LMFDB - Embedded newform 384.10.d.a.193.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,10,Mod(193,384)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("384.193");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	384.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$197.773761087$
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} + 13062x^{6} + 45211107x^{4} + 45928424926x^{2} + 852972309225 $ x^8 + 13062x^6 + 45211107x^4 + 45928424926*x^2 + 852972309225
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{32}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			193.8
Root			$-43.3581i$ of defining polynomial
Character	$\chi$	$=$	384.193
Dual form			384.10.d.a.193.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+81.0000i q^{3} +1783.68i q^{5} -4965.03 q^{7} -6561.00 q^{9} +O(q^{10})$	$q+81.0000i q^{3} +1783.68i q^{5} -4965.03 q^{7} -6561.00 q^{9} +53705.0i q^{11} -54228.9i q^{13} -144478. q^{15} -554662. q^{17} +650852. i q^{19} -402168. i q^{21} +183775. q^{23} -1.22839e6 q^{25} -531441. i q^{27} +2.02445e6i q^{29} -3.43180e6 q^{31} -4.35011e6 q^{33} -8.85603e6i q^{35} -1.30084e6i q^{37} +4.39254e6 q^{39} +157393. q^{41} -2.10193e6i q^{43} -1.17027e7i q^{45} -4.73332e7 q^{47} -1.57021e7 q^{49} -4.49277e7i q^{51} +1.00735e8i q^{53} -9.57926e7 q^{55} -5.27190e7 q^{57} -1.80263e7i q^{59} -5.48498e7i q^{61} +3.25756e7 q^{63} +9.67270e7 q^{65} -2.13387e8i q^{67} +1.48858e7i q^{69} -1.64615e8 q^{71} -7.58505e7 q^{73} -9.94998e7i q^{75} -2.66647e8i q^{77} +4.25335e8 q^{79} +4.30467e7 q^{81} -1.06618e8i q^{83} -9.89341e8i q^{85} -1.63981e8 q^{87} -1.02792e9 q^{89} +2.69248e8i q^{91} -2.77976e8i q^{93} -1.16091e9 q^{95} +6.24259e8 q^{97} -3.52359e8i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 13632 q^{7} - 52488 q^{9}+O(q^{10}) $ 8 * q - 13632 * q^7 - 52488 * q^9	$ 8 q - 13632 q^{7} - 52488 q^{9} - 90720 q^{15} + 8304 q^{17} - 4612608 q^{23} - 4754904 q^{25} - 7499328 q^{31} - 6213024 q^{33} - 23211360 q^{39} - 43518896 q^{41} + 49382016 q^{47} - 74106808 q^{49} + 19030656 q^{55} + 38141280 q^{57} + 89439552 q^{63} - 110270336 q^{65} - 741751296 q^{71} - 1507903440 q^{73} + 1008373440 q^{79} + 344373768 q^{81} + 423468000 q^{87} - 1337034448 q^{89} - 543950208 q^{95} - 904817936 q^{97}+O(q^{100}) $ 8 * q - 13632 * q^7 - 52488 * q^9 - 90720 * q^15 + 8304 * q^17 - 4612608 * q^23 - 4754904 * q^25 - 7499328 * q^31 - 6213024 * q^33 - 23211360 * q^39 - 43518896 * q^41 + 49382016 * q^47 - 74106808 * q^49 + 19030656 * q^55 + 38141280 * q^57 + 89439552 * q^63 - 110270336 * q^65 - 741751296 * q^71 - 1507903440 * q^73 + 1008373440 * q^79 + 344373768 * q^81 + 423468000 * q^87 - 1337034448 * q^89 - 543950208 * q^95 - 904817936 * 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
$3$	81.0000i	0.577350i
$3$	81.0000i	0.577350i
$4$	0	0
$5$	1783.68i	1.27630i	0.769913	+	0.638149i	$0.220300\pi$
$5$	1783.68i	1.27630i	−0.769913	+	0.638149i	$0.779700\pi$
$6$	0	0
$7$	−4965.03	−0.781593	−0.390797	−	0.920477i	$-0.627800\pi$
$7$	−4965.03	−0.781593	−0.390797	+	0.920477i	$0.627800\pi$
$8$	0	0
$9$	−6561.00	−0.333333
$10$	0	0
$11$	53705.0i	1.10598i	0.833187	+	0.552991i	$0.186513\pi$
$11$	53705.0i	1.10598i	−0.833187	+	0.552991i	$0.813487\pi$
$12$	0	0
$13$	− 54228.9i	− 0.526606i	−0.964713	−	0.263303i	$-0.915188\pi$
$13$	− 54228.9i	− 0.526606i	0.964713	−	0.263303i	$-0.0848118\pi$
$14$	0	0
$15$	−144478.	−0.736871
$16$	0	0
$17$	−554662.	−1.61068	−0.805339	−	0.592815i	$-0.798017\pi$
$17$	−554662.	−1.61068	−0.805339	+	0.592815i	$0.798017\pi$
$18$	0	0
$19$	650852.i	1.14575i	0.819642	+	0.572876i	$0.194172\pi$
$19$	650852.i	1.14575i	−0.819642	+	0.572876i	$0.805828\pi$
$20$	0	0
$21$	− 402168.i	− 0.451253i
$22$	0	0
$23$	183775.	0.136934	0.0684669	−	0.997653i	$-0.478189\pi$
$23$	183775.	0.136934	0.0684669	+	0.997653i	$0.478189\pi$
$24$	0	0
$25$	−1.22839e6	−0.628937
$26$	0	0
$27$	− 531441.i	− 0.192450i
$28$	0	0
$29$	2.02445e6i	0.531516i	0.964040	+	0.265758i	$0.0856222\pi$
$29$	2.02445e6i	0.531516i	−0.964040	+	0.265758i	$0.914378\pi$
$30$	0	0
$31$	−3.43180e6	−0.667413	−0.333706	−	0.942677i	$-0.608299\pi$
$31$	−3.43180e6	−0.667413	−0.333706	+	0.942677i	$0.608299\pi$
$32$	0	0
$33$	−4.35011e6	−0.638539
$34$	0	0
$35$	− 8.85603e6i	− 0.997546i
$36$	0	0
$37$	− 1.30084e6i	− 0.114108i	−0.998371	−	0.0570542i	$-0.981829\pi$
$37$	− 1.30084e6i	− 0.114108i	0.998371	−	0.0570542i	$-0.0181708\pi$
$38$	0	0
$39$	4.39254e6	0.304036
$40$	0	0
$41$	157393.	0.00869877	0.00434938	−	0.999991i	$-0.498616\pi$
$41$	157393.	0.00869877	0.00434938	+	0.999991i	$0.498616\pi$
$42$	0	0
$43$	− 2.10193e6i	− 0.0937583i	−0.998901	−	0.0468792i	$-0.985072\pi$
$43$	− 2.10193e6i	− 0.0937583i	0.998901	−	0.0468792i	$-0.0149276\pi$
$44$	0	0
$45$	− 1.17027e7i	− 0.425433i
$46$	0	0
$47$	−4.73332e7	−1.41490	−0.707449	−	0.706764i	$-0.750154\pi$
$47$	−4.73332e7	−1.41490	−0.707449	+	0.706764i	$0.750154\pi$
$48$	0	0
$49$	−1.57021e7	−0.389112
$50$	0	0
$51$	− 4.49277e7i	− 0.929925i
$52$	0	0
$53$	1.00735e8i	1.75364i	0.480820	+	0.876819i	$0.340339\pi$
$53$	1.00735e8i	1.75364i	−0.480820	+	0.876819i	$0.659661\pi$
$54$	0	0
$55$	−9.57926e7	−1.41156
$56$	0	0
$57$	−5.27190e7	−0.661501
$58$	0	0
