LMFDB - Embedded newform 441.7.d.a.244.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,7,Mod(244,441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("441.244");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 441 = 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	441.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:	$101.453850876$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2\cdot 7 $
Twist minimal:	no (minimal twist has level 7)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			244.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	441.244
Dual form			441.7.d.a.244.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+12.0000 q^{2} +80.0000 q^{4} -181.865i q^{5} +192.000 q^{8} +O(q^{10})$	$q+12.0000 q^{2} +80.0000 q^{4} -181.865i q^{5} +192.000 q^{8} -2182.38i q^{10} -1479.00 q^{11} +484.974i q^{13} -2816.00 q^{16} -3018.96i q^{17} +6874.51i q^{19} -14549.2i q^{20} -17748.0 q^{22} +5913.00 q^{23} -17450.0 q^{25} +5819.69i q^{26} -3978.00 q^{29} +12815.4i q^{31} -46080.0 q^{32} -36227.6i q^{34} -61577.0 q^{37} +82494.1i q^{38} -34918.1i q^{40} +110574. i q^{41} -17414.0 q^{43} -118320. q^{44} +70956.0 q^{46} +30662.5i q^{47} -209400. q^{50} +38797.9i q^{52} +60513.0 q^{53} +268979. i q^{55} -47736.0 q^{58} -215729. i q^{59} +162745. i q^{61} +153785. i q^{62} -372736. q^{64} +88200.0 q^{65} -268777. q^{67} -241517. i q^{68} -101922. q^{71} -317646. i q^{73} -738924. q^{74} +549961. i q^{76} +362231. q^{79} +512133. i q^{80} +1.32689e6i q^{82} -216783. i q^{83} -549045. q^{85} -208968. q^{86} -283968. q^{88} +1.33456e6i q^{89} +473040. q^{92} +367950. i q^{94} +1.25024e6 q^{95} +1.51409e6i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 24 q^{2} + 160 q^{4} + 384 q^{8}+O(q^{10}) $ 2 * q + 24 * q^2 + 160 * q^4 + 384 * q^8	$ 2 q + 24 q^{2} + 160 q^{4} + 384 q^{8} - 2958 q^{11} - 5632 q^{16} - 35496 q^{22} + 11826 q^{23} - 34900 q^{25} - 7956 q^{29} - 92160 q^{32} - 123154 q^{37} - 34828 q^{43} - 236640 q^{44} + 141912 q^{46} - 418800 q^{50} + 121026 q^{53} - 95472 q^{58} - 745472 q^{64} + 176400 q^{65} - 537554 q^{67} - 203844 q^{71} - 1477848 q^{74} + 724462 q^{79} - 1098090 q^{85} - 417936 q^{86} - 567936 q^{88} + 946080 q^{92} + 2500470 q^{95}+O(q^{100}) $ 2 * q + 24 * q^2 + 160 * q^4 + 384 * q^8 - 2958 * q^11 - 5632 * q^16 - 35496 * q^22 + 11826 * q^23 - 34900 * q^25 - 7956 * q^29 - 92160 * q^32 - 123154 * q^37 - 34828 * q^43 - 236640 * q^44 + 141912 * q^46 - 418800 * q^50 + 121026 * q^53 - 95472 * q^58 - 745472 * q^64 + 176400 * q^65 - 537554 * q^67 - 203844 * q^71 - 1477848 * q^74 + 724462 * q^79 - 1098090 * q^85 - 417936 * q^86 - 567936 * q^88 + 946080 * q^92 + 2500470 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/441\mathbb{Z}\right)^\times$.

$n$	$199$	$344$
$\chi(n)$	$-1$	$1$

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 $

$2$	12.0000	1.50000	0.750000	−	0.661438i	$-0.230053\pi$
$2$	12.0000	1.50000	0.750000	+	0.661438i	$0.230053\pi$
$3$	0	0
$3$	0	0
$4$	80.0000	1.25000
$5$	− 181.865i	− 1.45492i	−0.686149	−	0.727461i	$-0.740700\pi$
$5$	− 181.865i	− 1.45492i	0.686149	−	0.727461i	$-0.259300\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	192.000	0.375000
$9$	0	0
$10$	− 2182.38i	− 2.18238i
$11$	−1479.00	−1.11119	−0.555597	−	0.831452i	$-0.687510\pi$
$11$	−1479.00	−1.11119	−0.555597	+	0.831452i	$0.687510\pi$
$12$	0	0
$13$	484.974i	0.220744i	0.993890	+	0.110372i	$0.0352042\pi$
$13$	484.974i	0.220744i	−0.993890	+	0.110372i	$0.964796\pi$
$14$	0	0
$15$	0	0
$16$	−2816.00	−0.687500
