LMFDB - Embedded newform 384.6.d.g.193.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,6,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, 6, names="a")

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

chi := DirichletCharacter("384.193");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 6 $
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:	$61.5873868082$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{61})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 31x^{2} + 225 $ x^4 + 31*x^2 + 225
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			193.2
Root			$4.40512i$ of defining polynomial
Character	$\chi$	$=$	384.193
Dual form			384.6.d.g.193.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-9.00000i q^{3} +51.2410i q^{5} -164.205 q^{7} -81.0000 q^{9} +O(q^{10})$	$q-9.00000i q^{3} +51.2410i q^{5} -164.205 q^{7} -81.0000 q^{9} +296.964i q^{11} +329.374i q^{13} +461.169 q^{15} +2086.53 q^{17} +1142.75i q^{19} +1477.84i q^{21} -2479.30 q^{23} +499.360 q^{25} +729.000i q^{27} -6292.18i q^{29} -6485.39 q^{31} +2672.68 q^{33} -8414.03i q^{35} -6679.52i q^{37} +2964.37 q^{39} -11613.3 q^{41} +14026.4i q^{43} -4150.52i q^{45} +7021.82 q^{47} +10156.3 q^{49} -18778.8i q^{51} -546.953i q^{53} -15216.7 q^{55} +10284.7 q^{57} -37901.1i q^{59} +6773.95i q^{61} +13300.6 q^{63} -16877.5 q^{65} -49749.0i q^{67} +22313.7i q^{69} -22077.0 q^{71} +61034.9 q^{73} -4494.24i q^{75} -48763.0i q^{77} -73710.0 q^{79} +6561.00 q^{81} +34602.9i q^{83} +106916. i q^{85} -56629.6 q^{87} -42428.8 q^{89} -54084.9i q^{91} +58368.5i q^{93} -58555.5 q^{95} -138179. q^{97} -24054.1i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 32 q^{7} - 324 q^{9}+O(q^{10}) $ 4 * q - 32 * q^7 - 324 * q^9	$ 4 q - 32 q^{7} - 324 q^{9} + 720 q^{15} + 1848 q^{17} - 7168 q^{23} + 6996 q^{25} - 7072 q^{31} + 6192 q^{33} - 3888 q^{39} + 5032 q^{41} - 9152 q^{47} + 30628 q^{49} - 29376 q^{55} + 9648 q^{57} + 2592 q^{63} - 46016 q^{65} - 151040 q^{71} + 34200 q^{73} - 200992 q^{79} + 26244 q^{81} - 155664 q^{87} - 180712 q^{89} - 130752 q^{95} - 199816 q^{97}+O(q^{100}) $ 4 * q - 32 * q^7 - 324 * q^9 + 720 * q^15 + 1848 * q^17 - 7168 * q^23 + 6996 * q^25 - 7072 * q^31 + 6192 * q^33 - 3888 * q^39 + 5032 * q^41 - 9152 * q^47 + 30628 * q^49 - 29376 * q^55 + 9648 * q^57 + 2592 * q^63 - 46016 * q^65 - 151040 * q^71 + 34200 * q^73 - 200992 * q^79 + 26244 * q^81 - 155664 * q^87 - 180712 * q^89 - 130752 * q^95 - 199816 * 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
$3$	− 9.00000i	− 0.577350i
$3$	− 9.00000i	− 0.577350i
$4$	0	0
$5$	51.2410i	0.916627i	0.888791	+	0.458313i	$0.151546\pi$
$5$	51.2410i	0.916627i	−0.888791	+	0.458313i	$0.848454\pi$
$6$	0	0
$7$	−164.205	−1.26661	−0.633303	−	0.773904i	$-0.718301\pi$
$7$	−164.205	−1.26661	−0.633303	+	0.773904i	$0.718301\pi$
$8$	0	0
$9$	−81.0000	−0.333333
$10$	0	0
$11$	296.964i	0.739984i	0.929035	+	0.369992i	$0.120640\pi$
$11$	296.964i	0.739984i	−0.929035	+	0.369992i	$0.879360\pi$
$12$	0	0
$13$	329.374i	0.540544i	0.962784	+	0.270272i	$0.0871136\pi$
$13$	329.374i	0.540544i	−0.962784	+	0.270272i	$0.912886\pi$
$14$	0	0
$15$	461.169	0.529215
$16$	0	0
$17$	2086.53	1.75107	0.875533	−	0.483158i	$-0.160510\pi$
$17$	2086.53	1.75107	0.875533	+	0.483158i	$0.160510\pi$
$18$	0	0
$19$	1142.75i	0.726217i	0.931747	+	0.363109i	$0.118285\pi$
$19$	1142.75i	0.726217i	−0.931747	+	0.363109i	$0.881715\pi$
$20$	0	0
$21$	1477.84i	0.731275i
$22$	0	0
$23$	−2479.30	−0.977259	−0.488630	−	0.872491i	$-0.662503\pi$
$23$	−2479.30	−0.977259	−0.488630	+	0.872491i	$0.662503\pi$
$24$	0	0
$25$	499.360	0.159795
$26$	0	0
$27$	729.000i	0.192450i
$28$	0	0
$29$	− 6292.18i	− 1.38933i	−0.719332	−	0.694666i	$-0.755552\pi$
$29$	− 6292.18i	− 1.38933i	0.719332	−	0.694666i	$-0.244448\pi$
$30$	0	0
$31$	−6485.39	−1.21208	−0.606041	−	0.795433i	$-0.707243\pi$
$31$	−6485.39	−1.21208	−0.606041	+	0.795433i	$0.707243\pi$
$32$	0	0
$33$	2672.68	0.427230
$34$	0	0
$35$	− 8414.03i	− 1.16100i
$36$	0	0
$37$	− 6679.52i	− 0.802123i	−0.916051	−	0.401062i	$-0.868641\pi$
$37$	− 6679.52i	− 0.802123i	0.916051	−	0.401062i	$-0.131359\pi$
$38$	0	0
$39$	2964.37	0.312083
$40$	0	0
$41$	−11613.3	−1.07894	−0.539468	−	0.842006i	$-0.681375\pi$
$41$	−11613.3	−1.07894	−0.539468	+	0.842006i	$0.681375\pi$
$42$	0	0
$43$	14026.4i	1.15685i	0.815737	+	0.578424i	$0.196332\pi$
$43$	14026.4i	1.15685i	−0.815737	+	0.578424i	$0.803668\pi$
$44$	0	0
$45$	− 4150.52i	− 0.305542i
$46$	0	0
$47$	7021.82	0.463666	0.231833	−	0.972756i	$-0.425528\pi$
$47$	7021.82	0.463666	0.231833	+	0.972756i	$0.425528\pi$
$48$	0	0
$49$	10156.3	0.604289
$50$	0	0
$51$	− 18778.8i	− 1.01098i
$52$	0	0
$53$	− 546.953i	− 0.0267461i	−0.999911	−	0.0133730i	$-0.995743\pi$
$53$	− 546.953i	− 0.0267461i	0.999911	−	0.0133730i	$-0.00425690\pi$
$54$	0	0
$55$	−15216.7	−0.678289
$56$	0	0
$57$	10284.7	0.419282
$58$	0	0
$59$	− 37901.1i	− 1.41750i	−0.705462	−	0.708748i	$-0.749260\pi$
$59$	− 37901.1i	− 1.41750i	0.705462	−	0.708748i	$-0.250740\pi$
$60$	0	0
