LMFDB - Embedded newform 320.6.d.d.161.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [320,6,Mod(161,320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(320, 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("320.161");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 320 = 2^{6} \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	320.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:	$51.3228223402$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 343 x^{10} - 696 x^{9} + 44406 x^{8} + 179640 x^{7} - 2401691 x^{6} - 15554592 x^{5} + \cdots + 653157349 $ x^12 - 343x^10 - 696x^9 + 44406x^8 + 179640x^7 - 2401691x^6 - 15554592x^5 + 26901210x^4 + 434775816x^3 + 1271335685x^2 + 1475231592x + 653157349
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{30} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.3
Root			$10.2435 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	320.161
Dual form			320.6.d.d.161.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-16.5588i q^{3} -25.0000i q^{5} +55.4690 q^{7} -31.1927 q^{9} +O(q^{10})$	$q-16.5588i q^{3} -25.0000i q^{5} +55.4690 q^{7} -31.1927 q^{9} +153.628i q^{11} -57.3312i q^{13} -413.969 q^{15} +1783.25 q^{17} -1595.98i q^{19} -918.498i q^{21} +4115.40 q^{23} -625.000 q^{25} -3507.27i q^{27} +5003.52i q^{29} +1823.88 q^{31} +2543.90 q^{33} -1386.72i q^{35} -1800.07i q^{37} -949.334 q^{39} +5455.32 q^{41} -8258.91i q^{43} +779.817i q^{45} +8409.69 q^{47} -13730.2 q^{49} -29528.5i q^{51} +6743.29i q^{53} +3840.71 q^{55} -26427.5 q^{57} -17.5455i q^{59} -18829.3i q^{61} -1730.23 q^{63} -1433.28 q^{65} +5972.99i q^{67} -68145.9i q^{69} -16857.0 q^{71} -56634.5 q^{73} +10349.2i q^{75} +8521.61i q^{77} +37295.3 q^{79} -65655.8 q^{81} -57832.9i q^{83} -44581.3i q^{85} +82852.2 q^{87} -93163.3 q^{89} -3180.10i q^{91} -30201.2i q^{93} -39899.6 q^{95} -91299.9 q^{97} -4792.08i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 268 q^{7} - 428 q^{9}+O(q^{10}) $ 12 * q + 268 * q^7 - 428 * q^9	$ 12 q + 268 q^{7} - 428 q^{9} - 900 q^{15} + 2400 q^{17} + 8108 q^{23} - 7500 q^{25} + 7976 q^{31} - 20776 q^{33} + 40984 q^{39} - 56408 q^{41} + 21172 q^{47} - 5540 q^{49} + 18400 q^{55} + 39992 q^{57} + 179516 q^{63} + 44000 q^{65} + 367704 q^{71} + 58736 q^{73} + 26192 q^{79} + 411692 q^{81} + 183200 q^{87} + 87672 q^{89} + 121000 q^{95} - 172336 q^{97}+O(q^{100}) $ 12 * q + 268 * q^7 - 428 * q^9 - 900 * q^15 + 2400 * q^17 + 8108 * q^23 - 7500 * q^25 + 7976 * q^31 - 20776 * q^33 + 40984 * q^39 - 56408 * q^41 + 21172 * q^47 - 5540 * q^49 + 18400 * q^55 + 39992 * q^57 + 179516 * q^63 + 44000 * q^65 + 367704 * q^71 + 58736 * q^73 + 26192 * q^79 + 411692 * q^81 + 183200 * q^87 + 87672 * q^89 + 121000 * q^95 - 172336 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/320\mathbb{Z}\right)^\times$.

$n$	$191$	$257$	$261$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	− 16.5588i	− 1.06225i	−0.847295	−	0.531123i	$-0.821770\pi$
$3$	− 16.5588i	− 1.06225i	0.847295	−	0.531123i	$-0.178230\pi$
$4$	0	0
$5$	− 25.0000i	− 0.447214i
$5$	− 25.0000i	− 0.447214i
$6$	0	0
$7$	55.4690	0.427863	0.213932	−	0.976849i	$-0.431373\pi$
$7$	55.4690	0.427863	0.213932	+	0.976849i	$0.431373\pi$
$8$	0	0
$9$	−31.1927	−0.128365
$10$	0	0
$11$	153.628i	0.382816i	0.981511	+	0.191408i	$0.0613053\pi$
$11$	153.628i	0.382816i	−0.981511	+	0.191408i	$0.938695\pi$
$12$	0	0
$13$	− 57.3312i	− 0.0940877i	−0.998893	−	0.0470438i	$-0.985020\pi$
$13$	− 57.3312i	− 0.0940877i	0.998893	−	0.0470438i	$-0.0149800\pi$
$14$	0	0
$15$	−413.969	−0.475051
$16$	0	0
$17$	1783.25	1.49655	0.748274	−	0.663389i	$-0.230883\pi$
$17$	1783.25	1.49655	0.748274	+	0.663389i	$0.230883\pi$
$18$	0	0
$19$	− 1595.98i	− 1.01425i	−0.861873	−	0.507124i	$-0.830709\pi$
$19$	− 1595.98i	− 1.01425i	0.861873	−	0.507124i	$-0.169291\pi$
$20$	0	0
$21$	− 918.498i	− 0.454496i
$22$	0	0
$23$	4115.40	1.62215	0.811077	−	0.584939i	$-0.198882\pi$
$23$	4115.40	1.62215	0.811077	+	0.584939i	$0.198882\pi$
$24$	0	0
$25$	−625.000	−0.200000
$26$	0	0
$27$	− 3507.27i	− 0.925890i
$28$	0	0
$29$	5003.52i	1.10479i	0.833581	+	0.552397i	$0.186287\pi$
$29$	5003.52i	1.10479i	−0.833581	+	0.552397i	$0.813713\pi$
$30$	0	0
$31$	1823.88	0.340873	0.170436	−	0.985369i	$-0.445482\pi$
$31$	1823.88	0.340873	0.170436	+	0.985369i	$0.445482\pi$
$32$	0	0
$33$	2543.90	0.406644
$34$	0	0
$35$	− 1386.72i	− 0.191346i
$36$	0	0
$37$	− 1800.07i	− 0.216165i	−0.994142	−	0.108083i	$-0.965529\pi$
$37$	− 1800.07i	− 0.216165i	0.994142	−	0.108083i	$-0.0344711\pi$
$38$	0	0
$39$	−949.334	−0.0999442
$40$	0	0
$41$	5455.32	0.506828	0.253414	−	0.967358i	$-0.418447\pi$
$41$	5455.32	0.506828	0.253414	+	0.967358i	$0.418447\pi$
$42$	0	0
$43$	− 8258.91i	− 0.681164i	−0.940215	−	0.340582i	$-0.889376\pi$
$43$	− 8258.91i	− 0.681164i	0.940215	−	0.340582i	$-0.110624\pi$
$44$	0	0
$45$	779.817i	0.0574066i
$46$	0	0
$47$	8409.69	0.555310	0.277655	−	0.960681i	$-0.410443\pi$
$47$	8409.69	0.555310	0.277655	+	0.960681i	$0.410443\pi$
$48$	0	0
$49$	−13730.2	−0.816933
$50$	0	0
$51$	− 29528.5i	− 1.58970i
