−1.32200
$806$	0	0
$807$	− 1.56573e8i	− 0.297917i
$808$	0	0
$809$	5.17580e8	0.977534	0.488767	−	0.872414i	$-0.337447\pi$
$809$	5.17580e8	0.977534	0.488767	+	0.872414i	$0.337447\pi$
$810$	0	0
$811$	− 9.43642e8i	− 1.76907i	−0.466476	−	0.884534i	$-0.654476\pi$
$811$	− 9.43642e8i	− 1.76907i	0.466476	−	0.884534i	$-0.345524\pi$
$812$	0	0
$813$	1.03212e8	0.192069
$814$	0	0
$815$	3.00908e8i	0.555855i
$816$	0	0
$817$	2.00071e8	0.366875
$818$	0	0
$819$	− 8.40744e7i	− 0.153042i
$820$	0	0
$821$	−6.51239e8	−1.17682	−0.588411	−	0.808562i	$-0.700246\pi$
$821$	−6.51239e8	−1.17682	−0.588411	+	0.808562i	$0.700246\pi$
$822$	0	0
$823$	− 1.12175e8i	− 0.201231i	−0.994925	−	0.100615i	$-0.967919\pi$
$823$	− 1.12175e8i	− 0.201231i	0.994925	−	0.100615i	$-0.0320812\pi$
$824$	0	0
$825$	−6.14722e8	−1.09476
$826$	0	0
$827$	− 6.24440e8i	− 1.10401i	−0.833840	−	0.552007i	$-0.813862\pi$
$827$	− 6.24440e8i	− 1.10401i	0.833840	−	0.552007i	$-0.186138\pi$
$828$	0	0
$829$	5.96016e8	1.04615	0.523076	−	0.852286i	$-0.324784\pi$
$829$	5.96016e8	1.04615	0.523076	+	0.852286i	$0.324784\pi$
$830$	0	0
$831$	3.80915e8i	0.663782i
$832$	0	0
$833$	2.79924e8	0.484289
$834$	0	0
$835$	1.49658e9i	2.57064i
$836$	0	0
$837$	1.66445e8	0.283854
$838$	0	0
$839$	− 4.46780e7i	− 0.0756497i	−0.999284	−	0.0378249i	$-0.987957\pi$
$839$	− 4.46780e7i	− 0.0756497i	0.999284	−	0.0378249i	$-0.0120429\pi$
$840$	0	0
$841$	−5.26291e8	−0.884785
$842$	0	0
$843$	2.17656e8i	0.363320i
$844$	0	0
$845$	−6.38892e8	−1.05891
$846$	0	0
$847$	3.61446e8i	0.594830i
$848$	0	0
$849$	6.68485e8	1.09237
$850$	0	0
$851$	1.55116e8i	0.251691i
$852$	0	0
$853$	1.12761e9	1.81683	0.908414	−	0.418072i	$-0.137294\pi$
$853$	1.12761e9	1.81683	0.908414	+	0.418072i	$0.137294\pi$
$854$	0	0
$855$	− 2.08825e8i	− 0.334105i
$856$	0	0
$857$	−5.47949e8	−0.870557	−0.435279	−	0.900296i	$-0.643350\pi$
$857$	−5.47949e8	−0.870557	−0.435279	+	0.900296i	$0.643350\pi$
$858$	0	0
$859$	6.34544e8i	1.00111i	0.865704	+	0.500556i	$0.166871\pi$
$859$	6.34544e8i	1.00111i	−0.865704	+	0.500556i	$0.833129\pi$
$860$	0	0
$861$	2.36801e8	0.371000
$862$	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
$867$	3.63396e8i	0.557600i
$868$	0	0
$869$	7.63340e8	1.16321
$870$	0	0
$871$	4.65160e8i	0.703960i
$872$	0	0
$873$	−2.13118e8	−0.320316
$874$	0	0
$875$	− 3.72036e8i	− 0.555342i
$876$	0	0
$877$	−6.11977e6	−0.00907270	−0.00453635	−	0.999990i	$-0.501444\pi$
$877$	−6.11977e6	−0.00907270	−0.00453635	+	0.999990i	$0.501444\pi$
$878$	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
$883$	4.91469e8i	0.713861i	0.934131	+	0.356931i	$0.116177\pi$
$883$	4.91469e8i	0.713861i	−0.934131	+	0.356931i	$0.883823\pi$
$884$	0	0
$885$	−6.48778e8	−0.935980
$886$	0	0
$887$	− 6.95698e8i	− 0.996897i	−0.866919	−	0.498448i	$-0.833903\pi$
$887$	− 6.95698e8i	− 0.996897i	0.866919	−	0.498448i	$-0.166097\pi$
$888$	0	0
$889$	1.85984e8	0.264710
$890$	0	0
$891$	1.03530e8i	0.146364i
$892$	0	0
$893$	−6.70534e8	−0.941600
$894$	0	0
$895$	− 1.72450e9i	− 2.40544i
$896$	0	0
$897$	−2.47380e8	−0.342758
$898$	0	0
$899$	− 3.63756e8i	− 0.500646i
$900$	0	0
$901$	−1.87742e9	−2.56678
$902$	0	0
$903$	− 1.96629e8i	− 0.267045i
$904$	0	0
$905$	1.17159e9	1.58063
$906$	0	0
$907$	− 1.10192e9i	− 1.47682i	−0.674351	−	0.738411i	$-0.735577\pi$
$907$	− 1.10192e9i	− 1.47682i	0.674351	−	0.738411i	$-0.264423\pi$
$908$	0	0
$909$	4.38592e8	0.583942
$910$	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
$915$	− 2.55540e8i	− 0.333576i
$916$	0	0
$917$	−4.80458e8	−0.623085
$918$	0	0
$919$	− 1.13853e9i	− 1.46690i	−0.679746	−	0.733448i	$-0.737910\pi$
$919$	− 1.13853e9i	− 1.46690i	0.679746	−	0.733448i	$-0.262090\pi$
$920$	0	0
$921$	−4.26360e8	−0.545754
$922$	0	0
$923$	− 8.32594e8i	− 1.05883i
$924$	0	0