$17$	− 3018.96i	− 0.614485i	−0.951631	−	0.307242i	$-0.900594\pi$
$17$	− 3018.96i	− 0.614485i	0.951631	−	0.307242i	$-0.0994063\pi$
$18$	0	0
$19$	6874.51i	1.00226i	0.865372	+	0.501131i	$0.167082\pi$
$19$	6874.51i	1.00226i	−0.865372	+	0.501131i	$0.832918\pi$
$20$	− 14549.2i	− 1.81865i
$21$	0	0
$22$	−17748.0	−1.66679
$23$	5913.00	0.485987	0.242993	−	0.970028i	$-0.421871\pi$
$23$	5913.00	0.485987	0.242993	+	0.970028i	$0.421871\pi$
$24$	0	0
$25$	−17450.0	−1.11680
$26$	5819.69i	0.331116i
$27$	0	0
$28$	0	0
$29$	−3978.00	−0.163106	−0.0815532	−	0.996669i	$-0.525988\pi$
$29$	−3978.00	−0.163106	−0.0815532	+	0.996669i	$0.525988\pi$
$30$	0	0
$31$	12815.4i	0.430178i	0.976594	+	0.215089i	$0.0690042\pi$
$31$	12815.4i	0.430178i	−0.976594	+	0.215089i	$0.930996\pi$
$32$	−46080.0	−1.40625
$33$	0	0
$34$	− 36227.6i	− 0.921727i
$35$	0	0
$36$	0	0
$37$	−61577.0	−1.21566	−0.607832	−	0.794066i	$-0.707960\pi$
$37$	−61577.0	−1.21566	−0.607832	+	0.794066i	$0.707960\pi$
$38$	82494.1i	1.50339i
$39$	0	0
$40$	− 34918.1i	− 0.545596i
$41$	110574.i	1.60436i	0.597082	+	0.802180i	$0.296327\pi$
$41$	110574.i	1.60436i	−0.597082	+	0.802180i	$0.703673\pi$
$42$	0	0
$43$	−17414.0	−0.219025	−0.109512	−	0.993985i	$-0.534929\pi$
$43$	−17414.0	−0.219025	−0.109512	+	0.993985i	$0.534929\pi$
$44$	−118320.	−1.38899
$45$	0	0
$46$	70956.0	0.728980
$47$	30662.5i	0.295334i	0.989037	+	0.147667i	$0.0471764\pi$
$47$	30662.5i	0.295334i	−0.989037	+	0.147667i	$0.952824\pi$
$48$	0	0
$49$	0	0
$50$	−209400.	−1.67520
$51$	0	0
$52$	38797.9i	0.275930i
$53$	60513.0	0.406463	0.203232	−	0.979131i	$-0.434856\pi$
$53$	60513.0	0.406463	0.203232	+	0.979131i	$0.434856\pi$
$54$	0	0
$55$	268979.i	1.61670i
$56$	0	0
$57$	0	0
$58$	−47736.0	−0.244659
$59$	− 215729.i	− 1.05039i	−0.850981	−	0.525196i	$-0.823992\pi$
$59$	− 215729.i	− 1.05039i	0.850981	−	0.525196i	$-0.176008\pi$
$60$	0	0
$61$	162745.i	0.716999i	0.933530	+	0.358500i	$0.116712\pi$
$61$	162745.i	0.716999i	−0.933530	+	0.358500i	$0.883288\pi$
$62$	153785.i	0.645268i
$63$	0	0
$64$	−372736.	−1.42188
$65$	88200.0	0.321165
$66$	0	0
$67$	−268777.	−0.893650	−0.446825	−	0.894621i	$-0.647445\pi$
$67$	−268777.	−0.893650	−0.446825	+	0.894621i	$0.647445\pi$
$68$	− 241517.i	− 0.768106i
$69$	0	0
$70$	0	0
$71$	−101922.	−0.284769	−0.142385	−	0.989811i	$-0.545477\pi$
$71$	−101922.	−0.284769	−0.142385	+	0.989811i	$0.545477\pi$
$72$	0	0
$73$	− 317646.i	− 0.816535i	−0.912862	−	0.408267i	$-0.866133\pi$
$73$	− 317646.i	− 0.816535i	0.912862	−	0.408267i	$-0.133867\pi$
$74$	−738924.	−1.82350
$75$	0	0
$76$	549961.i	1.25283i
$77$	0	0
$78$	0	0
$79$	362231.	0.734690	0.367345	−	0.930085i	$-0.380267\pi$
$79$	362231.	0.734690	0.367345	+	0.930085i	$0.380267\pi$
$80$	512133.i	1.00026i
$81$	0	0
$82$	1.32689e6i	2.40654i
$83$	− 216783.i	− 0.379133i	−0.981868	−	0.189567i	$-0.939292\pi$
$83$	− 216783.i	− 0.379133i	0.981868	−	0.189567i	$-0.0607083\pi$
$84$	0	0
$85$	−549045.	−0.894028
$86$	−208968.	−0.328537
$87$	0	0
$88$	−283968.	−0.416698
$89$	1.33456e6i	1.89308i	0.322583	+	0.946541i	$0.395449\pi$
$89$	1.33456e6i	1.89308i	−0.322583	+	0.946541i	$0.604551\pi$
$90$	0	0
$91$	0	0
$92$	473040.	0.607483
$93$	0	0
$94$	367950.i	0.443001i
$95$	1.25024e6	1.45821
$96$	0	0
$97$	1.51409e6i	1.65896i	0.558535	+	0.829481i	$0.311364\pi$
$97$	1.51409e6i	1.65896i	−0.558535	+	0.829481i	$0.688636\pi$
$98$	0	0
$99$	0	0
$100$	−1.39600e6	−1.39600
$101$	− 1.32467e6i	− 1.28571i	−0.765987	−	0.642856i	$-0.777749\pi$
$101$	− 1.32467e6i	− 1.28571i	0.765987	−	0.642856i	$-0.222251\pi$
$102$	0	0
$103$	64004.5i	0.0585732i	0.999571	+	0.0292866i	$0.00932354\pi$