$52$	0	0
$53$	6743.29i	0.329748i	0.986315	+	0.164874i	$0.0527217\pi$
$53$	6743.29i	0.329748i	−0.986315	+	0.164874i	$0.947278\pi$
$54$	0	0
$55$	3840.71	0.171200
$56$	0	0
$57$	−26427.5	−1.07738
$58$	0	0
$59$	− 17.5455i	0 0.000656201i	−1.00000		0.000328100i	$-0.999896\pi$
$59$	− 17.5455i	0 0.000656201i	1.00000		0.000328100i	$-0.000104438\pi$
$60$	0	0
$61$	− 18829.3i	− 0.647901i	−0.946074	−	0.323951i	$-0.894989\pi$
$61$	− 18829.3i	− 0.647901i	0.946074	−	0.323951i	$-0.105011\pi$
$62$	0	0
$63$	−1730.23	−0.0549227
$64$	0	0
$65$	−1433.28	−0.0420773
$66$	0	0
$67$	5972.99i	0.162557i	0.996691	+	0.0812783i	$0.0259003\pi$
$67$	5972.99i	0.162557i	−0.996691	+	0.0812783i	$0.974100\pi$
$68$	0	0
$69$	− 68145.9i	− 1.72313i
$70$	0	0
$71$	−16857.0	−0.396857	−0.198428	−	0.980115i	$-0.563584\pi$
$71$	−16857.0	−0.396857	−0.198428	+	0.980115i	$0.563584\pi$
$72$	0	0
$73$	−56634.5	−1.24387	−0.621933	−	0.783070i	$-0.713653\pi$
$73$	−56634.5	−1.24387	−0.621933	+	0.783070i	$0.713653\pi$
$74$	0	0
$75$	10349.2i	0.212449i
$76$	0	0
$77$	8521.61i	0.163793i
$78$	0	0
$79$	37295.3	0.672336	0.336168	−	0.941802i	$-0.390869\pi$
$79$	37295.3	0.672336	0.336168	+	0.941802i	$0.390869\pi$
$80$	0	0
$81$	−65655.8	−1.11189
$82$	0	0
$83$	− 57832.9i	− 0.921466i	−0.887539	−	0.460733i	$-0.847587\pi$
$83$	− 57832.9i	− 0.921466i	0.887539	−	0.460733i	$-0.152413\pi$
$84$	0	0
$85$	− 44581.3i	− 0.669277i
$86$	0	0
$87$	82852.2	1.17356
$88$	0	0
$89$	−93163.3	−1.24672	−0.623361	−	0.781934i	$-0.714233\pi$
$89$	−93163.3	−1.24672	−0.623361	+	0.781934i	$0.714233\pi$
$90$	0	0
$91$	− 3180.10i	− 0.0402567i
$92$	0	0
$93$	− 30201.2i	− 0.362090i
$94$	0	0
$95$	−39899.6	−0.453585
$96$	0	0
$97$	−91299.9	−0.985238	−0.492619	−	0.870245i	$-0.663960\pi$
$97$	−91299.9	−0.985238	−0.492619	+	0.870245i	$0.663960\pi$
$98$	0	0
$99$	− 4792.08i	− 0.0491402i
$100$	0	0
$101$	− 167055.i	− 1.62951i	−0.579807	−	0.814754i	$-0.696872\pi$
$101$	− 167055.i	− 1.62951i	0.579807	−	0.814754i	$-0.303128\pi$
$102$	0	0
$103$	66663.7	0.619151	0.309575	−	0.950875i	$-0.399813\pi$
$103$	66663.7	0.619151	0.309575	+	0.950875i	$0.399813\pi$
$104$	0	0
$105$	−22962.4	−0.203257
$106$	0	0
$107$	− 102568.i	− 0.866072i	−0.901377	−	0.433036i	$-0.857442\pi$
$107$	− 102568.i	− 0.866072i	0.901377	−	0.433036i	$-0.142558\pi$
$108$	0	0
$109$	151278.i	1.21958i	0.792563	+	0.609790i	$0.208746\pi$
$109$	151278.i	1.21958i	−0.792563	+	0.609790i	$0.791254\pi$
$110$	0	0
$111$	−29807.0	−0.229621
$112$	0	0
$113$	−27795.5	−0.204776	−0.102388	−	0.994745i	$-0.532648\pi$
$113$	−27795.5	−0.204776	−0.102388	+	0.994745i	$0.532648\pi$
$114$	0	0
$115$	− 102885.i	− 0.725449i
$116$	0	0
$117$	1788.31i	0.0120776i
$118$	0	0
$119$	98915.3	0.640318
$120$	0	0
$121$	137449.	0.853452
$122$	0	0
$123$	− 90333.4i	− 0.538376i
$124$	0	0
$125$	15625.0i	0.0894427i
$126$	0	0
$127$	134073.	0.737620	0.368810	−	0.929505i	$-0.379765\pi$
$127$	134073.	0.737620	0.368810	+	0.929505i	$0.379765\pi$
$128$	0	0
$129$	−136757.	−0.723563
$130$	0	0
$131$	− 372871.i	− 1.89837i	−0.314723	−	0.949184i	$-0.601912\pi$
$131$	− 372871.i	− 1.89837i	0.314723	−	0.949184i	$-0.398088\pi$
$132$	0	0
$133$	− 88527.5i	− 0.433959i
$134$	0	0
$135$	−87681.7	−0.414071
$136$	0	0
$137$	−146243.	−0.665690	−0.332845	−	0.942981i	$-0.608009\pi$
$137$	−146243.	−0.665690	−0.332845	+	0.942981i	$0.608009\pi$
$138$	0	0
$139$	− 226750.i	− 0.995428i	−0.867341	−	0.497714i	$-0.834173\pi$
$139$	− 226750.i	− 0.995428i	0.867341	−	0.497714i	$-0.165827\pi$
$140$	0	0
$141$	− 139254.i	− 0.589875i
$142$	0	0
$143$	8807.71	0.0360183
$144$	0	0
$145$	125088.	0.494079
$146$	0	0
$147$	227355.i	0.867783i
$148$	0	0
$149$	4820.91i	0.0177895i	0.999960	+	0.00889474i	$0.00283132\pi$
$149$	4820.91i	0.0177895i	−0.999960	+	0.00889474i	$0.997169\pi$
$150$	0	0
$151$	−7260.42	−0.0259131	−0.0129566	−	0.999916i	$-0.504124\pi$
$151$	−7260.42	−0.0259131	−0.0129566	+	0.999916i	$0.504124\pi$
$152$	0	0
$153$	−55624.5	−0.192104
$154$	0	0
$155$	− 45597.0i	− 0.152443i
$156$	0	0
$157$	420038.i	1.36000i	0.733211	+	0.680001i	$0.238021\pi$
$157$	420038.i	1.36000i	−0.733211	+	0.680001i	$0.761979\pi$
$158$	0	0
$159$	111661.	0.350273
$160$	0	0
$161$	228277.	0.694060
$162$	0	0
$163$	34691.5i	0.102271i	0.998692	+	0.0511356i	$0.0162841\pi$
$163$	34691.5i	0.102271i	−0.998692	+	0.0511356i	$0.983716\pi$
$164$	0	0
$165$	− 63597.4i	− 0.181857i
$166$	0	0
$167$	519139.	1.44043	0.720216	−	0.693750i	$-0.244043\pi$
$167$	519139.	1.44043	0.720216	+	0.693750i	$0.244043\pi$
$168$	0	0
$169$	368006.	0.991148
$170$	0	0
$171$	49783.0i	0.130194i
$172$	0	0
$173$	579939.i	1.47322i	0.676319	+	0.736609i	$0.263574\pi$
$173$	579939.i	1.47322i	−0.676319	+	0.736609i	$0.736426\pi$
$174$	0	0
$175$	−34668.1	−0.0855727
$176$	0	0
$177$	−290.532	−0.000697046 0
$178$	0	0
$179$	280387.i	0.654072i	0.945012	+	0.327036i	$0.106050\pi$
$179$	280387.i	0.654072i	−0.945012	+	0.327036i	$0.893950\pi$
$180$	0	0
$181$	− 877810.i	− 1.99161i	−0.0915109	−	0.995804i	$-0.529170\pi$
$181$	− 877810.i	− 1.99161i	0.0915109	−	0.995804i	$-0.470830\pi$
$182$	0	0
$183$	−311789.	−0.688230
$184$	0	0
