LMFDB - Embedded newform 256.9.c.n.255.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [256,9,Mod(255,256)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(256, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("256.255");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 256 = 2^{8} $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	256.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$104.288924176$
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} - 155x^{10} + 13530x^{8} - 838070x^{6} + 36386725x^{4} - 1689017631x^{2} + 57111440400 $ x^12 - 155x^10 + 13530x^8 - 838070x^6 + 36386725x^4 - 1689017631*x^2 + 57111440400
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{100} $
Twist minimal:	no (minimal twist has level 8)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			255.6
Root			$-4.16154 + 5.83239i$ of defining polynomial
Character	$\chi$	$=$	256.255
Dual form			256.9.c.n.255.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-17.4864i q^{3} +796.785 q^{5} -1779.55i q^{7} +6255.23 q^{9} +O(q^{10})$	$q-17.4864i q^{3} +796.785 q^{5} -1779.55i q^{7} +6255.23 q^{9} -1458.80i q^{11} -30485.3 q^{13} -13932.9i q^{15} -82919.0 q^{17} +214604. i q^{19} -31118.0 q^{21} +488296. i q^{23} +244241. q^{25} -224110. i q^{27} -499553. q^{29} +1.35347e6i q^{31} -25509.1 q^{33} -1.41792e6i q^{35} -334053. q^{37} +533079. i q^{39} -3.11641e6 q^{41} +1.81268e6i q^{43} +4.98407e6 q^{45} +773477. i q^{47} +2.59799e6 q^{49} +1.44996e6i q^{51} +1.78592e6 q^{53} -1.16235e6i q^{55} +3.75265e6 q^{57} +1.59466e6i q^{59} -1.18438e7 q^{61} -1.11315e7i q^{63} -2.42902e7 q^{65} +2.22687e7i q^{67} +8.53854e6 q^{69} -2.06234e7i q^{71} +4.21796e6 q^{73} -4.27090e6i q^{75} -2.59601e6 q^{77} -4.96781e7i q^{79} +3.71217e7 q^{81} -1.13729e7i q^{83} -6.60686e7 q^{85} +8.73538e6i q^{87} -2.48076e7 q^{89} +5.42502e7i q^{91} +2.36672e7 q^{93} +1.70993e8i q^{95} +1.14484e7 q^{97} -9.12511e6i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 32676 q^{9}+O(q^{10}) $ 12 * q - 32676 * q^9	$ 12 q - 32676 q^{9} - 402152 q^{17} + 895860 q^{25} + 3473712 q^{33} - 7634200 q^{41} - 25310964 q^{49} - 15047472 q^{57} + 54120960 q^{65} + 110969832 q^{73} - 158290884 q^{81} - 284346904 q^{89} + 250387224 q^{97}+O(q^{100}) $ 12 * q - 32676 * q^9 - 402152 * q^17 + 895860 * q^25 + 3473712 * q^33 - 7634200 * q^41 - 25310964 * q^49 - 15047472 * q^57 + 54120960 * q^65 + 110969832 * q^73 - 158290884 * q^81 - 284346904 * q^89 + 250387224 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/256\mathbb{Z}\right)^\times$.

$n$	$5$	$255$
$\chi(n)$	$1$	$-1$

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
$3$	− 17.4864i	− 0.215882i	−0.994157	−	0.107941i	$-0.965574\pi$
$3$	− 17.4864i	− 0.215882i	0.994157	−	0.107941i	$-0.0344257\pi$
$4$	0	0
$5$	796.785	1.27486	0.637428	−	0.770510i	$-0.279998\pi$
$5$	796.785	1.27486	0.637428	+	0.770510i	$0.279998\pi$
$6$	0	0
$7$	− 1779.55i	− 0.741171i	−0.928798	−	0.370586i	$-0.879157\pi$
$7$	− 1779.55i	− 0.741171i	0.928798	−	0.370586i	$-0.120843\pi$
$8$	0	0
$9$	6255.23	0.953395
$10$	0	0
$11$	− 1458.80i	− 0.0996379i	−0.998758	−	0.0498189i	$-0.984136\pi$
$11$	− 1458.80i	− 0.0996379i	0.998758	−	0.0498189i	$-0.0158644\pi$
$12$	0	0
$13$	−30485.3	−1.06738	−0.533688	−	0.845682i	$-0.679194\pi$
$13$	−30485.3	−1.06738	−0.533688	+	0.845682i	$0.679194\pi$
$14$	0	0
$15$	− 13932.9i	− 0.275218i
$16$	0	0
$17$	−82919.0	−0.992792	−0.496396	−	0.868096i	$-0.665344\pi$
$17$	−82919.0	−0.992792	−0.496396	+	0.868096i	$0.665344\pi$
$18$	0	0
$19$	214604.i	1.64673i	0.567510	+	0.823366i	$0.307907\pi$
$19$	214604.i	1.64673i	−0.567510	+	0.823366i	$0.692093\pi$
$20$	0	0
$21$	−31118.0	−0.160005
$22$	0	0
$23$	488296.i	1.74490i	0.488700	+	0.872452i	$0.337471\pi$
$23$	488296.i	1.74490i	−0.488700	+	0.872452i	$0.662529\pi$
$24$	0	0
$25$	244241.	0.625257
$26$	0	0
$27$	− 224110.i	− 0.421702i
$28$	0	0
$29$	−499553.	−0.706300	−0.353150	−	0.935567i	$-0.614890\pi$
$29$	−499553.	−0.706300	−0.353150	+	0.935567i	$0.614890\pi$
$30$	0	0
$31$	1.35347e6i	1.46555i	0.680472	+	0.732774i	$0.261775\pi$
$31$	1.35347e6i	1.46555i	−0.680472	+	0.732774i	$0.738225\pi$
$32$	0	0
$33$	−25509.1	−0.0215100
$34$	0	0
$35$	− 1.41792e6i	− 0.944887i
$36$	0	0
$37$	−334053.	−0.178241	−0.0891206	−	0.996021i	$-0.528406\pi$
$37$	−334053.	−0.178241	−0.0891206	+	0.996021i	$0.528406\pi$
$38$	0	0
$39$	533079.i	0.230427i
$40$	0	0
$41$	−3.11641e6	−1.10286	−0.551428	−	0.834223i	$-0.685917\pi$
$41$	−3.11641e6	−1.10286	−0.551428	+	0.834223i	$0.685917\pi$
$42$	0	0
$43$	1.81268e6i	0.530208i	0.964220	+	0.265104i	$0.0854064\pi$
$43$	1.81268e6i	0.530208i	−0.964220	+	0.265104i	$0.914594\pi$
$44$	0	0
$45$	4.98407e6	1.21544
$46$	0	0
$47$	773477.i	0.158510i	0.996854	+	0.0792548i	$0.0252541\pi$
$47$	773477.i	0.158510i	−0.996854	+	0.0792548i	$0.974746\pi$
$48$	0	0
$49$	2.59799e6	0.450665
$50$	0	0
$51$	1.44996e6i	0.214326i
$52$	0	0
$53$	1.78592e6	0.226339	0.113169	−	0.993576i	$-0.463900\pi$
$53$	1.78592e6	0.226339	0.113169	+	0.993576i	$0.463900\pi$
$54$	0	0
$55$	− 1.16235e6i	− 0.127024i
$56$	0	0
$57$	3.75265e6	0.355499
$58$	0	0
$59$	1.59466e6i	0.131602i	0.997833	+	0.0658008i	$0.0209602\pi$