$59$	− 1.80263e7i	− 0.193674i	−0.995300	−	0.0968372i	$-0.969127\pi$
$59$	− 1.80263e7i	− 0.193674i	0.995300	−	0.0968372i	$-0.0308726\pi$
$60$	0	0
$61$	− 5.48498e7i	− 0.507214i	−0.967307	−	0.253607i	$-0.918383\pi$
$61$	− 5.48498e7i	− 0.507214i	0.967307	−	0.253607i	$-0.0816169\pi$
$62$	0	0
$63$	3.25756e7	0.260531
$64$	0	0
$65$	9.67270e7	0.672106
$66$	0	0
$67$	− 2.13387e8i	− 1.29369i	−0.762621	−	0.646846i	$-0.776088\pi$
$67$	− 2.13387e8i	− 1.29369i	0.762621	−	0.646846i	$-0.223912\pi$
$68$	0	0
$69$	1.48858e7i	0.0790588i
$70$	0	0
$71$	−1.64615e8	−0.768786	−0.384393	−	0.923169i	$-0.625589\pi$
$71$	−1.64615e8	−0.768786	−0.384393	+	0.923169i	$0.625589\pi$
$72$	0	0
$73$	−7.58505e7	−0.312612	−0.156306	−	0.987709i	$-0.549959\pi$
$73$	−7.58505e7	−0.312612	−0.156306	+	0.987709i	$0.549959\pi$
$74$	0	0
$75$	− 9.94998e7i	− 0.363117i
$76$	0	0
$77$	− 2.66647e8i	− 0.864428i
$78$	0	0
$79$	4.25335e8	1.22860	0.614298	−	0.789074i	$-0.289439\pi$
$79$	4.25335e8	1.22860	0.614298	+	0.789074i	$0.289439\pi$
$80$	0	0
$81$	4.30467e7	0.111111
$82$	0	0
$83$	− 1.06618e8i	− 0.246592i	−0.992370	−	0.123296i	$-0.960654\pi$
$83$	− 1.06618e8i	− 0.246592i	0.992370	−	0.123296i	$-0.0393465\pi$
$84$	0	0
$85$	− 9.89341e8i	− 2.05570i
$86$	0	0
$87$	−1.63981e8	−0.306871
$88$	0	0
$89$	−1.02792e9	−1.73662	−0.868308	−	0.496025i	$-0.834793\pi$
$89$	−1.02792e9	−1.73662	−0.868308	+	0.496025i	$0.834793\pi$
$90$	0	0
$91$	2.69248e8i	0.411591i
$92$	0	0
$93$	− 2.77976e8i	− 0.385331i
$94$	0	0
$95$	−1.16091e9	−1.46232
$96$	0	0
$97$	6.24259e8	0.715966	0.357983	−	0.933728i	$-0.383465\pi$
$97$	6.24259e8	0.715966	0.357983	+	0.933728i	$0.383465\pi$
$98$	0	0
$99$	− 3.52359e8i	− 0.368660i
$100$	0	0
$101$	− 5.25002e8i	− 0.502013i	−0.967985	−	0.251006i	$-0.919239\pi$
$101$	− 5.25002e8i	− 0.502013i	0.967985	−	0.251006i	$-0.0807615\pi$
$102$	0	0
$103$	1.09299e9	0.956859	0.478429	−	0.878126i	$-0.341206\pi$
$103$	1.09299e9	0.956859	0.478429	+	0.878126i	$0.341206\pi$
$104$	0	0
$105$	7.17339e8	0.575933
$106$	0	0
$107$	2.50815e9i	1.84981i	0.380202	+	0.924904i	$0.375855\pi$
$107$	2.50815e9i	1.84981i	−0.380202	+	0.924904i	$0.624145\pi$
$108$	0	0
$109$	1.99633e9i	1.35461i	0.735703	+	0.677304i	$0.236852\pi$
$109$	1.99633e9i	1.35461i	−0.735703	+	0.677304i	$0.763148\pi$
$110$	0	0
$111$	1.05368e8	0.0658805
$112$	0	0
$113$	−1.75634e9	−1.01334	−0.506670	−	0.862140i	$-0.669124\pi$
$113$	−1.75634e9	−1.01334	−0.506670	+	0.862140i	$0.669124\pi$
$114$	0	0
$115$	3.27796e8i	0.174768i
$116$	0	0
$117$	3.55796e8i	0.175535i
$118$	0	0
$119$	2.75392e9	1.25889
$120$	0	0
$121$	−5.26281e8	−0.223195
$122$	0	0
$123$	1.27488e7i	0.00502224i
$124$	0	0
$125$	1.29269e9i	0.473587i
$126$	0	0
$127$	4.66916e9	1.59266	0.796328	−	0.604865i	$-0.206773\pi$
$127$	4.66916e9	1.59266	0.796328	+	0.604865i	$0.206773\pi$
$128$	0	0
$129$	1.70256e8	0.0541314
$130$	0	0
$131$	− 5.48941e9i	− 1.62857i	−0.580468	−	0.814283i	$-0.697131\pi$
$131$	− 5.48941e9i	− 1.62857i	0.580468	−	0.814283i	$-0.302869\pi$
$132$	0	0
$133$	− 3.23150e9i	− 0.895513i
$134$	0	0
$135$	9.47921e8	0.245624
$136$	0	0
$137$	−3.05978e9	−0.742075	−0.371037	−	0.928618i	$-0.620998\pi$
$137$	−3.05978e9	−0.742075	−0.371037	+	0.928618i	$0.620998\pi$
$138$	0	0
$139$	8.06711e9i	1.83295i	0.400089	+	0.916476i	$0.368979\pi$
$139$	8.06711e9i	1.83295i	−0.400089	+	0.916476i	$0.631021\pi$
$140$	0	0
$141$	− 3.83399e9i	− 0.816892i
$142$	0	0
$143$	2.91236e9	0.582416
$144$	0	0
$145$	−3.61097e9	−0.678373
$146$	0	0
$147$	− 1.27187e9i	− 0.224654i
$148$	0	0
$149$	6.06220e9i	1.00761i	0.863817	+	0.503805i	$0.168067\pi$
$149$	6.06220e9i	1.00761i	−0.863817	+	0.503805i	$0.831933\pi$
$150$	0	0
$151$	−5.92499e9	−0.927451	−0.463726	−	0.885979i	$-0.653488\pi$
$151$	−5.92499e9	−0.927451	−0.463726	+	0.885979i	$0.653488\pi$
$152$	0	0
$153$	3.63914e9	0.536892
$154$	0	0
$155$	− 6.12124e9i	− 0.851818i
$156$	0	0
$157$	− 5.68760e9i	− 0.747103i	−0.927609	−	0.373551i	$-0.878140\pi$
$157$	− 5.68760e9i	− 0.747103i	0.927609	−	0.373551i	$-0.121860\pi$
$158$	0	0
$159$	−8.15956e9	−1.01246
$160$	0	0
$161$	−9.12448e8	−0.107027
$162$	0	0
$163$	− 4.56247e9i	− 0.506239i	−0.967435	−	0.253120i	$-0.918543\pi$
$163$	− 4.56247e9i	− 0.506239i	0.967435	−	0.253120i	$-0.0814566\pi$
$164$	0	0
$165$	− 7.75920e9i	− 0.814966i
$166$	0	0
$167$	1.22032e9	0.121409	0.0607044	−	0.998156i	$-0.480665\pi$
$167$	1.22032e9	0.121409	0.0607044	+	0.998156i	$0.480665\pi$
$168$	0	0
$169$	7.66373e9	0.722687
$170$	0	0
$171$	− 4.27024e9i	− 0.381918i
$172$	0	0
$173$	2.24268e10i	1.90353i	0.306821	+	0.951767i	$0.400735\pi$
$173$	2.24268e10i	1.90353i	−0.306821	+	0.951767i	$0.599265\pi$
$174$	0	0
$175$	6.09901e9	0.491573
$176$	0	0
$177$	1.46013e9	0.111818
$178$	0	0
$179$	− 1.72440e10i	− 1.25545i	−0.778435	−	0.627725i	$-0.783986\pi$
$179$	− 1.72440e10i	− 1.25545i	0.778435	−	0.627725i	$-0.216014\pi$
$180$	0	0
$181$	− 2.90161e9i	− 0.200949i	−0.994940	−	0.100474i	$-0.967964\pi$
$181$	− 2.90161e9i	− 0.200949i	0.994940	−	0.100474i	$-0.0320361\pi$
$182$	0	0
$183$	4.44283e9	0.292840
$184$	0	0
$185$	2.32029e9	0.145636
$186$	0	0
$187$	− 2.97882e10i	− 1.78138i
$188$	0	0
$189$	2.63862e9i	0.150418i
$190$	0	0
$191$	1.14739e9	0.0623824	0.0311912	−	0.999513i	$-0.490070\pi$