$103$	64004.5i	0.0585732i	−0.999571	+	0.0292866i	$0.990676\pi$
$104$	93115.1i	0.0827789i
$105$	0	0
$106$	726156.	0.609695
$107$	−660543.	−0.539200	−0.269600	−	0.962972i	$-0.586891\pi$
$107$	−660543.	−0.539200	−0.269600	+	0.962972i	$0.586891\pi$
$108$	0	0
$109$	−73169.0	−0.0564999	−0.0282499	−	0.999601i	$-0.508993\pi$
$109$	−73169.0	−0.0564999	−0.0282499	+	0.999601i	$0.508993\pi$
$110$	3.22775e6i	2.42505i
$111$	0	0
$112$	0	0
$113$	−1.60351e6	−1.11131	−0.555655	−	0.831413i	$-0.687532\pi$
$113$	−1.60351e6	−1.11131	−0.555655	+	0.831413i	$0.687532\pi$
$114$	0	0
$115$	− 1.07537e6i	− 0.707073i
$116$	−318240.	−0.203883
$117$	0	0
$118$	− 2.58874e6i	− 1.57559i
$119$	0	0
$120$	0	0
$121$	415880.	0.234753
$122$	1.95294e6i	1.07550i
$123$	0	0
$124$	1.02524e6i	0.537723i
$125$	331904.i	0.169935i
$126$	0	0
$127$	−3.22997e6	−1.57684	−0.788418	−	0.615139i	$-0.789100\pi$
$127$	−3.22997e6	−1.57684	−0.788418	+	0.615139i	$0.789100\pi$
$128$	−1.52371e6	−0.726562
$129$	0	0
$130$	1.05840e6	0.481748
$131$	− 1.99546e6i	− 0.887625i	−0.896120	−	0.443813i	$-0.853626\pi$
$131$	− 1.99546e6i	− 0.887625i	0.896120	−	0.443813i	$-0.146374\pi$
$132$	0	0
$133$	0	0
$134$	−3.22532e6	−1.34048
$135$	0	0
$136$	− 579641.i	− 0.230432i
$137$	−1.42158e6	−0.552854	−0.276427	−	0.961035i	$-0.589150\pi$
$137$	−1.42158e6	−0.552854	−0.276427	+	0.961035i	$0.589150\pi$
$138$	0	0
$139$	− 2.43603e6i	− 0.907063i	−0.891240	−	0.453531i	$-0.850164\pi$
$139$	− 2.43603e6i	− 0.907063i	0.891240	−	0.453531i	$-0.149836\pi$
$140$	0	0
$141$	0	0
$142$	−1.22306e6	−0.427154
$143$	− 717277.i	− 0.245289i
$144$	0	0
$145$	723460.i	0.237307i
$146$	− 3.81175e6i	− 1.22480i
$147$	0	0
$148$	−4.92616e6	−1.51958
$149$	−3.14375e6	−0.950363	−0.475181	−	0.879888i	$-0.657618\pi$
$149$	−3.14375e6	−0.950363	−0.475181	+	0.879888i	$0.657618\pi$
$150$	0	0
$151$	−2.28220e6	−0.662862	−0.331431	−	0.943479i	$-0.607531\pi$
$151$	−2.28220e6	−0.662862	−0.331431	+	0.943479i	$0.607531\pi$
$152$	1.31991e6i	0.375848i
$153$	0	0
$154$	0	0
$155$	2.33068e6	0.625876
$156$	0	0
$157$	148511.i	0.0383761i	0.999816	+	0.0191880i	$0.00610811\pi$
$157$	148511.i	0.0383761i	−0.999816	+	0.0191880i	$0.993892\pi$
$158$	4.34677e6	1.10204
$159$	0	0
$160$	8.38035e6i	2.04599i
$161$	0	0
$162$	0	0
$163$	−7.09747e6	−1.63886	−0.819428	−	0.573182i	$-0.805709\pi$
$163$	−7.09747e6	−1.63886	−0.819428	+	0.573182i	$0.805709\pi$
$164$	8.84593e6i	2.00545i
$165$	0	0
$166$	− 2.60140e6i	− 0.568700i
$167$	645986.i	0.138699i	0.997592	+	0.0693495i	$0.0220924\pi$
$167$	645986.i	0.138699i	−0.997592	+	0.0693495i	$0.977908\pi$
$168$	0	0
$169$	4.59161e6	0.951272
$170$	−6.58854e6	−1.34104
$171$	0	0
$172$	−1.39312e6	−0.273781
$173$	− 3.13583e6i	− 0.605640i	−0.953048	−	0.302820i	$-0.902072\pi$
$173$	− 3.13583e6i	− 0.605640i	0.953048	−	0.302820i	$-0.0979281\pi$
$174$	0	0
$175$	0	0
$176$	4.16486e6	0.763946
$177$	0	0
$178$	1.60148e7i	2.83962i
$179$	−1.01628e7	−1.77196	−0.885980	−	0.463724i	$-0.846513\pi$
$179$	−1.01628e7	−1.77196	−0.885980	+	0.463724i	$0.846513\pi$
$180$	0	0
$181$	− 8.72517e6i	− 1.47143i	−0.677294	−	0.735713i	$-0.736847\pi$
$181$	− 8.72517e6i	− 1.47143i	0.677294	−	0.735713i	$-0.263153\pi$
$182$	0	0
$183$	0	0
$184$	1.13530e6	0.182245
$185$	1.11987e7i	1.76870i
$186$	0	0
$187$	4.46505e6i	0.682812i
$188$	2.45300e6i	0.369168i
$189$	0	0
$190$	1.50028e7	2.18732
$191$	−4.23844e6	−0.608283	−0.304142	−	0.952627i	$-0.598370\pi$
$191$	−4.23844e6	−0.608283	−0.304142	+	0.952627i	$0.598370\pi$
$192$	0	0
$193$	6.14821e6	0.855217	0.427609	−	0.903964i	$-0.359356\pi$
$193$	6.14821e6	0.855217	0.427609	+	0.903964i	$0.359356\pi$