$59$	1.59466e6i	0.131602i	−0.997833	+	0.0658008i	$0.979040\pi$
$60$	0	0
$61$	−1.18438e7	−0.855406	−0.427703	−	0.903919i	$-0.640677\pi$
$61$	−1.18438e7	−0.855406	−0.427703	+	0.903919i	$0.640677\pi$
$62$	0	0
$63$	− 1.11315e7i	− 0.706629i
$64$	0	0
$65$	−2.42902e7	−1.36075
$66$	0	0
$67$	2.22687e7i	1.10509i	0.833485	+	0.552543i	$0.186342\pi$
$67$	2.22687e7i	1.10509i	−0.833485	+	0.552543i	$0.813658\pi$
$68$	0	0
$69$	8.53854e6	0.376693
$70$	0	0
$71$	− 2.06234e7i	− 0.811572i	−0.913968	−	0.405786i	$-0.866998\pi$
$71$	− 2.06234e7i	− 0.811572i	0.913968	−	0.405786i	$-0.133002\pi$
$72$	0	0
$73$	4.21796e6	0.148529	0.0742644	−	0.997239i	$-0.476339\pi$
$73$	4.21796e6	0.148529	0.0742644	+	0.997239i	$0.476339\pi$
$74$	0	0
$75$	− 4.27090e6i	− 0.134981i
$76$	0	0
$77$	−2.59601e6	−0.0738487
$78$	0	0
$79$	− 4.96781e7i	− 1.27543i	−0.770272	−	0.637715i	$-0.779880\pi$
$79$	− 4.96781e7i	− 1.27543i	0.770272	−	0.637715i	$-0.220120\pi$
$80$	0	0
$81$	3.71217e7	0.862357
$82$	0	0
$83$	− 1.13729e7i	− 0.239640i	−0.992796	−	0.119820i	$-0.961768\pi$
$83$	− 1.13729e7i	− 0.239640i	0.992796	−	0.119820i	$-0.0382317\pi$
$84$	0	0
$85$	−6.60686e7	−1.26567
$86$	0	0
$87$	8.73538e6i	0.152477i
$88$	0	0
$89$	−2.48076e7	−0.395389	−0.197695	−	0.980264i	$-0.563345\pi$
$89$	−2.48076e7	−0.395389	−0.197695	+	0.980264i	$0.563345\pi$
$90$	0	0
$91$	5.42502e7i	0.791108i
$92$	0	0
$93$	2.36672e7	0.316385
$94$	0	0
$95$	1.70993e8i	2.09935i
$96$	0	0
$97$	1.14484e7	0.129318	0.0646589	−	0.997907i	$-0.479404\pi$
$97$	1.14484e7	0.129318	0.0646589	+	0.997907i	$0.479404\pi$
$98$	0	0
$99$	− 9.12511e6i	− 0.0949943i
$100$	0	0
$101$	1.47673e8	1.41910	0.709552	−	0.704653i	$-0.248897\pi$
$101$	1.47673e8	1.41910	0.709552	+	0.704653i	$0.248897\pi$
$102$	0	0
$103$	5.72198e7i	0.508390i	0.967153	+	0.254195i	$0.0818106\pi$
$103$	5.72198e7i	0.508390i	−0.967153	+	0.254195i	$0.918189\pi$
$104$	0	0
$105$	−2.47943e7	−0.203984
$106$	0	0
$107$	1.90189e8i	1.45094i	0.688252	+	0.725471i	$0.258378\pi$
$107$	1.90189e8i	1.45094i	−0.688252	+	0.725471i	$0.741622\pi$
$108$	0	0
$109$	5.09345e7	0.360833	0.180416	−	0.983590i	$-0.442256\pi$
$109$	5.09345e7	0.360833	0.180416	+	0.983590i	$0.442256\pi$
$110$	0	0
$111$	5.84138e6i	0.0384790i
$112$	0	0
$113$	−7.76455e6	−0.0476214	−0.0238107	−	0.999716i	$-0.507580\pi$
$113$	−7.76455e6	−0.0476214	−0.0238107	+	0.999716i	$0.507580\pi$
$114$	0	0
$115$	3.89067e8i	2.22450i
$116$	0	0
$117$	−1.90693e8	−1.01763
$118$	0	0
$119$	1.47559e8i	0.735829i
$120$	0	0
$121$	2.12231e8	0.990072
$122$	0	0
$123$	5.44948e7i	0.238086i
$124$	0	0
$125$	−1.16637e8	−0.477743
$126$	0	0
$127$	− 2.54116e8i	− 0.976826i	−0.872612	−	0.488413i	$-0.837576\pi$
$127$	− 2.54116e8i	− 0.976826i	0.872612	−	0.488413i	$-0.162424\pi$
$128$	0	0
$129$	3.16972e7	0.114462
$130$	0	0
$131$	2.72967e8i	0.926885i	0.886127	+	0.463442i	$0.153386\pi$
$131$	2.72967e8i	0.926885i	−0.886127	+	0.463442i	$0.846614\pi$
$132$	0	0
$133$	3.81899e8	1.22051
$134$	0	0
$135$	− 1.78567e8i	− 0.537609i
$136$	0	0
$137$	3.72346e7	0.105698	0.0528488	−	0.998603i	$-0.483170\pi$
$137$	3.72346e7	0.105698	0.0528488	+	0.998603i	$0.483170\pi$
$138$	0	0
$139$	1.05417e8i	0.282392i	0.989982	+	0.141196i	$0.0450947\pi$
$139$	1.05417e8i	0.282392i	−0.989982	+	0.141196i	$0.954905\pi$
$140$	0	0
$141$	1.35253e7	0.0342193
$142$	0	0
$143$	4.44719e7i	0.106351i
$144$	0	0
$145$	−3.98036e8	−0.900431
$146$	0	0
$147$	− 4.54296e7i	− 0.0972903i
$148$	0	0
$149$	−6.08521e8	−1.23461	−0.617306	−	0.786723i	$-0.711776\pi$
$149$	−6.08521e8	−1.23461	−0.617306	+	0.786723i	$0.711776\pi$
$150$	0	0
$151$	− 4.73729e8i	− 0.911219i	−0.890180	−	0.455609i	$-0.849421\pi$
$151$	− 4.73729e8i	− 0.911219i	0.890180	−	0.455609i	$-0.150579\pi$
$152$	0	0
$153$	−5.18677e8	−0.946523
$154$	0	0
$155$	1.07842e9i	1.86836i
$156$	0	0
$157$	−4.88576e8	−0.804144	−0.402072	−	0.915608i	$-0.631710\pi$
$157$	−4.88576e8	−0.804144	−0.402072	+	0.915608i	$0.631710\pi$
$158$	0	0
$159$	− 3.12294e7i	− 0.0488624i
$160$	0	0
$161$	8.68948e8	1.29327
$162$	0	0
$163$	8.94547e8i	1.26722i	0.773651	+	0.633611i	$0.218428\pi$
$163$	8.94547e8i	1.26722i	−0.773651	+	0.633611i	$0.781572\pi$
$164$	0	0
$165$	−2.03253e7	−0.0274221
$166$	0	0
$167$	− 7.04084e7i	− 0.0905229i	−0.998975	−	0.0452615i	$-0.985588\pi$
$167$	− 7.04084e7i	− 0.0905229i	0.998975	−	0.0452615i	$-0.0144121\pi$
$168$	0	0
$169$	1.13624e8	0.139291
$170$	0	0
$171$	1.34240e9i	1.56999i
$172$	0	0
$173$	1.52030e9	1.69725	0.848623	−	0.528999i	$-0.177432\pi$
$173$	1.52030e9	1.69725	0.848623	+	0.528999i	$0.177432\pi$
$174$	0	0
$175$	− 4.34640e8i	− 0.463423i
$176$	0	0
$177$	2.78850e7	0.0284104
$178$	0	0
$179$	1.46877e9i	1.43068i	0.698776	+	0.715341i	$0.253729\pi$
$179$	1.46877e9i	1.43068i	−0.698776	+	0.715341i	$0.746271\pi$
$180$	0	0
$181$	4.02425e8	0.374948	0.187474	−	0.982270i	$-0.439970\pi$
$181$	4.02425e8	0.374948	0.187474	+	0.982270i	$0.439970\pi$
$182$	0	0
$183$	2.07106e8i	0.184667i
$184$	0	0
$185$	−2.66168e8	−0.227232
$186$	0	0
$187$	1.20962e8i	0.0989197i
$188$	0	0
$189$	−3.98815e8	−0.312554
$190$	0	0
$191$	3.03920e8i	0.228363i	0.993460	+	0.114182i	$0.0364246\pi$
$191$	3.03920e8i	0.228363i	−0.993460	+	0.114182i	$0.963575\pi$
$192$	0	0