$925$	−2.74092e8	−0.346315
$926$	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
$931$	1.78872e8i	0.221663i
$932$	0	0
$933$	5.25900e8	0.647528
$934$	0	0
$935$	2.35791e9i	2.88464i
$936$	0	0
$937$	6.16456e7	0.0749348	0.0374674	−	0.999298i	$-0.488071\pi$
$937$	6.16456e7	0.0749348	0.0374674	+	0.999298i	$0.488071\pi$
$938$	0	0
$939$	− 7.66782e8i	− 0.926137i
$940$	0	0
$941$	3.10725e8	0.372913	0.186457	−	0.982463i	$-0.440300\pi$
$941$	3.10725e8	0.372913	0.186457	+	0.982463i	$0.440300\pi$
$942$	0	0
$943$	− 6.96763e8i	− 0.830902i
$944$	0	0
$945$	−2.05232e8	−0.243192
$946$	0	0
$947$	1.13722e8i	0.133904i	0.997756	+	0.0669522i	$0.0213275\pi$
$947$	1.13722e8i	0.133904i	−0.997756	+	0.0669522i	$0.978672\pi$
$948$	0	0
$949$	−4.98023e8	−0.582707
$950$	0	0
$951$	3.07234e8i	0.357214i
$952$	0	0
$953$	5.25518e8	0.607168	0.303584	−	0.952805i	$-0.401817\pi$
$953$	5.25518e8	0.607168	0.303584	+	0.952805i	$0.401817\pi$
$954$	0	0
$955$	1.17743e8i	0.135184i
$956$	0	0
$957$	2.26258e8	0.258148
$958$	0	0
$959$	− 7.69687e8i	− 0.872686i
$960$	0	0
$961$	−1.04323e9	−1.17547
$962$	0	0
$963$	− 3.27045e8i	− 0.366209i
$964$	0	0
$965$	1.51423e9	1.68504
$966$	0	0
$967$	− 5.50448e8i	− 0.608747i	−0.952553	−	0.304373i	$-0.901553\pi$
$967$	− 5.50448e8i	− 0.608747i	0.952553	−	0.304373i	$-0.0984470\pi$
$968$	0	0
$969$	−4.72647e8	−0.519476
$970$	0	0
$971$	− 1.54851e9i	− 1.69144i	−0.533624	−	0.845722i	$-0.679170\pi$
$971$	− 1.54851e9i	− 1.69144i	0.533624	−	0.845722i	$-0.320830\pi$
$972$	0	0
$973$	1.19618e9	1.29855
$974$	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
$979$	− 1.41959e8i	− 0.151292i
$980$	0	0
$981$	−1.71223e8	−0.181366
$982$	0	0
$983$	− 7.96734e8i	− 0.838789i	−0.907804	−	0.419395i	$-0.862242\pi$
$983$	− 7.96734e8i	− 0.838789i	0.907804	−	0.419395i	$-0.137758\pi$
$984$	0	0
$985$	3.74402e8	0.391769
$986$	0	0
$987$	6.58999e8i	0.685383i
$988$	0	0
$989$	−5.78562e8	−0.598082
$990$	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
$995$	1.98054e9i	2.01054i
$996$	0	0
$997$	4.81174e8	0.485531	0.242765	−	0.970085i	$-0.421945\pi$
$997$	4.81174e8	0.485531	0.242765	+	0.970085i	$0.421945\pi$
$998$	0	0
$999$	4.61619e7i	0.0463007i

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.7.b.c.319.1	✓	8	16.13	even	4
384.7.b.c.319.4	yes	8	16.11	odd	4
384.7.b.c.319.5	yes	8	16.3	odd	4
384.7.b.c.319.8	yes	8	16.5	even	4
768.7.g.g.511.1		8	8.5	even	2	inner
768.7.g.g.511.4		8	4.3	odd	2	inner
768.7.g.g.511.5		8	8.3	odd	2	inner
768.7.g.g.511.8		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:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	768.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+15.5885i q^{3} +195.235 q^{5} -277.510i q^{7} -243.000 q^{9} +O(q^{10})\)	\(q+15.5885i q^{3} +195.235 q^{5} -277.510i q^{7} -243.000 q^{9} +1753.29i q^{11} -1246.75 q^{13} +3043.41i q^{15} +6888.35 q^{17} +4401.67i q^{19} +4325.95 q^{21} -12728.7i q^{23} +22491.7 q^{25} -3788.00i q^{27} -8278.44 q^{29} +43940.1i q^{31} -27331.1 q^{33} -54179.6i q^{35} -12186.4 q^{37} -19434.9i q^{39} +54739.6 q^{41} -45453.4i q^{43} -47442.1 q^{45} +152336. i q^{47} +40637.3 q^{49} +107379. i q^{51} -272550. q^{53} +342303. i q^{55} -68615.3 q^{57} +213175. i q^{59} -83964.8 q^{61} +67434.9i q^{63} -243409. q^{65} -373099. i q^{67} +198420. q^{69} +667812. i q^{71} +399457. q^{73} +350611. i q^{75} +486555. q^{77} -435376. i q^{79} +59049.0 q^{81} +246583. i q^{83} +1.34485e6 q^{85} -129048. i q^{87} -80967.5 q^{89} +345985. i q^{91} -684959. q^{93} +859361. i q^{95} +877030. q^{97} -426049. i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 1944 q^{9}+O(q^{10}) \) 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

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