$61$	6773.95i	0.233087i	0.993186	+	0.116543i	$0.0371814\pi$
$61$	6773.95i	0.233087i	−0.993186	+	0.116543i	$0.962819\pi$
$62$	0	0
$63$	13300.6	0.422202
$64$	0	0
$65$	−16877.5	−0.495477
$66$	0	0
$67$	− 49749.0i	− 1.35393i	−0.736014	−	0.676966i	$-0.763294\pi$
$67$	− 49749.0i	− 1.35393i	0.736014	−	0.676966i	$-0.236706\pi$
$68$	0	0
$69$	22313.7i	0.564221i
$70$	0	0
$71$	−22077.0	−0.519750	−0.259875	−	0.965642i	$-0.583681\pi$
$71$	−22077.0	−0.519750	−0.259875	+	0.965642i	$0.583681\pi$
$72$	0	0
$73$	61034.9	1.34051	0.670256	−	0.742130i	$-0.266184\pi$
$73$	61034.9	1.34051	0.670256	+	0.742130i	$0.266184\pi$
$74$	0	0
$75$	− 4494.24i	− 0.0922578i
$76$	0	0
$77$	− 48763.0i	− 0.937267i
$78$	0	0
$79$	−73710.0	−1.32880	−0.664398	−	0.747379i	$-0.731312\pi$
$79$	−73710.0	−1.32880	−0.664398	+	0.747379i	$0.731312\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	34602.9i	0.551337i	0.961253	+	0.275668i	$0.0888992\pi$
$83$	34602.9i	0.551337i	−0.961253	+	0.275668i	$0.911101\pi$
$84$	0	0
$85$	106916.i	1.60507i
$86$	0	0
$87$	−56629.6	−0.802132
$88$	0	0
$89$	−42428.8	−0.567787	−0.283894	−	0.958856i	$-0.591626\pi$
$89$	−42428.8	−0.567787	−0.283894	+	0.958856i	$0.591626\pi$
$90$	0	0
$91$	− 54084.9i	− 0.684656i
$92$	0	0
$93$	58368.5i	0.699796i
$94$	0	0
$95$	−58555.5	−0.665670
$96$	0	0
$97$	−138179.	−1.49112	−0.745558	−	0.666441i	$-0.767817\pi$
$97$	−138179.	−1.49112	−0.745558	+	0.666441i	$0.767817\pi$
$98$	0	0
$99$	− 24054.1i	− 0.246661i
$100$	0	0
$101$	− 13797.7i	− 0.134587i	−0.997733	−	0.0672934i	$-0.978564\pi$
$101$	− 13797.7i	− 0.134587i	0.997733	−	0.0672934i	$-0.0214364\pi$
$102$	0	0
$103$	−137419.	−1.27630	−0.638152	−	0.769911i	$-0.720301\pi$
$103$	−137419.	−1.27630	−0.638152	+	0.769911i	$0.720301\pi$
$104$	0	0
$105$	−75726.3	−0.670306
$106$	0	0
$107$	− 195369.i	− 1.64967i	−0.565375	−	0.824834i	$-0.691268\pi$
$107$	− 195369.i	− 1.64967i	0.565375	−	0.824834i	$-0.308732\pi$
$108$	0	0
$109$	− 171514.i	− 1.38272i	−0.722511	−	0.691360i	$-0.757012\pi$
$109$	− 171514.i	− 1.38272i	0.722511	−	0.691360i	$-0.242988\pi$
$110$	0	0
$111$	−60115.7	−0.463106
$112$	0	0
$113$	−35205.4	−0.259366	−0.129683	−	0.991555i	$-0.541396\pi$
$113$	−35205.4	−0.259366	−0.129683	+	0.991555i	$0.541396\pi$
$114$	0	0
$115$	− 127042.i	− 0.895782i
$116$	0	0
$117$	− 26679.3i	− 0.180181i
$118$	0	0
$119$	−342619.	−2.21791
$120$	0	0
$121$	72863.4	0.452424
$122$	0	0
$123$	104520.i	0.622924i
$124$	0	0
$125$	185716.i	1.06310i
$126$	0	0
$127$	272003.	1.49645	0.748227	−	0.663442i	$-0.230905\pi$
$127$	272003.	1.49645	0.748227	+	0.663442i	$0.230905\pi$
$128$	0	0
$129$	126238.	0.667906
$130$	0	0
$131$	28604.6i	0.145632i	0.997345	+	0.0728161i	$0.0231986\pi$
$131$	28604.6i	0.145632i	−0.997345	+	0.0728161i	$0.976801\pi$
$132$	0	0
$133$	− 187645.i	− 0.919830i
$134$	0	0
$135$	−37354.7	−0.176405
$136$	0	0
$137$	−120391.	−0.548016	−0.274008	−	0.961727i	$-0.588350\pi$
$137$	−120391.	−0.548016	−0.274008	+	0.961727i	$0.588350\pi$
$138$	0	0
$139$	− 374434.i	− 1.64376i	−0.569662	−	0.821879i	$-0.692926\pi$
$139$	− 374434.i	− 1.64376i	0.569662	−	0.821879i	$-0.307074\pi$
$140$	0	0
$141$	− 63196.4i	− 0.267697i
$142$	0	0
$143$	−97812.2	−0.399994
$144$	0	0
$145$	322418.	1.27350
$146$	0	0
$147$	− 91406.5i	− 0.348886i
$148$	0	0
$149$	182441.i	0.673221i	0.941644	+	0.336610i	$0.109281\pi$
$149$	182441.i	0.673221i	−0.941644	+	0.336610i	$0.890719\pi$
$150$	0	0
$151$	−258575.	−0.922876	−0.461438	−	0.887172i	$-0.652666\pi$
$151$	−258575.	−0.922876	−0.461438	+	0.887172i	$0.652666\pi$
$152$	0	0
$153$	−169009.	−0.583689
$154$	0	0
$155$	− 332318.i	− 1.11103i
$156$	0	0
$157$	172230.i	0.557646i	0.960343	+	0.278823i	$0.0899443\pi$
$157$	172230.i	0.557646i	−0.960343	+	0.278823i	$0.910056\pi$
$158$	0	0
$159$	−4922.58	−0.0154419
$160$	0	0
$161$	407114.	1.23780
$162$	0	0
$163$	− 322054.i	− 0.949422i	−0.880142	−	0.474711i	$-0.842553\pi$
$163$	− 322054.i	− 0.949422i	0.880142	−	0.474711i	$-0.157447\pi$
$164$	0	0
$165$	136951.i	0.391610i
$166$	0	0
$167$	−143101.	−0.397055	−0.198527	−	0.980095i	$-0.563616\pi$
$167$	−143101.	−0.397055	−0.198527	+	0.980095i	$0.563616\pi$
$168$	0	0
$169$	262806.	0.707812
$170$	0	0
$171$	− 92562.6i	− 0.242072i
$172$	0	0
$173$	78952.5i	0.200563i	0.994959	+	0.100281i	$0.0319743\pi$
$173$	78952.5i	0.200563i	−0.994959	+	0.100281i	$0.968026\pi$
$174$	0	0
$175$	−81997.4	−0.202397
$176$	0	0
$177$	−341110.	−0.818391
$178$	0	0
$179$	− 401034.i	− 0.935509i	−0.883858	−	0.467755i	$-0.845063\pi$
$179$	− 401034.i	− 0.935509i	0.883858	−	0.467755i	$-0.154937\pi$
$180$	0	0
$181$	− 716326.i	− 1.62523i	−0.582802	−	0.812614i	$-0.698044\pi$
$181$	− 716326.i	− 1.62523i	0.582802	−	0.812614i	$-0.301956\pi$
$182$	0	0
$183$	60965.6	0.134573
$184$	0	0
$185$	342265.	0.735248
$186$	0	0
$187$	619625.i	1.29576i
$188$	0	0
$189$	− 119705.i	− 0.243758i
$190$	0	0
$191$	902895.	1.79083	0.895414	−	0.445235i	$-0.146880\pi$
$191$	902895.	1.79083	0.895414	+	0.445235i	$0.146880\pi$
$192$	0	0
$193$	−577366.	−1.11573	−0.557864	−	0.829933i	$-0.688379\pi$
$193$	−577366.	−1.11573	−0.557864	+	0.829933i	$0.688379\pi$