$193$	−2.02929e9	−1.46257	−0.731284	−	0.682073i	$-0.761078\pi$
$193$	−2.02929e9	−1.46257	−0.731284	+	0.682073i	$0.761078\pi$
$194$	0	0
$195$	4.24749e8i	0.293761i
$196$	0	0
$197$	1.17091e9	0.777422	0.388711	−	0.921360i	$-0.372920\pi$
$197$	1.17091e9	0.777422	0.388711	+	0.921360i	$0.372920\pi$
$198$	0	0
$199$	1.40831e9i	0.898020i	0.893527	+	0.449010i	$0.148223\pi$
$199$	1.40831e9i	0.898020i	−0.893527	+	0.449010i	$0.851777\pi$
$200$	0	0
$201$	3.89400e8	0.238568
$202$	0	0
$203$	8.88980e8i	0.523489i
$204$	0	0
$205$	−2.48311e9	−1.40598
$206$	0	0
$207$	3.05440e9i	1.66358i
$208$	0	0
$209$	3.13064e8	0.164077
$210$	0	0
$211$	2.76916e9i	1.39707i	0.715575	+	0.698535i	$0.246165\pi$
$211$	2.76916e9i	1.39707i	−0.715575	+	0.698535i	$0.753835\pi$
$212$	0	0
$213$	−3.60629e8	−0.175203
$214$	0	0
$215$	1.44431e9i	0.675939i
$216$	0	0
$217$	2.40856e9	1.08622
$218$	0	0
$219$	− 7.37569e7i	− 0.0320646i
$220$	0	0
$221$	2.52781e9	1.05968
$222$	0	0
$223$	− 2.53904e9i	− 1.02672i	−0.858174	−	0.513358i	$-0.828401\pi$
$223$	− 2.53904e9i	− 1.02672i	0.858174	−	0.513358i	$-0.171599\pi$
$224$	0	0
$225$	1.52778e9	0.596117
$226$	0	0
$227$	− 1.37848e9i	− 0.519153i	−0.965723	−	0.259577i	$-0.916417\pi$
$227$	− 1.37848e9i	− 0.519153i	0.965723	−	0.259577i	$-0.0835830\pi$
$228$	0	0
$229$	−3.26124e9	−1.18588	−0.592941	−	0.805246i	$-0.702033\pi$
$229$	−3.26124e9	−1.18588	−0.592941	+	0.805246i	$0.702033\pi$
$230$	0	0
$231$	4.53949e7i	0.0159426i
$232$	0	0
$233$	−3.98642e9	−1.35257	−0.676285	−	0.736640i	$-0.736411\pi$
$233$	−3.98642e9	−1.35257	−0.676285	+	0.736640i	$0.736411\pi$
$234$	0	0
$235$	6.16295e8i	0.202077i
$236$	0	0
$237$	−8.68692e8	−0.275342
$238$	0	0
$239$	− 9.59043e8i	− 0.293932i	−0.989142	−	0.146966i	$-0.953049\pi$
$239$	− 9.59043e8i	− 0.293932i	0.989142	−	0.146966i	$-0.0469507\pi$
$240$	0	0
$241$	1.65969e9	0.491993	0.245996	−	0.969271i	$-0.420885\pi$
$241$	1.65969e9	0.491993	0.245996	+	0.969271i	$0.420885\pi$
$242$	0	0
$243$	− 2.11951e9i	− 0.607869i
$244$	0	0
$245$	2.07004e9	0.574533
$246$	0	0
$247$	− 6.54227e9i	− 1.75768i
$248$	0	0
$249$	−1.98871e8	−0.0517338
$250$	0	0
$251$	2.93567e9i	0.739627i	0.929106	+	0.369814i	$0.120578\pi$
$251$	2.93567e9i	0.739627i	−0.929106	+	0.369814i	$0.879422\pi$
$252$	0	0
$253$	7.12325e8	0.173859
$254$	0	0
$255$	1.15530e9i	0.273234i
$256$	0	0
$257$	−9.42169e8	−0.215971	−0.107986	−	0.994152i	$-0.534440\pi$
$257$	−9.42169e8	−0.215971	−0.107986	+	0.994152i	$0.534440\pi$
$258$	0	0
$259$	5.94464e8i	0.132107i
$260$	0	0
$261$	−3.12481e9	−0.673383
$262$	0	0
$263$	− 4.35300e9i	− 0.909841i	−0.890532	−	0.454920i	$-0.849668\pi$
$263$	− 4.35300e9i	− 0.909841i	0.890532	−	0.454920i	$-0.150332\pi$
$264$	0	0
$265$	1.42300e9	0.288549
$266$	0	0
$267$	4.33796e8i	0.0853573i
$268$	0	0
$269$	−1.63941e9	−0.313098	−0.156549	−	0.987670i	$-0.550037\pi$
$269$	−1.63941e9	−0.313098	−0.156549	+	0.987670i	$0.550037\pi$
$270$	0	0
$271$	− 6.53537e9i	− 1.21169i	−0.795581	−	0.605847i	$-0.792834\pi$
$271$	− 6.53537e9i	− 1.21169i	0.795581	−	0.605847i	$-0.207166\pi$
$272$	0	0
$273$	9.48642e8	0.170786
$274$	0	0
$275$	− 3.56298e8i	− 0.0622993i
$276$	0	0
$277$	8.55840e9	1.45369	0.726847	−	0.686799i	$-0.240985\pi$
$277$	8.55840e9	1.45369	0.726847	+	0.686799i	$0.240985\pi$
$278$	0	0
$279$	8.46623e9i	1.39725i
$280$	0	0
$281$	5.19165e9	0.832683	0.416342	−	0.909208i	$-0.363312\pi$
$281$	5.19165e9	0.832683	0.416342	+	0.909208i	$0.363312\pi$
$282$	0	0
$283$	− 7.13116e9i	− 1.11177i	−0.831259	−	0.555885i	$-0.812380\pi$
$283$	− 7.13116e9i	− 1.11177i	0.831259	−	0.555885i	$-0.187620\pi$
$284$	0	0
$285$	2.99006e9	0.453210
$286$	0	0
$287$	5.54581e9i	0.817405i
$288$	0	0
$289$	−1.00194e8	−0.0143632
$290$	0	0
$291$	− 2.00192e8i	− 0.0279173i
$292$	0	0
$293$	−2.75986e9	−0.374470	−0.187235	−	0.982315i	$-0.559953\pi$
$293$	−2.75986e9	−0.374470	−0.187235	+	0.982315i	$0.559953\pi$
$294$	0	0
$295$	1.27060e9i	0.167773i
$296$	0	0
$297$	−3.26931e8	−0.0420175
$298$	0	0
$299$	− 1.48858e10i	− 1.86247i
$300$	0	0
$301$	3.22575e9	0.392975
$302$	0	0
$303$	− 2.58226e9i	− 0.306359i
$304$	0	0
$305$	−9.43698e9	−1.09052
$306$	0	0
$307$	− 1.03453e10i	− 1.16463i	−0.812963	−	0.582316i	$-0.802147\pi$
$307$	− 1.03453e10i	− 1.16463i	0.812963	−	0.582316i	$-0.197853\pi$
$308$	0	0
$309$	1.00057e9	0.109752
$310$	0	0
$311$	6.19845e9i	0.662585i	0.943528	+	0.331292i	$0.107485\pi$
$311$	6.19845e9i	0.662585i	−0.943528	+	0.331292i	$0.892515\pi$
$312$	0	0
$313$	−7.26539e9	−0.756975	−0.378488	−	0.925606i	$-0.623556\pi$
$313$	−7.26539e9	−0.756975	−0.378488	+	0.925606i	$0.623556\pi$
$314$	0	0
$315$	− 8.86941e9i	− 0.900850i
$316$	0	0
$317$	−1.04449e10	−1.03435	−0.517175	−	0.855879i	$-0.673017\pi$
$317$	−1.04449e10	−1.03435	−0.517175	+	0.855879i	$0.673017\pi$
$318$	0	0
$319$	7.28747e8i	0.0703743i
$320$	0	0
$321$	3.32572e9	0.313232
$322$	0	0
$323$	− 1.77947e10i	− 1.63486i
$324$	0	0
$325$	−7.44576e9	−0.667384
$326$	0	0
$327$	− 8.90661e8i	− 0.0778971i
$328$	0	0
$329$	1.37644e9	0.117483
$330$	0	0
$331$	4.99859e9i	0.416424i	0.978084	+	0.208212i	$0.0667644\pi$
$331$	4.99859e9i	0.416424i	−0.978084	+	0.208212i	$0.933236\pi$
$332$	0	0
$333$	−2.08958e9	−0.169934
$334$	0	0
$335$	1.77434e10i	1.40882i
$336$	0	0