$194$	1.81691e7i	2.48844i
$195$	0	0
$196$	0	0
$197$	−790554.	−0.103403	−0.0517015	−	0.998663i	$-0.516464\pi$
$197$	−790554.	−0.103403	−0.0517015	+	0.998663i	$0.516464\pi$
$198$	0	0
$199$	− 8.12583e6i	− 1.03112i	−0.856854	−	0.515559i	$-0.827584\pi$
$199$	− 8.12583e6i	− 1.03112i	0.856854	−	0.515559i	$-0.172416\pi$
$200$	−3.35040e6	−0.418800
$201$	0	0
$202$	− 1.58960e7i	− 1.92857i
$203$	0	0
$204$	0	0
$205$	2.01096e7	2.33422
$206$	768054.i	0.0878597i
$207$	0	0
$208$	− 1.36569e6i	− 0.151761i
$209$	− 1.01674e7i	− 1.11371i
$210$	0	0
$211$	1.16724e7	1.24254	0.621271	−	0.783595i	$-0.286616\pi$
$211$	1.16724e7	1.24254	0.621271	+	0.783595i	$0.286616\pi$
$212$	4.84104e6	0.508079
$213$	0	0
$214$	−7.92652e6	−0.808800
$215$	3.16700e6i	0.318664i
$216$	0	0
$217$	0	0
$218$	−878028.	−0.0847498
$219$	0	0
$220$	2.15183e7i	2.02088i
$221$	1.46412e6	0.135644
$222$	0	0
$223$	− 2.89821e6i	− 0.261345i	−0.991426	−	0.130673i	$-0.958286\pi$
$223$	− 2.89821e6i	− 0.261345i	0.991426	−	0.130673i	$-0.0417137\pi$
$224$	0	0
$225$	0	0
$226$	−1.92421e7	−1.66697
$227$	− 1.67710e7i	− 1.43378i	−0.697188	−	0.716888i	$-0.745566\pi$
$227$	− 1.67710e7i	− 1.43378i	0.697188	−	0.716888i	$-0.254434\pi$
$228$	0	0
$229$	− 1.74779e7i	− 1.45540i	−0.685895	−	0.727701i	$-0.740589\pi$
$229$	− 1.74779e7i	− 1.45540i	0.685895	−	0.727701i	$-0.259411\pi$
$230$	− 1.29044e7i	− 1.06061i
$231$	0	0
$232$	−763776.	−0.0611649
$233$	1.02044e7	0.806711	0.403355	−	0.915043i	$-0.367844\pi$
$233$	1.02044e7	0.806711	0.403355	+	0.915043i	$0.367844\pi$
$234$	0	0
$235$	5.57644e6	0.429689
$236$	− 1.72583e7i	− 1.31299i
$237$	0	0
$238$	0	0
$239$	−1.43114e6	−0.104831	−0.0524153	−	0.998625i	$-0.516692\pi$
$239$	−1.43114e6	−0.104831	−0.0524153	+	0.998625i	$0.516692\pi$
$240$	0	0
$241$	− 1.55039e7i	− 1.10762i	−0.832643	−	0.553810i	$-0.813173\pi$
$241$	− 1.55039e7i	− 1.10762i	0.832643	−	0.553810i	$-0.186827\pi$
$242$	4.99056e6	0.352130
$243$	0	0
$244$	1.30196e7i	0.896249i
$245$	0	0
$246$	0	0
$247$	−3.33396e6	−0.221243
$248$	2.46057e6i	0.161317i
$249$	0	0
$250$	3.98285e6i	0.254902i
$251$	− 3.44089e6i	− 0.217595i	−0.994064	−	0.108798i	$-0.965300\pi$
$251$	− 3.44089e6i	− 0.217595i	0.994064	−	0.108798i	$-0.0347001\pi$
$252$	0	0
$253$	−8.74533e6	−0.540026
$254$	−3.87596e7	−2.36526
$255$	0	0
$256$	5.57056e6	0.332031
$257$	2.32527e7i	1.36985i	0.728613	+	0.684926i	$0.240166\pi$
$257$	2.32527e7i	1.36985i	−0.728613	+	0.684926i	$0.759834\pi$
$258$	0	0
$259$	0	0
$260$	7.05600e6	0.401457
$261$	0	0
$262$	− 2.39456e7i	− 1.33144i
$263$	2.82618e7	1.55358	0.776789	−	0.629761i	$-0.216847\pi$
$263$	2.82618e7	1.55358	0.776789	+	0.629761i	$0.216847\pi$
$264$	0	0
$265$	− 1.10052e7i	− 0.591372i
$266$	0	0
$267$	0	0
$268$	−2.15022e7	−1.11706
$269$	1.58102e7i	0.812235i	0.913821	+	0.406118i	$0.133118\pi$
$269$	1.58102e7i	0.812235i	−0.913821	+	0.406118i	$0.866882\pi$
$270$	0	0
$271$	3.65028e7i	1.83408i	0.398793	+	0.917041i	$0.369429\pi$
$271$	3.65028e7i	1.83408i	−0.398793	+	0.917041i	$0.630571\pi$
$272$	8.50140e6i	0.422458i
$273$	0	0
$274$	−1.70590e7	−0.829281
$275$	2.58086e7	1.24098
$276$	0	0
$277$	3.14807e7	1.48117	0.740585	−	0.671962i	$-0.234548\pi$
$277$	3.14807e7	1.48117	0.740585	+	0.671962i	$0.234548\pi$
$278$	− 2.92323e7i	− 1.36059i
$279$	0	0
$280$	0	0
$281$	−6.62368e6	−0.298525	−0.149262	−	0.988798i	$-0.547690\pi$
$281$	−6.62368e6	−0.298525	−0.149262	+	0.988798i	$0.547690\pi$
$282$	0	0
$283$	4.02480e7i	1.77576i	0.460071	+	0.887882i	$0.347824\pi$
$283$	4.02480e7i	1.77576i	−0.460071	+	0.887882i	$0.652176\pi$
$284$	−8.15376e6	−0.355961