$194$	0	0
$195$	151897.i	0.286064i
$196$	0	0
$197$	− 258415.i	− 0.474408i	−0.971460	−	0.237204i	$-0.923769\pi$
$197$	− 258415.i	− 0.474408i	0.971460	−	0.237204i	$-0.0762310\pi$
$198$	0	0
$199$	−953539.	−1.70689	−0.853445	−	0.521182i	$-0.825491\pi$
$199$	−953539.	−1.70689	−0.853445	+	0.521182i	$0.825491\pi$
$200$	0	0
$201$	−447741.	−0.781694
$202$	0	0
$203$	1.03321e6i	1.75974i
$204$	0	0
$205$	− 595077.i	− 0.988982i
$206$	0	0
$207$	200823.	0.325753
$208$	0	0
$209$	−339355.	−0.537389
$210$	0	0
$211$	677543.i	1.04769i	0.851815	+	0.523843i	$0.175502\pi$
$211$	677543.i	1.04769i	−0.851815	+	0.523843i	$0.824498\pi$
$212$	0	0
$213$	198693.i	0.300078i
$214$	0	0
$215$	−718728.	−1.06040
$216$	0	0
$217$	1.06493e6	1.53523
$218$	0	0
$219$	− 549314.i	− 0.773945i
$220$	0	0
$221$	687249.i	0.946528i
$222$	0	0
$223$	908498.	1.22338	0.611691	−	0.791097i	$-0.290490\pi$
$223$	908498.	1.22338	0.611691	+	0.791097i	$0.290490\pi$
$224$	0	0
$225$	−40448.2	−0.0532651
$226$	0	0
$227$	− 243977.i	− 0.314256i	−0.987578	−	0.157128i	$-0.949776\pi$
$227$	− 243977.i	− 0.314256i	0.987578	−	0.157128i	$-0.0502235\pi$
$228$	0	0
$229$	− 97378.2i	− 0.122708i	−0.998116	−	0.0613540i	$-0.980458\pi$
$229$	− 97378.2i	− 0.122708i	0.998116	−	0.0613540i	$-0.0195419\pi$
$230$	0	0
$231$	−438867.	−0.541131
$232$	0	0
$233$	−1.03417e6	−1.24797	−0.623983	−	0.781438i	$-0.714487\pi$
$233$	−1.03417e6	−1.24797	−0.623983	+	0.781438i	$0.714487\pi$
$234$	0	0
$235$	359805.i	0.425008i
$236$	0	0
$237$	663390.i	0.767181i
$238$	0	0
$239$	737319.	0.834950	0.417475	−	0.908688i	$-0.362915\pi$
$239$	737319.	0.834950	0.417475	+	0.908688i	$0.362915\pi$
$240$	0	0
$241$	695861.	0.771756	0.385878	−	0.922550i	$-0.373899\pi$
$241$	695861.	0.771756	0.385878	+	0.922550i	$0.373899\pi$
$242$	0	0
$243$	− 59049.0i	− 0.0641500i
$244$	0	0
$245$	520418.i	0.553907i
$246$	0	0
$247$	−376391.	−0.392552
$248$	0	0
$249$	311426.	0.318314
$250$	0	0
$251$	914370.i	0.916089i	0.888929	+	0.458044i	$0.151450\pi$
$251$	914370.i	0.916089i	−0.888929	+	0.458044i	$0.848550\pi$
$252$	0	0
$253$	− 736263.i	− 0.723156i
$254$	0	0
$255$	962244.	0.926690
$256$	0	0
$257$	−1.08160e6	−1.02149	−0.510743	−	0.859733i	$-0.670630\pi$
$257$	−1.08160e6	−1.02149	−0.510743	+	0.859733i	$0.670630\pi$
$258$	0	0
$259$	1.09681e6i	1.01597i
$260$	0	0
$261$	509667.i	0.463111i
$262$	0	0
$263$	1.08575e6	0.967923	0.483961	−	0.875089i	$-0.339198\pi$
$263$	1.08575e6	0.967923	0.483961	+	0.875089i	$0.339198\pi$
$264$	0	0
$265$	28026.4	0.0245162
$266$	0	0
$267$	381859.i	0.327812i
$268$	0	0
$269$	2.15191e6i	1.81319i	0.421999	+	0.906596i	$0.361329\pi$
$269$	2.15191e6i	1.81319i	−0.421999	+	0.906596i	$0.638671\pi$
$270$	0	0
$271$	1.47928e6	1.22357	0.611783	−	0.791026i	$-0.290453\pi$
$271$	1.47928e6	1.22357	0.611783	+	0.791026i	$0.290453\pi$
$272$	0	0
$273$	−486764.	−0.395286
$274$	0	0
$275$	148292.i	0.118246i
$276$	0	0
$277$	97604.6i	0.0764312i	0.999270	+	0.0382156i	$0.0121674\pi$
$277$	97604.6i	0.0764312i	−0.999270	+	0.0382156i	$0.987833\pi$
$278$	0	0
$279$	525317.	0.404027
$280$	0	0
$281$	−582276.	−0.439909	−0.219955	−	0.975510i	$-0.570591\pi$
$281$	−582276.	−0.439909	−0.219955	+	0.975510i	$0.570591\pi$
$282$	0	0
$283$	− 796049.i	− 0.590845i	−0.955367	−	0.295423i	$-0.904540\pi$
$283$	− 796049.i	− 0.590845i	0.955367	−	0.295423i	$-0.0954605\pi$
$284$	0	0
$285$	527000.i	0.384325i
$286$	0	0
$287$	1.90696e6	1.36659
$288$	0	0
$289$	2.93376e6	2.06624
$290$	0	0
$291$	1.24361e6i	0.860896i
$292$	0	0
$293$	− 804512.i	− 0.547474i	−0.961805	−	0.273737i	$-0.911740\pi$
$293$	− 804512.i	− 0.547474i	0.961805	−	0.273737i	$-0.0882598\pi$
$294$	0	0
$295$	1.94209e6	1.29931
$296$	0	0
$297$	−216487.	−0.142410
$298$	0	0
$299$	− 816618.i	− 0.528252i
$300$	0	0
$301$	− 2.30321e6i	− 1.46527i
$302$	0	0
$303$	−124179.	−0.0777038
$304$	0	0
$305$	−347104.	−0.213654
$306$	0	0
$307$	1.07397e6i	0.650349i	0.945654	+	0.325175i	$0.105423\pi$
$307$	1.07397e6i	0.650349i	−0.945654	+	0.325175i	$0.894577\pi$
$308$	0	0
$309$	1.23677e6i	0.736874i
$310$	0	0
$311$	−2.77894e6	−1.62921	−0.814607	−	0.580014i	$-0.803047\pi$
$311$	−2.77894e6	−1.62921	−0.814607	+	0.580014i	$0.803047\pi$
$312$	0	0
$313$	−32104.7	−0.0185228	−0.00926142	−	0.999957i	$-0.502948\pi$
$313$	−32104.7	−0.0185228	−0.00926142	+	0.999957i	$0.502948\pi$
$314$	0	0
$315$	681536.i	0.387001i
$316$	0	0
$317$	− 3.09756e6i	− 1.73130i	−0.500652	−	0.865649i	$-0.666906\pi$
$317$	− 3.09756e6i	− 1.73130i	0.500652	−	0.865649i	$-0.333094\pi$
$318$	0	0
$319$	1.86855e6	1.02808
$320$	0	0
$321$	−1.75832e6	−0.952436
$322$	0	0
$323$	2.38438e6i	1.27165i
$324$	0	0
$325$	164476.i	0.0863763i
$326$	0	0
$327$	−1.54363e6	−0.798313
$328$	0	0
$329$	−1.15302e6	−0.587281
$330$	0	0
$331$	− 642006.i	− 0.322084i	−0.986948	−	0.161042i	$-0.948515\pi$
$331$	− 642006.i	− 0.322084i	0.986948	−	0.161042i	$-0.0514854\pi$
$332$	0	0
$333$	541041.i	0.267374i
$334$	0	0
$335$	2.54919e6	1.24105
$336$	0	0
$337$	3.15555e6	1.51356	0.756781	−	0.653669i	$-0.226771\pi$
$337$	3.15555e6	1.51356	0.756781	+	0.653669i	$0.226771\pi$
$338$	0	0