$185$	−45001.9	−0.0966721
$186$	0	0
$187$	273959.i	0.572903i
$188$	0	0
$189$	− 194545.i	− 0.396154i
$190$	0	0
$191$	−998587.	−1.98063	−0.990313	−	0.138852i	$-0.955659\pi$
$191$	−998587.	−1.98063	−0.990313	+	0.138852i	$0.955659\pi$
$192$	0	0
$193$	−483429.	−0.934199	−0.467100	−	0.884205i	$-0.654701\pi$
$193$	−483429.	−0.934199	−0.467100	+	0.884205i	$0.654701\pi$
$194$	0	0
$195$	23733.3i	0.0446964i
$196$	0	0
$197$	219120.i	0.402269i	0.979564	+	0.201135i	$0.0644629\pi$
$197$	219120.i	0.402269i	−0.979564	+	0.201135i	$0.935537\pi$
$198$	0	0
$199$	492419.	0.881460	0.440730	−	0.897640i	$-0.354720\pi$
$199$	492419.	0.881460	0.440730	+	0.897640i	$0.354720\pi$
$200$	0	0
$201$	98905.3	0.172675
$202$	0	0
$203$	277540.i	0.472700i
$204$	0	0
$205$	− 136383.i	− 0.226660i
$206$	0	0
$207$	−128370.	−0.208228
$208$	0	0
$209$	245188.	0.388270
$210$	0	0
$211$	940463.i	1.45424i	0.686511	+	0.727119i	$0.259141\pi$
$211$	940463.i	1.45424i	−0.686511	+	0.727119i	$0.740859\pi$
$212$	0	0
$213$	279131.i	0.421559i
$214$	0	0
$215$	−206473.	−0.304626
$216$	0	0
$217$	101169.	0.145847
$218$	0	0
$219$	937797.i	1.32129i
$220$	0	0
$221$	− 102236.i	− 0.140807i
$222$	0	0
$223$	−1.11376e6	−1.49979	−0.749895	−	0.661557i	$-0.769896\pi$
$223$	−1.11376e6	−1.49979	−0.749895	+	0.661557i	$0.769896\pi$
$224$	0	0
$225$	19495.4	0.0256730
$226$	0	0
$227$	605351.i	0.779728i	0.920873	+	0.389864i	$0.127478\pi$
$227$	605351.i	0.779728i	−0.920873	+	0.389864i	$0.872522\pi$
$228$	0	0
$229$	1.24526e6i	1.56917i	0.620020	+	0.784586i	$0.287125\pi$
$229$	1.24526e6i	1.56917i	−0.620020	+	0.784586i	$0.712875\pi$
$230$	0	0
$231$	141107.	0.173988
$232$	0	0
$233$	457782.	0.552420	0.276210	−	0.961097i	$-0.410921\pi$
$233$	457782.	0.552420	0.276210	+	0.961097i	$0.410921\pi$
$234$	0	0
$235$	− 210242.i	− 0.248342i
$236$	0	0
$237$	− 617565.i	− 0.714186i
$238$	0	0
$239$	−984849.	−1.11526	−0.557629	−	0.830091i	$-0.688289\pi$
$239$	−984849.	−1.11526	−0.557629	+	0.830091i	$0.688289\pi$
$240$	0	0
$241$	1.38553e6	1.53665	0.768324	−	0.640061i	$-0.221091\pi$
$241$	1.38553e6	1.53665	0.768324	+	0.640061i	$0.221091\pi$
$242$	0	0
$243$	234914.i	0.255207i
$244$	0	0
$245$	343255.i	0.365344i
$246$	0	0
$247$	−91499.6	−0.0954282
$248$	0	0
$249$	−957641.	−0.978823
$250$	0	0
$251$	− 534745.i	− 0.535750i	−0.963454	−	0.267875i	$-0.913679\pi$
$251$	− 534745.i	− 0.535750i	0.963454	−	0.267875i	$-0.0863214\pi$
$252$	0	0
$253$	632242.i	0.620986i
$254$	0	0
$255$	−738212.	−0.710936
$256$	0	0
$257$	855941.	0.808372	0.404186	−	0.914677i	$-0.367555\pi$
$257$	855941.	0.808372	0.404186	+	0.914677i	$0.367555\pi$
$258$	0	0
$259$	− 99848.3i	− 0.0924892i
$260$	0	0
$261$	− 156073.i	− 0.141817i
$262$	0	0
$263$	666515.	0.594183	0.297092	−	0.954849i	$-0.403983\pi$
$263$	666515.	0.594183	0.297092	+	0.954849i	$0.403983\pi$
$264$	0	0
$265$	168582.	0.147468
$266$	0	0
$267$	1.54267e6i	1.32432i
$268$	0	0
$269$	− 189848.i	− 0.159965i	−0.996796	−	0.0799826i	$-0.974514\pi$
$269$	− 189848.i	− 0.159965i	0.996796	−	0.0799826i	$-0.0254865\pi$
$270$	0	0
$271$	1.45698e6	1.20512	0.602559	−	0.798075i	$-0.294148\pi$
$271$	1.45698e6	1.20512	0.602559	+	0.798075i	$0.294148\pi$
$272$	0	0
$273$	−52658.6	−0.0427624
$274$	0	0
$275$	− 96017.8i	− 0.0765632i
$276$	0	0
$277$	1.08051e6i	0.846115i	0.906103	+	0.423057i	$0.139043\pi$
$277$	1.08051e6i	0.846115i	−0.906103	+	0.423057i	$0.860957\pi$
$278$	0	0
$279$	−56891.7	−0.0437561
$280$	0	0
$281$	1.48051e6	1.11852	0.559261	−	0.828991i	$-0.311085\pi$
$281$	1.48051e6	1.11852	0.559261	+	0.828991i	$0.311085\pi$
$282$	0	0
$283$	− 479977.i	− 0.356250i	−0.984008	−	0.178125i	$-0.942997\pi$
$283$	− 479977.i	− 0.356250i	0.984008	−	0.178125i	$-0.0570031\pi$
$284$	0	0
$285$	660687.i	0.481819i
$286$	0	0
$287$	302601.	0.216853
$288$	0	0
$289$	1.76014e6	1.23966
$290$	0	0
$291$	1.51181e6i	1.04656i
$292$	0	0
$293$	2.07194e6i	1.40996i	0.709226	+	0.704981i	$0.249044\pi$
$293$	2.07194e6i	1.40996i	−0.709226	+	0.704981i	$0.750956\pi$
$294$	0	0
$295$	−438.639	−0.000293462 0
$296$	0	0
$297$	538816.	0.354445
$298$	0	0
$299$	− 235941.i	− 0.152625i
$300$	0	0
$301$	− 458113.i	− 0.291445i
$302$	0	0
$303$	−2.76623e6	−1.73094
$304$	0	0
$305$	−470732.	−0.289750
$306$	0	0
$307$	− 2.04780e6i	− 1.24006i	−0.784579	−	0.620029i	$-0.787121\pi$
$307$	− 2.04780e6i	− 1.24006i	0.784579	−	0.620029i	$-0.212879\pi$
$308$	0	0
$309$	− 1.10387e6i	− 0.657690i
$310$	0	0
$311$	2.77842e6	1.62891	0.814455	−	0.580227i	$-0.197036\pi$
$311$	2.77842e6	1.62891	0.814455	+	0.580227i	$0.197036\pi$
$312$	0	0
$313$	1.78088e6	1.02748	0.513741	−	0.857945i	$-0.328259\pi$
$313$	1.78088e6	1.02748	0.513741	+	0.857945i	$0.328259\pi$
$314$	0	0
$315$	43255.7i	0.0245622i
$316$	0	0
$317$	− 1.67386e6i	− 0.935560i	−0.883845	−	0.467780i	$-0.845054\pi$
$317$	− 1.67386e6i	− 0.935560i	0.883845	−	0.467780i	$-0.154946\pi$
$318$	0	0
$319$	−768684.	−0.422932
$320$	0	0
$321$	−1.69841e6	−0.919981
$322$	0	0
$323$	− 2.84604e6i	− 1.51787i
$324$	0	0
$325$	35832.0i	0.0188175i
$326$	0	0
$327$	2.50498e6	1.29549
$328$	0	0
$329$	466477.	0.237597
$330$	0	0
$331$	1.33416e6i	0.669325i	0.942338	+	0.334663i	$0.108622\pi$