$191$	1.14739e9	0.0623824	0.0311912	+	0.999513i	$0.490070\pi$
$192$	0	0
$193$	3.19936e10	1.65980	0.829900	−	0.557913i	$-0.188398\pi$
$193$	3.19936e10	1.65980	0.829900	+	0.557913i	$0.188398\pi$
$194$	0	0
$195$	7.83489e9i	0.388040i
$196$	0	0
$197$	3.38558e9i	0.160153i	0.996789	+	0.0800764i	$0.0255164\pi$
$197$	3.38558e9i	0.160153i	−0.996789	+	0.0800764i	$0.974484\pi$
$198$	0	0
$199$	7.28755e9	0.329415	0.164707	−	0.986342i	$-0.447332\pi$
$199$	7.28755e9	0.329415	0.164707	+	0.986342i	$0.447332\pi$
$200$	0	0
$201$	1.72843e10	0.746913
$202$	0	0
$203$	− 1.00515e10i	− 0.415429i
$204$	0	0
$205$	2.80739e8i	0.0111022i
$206$	0	0
$207$	−1.20575e9	−0.0456446
$208$	0	0
$209$	−3.49540e10	−1.26718
$210$	0	0
$211$	− 1.58433e9i	− 0.0550268i	−0.999621	−	0.0275134i	$-0.991241\pi$
$211$	− 1.58433e9i	− 0.0550268i	0.999621	−	0.0275134i	$-0.00875889\pi$
$212$	0	0
$213$	− 1.33338e10i	− 0.443859i
$214$	0	0
$215$	3.74917e9	0.119664
$216$	0	0
$217$	1.70390e10	0.521645
$218$	0	0
$219$	− 6.14389e9i	− 0.180487i
$220$	0	0
$221$	3.00787e10i	0.848192i
$222$	0	0
$223$	2.82233e10	0.764250	0.382125	−	0.924111i	$-0.375192\pi$
$223$	2.82233e10	0.764250	0.382125	+	0.924111i	$0.375192\pi$
$224$	0	0
$225$	8.05948e9	0.209646
$226$	0	0
$227$	1.00872e10i	0.252147i	0.992021	+	0.126073i	$0.0402375\pi$
$227$	1.00872e10i	0.252147i	−0.992021	+	0.126073i	$0.959763\pi$
$228$	0	0
$229$	3.06314e9i	0.0736051i	0.999323	+	0.0368025i	$0.0117173\pi$
$229$	3.06314e9i	0.0736051i	−0.999323	+	0.0368025i	$0.988283\pi$
$230$	0	0
$231$	2.15984e10	0.499077
$232$	0	0
$233$	−4.21496e10	−0.936896	−0.468448	−	0.883491i	$-0.655187\pi$
$233$	−4.21496e10	−0.936896	−0.468448	+	0.883491i	$0.655187\pi$
$234$	0	0
$235$	− 8.44273e10i	− 1.80583i
$236$	0	0
$237$	3.44521e10i	0.709330i
$238$	0	0
$239$	8.33007e10	1.65142	0.825711	−	0.564093i	$-0.190774\pi$
$239$	8.33007e10	1.65142	0.825711	+	0.564093i	$0.190774\pi$
$240$	0	0
$241$	1.35832e10	0.259373	0.129686	−	0.991555i	$-0.458603\pi$
$241$	1.35832e10	0.259373	0.129686	+	0.991555i	$0.458603\pi$
$242$	0	0
$243$	3.48678e9i	0.0641500i
$244$	0	0
$245$	− 2.80075e10i	− 0.496623i
$246$	0	0
$247$	3.52950e10	0.603360
$248$	0	0
$249$	8.63606e9	0.142370
$250$	0	0
$251$	9.40336e9i	0.149538i	0.997201	+	0.0747690i	$0.0238219\pi$
$251$	9.40336e9i	0.149538i	−0.997201	+	0.0747690i	$0.976178\pi$
$252$	0	0
$253$	9.86963e9i	0.151446i
$254$	0	0
$255$	8.01366e10	1.18686
$256$	0	0
$257$	−3.20439e10	−0.458190	−0.229095	−	0.973404i	$-0.573577\pi$
$257$	−3.20439e10	−0.458190	−0.229095	+	0.973404i	$0.573577\pi$
$258$	0	0
$259$	6.45874e9i	0.0891864i
$260$	0	0
$261$	− 1.32824e10i	− 0.177172i
$262$	0	0
$263$	8.93217e10	1.15121	0.575607	−	0.817726i	$-0.304766\pi$
$263$	8.93217e10	1.15121	0.575607	+	0.817726i	$0.304766\pi$
$264$	0	0
$265$	−1.79680e11	−2.23817
$266$	0	0
$267$	− 8.32615e10i	− 1.00264i
$268$	0	0
$269$	3.70480e10i	0.431400i	0.976460	+	0.215700i	$0.0692033\pi$
$269$	3.70480e10i	0.431400i	−0.976460	+	0.215700i	$0.930797\pi$
$270$	0	0
$271$	1.10018e11	1.23909	0.619546	−	0.784960i	$-0.287317\pi$
$271$	1.10018e11	1.23909	0.619546	+	0.784960i	$0.287317\pi$
$272$	0	0
$273$	−2.18091e10	−0.237632
$274$	0	0
$275$	− 6.59709e10i	− 0.695593i
$276$	0	0
$277$	− 8.60128e10i	− 0.877818i	−0.898532	−	0.438909i	$-0.855365\pi$
$277$	− 8.60128e10i	− 0.877818i	0.898532	−	0.438909i	$-0.144635\pi$
$278$	0	0
$279$	2.25160e10	0.222471
$280$	0	0
$281$	1.40155e11	1.34101	0.670503	−	0.741907i	$-0.266078\pi$
$281$	1.40155e11	1.34101	0.670503	+	0.741907i	$0.266078\pi$
$282$	0	0
$283$	− 6.86732e10i	− 0.636426i	−0.948019	−	0.318213i	$-0.896917\pi$
$283$	− 6.86732e10i	− 0.636426i	0.948019	−	0.318213i	$-0.103083\pi$
$284$	0	0
$285$	− 9.40339e10i	− 0.844272i
$286$	0	0
$287$	−7.81461e8	−0.00679890
$288$	0	0
$289$	1.89063e11	1.59428
$290$	0	0
$291$	5.05650e10i	0.413363i
$292$	0	0
$293$	3.90112e10i	0.309233i	0.987975	+	0.154616i	$0.0494141\pi$
$293$	3.90112e10i	0.309233i	−0.987975	+	0.154616i	$0.950586\pi$
$294$	0	0
$295$	3.21531e10	0.247186
$296$	0	0
$297$	2.85410e10	0.212846
$298$	0	0
$299$	− 9.96590e9i	− 0.0721101i
$300$	0	0
$301$	1.04361e10i	0.0732809i
$302$	0	0
$303$	4.25251e10	0.289837
$304$	0	0
$305$	9.78346e10	0.647356
$306$	0	0
$307$	− 1.94107e11i	− 1.24715i	−0.781765	−	0.623573i	$-0.785680\pi$
$307$	− 1.94107e11i	− 1.24715i	0.781765	−	0.623573i	$-0.214320\pi$
$308$	0	0
$309$	8.85320e10i	0.552443i
$310$	0	0
$311$	1.41999e11	0.860726	0.430363	−	0.902656i	$-0.358386\pi$
$311$	1.41999e11	0.860726	0.430363	+	0.902656i	$0.358386\pi$
$312$	0	0
$313$	−1.96211e11	−1.15551	−0.577756	−	0.816209i	$-0.696072\pi$
$313$	−1.96211e11	−1.15551	−0.577756	+	0.816209i	$0.696072\pi$
$314$	0	0
$315$	5.81044e10i	0.332515i
$316$	0	0
$317$	− 3.32729e11i	− 1.85065i	−0.379177	−	0.925324i	$-0.623793\pi$
$317$	− 3.32729e11i	− 1.85065i	0.379177	−	0.925324i	$-0.376207\pi$
$318$	0	0
$319$	−1.08723e11	−0.587847
$320$	0	0
$321$	−2.03160e11	−1.06799
$322$	0	0
$323$	− 3.61003e11i	− 1.84544i
$324$	0	0
$325$	6.66143e10i	0.331202i
$326$	0	0
$327$	−1.61703e11	−0.782084
$328$	0	0
$329$	2.35011e11	1.10587
$330$	0	0
$331$	− 2.82155e11i	− 1.29200i	−0.763338	−	0.646000i	$-0.776441\pi$
$331$	− 2.82155e11i	− 1.29200i	0.763338	−	0.646000i	$-0.223559\pi$
$332$	0	0