$285$	0	0
$286$	− 8.60732e6i	− 0.367934i
$287$	0	0
$288$	0	0
$289$	1.50234e7	0.622408
$290$	8.68152e6i	0.355961i
$291$	0	0
$292$	− 2.54117e7i	− 1.02067i
$293$	− 1.26797e7i	− 0.504086i	−0.967716	−	0.252043i	$-0.918898\pi$
$293$	− 1.26797e7i	− 0.504086i	0.967716	−	0.252043i	$-0.0811024\pi$
$294$	0	0
$295$	−3.92336e7	−1.52824
$296$	−1.18228e7	−0.455874
$297$	0	0
$298$	−3.77250e7	−1.42554
$299$	2.86765e6i	0.107279i
$300$	0	0
$301$	0	0
$302$	−2.73864e7	−0.994293
$303$	0	0
$304$	− 1.93586e7i	− 0.689055i
$305$	2.95977e7	1.04318
$306$	0	0
$307$	1.77258e6i	0.0612620i	0.999531	+	0.0306310i	$0.00975167\pi$
$307$	1.77258e6i	0.0612620i	−0.999531	+	0.0306310i	$0.990248\pi$
$308$	0	0
$309$	0	0
$310$	2.79682e7	0.938814
$311$	− 3.65265e7i	− 1.21430i	−0.794586	−	0.607152i	$-0.792312\pi$
$311$	− 3.65265e7i	− 1.21430i	0.794586	−	0.607152i	$-0.207688\pi$
$312$	0	0
$313$	1.16049e7i	0.378451i	0.981934	+	0.189226i	$0.0605977\pi$
$313$	1.16049e7i	0.378451i	−0.981934	+	0.189226i	$0.939402\pi$
$314$	1.78213e6i	0.0575641i
$315$	0	0
$316$	2.89785e7	0.918363
$317$	−2.11641e7	−0.664388	−0.332194	−	0.943211i	$-0.607789\pi$
$317$	−2.11641e7	−0.664388	−0.332194	+	0.943211i	$0.607789\pi$
$318$	0	0
$319$	5.88346e6	0.181243
$320$	6.77878e7i	2.06872i
$321$	0	0
$322$	0	0
$323$	2.07539e7	0.615874
$324$	0	0
$325$	− 8.46280e6i	− 0.246527i
$326$	−8.51697e7	−2.45828
$327$	0	0
$328$	2.12302e7i	0.601635i
$329$	0	0
$330$	0	0
$331$	−5.92199e7	−1.63299	−0.816496	−	0.577352i	$-0.804086\pi$
$331$	−5.92199e7	−1.63299	−0.816496	+	0.577352i	$0.804086\pi$
$332$	− 1.73427e7i	− 0.473917i
$333$	0	0
$334$	7.75183e6i	0.208049i
$335$	4.88812e7i	1.30019i
$336$	0	0
$337$	3.67798e7	0.960992	0.480496	−	0.876997i	$-0.340457\pi$
$337$	3.67798e7	0.960992	0.480496	+	0.876997i	$0.340457\pi$
$338$	5.50993e7	1.42691
$339$	0	0
$340$	−4.39236e7	−1.11754
$341$	− 1.89540e7i	− 0.478012i
$342$	0	0
$343$	0	0
$344$	−3.34349e6	−0.0821343
$345$	0	0
$346$	− 3.76300e7i	− 0.908460i
$347$	−3.39775e7	−0.813210	−0.406605	−	0.913604i	$-0.633287\pi$
$347$	−3.39775e7	−0.813210	−0.406605	+	0.913604i	$0.633287\pi$
$348$	0	0
$349$	5.48045e7i	1.28926i	0.764495	+	0.644629i	$0.222988\pi$
$349$	5.48045e7i	1.28926i	−0.764495	+	0.644629i	$0.777012\pi$
$350$	0	0
$351$	0	0
$352$	6.81523e7	1.56262
$353$	− 4.76003e6i	− 0.108215i	−0.998535	−	0.0541073i	$-0.982769\pi$
$353$	− 4.76003e6i	− 0.108215i	0.998535	−	0.0541073i	$-0.0172313\pi$
$354$	0	0
$355$	1.85361e7i	0.414317i
$356$	1.06765e8i	2.36635i
$357$	0	0
$358$	−1.21953e8	−2.65794
$359$	5.31169e6	0.114802	0.0574010	−	0.998351i	$-0.481719\pi$
$359$	5.31169e6	0.114802	0.0574010	+	0.998351i	$0.481719\pi$
$360$	0	0
$361$	−213002.	−0.00452754
$362$	− 1.04702e8i	− 2.20714i
$363$	0	0
$364$	0	0
$365$	−5.77688e7	−1.18800
$366$	0	0
$367$	− 7.98359e7i	− 1.61510i	−0.589798	−	0.807551i	$-0.700793\pi$
$367$	− 7.98359e7i	− 1.61510i	0.589798	−	0.807551i	$-0.299207\pi$
$368$	−1.66510e7	−0.334116
$369$	0	0
$370$	1.34385e8i	2.65304i
$371$	0	0
$372$	0	0
$373$	4.22601e7	0.814336	0.407168	−	0.913353i	$-0.366516\pi$
$373$	4.22601e7	0.814336	0.407168	+	0.913353i	$0.366516\pi$
$374$	5.35806e7i	1.02422i
$375$	0	0
$376$	5.88720e6i	0.110750i
$377$	− 1.92923e6i	− 0.0360047i
$378$	0	0
$379$	1.28840e7	0.236664	0.118332	−	0.992974i	$-0.462245\pi$
$379$	1.28840e7	0.236664	0.118332	+	0.992974i	$0.462245\pi$
$380$	1.00019e8	1.82277
$381$	0	0
$382$	−5.08613e7	−0.912425
$383$	2.67570e7i	0.476257i	0.971234	+	0.238128i	$0.0765339\pi$
$383$	2.67570e7i	0.476257i	−0.971234	+	0.238128i	$0.923466\pi$
$384$	0	0
$385$	0	0
$386$	7.37785e7	1.28283