$337$	3.29299e9	0.255312	0.127656	−	0.991819i	$-0.459255\pi$
$337$	3.29299e9	0.255312	0.127656	+	0.991819i	$0.459255\pi$
$338$	0	0
$339$	1.35774e8i	0.0102806i
$340$	0	0
$341$	1.97443e9	0.146024
$342$	0	0
$343$	− 1.48820e10i	− 1.07519i
$344$	0	0
$345$	6.80338e9	0.480229
$346$	0	0
$347$	− 2.09239e9i	− 0.144319i	−0.997393	−	0.0721596i	$-0.977011\pi$
$347$	− 2.09239e9i	− 0.144319i	0.997393	−	0.0721596i	$-0.0229891\pi$
$348$	0	0
$349$	1.00832e10	0.679665	0.339833	−	0.940486i	$-0.389630\pi$
$349$	1.00832e10	0.679665	0.339833	+	0.940486i	$0.389630\pi$
$350$	0	0
$351$	6.83206e9i	0.450115i
$352$	0	0
$353$	1.75112e10	1.12776	0.563879	−	0.825857i	$-0.309308\pi$
$353$	1.75112e10	1.12776	0.563879	+	0.825857i	$0.309308\pi$
$354$	0	0
$355$	− 1.64324e10i	− 1.03464i
$356$	0	0
$357$	2.58027e9	0.158852
$358$	0	0
$359$	2.12923e10i	1.28187i	0.767594	+	0.640937i	$0.221454\pi$
$359$	2.12923e10i	1.28187i	−0.767594	+	0.640937i	$0.778546\pi$
$360$	0	0
$361$	−2.90713e10	−1.71173
$362$	0	0
$363$	− 3.71115e9i	− 0.213738i
$364$	0	0
$365$	3.36080e9	0.189353
$366$	0	0
$367$	− 1.81508e10i	− 1.00054i	−0.865871	−	0.500268i	$-0.833235\pi$
$367$	− 1.81508e10i	− 1.00054i	0.865871	−	0.500268i	$-0.166765\pi$
$368$	0	0
$369$	−1.94938e10	−1.05146
$370$	0	0
$371$	− 3.17814e9i	− 0.167756i
$372$	0	0
$373$	−2.88943e10	−1.49271	−0.746356	−	0.665547i	$-0.768198\pi$
$373$	−2.88943e10	−1.49271	−0.746356	+	0.665547i	$0.768198\pi$
$374$	0	0
$375$	2.03955e9i	0.103136i
$376$	0	0
$377$	1.52290e10	0.753888
$378$	0	0
$379$	− 1.74756e10i	− 0.846983i	−0.905900	−	0.423492i	$-0.860804\pi$
$379$	− 1.74756e10i	− 0.846983i	0.905900	−	0.423492i	$-0.139196\pi$
$380$	0	0
$381$	−4.44358e9	−0.210879
$382$	0	0
$383$	− 2.01489e10i	− 0.936389i	−0.883625	−	0.468195i	$-0.844905\pi$
$383$	− 2.01489e10i	− 0.936389i	0.883625	−	0.468195i	$-0.155095\pi$
$384$	0	0
$385$	−2.06846e9	−0.0941465
$386$	0	0
$387$	1.13387e10i	0.505498i
$388$	0	0
$389$	3.27161e10	1.42877	0.714387	−	0.699751i	$-0.246706\pi$
$389$	3.27161e10	1.42877	0.714387	+	0.699751i	$0.246706\pi$
$390$	0	0
$391$	− 4.04890e10i	− 1.73233i
$392$	0	0
$393$	4.77322e9	0.200097
$394$	0	0
$395$	− 3.95828e10i	− 1.62599i
$396$	0	0
$397$	−2.27039e9	−0.0913985	−0.0456992	−	0.998955i	$-0.514552\pi$
$397$	−2.27039e9	−0.0913985	−0.0456992	+	0.998955i	$0.514552\pi$
$398$	0	0
$399$	− 6.67804e9i	− 0.263486i
$400$	0	0
$401$	−1.40935e10	−0.545055	−0.272528	−	0.962148i	$-0.587860\pi$
$401$	−1.40935e10	−0.545055	−0.272528	+	0.962148i	$0.587860\pi$
$402$	0	0
$403$	− 4.12608e10i	− 1.56429i
$404$	0	0
$405$	2.95780e10	1.09938
$406$	0	0
$407$	4.87316e8i	0.0177596i
$408$	0	0
$409$	3.82460e10	1.36676	0.683380	−	0.730063i	$-0.260509\pi$
$409$	3.82460e10	1.36676	0.683380	+	0.730063i	$0.260509\pi$
$410$	0	0
$411$	− 6.51100e8i	− 0.0228182i
$412$	0	0
$413$	2.83779e9	0.0975394
$414$	0	0
$415$	− 9.06175e9i	− 0.305506i
$416$	0	0
$417$	1.84337e9	0.0609632
$418$	0	0
$419$	2.64682e10i	0.858754i	0.903125	+	0.429377i	$0.141267\pi$
$419$	2.64682e10i	0.858754i	−0.903125	+	0.429377i	$0.858733\pi$
$420$	0	0
$421$	2.77128e10	0.882170	0.441085	−	0.897465i	$-0.354594\pi$
$421$	2.77128e10	0.882170	0.441085	+	0.897465i	$0.354594\pi$
$422$	0	0
$423$	4.83827e9i	0.151122i
$424$	0	0
$425$	−2.02522e10	−0.620750
$426$	0	0
$427$	2.10767e10i	0.634003i
$428$	0	0
$429$	7.77654e8	0.0229592
$430$	0	0
$431$	6.54522e9i	0.189677i	0.995493	+	0.0948386i	$0.0302335\pi$
$431$	6.54522e9i	0.189677i	−0.995493	+	0.0948386i	$0.969767\pi$
$432$	0	0
$433$	3.87616e10	1.10268	0.551341	−	0.834280i	$-0.314116\pi$
$433$	3.87616e10	1.10268	0.551341	+	0.834280i	$0.314116\pi$
$434$	0	0
$435$	6.96022e9i	0.194386i
$436$	0	0
$437$	−1.04790e11	−2.87339
$438$	0	0
$439$	5.32803e9i	0.143453i	0.997424	+	0.0717264i	$0.0228508\pi$
$439$	5.32803e9i	0.143453i	−0.997424	+	0.0717264i	$0.977149\pi$
$440$	0	0
$441$	1.62510e10	0.429662
$442$	0	0
$443$	2.40425e10i	0.624258i	0.950040	+	0.312129i	$0.101042\pi$
$443$	2.40425e10i	0.624258i	−0.950040	+	0.312129i	$0.898958\pi$
$444$	0	0
$445$	−1.97663e10	−0.504065
$446$	0	0
$447$	1.06408e10i	0.266530i
$448$	0	0
$449$	−1.39499e10	−0.343231	−0.171616	−	0.985164i	$-0.554899\pi$
$449$	−1.39499e10	−0.343231	−0.171616	+	0.985164i	$0.554899\pi$
$450$	0	0
$451$	4.54621e9i	0.109886i
$452$	0	0
$453$	−8.28383e9	−0.196715
$454$	0	0
$455$	4.32258e10i	1.00855i
$456$	0	0
$457$	−7.15922e10	−1.64135	−0.820675	−	0.571396i	$-0.806402\pi$
$457$	−7.15922e10	−1.64135	−0.820675	+	0.571396i	$0.806402\pi$
$458$	0	0
$459$	1.85830e10i	0.418663i
$460$	0	0
$461$	1.49659e10	0.331360	0.165680	−	0.986180i	$-0.447018\pi$
$461$	1.49659e10	0.331360	0.165680	+	0.986180i	$0.447018\pi$
$462$	0	0
$463$	8.49544e10i	1.84868i	0.381568	+	0.924341i	$0.375384\pi$
$463$	8.49544e10i	1.84868i	−0.381568	+	0.924341i	$0.624616\pi$
$464$	0	0
$465$	1.88577e10	0.403345
$466$	0	0
$467$	2.02009e10i	0.424721i	0.977191	+	0.212361i	$0.0681151\pi$
$467$	2.02009e10i	0.424721i	−0.977191	+	0.212361i	$0.931885\pi$
$468$	0	0
$469$	3.96283e10	0.819058
$470$	0	0
$471$	8.54344e9i	0.173600i
$472$	0	0
$473$	2.64433e9	0.0528288
$474$	0	0
$475$	5.24151e10i	1.02963i
$476$	0	0
$477$	1.11713e10	0.215790
$478$	0	0