$339$	316849.i	0.149745i
$340$	0	0
$341$	− 1.92593e6i	− 0.896921i
$342$	0	0
$343$	1.09208e6	0.501210
$344$	0	0
$345$	−1.14338e6	−0.517180
$346$	0	0
$347$	3.66352e6i	1.63333i	0.577110	+	0.816667i	$0.304180\pi$
$347$	3.66352e6i	1.63333i	−0.577110	+	0.816667i	$0.695820\pi$
$348$	0	0
$349$	− 2.51155e6i	− 1.10377i	−0.833921	−	0.551884i	$-0.813909\pi$
$349$	− 2.51155e6i	− 1.10377i	0.833921	−	0.551884i	$-0.186091\pi$
$350$	0	0
$351$	−240114.	−0.104028
$352$	0	0
$353$	−3.02557e6	−1.29232	−0.646161	−	0.763201i	$-0.723626\pi$
$353$	−3.02557e6	−1.29232	−0.646161	+	0.763201i	$0.723626\pi$
$354$	0	0
$355$	− 1.13125e6i	− 0.476417i
$356$	0	0
$357$	3.08357e6i	1.28051i
$358$	0	0
$359$	797992.	0.326785	0.163393	−	0.986561i	$-0.447756\pi$
$359$	797992.	0.326785	0.163393	+	0.986561i	$0.447756\pi$
$360$	0	0
$361$	1.17023e6	0.472609
$362$	0	0
$363$	− 655770.i	− 0.261207i
$364$	0	0
$365$	3.12749e6i	1.22875i
$366$	0	0
$367$	798457.	0.309447	0.154723	−	0.987958i	$-0.450551\pi$
$367$	798457.	0.309447	0.154723	+	0.987958i	$0.450551\pi$
$368$	0	0
$369$	940677.	0.359645
$370$	0	0
$371$	89812.4i	0.0338767i
$372$	0	0
$373$	− 2.56107e6i	− 0.953122i	−0.879141	−	0.476561i	$-0.841883\pi$
$373$	− 2.56107e6i	− 0.953122i	0.879141	−	0.476561i	$-0.158117\pi$
$374$	0	0
$375$	1.67144e6	0.613781
$376$	0	0
$377$	2.07248e6	0.750995
$378$	0	0
$379$	3.95135e6i	1.41302i	0.707705	+	0.706508i	$0.249731\pi$
$379$	3.95135e6i	1.41302i	−0.707705	+	0.706508i	$0.750269\pi$
$380$	0	0
$381$	− 2.44802e6i	− 0.863979i
$382$	0	0
$383$	−5.68872e6	−1.98161	−0.990803	−	0.135313i	$-0.956796\pi$
$383$	−5.68872e6	−1.98161	−0.990803	+	0.135313i	$0.956796\pi$
$384$	0	0
$385$	2.49866e6	0.859124
$386$	0	0
$387$	− 1.13614e6i	− 0.385616i
$388$	0	0
$389$	4.91032e6i	1.64526i	0.568574	+	0.822632i	$0.307495\pi$
$389$	4.91032e6i	1.64526i	−0.568574	+	0.822632i	$0.692505\pi$
$390$	0	0
$391$	−5.17314e6	−1.71125
$392$	0	0
$393$	257441.	0.0840808
$394$	0	0
$395$	− 3.77697e6i	− 1.21801i
$396$	0	0
$397$	− 1.25380e6i	− 0.399256i	−0.979872	−	0.199628i	$-0.936027\pi$
$397$	− 1.25380e6i	− 0.399256i	0.979872	−	0.199628i	$-0.0639734\pi$
$398$	0	0
$399$	−1.68880e6	−0.531064
$400$	0	0
$401$	17426.6	0.00541194	0.00270597	−	0.999996i	$-0.499139\pi$
$401$	17426.6	0.00541194	0.00270597	+	0.999996i	$0.499139\pi$
$402$	0	0
$403$	− 2.13612e6i	− 0.655183i
$404$	0	0
$405$	336192.i	0.101847i
$406$	0	0
$407$	1.98358e6	0.593558
$408$	0	0
$409$	6.38023e6	1.88594	0.942970	−	0.332877i	$-0.108019\pi$
$409$	6.38023e6	1.88594	0.942970	+	0.332877i	$0.108019\pi$
$410$	0	0
$411$	1.08352e6i	0.316397i
$412$	0	0
$413$	6.22355e6i	1.79541i
$414$	0	0
$415$	−1.77309e6	−0.505370
$416$	0	0
$417$	−3.36990e6	−0.949025
$418$	0	0
$419$	1.98169e6i	0.551443i	0.961238	+	0.275721i	$0.0889167\pi$
$419$	1.98169e6i	0.551443i	−0.961238	+	0.275721i	$0.911083\pi$
$420$	0	0
$421$	− 6.90614e6i	− 1.89902i	−0.313733	−	0.949511i	$-0.601580\pi$
$421$	− 6.90614e6i	− 1.89902i	0.313733	−	0.949511i	$-0.398420\pi$
$422$	0	0
$423$	−568767.	−0.154555
$424$	0	0
$425$	1.04193e6	0.279812
$426$	0	0
$427$	− 1.11232e6i	− 0.295229i
$428$	0	0
$429$	880310.i	0.230936i
$430$	0	0
$431$	4.29270e6	1.11311	0.556554	−	0.830811i	$-0.312123\pi$
$431$	4.29270e6	1.11311	0.556554	+	0.830811i	$0.312123\pi$
$432$	0	0
$433$	−1.75102e6	−0.448818	−0.224409	−	0.974495i	$-0.572045\pi$
$433$	−1.75102e6	−0.448818	−0.224409	+	0.974495i	$0.572045\pi$
$434$	0	0
$435$	− 2.90176e6i	− 0.735256i
$436$	0	0
$437$	− 2.83322e6i	− 0.709702i
$438$	0	0
$439$	796336.	0.197213	0.0986064	−	0.995127i	$-0.468562\pi$
$439$	796336.	0.197213	0.0986064	+	0.995127i	$0.468562\pi$
$440$	0	0
$441$	−822659.	−0.201430
$442$	0	0
$443$	− 5.15502e6i	− 1.24802i	−0.781417	−	0.624009i	$-0.785503\pi$
$443$	− 5.15502e6i	− 1.24802i	0.781417	−	0.624009i	$-0.214497\pi$
$444$	0	0
$445$	− 2.17409e6i	− 0.520449i
$446$	0	0
$447$	1.64197e6	0.388684
$448$	0	0
$449$	−2.01303e6	−0.471232	−0.235616	−	0.971846i	$-0.575711\pi$
$449$	−2.01303e6	−0.471232	−0.235616	+	0.971846i	$0.575711\pi$
$450$	0	0
$451$	− 3.44873e6i	− 0.798395i
$452$	0	0
$453$	2.32717e6i	0.532823i
$454$	0	0
$455$	2.77136e6	0.627574
$456$	0	0
$457$	−6.66508e6	−1.49285	−0.746423	−	0.665472i	$-0.768230\pi$
$457$	−6.66508e6	−1.49285	−0.746423	+	0.665472i	$0.768230\pi$
$458$	0	0
$459$	1.52108e6i	0.336993i
$460$	0	0
$461$	4.94928e6i	1.08465i	0.840169	+	0.542325i	$0.182456\pi$
$461$	4.94928e6i	1.08465i	−0.840169	+	0.542325i	$0.817544\pi$
$462$	0	0
$463$	−5.00463e6	−1.08497	−0.542487	−	0.840064i	$-0.682517\pi$
$463$	−5.00463e6	−1.08497	−0.542487	+	0.840064i	$0.682517\pi$
$464$	0	0
$465$	−2.99086e6	−0.641452
$466$	0	0
$467$	− 5.44773e6i	− 1.15591i	−0.816069	−	0.577954i	$-0.803851\pi$
$467$	− 5.44773e6i	− 1.15591i	0.816069	−	0.577954i	$-0.196149\pi$
$468$	0	0
$469$	8.16903e6i	1.71490i
$470$	0	0
$471$	1.55007e6	0.321957
$472$	0	0
$473$	−4.16534e6	−0.856048
$474$	0	0
$475$	570643.i	0.116046i
$476$	0	0
$477$	44303.2i	0.00891536i
$478$	0	0
$479$	−7.60147e6	−1.51377	−0.756883	−	0.653550i	$-0.773279\pi$
$479$	−7.60147e6	−1.51377	−0.756883	+	0.653550i	$0.773279\pi$