$331$	1.33416e6i	0.669325i	−0.942338	+	0.334663i	$0.891378\pi$
$332$	0	0
$333$	56149.1i	0.0277481i
$334$	0	0
$335$	149325.	0.0726975
$336$	0	0
$337$	1.18056e6	0.566257	0.283128	−	0.959082i	$-0.408628\pi$
$337$	1.18056e6	0.566257	0.283128	+	0.959082i	$0.408628\pi$
$338$	0	0
$339$	460260.i	0.217522i
$340$	0	0
$341$	280200.i	0.130491i
$342$	0	0
$343$	−1.69387e6	−0.777399
$344$	0	0
$345$	−1.70365e6	−0.770605
$346$	0	0
$347$	− 669989.i	− 0.298706i	−0.988784	−	0.149353i	$-0.952281\pi$
$347$	− 669989.i	− 0.298706i	0.988784	−	0.149353i	$-0.0477191\pi$
$348$	0	0
$349$	− 1.64640e6i	− 0.723554i	−0.932265	−	0.361777i	$-0.882170\pi$
$349$	− 1.64640e6i	− 0.723554i	0.932265	−	0.361777i	$-0.117830\pi$
$350$	0	0
$351$	−201076.	−0.0871148
$352$	0	0
$353$	−4.33852e6	−1.85312	−0.926562	−	0.376143i	$-0.877250\pi$
$353$	−4.33852e6	−1.85312	−0.926562	+	0.376143i	$0.877250\pi$
$354$	0	0
$355$	421425.i	0.177480i
$356$	0	0
$357$	− 1.63791e6i	− 0.680175i
$358$	0	0
$359$	3.33513e6	1.36577	0.682883	−	0.730527i	$-0.260726\pi$
$359$	3.33513e6	1.36577	0.682883	+	0.730527i	$0.260726\pi$
$360$	0	0
$361$	−71060.5	−0.0286986
$362$	0	0
$363$	− 2.27599e6i	− 0.906575i
$364$	0	0
$365$	1.41586e6i	0.556274i
$366$	0	0
$367$	−4.53771e6	−1.75862	−0.879309	−	0.476252i	$-0.841995\pi$
$367$	−4.53771e6	−1.75862	−0.879309	+	0.476252i	$0.841995\pi$
$368$	0	0
$369$	−170166.	−0.0650590
$370$	0	0
$371$	374043.i	0.141087i
$372$	0	0
$373$	4.47262e6i	1.66452i	0.554383	+	0.832262i	$0.312954\pi$
$373$	4.47262e6i	1.66452i	−0.554383	+	0.832262i	$0.687046\pi$
$374$	0	0
$375$	258731.	0.0950101
$376$	0	0
$377$	286858.	0.103947
$378$	0	0
$379$	189146.i	0.0676391i	0.999428	+	0.0338196i	$0.0107672\pi$
$379$	189146.i	0.0676391i	−0.999428	+	0.0338196i	$0.989233\pi$
$380$	0	0
$381$	− 2.22009e6i	− 0.783534i
$382$	0	0
$383$	−3.16389e6	−1.10211	−0.551054	−	0.834469i	$-0.685774\pi$
$383$	−3.16389e6	−1.10211	−0.551054	+	0.834469i	$0.685774\pi$
$384$	0	0
$385$	213040.	0.0732504
$386$	0	0
$387$	257618.i	0.0874376i
$388$	0	0
$389$	− 2.39411e6i	− 0.802178i	−0.916039	−	0.401089i	$-0.868632\pi$
$389$	− 2.39411e6i	− 0.802178i	0.916039	−	0.401089i	$-0.131368\pi$
$390$	0	0
$391$	7.33880e6	2.42763
$392$	0	0
$393$	−6.17428e6	−2.01653
$394$	0	0
$395$	− 932383.i	− 0.300678i
$396$	0	0
$397$	− 5.57288e6i	− 1.77461i	−0.461182	−	0.887306i	$-0.652574\pi$
$397$	− 5.57288e6i	− 1.77461i	0.461182	−	0.887306i	$-0.347426\pi$
$398$	0	0
$399$	−1.46591e6	−0.460971
$400$	0	0
$401$	−1.95214e6	−0.606248	−0.303124	−	0.952951i	$-0.598030\pi$
$401$	−1.95214e6	−0.606248	−0.303124	+	0.952951i	$0.598030\pi$
$402$	0	0
$403$	− 104565.i	− 0.0320719i
$404$	0	0
$405$	1.64140e6i	0.497251i
$406$	0	0
$407$	276543.	0.0827515
$408$	0	0
$409$	−5.80819e6	−1.71685	−0.858425	−	0.512939i	$-0.828557\pi$
$409$	−5.80819e6	−1.71685	−0.858425	+	0.512939i	$0.828557\pi$
$410$	0	0
$411$	2.42160e6i	0.707127i
$412$	0	0
$413$	− 973.233i	0 0.000280764i
$414$	0	0
$415$	−1.44582e6	−0.412092
$416$	0	0
$417$	−3.75470e6	−1.05739
$418$	0	0
$419$	− 3.04430e6i	− 0.847133i	−0.905865	−	0.423567i	$-0.860778\pi$
$419$	− 3.04430e6i	− 0.847133i	0.905865	−	0.423567i	$-0.139222\pi$
$420$	0	0
$421$	− 1.07953e6i	− 0.296846i	−0.988924	−	0.148423i	$-0.952580\pi$
$421$	− 1.07953e6i	− 0.296846i	0.988924	−	0.148423i	$-0.0474197\pi$
$422$	0	0
$423$	−262321.	−0.0712823
$424$	0	0
$425$	−1.11453e6	−0.299310
$426$	0	0
$427$	− 1.04444e6i	− 0.277213i
$428$	0	0
$429$	− 145845.i	− 0.0382602i
$430$	0	0
$431$	6.45588e6	1.67403	0.837013	−	0.547183i	$-0.184300\pi$
$431$	6.45588e6	1.67403	0.837013	+	0.547183i	$0.184300\pi$
$432$	0	0
$433$	−1.74868e6	−0.448221	−0.224110	−	0.974564i	$-0.571948\pi$
$433$	−1.74868e6	−0.448221	−0.224110	+	0.974564i	$0.571948\pi$
$434$	0	0
$435$	− 2.07130e6i	− 0.524833i
$436$	0	0
$437$	− 6.56810e6i	− 1.64527i
$438$	0	0
$439$	−1.58712e6	−0.393049	−0.196525	−	0.980499i	$-0.562966\pi$
$439$	−1.58712e6	−0.393049	−0.196525	+	0.980499i	$0.562966\pi$
$440$	0	0
$441$	428282.	0.104866
$442$	0	0
$443$	− 2.17901e6i	− 0.527534i	−0.964586	−	0.263767i	$-0.915035\pi$
$443$	− 2.17901e6i	− 0.527534i	0.964586	−	0.263767i	$-0.0849650\pi$
$444$	0	0
$445$	2.32908e6i	0.557551i
$446$	0	0
$447$	79828.3	0.0188968
$448$	0	0
$449$	3.99854e6	0.936021	0.468010	−	0.883723i	$-0.344971\pi$
$449$	3.99854e6	0.936021	0.468010	+	0.883723i	$0.344971\pi$
$450$	0	0
$451$	838092.i	0.194022i
$452$	0	0
$453$	120224.i	0.0275261i
$454$	0	0
$455$	−79502.6	−0.0180033
$456$	0	0
$457$	5.36373e6	1.20137	0.600685	−	0.799486i	$-0.294895\pi$
$457$	5.36373e6	1.20137	0.600685	+	0.799486i	$0.294895\pi$
$458$	0	0
$459$	− 6.25435e6i	− 1.38564i
$460$	0	0
$461$	− 2.44852e6i	− 0.536601i	−0.963335	−	0.268300i	$-0.913538\pi$
$461$	− 2.44852e6i	− 0.536601i	0.963335	−	0.268300i	$-0.0864620\pi$
$462$	0	0
$463$	−2.81552e6	−0.610389	−0.305195	−	0.952290i	$-0.598722\pi$
$463$	−2.81552e6	−0.610389	−0.305195	+	0.952290i	$0.598722\pi$
$464$	0	0
$465$	−755030.	−0.161932
$466$	0	0
$467$	− 1.24614e6i	− 0.264408i	−0.991223	−	0.132204i	$-0.957795\pi$
$467$	− 1.24614e6i	− 0.264408i	0.991223	−	0.132204i	$-0.0422053\pi$