$479$	− 9.48673e9i	− 0.180208i	−0.995932	−	0.0901041i	$-0.971280\pi$
$479$	− 9.48673e9i	− 0.180208i	0.995932	−	0.0901041i	$-0.0287200\pi$
$480$	0	0
$481$	1.01837e10	0.190250
$482$	0	0
$483$	− 1.51948e10i	− 0.279194i
$484$	0	0
$485$	9.12192e9	0.164862
$486$	0	0
$487$	2.60799e10i	0.463650i	0.972758	+	0.231825i	$0.0744697\pi$
$487$	2.60799e10i	0.463650i	−0.972758	+	0.231825i	$0.925530\pi$
$488$	0	0
$489$	1.56424e10	0.273570
$490$	0	0
$491$	5.60323e10i	0.964079i	0.876150	+	0.482039i	$0.160104\pi$
$491$	5.60323e10i	0.964079i	−0.876150	+	0.482039i	$0.839896\pi$
$492$	0	0
$493$	4.14224e10	0.701210
$494$	0	0
$495$	− 7.27075e9i	− 0.121104i
$496$	0	0
$497$	−3.67004e10	−0.601514
$498$	0	0
$499$	1.12235e11i	1.81021i	0.425192	+	0.905103i	$0.360206\pi$
$499$	1.12235e11i	1.81021i	−0.425192	+	0.905103i	$0.639794\pi$
$500$	0	0
$501$	−1.23119e9	−0.0195422
$502$	0	0
$503$	− 8.45980e10i	− 1.32156i	−0.750578	−	0.660782i	$-0.770225\pi$
$503$	− 8.45980e10i	− 1.32156i	0.750578	−	0.660782i	$-0.229775\pi$
$504$	0	0
$505$	1.17663e11	1.80915
$506$	0	0
$507$	− 1.98687e9i	− 0.0300703i
$508$	0	0
$509$	−1.13293e11	−1.68784	−0.843919	−	0.536471i	$-0.819757\pi$
$509$	−1.13293e11	−1.68784	−0.843919	+	0.536471i	$0.819757\pi$
$510$	0	0
$511$	− 7.50607e9i	− 0.110085i
$512$	0	0
$513$	4.80948e10	0.694431
$514$	0	0
$515$	4.55919e10i	0.648124i
$516$	0	0
$517$	1.12835e9	0.0157936
$518$	0	0
$519$	− 2.65846e10i	− 0.366404i
$520$	0	0
$521$	−1.03318e11	−1.40225	−0.701125	−	0.713038i	$-0.747319\pi$
$521$	−1.03318e11	−1.40225	−0.701125	+	0.713038i	$0.747319\pi$
$522$	0	0
$523$	3.55661e8i	0.00475367i	0.999997	+	0.00237684i	$0.000756572\pi$
$523$	3.55661e8i	0.00475367i	−0.999997	+	0.00237684i	$0.999243\pi$
$524$	0	0
$525$	−7.60029e9	−0.100044
$526$	0	0
$527$	− 1.12228e11i	− 1.45499i
$528$	0	0
$529$	−1.60122e11	−2.04469
$530$	0	0
$531$	9.97498e9i	0.125468i
$532$	0	0
$533$	9.50047e10	1.17716
$534$	0	0
$535$	1.51540e11i	1.84974i
$536$	0	0
$537$	2.56836e10	0.308858
$538$	0	0
$539$	− 3.78995e9i	− 0.0449033i
$540$	0	0
$541$	1.30241e11	1.52041	0.760204	−	0.649685i	$-0.225099\pi$
$541$	1.30241e11	1.52041	0.760204	+	0.649685i	$0.225099\pi$
$542$	0	0
$543$	− 7.03697e9i	− 0.0809444i
$544$	0	0
$545$	4.05838e10	0.460009
$546$	0	0
$547$	1.43558e11i	1.60353i	0.597638	+	0.801766i	$0.296106\pi$
$547$	1.43558e11i	1.60353i	−0.597638	+	0.801766i	$0.703894\pi$
$548$	0	0
$549$	−7.40858e10	−0.815540
$550$	0	0
$551$	− 1.07206e11i	− 1.16309i
$552$	0	0
$553$	−8.84048e10	−0.945313
$554$	0	0
$555$	4.65433e9i	0.0490552i
$556$	0	0
$557$	2.53322e10	0.263180	0.131590	−	0.991304i	$-0.457992\pi$
$557$	2.53322e10	0.263180	0.131590	+	0.991304i	$0.457992\pi$
$558$	0	0
$559$	− 5.52600e10i	− 0.565932i
$560$	0	0
$561$	2.11519e9	0.0213550
$562$	0	0
$563$	− 9.04001e10i	− 0.899778i	−0.893084	−	0.449889i	$-0.851464\pi$
$563$	− 9.04001e10i	− 0.899778i	0.893084	−	0.449889i	$-0.148536\pi$
$564$	0	0
$565$	−6.18668e9	−0.0607105
$566$	0	0
$567$	− 6.60599e10i	− 0.639155i
$568$	0	0
$569$	1.46686e11	1.39939	0.699696	−	0.714441i	$-0.253319\pi$
$569$	1.46686e11	1.39939	0.699696	+	0.714441i	$0.253319\pi$
$570$	0	0
$571$	1.32868e11i	1.24990i	0.780666	+	0.624949i	$0.214880\pi$
$571$	1.32868e11i	1.24990i	−0.780666	+	0.624949i	$0.785120\pi$
$572$	0	0
$573$	5.31447e9	0.0492994
$574$	0	0
$575$	1.19262e11i	1.09101i
$576$	0	0
$577$	1.26715e11	1.14321	0.571604	−	0.820529i	$-0.306321\pi$
$577$	1.26715e11	1.14321	0.571604	+	0.820529i	$0.306321\pi$
$578$	0	0
$579$	3.54851e10i	0.315741i
$580$	0	0
$581$	−2.02387e10	−0.177614
$582$	0	0
$583$	− 2.60530e9i	− 0.0225519i
$584$	0	0
$585$	−1.51941e11	−1.29733
$586$	0	0
$587$	2.10292e11i	1.77121i	0.464438	+	0.885606i	$0.346256\pi$
$587$	2.10292e11i	1.77121i	−0.464438	+	0.885606i	$0.653744\pi$
$588$	0	0
$589$	−2.90459e11	−2.41337
$590$	0	0
$591$	− 2.04749e10i	− 0.167831i
$592$	0	0
$593$	1.03227e10	0.0834789	0.0417394	−	0.999129i	$-0.486710\pi$
$593$	1.03227e10	0.0834789	0.0417394	+	0.999129i	$0.486710\pi$
$594$	0	0
$595$	1.17573e11i	0.938076i
$596$	0	0
$597$	2.46263e10	0.193866
$598$	0	0
$599$	− 1.38904e11i	− 1.07897i	−0.841996	−	0.539483i	$-0.818620\pi$
$599$	− 1.38904e11i	− 1.07897i	0.841996	−	0.539483i	$-0.181380\pi$
$600$	0	0
$601$	6.88691e10	0.527870	0.263935	−	0.964540i	$-0.414980\pi$
$601$	6.88691e10	0.527870	0.263935	+	0.964540i	$0.414980\pi$
$602$	0	0
$603$	1.39296e11i	1.05358i
$604$	0	0
$605$	1.69102e11	1.26220
$606$	0	0
$607$	4.98379e10i	0.367117i	0.983009	+	0.183559i	$0.0587617\pi$
$607$	4.98379e10i	0.367117i	−0.983009	+	0.183559i	$0.941238\pi$
$608$	0	0
$609$	1.55451e10	0.113012
$610$	0	0
$611$	− 2.35797e10i	− 0.169189i
$612$	0	0
$613$	−1.64085e11	−1.16205	−0.581027	−	0.813884i	$-0.697349\pi$
$613$	−1.64085e11	−1.16205	−0.581027	+	0.813884i	$0.697349\pi$
$614$	0	0
$615$	4.34206e10i	0.303526i
$616$	0	0
$617$	−1.16747e11	−0.805576	−0.402788	−	0.915293i	$-0.631959\pi$
$617$	−1.16747e11	−0.805576	−0.402788	+	0.915293i	$0.631959\pi$
$618$	0	0
$619$	− 1.51634e11i	− 1.03285i	−0.856334	−	0.516423i	$-0.827263\pi$
$619$	− 1.51634e11i	− 1.03285i	0.856334	−	0.516423i	$-0.172737\pi$
$620$	0	0
$621$	1.09432e11	0.735830
$622$	0	0
$623$	4.41465e10i	0.293051i
$624$	0	0
$625$	−1.88341e11	−1.23431