$333$	8.53484e9i	0.0380361i
$334$	0	0
$335$	3.80614e11	1.65114
$336$	0	0
$337$	−2.98725e11	−1.26164	−0.630822	−	0.775928i	$-0.717282\pi$
$337$	−2.98725e11	−1.26164	−0.630822	+	0.775928i	$0.717282\pi$
$338$	0	0
$339$	− 1.42263e11i	− 0.585052i
$340$	0	0
$341$	− 1.84305e11i	− 0.738146i
$342$	0	0
$343$	2.78318e11	1.08572
$344$	0	0
$345$	−2.65515e10	−0.100903
$346$	0	0
$347$	− 4.84025e11i	− 1.79219i	−0.443858	−	0.896097i	$-0.646391\pi$
$347$	− 4.84025e11i	− 1.79219i	0.443858	−	0.896097i	$-0.353609\pi$
$348$	0	0
$349$	− 2.44115e10i	− 0.0880806i	−0.999030	−	0.0440403i	$-0.985977\pi$
$349$	− 2.44115e10i	− 0.0880806i	0.999030	−	0.0440403i	$-0.0140230\pi$
$350$	0	0
$351$	−2.88194e10	−0.101345
$352$	0	0
$353$	−3.28670e11	−1.12661	−0.563306	−	0.826248i	$-0.690471\pi$
$353$	−3.28670e11	−1.12661	−0.563306	+	0.826248i	$0.690471\pi$
$354$	0	0
$355$	− 2.93620e11i	− 0.981201i
$356$	0	0
$357$	2.23067e11i	0.726823i
$358$	0	0
$359$	−4.58376e11	−1.45646	−0.728228	−	0.685335i	$-0.759656\pi$
$359$	−4.58376e11	−1.45646	−0.728228	+	0.685335i	$0.759656\pi$
$360$	0	0
$361$	−1.00920e11	−0.312749
$362$	0	0
$363$	− 4.26288e10i	− 0.128862i
$364$	0	0
$365$	− 1.35293e11i	− 0.398986i
$366$	0	0
$367$	5.88051e11	1.69207	0.846034	−	0.533128i	$-0.178984\pi$
$367$	5.88051e11	1.69207	0.846034	+	0.533128i	$0.178984\pi$
$368$	0	0
$369$	−1.03265e9	−0.00289959
$370$	0	0
$371$	− 5.00154e11i	− 1.37063i
$372$	0	0
$373$	− 6.61776e9i	− 0.0177020i	−0.999961	−	0.00885098i	$-0.997183\pi$
$373$	− 6.61776e9i	− 0.0177020i	0.999961	−	0.00885098i	$-0.00281739\pi$
$374$	0	0
$375$	−1.04708e11	−0.273426
$376$	0	0
$377$	1.09784e11	0.279899
$378$	0	0
$379$	5.36022e11i	1.33446i	0.744851	+	0.667231i	$0.232521\pi$
$379$	5.36022e11i	1.33446i	−0.744851	+	0.667231i	$0.767479\pi$
$380$	0	0
$381$	3.78202e11i	0.919520i
$382$	0	0
$383$	5.39762e11	1.28176	0.640882	−	0.767640i	$-0.278569\pi$
$383$	5.39762e11	1.28176	0.640882	+	0.767640i	$0.278569\pi$
$384$	0	0
$385$	4.75613e11	1.10327
$386$	0	0
$387$	1.37907e10i	0.0312528i
$388$	0	0
$389$	1.54973e11i	0.343150i	0.985171	+	0.171575i	$0.0548855\pi$
$389$	1.54973e11i	0.343150i	−0.985171	+	0.171575i	$0.945114\pi$
$390$	0	0
$391$	−1.01933e11	−0.220556
$392$	0	0
$393$	4.44643e11	0.940253
$394$	0	0
$395$	7.58662e11i	1.56806i
$396$	0	0
$397$	4.94662e11i	0.999427i	0.866191	+	0.499714i	$0.166561\pi$
$397$	4.94662e11i	0.999427i	−0.866191	+	0.499714i	$0.833439\pi$
$398$	0	0
$399$	2.61751e11	0.517024
$400$	0	0
$401$	4.84391e11	0.935506	0.467753	−	0.883859i	$-0.345064\pi$
$401$	4.84391e11	0.935506	0.467753	+	0.883859i	$0.345064\pi$
$402$	0	0
$403$	1.86103e11i	0.351463i
$404$	0	0
$405$	7.67816e10i	0.141811i
$406$	0	0
$407$	6.98619e10	0.126202
$408$	0	0
$409$	−3.98035e11	−0.703341	−0.351671	−	0.936124i	$-0.614386\pi$
$409$	−3.98035e11	−0.703341	−0.351671	+	0.936124i	$0.614386\pi$
$410$	0	0
$411$	− 2.47842e11i	− 0.428437i
$412$	0	0
$413$	8.95011e10i	0.151375i
$414$	0	0
$415$	1.90172e11	0.314725
$416$	0	0
$417$	−6.53436e11	−1.05826
$418$	0	0
$419$	6.97991e10i	0.110633i	0.998469	+	0.0553167i	$0.0176169\pi$
$419$	6.97991e10i	0.110633i	−0.998469	+	0.0553167i	$0.982383\pi$
$420$	0	0
$421$	3.17674e11i	0.492847i	0.969162	+	0.246423i	$0.0792554\pi$
$421$	3.17674e11i	0.492847i	−0.969162	+	0.246423i	$0.920745\pi$
$422$	0	0
$423$	3.10553e11	0.471633
$424$	0	0
$425$	6.81343e11	1.01301
$426$	0	0
$427$	2.72331e11i	0.396435i
$428$	0	0
$429$	2.35901e11i	0.336258i
$430$	0	0
$431$	7.99506e11	1.11603	0.558013	−	0.829832i	$-0.311564\pi$
$431$	7.99506e11	1.11603	0.558013	+	0.829832i	$0.311564\pi$
$432$	0	0
$433$	6.63025e11	0.906430	0.453215	−	0.891401i	$-0.350277\pi$
$433$	6.63025e11	0.906430	0.453215	+	0.891401i	$0.350277\pi$
$434$	0	0
$435$	− 2.92489e11i	− 0.391659i
$436$	0	0
$437$	1.19610e11i	0.156892i
$438$	0	0
$439$	−4.31042e10	−0.0553898	−0.0276949	−	0.999616i	$-0.508817\pi$
$439$	−4.31042e10	−0.0553898	−0.0276949	+	0.999616i	$0.508817\pi$
$440$	0	0
$441$	1.03021e11	0.129704
$442$	0	0
$443$	− 6.49651e11i	− 0.801426i	−0.916204	−	0.400713i	$-0.868762\pi$
$443$	− 6.49651e11i	− 0.801426i	0.916204	−	0.400713i	$-0.131238\pi$
$444$	0	0
$445$	− 1.83348e12i	− 2.21644i
$446$	0	0
$447$	−4.91039e11	−0.581744
$448$	0	0
$449$	2.28210e10	0.0264988	0.0132494	−	0.999912i	$-0.495782\pi$
$449$	2.28210e10	0.0264988	0.0132494	+	0.999912i	$0.495782\pi$
$450$	0	0
$451$	8.45279e9i	0.00962067i
$452$	0	0
$453$	− 4.79924e11i	− 0.535464i
$454$	0	0
$455$	−4.80253e11	−0.525313
$456$	0	0
$457$	−1.44749e12	−1.55237	−0.776183	−	0.630508i	$-0.782847\pi$
$457$	−1.44749e12	−1.55237	−0.776183	+	0.630508i	$0.782847\pi$
$458$	0	0
$459$	2.94770e11i	0.309975i
$460$	0	0
$461$	1.28905e11i	0.132927i	0.997789	+	0.0664637i	$0.0211717\pi$
$461$	1.28905e11i	0.132927i	−0.997789	+	0.0664637i	$0.978828\pi$
$462$	0	0
$463$	1.71037e12	1.72972	0.864860	−	0.502013i	$-0.167407\pi$
$463$	1.71037e12	1.72972	0.864860	+	0.502013i	$0.167407\pi$
$464$	0	0
$465$	4.95820e11	0.491797
$466$	0	0
$467$	1.91054e12i	1.85879i	0.369092	+	0.929393i	$0.379669\pi$
$467$	1.91054e12i	1.85879i	−0.369092	+	0.929393i	$0.620331\pi$
$468$	0	0
$469$	1.05947e12i	1.01114i
$470$	0	0
$471$	4.60695e11	0.431340
$472$	0	0
$473$	1.12884e11	0.103695
$474$	0	0
