LMFDB - Embedded newform 1248.2.g.b.625.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1248,2,Mod(625,1248)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1248, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1248.625");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1248 = 2^{5} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1248.g (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$9.96533017226$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} + 2 x^{14} - 4 x^{13} + 9 x^{12} - 10 x^{11} + 2 x^{10} - 8 x^{9} + 28 x^{8} + \cdots + 256 $ x^16 - 2x^15 + 2x^14 - 4x^13 + 9x^12 - 10x^11 + 2x^10 - 8x^9 + 28x^8 - 16x^7 + 8x^6 - 80x^5 + 144x^4 - 128x^3 + 128x^2 - 256*x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{20} $
Twist minimal:	no (minimal twist has level 312)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			625.8
Root			$-1.32561 + 0.492712i$ of defining polynomial
Character	$\chi$	$=$	1248.625
Dual form			1248.2.g.b.625.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +4.33571i q^{5} +2.30442 q^{7} -1.00000 q^{9} +0.582255i q^{11} -1.00000i q^{13} +4.33571 q^{15} +3.40894 q^{17} -0.0627418i q^{19} -2.30442i q^{21} +6.65973 q^{23} -13.7984 q^{25} +1.00000i q^{27} +2.41805i q^{29} -5.63600 q^{31} +0.582255 q^{33} +9.99131i q^{35} +9.27328i q^{37} -1.00000 q^{39} +4.75376 q^{41} +7.86434i q^{43} -4.33571i q^{45} +0.0312900 q^{47} -1.68964 q^{49} -3.40894i q^{51} +13.1656i q^{53} -2.52449 q^{55} -0.0627418 q^{57} -4.75759i q^{59} +3.29149i q^{61} -2.30442 q^{63} +4.33571 q^{65} -11.8777i q^{67} -6.65973i q^{69} -13.6251 q^{71} +0.437175 q^{73} +13.7984i q^{75} +1.34176i q^{77} +6.54503 q^{79} +1.00000 q^{81} -6.62844i q^{83} +14.7802i q^{85} +2.41805 q^{87} +8.23880 q^{89} -2.30442i q^{91} +5.63600i q^{93} +0.272031 q^{95} -0.664432 q^{97} -0.582255i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 4 q^{7} - 16 q^{9} + 4 q^{15} + 16 q^{17} + 8 q^{23} - 32 q^{25} + 4 q^{31} - 16 q^{39} - 36 q^{41} - 24 q^{47} + 48 q^{49} - 24 q^{55} - 12 q^{57} + 4 q^{63} + 4 q^{65} - 32 q^{73} + 16 q^{81} - 8 q^{87}+ \cdots + 40 q^{97}+O(q^{100}) $ 16 * q - 4 * q^7 - 16 * q^9 + 4 * q^15 + 16 * q^17 + 8 * q^23 - 32 * q^25 + 4 * q^31 - 16 * q^39 - 36 * q^41 - 24 * q^47 + 48 * q^49 - 24 * q^55 - 12 * q^57 + 4 * q^63 + 4 * q^65 - 32 * q^73 + 16 * q^81 - 8 * q^87 - 60 * q^89 + 24 * q^95 + 40 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1248\mathbb{Z}\right)^\times$.

$n$	$703$	$769$	$833$	$1093$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	0	0
$5$	4.33571i	1.93899i	0.245112	+	0.969495i	$0.421175\pi$
$5$	4.33571i	1.93899i	−0.245112	+	0.969495i	$0.578825\pi$
$6$	0	0
$7$	2.30442	0.870990	0.435495	−	0.900191i	$-0.356573\pi$
$7$	2.30442	0.870990	0.435495	+	0.900191i	$0.356573\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	0.582255i	0.175557i	0.996140	+	0.0877783i	$0.0279767\pi$
$11$	0.582255i	0.175557i	−0.996140	+	0.0877783i	$0.972023\pi$
$12$	0	0
$13$	− 1.00000i	− 0.277350i
$13$	− 1.00000i	− 0.277350i
$14$	0	0
$15$	4.33571	1.11948
$16$	0	0
$17$	3.40894	0.826790	0.413395	−	0.910552i	$-0.364343\pi$
$17$	3.40894	0.826790	0.413395	+	0.910552i	$0.364343\pi$
$18$	0	0
$19$	− 0.0627418i	− 0.0143940i	−0.999974	−	0.00719698i	$-0.997709\pi$
$19$	− 0.0627418i	− 0.0143940i	0.999974	−	0.00719698i	$-0.00229089\pi$
$20$	0	0
$21$	− 2.30442i	− 0.502866i
$22$	0	0
$23$	6.65973	1.38865	0.694325	−	0.719662i	$-0.255703\pi$
$23$	6.65973	1.38865	0.694325	+	0.719662i	$0.255703\pi$
$24$	0	0
$25$	−13.7984	−2.75968
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	2.41805i	0.449021i	0.974472	+	0.224510i	$0.0720783\pi$
$29$	2.41805i	0.449021i	−0.974472	+	0.224510i	$0.927922\pi$
$30$	0	0
$31$	−5.63600	−1.01226	−0.506128	−	0.862458i	$-0.668924\pi$
$31$	−5.63600	−1.01226	−0.506128	+	0.862458i	$0.668924\pi$
$32$	0	0
$33$	0.582255	0.101358
$34$	0	0
$35$	9.99131i	1.68884i
$36$	0	0
$37$	9.27328i	1.52452i	0.647272	+	0.762259i	$0.275910\pi$
$37$	9.27328i	1.52452i	−0.647272	+	0.762259i	$0.724090\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	4.75376	0.742413	0.371207	−	0.928550i	$-0.378944\pi$
$41$	4.75376	0.742413	0.371207	+	0.928550i	$0.378944\pi$
$42$	0	0
$43$	7.86434i	1.19930i	0.800262	+	0.599650i	$0.204694\pi$
$43$	7.86434i	1.19930i	−0.800262	+	0.599650i	$0.795306\pi$
$44$	0	0
$45$	− 4.33571i	− 0.646330i
$46$	0	0
$47$	0.0312900	0.00456412	0.00228206	−	0.999997i	$-0.499274\pi$
$47$	0.0312900	0.00456412	0.00228206	+	0.999997i	$0.499274\pi$
$48$	0	0
$49$	−1.68964	−0.241377
$50$	0	0
$51$	− 3.40894i	− 0.477347i
$52$	0	0
$53$	13.1656i	1.80843i	0.427078	+	0.904215i	$0.359543\pi$
$53$	13.1656i	1.80843i	−0.427078	+	0.904215i	$0.640457\pi$
$54$	0	0
$55$	−2.52449	−0.340402
$56$	0	0
$57$	−0.0627418	−0.00831036
$58$	0	0
$59$	− 4.75759i	− 0.619385i	−0.950837	−	0.309693i	$-0.899774\pi$
$59$	− 4.75759i	− 0.619385i	0.950837	−	0.309693i	$-0.100226\pi$