$468$	0	0
$469$	331316.i	0.0695520i
$470$	0	0
$471$	6.95531e6	1.44466
$472$	0	0
$473$	1.26880e6	0.260760
$474$	0	0
$475$	997489.i	0.202850i
$476$	0	0
$477$	− 210341.i	− 0.0423281i
$478$	0	0
$479$	−2.01494e6	−0.401258	−0.200629	−	0.979667i	$-0.564299\pi$
$479$	−2.01494e6	−0.401258	−0.200629	+	0.979667i	$0.564299\pi$
$480$	0	0
$481$	−103200.	−0.0203385
$482$	0	0
$483$	− 3.77998e6i	− 0.737262i
$484$	0	0
$485$	2.28250e6i	0.440612i
$486$	0	0
$487$	8.40493e6	1.60587	0.802937	−	0.596064i	$-0.203270\pi$
$487$	8.40493e6	1.60587	0.802937	+	0.596064i	$0.203270\pi$
$488$	0	0
$489$	574448.	0.108637
$490$	0	0
$491$	4.54738e6i	0.851250i	0.904900	+	0.425625i	$0.139946\pi$
$491$	4.54738e6i	0.851250i	−0.904900	+	0.425625i	$0.860054\pi$
$492$	0	0
$493$	8.92255e6i	1.65338i
$494$	0	0
$495$	−119802.	−0.0219761
$496$	0	0
$497$	−935040.	−0.169801
$498$	0	0
$499$	3.00176e6i	0.539665i	0.962907	+	0.269833i	$0.0869684\pi$
$499$	3.00176e6i	0.539665i	−0.962907	+	0.269833i	$0.913032\pi$
$500$	0	0
$501$	− 8.59630e6i	− 1.53009i
$502$	0	0
$503$	−6.17709e6	−1.08859	−0.544294	−	0.838894i	$-0.683202\pi$
$503$	−6.17709e6	−1.08859	−0.544294	+	0.838894i	$0.683202\pi$
$504$	0	0
$505$	−4.17638e6	−0.728738
$506$	0	0
$507$	− 6.09373e6i	− 1.05284i
$508$	0	0
$509$	− 628764.i	− 0.107571i	−0.998553	−	0.0537853i	$-0.982871\pi$
$509$	− 628764.i	− 0.107571i	0.998553	−	0.0537853i	$-0.0171287\pi$
$510$	0	0
$511$	−3.14146e6	−0.532205
$512$	0	0
$513$	−5.59754e6	−0.939082
$514$	0	0
$515$	− 1.66659e6i	− 0.276893i
$516$	0	0
$517$	1.29197e6i	0.212581i
$518$	0	0
$519$	9.60307e6	1.56492
$520$	0	0
$521$	3.46406e6	0.559102	0.279551	−	0.960131i	$-0.409814\pi$
$521$	3.46406e6	0.559102	0.279551	+	0.960131i	$0.409814\pi$
$522$	0	0
$523$	− 7.86718e6i	− 1.25766i	−0.777541	−	0.628832i	$-0.783533\pi$
$523$	− 7.86718e6i	− 1.25766i	0.777541	−	0.628832i	$-0.216467\pi$
$524$	0	0
$525$	574061.i	0.0908992i
$526$	0	0
$527$	3.25244e6	0.510133
$528$	0	0
$529$	1.05001e7	1.63138
$530$	0	0
$531$	547.293i	0 8.42332e-5i
$532$	0	0
$533$	− 312760.i	− 0.0476863i
$534$	0	0
$535$	−2.56421e6	−0.387319
$536$	0	0
$537$	4.64286e6	0.694785
$538$	0	0
$539$	− 2.10935e6i	− 0.312735i
$540$	0	0
$541$	5.36006e6i	0.787366i	0.919246	+	0.393683i	$0.128799\pi$
$541$	5.36006e6i	0.787366i	−0.919246	+	0.393683i	$0.871201\pi$
$542$	0	0
$543$	−1.45354e7	−2.11558
$544$	0	0
$545$	3.78196e6	0.545413
$546$	0	0
$547$	9.71604e6i	1.38842i	0.719773	+	0.694210i	$0.244246\pi$
$547$	9.71604e6i	1.38842i	−0.719773	+	0.694210i	$0.755754\pi$
$548$	0	0
$549$	587335.i	0.0831678i
$550$	0	0
$551$	7.98554e6	1.12053
$552$	0	0
$553$	2.06873e6	0.287668
$554$	0	0
$555$	745175.i	0.102689i
$556$	0	0
$557$	6.34509e6i	0.866562i	0.901259	+	0.433281i	$0.142644\pi$
$557$	6.34509e6i	0.866562i	−0.901259	+	0.433281i	$0.857356\pi$
$558$	0	0
$559$	−473493.	−0.0640891
$560$	0	0
$561$	4.53642e6	0.608563
$562$	0	0
$563$	− 2.79024e6i	− 0.370997i	−0.982645	−	0.185498i	$-0.940610\pi$
$563$	− 2.79024e6i	− 0.370997i	0.982645	−	0.185498i	$-0.0593899\pi$
$564$	0	0
$565$	694889.i	0.0915786i
$566$	0	0
$567$	−3.64186e6	−0.475736
$568$	0	0
$569$	−1.13522e7	−1.46994	−0.734972	−	0.678098i	$-0.762805\pi$
$569$	−1.13522e7	−1.46994	−0.734972	+	0.678098i	$0.762805\pi$
$570$	0	0
$571$	6.20109e6i	0.795935i	0.917400	+	0.397967i	$0.130284\pi$
$571$	6.20109e6i	0.795935i	−0.917400	+	0.397967i	$0.869716\pi$
$572$	0	0
$573$	1.65354e7i	2.10391i
$574$	0	0
$575$	−2.57212e6	−0.324431
$576$	0	0
$577$	7.88135e6	0.985510	0.492755	−	0.870168i	$-0.335990\pi$
$577$	7.88135e6	0.985510	0.492755	+	0.870168i	$0.335990\pi$
$578$	0	0
$579$	8.00499e6i	0.992349i
$580$	0	0
$581$	− 3.20793e6i	− 0.394262i
$582$	0	0
$583$	−1.03596e6	−0.126233
$584$	0	0
$585$	44707.9	0.00540125
$586$	0	0
$587$	1.32264e7i	1.58433i	0.610306	+	0.792166i	$0.291047\pi$
$587$	1.32264e7i	1.58433i	−0.610306	+	0.792166i	$0.708953\pi$
$588$	0	0
$589$	− 2.91088e6i	− 0.345729i
$590$	0	0
$591$	3.62836e6	0.427309
$592$	0	0
$593$	−1.27301e7	−1.48660	−0.743301	−	0.668957i	$-0.766741\pi$
$593$	−1.27301e7	−1.48660	−0.743301	+	0.668957i	$0.766741\pi$
$594$	0	0
$595$	− 2.47288e6i	− 0.286359i
$596$	0	0
$597$	− 8.15386e6i	− 0.936326i
$598$	0	0
$599$	−4.85153e6	−0.552474	−0.276237	−	0.961090i	$-0.589087\pi$
$599$	−4.85153e6	−0.552474	−0.276237	+	0.961090i	$0.589087\pi$
$600$	0	0
$601$	1.36513e7	1.54166	0.770829	−	0.637042i	$-0.219842\pi$
$601$	1.36513e7	1.54166	0.770829	+	0.637042i	$0.219842\pi$
$602$	0	0
$603$	− 186314.i	− 0.0208666i
$604$	0	0
$605$	− 3.43623e6i	− 0.381675i
$606$	0	0
$607$	−3.82670e6	−0.421554	−0.210777	−	0.977534i	$-0.567599\pi$
$607$	−3.82670e6	−0.421554	−0.210777	+	0.977534i	$0.567599\pi$
$608$	0	0
$609$	4.59573e6	0.502124
$610$	0	0
$611$	− 482138.i	− 0.0522478i
$612$	0	0
$613$	8.97723e6i	0.964920i	0.875918	+	0.482460i	$0.160257\pi$
$613$	8.97723e6i	0.964920i	−0.875918	+	0.482460i	$0.839743\pi$
$614$	0	0
$615$	−2.25833e6	−0.240769
$616$	0	0
$617$	−1.21136e7	−1.28103	−0.640514	−	0.767946i	$-0.721279\pi$
$617$	−1.21136e7	−1.28103	−0.640514	+	0.767946i	$0.721279\pi$
$618$	0	0