$387$	0	0
$388$	1.21127e8i	2.07370i
$389$	1.07557e7	0.182722	0.0913610	−	0.995818i	$-0.470878\pi$
$389$	1.07557e7	0.182722	0.0913610	+	0.995818i	$0.470878\pi$
$390$	0	0
$391$	− 1.78511e7i	− 0.298632i
$392$	0	0
$393$	0	0
$394$	−9.48665e6	−0.155104
$395$	− 6.58773e7i	− 1.06892i
$396$	0	0
$397$	9.52333e6i	0.152201i	0.997100	+	0.0761005i	$0.0242470\pi$
$397$	9.52333e6i	0.152201i	−0.997100	+	0.0761005i	$0.975753\pi$
$398$	− 9.75099e7i	− 1.54668i
$399$	0	0
$400$	4.91392e7	0.767800
$401$	−7.67884e7	−1.19086	−0.595432	−	0.803405i	$-0.703019\pi$
$401$	−7.67884e7	−1.19086	−0.595432	+	0.803405i	$0.703019\pi$
$402$	0	0
$403$	−6.21516e6	−0.0949592
$404$	− 1.05974e8i	− 1.60714i
$405$	0	0
$406$	0	0
$407$	9.10724e7	1.35084
$408$	0	0
$409$	− 3.00588e7i	− 0.439341i	−0.975574	−	0.219670i	$-0.929502\pi$
$409$	− 3.00588e7i	− 0.439341i	0.975574	−	0.219670i	$-0.0704982\pi$
$410$	2.41315e8	3.50133
$411$	0	0
$412$	5.12036e6i	0.0732165i
$413$	0	0
$414$	0	0
$415$	−3.94254e7	−0.551610
$416$	− 2.23476e7i	− 0.310421i
$417$	0	0
$418$	− 1.22009e8i	− 1.67056i
$419$	4.76385e7i	0.647614i	0.946123	+	0.323807i	$0.104963\pi$
$419$	4.76385e7i	0.647614i	−0.946123	+	0.323807i	$0.895037\pi$
$420$	0	0
$421$	−2.79191e7	−0.374158	−0.187079	−	0.982345i	$-0.559902\pi$
$421$	−2.79191e7	−0.374158	−0.187079	+	0.982345i	$0.559902\pi$
$422$	1.40068e8	1.86381
$423$	0	0
$424$	1.16185e7	0.152424
$425$	5.26809e7i	0.686257i
$426$	0	0
$427$	0	0
$428$	−5.28434e7	−0.674000
$429$	0	0
$430$	3.80040e7i	0.477996i
$431$	−3.31444e6	−0.0413979	−0.0206989	−	0.999786i	$-0.506589\pi$
$431$	−3.31444e6	−0.0413979	−0.0206989	+	0.999786i	$0.506589\pi$
$432$	0	0
$433$	− 4.59746e7i	− 0.566310i	−0.959074	−	0.283155i	$-0.908619\pi$
$433$	− 4.59746e7i	− 0.566310i	0.959074	−	0.283155i	$-0.0913811\pi$
$434$	0	0
$435$	0	0
$436$	−5.85352e6	−0.0706249
$437$	4.06490e7i	0.487086i
$438$	0	0
$439$	5.44399e7i	0.643463i	0.946831	+	0.321732i	$0.104265\pi$
$439$	5.44399e7i	0.643463i	−0.946831	+	0.321732i	$0.895735\pi$
$440$	5.16439e7i	0.606263i
$441$	0	0
$442$	1.75694e7	0.203466
$443$	−1.05887e8	−1.21796	−0.608978	−	0.793187i	$-0.708420\pi$
$443$	−1.05887e8	−1.21796	−0.608978	+	0.793187i	$0.708420\pi$
$444$	0	0
$445$	2.42711e8	2.75429
$446$	− 3.47785e7i	− 0.392018i
$447$	0	0
$448$	0	0
$449$	−1.14053e8	−1.26000	−0.629998	−	0.776597i	$-0.716944\pi$
$449$	−1.14053e8	−1.26000	−0.629998	+	0.776597i	$0.716944\pi$
$450$	0	0
$451$	− 1.63539e8i	− 1.78276i
$452$	−1.28280e8	−1.38914
$453$	0	0
$454$	− 2.01252e8i	− 2.15066i
$455$	0	0
$456$	0	0
$457$	9.84729e7	1.03174	0.515868	−	0.856668i	$-0.327470\pi$
$457$	9.84729e7	1.03174	0.515868	+	0.856668i	$0.327470\pi$
$458$	− 2.09735e8i	− 2.18310i
$459$	0	0
$460$	− 8.60296e7i	− 0.883841i
$461$	3.58333e7i	0.365750i	0.983136	+	0.182875i	$0.0585403\pi$
$461$	3.58333e7i	0.365750i	−0.983136	+	0.182875i	$0.941460\pi$
$462$	0	0
$463$	1.81000e8	1.82362	0.911810	−	0.410612i	$-0.134685\pi$
$463$	1.81000e8	1.82362	0.911810	+	0.410612i	$0.134685\pi$
$464$	1.12020e7	0.112136
$465$	0	0
$466$	1.22452e8	1.21007
$467$	− 8.48539e7i	− 0.833146i	−0.909102	−	0.416573i	$-0.863231\pi$
$467$	− 8.48539e7i	− 0.833146i	0.909102	−	0.416573i	$-0.136769\pi$
$468$	0	0
$469$	0	0
$470$	6.69173e7	0.644533
$471$	0	0
$472$	− 4.14199e7i	− 0.393897i
$473$	2.57553e7	0.243379
$474$	0	0
$475$	− 1.19960e8i	− 1.11933i
$476$	0	0
$477$	0	0
$478$	−1.71737e7	−0.157246
$479$	− 1.59224e8i	− 1.44878i	−0.689390	−	0.724390i	$-0.742121\pi$
$479$	− 1.59224e8i	− 1.44878i	0.689390	−	0.724390i	$-0.257879\pi$
$480$	0	0
$481$	− 2.98633e7i	− 0.268350i
$482$	− 1.86047e8i	− 1.66143i