$60$	0	0
$61$	3.29149i	0.421432i	0.977547	+	0.210716i	$0.0675795\pi$
$61$	3.29149i	0.421432i	−0.977547	+	0.210716i	$0.932420\pi$
$62$	0	0
$63$	−2.30442	−0.290330
$64$	0	0
$65$	4.33571	0.537779
$66$	0	0
$67$	− 11.8777i	− 1.45109i	−0.688175	−	0.725545i	$-0.741588\pi$
$67$	− 11.8777i	− 1.45109i	0.688175	−	0.725545i	$-0.258412\pi$
$68$	0	0
$69$	− 6.65973i	− 0.801737i
$70$	0	0
$71$	−13.6251	−1.61700	−0.808500	−	0.588496i	$-0.799720\pi$
$71$	−13.6251	−1.61700	−0.808500	+	0.588496i	$0.799720\pi$
$72$	0	0
$73$	0.437175	0.0511675	0.0255837	−	0.999673i	$-0.491856\pi$
$73$	0.437175	0.0511675	0.0255837	+	0.999673i	$0.491856\pi$
$74$	0	0
$75$	13.7984i	1.59330i
$76$	0	0
$77$	1.34176i	0.152908i
$78$	0	0
$79$	6.54503	0.736373	0.368187	−	0.929752i	$-0.379979\pi$
$79$	6.54503	0.736373	0.368187	+	0.929752i	$0.379979\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 6.62844i	− 0.727566i	−0.931484	−	0.363783i	$-0.881485\pi$
$83$	− 6.62844i	− 0.727566i	0.931484	−	0.363783i	$-0.118515\pi$
$84$	0	0
$85$	14.7802i	1.60314i
$86$	0	0
$87$	2.41805	0.259242
$88$	0	0
$89$	8.23880	0.873311	0.436655	−	0.899629i	$-0.356163\pi$
$89$	8.23880	0.873311	0.436655	+	0.899629i	$0.356163\pi$
$90$	0	0
$91$	− 2.30442i	− 0.241569i
$92$	0	0
$93$	5.63600i	0.584426i
$94$	0	0
$95$	0.272031	0.0279097
$96$	0	0
$97$	−0.664432	−0.0674629	−0.0337314	−	0.999431i	$-0.510739\pi$
$97$	−0.664432	−0.0674629	−0.0337314	+	0.999431i	$0.510739\pi$
$98$	0	0
$99$	− 0.582255i	− 0.0585189i
$100$	0	0
$101$	11.3867i	1.13302i	0.824055	+	0.566510i	$0.191707\pi$
$101$	11.3867i	1.13302i	−0.824055	+	0.566510i	$0.808293\pi$
$102$	0	0
$103$	14.9131	1.46943	0.734716	−	0.678375i	$-0.237315\pi$
$103$	14.9131	1.46943	0.734716	+	0.678375i	$0.237315\pi$
$104$	0	0
$105$	9.99131	0.975052
$106$	0	0
$107$	7.39073i	0.714489i	0.934011	+	0.357244i	$0.116284\pi$
$107$	7.39073i	0.714489i	−0.934011	+	0.357244i	$0.883716\pi$
$108$	0	0
$109$	− 17.4880i	− 1.67505i	−0.546399	−	0.837525i	$-0.684002\pi$
$109$	− 17.4880i	− 1.67505i	0.546399	−	0.837525i	$-0.315998\pi$
$110$	0	0
$111$	9.27328	0.880181
$112$	0	0
$113$	10.4662	0.984576	0.492288	−	0.870432i	$-0.336161\pi$
$113$	10.4662	0.984576	0.492288	+	0.870432i	$0.336161\pi$
$114$	0	0
$115$	28.8747i	2.69258i
$116$	0	0
$117$	1.00000i	0.0924500i
$118$	0	0
$119$	7.85564	0.720126
$120$	0	0
$121$	10.6610	0.969180
$122$	0	0
$123$	− 4.75376i	− 0.428632i
$124$	0	0
$125$	− 38.1473i	− 3.41200i
$126$	0	0
$127$	−20.9445	−1.85852	−0.929262	−	0.369420i	$-0.879556\pi$
$127$	−20.9445	−1.85852	−0.929262	+	0.369420i	$0.879556\pi$
$128$	0	0
$129$	7.86434	0.692416
$130$	0	0
$131$	− 3.39073i	− 0.296249i	−0.988969	−	0.148125i	$-0.952676\pi$
$131$	− 3.39073i	− 0.296249i	0.988969	−	0.148125i	$-0.0473237\pi$
$132$	0	0
$133$	− 0.144584i	− 0.0125370i
$134$	0	0
$135$	−4.33571	−0.373159
$136$	0	0
$137$	−5.78852	−0.494546	−0.247273	−	0.968946i	$-0.579534\pi$
$137$	−5.78852	−0.494546	−0.247273	+	0.968946i	$0.579534\pi$
$138$	0	0
$139$	18.0874i	1.53415i	0.641558	+	0.767075i	$0.278289\pi$
$139$	18.0874i	1.53415i	−0.641558	+	0.767075i	$0.721711\pi$
$140$	0	0
$141$	− 0.0312900i	− 0.00263509i
$142$	0	0
$143$	0.582255	0.0486906
$144$	0	0
$145$	−10.4840	−0.870647
$146$	0	0
$147$	1.68964i	0.139359i
$148$	0	0
$149$	− 9.21251i	− 0.754719i	−0.926067	−	0.377359i	$-0.876832\pi$
$149$	− 9.21251i	− 0.754719i	0.926067	−	0.377359i	$-0.123168\pi$
$150$	0	0
$151$	2.49249	0.202836	0.101418	−	0.994844i	$-0.467662\pi$
$151$	2.49249	0.202836	0.101418	+	0.994844i	$0.467662\pi$
$152$	0	0
$153$	−3.40894	−0.275597
$154$	0	0
$155$	− 24.4361i	− 1.96275i
$156$	0	0
$157$	− 4.18956i	− 0.334363i	−0.985926	−	0.167182i	$-0.946533\pi$
$157$	− 4.18956i	− 0.334363i	0.985926	−	0.167182i	$-0.0534666\pi$
$158$	0	0
$159$	13.1656	1.04410
$160$	0	0
$161$	15.3468	1.20950
$162$	0	0
$163$	− 10.8387i	− 0.848949i	−0.905440	−	0.424475i	$-0.860459\pi$
$163$	− 10.8387i	− 0.848949i	0.905440	−	0.424475i	$-0.139541\pi$
$164$	0	0
$165$	2.52449i	0.196531i
$166$	0	0
$167$	13.5345	1.04733	0.523667	−	0.851923i	$-0.324564\pi$
$167$	13.5345	1.04733	0.523667	+	0.851923i	$0.324564\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	0.0627418i	0.00479799i
$172$	0	0
$173$	11.3922i	0.866134i	0.901362	+	0.433067i	$0.142569\pi$
$173$	11.3922i	0.866134i	−0.901362	+	0.433067i	$0.857431\pi$
$174$	0	0
$175$	−31.7974	−2.40365
$176$	0	0
$177$	−4.75759	−0.357602
$178$	0	0
$179$	− 9.99131i	− 0.746786i	−0.927673	−	0.373393i	$-0.878194\pi$
$179$	− 9.99131i	− 0.746786i	0.927673	−	0.373393i	$-0.121806\pi$
$180$	0	0
$181$	− 24.2251i	− 1.80064i	−0.435229	−	0.900320i	$-0.643332\pi$
$181$	− 24.2251i	− 1.80064i	0.435229	−	0.900320i	$-0.356668\pi$
$182$	0	0
$183$	3.29149	0.243314
$184$	0	0
$185$	−40.2063	−2.95602
$186$	0	0
$187$	1.98487i	0.145148i
$188$	0	0
$189$	2.30442i	0.167622i
$190$	0	0
$191$	14.2745	1.03287	0.516434	−	0.856327i	$-0.327259\pi$
$191$	14.2745	1.03287	0.516434	+	0.856327i	$0.327259\pi$
$192$	0	0