$619$	1.72513e7i	1.80965i	0.425779	+	0.904827i	$0.360000\pi$
$619$	1.72513e7i	1.80965i	−0.425779	+	0.904827i	$0.640000\pi$
$620$	0	0
$621$	− 1.44338e7i	− 1.50194i
$622$	0	0
$623$	−5.16767e6	−0.533427
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	− 4.06002e6i	− 0.412438i
$628$	0	0
$629$	− 3.20999e6i	− 0.323502i
$630$	0	0
$631$	−1.60671e7	−1.60644	−0.803220	−	0.595683i	$-0.796881\pi$
$631$	−1.60671e7	−1.60644	−0.803220	+	0.595683i	$0.796881\pi$
$632$	0	0
$633$	1.55729e7	1.54476
$634$	0	0
$635$	− 3.35183e6i	− 0.329874i
$636$	0	0
$637$	787169.i	0.0768633i
$638$	0	0
$639$	525815.	0.0509425
$640$	0	0
$641$	4.51328e6	0.433857	0.216929	−	0.976187i	$-0.430396\pi$
$641$	4.51328e6	0.433857	0.216929	+	0.976187i	$0.430396\pi$
$642$	0	0
$643$	1.72356e7i	1.64399i	0.569495	+	0.821995i	$0.307139\pi$
$643$	1.72356e7i	1.64399i	−0.569495	+	0.821995i	$0.692861\pi$
$644$	0	0
$645$	3.41893e6i	0.323587i
$646$	0	0
$647$	4.25296e6	0.399421	0.199711	−	0.979855i	$-0.436000\pi$
$647$	4.25296e6	0.399421	0.199711	+	0.979855i	$0.436000\pi$
$648$	0	0
$649$	2695.49	0.000251204 0
$650$	0	0
$651$	− 1.67523e6i	− 0.154925i
$652$	0	0
$653$	2.28808e6i	0.209985i	0.994473	+	0.104992i	$0.0334818\pi$
$653$	2.28808e6i	0.209985i	−0.994473	+	0.104992i	$0.966518\pi$
$654$	0	0
$655$	−9.32177e6	−0.848976
$656$	0	0
$657$	1.76658e6	0.159669
$658$	0	0
$659$	6.31745e6i	0.566668i	0.959021	+	0.283334i	$0.0914404\pi$
$659$	6.31745e6i	0.566668i	−0.959021	+	0.283334i	$0.908560\pi$
$660$	0	0
$661$	− 9.62603e6i	− 0.856926i	−0.903559	−	0.428463i	$-0.859055\pi$
$661$	− 9.62603e6i	− 0.856926i	0.903559	−	0.428463i	$-0.140945\pi$
$662$	0	0
$663$	−1.69290e6	−0.149571
$664$	0	0
$665$	−2.21319e6	−0.194073
$666$	0	0
$667$	2.05915e7i	1.79214i
$668$	0	0
$669$	1.84425e7i	1.59314i
$670$	0	0
$671$	2.89271e6	0.248027
$672$	0	0
$673$	−5.75251e6	−0.489575	−0.244788	−	0.969577i	$-0.578718\pi$
$673$	−5.75251e6	−0.489575	−0.244788	+	0.969577i	$0.578718\pi$
$674$	0	0
$675$	2.19204e6i	0.185178i
$676$	0	0
$677$	6.21796e6i	0.521406i	0.965419	+	0.260703i	$0.0839543\pi$
$677$	6.21796e6i	0.521406i	−0.965419	+	0.260703i	$0.916046\pi$
$678$	0	0
$679$	−5.06431e6	−0.421547
$680$	0	0
$681$	1.00239e7	0.828262
$682$	0	0
$683$	1.93186e6i	0.158461i	0.996856	+	0.0792306i	$0.0252463\pi$
$683$	1.93186e6i	0.158461i	−0.996856	+	0.0792306i	$0.974754\pi$
$684$	0	0
$685$	3.65606e6i	0.297706i
$686$	0	0
$687$	2.06199e7	1.66685
$688$	0	0
$689$	386601.	0.0310252
$690$	0	0
$691$	2.04595e7i	1.63005i	0.579429	+	0.815023i	$0.303276\pi$
$691$	2.04595e7i	1.63005i	−0.579429	+	0.815023i	$0.696724\pi$
$692$	0	0
$693$	− 265812.i	− 0.0210253i
$694$	0	0
$695$	−5.66874e6	−0.445169
$696$	0	0
$697$	9.72822e6	0.758493
$698$	0	0
$699$	− 7.58031e6i	− 0.586805i
$700$	0	0
$701$	− 2.02793e7i	− 1.55868i	−0.626599	−	0.779342i	$-0.715553\pi$
$701$	− 2.02793e7i	− 1.55868i	0.626599	−	0.779342i	$-0.284447\pi$
$702$	0	0
$703$	−2.87289e6	−0.219245
$704$	0	0
$705$	−3.48135e6	−0.263800
$706$	0	0
$707$	− 9.26637e6i	− 0.697206i
$708$	0	0
$709$	7.36145e6i	0.549981i	0.961447	+	0.274991i	$0.0886747\pi$
$709$	7.36145e6i	0.549981i	−0.961447	+	0.274991i	$0.911325\pi$
$710$	0	0
$711$	−1.16334e6	−0.0863044
$712$	0	0
$713$	7.50599e6	0.552948
$714$	0	0
$715$	− 220193.i	− 0.0161079i
$716$	0	0
$717$	1.63079e7i	1.18468i
$718$	0	0
$719$	325405.	0.0234748	0.0117374	−	0.999931i	$-0.496264\pi$
$719$	325405.	0.0234748	0.0117374	+	0.999931i	$0.496264\pi$
$720$	0	0
$721$	3.69777e6	0.264912
$722$	0	0
$723$	− 2.29427e7i	− 1.63230i
$724$	0	0
$725$	− 3.12720e6i	− 0.220959i
$726$	0	0
$727$	7.04688e6	0.494494	0.247247	−	0.968953i	$-0.420474\pi$
$727$	7.04688e6	0.494494	0.247247	+	0.968953i	$0.420474\pi$
$728$	0	0
$729$	−1.20645e7	−0.840795
$730$	0	0
$731$	− 1.47277e7i	− 1.01939i
$732$	0	0
$733$	− 705266.i	− 0.0484834i	−0.999706	−	0.0242417i	$-0.992283\pi$
$733$	− 705266.i	− 0.0484834i	0.999706	−	0.0242417i	$-0.00771713\pi$
$734$	0	0
$735$	5.68388e6	0.388084
$736$	0	0
$737$	−917621.	−0.0622293
$738$	0	0
$739$	− 4.61861e6i	− 0.311100i	−0.987828	−	0.155550i	$-0.950285\pi$
$739$	− 4.61861e6i	− 0.311100i	0.987828	−	0.155550i	$-0.0497150\pi$
$740$	0	0
$741$	1.51512e6i	0.101368i
$742$	0	0
$743$	−4.68905e6	−0.311611	−0.155805	−	0.987788i	$-0.549797\pi$
$743$	−4.68905e6	−0.311611	−0.155805	+	0.987788i	$0.549797\pi$
$744$	0	0
$745$	120523.	0.00795570
$746$	0	0
$747$	1.80396e6i	0.118284i
$748$	0	0
$749$	− 5.68936e6i	− 0.370560i
$750$	0	0
$751$	−1.71512e7	−1.10967	−0.554836	−	0.831960i	$-0.687219\pi$
$751$	−1.71512e7	−1.10967	−0.554836	+	0.831960i	$0.687219\pi$
$752$	0	0
$753$	−8.85471e6	−0.569098
$754$	0	0
$755$	181511.i	0.0115887i
$756$	0	0
$757$	1.45909e7i	0.925427i	0.886508	+	0.462713i	$0.153124\pi$
$757$	1.45909e7i	0.925427i	−0.886508	+	0.462713i	$0.846876\pi$
$758$	0	0
$759$	1.04691e7	0.659640
$760$	0	0
$761$	6.25243e6	0.391370	0.195685	−	0.980667i	$-0.437307\pi$
$761$	6.25243e6	0.391370	0.195685	+	0.980667i	$0.437307\pi$
$762$	0	0
$763$	8.39126e6i	0.521814i
$764$	0	0
$765$	1.39061e6i	0.0859117i
$766$	0	0
$767$	−1005.91	−6.17404e−5 0
$768$	0	0
$769$	2.81919e7	1.71913	0.859566	−	0.511025i	$-0.170734\pi$