$475$	− 7.99502e11i	− 0.720606i
$476$	0	0
$477$	− 6.60924e11i	− 0.584546i
$478$	0	0
$479$	−1.92498e12	−1.67077	−0.835385	−	0.549665i	$-0.814756\pi$
$479$	−1.92498e12	−1.67077	−0.835385	+	0.549665i	$0.814756\pi$
$480$	0	0
$481$	−7.05433e10	−0.0600901
$482$	0	0
$483$	− 7.39083e10i	− 0.0617918i
$484$	0	0
$485$	1.11348e12i	0.913785i
$486$	0	0
$487$	−2.29897e12	−1.85205	−0.926026	−	0.377460i	$-0.876798\pi$
$487$	−2.29897e12	−1.85205	−0.926026	+	0.377460i	$0.876798\pi$
$488$	0	0
$489$	3.69560e11	0.292277
$490$	0	0
$491$	− 5.35110e11i	− 0.415505i	−0.978181	−	0.207753i	$-0.933385\pi$
$491$	− 5.35110e11i	− 0.415505i	0.978181	−	0.207753i	$-0.0666149\pi$
$492$	0	0
$493$	− 1.12289e12i	− 0.856101i
$494$	0	0
$495$	6.28495e11	0.470521
$496$	0	0
$497$	8.17317e11	0.600878
$498$	0	0
$499$	1.06412e12i	0.768312i	0.923268	+	0.384156i	$0.125508\pi$
$499$	1.06412e12i	0.768312i	−0.923268	+	0.384156i	$0.874492\pi$
$500$	0	0
$501$	9.88461e10i	0.0700954i
$502$	0	0
$503$	−2.01345e12	−1.40244	−0.701220	−	0.712945i	$-0.747361\pi$
$503$	−2.01345e12	−1.40244	−0.701220	+	0.712945i	$0.747361\pi$
$504$	0	0
$505$	9.36436e11	0.640718
$506$	0	0
$507$	6.20762e11i	0.417243i
$508$	0	0
$509$	1.34279e12i	0.886701i	0.896348	+	0.443350i	$0.146210\pi$
$509$	1.34279e12i	0.886701i	−0.896348	+	0.443350i	$0.853790\pi$
$510$	0	0
$511$	3.76600e11	0.244336
$512$	0	0
$513$	3.45889e11	0.220500
$514$	0	0
$515$	1.94954e12i	1.22124i
$516$	0	0
$517$	− 2.54203e12i	− 1.56485i
$518$	0	0
$519$	−1.81657e12	−1.09901
$520$	0	0
$521$	−1.74500e12	−1.03759	−0.518796	−	0.854898i	$-0.673619\pi$
$521$	−1.74500e12	−1.03759	−0.518796	+	0.854898i	$0.673619\pi$
$522$	0	0
$523$	− 1.62381e12i	− 0.949026i	−0.880249	−	0.474513i	$-0.842624\pi$
$523$	− 1.62381e12i	− 0.949026i	0.880249	−	0.474513i	$-0.157376\pi$
$524$	0	0
$525$	4.94020e11i	0.283810i
$526$	0	0
$527$	1.90349e12	1.07499
$528$	0	0
$529$	−1.76738e12	−0.981249
$530$	0	0
$531$	1.18270e11i	0.0645581i
$532$	0	0
$533$	− 8.53524e9i	− 0.00458082i
$534$	0	0
$535$	−4.47374e12	−2.36091
$536$	0	0
$537$	1.39676e12	0.724835
$538$	0	0
$539$	− 8.43280e11i	− 0.430351i
$540$	0	0
$541$	2.26145e12i	1.13501i	0.823370	+	0.567505i	$0.192091\pi$
$541$	2.26145e12i	1.13501i	−0.823370	+	0.567505i	$0.807909\pi$
$542$	0	0
$543$	2.35031e11	0.116018
$544$	0	0
$545$	−3.56082e12	−1.72888
$546$	0	0
$547$	− 2.08228e12i	− 0.994480i	−0.867613	−	0.497240i	$-0.834347\pi$
$547$	− 2.08228e12i	− 0.994480i	0.867613	−	0.497240i	$-0.165653\pi$
$548$	0	0
$549$	3.59870e11i	0.169071i
$550$	0	0
$551$	−1.31762e12	−0.608986
$552$	0	0
$553$	−2.11180e12	−0.960263
$554$	0	0
$555$	1.87944e11i	0.0840832i
$556$	0	0
$557$	1.69657e12i	0.746831i	0.927664	+	0.373415i	$0.121813\pi$
$557$	1.69657e12i	0.746831i	−0.927664	+	0.373415i	$0.878187\pi$
$558$	0	0
$559$	−1.13985e11	−0.0493736
$560$	0	0
$561$	2.41284e12	1.02848
$562$	0	0
$563$	− 1.48379e11i	− 0.0622423i	−0.999516	−	0.0311212i	$-0.990092\pi$
$563$	− 1.48379e11i	− 0.0622423i	0.999516	−	0.0311212i	$-0.00990778\pi$
$564$	0	0
$565$	− 3.13275e12i	− 1.29332i
$566$	0	0
$567$	−2.13728e11	−0.0868437
$568$	0	0
$569$	−9.31992e11	−0.372741	−0.186370	−	0.982480i	$-0.559672\pi$
$569$	−9.31992e11	−0.372741	−0.186370	+	0.982480i	$0.559672\pi$
$570$	0	0
$571$	4.97476e12i	1.95844i	0.202807	+	0.979219i	$0.434993\pi$
$571$	4.97476e12i	1.95844i	−0.202807	+	0.979219i	$0.565007\pi$
$572$	0	0
$573$	9.29388e10i	0.0360165i
$574$	0	0
$575$	−2.25748e11	−0.0861228
$576$	0	0
$577$	7.93519e11	0.298034	0.149017	−	0.988835i	$-0.452389\pi$
$577$	7.93519e11	0.298034	0.149017	+	0.988835i	$0.452389\pi$
$578$	0	0
$579$	2.59148e12i	0.958286i
$580$	0	0
$581$	5.29362e11i	0.192735i
$582$	0	0
$583$	−5.40999e12	−1.93949
$584$	0	0
$585$	−6.34626e11	−0.224035
$586$	0	0
$587$	4.52536e11i	0.157319i	0.996902	+	0.0786596i	$0.0250640\pi$
$587$	4.52536e11i	0.157319i	−0.996902	+	0.0786596i	$0.974936\pi$
$588$	0	0
$589$	− 2.23359e12i	− 0.764690i
$590$	0	0
$591$	−2.74232e11	−0.0924643
$592$	0	0
$593$	−2.09626e12	−0.696144	−0.348072	−	0.937468i	$-0.613164\pi$
$593$	−2.09626e12	−0.696144	−0.348072	+	0.937468i	$0.613164\pi$
$594$	0	0
$595$	4.91211e12i	1.60672i
$596$	0	0
$597$	5.90292e11i	0.190188i
$598$	0	0
$599$	−2.30531e12	−0.731659	−0.365829	−	0.930682i	$-0.619215\pi$
$599$	−2.30531e12	−0.731659	−0.365829	+	0.930682i	$0.619215\pi$
$600$	0	0
$601$	2.13269e12	0.666794	0.333397	−	0.942787i	$-0.391805\pi$
$601$	2.13269e12	0.666794	0.333397	+	0.942787i	$0.391805\pi$
$602$	0	0
$603$	1.40003e12i	0.431230i
$604$	0	0
$605$	− 9.38718e11i	− 0.284863i
$606$	0	0
$607$	−3.72283e12	−1.11308	−0.556538	−	0.830822i	$-0.687870\pi$
$607$	−3.72283e12	−1.11308	−0.556538	+	0.830822i	$0.687870\pi$
$608$	0	0
$609$	8.14168e11	0.239848
$610$	0	0
$611$	2.56682e12i	0.745093i
$612$	0	0
$613$	− 6.70140e12i	− 1.91687i	−0.285305	−	0.958437i	$-0.592095\pi$
$613$	− 6.70140e12i	− 1.91687i	0.285305	−	0.958437i	$-0.407905\pi$
$614$	0	0
$615$	−2.27398e10	−0.00640987
$616$	0	0
$617$	1.02429e12	0.284537	0.142268	−	0.989828i	$-0.454560\pi$
$617$	1.02429e12	0.284537	0.142268	+	0.989828i	$0.454560\pi$
$618$	0	0
$619$	3.15002e12i	0.862393i	0.902258	+	0.431196i	$0.141908\pi$
$619$	3.15002e12i	0.862393i	−0.902258	+	0.431196i	$0.858092\pi$
$620$	0	0