$483$	0	0
$484$	3.32704e7	0.293442
$485$	2.75360e8	2.41366
$486$	0	0
$487$	−8.06142e7	−0.697951	−0.348975	−	0.937132i	$-0.613470\pi$
$487$	−8.06142e7	−0.697951	−0.348975	+	0.937132i	$0.613470\pi$
$488$	3.12471e7i	0.268875i
$489$	0	0
$490$	0	0
$491$	1.46544e7	0.123800	0.0619002	−	0.998082i	$-0.480284\pi$
$491$	1.46544e7	0.123800	0.0619002	+	0.998082i	$0.480284\pi$
$492$	0	0
$493$	1.20094e7i	0.100226i
$494$	−4.00075e7	−0.331864
$495$	0	0
$496$	− 3.60883e7i	− 0.295748i
$497$	0	0
$498$	0	0
$499$	−3.12556e7	−0.251551	−0.125776	−	0.992059i	$-0.540142\pi$
$499$	−3.12556e7	−0.251551	−0.125776	+	0.992059i	$0.540142\pi$
$500$	2.65523e7i	0.212419i
$501$	0	0
$502$	− 4.12907e7i	− 0.326393i
$503$	− 1.42620e8i	− 1.12067i	−0.828267	−	0.560334i	$-0.810673\pi$
$503$	− 1.42620e8i	− 1.12067i	0.828267	−	0.560334i	$-0.189327\pi$
$504$	0	0
$505$	−2.40912e8	−1.87061
$506$	−1.04944e8	−0.810039
$507$	0	0
$508$	−2.58397e8	−1.97105
$509$	1.04526e7i	0.0792628i	0.999214	+	0.0396314i	$0.0126184\pi$
$509$	1.04526e7i	0.0792628i	−0.999214	+	0.0396314i	$0.987382\pi$
$510$	0	0
$511$	0	0
$512$	1.64364e8	1.22461
$513$	0	0
$514$	2.79032e8i	2.05478i
$515$	1.16402e7	0.0852194
$516$	0	0
$517$	− 4.53498e7i	− 0.328174i
$518$	0	0
$519$	0	0
$520$	1.69344e7	0.120437
$521$	2.72001e6i	0.0192335i	0.999954	+	0.00961674i	$0.00306115\pi$
$521$	2.72001e6i	0.0192335i	−0.999954	+	0.00961674i	$0.996939\pi$
$522$	0	0
$523$	− 2.29072e8i	− 1.60128i	−0.599145	−	0.800640i	$-0.704493\pi$
$523$	− 2.29072e8i	− 1.60128i	0.599145	−	0.800640i	$-0.295507\pi$
$524$	− 1.59637e8i	− 1.10953i
$525$	0	0
$526$	3.39142e8	2.33037
$527$	3.86894e7	0.264338
$528$	0	0
$529$	−1.13072e8	−0.763817
$530$	− 1.32063e8i	− 0.887058i
$531$	0	0
$532$	0	0
$533$	−5.36256e7	−0.354153
$534$	0	0
$535$	1.20130e8i	0.784494i
$536$	−5.16052e7	−0.335119
$537$	0	0
$538$	1.89723e8i	1.21835i
$539$	0	0
$540$	0	0
$541$	2.29389e8	1.44871	0.724355	−	0.689428i	$-0.242138\pi$
$541$	2.29389e8	1.44871	0.724355	+	0.689428i	$0.242138\pi$
$542$	4.38034e8i	2.75112i
$543$	0	0
$544$	1.39114e8i	0.864119i
$545$	1.33069e7i	0.0822030i
$546$	0	0
$547$	7.87986e7	0.481456	0.240728	−	0.970593i	$-0.422614\pi$
$547$	7.87986e7	0.481456	0.240728	+	0.970593i	$0.422614\pi$
$548$	−1.13727e8	−0.691068
$549$	0	0
$550$	3.09703e8	1.86147
$551$	− 2.73468e7i	− 0.163475i
$552$	0	0
$553$	0	0
$554$	3.77768e8	2.22176
$555$	0	0
$556$	− 1.94882e8i	− 1.13383i
$557$	1.07355e8	0.621236	0.310618	−	0.950535i	$-0.399464\pi$
$557$	1.07355e8	0.621236	0.310618	+	0.950535i	$0.399464\pi$
$558$	0	0
$559$	− 8.44534e6i	− 0.0483484i
$560$	0	0
$561$	0	0
$562$	−7.94842e7	−0.447787
$563$	2.29315e8i	1.28501i	0.766280	+	0.642507i	$0.222106\pi$
$563$	2.29315e8i	1.28501i	−0.766280	+	0.642507i	$0.777894\pi$
$564$	0	0
$565$	2.91622e8i	1.61687i
$566$	4.82976e8i	2.66365i
$567$	0	0
$568$	−1.95690e7	−0.106788
$569$	1.47105e8	0.798526	0.399263	−	0.916836i	$-0.369266\pi$
$569$	1.47105e8	0.798526	0.399263	+	0.916836i	$0.369266\pi$
$570$	0	0
$571$	2.75556e6	0.0148014	0.00740068	−	0.999973i	$-0.497644\pi$
$571$	2.75556e6	0.0148014	0.00740068	+	0.999973i	$0.497644\pi$
$572$	− 5.73822e7i	− 0.306612i
$573$	0	0
$574$	0	0
$575$	−1.03182e8	−0.542750
$576$	0	0
$577$	3.19900e6i	0.0166528i	0.999965	+	0.00832639i	$0.00265040\pi$
$577$	3.19900e6i	0.0166528i	−0.999965	+	0.00832639i	$0.997350\pi$
$578$	1.80281e8	0.933612
$579$	0	0
$580$	5.78768e7i	0.296634i
$581$	0	0
$582$	0	0
$583$	−8.94987e7	−0.451660
$584$	− 6.09880e7i	− 0.306201i
$585$	0	0
$586$	− 1.52156e8i	− 0.756129i
$587$	2.39854e8i	1.18586i	0.805254	+	0.592929i	$0.202029\pi$