$480$	0	0
$481$	2.20006e6	0.433583
$482$	0	0
$483$	− 3.66402e6i	− 0.714645i
$484$	0	0
$485$	− 7.08041e6i	− 1.36680i
$486$	0	0
$487$	−2.88190e6	−0.550625	−0.275313	−	0.961355i	$-0.588781\pi$
$487$	−2.88190e6	−0.550625	−0.275313	+	0.961355i	$0.588781\pi$
$488$	0	0
$489$	−2.89848e6	−0.548149
$490$	0	0
$491$	− 4.75473e6i	− 0.890066i	−0.895514	−	0.445033i	$-0.853192\pi$
$491$	− 4.75473e6i	− 0.890066i	0.895514	−	0.445033i	$-0.146808\pi$
$492$	0	0
$493$	− 1.31288e7i	− 2.43281i
$494$	0	0
$495$	1.23256e6	0.226096
$496$	0	0
$497$	3.62516e6	0.658318
$498$	0	0
$499$	− 337011.i	− 0.0605888i	−0.999541	−	0.0302944i	$-0.990356\pi$
$499$	− 337011.i	− 0.0605888i	0.999541	−	0.0302944i	$-0.00964448\pi$
$500$	0	0
$501$	1.28791e6i	0.229240i
$502$	0	0
$503$	−5.97808e6	−1.05352	−0.526758	−	0.850015i	$-0.676593\pi$
$503$	−5.97808e6	−1.05352	−0.526758	+	0.850015i	$0.676593\pi$
$504$	0	0
$505$	707007.	0.123366
$506$	0	0
$507$	− 2.36525e6i	− 0.408656i
$508$	0	0
$509$	8.71584e6i	1.49113i	0.666434	+	0.745564i	$0.267820\pi$
$509$	8.71584e6i	1.49113i	−0.666434	+	0.745564i	$0.732180\pi$
$510$	0	0
$511$	−1.00222e7	−1.69790
$512$	0	0
$513$	−833063.	−0.139761
$514$	0	0
$515$	− 7.04149e6i	− 1.16989i
$516$	0	0
$517$	2.08523e6i	0.343105i
$518$	0	0
$519$	710572.	0.115795
$520$	0	0
$521$	−2.18008e6	−0.351867	−0.175934	−	0.984402i	$-0.556294\pi$
$521$	−2.18008e6	−0.351867	−0.175934	+	0.984402i	$0.556294\pi$
$522$	0	0
$523$	− 5.08989e6i	− 0.813680i	−0.913499	−	0.406840i	$-0.866631\pi$
$523$	− 5.08989e6i	− 0.813680i	0.913499	−	0.406840i	$-0.133369\pi$
$524$	0	0
$525$	737977.i	0.116854i
$526$	0	0
$527$	−1.35320e7	−2.12244
$528$	0	0
$529$	−289405.	−0.0449642
$530$	0	0
$531$	3.06999e6i	0.472498i
$532$	0	0
$533$	− 3.82512e6i	− 0.583212i
$534$	0	0
$535$	1.00109e7	1.51213
$536$	0	0
$537$	−3.60930e6	−0.540117
$538$	0	0
$539$	3.01605e6i	0.447164i
$540$	0	0
$541$	9.48126e6i	1.39275i	0.717679	+	0.696374i	$0.245205\pi$
$541$	9.48126e6i	1.39275i	−0.717679	+	0.696374i	$0.754795\pi$
$542$	0	0
$543$	−6.44693e6	−0.938326
$544$	0	0
$545$	8.78856e6	1.26744
$546$	0	0
$547$	7.78849e6i	1.11297i	0.830856	+	0.556487i	$0.187851\pi$
$547$	7.78849e6i	1.11297i	−0.830856	+	0.556487i	$0.812149\pi$
$548$	0	0
$549$	− 548690.i	− 0.0776956i
$550$	0	0
$551$	7.19038e6	1.00896
$552$	0	0
$553$	1.21035e7	1.68306
$554$	0	0
$555$	− 3.08039e6i	− 0.424496i
$556$	0	0
$557$	6.32393e6i	0.863673i	0.901952	+	0.431836i	$0.142134\pi$
$557$	6.32393e6i	0.863673i	−0.901952	+	0.431836i	$0.857866\pi$
$558$	0	0
$559$	−4.61994e6	−0.625327
$560$	0	0
$561$	5.57662e6	0.748108
$562$	0	0
$563$	8.84257e6i	1.17573i	0.808959	+	0.587865i	$0.200031\pi$
$563$	8.84257e6i	1.17573i	−0.808959	+	0.587865i	$0.799969\pi$
$564$	0	0
$565$	− 1.80396e6i	− 0.237742i
$566$	0	0
$567$	−1.07735e6	−0.140734
$568$	0	0
$569$	−1.63627e6	−0.211872	−0.105936	−	0.994373i	$-0.533784\pi$
$569$	−1.63627e6	−0.211872	−0.105936	+	0.994373i	$0.533784\pi$
$570$	0	0
$571$	− 2.33632e6i	− 0.299876i	−0.988695	−	0.149938i	$-0.952093\pi$
$571$	− 2.33632e6i	− 0.299876i	0.988695	−	0.149938i	$-0.0479074\pi$
$572$	0	0
$573$	− 8.12605e6i	− 1.03393i
$574$	0	0
$575$	−1.23806e6	−0.156161
$576$	0	0
$577$	1.12142e6	0.140225	0.0701127	−	0.997539i	$-0.477664\pi$
$577$	1.12142e6	0.140225	0.0701127	+	0.997539i	$0.477664\pi$
$578$	0	0
$579$	5.19629e6i	0.644165i
$580$	0	0
$581$	− 5.68196e6i	− 0.698326i
$582$	0	0
$583$	162425.	0.0197917
$584$	0	0
$585$	1.36707e6	0.165159
$586$	0	0
$587$	− 4.39806e6i	− 0.526824i	−0.964683	−	0.263412i	$-0.915152\pi$
$587$	− 4.39806e6i	− 0.526824i	0.964683	−	0.263412i	$-0.0848479\pi$
$588$	0	0
$589$	− 7.41117e6i	− 0.880234i
$590$	0	0
$591$	−2.32574e6	−0.273900
$592$	0	0
$593$	−1.05010e7	−1.22630	−0.613149	−	0.789968i	$-0.710097\pi$
$593$	−1.05010e7	−1.22630	−0.613149	+	0.789968i	$0.710097\pi$
$594$	0	0
$595$	− 1.75561e7i	− 2.03300i
$596$	0	0
$597$	8.58185e6i	0.985474i
$598$	0	0
$599$	−3.41029e6	−0.388351	−0.194175	−	0.980967i	$-0.562203\pi$
$599$	−3.41029e6	−0.388351	−0.194175	+	0.980967i	$0.562203\pi$
$600$	0	0
$601$	1.14158e6	0.128920	0.0644601	−	0.997920i	$-0.479467\pi$
$601$	1.14158e6	0.128920	0.0644601	+	0.997920i	$0.479467\pi$
$602$	0	0
$603$	4.02967e6i	0.451311i
$604$	0	0
$605$	3.73359e6i	0.414704i
$606$	0	0
$607$	−3.01017e6	−0.331603	−0.165802	−	0.986159i	$-0.553021\pi$
$607$	−3.01017e6	−0.331603	−0.165802	+	0.986159i	$0.553021\pi$
$608$	0	0
$609$	9.29887e6	1.01598
$610$	0	0
$611$	2.31280e6i	0.250632i
$612$	0	0
$613$	− 921343.i	− 0.0990307i	−0.998773	−	0.0495154i	$-0.984232\pi$
$613$	− 921343.i	− 0.0990307i	0.998773	−	0.0495154i	$-0.0157677\pi$
$614$	0	0
$615$	−5.35569e6	−0.570989
$616$	0	0
$617$	−6.38884e6	−0.675631	−0.337815	−	0.941212i	$-0.609688\pi$
$617$	−6.38884e6	−0.675631	−0.337815	+	0.941212i	$0.609688\pi$
$618$	0	0
$619$	− 1.60064e7i	− 1.67907i	−0.543308	−	0.839534i	$-0.682828\pi$
$619$	− 1.60064e7i	− 1.67907i	0.543308	−	0.839534i	$-0.317172\pi$
$620$	0	0
$621$	− 1.80741e6i	− 0.188074i
$622$	0	0
$623$	6.96702e6	0.719162
$624$	0	0
$625$	−7.95576e6	−0.814670