$626$	0	0
$627$	− 5.47436e9i	− 0.0354212i
$628$	0	0
$629$	2.76993e10	0.176957
$630$	0	0
$631$	2.66911e11i	1.68364i	0.539757	+	0.841821i	$0.318516\pi$
$631$	2.66911e11i	1.68364i	−0.539757	+	0.841821i	$0.681484\pi$
$632$	0	0
$633$	4.84227e10	0.301602
$634$	0	0
$635$	− 2.02476e11i	− 1.24531i
$636$	0	0
$637$	−7.92007e10	−0.481029
$638$	0	0
$639$	− 1.29004e11i	− 0.773748i
$640$	0	0
$641$	1.55457e11	0.920827	0.460414	−	0.887705i	$-0.347701\pi$
$641$	1.55457e11	0.920827	0.460414	+	0.887705i	$0.347701\pi$
$642$	0	0
$643$	− 1.27487e11i	− 0.745800i	−0.927871	−	0.372900i	$-0.878363\pi$
$643$	− 1.27487e11i	− 0.745800i	0.927871	−	0.372900i	$-0.121637\pi$
$644$	0	0
$645$	2.52559e10	0.145923
$646$	0	0
$647$	1.80017e11i	1.02730i	0.858000	+	0.513649i	$0.171707\pi$
$647$	1.80017e11i	1.02730i	−0.858000	+	0.513649i	$0.828293\pi$
$648$	0	0
$649$	2.32629e9	0.0131125
$650$	0	0
$651$	− 4.21171e10i	− 0.234496i
$652$	0	0
$653$	9.62574e9	0.0529397	0.0264699	−	0.999650i	$-0.491573\pi$
$653$	9.62574e9	0.0529397	0.0264699	+	0.999650i	$0.491573\pi$
$654$	0	0
$655$	2.17496e11i	1.18164i
$656$	0	0
$657$	2.63843e10	0.141607
$658$	0	0
$659$	4.10033e10i	0.217409i	0.994074	+	0.108704i	$0.0346702\pi$
$659$	4.10033e10i	0.217409i	−0.994074	+	0.108704i	$0.965330\pi$
$660$	0	0
$661$	2.41101e11	1.26297	0.631486	−	0.775387i	$-0.282445\pi$
$661$	2.41101e11	1.26297	0.631486	+	0.775387i	$0.282445\pi$
$662$	0	0
$663$	− 4.42024e10i	− 0.228766i
$664$	0	0
$665$	3.04291e11	1.55598
$666$	0	0
$667$	− 2.43929e11i	− 1.23243i
$668$	0	0
$669$	−4.43987e10	−0.221649
$670$	0	0
$671$	1.72777e10i	0.0852309i
$672$	0	0
$673$	−1.07571e11	−0.524367	−0.262183	−	0.965018i	$-0.584443\pi$
$673$	−1.07571e11	−0.524367	−0.262183	+	0.965018i	$0.584443\pi$
$674$	0	0
$675$	− 5.47368e10i	− 0.263672i
$676$	0	0
$677$	2.01309e9	0.00958316	0.00479158	−	0.999989i	$-0.498475\pi$
$677$	2.01309e9	0.00958316	0.00479158	+	0.999989i	$0.498475\pi$
$678$	0	0
$679$	− 2.03730e10i	− 0.0958466i
$680$	0	0
$681$	−2.41046e10	−0.112076
$682$	0	0
$683$	− 7.91693e10i	− 0.363810i	−0.983316	−	0.181905i	$-0.941774\pi$
$683$	− 7.91693e10i	− 0.363810i	0.983316	−	0.181905i	$-0.0582263\pi$
$684$	0	0
$685$	2.96680e10	0.134749
$686$	0	0
$687$	5.70275e10i	0.256010i
$688$	0	0
$689$	−5.44444e10	−0.241589
$690$	0	0
$691$	− 4.38242e10i	− 0.192222i	−0.995371	−	0.0961108i	$-0.969360\pi$
$691$	− 4.38242e10i	− 0.192222i	0.995371	−	0.0961108i	$-0.0306403\pi$
$692$	0	0
$693$	−1.62386e10	−0.0704070
$694$	0	0
$695$	8.39948e10i	0.360009i
$696$	0	0
$697$	2.58409e11	1.09491
$698$	0	0
$699$	6.97083e10i	0.291995i
$700$	0	0
$701$	−1.77030e10	−0.0733118	−0.0366559	−	0.999328i	$-0.511671\pi$
$701$	−1.77030e10	−0.0733118	−0.0366559	+	0.999328i	$0.511671\pi$
$702$	0	0
$703$	− 7.16890e10i	− 0.293516i
$704$	0	0
$705$	1.07768e10	0.0436247
$706$	0	0
$707$	− 2.62791e11i	− 1.05180i
$708$	0	0
$709$	6.68754e10	0.264656	0.132328	−	0.991206i	$-0.457755\pi$
$709$	6.68754e10	0.264656	0.132328	+	0.991206i	$0.457755\pi$
$710$	0	0
$711$	− 3.10748e11i	− 1.21599i
$712$	0	0
$713$	−6.60891e11	−2.55724
$714$	0	0
$715$	3.54346e10i	0.135582i
$716$	0	0
$717$	−1.67702e10	−0.0634544
$718$	0	0
$719$	− 4.12968e11i	− 1.54526i	−0.634857	−	0.772630i	$-0.718941\pi$
$719$	− 4.12968e11i	− 1.54526i	0.634857	−	0.772630i	$-0.281059\pi$
$720$	0	0
$721$	1.01826e11	0.376804
$722$	0	0
$723$	− 2.90220e10i	− 0.106212i
$724$	0	0
$725$	−1.22011e11	−0.441619
$726$	0	0
$727$	− 4.23681e11i	− 1.51671i	−0.651843	−	0.758354i	$-0.726004\pi$
$727$	− 4.23681e11i	− 1.51671i	0.651843	−	0.758354i	$-0.273996\pi$
$728$	0	0
$729$	2.06493e11	0.731130
$730$	0	0
$731$	− 1.50305e11i	− 0.526387i
$732$	0	0
$733$	9.12513e10	0.316099	0.158050	−	0.987431i	$-0.449479\pi$
$733$	9.12513e10	0.316099	0.158050	+	0.987431i	$0.449479\pi$
$734$	0	0
$735$	− 3.61976e10i	− 0.124031i
$736$	0	0
$737$	3.24855e10	0.110108
$738$	0	0
$739$	− 2.27721e11i	− 0.763527i	−0.924260	−	0.381764i	$-0.875317\pi$
$739$	− 2.27721e11i	− 0.763527i	0.924260	−	0.381764i	$-0.124683\pi$
$740$	0	0
$741$	−1.14401e11	−0.379451
$742$	0	0
$743$	2.30762e10i	0.0757196i	0.999283	+	0.0378598i	$0.0120540\pi$
$743$	2.30762e10i	0.0757196i	−0.999283	+	0.0378598i	$0.987946\pi$
$744$	0	0
$745$	−4.84860e11	−1.57395
$746$	0	0
$747$	− 7.11401e10i	− 0.228471i
$748$	0	0
$749$	3.38451e11	1.07540
$750$	0	0
$751$	8.42681e10i	0.264913i	0.991189	+	0.132457i	$0.0422865\pi$
$751$	8.42681e10i	0.264913i	−0.991189	+	0.132457i	$0.957713\pi$
$752$	0	0
$753$	5.13344e10	0.159672
$754$	0	0
$755$	− 3.77460e11i	− 1.16167i
$756$	0	0
$757$	−4.53458e11	−1.38087	−0.690436	−	0.723393i	$-0.742581\pi$
$757$	−4.53458e11	−1.38087	−0.690436	+	0.723393i	$0.742581\pi$
$758$	0	0
$759$	− 1.24560e10i	− 0.0375329i
$760$	0	0
$761$	3.05406e11	0.910623	0.455311	−	0.890332i	$-0.349528\pi$
$761$	3.05406e11	0.910623	0.455311	+	0.890332i	$0.349528\pi$
$762$	0	0
$763$	− 9.06406e10i	− 0.267439i
$764$	0	0
$765$	−4.13274e11	−1.20668
$766$	0	0
$767$	− 4.86138e10i	− 0.140468i
$768$	0	0
$769$	3.54824e11	1.01463	0.507316	−	0.861760i	$-0.330638\pi$
$769$	3.54824e11	1.01463	0.507316	+	0.861760i	$0.330638\pi$
$770$	0	0
$771$	1.64751e10i	0.0466242i
$772$	0	0
$773$	1.47448e11	0.412971	0.206485	−	0.978450i	$-0.433797\pi$