$193$	4.98894	0.359112	0.179556	−	0.983748i	$-0.442534\pi$
$193$	4.98894	0.359112	0.179556	+	0.983748i	$0.442534\pi$
$194$	0	0
$195$	− 4.33571i	− 0.310487i
$196$	0	0
$197$	− 10.9989i	− 0.783637i	−0.920042	−	0.391819i	$-0.871846\pi$
$197$	− 10.9989i	− 0.783637i	0.920042	−	0.391819i	$-0.128154\pi$
$198$	0	0
$199$	−9.13566	−0.647609	−0.323805	−	0.946124i	$-0.604962\pi$
$199$	−9.13566	−0.647609	−0.323805	+	0.946124i	$0.604962\pi$
$200$	0	0
$201$	−11.8777	−0.837787
$202$	0	0
$203$	5.57221i	0.391093i
$204$	0	0
$205$	20.6110i	1.43953i
$206$	0	0
$207$	−6.65973	−0.462883
$208$	0	0
$209$	0.0365318	0.00252696
$210$	0	0
$211$	1.06551i	0.0733526i	0.999327	+	0.0366763i	$0.0116770\pi$
$211$	1.06551i	0.0733526i	−0.999327	+	0.0366763i	$0.988323\pi$
$212$	0	0
$213$	13.6251i	0.933576i
$214$	0	0
$215$	−34.0975	−2.32543
$216$	0	0
$217$	−12.9877	−0.881665
$218$	0	0
$219$	− 0.437175i	− 0.0295416i
$220$	0	0
$221$	− 3.40894i	− 0.229310i
$222$	0	0
$223$	−8.91055	−0.596695	−0.298347	−	0.954457i	$-0.596435\pi$
$223$	−8.91055	−0.596695	−0.298347	+	0.954457i	$0.596435\pi$
$224$	0	0
$225$	13.7984	0.919894
$226$	0	0
$227$	− 17.1895i	− 1.14091i	−0.821330	−	0.570454i	$-0.806767\pi$
$227$	− 17.1895i	− 1.14091i	0.821330	−	0.570454i	$-0.193233\pi$
$228$	0	0
$229$	− 10.3375i	− 0.683122i	−0.939860	−	0.341561i	$-0.889044\pi$
$229$	− 10.3375i	− 0.683122i	0.939860	−	0.341561i	$-0.110956\pi$
$230$	0	0
$231$	1.34176	0.0882815
$232$	0	0
$233$	0.475100	0.0311248	0.0155624	−	0.999879i	$-0.495046\pi$
$233$	0.475100	0.0311248	0.0155624	+	0.999879i	$0.495046\pi$
$234$	0	0
$235$	0.135664i	0.00884977i
$236$	0	0
$237$	− 6.54503i	− 0.425145i
$238$	0	0
$239$	−8.74191	−0.565467	−0.282734	−	0.959198i	$-0.591241\pi$
$239$	−8.74191	−0.565467	−0.282734	+	0.959198i	$0.591241\pi$
$240$	0	0
$241$	11.6534	0.750660	0.375330	−	0.926891i	$-0.377529\pi$
$241$	11.6534	0.750660	0.375330	+	0.926891i	$0.377529\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	− 7.32578i	− 0.468027i
$246$	0	0
$247$	−0.0627418	−0.00399217
$248$	0	0
$249$	−6.62844	−0.420060
$250$	0	0
$251$	5.41847i	0.342011i	0.985270	+	0.171005i	$0.0547015\pi$
$251$	5.41847i	0.342011i	−0.985270	+	0.171005i	$0.945298\pi$
$252$	0	0
$253$	3.87767i	0.243787i
$254$	0	0
$255$	14.7802	0.925571
$256$	0	0
$257$	1.09367	0.0682212	0.0341106	−	0.999418i	$-0.489140\pi$
$257$	1.09367	0.0682212	0.0341106	+	0.999418i	$0.489140\pi$
$258$	0	0
$259$	21.3695i	1.32784i
$260$	0	0
$261$	− 2.41805i	− 0.149674i
$262$	0	0
$263$	−8.89318	−0.548377	−0.274189	−	0.961676i	$-0.588409\pi$
$263$	−8.89318	−0.548377	−0.274189	+	0.961676i	$0.588409\pi$
$264$	0	0
$265$	−57.0821	−3.50653
$266$	0	0
$267$	− 8.23880i	− 0.504206i
$268$	0	0
$269$	15.6451i	0.953901i	0.878930	+	0.476951i	$0.158258\pi$
$269$	15.6451i	0.953901i	−0.878930	+	0.476951i	$0.841742\pi$
$270$	0	0
$271$	−17.2508	−1.04791	−0.523956	−	0.851746i	$-0.675544\pi$
$271$	−17.2508	−1.04791	−0.523956	+	0.851746i	$0.675544\pi$
$272$	0	0
$273$	−2.30442	−0.139470
$274$	0	0
$275$	− 8.03420i	− 0.484480i
$276$	0	0
$277$	− 5.22709i	− 0.314065i	−0.987593	−	0.157033i	$-0.949807\pi$
$277$	− 5.22709i	− 0.314065i	0.987593	−	0.157033i	$-0.0501928\pi$
$278$	0	0
$279$	5.63600	0.337419
$280$	0	0
$281$	23.5272	1.40351	0.701757	−	0.712416i	$-0.252399\pi$
$281$	23.5272	1.40351	0.701757	+	0.712416i	$0.252399\pi$
$282$	0	0
$283$	− 6.22302i	− 0.369920i	−0.982746	−	0.184960i	$-0.940784\pi$
$283$	− 6.22302i	− 0.369920i	0.982746	−	0.184960i	$-0.0592156\pi$
$284$	0	0
$285$	− 0.272031i	− 0.0161137i
$286$	0	0
$287$	10.9547	0.646634
$288$	0	0
$289$	−5.37912	−0.316419
$290$	0	0
$291$	0.664432i	0.0389497i
$292$	0	0
$293$	13.3240i	0.778396i	0.921154	+	0.389198i	$0.127248\pi$
$293$	13.3240i	0.778396i	−0.921154	+	0.389198i	$0.872752\pi$
$294$	0	0
$295$	20.6275	1.20098
$296$	0	0
$297$	−0.582255	−0.0337859
$298$	0	0
$299$	− 6.65973i	− 0.385142i
$300$	0	0
$301$	18.1228i	1.04458i
$302$	0	0
$303$	11.3867	0.654149
$304$	0	0
$305$	−14.2710	−0.817152
$306$	0	0
$307$	− 28.7588i	− 1.64135i	−0.571396	−	0.820675i	$-0.693598\pi$
$307$	− 28.7588i	− 1.64135i	0.571396	−	0.820675i	$-0.306402\pi$
$308$	0	0
$309$	− 14.9131i	− 0.848377i
$310$	0	0
$311$	−21.9394	−1.24407	−0.622034	−	0.782990i	$-0.713694\pi$
$311$	−21.9394	−1.24407	−0.622034	+	0.782990i	$0.713694\pi$
$312$	0	0
$313$	−2.60950	−0.147498	−0.0737488	−	0.997277i	$-0.523496\pi$
$313$	−2.60950	−0.147498	−0.0737488	+	0.997277i	$0.523496\pi$
$314$	0	0
$315$	− 9.99131i	− 0.562947i
$316$	0	0
$317$	− 16.0656i	− 0.902334i	−0.892439	−	0.451167i	$-0.851008\pi$
$317$	− 16.0656i	− 0.902334i	0.892439	−	0.451167i	$-0.148992\pi$
$318$	0	0
$319$	−1.40792	−0.0788286
$320$	0	0
$321$	7.39073	0.412510
$322$	0	0
$323$	− 0.213883i	− 0.0119008i
$324$	0	0
$325$	13.7984i	0.765398i
$326$	0	0
$327$	−17.4880	−0.967091
$328$	0	0
$329$	0.0721054	0.00397530
$330$	0	0
$331$	− 4.12863i	− 0.226930i	−0.993542	−	0.113465i	$-0.963805\pi$
$331$	− 4.12863i	− 0.226930i	0.993542	−	0.113465i	$-0.0361950\pi$