$626$	0	0
$627$	3.05420e6i	0.310262i
$628$	0	0
$629$	− 1.39370e7i	− 1.40457i
$630$	0	0
$631$	9.64983e6	0.964820	0.482410	−	0.875946i	$-0.339762\pi$
$631$	9.64983e6	0.964820	0.482410	+	0.875946i	$0.339762\pi$
$632$	0	0
$633$	6.09789e6	0.604881
$634$	0	0
$635$	1.39377e7i	1.37169i
$636$	0	0
$637$	3.34521e6i	0.326645i
$638$	0	0
$639$	1.78824e6	0.173250
$640$	0	0
$641$	−2.73675e6	−0.263081	−0.131540	−	0.991311i	$-0.541992\pi$
$641$	−2.73675e6	−0.263081	−0.131540	+	0.991311i	$0.541992\pi$
$642$	0	0
$643$	1.28018e7i	1.22108i	0.791986	+	0.610539i	$0.209047\pi$
$643$	1.28018e7i	1.22108i	−0.791986	+	0.610539i	$0.790953\pi$
$644$	0	0
$645$	6.46855e6i	0.612221i
$646$	0	0
$647$	4.78055e6	0.448970	0.224485	−	0.974478i	$-0.427930\pi$
$647$	4.78055e6	0.448970	0.224485	+	0.974478i	$0.427930\pi$
$648$	0	0
$649$	1.12553e7	1.04892
$650$	0	0
$651$	− 9.58440e6i	− 0.886365i
$652$	0	0
$653$	− 5.63674e6i	− 0.517303i	−0.965971	−	0.258652i	$-0.916722\pi$
$653$	− 5.63674e6i	− 0.517303i	0.965971	−	0.258652i	$-0.0832782\pi$
$654$	0	0
$655$	−1.46573e6	−0.133490
$656$	0	0
$657$	−4.94383e6	−0.446838
$658$	0	0
$659$	− 1.18336e7i	− 1.06146i	−0.847540	−	0.530731i	$-0.821917\pi$
$659$	− 1.18336e7i	− 1.06146i	0.847540	−	0.530731i	$-0.178083\pi$
$660$	0	0
$661$	9.73159e6i	0.866324i	0.901316	+	0.433162i	$0.142602\pi$
$661$	9.73159e6i	0.866324i	−0.901316	+	0.433162i	$0.857398\pi$
$662$	0	0
$663$	6.18524e6	0.546478
$664$	0	0
$665$	9.61511e6	0.843141
$666$	0	0
$667$	1.56002e7i	1.35774i
$668$	0	0
$669$	− 8.17649e6i	− 0.706320i
$670$	0	0
$671$	−2.01162e6	−0.172480
$672$	0	0
$673$	−2.17026e7	−1.84703	−0.923513	−	0.383566i	$-0.874696\pi$
$673$	−2.17026e7	−1.84703	−0.923513	+	0.383566i	$0.874696\pi$
$674$	0	0
$675$	364033.i	0.0307526i
$676$	0	0
$677$	− 1.47841e7i	− 1.23972i	−0.784711	−	0.619861i	$-0.787189\pi$
$677$	− 1.47841e7i	− 1.23972i	0.784711	−	0.619861i	$-0.212811\pi$
$678$	0	0
$679$	2.26896e7	1.88866
$680$	0	0
$681$	−2.19579e6	−0.181436
$682$	0	0
$683$	2.77071e6i	0.227269i	0.993523	+	0.113634i	$0.0362492\pi$
$683$	2.77071e6i	0.227269i	−0.993523	+	0.113634i	$0.963751\pi$
$684$	0	0
$685$	− 6.16897e6i	− 0.502327i
$686$	0	0
$687$	−876404.	−0.0708455
$688$	0	0
$689$	180152.	0.0144574
$690$	0	0
$691$	− 2.22984e6i	− 0.177655i	−0.996047	−	0.0888277i	$-0.971688\pi$
$691$	− 2.22984e6i	− 0.177655i	0.996047	−	0.0888277i	$-0.0283121\pi$
$692$	0	0
$693$	3.94980e6i	0.312422i
$694$	0	0
$695$	1.91864e7	1.50671
$696$	0	0
$697$	−2.42315e7	−1.88929
$698$	0	0
$699$	9.30754e6i	0.720514i
$700$	0	0
$701$	− 1.16646e7i	− 0.896553i	−0.893895	−	0.448277i	$-0.852038\pi$
$701$	− 1.16646e7i	− 0.896553i	0.893895	−	0.448277i	$-0.147962\pi$
$702$	0	0
$703$	7.63301e6	0.582516
$704$	0	0
$705$	3.23824e6	0.245379
$706$	0	0
$707$	2.26565e6i	0.170468i
$708$	0	0
$709$	1.82422e7i	1.36290i	0.731867	+	0.681448i	$0.238649\pi$
$709$	1.82422e7i	1.36290i	−0.731867	+	0.681448i	$0.761351\pi$
$710$	0	0
$711$	5.97051e6	0.442932
$712$	0	0
$713$	1.60792e7	1.18452
$714$	0	0
$715$	− 5.01200e6i	− 0.366645i
$716$	0	0
$717$	− 6.63587e6i	− 0.482059i
$718$	0	0
$719$	−1.44884e7	−1.04519	−0.522597	−	0.852580i	$-0.675037\pi$
$719$	−1.44884e7	−1.04519	−0.522597	+	0.852580i	$0.675037\pi$
$720$	0	0
$721$	2.25649e7	1.61657
$722$	0	0
$723$	− 6.26275e6i	− 0.445574i
$724$	0	0
$725$	− 3.14206e6i	− 0.222009i
$726$	0	0
$727$	−1.16435e7	−0.817045	−0.408522	−	0.912748i	$-0.633956\pi$
$727$	−1.16435e7	−0.817045	−0.408522	+	0.912748i	$0.633956\pi$
$728$	0	0
$729$	−531441.	−0.0370370
$730$	0	0
$731$	2.92666e7i	2.02572i
$732$	0	0
$733$	1.36982e7i	0.941683i	0.882218	+	0.470841i	$0.156050\pi$
$733$	1.36982e7i	0.941683i	−0.882218	+	0.470841i	$0.843950\pi$
$734$	0	0
$735$	4.68376e6	0.319798
$736$	0	0
$737$	1.47737e7	1.00189
$738$	0	0
$739$	1.29727e7i	0.873817i	0.899506	+	0.436908i	$0.143927\pi$
$739$	1.29727e7i	0.873817i	−0.899506	+	0.436908i	$0.856073\pi$
$740$	0	0
$741$	3.38752e6i	0.226640i
$742$	0	0
$743$	1.26100e7	0.837998	0.418999	−	0.907987i	$-0.362381\pi$
$743$	1.26100e7	0.837998	0.418999	+	0.907987i	$0.362381\pi$
$744$	0	0
$745$	−9.34848e6	−0.617092
$746$	0	0
$747$	− 2.80283e6i	− 0.183779i
$748$	0	0
$749$	3.20806e7i	2.08948i
$750$	0	0
$751$	6.59103e6	0.426435	0.213218	−	0.977005i	$-0.431606\pi$
$751$	6.59103e6	0.426435	0.213218	+	0.977005i	$0.431606\pi$
$752$	0	0
$753$	8.22933e6	0.528904
$754$	0	0
$755$	− 1.32496e7i	− 0.845933i
$756$	0	0
$757$	3.08973e7i	1.95966i	0.199838	+	0.979829i	$0.435959\pi$
$757$	3.08973e7i	1.95966i	−0.199838	+	0.979829i	$0.564041\pi$
$758$	0	0
$759$	−6.62637e6	−0.417514
$760$	0	0
$761$	−1.81415e7	−1.13556	−0.567781	−	0.823180i	$-0.692198\pi$
$761$	−1.81415e7	−1.13556	−0.567781	+	0.823180i	$0.692198\pi$
$762$	0	0
$763$	2.81635e7i	1.75136i
$764$	0	0
$765$	− 8.66019e6i	− 0.535025i
$766$	0	0
$767$	1.24836e7	0.766218
$768$	0	0
$769$	−1.65019e7	−1.00628	−0.503139	−	0.864206i	$-0.667822\pi$
$769$	−1.65019e7	−1.00628	−0.503139	+	0.864206i	$0.667822\pi$
$770$	0	0
$771$	9.73437e6i	0.589755i
$772$	0	0
$773$	2.75342e6i	0.165739i	0.996560	+	0.0828693i	$0.0264084\pi$