$621$	− 9.76655e10i	− 0.0263529i
$622$	0	0
$623$	5.10365e12	1.35733
$624$	0	0
$625$	−4.70495e12	−1.23338
$626$	0	0
$627$	− 2.83127e12i	− 0.731607i
$628$	0	0
$629$	7.21530e11i	0.183792i
$630$	0	0
$631$	1.99605e12	0.501233	0.250617	−	0.968086i	$-0.419367\pi$
$631$	1.99605e12	0.501233	0.250617	+	0.968086i	$0.419367\pi$
$632$	0	0
$633$	1.28331e11	0.0317697
$634$	0	0
$635$	8.32829e12i	2.03270i
$636$	0	0
$637$	8.51506e11i	0.204909i
$638$	0	0
$639$	1.08004e12	0.256262
$640$	0	0
$641$	7.10597e12	1.66250	0.831252	−	0.555896i	$-0.187625\pi$
$641$	7.10597e12	1.66250	0.831252	+	0.555896i	$0.187625\pi$
$642$	0	0
$643$	1.13065e12i	0.260843i	0.991459	+	0.130422i	$0.0416331\pi$
$643$	1.13065e12i	0.260843i	−0.991459	+	0.130422i	$0.958367\pi$
$644$	0	0
$645$	3.03683e11i	0.0690878i
$646$	0	0
$647$	−5.14088e12	−1.15337	−0.576685	−	0.816967i	$-0.695654\pi$
$647$	−5.14088e12	−1.15337	−0.576685	+	0.816967i	$0.695654\pi$
$648$	0	0
$649$	9.68102e11	0.214200
$650$	0	0
$651$	1.38016e12i	0.301172i
$652$	0	0
$653$	5.23628e11i	0.112697i	0.998411	+	0.0563486i	$0.0179458\pi$
$653$	5.23628e11i	0.112697i	−0.998411	+	0.0563486i	$0.982054\pi$
$654$	0	0
$655$	9.79136e12	2.07854
$656$	0	0
$657$	4.97655e11	0.104204
$658$	0	0
$659$	− 2.73828e12i	− 0.565579i	−0.959182	−	0.282789i	$-0.908740\pi$
$659$	− 2.73828e12i	− 0.565579i	0.959182	−	0.282789i	$-0.0912598\pi$
$660$	0	0
$661$	− 1.53582e12i	− 0.312920i	−0.987684	−	0.156460i	$-0.949992\pi$
$661$	− 1.53582e12i	− 0.312920i	0.987684	−	0.156460i	$-0.0500082\pi$
$662$	0	0
$663$	−2.43638e12	−0.489704
$664$	0	0
$665$	5.76396e12	1.14294
$666$	0	0
$667$	3.72043e11i	0.0727825i
$668$	0	0
$669$	2.28609e12i	0.441240i
$670$	0	0
$671$	2.94571e12	0.560969
$672$	0	0
$673$	−9.06140e12	−1.70266	−0.851329	−	0.524632i	$-0.824203\pi$
$673$	−9.06140e12	−1.70266	−0.851329	+	0.524632i	$0.824203\pi$
$674$	0	0
$675$	6.52818e11i	0.121039i
$676$	0	0
$677$	8.23814e12i	1.50723i	0.657315	+	0.753616i	$0.271692\pi$
$677$	8.23814e12i	1.50723i	−0.657315	+	0.753616i	$0.728308\pi$
$678$	0	0
$679$	−3.09947e12	−0.559594
$680$	0	0
$681$	−8.17061e11	−0.145577
$682$	0	0
$683$	1.99109e12i	0.350104i	0.984559	+	0.175052i	$0.0560094\pi$
$683$	1.99109e12i	0.350104i	−0.984559	+	0.175052i	$0.943991\pi$
$684$	0	0
$685$	− 5.45767e12i	− 0.947109i
$686$	0	0
$687$	−2.48115e11	−0.0424959
$688$	0	0
$689$	5.46276e12	0.923476
$690$	0	0
$691$	6.48604e11i	0.108225i	0.998535	+	0.0541126i	$0.0172330\pi$
$691$	6.48604e11i	0.108225i	−0.998535	+	0.0541126i	$0.982767\pi$
$692$	0	0
$693$	1.74947e12i	0.288143i
$694$	0	0
$695$	−1.43891e13	−2.33939
$696$	0	0
$697$	−8.72999e10	−0.0140109
$698$	0	0
$699$	− 3.41412e12i	− 0.540917i
$700$	0	0
$701$	9.56476e12i	1.49604i	0.663677	+	0.748019i	$0.268995\pi$
$701$	9.56476e12i	1.49604i	−0.663677	+	0.748019i	$0.731005\pi$
$702$	0	0
$703$	8.46657e11	0.130740
$704$	0	0
$705$	6.83861e12	1.04260
$706$	0	0
$707$	2.60665e12i	0.392370i
$708$	0	0
$709$	5.02818e12i	0.747314i	0.927567	+	0.373657i	$0.121896\pi$
$709$	5.02818e12i	0.747314i	−0.927567	+	0.373657i	$0.878104\pi$
$710$	0	0
$711$	−2.79062e12	−0.409532
$712$	0	0
$713$	−6.30679e11	−0.0913914
$714$	0	0
$715$	5.19473e12i	0.743336i
$716$	0	0
$717$	6.74736e12i	0.953449i
$718$	0	0
$719$	−3.87923e11	−0.0541334	−0.0270667	−	0.999634i	$-0.508617\pi$
$719$	−3.87923e11	−0.0541334	−0.0270667	+	0.999634i	$0.508617\pi$
$720$	0	0
$721$	−5.42672e12	−0.747874
$722$	0	0
$723$	1.10024e12i	0.149749i
$724$	0	0
$725$	− 2.48682e12i	− 0.334290i
$726$	0	0
$727$	1.21943e12	0.161902	0.0809510	−	0.996718i	$-0.474204\pi$
$727$	1.21943e12	0.161902	0.0809510	+	0.996718i	$0.474204\pi$
$728$	0	0
$729$	−2.82430e11	−0.0370370
$730$	0	0
$731$	1.16586e12i	0.151014i
$732$	0	0
$733$	− 1.25774e12i	− 0.160925i	−0.996758	−	0.0804626i	$-0.974360\pi$
$733$	− 1.25774e12i	− 0.160925i	0.996758	−	0.0804626i	$-0.0256397\pi$
$734$	0	0
$735$	2.26861e12	0.286725
$736$	0	0
$737$	1.14599e13	1.43080
$738$	0	0
$739$	− 1.08922e13i	− 1.34343i	−0.740812	−	0.671713i	$-0.765559\pi$
$739$	− 1.08922e13i	− 1.34343i	0.740812	−	0.671713i	$-0.234441\pi$
$740$	0	0
$741$	2.85889e12i	0.348350i
$742$	0	0
$743$	−7.60817e12	−0.915863	−0.457932	−	0.888987i	$-0.651410\pi$
$743$	−7.60817e12	−0.915863	−0.457932	+	0.888987i	$0.651410\pi$
$744$	0	0
$745$	−1.08130e13	−1.28601
$746$	0	0
$747$	6.99521e11i	0.0821974i
$748$	0	0
$749$	− 1.24530e13i	− 1.44580i
$750$	0	0
$751$	−1.34422e13	−1.54202	−0.771010	−	0.636823i	$-0.780248\pi$
$751$	−1.34422e13	−1.54202	−0.771010	+	0.636823i	$0.780248\pi$
$752$	0	0
$753$	−7.61672e11	−0.0863358
$754$	0	0
$755$	− 1.05683e13i	− 1.18370i
$756$	0	0
$757$	1.22512e12i	0.135597i	0.997699	+	0.0677983i	$0.0215974\pi$
$757$	1.22512e12i	0.135597i	−0.997699	+	0.0677983i	$0.978403\pi$
$758$	0	0
$759$	−7.99440e11	−0.0874375
$760$	0	0
$761$	1.86163e12	0.201216	0.100608	−	0.994926i	$-0.467921\pi$
$761$	1.86163e12	0.201216	0.100608	+	0.994926i	$0.467921\pi$
$762$	0	0
$763$	− 9.91186e12i	− 1.05875i
$764$	0	0
$765$	6.49106e12i	0.685235i
$766$	0	0
$767$	−9.77545e11	−0.101990
$768$	0	0
$769$	−1.61673e13	−1.66713	−0.833564	−	0.552422i	$-0.813704\pi$
$769$	−1.61673e13	−1.66713	−0.833564	+	0.552422i	$0.813704\pi$
$770$	0	0
$771$	− 2.59555e12i	− 0.264536i
$772$	0	0