$332$	0	0
$333$	− 9.27328i	− 0.508172i
$334$	0	0
$335$	51.4982	2.81365
$336$	0	0
$337$	6.01639	0.327734	0.163867	−	0.986482i	$-0.447603\pi$
$337$	6.01639	0.327734	0.163867	+	0.986482i	$0.447603\pi$
$338$	0	0
$339$	− 10.4662i	− 0.568445i
$340$	0	0
$341$	− 3.28159i	− 0.177708i
$342$	0	0
$343$	−20.0246	−1.08123
$344$	0	0
$345$	28.8747	1.55456
$346$	0	0
$347$	− 1.60019i	− 0.0859029i	−0.999077	−	0.0429514i	$-0.986324\pi$
$347$	− 1.60019i	− 0.0859029i	0.999077	−	0.0429514i	$-0.0136761\pi$
$348$	0	0
$349$	− 0.755303i	− 0.0404305i	−0.999796	−	0.0202152i	$-0.993565\pi$
$349$	− 0.755303i	− 0.0404305i	0.999796	−	0.0202152i	$-0.00643515\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	23.2960	1.23992	0.619962	−	0.784632i	$-0.287148\pi$
$353$	23.2960	1.23992	0.619962	+	0.784632i	$0.287148\pi$
$354$	0	0
$355$	− 59.0745i	− 3.13535i
$356$	0	0
$357$	− 7.85564i	− 0.415765i
$358$	0	0
$359$	32.9111	1.73698	0.868491	−	0.495706i	$-0.165090\pi$
$359$	32.9111	1.73698	0.868491	+	0.495706i	$0.165090\pi$
$360$	0	0
$361$	18.9961	0.999793
$362$	0	0
$363$	− 10.6610i	− 0.559556i
$364$	0	0
$365$	1.89547i	0.0992132i
$366$	0	0
$367$	−2.51714	−0.131394	−0.0656969	−	0.997840i	$-0.520927\pi$
$367$	−2.51714	−0.131394	−0.0656969	+	0.997840i	$0.520927\pi$
$368$	0	0
$369$	−4.75376	−0.247471
$370$	0	0
$371$	30.3390i	1.57512i
$372$	0	0
$373$	− 25.9826i	− 1.34533i	−0.739947	−	0.672665i	$-0.765150\pi$
$373$	− 25.9826i	− 1.34533i	0.739947	−	0.672665i	$-0.234850\pi$
$374$	0	0
$375$	−38.1473	−1.96992
$376$	0	0
$377$	2.41805	0.124536
$378$	0	0
$379$	− 25.5104i	− 1.31038i	−0.755464	−	0.655190i	$-0.772589\pi$
$379$	− 25.5104i	− 1.31038i	0.755464	−	0.655190i	$-0.227411\pi$
$380$	0	0
$381$	20.9445i	1.07302i
$382$	0	0
$383$	25.1320	1.28419	0.642093	−	0.766627i	$-0.278066\pi$
$383$	25.1320	1.28419	0.642093	+	0.766627i	$0.278066\pi$
$384$	0	0
$385$	−5.81750	−0.296487
$386$	0	0
$387$	− 7.86434i	− 0.399767i
$388$	0	0
$389$	− 21.2888i	− 1.07938i	−0.841862	−	0.539692i	$-0.818541\pi$
$389$	− 21.2888i	− 1.07938i	0.841862	−	0.539692i	$-0.181459\pi$
$390$	0	0
$391$	22.7026	1.14812
$392$	0	0
$393$	−3.39073	−0.171040
$394$	0	0
$395$	28.3774i	1.42782i
$396$	0	0
$397$	− 12.1291i	− 0.608741i	−0.952554	−	0.304371i	$-0.901554\pi$
$397$	− 12.1291i	− 0.608741i	0.952554	−	0.304371i	$-0.0984461\pi$
$398$	0	0
$399$	−0.144584	−0.00723824
$400$	0	0
$401$	10.4114	0.519919	0.259959	−	0.965620i	$-0.416291\pi$
$401$	10.4114	0.519919	0.259959	+	0.965620i	$0.416291\pi$
$402$	0	0
$403$	5.63600i	0.280749i
$404$	0	0
$405$	4.33571i	0.215443i
$406$	0	0
$407$	−5.39942	−0.267639
$408$	0	0
$409$	−2.94848	−0.145793	−0.0728965	−	0.997340i	$-0.523224\pi$
$409$	−2.94848	−0.145793	−0.0728965	+	0.997340i	$0.523224\pi$
$410$	0	0
$411$	5.78852i	0.285526i
$412$	0	0
$413$	− 10.9635i	− 0.539478i
$414$	0	0
$415$	28.7390	1.41074
$416$	0	0
$417$	18.0874	0.885742
$418$	0	0
$419$	26.3514i	1.28735i	0.765299	+	0.643675i	$0.222591\pi$
$419$	26.3514i	1.28735i	−0.765299	+	0.643675i	$0.777409\pi$
$420$	0	0
$421$	4.54054i	0.221292i	0.993860	+	0.110646i	$0.0352920\pi$
$421$	4.54054i	0.221292i	−0.993860	+	0.110646i	$0.964708\pi$
$422$	0	0
$423$	−0.0312900	−0.00152137
$424$	0	0
$425$	−47.0380	−2.28168
$426$	0	0
$427$	7.58498i	0.367063i
$428$	0	0
$429$	− 0.582255i	− 0.0281116i
$430$	0	0
$431$	−15.3547	−0.739608	−0.369804	−	0.929110i	$-0.620575\pi$
$431$	−15.3547	−0.739608	−0.369804	+	0.929110i	$0.620575\pi$
$432$	0	0
$433$	27.9869	1.34497	0.672483	−	0.740113i	$-0.265228\pi$
$433$	27.9869	1.34497	0.672483	+	0.740113i	$0.265228\pi$
$434$	0	0
$435$	10.4840i	0.502668i
$436$	0	0
$437$	− 0.417844i	− 0.0199882i
$438$	0	0
$439$	6.64728	0.317257	0.158629	−	0.987338i	$-0.449293\pi$
$439$	6.64728	0.317257	0.158629	+	0.987338i	$0.449293\pi$
$440$	0	0
$441$	1.68964	0.0804589
$442$	0	0
$443$	− 32.6534i	− 1.55141i	−0.631096	−	0.775705i	$-0.717395\pi$
$443$	− 32.6534i	− 1.55141i	0.631096	−	0.775705i	$-0.282605\pi$
$444$	0	0
$445$	35.7211i	1.69334i
$446$	0	0
$447$	−9.21251	−0.435737
$448$	0	0
$449$	−21.1274	−0.997063	−0.498531	−	0.866872i	$-0.666127\pi$
$449$	−21.1274	−0.997063	−0.498531	+	0.866872i	$0.666127\pi$
$450$	0	0
$451$	2.76790i	0.130336i
$452$	0	0
$453$	− 2.49249i	− 0.117107i
$454$	0	0
$455$	9.99131	0.468400
$456$	0	0
$457$	18.2163	0.852121	0.426060	−	0.904695i	$-0.359901\pi$
$457$	18.2163	0.852121	0.426060	+	0.904695i	$0.359901\pi$
$458$	0	0
$459$	3.40894i	0.159116i
$460$	0	0
$461$	20.4992i	0.954741i	0.878702	+	0.477371i	$0.158410\pi$
$461$	20.4992i	0.954741i	−0.878702	+	0.477371i	$0.841590\pi$
$462$	0	0
$463$	5.45645	0.253583	0.126791	−	0.991929i	$-0.459532\pi$
$463$	5.45645	0.253583	0.126791	+	0.991929i	$0.459532\pi$
$464$	0	0
$465$	−24.4361	−1.13320
$466$	0	0
$467$	33.7488i	1.56171i	0.624714	+	0.780853i	$0.285215\pi$
$467$	33.7488i	1.56171i	−0.624714	+	0.780853i	$0.714785\pi$
$468$	0	0
$469$	− 27.3712i	− 1.26388i
$470$	0	0
$471$	−4.18956	−0.193045