$773$	1.47448e11	0.412971	0.206485	+	0.978450i	$0.433797\pi$
$774$	0	0
$775$	3.30572e11i	0.916345i
$776$	0	0
$777$	1.03950e10	0.0285195
$778$	0	0
$779$	− 6.68793e11i	− 1.81611i
$780$	0	0
$781$	−3.00854e10	−0.0808633
$782$	0	0
$783$	1.11955e11i	0.297848i
$784$	0	0
$785$	−3.89290e11	−1.02517
$786$	0	0
$787$	− 5.12163e11i	− 1.33509i	−0.744572	−	0.667543i	$-0.767346\pi$
$787$	− 5.12163e11i	− 1.33509i	0.744572	−	0.667543i	$-0.232654\pi$
$788$	0	0
$789$	−7.61183e10	−0.196418
$790$	0	0
$791$	1.38174e10i	0.0352956i
$792$	0	0
$793$	3.61063e11	0.913040
$794$	0	0
$795$	− 2.48831e10i	− 0.0622925i
$796$	0	0
$797$	−3.39499e11	−0.841404	−0.420702	−	0.907199i	$-0.638216\pi$
$797$	−3.39499e11	−0.841404	−0.420702	+	0.907199i	$0.638216\pi$
$798$	0	0
$799$	− 6.41359e10i	− 0.157367i
$800$	0	0
$801$	−1.55177e11	−0.376962
$802$	0	0
$803$	− 6.15315e9i	− 0.0147991i
$804$	0	0
$805$	6.92364e11	1.64874
$806$	0	0
$807$	2.86675e10i	0.0675920i
$808$	0	0
$809$	−1.13991e11	−0.266120	−0.133060	−	0.991108i	$-0.542480\pi$
$809$	−1.13991e11	−0.266120	−0.133060	+	0.991108i	$0.542480\pi$
$810$	0	0
$811$	5.19425e10i	0.120071i	0.998196	+	0.0600357i	$0.0191215\pi$
$811$	5.19425e10i	0.120071i	−0.998196	+	0.0600357i	$0.980879\pi$
$812$	0	0
$813$	−1.14280e11	−0.261582
$814$	0	0
$815$	7.12762e11i	1.61553i
$816$	0	0
$817$	−3.89007e11	−0.873112
$818$	0	0
$819$	3.39347e11i	0.754239i
$820$	0	0
$821$	1.77079e11	0.389757	0.194878	−	0.980827i	$-0.437569\pi$
$821$	1.77079e11	0.389757	0.194878	+	0.980827i	$0.437569\pi$
$822$	0	0
$823$	6.99908e9i	0.0152560i	0.999971	+	0.00762802i	$0.00242810\pi$
$823$	6.99908e9i	0.0152560i	−0.999971	+	0.00762802i	$0.997572\pi$
$824$	0	0
$825$	−6.23038e9	−0.0134493
$826$	0	0
$827$	− 7.66261e11i	− 1.63815i	−0.573683	−	0.819077i	$-0.694486\pi$
$827$	− 7.66261e11i	− 1.63815i	0.573683	−	0.819077i	$-0.305514\pi$
$828$	0	0
$829$	6.81365e11	1.44265	0.721327	−	0.692595i	$-0.243533\pi$
$829$	6.81365e11	1.44265	0.721327	+	0.692595i	$0.243533\pi$
$830$	0	0
$831$	− 1.49656e11i	− 0.313826i
$832$	0	0
$833$	−2.15423e11	−0.447417
$834$	0	0
$835$	− 5.61003e10i	− 0.115404i
$836$	0	0
$837$	3.03325e11	0.618025
$838$	0	0
$839$	2.98466e11i	0.602348i	0.953569	+	0.301174i	$0.0973785\pi$
$839$	2.98466e11i	0.602348i	−0.953569	+	0.301174i	$0.902622\pi$
$840$	0	0
$841$	−2.50693e11	−0.501140
$842$	0	0
$843$	− 9.07833e10i	− 0.179761i
$844$	0	0
$845$	9.05337e10	0.177576
$846$	0	0
$847$	− 3.77676e11i	− 0.733813i
$848$	0	0
$849$	−1.24698e11	−0.240011
$850$	0	0
$851$	− 1.63117e11i	− 0.311014i
$852$	0	0
$853$	−8.28052e11	−1.56409	−0.782045	−	0.623223i	$-0.785823\pi$
$853$	−8.28052e11	−1.56409	−0.782045	+	0.623223i	$0.785823\pi$
$854$	0	0
$855$	1.06960e12i	2.00151i
$856$	0	0
$857$	3.13054e11	0.580357	0.290179	−	0.956973i	$-0.406285\pi$
$857$	3.13054e11	0.580357	0.290179	+	0.956973i	$0.406285\pi$
$858$	0	0
$859$	5.33113e11i	0.979145i	0.871963	+	0.489572i	$0.162847\pi$
$859$	5.33113e11i	0.979145i	−0.871963	+	0.489572i	$0.837153\pi$
$860$	0	0
$861$	9.69763e10	0.176463
$862$	0	0
$863$	− 1.76376e11i	− 0.317977i	−0.987280	−	0.158989i	$-0.949177\pi$
$863$	− 1.76376e11i	− 0.317977i	0.987280	−	0.158989i	$-0.0508233\pi$
$864$	0	0
$865$	1.21135e12	2.16374
$866$	0	0
$867$	1.75203e9i	0.00310075i
$868$	0	0
$869$	−7.24703e10	−0.127081
$870$	0	0
$871$	− 6.78869e11i	− 1.17954i
$872$	0	0
$873$	7.16124e10	0.123291
$874$	0	0
$875$	2.07561e11i	0.354090i
$876$	0	0
$877$	−4.62552e11	−0.781920	−0.390960	−	0.920408i	$-0.627857\pi$
$877$	−4.62552e11	−0.781920	−0.390960	+	0.920408i	$0.627857\pi$
$878$	0	0
$879$	4.82601e10i	0.0808412i
$880$	0	0
$881$	1.48349e10	0.0246253	0.0123127	−	0.999924i	$-0.496081\pi$
$881$	1.48349e10	0.0246253	0.0123127	+	0.999924i	$0.496081\pi$
$882$	0	0
$883$	6.64762e11i	1.09351i	0.837292	+	0.546756i	$0.184137\pi$
$883$	6.64762e11i	1.09351i	−0.837292	+	0.546756i	$0.815863\pi$
$884$	0	0
$885$	2.22183e10	0.0362191
$886$	0	0
$887$	− 5.25943e11i	− 0.849658i	−0.905274	−	0.424829i	$-0.860334\pi$
$887$	− 5.25943e11i	− 0.849658i	0.905274	−	0.424829i	$-0.139666\pi$
$888$	0	0
$889$	−4.52213e11	−0.723996
$890$	0	0
$891$	− 5.41530e10i	− 0.0859235i
$892$	0	0
$893$	−1.65991e11	−0.261023
$894$	0	0
$895$	1.17030e12i	1.82391i
$896$	0	0
$897$	−2.60300e11	−0.402073
$898$	0	0
$899$	− 6.76127e11i	− 1.03512i
$900$	0	0
$901$	−1.48087e11	−0.224707
$902$	0	0
$903$	− 5.64069e10i	− 0.0848361i
$904$	0	0
$905$	3.20646e11	0.478004
$906$	0	0
$907$	1.11825e12i	1.65239i	0.563388	+	0.826193i	$0.309498\pi$
$907$	1.11825e12i	1.65239i	−0.563388	+	0.826193i	$0.690502\pi$
$908$	0	0
$909$	9.23725e11	1.35297
$910$	0	0
$911$	− 3.98496e11i	− 0.578563i	−0.957244	−	0.289282i	$-0.906584\pi$
$911$	− 3.98496e11i	− 0.578563i	0.957244	−	0.289282i	$-0.0934164\pi$
$912$	0	0
$913$	−1.65908e10	−0.0238772
$914$	0	0
$915$	1.65019e11i	0.235423i
$916$	0	0
$917$	4.85760e11	0.686980
$918$	0	0
$919$	− 6.28911e11i	− 0.881713i	−0.897578	−	0.440857i	$-0.854675\pi$
$919$	− 6.28911e11i	− 0.881713i	0.897578	−	0.440857i	$-0.145325\pi$
$920$	0	0
$921$	−1.80902e11	−0.251423
$922$	0	0
$923$	6.28711e11i	0.866252i
$924$	0	0
$925$	−8.15894e10	−0.111447
$926$	0	0
$927$	3.57923e11i	0.484697i