$773$	2.75342e6i	0.165739i	−0.996560	+	0.0828693i	$0.973592\pi$
$774$	0	0
$775$	−3.23855e6	−0.193685
$776$	0	0
$777$	9.87130e6	0.586573
$778$	0	0
$779$	− 1.32711e7i	− 0.783542i
$780$	0	0
$781$	− 6.55608e6i	− 0.384607i
$782$	0	0
$783$	4.58700e6	0.267377
$784$	0	0
$785$	−8.82521e6	−0.511153
$786$	0	0
$787$	3.33836e7i	1.92131i	0.277749	+	0.960654i	$0.410412\pi$
$787$	3.33836e7i	1.92131i	−0.277749	+	0.960654i	$0.589588\pi$
$788$	0	0
$789$	− 9.77175e6i	− 0.558830i
$790$	0	0
$791$	5.78091e6	0.328515
$792$	0	0
$793$	−2.23116e6	−0.125994
$794$	0	0
$795$	− 252238.i	− 0.0141544i
$796$	0	0
$797$	1.58160e7i	0.881966i	0.897515	+	0.440983i	$0.145370\pi$
$797$	1.58160e7i	0.881966i	−0.897515	+	0.440983i	$0.854630\pi$
$798$	0	0
$799$	1.46512e7	0.811910
$800$	0	0
$801$	3.43673e6	0.189262
$802$	0	0
$803$	1.81252e7i	0.991957i
$804$	0	0
$805$	2.08609e7i	1.13460i
$806$	0	0
$807$	1.93672e7	1.04685
$808$	0	0
$809$	−5.87717e6	−0.315716	−0.157858	−	0.987462i	$-0.550459\pi$
$809$	−5.87717e6	−0.315716	−0.157858	+	0.987462i	$0.550459\pi$
$810$	0	0
$811$	2.48561e7i	1.32703i	0.748163	+	0.663515i	$0.230936\pi$
$811$	2.48561e7i	1.32703i	−0.748163	+	0.663515i	$0.769064\pi$
$812$	0	0
$813$	− 1.33135e7i	− 0.706426i
$814$	0	0
$815$	1.65024e7	0.870266
$816$	0	0
$817$	−1.60287e7	−0.840122
$818$	0	0
$819$	4.38087e6i	0.228219i
$820$	0	0
$821$	− 3.62723e7i	− 1.87809i	−0.343793	−	0.939046i	$-0.611712\pi$
$821$	− 3.62723e7i	− 1.87809i	0.343793	−	0.939046i	$-0.388288\pi$
$822$	0	0
$823$	−3.11177e7	−1.60143	−0.800714	−	0.599046i	$-0.795547\pi$
$823$	−3.11177e7	−1.60143	−0.800714	+	0.599046i	$0.795547\pi$
$824$	0	0
$825$	1.33463e6	0.0682693
$826$	0	0
$827$	3.16856e6i	0.161101i	0.996751	+	0.0805505i	$0.0256678\pi$
$827$	3.16856e6i	0.161101i	−0.996751	+	0.0805505i	$0.974332\pi$
$828$	0	0
$829$	− 2.04472e7i	− 1.03335i	−0.856182	−	0.516675i	$-0.827169\pi$
$829$	− 2.04472e7i	− 1.03335i	0.856182	−	0.516675i	$-0.172831\pi$
$830$	0	0
$831$	878441.	0.0441276
$832$	0	0
$833$	2.11914e7	1.05815
$834$	0	0
$835$	− 7.33262e6i	− 0.363951i
$836$	0	0
$837$	− 4.72785e6i	− 0.233265i
$838$	0	0
$839$	−3.56467e7	−1.74829	−0.874146	−	0.485663i	$-0.838578\pi$
$839$	−3.56467e7	−1.74829	−0.874146	+	0.485663i	$0.838578\pi$
$840$	0	0
$841$	−1.90804e7	−0.930246
$842$	0	0
$843$	5.24048e6i	0.253982i
$844$	0	0
$845$	1.34664e7i	0.648800i
$846$	0	0
$847$	−1.19645e7	−0.573043
$848$	0	0
$849$	−7.16444e6	−0.341125
$850$	0	0
$851$	1.65606e7i	0.783883i
$852$	0	0
$853$	− 5.41977e6i	− 0.255040i	−0.991836	−	0.127520i	$-0.959298\pi$
$853$	− 5.41977e6i	− 0.255040i	0.991836	−	0.127520i	$-0.0407017\pi$
$854$	0	0
$855$	4.74300e6	0.221890
$856$	0	0
$857$	1.62824e7	0.757298	0.378649	−	0.925540i	$-0.376389\pi$
$857$	1.62824e7	0.757298	0.378649	+	0.925540i	$0.376389\pi$
$858$	0	0
$859$	− 1.02620e7i	− 0.474512i	−0.971447	−	0.237256i	$-0.923752\pi$
$859$	− 1.02620e7i	− 0.474512i	0.971447	−	0.237256i	$-0.0762480\pi$
$860$	0	0
$861$	− 1.71626e7i	− 0.788999i
$862$	0	0
$863$	3.44259e7	1.57347	0.786734	−	0.617293i	$-0.211771\pi$
$863$	3.44259e7	1.57347	0.786734	+	0.617293i	$0.211771\pi$
$864$	0	0
$865$	−4.04560e6	−0.183841
$866$	0	0
$867$	− 2.64038e7i	− 1.19294i
$868$	0	0
$869$	− 2.18892e7i	− 0.983288i
$870$	0	0
$871$	1.63860e7	0.731860
$872$	0	0
$873$	1.11925e7	0.497039
$874$	0	0
$875$	− 3.04955e7i	− 1.34653i
$876$	0	0
$877$	1.77865e7i	0.780893i	0.920626	+	0.390447i	$0.127679\pi$
$877$	1.77865e7i	0.780893i	−0.920626	+	0.390447i	$0.872321\pi$
$878$	0	0
$879$	−7.24061e6	−0.316084
$880$	0	0
$881$	2.69565e7	1.17010	0.585050	−	0.810997i	$-0.301075\pi$
$881$	2.69565e7	1.17010	0.585050	+	0.810997i	$0.301075\pi$
$882$	0	0
$883$	− 4.51936e7i	− 1.95063i	−0.220820	−	0.975315i	$-0.570873\pi$
$883$	− 4.51936e7i	− 1.95063i	0.220820	−	0.975315i	$-0.429127\pi$
$884$	0	0
$885$	− 1.74788e7i	− 0.750159i
$886$	0	0
$887$	2.51019e7	1.07127	0.535634	−	0.844450i	$-0.320073\pi$
$887$	2.51019e7	1.07127	0.535634	+	0.844450i	$0.320073\pi$
$888$	0	0
$889$	−4.46642e7	−1.89542
$890$	0	0
$891$	1.94838e6i	0.0822204i
$892$	0	0
$893$	8.02417e6i	0.336722i
$894$	0	0
$895$	2.05494e7	0.857513
$896$	0	0
$897$	−7.34956e6	−0.304986
$898$	0	0
$899$	4.08073e7i	1.68399i
$900$	0	0
$901$	− 1.14123e6i	− 0.0468342i
$902$	0	0
$903$	−2.07289e7	−0.845973
$904$	0	0
$905$	3.67053e7	1.48973
$906$	0	0
$907$	− 4.08270e7i	− 1.64790i	−0.566665	−	0.823948i	$-0.691767\pi$
$907$	− 4.08270e7i	− 1.64790i	0.566665	−	0.823948i	$-0.308233\pi$
$908$	0	0
$909$	1.11761e6i	0.0448623i
$910$	0	0
$911$	−1.34261e7	−0.535988	−0.267994	−	0.963421i	$-0.586361\pi$
$911$	−1.34261e7	−0.535988	−0.267994	+	0.963421i	$0.586361\pi$
$912$	0	0
$913$	−1.02758e7	−0.407980
$914$	0	0
$915$	3.12394e6i	0.123353i
$916$	0	0
$917$	− 4.69702e6i	− 0.184458i
$918$	0	0
$919$	1.72734e7	0.674666	0.337333	−	0.941385i	$-0.390475\pi$
$919$	1.72734e7	0.674666	0.337333	+	0.941385i	$0.390475\pi$
$920$	0	0
$921$	9.66574e6	0.375479
$922$	0	0
$923$	− 7.27160e6i	− 0.280948i
$924$	0	0
$925$	− 3.33549e6i	− 0.128175i
$926$	0	0
$927$	1.11309e7	0.425435
$928$	0	0