$472$	0	0
$473$	−4.57905	−0.210545
$474$	0	0
$475$	0.865737i	0.0397227i
$476$	0	0
$477$	− 13.1656i	− 0.602810i
$478$	0	0
$479$	7.19006	0.328522	0.164261	−	0.986417i	$-0.447476\pi$
$479$	7.19006	0.328522	0.164261	+	0.986417i	$0.447476\pi$
$480$	0	0
$481$	9.27328	0.422825
$482$	0	0
$483$	− 15.3468i	− 0.698305i
$484$	0	0
$485$	− 2.88079i	− 0.130810i
$486$	0	0
$487$	9.72505	0.440684	0.220342	−	0.975423i	$-0.429283\pi$
$487$	9.72505	0.440684	0.220342	+	0.975423i	$0.429283\pi$
$488$	0	0
$489$	−10.8387	−0.490141
$490$	0	0
$491$	− 7.93742i	− 0.358211i	−0.983830	−	0.179105i	$-0.942680\pi$
$491$	− 7.93742i	− 0.358211i	0.983830	−	0.179105i	$-0.0573203\pi$
$492$	0	0
$493$	8.24300i	0.371246i
$494$	0	0
$495$	2.52449	0.113467
$496$	0	0
$497$	−31.3980	−1.40839
$498$	0	0
$499$	− 10.1702i	− 0.455282i	−0.973745	−	0.227641i	$-0.926899\pi$
$499$	− 10.1702i	− 0.455282i	0.973745	−	0.227641i	$-0.0731013\pi$
$500$	0	0
$501$	− 13.5345i	− 0.604679i
$502$	0	0
$503$	22.4670	1.00176	0.500878	−	0.865518i	$-0.333011\pi$
$503$	22.4670	1.00176	0.500878	+	0.865518i	$0.333011\pi$
$504$	0	0
$505$	−49.3695	−2.19691
$506$	0	0
$507$	1.00000i	0.0444116i
$508$	0	0
$509$	2.95818i	0.131119i	0.997849	+	0.0655595i	$0.0208832\pi$
$509$	2.95818i	0.131119i	−0.997849	+	0.0655595i	$0.979117\pi$
$510$	0	0
$511$	1.00744	0.0445663
$512$	0	0
$513$	0.0627418	0.00277012
$514$	0	0
$515$	64.6589i	2.84921i
$516$	0	0
$517$	0.0182188i	0 0.000801261i
$518$	0	0
$519$	11.3922	0.500063
$520$	0	0
$521$	−38.3759	−1.68128	−0.840640	−	0.541594i	$-0.817821\pi$
$521$	−38.3759	−1.68128	−0.840640	+	0.541594i	$0.817821\pi$
$522$	0	0
$523$	20.7001i	0.905153i	0.891726	+	0.452576i	$0.149495\pi$
$523$	20.7001i	0.905153i	−0.891726	+	0.452576i	$0.850505\pi$
$524$	0	0
$525$	31.7974i	1.38775i
$526$	0	0
$527$	−19.2128	−0.836923
$528$	0	0
$529$	21.3520	0.928349
$530$	0	0
$531$	4.75759i	0.206462i
$532$	0	0
$533$	− 4.75376i	− 0.205908i
$534$	0	0
$535$	−32.0441	−1.38539
$536$	0	0
$537$	−9.99131	−0.431157
$538$	0	0
$539$	− 0.983800i	− 0.0423753i
$540$	0	0
$541$	− 0.180587i	− 0.00776404i	−0.999992	−	0.00388202i	$-0.998764\pi$
$541$	− 0.180587i	− 0.00776404i	0.999992	−	0.00388202i	$-0.00123569\pi$
$542$	0	0
$543$	−24.2251	−1.03960
$544$	0	0
$545$	75.8231	3.24791
$546$	0	0
$547$	− 11.6587i	− 0.498489i	−0.968441	−	0.249244i	$-0.919818\pi$
$547$	− 11.6587i	− 0.498489i	0.968441	−	0.249244i	$-0.0801822\pi$
$548$	0	0
$549$	− 3.29149i	− 0.140477i
$550$	0	0
$551$	0.151713	0.00646319
$552$	0	0
$553$	15.0825	0.641374
$554$	0	0
$555$	40.2063i	1.70666i
$556$	0	0
$557$	13.1212i	0.555961i	0.960587	+	0.277981i	$0.0896651\pi$
$557$	13.1212i	0.555961i	−0.960587	+	0.277981i	$0.910335\pi$
$558$	0	0
$559$	7.86434	0.332626
$560$	0	0
$561$	1.98487	0.0838015
$562$	0	0
$563$	22.1529i	0.933633i	0.884354	+	0.466817i	$0.154599\pi$
$563$	22.1529i	0.933633i	−0.884354	+	0.466817i	$0.845401\pi$
$564$	0	0
$565$	45.3784i	1.90908i
$566$	0	0
$567$	2.30442	0.0967766
$568$	0	0
$569$	32.0592	1.34399	0.671997	−	0.740554i	$-0.265437\pi$
$569$	32.0592	1.34399	0.671997	+	0.740554i	$0.265437\pi$
$570$	0	0
$571$	19.0642i	0.797812i	0.916992	+	0.398906i	$0.130610\pi$
$571$	19.0642i	0.797812i	−0.916992	+	0.398906i	$0.869390\pi$
$572$	0	0
$573$	− 14.2745i	− 0.596327i
$574$	0	0
$575$	−91.8937	−3.83223
$576$	0	0
$577$	6.36039	0.264786	0.132393	−	0.991197i	$-0.457734\pi$
$577$	6.36039	0.264786	0.132393	+	0.991197i	$0.457734\pi$
$578$	0	0
$579$	− 4.98894i	− 0.207333i
$580$	0	0
$581$	− 15.2747i	− 0.633703i
$582$	0	0
$583$	−7.66572	−0.317482
$584$	0	0
$585$	−4.33571	−0.179260
$586$	0	0
$587$	16.8490i	0.695432i	0.937600	+	0.347716i	$0.113043\pi$
$587$	16.8490i	0.695432i	−0.937600	+	0.347716i	$0.886957\pi$
$588$	0	0
$589$	0.353613i	0.0145704i
$590$	0	0
$591$	−10.9989	−0.452433
$592$	0	0
$593$	−30.9202	−1.26974	−0.634871	−	0.772618i	$-0.718947\pi$
$593$	−30.9202	−1.26974	−0.634871	+	0.772618i	$0.718947\pi$
$594$	0	0
$595$	34.0598i	1.39632i
$596$	0	0
$597$	9.13566i	0.373897i
$598$	0	0
$599$	0.158853	0.00649057	0.00324528	−	0.999995i	$-0.498967\pi$
$599$	0.158853	0.00649057	0.00324528	+	0.999995i	$0.498967\pi$
$600$	0	0
$601$	−6.96910	−0.284276	−0.142138	−	0.989847i	$-0.545398\pi$
$601$	−6.96910	−0.284276	−0.142138	+	0.989847i	$0.545398\pi$
$602$	0	0
$603$	11.8777i	0.483697i
$604$	0	0
$605$	46.2229i	1.87923i
$606$	0	0
$607$	−43.8942	−1.78161	−0.890805	−	0.454385i	$-0.849859\pi$
$607$	−43.8942	−1.78161	−0.890805	+	0.454385i	$0.849859\pi$
$608$	0	0
$609$	5.57221	0.225797
$610$	0	0
$611$	− 0.0312900i	− 0.00126586i
$612$	0	0
$613$	0.317515i	0.0128243i	0.999979	+	0.00641216i	$0.00204107\pi$
$613$	0.317515i	0.0128243i	−0.999979	+	0.00641216i	$0.997959\pi$
$614$	0	0
$615$	20.6110	0.831114
$616$	0	0
$617$	2.54233	0.102350	0.0511752	−	0.998690i	$-0.483703\pi$
$617$	2.54233	0.102350	0.0511752	+	0.998690i	$0.483703\pi$
$618$	0	0
$619$	36.6748i	1.47409i	0.675846	+	0.737043i	$0.263778\pi$