$773$	− 1.51715e13i	− 1.52834i	−0.645012	−	0.764172i	$-0.723148\pi$
$773$	− 1.51715e13i	− 1.52834i	0.645012	−	0.764172i	$-0.276852\pi$
$774$	0	0
$775$	4.21560e12	0.419761
$776$	0	0
$777$	−5.23158e11	−0.0514918
$778$	0	0
$779$	1.02439e11i	0.00996664i
$780$	0	0
$781$	− 8.84063e12i	− 0.850263i
$782$	0	0
$783$	1.07588e12	0.102290
$784$	0	0
$785$	1.01449e13	0.953526
$786$	0	0
$787$	1.28839e13i	1.19718i	0.801055	+	0.598590i	$0.204272\pi$
$787$	1.28839e13i	1.19718i	−0.801055	+	0.598590i	$0.795728\pi$
$788$	0	0
$789$	7.23506e12i	0.664654i
$790$	0	0
$791$	8.72027e12	0.792019
$792$	0	0
$793$	−2.97444e12	−0.267101
$794$	0	0
$795$	− 1.45540e13i	− 1.29221i
$796$	0	0
$797$	− 4.41358e12i	− 0.387462i	−0.981055	−	0.193731i	$-0.937941\pi$
$797$	− 4.41358e12i	− 0.387462i	0.981055	−	0.193731i	$-0.0620589\pi$
$798$	0	0
$799$	2.62539e13	2.27894
$800$	0	0
$801$	6.74418e12	0.578872
$802$	0	0
$803$	− 4.07355e12i	− 0.345743i
$804$	0	0
$805$	− 1.62752e12i	− 0.136598i
$806$	0	0
$807$	−3.00089e12	−0.249069
$808$	0	0
$809$	−1.24867e13	−1.02490	−0.512449	−	0.858718i	$-0.671262\pi$
$809$	−1.24867e13	−1.02490	−0.512449	+	0.858718i	$0.671262\pi$
$810$	0	0
$811$	− 5.75694e12i	− 0.467302i	−0.972321	−	0.233651i	$-0.924933\pi$
$811$	− 5.75694e12i	− 0.467302i	0.972321	−	0.233651i	$-0.0750674\pi$
$812$	0	0
$813$	8.91150e12i	0.715391i
$814$	0	0
$815$	8.13799e12	0.646112
$816$	0	0
$817$	1.36804e12	0.107424
$818$	0	0
$819$	− 1.76654e12i	− 0.137197i
$820$	0	0
$821$	1.15601e13i	0.888009i	0.896025	+	0.444004i	$0.146443\pi$
$821$	1.15601e13i	0.888009i	−0.896025	+	0.444004i	$0.853557\pi$
$822$	0	0
$823$	−1.28159e13	−0.973758	−0.486879	−	0.873469i	$-0.661865\pi$
$823$	−1.28159e13	−0.973758	−0.486879	+	0.873469i	$0.661865\pi$
$824$	0	0
$825$	5.34364e12	0.401601
$826$	0	0
$827$	− 1.88099e13i	− 1.39834i	−0.714957	−	0.699169i	$-0.753554\pi$
$827$	− 1.88099e13i	− 1.39834i	0.714957	−	0.699169i	$-0.246446\pi$
$828$	0	0
$829$	− 1.05174e13i	− 0.773413i	−0.922203	−	0.386707i	$-0.873613\pi$
$829$	− 1.05174e13i	− 0.773413i	0.922203	−	0.386707i	$-0.126387\pi$
$830$	0	0
$831$	6.96704e12	0.506808
$832$	0	0
$833$	8.70935e12	0.626734
$834$	0	0
$835$	2.17667e12i	0.154954i
$836$	0	0
$837$	1.82380e12i	0.128444i
$838$	0	0
$839$	2.13441e13	1.48713	0.743564	−	0.668665i	$-0.233134\pi$
$839$	2.13441e13	1.48713	0.743564	+	0.668665i	$0.233134\pi$
$840$	0	0
$841$	1.04087e13	0.717491
$842$	0	0
$843$	1.13526e13i	0.774230i
$844$	0	0
$845$	1.36696e13i	0.922364i
$846$	0	0
$847$	2.61300e12	0.174447
$848$	0	0
$849$	5.56253e12	0.367441
$850$	0	0
$851$	− 2.39063e11i	− 0.0156253i
$852$	0	0
$853$	− 1.81634e13i	− 1.17470i	−0.809334	−	0.587348i	$-0.800172\pi$
$853$	− 1.81634e13i	− 1.17470i	0.809334	−	0.587348i	$-0.199828\pi$
$854$	0	0
$855$	7.61674e12	0.487441
$856$	0	0
$857$	2.86744e13	1.81585	0.907927	−	0.419128i	$-0.137664\pi$
$857$	2.86744e13	1.81585	0.907927	+	0.419128i	$0.137664\pi$
$858$	0	0
$859$	− 8.97255e12i	− 0.562273i	−0.959668	−	0.281136i	$-0.909289\pi$
$859$	− 8.97255e12i	− 0.562273i	0.959668	−	0.281136i	$-0.0907113\pi$
$860$	0	0
$861$	− 6.32983e10i	− 0.00392535i
$862$	0	0
$863$	1.96293e13	1.20463	0.602317	−	0.798257i	$-0.294244\pi$
$863$	1.96293e13	1.20463	0.602317	+	0.798257i	$0.294244\pi$
$864$	0	0
$865$	−4.00023e13	−2.42948
$866$	0	0
$867$	1.53141e13i	0.920459i
$868$	0	0
$869$	2.28426e13i	1.35880i
$870$	0	0
$871$	−1.15717e13	−0.681265
$872$	0	0
$873$	−4.09576e12	−0.238655
$874$	0	0
$875$	− 6.41825e12i	− 0.370152i
$876$	0	0
$877$	− 1.82741e13i	− 1.04313i	−0.853211	−	0.521565i	$-0.825348\pi$
$877$	− 1.82741e13i	− 1.04313i	0.853211	−	0.521565i	$-0.174652\pi$
$878$	0	0
$879$	−3.15991e12	−0.178535
$880$	0	0
$881$	1.46604e12	0.0819888	0.0409944	−	0.999159i	$-0.486947\pi$
$881$	1.46604e12	0.0819888	0.0409944	+	0.999159i	$0.486947\pi$
$882$	0	0
$883$	9.47248e11i	0.0524373i	0.999656	+	0.0262187i	$0.00834661\pi$
$883$	9.47248e11i	0.0524373i	−0.999656	+	0.0262187i	$0.991653\pi$
$884$	0	0
$885$	2.60440e12i	0.142713i
$886$	0	0
$887$	5.51475e12	0.299136	0.149568	−	0.988751i	$-0.452212\pi$
$887$	5.51475e12	0.299136	0.149568	+	0.988751i	$0.452212\pi$
$888$	0	0
$889$	−2.31825e13	−1.24481
$890$	0	0
$891$	2.31182e12i	0.122887i
$892$	0	0
$893$	− 3.08069e13i	− 1.62112i
$894$	0	0
$895$	3.07578e13	1.60233
$896$	0	0
$897$	8.07238e11	0.0416328
$898$	0	0
$899$	− 6.94751e12i	− 0.354741i
$900$	0	0
$901$	− 5.58741e13i	− 2.82455i
$902$	0	0
$903$	−8.45327e11	−0.0423087
$904$	0	0
$905$	5.17555e12	0.256471
$906$	0	0
$907$	− 3.24027e13i	− 1.58982i	−0.606725	−	0.794912i	$-0.707517\pi$
$907$	− 3.24027e13i	− 1.58982i	0.606725	−	0.794912i	$-0.292483\pi$
$908$	0	0
$909$	3.44454e12i	0.167338i
$910$	0	0
$911$	2.85811e12	0.137482	0.0687410	−	0.997635i	$-0.478102\pi$
$911$	2.85811e12	0.137482	0.0687410	+	0.997635i	$0.478102\pi$
$912$	0	0
$913$	5.72592e12	0.272726
$914$	0	0
$915$	7.92460e12i	0.373751i
$916$	0	0
$917$	2.72551e13i	1.27288i
$918$	0	0
$919$	1.84172e13	0.851734	0.425867	−	0.904786i	$-0.359969\pi$
$919$	1.84172e13	0.851734	0.425867	+	0.904786i	$0.359969\pi$
$920$	0	0
$921$	1.57226e13	0.720040
$922$	0	0
$923$	8.92686e12i	0.404847i
$924$	0	0
$925$	1.59795e12i	0.0717670i
$926$	0	0
$927$	−7.17109e12	−0.318953
$928$	0	0