$769$	2.81919e7	1.71913	0.859566	+	0.511025i	$0.170734\pi$
$770$	0	0
$771$	− 1.41733e7i	− 0.858689i
$772$	0	0
$773$	2.92638e7i	1.76150i	0.473583	+	0.880749i	$0.342960\pi$
$773$	2.92638e7i	1.76150i	−0.473583	+	0.880749i	$0.657040\pi$
$774$	0	0
$775$	−1.13993e6	−0.0681745
$776$	0	0
$777$	−1.65336e6	−0.0982462
$778$	0	0
$779$	− 8.70659e6i	− 0.514049i
$780$	0	0
$781$	− 2.58971e6i	− 0.151923i
$782$	0	0
$783$	1.75487e7	1.02292
$784$	0	0
$785$	1.05010e7	0.608211
$786$	0	0
$787$	765622.i	0.0440634i	0.999757	+	0.0220317i	$0.00701347\pi$
$787$	765622.i	0.0440634i	−0.999757	+	0.0220317i	$0.992987\pi$
$788$	0	0
$789$	− 1.10367e7i	− 0.631168i
$790$	0	0
$791$	−1.54179e6	−0.0876161
$792$	0	0
$793$	−1.07950e6	−0.0609595
$794$	0	0
$795$	− 2.79151e6i	− 0.156647i
$796$	0	0
$797$	− 1.13923e7i	− 0.635281i	−0.948211	−	0.317641i	$-0.897109\pi$
$797$	− 1.13923e7i	− 0.635281i	0.948211	−	0.317641i	$-0.102891\pi$
$798$	0	0
$799$	1.49966e7	0.831048
$800$	0	0
$801$	2.90601e6	0.160035
$802$	0	0
$803$	− 8.70067e6i	− 0.476172i
$804$	0	0
$805$	− 5.70692e6i	− 0.310393i
$806$	0	0
$807$	−3.14365e6	−0.169922
$808$	0	0
$809$	−1.46651e7	−0.787798	−0.393899	−	0.919154i	$-0.628874\pi$
$809$	−1.46651e7	−0.787798	−0.393899	+	0.919154i	$0.628874\pi$
$810$	0	0
$811$	− 8.79076e6i	− 0.469326i	−0.972077	−	0.234663i	$-0.924601\pi$
$811$	− 8.79076e6i	− 0.469326i	0.972077	−	0.234663i	$-0.0753986\pi$
$812$	0	0
$813$	− 2.41257e7i	− 1.28013i
$814$	0	0
$815$	867286.	0.0457371
$816$	0	0
$817$	−1.31811e7	−0.690869
$818$	0	0
$819$	99196.0i	0.00516754i
$820$	0	0
$821$	1.25031e7i	0.647383i	0.946163	+	0.323691i	$0.104924\pi$
$821$	1.25031e7i	0.647383i	−0.946163	+	0.323691i	$0.895076\pi$
$822$	0	0
$823$	−2.36472e7	−1.21697	−0.608486	−	0.793565i	$-0.708223\pi$
$823$	−2.36472e7	−1.21697	−0.608486	+	0.793565i	$0.708223\pi$
$824$	0	0
$825$	−1.58994e6	−0.0813289
$826$	0	0
$827$	4.91285e6i	0.249787i	0.992170	+	0.124893i	$0.0398589\pi$
$827$	4.91285e6i	0.249787i	−0.992170	+	0.124893i	$0.960141\pi$
$828$	0	0
$829$	− 4.52433e6i	− 0.228648i	−0.993443	−	0.114324i	$-0.963530\pi$
$829$	− 4.52433e6i	− 0.228648i	0.993443	−	0.114324i	$-0.0364703\pi$
$830$	0	0
$831$	1.78919e7	0.898781
$832$	0	0
$833$	−2.44844e7	−1.22258
$834$	0	0
$835$	− 1.29785e7i	− 0.644181i
$836$	0	0
$837$	− 6.39684e6i	− 0.315611i
$838$	0	0
$839$	−4.83585e6	−0.237175	−0.118587	−	0.992944i	$-0.537837\pi$
$839$	−4.83585e6	−0.237175	−0.118587	+	0.992944i	$0.537837\pi$
$840$	0	0
$841$	−4.52410e6	−0.220568
$842$	0	0
$843$	− 2.45154e7i	− 1.18815i
$844$	0	0
$845$	− 9.20015e6i	− 0.443255i
$846$	0	0
$847$	7.62417e6	0.365161
$848$	0	0
$849$	−7.94783e6	−0.378424
$850$	0	0
$851$	− 7.40802e6i	− 0.350654i
$852$	0	0
$853$	− 4.52210e6i	− 0.212798i	−0.994324	−	0.106399i	$-0.966068\pi$
$853$	− 4.52210e6i	− 0.212798i	0.994324	−	0.106399i	$-0.0339321\pi$
$854$	0	0
$855$	1.24457e6	0.0582245
$856$	0	0
$857$	−2.18075e7	−1.01427	−0.507135	−	0.861867i	$-0.669295\pi$
$857$	−2.18075e7	−1.01427	−0.507135	+	0.861867i	$0.669295\pi$
$858$	0	0
$859$	− 3.47558e6i	− 0.160711i	−0.996766	−	0.0803553i	$-0.974395\pi$
$859$	− 3.47558e6i	− 0.160711i	0.996766	−	0.0803553i	$-0.0256055\pi$
$860$	0	0
$861$	− 5.01070e6i	− 0.230351i
$862$	0	0
$863$	2.20281e7	1.00682	0.503409	−	0.864048i	$-0.332079\pi$
$863$	2.20281e7	1.00682	0.503409	+	0.864048i	$0.332079\pi$
$864$	0	0
$865$	1.44985e7	0.658843
$866$	0	0
$867$	− 2.91457e7i	− 1.31682i
$868$	0	0
$869$	5.72962e6i	0.257381i
$870$	0	0
$871$	342439.	0.0152946
$872$	0	0
$873$	2.84789e6	0.126470
$874$	0	0
$875$	866703.i	0.0382693i
$876$	0	0
$877$	− 2.22181e7i	− 0.975456i	−0.872996	−	0.487728i	$-0.837826\pi$
$877$	− 2.22181e7i	− 0.975456i	0.872996	−	0.487728i	$-0.162174\pi$
$878$	0	0
$879$	3.43087e7	1.49773
$880$	0	0
$881$	−3.40974e7	−1.48007	−0.740033	−	0.672570i	$-0.765190\pi$
$881$	−3.40974e7	−1.48007	−0.740033	+	0.672570i	$0.765190\pi$
$882$	0	0
$883$	3.76822e7i	1.62642i	0.581967	+	0.813212i	$0.302283\pi$
$883$	3.76822e7i	1.62642i	−0.581967	+	0.813212i	$0.697717\pi$
$884$	0	0
$885$	7263.31i	0 0.000311729i
$886$	0	0
$887$	−2.33279e6	−0.0995558	−0.0497779	−	0.998760i	$-0.515851\pi$
$887$	−2.33279e6	−0.0995558	−0.0497779	+	0.998760i	$0.515851\pi$
$888$	0	0
$889$	7.43691e6	0.315601
$890$	0	0
$891$	− 1.00866e7i	− 0.425648i
$892$	0	0
$893$	− 1.34217e7i	− 0.563222i
$894$	0	0
$895$	7.00968e6	0.292510
$896$	0	0
$897$	−3.90689e6	−0.162125
$898$	0	0
$899$	9.12583e6i	0.376594i
$900$	0	0
$901$	1.20250e7i	0.493484i
$902$	0	0
$903$	−7.58579e6	−0.309586
$904$	0	0
$905$	−2.19452e7	−0.890674
$906$	0	0
$907$	3.00605e6i	0.121333i	0.998158	+	0.0606664i	$0.0193226\pi$
$907$	3.00605e6i	0.121333i	−0.998158	+	0.0606664i	$0.980677\pi$
$908$	0	0
$909$	5.21090e6i	0.209172i
$910$	0	0
$911$	9.65671e6	0.385508	0.192754	−	0.981247i	$-0.438258\pi$
$911$	9.65671e6	0.385508	0.192754	+	0.981247i	$0.438258\pi$
$912$	0	0
$913$	8.88477e6	0.352752
$914$	0	0
$915$	7.79474e6i	0.307786i
$916$	0	0
$917$	− 2.06828e7i	− 0.812242i
$918$	0	0
$919$	−3.59667e7	−1.40479	−0.702396	−	0.711786i	$-0.747886\pi$