$619$	36.6748i	1.47409i	−0.675846	+	0.737043i	$0.736222\pi$
$620$	0	0
$621$	6.65973i	0.267246i
$622$	0	0
$623$	18.9857	0.760645
$624$	0	0
$625$	96.4039	3.85616
$626$	0	0
$627$	− 0.0365318i	− 0.00145894i
$628$	0	0
$629$	31.6121i	1.26046i
$630$	0	0
$631$	32.5523	1.29589	0.647943	−	0.761689i	$-0.275630\pi$
$631$	32.5523	1.29589	0.647943	+	0.761689i	$0.275630\pi$
$632$	0	0
$633$	1.06551	0.0423501
$634$	0	0
$635$	− 90.8094i	− 3.60366i
$636$	0	0
$637$	1.68964i	0.0669458i
$638$	0	0
$639$	13.6251	0.539000
$640$	0	0
$641$	17.0435	0.673178	0.336589	−	0.941652i	$-0.390727\pi$
$641$	17.0435	0.673178	0.336589	+	0.941652i	$0.390727\pi$
$642$	0	0
$643$	− 11.2883i	− 0.445168i	−0.974914	−	0.222584i	$-0.928551\pi$
$643$	− 11.2883i	− 0.445168i	0.974914	−	0.222584i	$-0.0714492\pi$
$644$	0	0
$645$	34.0975i	1.34259i
$646$	0	0
$647$	−22.0458	−0.866711	−0.433355	−	0.901223i	$-0.642670\pi$
$647$	−22.0458	−0.866711	−0.433355	+	0.901223i	$0.642670\pi$
$648$	0	0
$649$	2.77013	0.108737
$650$	0	0
$651$	12.9877i	0.509029i
$652$	0	0
$653$	− 28.8324i	− 1.12830i	−0.825673	−	0.564149i	$-0.809204\pi$
$653$	− 28.8324i	− 1.12830i	0.825673	−	0.564149i	$-0.190796\pi$
$654$	0	0
$655$	14.7012	0.574425
$656$	0	0
$657$	−0.437175	−0.0170558
$658$	0	0
$659$	− 44.5007i	− 1.73350i	−0.498742	−	0.866750i	$-0.666205\pi$
$659$	− 44.5007i	− 1.73350i	0.498742	−	0.866750i	$-0.333795\pi$
$660$	0	0
$661$	− 11.2482i	− 0.437503i	−0.975781	−	0.218752i	$-0.929802\pi$
$661$	− 11.2482i	− 0.437503i	0.975781	−	0.218752i	$-0.0701984\pi$
$662$	0	0
$663$	−3.40894	−0.132392
$664$	0	0
$665$	0.626873	0.0243091
$666$	0	0
$667$	16.1036i	0.623533i
$668$	0	0
$669$	8.91055i	0.344502i
$670$	0	0
$671$	−1.91649	−0.0739852
$672$	0	0
$673$	−45.9571	−1.77152	−0.885758	−	0.464147i	$-0.846361\pi$
$673$	−45.9571	−1.77152	−0.885758	+	0.464147i	$0.846361\pi$
$674$	0	0
$675$	− 13.7984i	− 0.531101i
$676$	0	0
$677$	− 42.4795i	− 1.63262i	−0.577613	−	0.816310i	$-0.696016\pi$
$677$	− 42.4795i	− 1.63262i	0.577613	−	0.816310i	$-0.303984\pi$
$678$	0	0
$679$	−1.53113	−0.0587595
$680$	0	0
$681$	−17.1895	−0.658703
$682$	0	0
$683$	− 11.8075i	− 0.451800i	−0.974150	−	0.225900i	$-0.927468\pi$
$683$	− 11.8075i	− 0.451800i	0.974150	−	0.225900i	$-0.0725323\pi$
$684$	0	0
$685$	− 25.0973i	− 0.958920i
$686$	0	0
$687$	−10.3375	−0.394401
$688$	0	0
$689$	13.1656	0.501568
$690$	0	0
$691$	23.5462i	0.895739i	0.894099	+	0.447870i	$0.147817\pi$
$691$	23.5462i	0.895739i	−0.894099	+	0.447870i	$0.852183\pi$
$692$	0	0
$693$	− 1.34176i	− 0.0509693i
$694$	0	0
$695$	−78.4216	−2.97470
$696$	0	0
$697$	16.2053	0.613820
$698$	0	0
$699$	− 0.475100i	− 0.0179699i
$700$	0	0
$701$	5.15813i	0.194820i	0.995244	+	0.0974099i	$0.0310558\pi$
$701$	5.15813i	0.194820i	−0.995244	+	0.0974099i	$0.968944\pi$
$702$	0	0
$703$	0.581822	0.0219438
$704$	0	0
$705$	0.135664	0.00510942
$706$	0	0
$707$	26.2398i	0.986849i
$708$	0	0
$709$	− 20.1043i	− 0.755032i	−0.926003	−	0.377516i	$-0.876778\pi$
$709$	− 20.1043i	− 0.755032i	0.926003	−	0.377516i	$-0.123222\pi$
$710$	0	0
$711$	−6.54503	−0.245458
$712$	0	0
$713$	−37.5343	−1.40567
$714$	0	0
$715$	2.52449i	0.0944107i
$716$	0	0
$717$	8.74191i	0.326473i
$718$	0	0
$719$	10.5580	0.393745	0.196873	−	0.980429i	$-0.436921\pi$
$719$	10.5580	0.393745	0.196873	+	0.980429i	$0.436921\pi$
$720$	0	0
$721$	34.3661	1.27986
$722$	0	0
$723$	− 11.6534i	− 0.433394i
$724$	0	0
$725$	− 33.3652i	− 1.23915i
$726$	0	0
$727$	51.2972	1.90251	0.951255	−	0.308407i	$-0.0997958\pi$
$727$	51.2972	1.90251	0.951255	+	0.308407i	$0.0997958\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	26.8091i	0.991569i
$732$	0	0
$733$	22.1820i	0.819313i	0.912240	+	0.409656i	$0.134351\pi$
$733$	22.1820i	0.819313i	−0.912240	+	0.409656i	$0.865649\pi$
$734$	0	0
$735$	−7.32578	−0.270215
$736$	0	0
$737$	6.91585	0.254748
$738$	0	0
$739$	− 19.5216i	− 0.718115i	−0.933315	−	0.359057i	$-0.883098\pi$
$739$	− 19.5216i	− 0.718115i	0.933315	−	0.359057i	$-0.116902\pi$
$740$	0	0
$741$	0.0627418i	0.00230488i
$742$	0	0
$743$	49.0478	1.79939	0.899695	−	0.436518i	$-0.143789\pi$
$743$	49.0478	1.79939	0.899695	+	0.436518i	$0.143789\pi$
$744$	0	0
$745$	39.9428	1.46339
$746$	0	0
$747$	6.62844i	0.242522i
$748$	0	0
$749$	17.0314i	0.622313i
$750$	0	0
$751$	16.3556	0.596825	0.298412	−	0.954437i	$-0.403543\pi$
$751$	16.3556	0.596825	0.298412	+	0.954437i	$0.403543\pi$
$752$	0	0
$753$	5.41847	0.197460
$754$	0	0
$755$	10.8067i	0.393296i
$756$	0	0
$757$	9.50390i	0.345425i	0.984972	+	0.172713i	$0.0552532\pi$
$757$	9.50390i	0.345425i	−0.984972	+	0.172713i	$0.944747\pi$
$758$	0	0
$759$	3.87767	0.140750
$760$	0	0
$761$	21.2994	0.772102	0.386051	−	0.922477i	$-0.373839\pi$
$761$	21.2994	0.772102	0.386051	+	0.922477i	$0.373839\pi$
$762$	0	0
$763$	− 40.2998i	− 1.45895i
$764$	0	0
$765$	− 14.7802i	− 0.534379i
$766$	0	0
$767$	−4.75759	−0.171787
$768$	0	0
$769$	−28.7969	−1.03844	−0.519222	−	0.854639i	$-0.673778\pi$
$769$	−28.7969	−1.03844	−0.519222	+	0.854639i	$0.673778\pi$
$770$	0	0