$929$	−3.37811e13	−1.48800	−0.744000	−	0.668179i	$-0.767074\pi$
$929$	−3.37811e13	−1.48800	−0.744000	+	0.668179i	$0.767074\pi$
$930$	0	0
$931$	− 1.02197e13i	− 0.445826i
$932$	0	0
$933$	1.15020e13i	0.496940i
$934$	0	0
$935$	5.31326e13	2.27357
$936$	0	0
$937$	−4.50959e13	−1.91121	−0.955606	−	0.294649i	$-0.904797\pi$
$937$	−4.50959e13	−1.91121	−0.955606	+	0.294649i	$0.904797\pi$
$938$	0	0
$939$	− 1.58931e13i	− 0.667136i
$940$	0	0
$941$	2.11150e13i	0.877887i	0.898515	+	0.438943i	$0.144647\pi$
$941$	2.11150e13i	0.877887i	−0.898515	+	0.438943i	$0.855353\pi$
$942$	0	0
$943$	2.89249e10	0.00119116
$944$	0	0
$945$	−4.70646e12	−0.191978
$946$	0	0
$947$	− 1.89150e12i	− 0.0764244i	−0.999270	−	0.0382122i	$-0.987834\pi$
$947$	− 1.89150e12i	− 0.0764244i	0.999270	−	0.0382122i	$-0.0121663\pi$
$948$	0	0
$949$	4.11329e12i	0.164623i
$950$	0	0
$951$	2.69510e13	1.06847
$952$	0	0
$953$	9.36004e12	0.367587	0.183793	−	0.982965i	$-0.441162\pi$
$953$	9.36004e12	0.367587	0.183793	+	0.982965i	$0.441162\pi$
$954$	0	0
$955$	2.04658e12i	0.0796186i
$956$	0	0
$957$	− 8.80658e12i	− 0.339393i
$958$	0	0
$959$	1.51919e13	0.580001
$960$	0	0
$961$	−1.46624e13	−0.554560
$962$	0	0
$963$	− 1.64560e13i	− 0.616602i
$964$	0	0
$965$	5.70664e13i	2.11840i
$966$	0	0
$967$	2.61523e13	0.961814	0.480907	−	0.876772i	$-0.340308\pi$
$967$	2.61523e13	0.961814	0.480907	+	0.876772i	$0.340308\pi$
$968$	0	0
$969$	2.92412e13	1.06546
$970$	0	0
$971$	− 1.18913e13i	− 0.429281i	−0.976693	−	0.214640i	$-0.931142\pi$
$971$	− 1.18913e13i	− 0.429281i	0.976693	−	0.214640i	$-0.0688579\pi$
$972$	0	0
$973$	− 4.00534e13i	− 1.43262i
$974$	0	0
$975$	−5.39576e12	−0.191219
$976$	0	0
$977$	3.37020e12	0.118340	0.0591699	−	0.998248i	$-0.481155\pi$
$977$	3.37020e12	0.118340	0.0591699	+	0.998248i	$0.481155\pi$
$978$	0	0
$979$	− 5.52044e13i	− 1.92067i
$980$	0	0
$981$	− 1.30979e13i	− 0.451536i
$982$	0	0
$983$	−4.94392e13	−1.68881	−0.844404	−	0.535706i	$-0.820045\pi$
$983$	−4.94392e13	−1.68881	−0.844404	+	0.535706i	$0.820045\pi$
$984$	0	0
$985$	−6.03879e12	−0.204403
$986$	0	0
$987$	1.90359e13i	0.638477i
$988$	0	0
$989$	− 3.86281e11i	− 0.0128387i
$990$	0	0
$991$	−3.33144e13	−1.09724	−0.548619	−	0.836073i	$-0.684846\pi$
$991$	−3.33144e13	−1.09724	−0.548619	+	0.836073i	$0.684846\pi$
$992$	0	0
$993$	2.28546e13	0.745936
$994$	0	0
$995$	1.29987e13i	0.420431i
$996$	0	0
$997$	1.18744e12i	0.0380612i	0.999819	+	0.0190306i	$0.00605799\pi$
$997$	1.18744e12i	0.0380612i	−0.999819	+	0.0190306i	$0.993942\pi$
$998$	0	0
$999$	−6.91322e11	−0.0219602

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.10.d.a.193.8	yes	8
4.3	odd	2		384.10.d.b.193.4	yes	8
8.3	odd	2		384.10.d.b.193.5	yes	8
8.5	even	2	inner	384.10.d.a.193.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.10.d.a.193.1	✓	8	8.5	even	2	inner
384.10.d.a.193.8	yes	8	1.1	even	1	trivial
384.10.d.b.193.4	yes	8	4.3	odd	2
384.10.d.b.193.5	yes	8	8.3	odd	2

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:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	384.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+81.0000i q^{3} +1783.68i q^{5} -4965.03 q^{7} -6561.00 q^{9} +O(q^{10})\)	\(q+81.0000i q^{3} +1783.68i q^{5} -4965.03 q^{7} -6561.00 q^{9} +53705.0i q^{11} -54228.9i q^{13} -144478. q^{15} -554662. q^{17} +650852. i q^{19} -402168. i q^{21} +183775. q^{23} -1.22839e6 q^{25} -531441. i q^{27} +2.02445e6i q^{29} -3.43180e6 q^{31} -4.35011e6 q^{33} -8.85603e6i q^{35} -1.30084e6i q^{37} +4.39254e6 q^{39} +157393. q^{41} -2.10193e6i q^{43} -1.17027e7i q^{45} -4.73332e7 q^{47} -1.57021e7 q^{49} -4.49277e7i q^{51} +1.00735e8i q^{53} -9.57926e7 q^{55} -5.27190e7 q^{57} -1.80263e7i q^{59} -5.48498e7i q^{61} +3.25756e7 q^{63} +9.67270e7 q^{65} -2.13387e8i q^{67} +1.48858e7i q^{69} -1.64615e8 q^{71} -7.58505e7 q^{73} -9.94998e7i q^{75} -2.66647e8i q^{77} +4.25335e8 q^{79} +4.30467e7 q^{81} -1.06618e8i q^{83} -9.89341e8i q^{85} -1.63981e8 q^{87} -1.02792e9 q^{89} +2.69248e8i q^{91} -2.77976e8i q^{93} -1.16091e9 q^{95} +6.24259e8 q^{97} -3.52359e8i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 13632 q^{7} - 52488 q^{9}+O(q^{10}) \) 8 * q - 13632 * q^7 - 52488 * q^9	\( 8 q - 13632 q^{7} - 52488 q^{9} - 90720 q^{15} + 8304 q^{17} - 4612608 q^{23} - 4754904 q^{25} - 7499328 q^{31} - 6213024 q^{33} - 23211360 q^{39} - 43518896 q^{41} + 49382016 q^{47} - 74106808 q^{49} + 19030656 q^{55} + 38141280 q^{57} + 89439552 q^{63} - 110270336 q^{65} - 741751296 q^{71} - 1507903440 q^{73} + 1008373440 q^{79} + 344373768 q^{81} + 423468000 q^{87} - 1337034448 q^{89} - 543950208 q^{95} - 904817936 q^{97}+O(q^{100}) \) 8 * q - 13632 * q^7 - 52488 * q^9 - 90720 * q^15 + 8304 * q^17 - 4612608 * q^23 - 4754904 * q^25 - 7499328 * q^31 - 6213024 * q^33 - 23211360 * q^39 - 43518896 * q^41 + 49382016 * q^47 - 74106808 * q^49 + 19030656 * q^55 + 38141280 * q^57 + 89439552 * q^63 - 110270336 * q^65 - 741751296 * q^71 - 1507903440 * q^73 + 1008373440 * q^79 + 344373768 * q^81 + 423468000 * q^87 - 1337034448 * q^89 - 543950208 * q^95 - 904817936 * q^97

Introduction

L-functions

Modular forms

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Database

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