$919$	−3.59667e7	−1.40479	−0.702396	+	0.711786i	$0.747886\pi$
$920$	0	0
$921$	−3.39091e7	−1.31725
$922$	0	0
$923$	966431.i	0.0373393i
$924$	0	0
$925$	1.12505e6i	0.0432331i
$926$	0	0
$927$	−2.07942e6	−0.0794773
$928$	0	0
$929$	−2.66167e7	−1.01185	−0.505923	−	0.862579i	$-0.668848\pi$
$929$	−2.66167e7	−1.01185	−0.505923	+	0.862579i	$0.668848\pi$
$930$	0	0
$931$	2.19131e7i	0.828572i
$932$	0	0
$933$	− 4.60072e7i	− 1.73030i
$934$	0	0
$935$	6.84896e6	0.256210
$936$	0	0
$937$	1.41039e7	0.524797	0.262399	−	0.964960i	$-0.415486\pi$
$937$	1.41039e7	0.524797	0.262399	+	0.964960i	$0.415486\pi$
$938$	0	0
$939$	− 2.94892e7i	− 1.09144i
$940$	0	0
$941$	− 2.94208e6i	− 0.108313i	−0.998532	−	0.0541565i	$-0.982753\pi$
$941$	− 2.94208e6i	− 0.108313i	0.998532	−	0.0541565i	$-0.0172470\pi$
$942$	0	0
$943$	2.24508e7	0.822153
$944$	0	0
$945$	−4.86361e6	−0.177166
$946$	0	0
$947$	− 1.97465e7i	− 0.715511i	−0.933815	−	0.357755i	$-0.883542\pi$
$947$	− 1.97465e7i	− 0.715511i	0.933815	−	0.357755i	$-0.116458\pi$
$948$	0	0
$949$	3.24692e6i	0.117033i
$950$	0	0
$951$	−2.77171e7	−0.993795
$952$	0	0
$953$	−1.51163e7	−0.539155	−0.269577	−	0.962979i	$-0.586884\pi$
$953$	−1.51163e7	−0.539155	−0.269577	+	0.962979i	$0.586884\pi$
$954$	0	0
$955$	2.49647e7i	0.885763i
$956$	0	0
$957$	1.27285e7i	0.449258i
$958$	0	0
$959$	−8.11192e6	−0.284825
$960$	0	0
$961$	−2.53026e7	−0.883806
$962$	0	0
$963$	3.19938e6i	0.111173i
$964$	0	0
$965$	1.20857e7i	0.417787i
$966$	0	0
$967$	−4.34383e7	−1.49385	−0.746925	−	0.664908i	$-0.768471\pi$
$967$	−4.34383e7	−1.49385	−0.746925	+	0.664908i	$0.768471\pi$
$968$	0	0
$969$	−4.71269e7	−1.61235
$970$	0	0
$971$	9.27361e6i	0.315646i	0.987467	+	0.157823i	$0.0504476\pi$
$971$	9.27361e6i	0.315646i	−0.987467	+	0.157823i	$0.949552\pi$
$972$	0	0
$973$	− 1.25776e7i	− 0.425907i
$974$	0	0
$975$	593334.	0.0199888
$976$	0	0
$977$	−6.80505e6	−0.228084	−0.114042	−	0.993476i	$-0.536380\pi$
$977$	−6.80505e6	−0.228084	−0.114042	+	0.993476i	$0.536380\pi$
$978$	0	0
$979$	− 1.43125e7i	− 0.477265i
$980$	0	0
$981$	− 4.71878e6i	− 0.156551i
$982$	0	0
$983$	4.94290e7	1.63154	0.815771	−	0.578375i	$-0.196313\pi$
$983$	4.94290e7	1.63154	0.815771	+	0.578375i	$0.196313\pi$
$984$	0	0
$985$	5.47800e6	0.179900
$986$	0	0
$987$	− 7.72428e6i	− 0.252386i
$988$	0	0
$989$	− 3.39887e7i	− 1.10495i
$990$	0	0
$991$	3.41846e7	1.10572	0.552862	−	0.833273i	$-0.313536\pi$
$991$	3.41846e7	1.10572	0.552862	+	0.833273i	$0.313536\pi$
$992$	0	0
$993$	2.20920e7	0.710988
$994$	0	0
$995$	− 1.23105e7i	− 0.394201i
$996$	0	0
$997$	4.99930e7i	1.59284i	0.604746	+	0.796418i	$0.293274\pi$
$997$	4.99930e7i	1.59284i	−0.604746	+	0.796418i	$0.706726\pi$
$998$	0	0
$999$	−6.31334e6	−0.200145

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.6.d.d.161.3	yes	12
4.3	odd	2		320.6.d.c.161.10	yes	12
8.3	odd	2		320.6.d.c.161.3	✓	12
8.5	even	2	inner	320.6.d.d.161.10	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.6.d.c.161.3	✓	12	8.3	odd	2
320.6.d.c.161.10	yes	12	4.3	odd	2
320.6.d.d.161.3	yes	12	1.1	even	1	trivial
320.6.d.d.161.10	yes	12	8.5	even	2	inner

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:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	320.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-16.5588i q^{3} -25.0000i q^{5} +55.4690 q^{7} -31.1927 q^{9} +O(q^{10})\)	\(q-16.5588i q^{3} -25.0000i q^{5} +55.4690 q^{7} -31.1927 q^{9} +153.628i q^{11} -57.3312i q^{13} -413.969 q^{15} +1783.25 q^{17} -1595.98i q^{19} -918.498i q^{21} +4115.40 q^{23} -625.000 q^{25} -3507.27i q^{27} +5003.52i q^{29} +1823.88 q^{31} +2543.90 q^{33} -1386.72i q^{35} -1800.07i q^{37} -949.334 q^{39} +5455.32 q^{41} -8258.91i q^{43} +779.817i q^{45} +8409.69 q^{47} -13730.2 q^{49} -29528.5i q^{51} +6743.29i q^{53} +3840.71 q^{55} -26427.5 q^{57} -17.5455i q^{59} -18829.3i q^{61} -1730.23 q^{63} -1433.28 q^{65} +5972.99i q^{67} -68145.9i q^{69} -16857.0 q^{71} -56634.5 q^{73} +10349.2i q^{75} +8521.61i q^{77} +37295.3 q^{79} -65655.8 q^{81} -57832.9i q^{83} -44581.3i q^{85} +82852.2 q^{87} -93163.3 q^{89} -3180.10i q^{91} -30201.2i q^{93} -39899.6 q^{95} -91299.9 q^{97} -4792.08i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 268 q^{7} - 428 q^{9}+O(q^{10}) \) 12 * q + 268 * q^7 - 428 * q^9	\( 12 q + 268 q^{7} - 428 q^{9} - 900 q^{15} + 2400 q^{17} + 8108 q^{23} - 7500 q^{25} + 7976 q^{31} - 20776 q^{33} + 40984 q^{39} - 56408 q^{41} + 21172 q^{47} - 5540 q^{49} + 18400 q^{55} + 39992 q^{57} + 179516 q^{63} + 44000 q^{65} + 367704 q^{71} + 58736 q^{73} + 26192 q^{79} + 411692 q^{81} + 183200 q^{87} + 87672 q^{89} + 121000 q^{95} - 172336 q^{97}+O(q^{100}) \) 12 * q + 268 * q^7 - 428 * q^9 - 900 * q^15 + 2400 * q^17 + 8108 * q^23 - 7500 * q^25 + 7976 * q^31 - 20776 * q^33 + 40984 * q^39 - 56408 * q^41 + 21172 * q^47 - 5540 * q^49 + 18400 * q^55 + 39992 * q^57 + 179516 * q^63 + 44000 * q^65 + 367704 * q^71 + 58736 * q^73 + 26192 * q^79 + 411692 * q^81 + 183200 * q^87 + 87672 * q^89 + 121000 * q^95 - 172336 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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