$771$	− 1.09367i	− 0.0393875i
$772$	0	0
$773$	14.0271i	0.504518i	0.967660	+	0.252259i	$0.0811735\pi$
$773$	14.0271i	0.504518i	−0.967660	+	0.252259i	$0.918826\pi$
$774$	0	0
$775$	77.7679	2.79350
$776$	0	0
$777$	21.3695	0.766628
$778$	0	0
$779$	− 0.298260i	− 0.0106863i
$780$	0	0
$781$	− 7.93328i	− 0.283875i
$782$	0	0
$783$	−2.41805	−0.0864141
$784$	0	0
$785$	18.1647	0.648327
$786$	0	0
$787$	6.25929i	0.223119i	0.993758	+	0.111560i	$0.0355846\pi$
$787$	6.25929i	0.223119i	−0.993758	+	0.111560i	$0.964415\pi$
$788$	0	0
$789$	8.89318i	0.316606i
$790$	0	0
$791$	24.1185	0.857556
$792$	0	0
$793$	3.29149	0.116884
$794$	0	0
$795$	57.0821i	2.02449i
$796$	0	0
$797$	− 17.5042i	− 0.620031i	−0.950732	−	0.310015i	$-0.899666\pi$
$797$	− 17.5042i	− 0.620031i	0.950732	−	0.310015i	$-0.100334\pi$
$798$	0	0
$799$	0.106666	0.00377356
$800$	0	0
$801$	−8.23880	−0.291104
$802$	0	0
$803$	0.254548i	0.00898279i
$804$	0	0
$805$	66.5395i	2.34521i
$806$	0	0
$807$	15.6451	0.550735
$808$	0	0
$809$	−28.5503	−1.00377	−0.501887	−	0.864933i	$-0.667361\pi$
$809$	−28.5503	−1.00377	−0.501887	+	0.864933i	$0.667361\pi$
$810$	0	0
$811$	− 17.7550i	− 0.623462i	−0.950170	−	0.311731i	$-0.899091\pi$
$811$	− 17.7550i	− 0.623462i	0.950170	−	0.311731i	$-0.100909\pi$
$812$	0	0
$813$	17.2508i	0.605012i
$814$	0	0
$815$	46.9933	1.64610
$816$	0	0
$817$	0.493423	0.0172627
$818$	0	0
$819$	2.30442i	0.0805230i
$820$	0	0
$821$	47.7170i	1.66534i	0.553773	+	0.832668i	$0.313188\pi$
$821$	47.7170i	1.66534i	−0.553773	+	0.832668i	$0.686812\pi$
$822$	0	0
$823$	−11.6508	−0.406122	−0.203061	−	0.979166i	$-0.565089\pi$
$823$	−11.6508	−0.406122	−0.203061	+	0.979166i	$0.565089\pi$
$824$	0	0
$825$	−8.03420	−0.279715
$826$	0	0
$827$	− 45.4863i	− 1.58171i	−0.612001	−	0.790857i	$-0.709635\pi$
$827$	− 45.4863i	− 1.58171i	0.612001	−	0.790857i	$-0.290365\pi$
$828$	0	0
$829$	− 35.2812i	− 1.22537i	−0.790328	−	0.612684i	$-0.790090\pi$
$829$	− 35.2812i	− 1.22537i	0.790328	−	0.612684i	$-0.209910\pi$
$830$	0	0
$831$	−5.22709	−0.181326
$832$	0	0
$833$	−5.75987	−0.199568
$834$	0	0
$835$	58.6819i	2.03077i
$836$	0	0
$837$	− 5.63600i	− 0.194809i
$838$	0	0
$839$	−21.3143	−0.735853	−0.367926	−	0.929855i	$-0.619932\pi$
$839$	−21.3143	−0.735853	−0.367926	+	0.929855i	$0.619932\pi$
$840$	0	0
$841$	23.1530	0.798380
$842$	0	0
$843$	− 23.5272i	− 0.810319i
$844$	0	0
$845$	− 4.33571i	− 0.149153i
$846$	0	0
$847$	24.5674	0.844146
$848$	0	0
$849$	−6.22302	−0.213574
$850$	0	0
$851$	61.7575i	2.11702i
$852$	0	0
$853$	9.99152i	0.342103i	0.985262	+	0.171052i	$0.0547165\pi$
$853$	9.99152i	0.342103i	−0.985262	+	0.171052i	$0.945284\pi$
$854$	0	0
$855$	−0.272031	−0.00930325
$856$	0	0
$857$	−49.7277	−1.69867	−0.849333	−	0.527857i	$-0.822996\pi$
$857$	−49.7277	−1.69867	−0.849333	+	0.527857i	$0.822996\pi$
$858$	0	0
$859$	− 16.8742i	− 0.575741i	−0.957669	−	0.287870i	$-0.907053\pi$
$859$	− 16.8742i	− 0.575741i	0.957669	−	0.287870i	$-0.0929472\pi$
$860$	0	0
$861$	− 10.9547i	− 0.373334i
$862$	0	0
$863$	−36.9205	−1.25679	−0.628394	−	0.777895i	$-0.716287\pi$
$863$	−36.9205	−1.25679	−0.628394	+	0.777895i	$0.716287\pi$
$864$	0	0
$865$	−49.3934	−1.67942
$866$	0	0
$867$	5.37912i	0.182684i
$868$	0	0
$869$	3.81088i	0.129275i
$870$	0	0
$871$	−11.8777	−0.402460
$872$	0	0
$873$	0.664432	0.0224876
$874$	0	0
$875$	− 87.9076i	− 2.97182i
$876$	0	0
$877$	4.90899i	0.165765i	0.996559	+	0.0828824i	$0.0264126\pi$
$877$	4.90899i	0.165765i	−0.996559	+	0.0828824i	$0.973587\pi$
$878$	0	0
$879$	13.3240	0.449407
$880$	0	0
$881$	−7.76972	−0.261769	−0.130884	−	0.991398i	$-0.541782\pi$
$881$	−7.76972	−0.261769	−0.130884	+	0.991398i	$0.541782\pi$
$882$	0	0
$883$	− 15.6142i	− 0.525459i	−0.964869	−	0.262730i	$-0.915377\pi$
$883$	− 15.6142i	− 0.525459i	0.964869	−	0.262730i	$-0.0846227\pi$
$884$	0	0
$885$	− 20.6275i	− 0.693387i
$886$	0	0
$887$	2.11858	0.0711349	0.0355674	−	0.999367i	$-0.488676\pi$
$887$	2.11858	0.0711349	0.0355674	+	0.999367i	$0.488676\pi$
$888$	0	0
$889$	−48.2650	−1.61876
$890$	0	0
$891$	0.582255i	0.0195063i
$892$	0	0
$893$	− 0.00196319i	0 6.56957e-5i
$894$	0	0
$895$	43.3195	1.44801
$896$	0	0
$897$	−6.65973	−0.222362
$898$	0	0
$899$	− 13.6281i	− 0.454524i
$900$	0	0
$901$	44.8807i	1.49519i
$902$	0	0
$903$	18.1228	0.603087
$904$	0	0
$905$	105.033	3.49142
$906$	0	0
$907$	− 28.8005i	− 0.956305i	−0.878277	−	0.478153i	$-0.841307\pi$
$907$	− 28.8005i	− 0.956305i	0.878277	−	0.478153i	$-0.158693\pi$
$908$	0	0
$909$	− 11.3867i	− 0.377673i
$910$	0	0
$911$	−4.72015	−0.156386	−0.0781929	−	0.996938i	$-0.524915\pi$
$911$	−4.72015	−0.156386	−0.0781929	+	0.996938i	$0.524915\pi$
$912$	0	0
$913$	3.85945	0.127729
$914$	0	0
$915$	14.2710i	0.471783i
$916$	0	0
$917$	− 7.81367i	− 0.258030i
$918$	0	0
$919$	−43.7254	−1.44237	−0.721185	−	0.692743i	$-0.756402\pi$
$919$	−43.7254	−1.44237	−0.721185	+	0.692743i	$0.756402\pi$
$920$	0	0
$921$	−28.7588	−0.947633
$922$	0	0
$923$	13.6251i	0.448475i
$924$	0	0
$925$	− 127.956i	− 4.20718i