$928$	0	0
$929$	7.31045e11	0.981480	0.490740	−	0.871306i	$-0.336727\pi$
$929$	7.31045e11	0.981480	0.490740	+	0.871306i	$0.336727\pi$
$930$	0	0
$931$	5.57540e11i	0.742125i
$932$	0	0
$933$	1.08389e11	0.143040
$934$	0	0
$935$	9.63808e10i	0.126108i
$936$	0	0
$937$	−7.48792e11	−0.971411	−0.485705	−	0.874123i	$-0.661437\pi$
$937$	−7.48792e11	−0.971411	−0.485705	+	0.874123i	$0.661437\pi$
$938$	0	0
$939$	1.27046e11i	0.163417i
$940$	0	0
$941$	−9.67709e11	−1.23420	−0.617101	−	0.786884i	$-0.711693\pi$
$941$	−9.67709e11	−1.23420	−0.617101	+	0.786884i	$0.711693\pi$
$942$	0	0
$943$	− 1.52173e12i	− 1.92438i
$944$	0	0
$945$	−3.17770e11	−0.398461
$946$	0	0
$947$	− 1.41847e11i	− 0.176368i	−0.996104	−	0.0881839i	$-0.971894\pi$
$947$	− 1.41847e11i	− 0.176368i	0.996104	−	0.0881839i	$-0.0281063\pi$
$948$	0	0
$949$	−1.28586e11	−0.158536
$950$	0	0
$951$	1.82644e11i	0.223297i
$952$	0	0
$953$	1.01418e12	1.22954	0.614771	−	0.788706i	$-0.289249\pi$
$953$	1.01418e12	1.22954	0.614771	+	0.788706i	$0.289249\pi$
$954$	0	0
$955$	2.42159e11i	0.291130i
$956$	0	0
$957$	1.27432e10	0.0151925
$958$	0	0
$959$	− 6.62610e10i	− 0.0783400i
$960$	0	0
$961$	−9.78977e11	−1.14783
$962$	0	0
$963$	1.18968e12i	1.38332i
$964$	0	0
$965$	−1.61691e12	−1.86456
$966$	0	0
$967$	− 4.83390e11i	− 0.552830i	−0.961038	−	0.276415i	$-0.910854\pi$
$967$	− 4.83390e11i	− 0.552830i	0.961038	−	0.276415i	$-0.0891464\pi$
$968$	0	0
$969$	−3.11166e11	−0.352937
$970$	0	0
$971$	− 1.16310e12i	− 1.30840i	−0.756323	−	0.654198i	$-0.773006\pi$
$971$	− 1.16310e12i	− 1.30840i	0.756323	−	0.654198i	$-0.226994\pi$
$972$	0	0
$973$	1.87595e11	0.209301
$974$	0	0
$975$	1.30200e11i	0.144076i
$976$	0	0
$977$	4.01473e11	0.440634	0.220317	−	0.975428i	$-0.429291\pi$
$977$	4.01473e11	0.440634	0.220317	+	0.975428i	$0.429291\pi$
$978$	0	0
$979$	3.61893e10i	0.0393958i
$980$	0	0
$981$	3.18607e11	0.344016
$982$	0	0
$983$	− 1.72803e11i	− 0.185071i	−0.995709	−	0.0925353i	$-0.970503\pi$
$983$	− 1.72803e11i	− 0.185071i	0.995709	−	0.0925353i	$-0.0294971\pi$
$984$	0	0
$985$	9.32960e11	0.991101
$986$	0	0
$987$	− 2.40690e10i	− 0.0253624i
$988$	0	0
$989$	−8.85122e11	−0.925163
$990$	0	0
$991$	− 1.68640e12i	− 1.74850i	−0.485472	−	0.874252i	$-0.661352\pi$
$991$	− 1.68640e12i	− 1.74850i	0.485472	−	0.874252i	$-0.338648\pi$
$992$	0	0
$993$	8.74075e10	0.0898983
$994$	0	0
$995$	1.12212e12i	1.14485i
$996$	0	0
$997$	5.49409e11	0.556052	0.278026	−	0.960574i	$-0.410320\pi$
$997$	5.49409e11	0.556052	0.278026	+	0.960574i	$0.410320\pi$
$998$	0	0
$999$	7.48645e10i	0.0751647i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	256.9.c.n.255.6		12
4.3	odd	2	inner	256.9.c.n.255.8		12
8.3	odd	2	inner	256.9.c.n.255.5		12
8.5	even	2	inner	256.9.c.n.255.7		12
16.3	odd	4		8.9.d.b.3.2	yes	6
16.5	even	4		8.9.d.b.3.1	✓	6
16.11	odd	4		32.9.d.b.15.4		6
16.13	even	4		32.9.d.b.15.3		6
48.5	odd	4		72.9.b.b.19.6		6
48.11	even	4		288.9.b.b.271.2		6
48.29	odd	4		288.9.b.b.271.5		6
48.35	even	4		72.9.b.b.19.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.9.d.b.3.1	✓	6	16.5	even	4
8.9.d.b.3.2	yes	6	16.3	odd	4
32.9.d.b.15.3		6	16.13	even	4
32.9.d.b.15.4		6	16.11	odd	4
72.9.b.b.19.5		6	48.35	even	4
72.9.b.b.19.6		6	48.5	odd	4
256.9.c.n.255.5		12	8.3	odd	2	inner
256.9.c.n.255.6		12	1.1	even	1	trivial
256.9.c.n.255.7		12	8.5	even	2	inner
256.9.c.n.255.8		12	4.3	odd	2	inner
288.9.b.b.271.2		6	48.11	even	4
288.9.b.b.271.5		6	48.29	odd	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:	\( N \)	\(=\)	\( 256 = 2^{8} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	256.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-17.4864i q^{3} +796.785 q^{5} -1779.55i q^{7} +6255.23 q^{9} +O(q^{10})\)	\(q-17.4864i q^{3} +796.785 q^{5} -1779.55i q^{7} +6255.23 q^{9} -1458.80i q^{11} -30485.3 q^{13} -13932.9i q^{15} -82919.0 q^{17} +214604. i q^{19} -31118.0 q^{21} +488296. i q^{23} +244241. q^{25} -224110. i q^{27} -499553. q^{29} +1.35347e6i q^{31} -25509.1 q^{33} -1.41792e6i q^{35} -334053. q^{37} +533079. i q^{39} -3.11641e6 q^{41} +1.81268e6i q^{43} +4.98407e6 q^{45} +773477. i q^{47} +2.59799e6 q^{49} +1.44996e6i q^{51} +1.78592e6 q^{53} -1.16235e6i q^{55} +3.75265e6 q^{57} +1.59466e6i q^{59} -1.18438e7 q^{61} -1.11315e7i q^{63} -2.42902e7 q^{65} +2.22687e7i q^{67} +8.53854e6 q^{69} -2.06234e7i q^{71} +4.21796e6 q^{73} -4.27090e6i q^{75} -2.59601e6 q^{77} -4.96781e7i q^{79} +3.71217e7 q^{81} -1.13729e7i q^{83} -6.60686e7 q^{85} +8.73538e6i q^{87} -2.48076e7 q^{89} +5.42502e7i q^{91} +2.36672e7 q^{93} +1.70993e8i q^{95} +1.14484e7 q^{97} -9.12511e6i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 32676 q^{9}+O(q^{10}) \) 12 * q - 32676 * q^9	\( 12 q - 32676 q^{9} - 402152 q^{17} + 895860 q^{25} + 3473712 q^{33} - 7634200 q^{41} - 25310964 q^{49} - 15047472 q^{57} + 54120960 q^{65} + 110969832 q^{73} - 158290884 q^{81} - 284346904 q^{89} + 250387224 q^{97}+O(q^{100}) \) 12 * q - 32676 * q^9 - 402152 * q^17 + 895860 * q^25 + 3473712 * q^33 - 7634200 * q^41 - 25310964 * q^49 - 15047472 * q^57 + 54120960 * q^65 + 110969832 * q^73 - 158290884 * q^81 - 284346904 * q^89 + 250387224 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more