$929$	−7.85244e6	−0.298514	−0.149257	−	0.988798i	$-0.547688\pi$
$929$	−7.85244e6	−0.298514	−0.149257	+	0.988798i	$0.547688\pi$
$930$	0	0
$931$	1.16061e7i	0.438845i
$932$	0	0
$933$	2.50104e7i	0.940627i
$934$	0	0
$935$	−3.17502e7	−1.18773
$936$	0	0
$937$	−1.05686e7	−0.393251	−0.196626	−	0.980479i	$-0.562998\pi$
$937$	−1.05686e7	−0.393251	−0.196626	+	0.980479i	$0.562998\pi$
$938$	0	0
$939$	288942.i	0.0106942i
$940$	0	0
$941$	2.93627e6i	0.108099i	0.998538	+	0.0540495i	$0.0172129\pi$
$941$	2.93627e6i	0.108099i	−0.998538	+	0.0540495i	$0.982787\pi$
$942$	0	0
$943$	2.87929e7	1.05440
$944$	0	0
$945$	6.13383e6	0.223435
$946$	0	0
$947$	9.78248e6i	0.354466i	0.984169	+	0.177233i	$0.0567145\pi$
$947$	9.78248e6i	0.354466i	−0.984169	+	0.177233i	$0.943285\pi$
$948$	0	0
$949$	2.01033e7i	0.724606i
$950$	0	0
$951$	−2.78780e7	−0.999565
$952$	0	0
$953$	4.19365e7	1.49575	0.747876	−	0.663838i	$-0.231074\pi$
$953$	4.19365e7	1.49575	0.747876	+	0.663838i	$0.231074\pi$
$954$	0	0
$955$	4.62652e7i	1.64152i
$956$	0	0
$957$	− 1.68170e7i	− 0.593564i
$958$	0	0
$959$	1.97688e7	0.694120
$960$	0	0
$961$	1.34311e7	0.469142
$962$	0	0
$963$	1.58249e7i	0.549889i
$964$	0	0
$965$	− 2.95848e7i	− 1.02271i
$966$	0	0
$967$	−1.67787e7	−0.577021	−0.288510	−	0.957477i	$-0.593160\pi$
$967$	−1.67787e7	−0.577021	−0.288510	+	0.957477i	$0.593160\pi$
$968$	0	0
$969$	2.14594e7	0.734190
$970$	0	0
$971$	− 3.82611e7i	− 1.30229i	−0.758951	−	0.651147i	$-0.774288\pi$
$971$	− 3.82611e7i	− 1.30229i	0.758951	−	0.651147i	$-0.225712\pi$
$972$	0	0
$973$	6.14839e7i	2.08199i
$974$	0	0
$975$	1.48029e6	0.0498694
$976$	0	0
$977$	−2.26502e7	−0.759163	−0.379581	−	0.925158i	$-0.623932\pi$
$977$	−2.26502e7	−0.759163	−0.379581	+	0.925158i	$0.623932\pi$
$978$	0	0
$979$	− 1.25998e7i	− 0.420153i
$980$	0	0
$981$	1.38927e7i	0.460906i
$982$	0	0
$983$	−4.49137e7	−1.48250	−0.741250	−	0.671229i	$-0.765767\pi$
$983$	−4.49137e7	−1.48250	−0.741250	+	0.671229i	$0.765767\pi$
$984$	0	0
$985$	1.32414e7	0.434855
$986$	0	0
$987$	1.03772e7i	0.339067i
$988$	0	0
$989$	− 3.47757e7i	− 1.13054i
$990$	0	0
$991$	−1.03855e7	−0.335925	−0.167963	−	0.985793i	$-0.553719\pi$
$991$	−1.03855e7	−0.335925	−0.167963	+	0.985793i	$0.553719\pi$
$992$	0	0
$993$	−5.77805e6	−0.185955
$994$	0	0
$995$	− 4.88603e7i	− 1.56458i
$996$	0	0
$997$	4.57977e6i	0.145917i	0.997335	+	0.0729585i	$0.0232441\pi$
$997$	4.57977e6i	0.145917i	−0.997335	+	0.0729585i	$0.976756\pi$
$998$	0	0
$999$	4.86937e6	0.154369

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.6.d.g.193.2	✓	4
4.3	odd	2		384.6.d.h.193.4	yes	4
8.3	odd	2		384.6.d.h.193.1	yes	4
8.5	even	2	inner	384.6.d.g.193.3	yes	4
16.3	odd	4		768.6.a.q.1.2		2
16.5	even	4		768.6.a.m.1.1		2
16.11	odd	4		768.6.a.r.1.1		2
16.13	even	4		768.6.a.v.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.6.d.g.193.2	✓	4	1.1	even	1	trivial
384.6.d.g.193.3	yes	4	8.5	even	2	inner
384.6.d.h.193.1	yes	4	8.3	odd	2
384.6.d.h.193.4	yes	4	4.3	odd	2
768.6.a.m.1.1		2	16.5	even	4
768.6.a.q.1.2		2	16.3	odd	4
768.6.a.r.1.1		2	16.11	odd	4
768.6.a.v.1.2		2	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:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	384.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-9.00000i q^{3} +51.2410i q^{5} -164.205 q^{7} -81.0000 q^{9} +O(q^{10})\)	\(q-9.00000i q^{3} +51.2410i q^{5} -164.205 q^{7} -81.0000 q^{9} +296.964i q^{11} +329.374i q^{13} +461.169 q^{15} +2086.53 q^{17} +1142.75i q^{19} +1477.84i q^{21} -2479.30 q^{23} +499.360 q^{25} +729.000i q^{27} -6292.18i q^{29} -6485.39 q^{31} +2672.68 q^{33} -8414.03i q^{35} -6679.52i q^{37} +2964.37 q^{39} -11613.3 q^{41} +14026.4i q^{43} -4150.52i q^{45} +7021.82 q^{47} +10156.3 q^{49} -18778.8i q^{51} -546.953i q^{53} -15216.7 q^{55} +10284.7 q^{57} -37901.1i q^{59} +6773.95i q^{61} +13300.6 q^{63} -16877.5 q^{65} -49749.0i q^{67} +22313.7i q^{69} -22077.0 q^{71} +61034.9 q^{73} -4494.24i q^{75} -48763.0i q^{77} -73710.0 q^{79} +6561.00 q^{81} +34602.9i q^{83} +106916. i q^{85} -56629.6 q^{87} -42428.8 q^{89} -54084.9i q^{91} +58368.5i q^{93} -58555.5 q^{95} -138179. q^{97} -24054.1i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 32 q^{7} - 324 q^{9}+O(q^{10}) \) 4 * q - 32 * q^7 - 324 * q^9	\( 4 q - 32 q^{7} - 324 q^{9} + 720 q^{15} + 1848 q^{17} - 7168 q^{23} + 6996 q^{25} - 7072 q^{31} + 6192 q^{33} - 3888 q^{39} + 5032 q^{41} - 9152 q^{47} + 30628 q^{49} - 29376 q^{55} + 9648 q^{57} + 2592 q^{63} - 46016 q^{65} - 151040 q^{71} + 34200 q^{73} - 200992 q^{79} + 26244 q^{81} - 155664 q^{87} - 180712 q^{89} - 130752 q^{95} - 199816 q^{97}+O(q^{100}) \) 4 * q - 32 * q^7 - 324 * q^9 + 720 * q^15 + 1848 * q^17 - 7168 * q^23 + 6996 * q^25 - 7072 * q^31 + 6192 * q^33 - 3888 * q^39 + 5032 * q^41 - 9152 * q^47 + 30628 * q^49 - 29376 * q^55 + 9648 * q^57 + 2592 * q^63 - 46016 * q^65 - 151040 * q^71 + 34200 * q^73 - 200992 * q^79 + 26244 * q^81 - 155664 * q^87 - 180712 * q^89 - 130752 * q^95 - 199816 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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