$926$	0	0
$927$	−14.9131	−0.489811
$928$	0	0
$929$	4.56638	0.149818	0.0749091	−	0.997190i	$-0.476133\pi$
$929$	4.56638	0.149818	0.0749091	+	0.997190i	$0.476133\pi$
$930$	0	0
$931$	0.106011i	0.00347437i
$932$	0	0
$933$	21.9394i	0.718263i
$934$	0	0
$935$	−8.60585	−0.281441
$936$	0	0
$937$	−14.8258	−0.484338	−0.242169	−	0.970234i	$-0.577859\pi$
$937$	−14.8258	−0.484338	−0.242169	+	0.970234i	$0.577859\pi$
$938$	0	0
$939$	2.60950i	0.0851578i
$940$	0	0
$941$	23.4674i	0.765015i	0.923952	+	0.382508i	$0.124939\pi$
$941$	23.4674i	0.765015i	−0.923952	+	0.382508i	$0.875061\pi$
$942$	0	0
$943$	31.6588	1.03095
$944$	0	0
$945$	−9.99131	−0.325017
$946$	0	0
$947$	− 6.85871i	− 0.222878i	−0.993771	−	0.111439i	$-0.964454\pi$
$947$	− 6.85871i	− 0.222878i	0.993771	−	0.111439i	$-0.0355460\pi$
$948$	0	0
$949$	− 0.437175i	− 0.0141913i
$950$	0	0
$951$	−16.0656	−0.520963
$952$	0	0
$953$	−33.6118	−1.08879	−0.544396	−	0.838828i	$-0.683241\pi$
$953$	−33.6118	−1.08879	−0.544396	+	0.838828i	$0.683241\pi$
$954$	0	0
$955$	61.8902i	2.00272i
$956$	0	0
$957$	1.40792i	0.0455117i
$958$	0	0
$959$	−13.3392	−0.430745
$960$	0	0
$961$	0.764545	0.0246627
$962$	0	0
$963$	− 7.39073i	− 0.238163i
$964$	0	0
$965$	21.6306i	0.696314i
$966$	0	0
$967$	−38.4327	−1.23591	−0.617956	−	0.786212i	$-0.712039\pi$
$967$	−38.4327	−1.23591	−0.617956	+	0.786212i	$0.712039\pi$
$968$	0	0
$969$	−0.213883	−0.00687092
$970$	0	0
$971$	38.2120i	1.22628i	0.789974	+	0.613141i	$0.210094\pi$
$971$	38.2120i	1.22628i	−0.789974	+	0.613141i	$0.789906\pi$
$972$	0	0
$973$	41.6809i	1.33623i
$974$	0	0
$975$	13.7984	0.441903
$976$	0	0
$977$	−3.28633	−0.105139	−0.0525695	−	0.998617i	$-0.516741\pi$
$977$	−3.28633	−0.105139	−0.0525695	+	0.998617i	$0.516741\pi$
$978$	0	0
$979$	4.79708i	0.153315i
$980$	0	0
$981$	17.4880i	0.558350i
$982$	0	0
$983$	13.0281	0.415530	0.207765	−	0.978179i	$-0.433381\pi$
$983$	13.0281	0.415530	0.207765	+	0.978179i	$0.433381\pi$
$984$	0	0
$985$	47.6880	1.51946
$986$	0	0
$987$	− 0.0721054i	− 0.00229514i
$988$	0	0
$989$	52.3744i	1.66541i
$990$	0	0
$991$	−11.7140	−0.372106	−0.186053	−	0.982540i	$-0.559570\pi$
$991$	−11.7140	−0.372106	−0.186053	+	0.982540i	$0.559570\pi$
$992$	0	0
$993$	−4.12863	−0.131018
$994$	0	0
$995$	− 39.6096i	− 1.25571i
$996$	0	0
$997$	− 61.2009i	− 1.93825i	−0.246566	−	0.969126i	$-0.579302\pi$
$997$	− 61.2009i	− 1.93825i	0.246566	−	0.969126i	$-0.420698\pi$
$998$	0	0
$999$	−9.27328	−0.293394

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1248.2.g.b.625.8		16
3.2	odd	2		3744.2.g.e.1873.1		16
4.3	odd	2		312.2.g.b.157.6	yes	16
8.3	odd	2		312.2.g.b.157.5	✓	16
8.5	even	2	inner	1248.2.g.b.625.9		16
12.11	even	2		936.2.g.e.469.11		16
16.3	odd	4		9984.2.a.bt.1.8		8
16.5	even	4		9984.2.a.bs.1.1		8
16.11	odd	4		9984.2.a.bu.1.1		8
16.13	even	4		9984.2.a.bv.1.8		8
24.5	odd	2		3744.2.g.e.1873.16		16
24.11	even	2		936.2.g.e.469.12		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.2.g.b.157.5	✓	16	8.3	odd	2
312.2.g.b.157.6	yes	16	4.3	odd	2
936.2.g.e.469.11		16	12.11	even	2
936.2.g.e.469.12		16	24.11	even	2
1248.2.g.b.625.8		16	1.1	even	1	trivial
1248.2.g.b.625.9		16	8.5	even	2	inner
3744.2.g.e.1873.1		16	3.2	odd	2
3744.2.g.e.1873.16		16	24.5	odd	2
9984.2.a.bs.1.1		8	16.5	even	4
9984.2.a.bt.1.8		8	16.3	odd	4
9984.2.a.bu.1.1		8	16.11	odd	4
9984.2.a.bv.1.8		8	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:	\( N \)	\(=\)	\( 1248 = 2^{5} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1248.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +4.33571i q^{5} +2.30442 q^{7} -1.00000 q^{9} +0.582255i q^{11} -1.00000i q^{13} +4.33571 q^{15} +3.40894 q^{17} -0.0627418i q^{19} -2.30442i q^{21} +6.65973 q^{23} -13.7984 q^{25} +1.00000i q^{27} +2.41805i q^{29} -5.63600 q^{31} +0.582255 q^{33} +9.99131i q^{35} +9.27328i q^{37} -1.00000 q^{39} +4.75376 q^{41} +7.86434i q^{43} -4.33571i q^{45} +0.0312900 q^{47} -1.68964 q^{49} -3.40894i q^{51} +13.1656i q^{53} -2.52449 q^{55} -0.0627418 q^{57} -4.75759i q^{59} +3.29149i q^{61} -2.30442 q^{63} +4.33571 q^{65} -11.8777i q^{67} -6.65973i q^{69} -13.6251 q^{71} +0.437175 q^{73} +13.7984i q^{75} +1.34176i q^{77} +6.54503 q^{79} +1.00000 q^{81} -6.62844i q^{83} +14.7802i q^{85} +2.41805 q^{87} +8.23880 q^{89} -2.30442i q^{91} +5.63600i q^{93} +0.272031 q^{95} -0.664432 q^{97} -0.582255i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{7} - 16 q^{9} + 4 q^{15} + 16 q^{17} + 8 q^{23} - 32 q^{25} + 4 q^{31} - 16 q^{39} - 36 q^{41} - 24 q^{47} + 48 q^{49} - 24 q^{55} - 12 q^{57} + 4 q^{63} + 4 q^{65} - 32 q^{73} + 16 q^{81} - 8 q^{87}+ \cdots + 40 q^{97}+O(q^{100}) \) 16 * q - 4 * q^7 - 16 * q^9 + 4 * q^15 + 16 * q^17 + 8 * q^23 - 32 * q^25 + 4 * q^31 - 16 * q^39 - 36 * q^41 - 24 * q^47 + 48 * q^49 - 24 * q^55 - 12 * q^57 + 4 * q^63 + 4 * q^65 - 32 * q^73 + 16 * q^81 - 8 * q^87 - 60 * q^89 + 24 * q^95 + 40 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more