$587$	2.39854e8i	1.18586i	−0.805254	+	0.592929i	$0.797971\pi$
$588$	0	0
$589$	−8.80999e7	−0.431151
$590$	−4.70803e8	−2.29236
$591$	0	0
$592$	1.73401e8	0.835769
$593$	1.89913e8i	0.910734i	0.890304	+	0.455367i	$0.150492\pi$
$593$	1.89913e8i	0.910734i	−0.890304	+	0.455367i	$0.849508\pi$
$594$	0	0
$595$	0	0
$596$	−2.51500e8	−1.18795
$597$	0	0
$598$	3.44118e7i	0.160918i
$599$	−3.85151e8	−1.79205	−0.896025	−	0.444004i	$-0.853558\pi$
$599$	−3.85151e8	−1.79205	−0.896025	+	0.444004i	$0.853558\pi$
$600$	0	0
$601$	2.97376e8i	1.36988i	0.728598	+	0.684941i	$0.240172\pi$
$601$	2.97376e8i	1.36988i	−0.728598	+	0.684941i	$0.759828\pi$
$602$	0	0
$603$	0	0
$604$	−1.82576e8	−0.828577
$605$	− 7.56342e7i	− 0.341548i
$606$	0	0
$607$	− 2.95316e8i	− 1.32045i	−0.751069	−	0.660223i	$-0.770462\pi$
$607$	− 2.95316e8i	− 1.32045i	0.751069	−	0.660223i	$-0.229538\pi$
$608$	− 3.16777e8i	− 1.40943i
$609$	0	0
$610$	3.55173e8	1.56477
$611$	−1.48705e7	−0.0651932
$612$	0	0
$613$	−3.37212e8	−1.46394	−0.731968	−	0.681339i	$-0.761398\pi$
$613$	−3.37212e8	−1.46394	−0.731968	+	0.681339i	$0.761398\pi$
$614$	2.12710e7i	0.0918929i
$615$	0	0
$616$	0	0
$617$	−3.68150e8	−1.56736	−0.783682	−	0.621163i	$-0.786661\pi$
$617$	−3.68150e8	−1.56736	−0.783682	+	0.621163i	$0.786661\pi$
$618$	0	0
$619$	3.10795e7i	0.131039i	0.997851	+	0.0655197i	$0.0208705\pi$
$619$	3.10795e7i	0.131039i	−0.997851	+	0.0655197i	$0.979129\pi$
$620$	1.86455e8	0.782345
$621$	0	0
$622$	− 4.38318e8i	− 1.82146i
$623$	0	0
$624$	0	0
$625$	−2.12294e8	−0.869558
$626$	1.39259e8i	0.567677i
$627$	0	0
$628$	1.18809e7i	0.0479701i
$629$	1.85899e8i	0.747007i
$630$	0	0
$631$	−3.25406e8	−1.29520	−0.647601	−	0.761980i	$-0.724228\pi$
$631$	−3.25406e8	−1.29520	−0.647601	+	0.761980i	$0.724228\pi$
$632$	6.95484e7	0.275509
$633$	0	0
$634$	−2.53969e8	−0.996583
$635$	5.87419e8i	2.29418i
$636$	0	0
$637$	0	0
$638$	7.06015e7	0.271864
$639$	0	0
$640$	2.77110e8i	1.05709i
$641$	5.55637e7	0.210968	0.105484	−	0.994421i	$-0.466361\pi$
$641$	5.55637e7	0.210968	0.105484	+	0.994421i	$0.466361\pi$
$642$	0	0
$643$	9.33357e7i	0.351087i	0.984472	+	0.175544i	$0.0561683\pi$
$643$	9.33357e7i	0.351087i	−0.984472	+	0.175544i	$0.943832\pi$
$644$	0	0
$645$	0	0
$646$	2.49047e8	0.923812
$647$	− 1.93195e8i	− 0.713317i	−0.934235	−	0.356659i	$-0.883916\pi$
$647$	− 1.93195e8i	− 0.713317i	0.934235	−	0.356659i	$-0.116084\pi$
$648$	0	0
$649$	3.19063e8i	1.16719i
$650$	− 1.01554e8i	− 0.369790i
$651$	0	0
$652$	−5.67798e8	−2.04857
$653$	−3.62447e8	−1.30168	−0.650842	−	0.759213i	$-0.725584\pi$
$653$	−3.62447e8	−1.30168	−0.650842	+	0.759213i	$0.725584\pi$
$654$	0	0
$655$	−3.62906e8	−1.29143
$656$	− 3.11377e8i	− 1.10300i
$657$	0	0
$658$	0	0
$659$	−2.39985e8	−0.838549	−0.419274	−	0.907860i	$-0.637715\pi$
$659$	−2.39985e8	−0.838549	−0.419274	+	0.907860i	$0.637715\pi$
$660$	0	0
$661$	1.93221e7i	0.0669035i	0.999440	+	0.0334518i	$0.0106500\pi$
$661$	1.93221e7i	0.0669035i	−0.999440	+	0.0334518i	$0.989350\pi$
$662$	−7.10639e8	−2.44949
$663$	0	0
$664$	− 4.16224e7i	− 0.142175i
$665$	0	0
$666$	0	0
$667$	−2.35219e7	−0.0792675
$668$	5.16789e7i	0.173374i
$669$	0	0
$670$	5.86575e8i	1.95029i
$671$	− 2.40700e8i	− 0.796726i
$672$	0	0
$673$	−4.27171e7	−0.140138	−0.0700692	−	0.997542i	$-0.522322\pi$
$673$	−4.27171e7	−0.140138	−0.0700692	+	0.997542i	$0.522322\pi$
$674$	4.41358e8	1.44149
$675$	0	0
$676$	3.67329e8	1.18909
$677$	2.90750e8i	0.937032i	0.883455	+	0.468516i	$0.155211\pi$
$677$	2.90750e8i	0.937032i	−0.883455	+	0.468516i	$0.844789\pi$
$678$	0	0
$679$	0	0
$680$	−1.05417e8	−0.335261
$681$	0	0
$682$	− 2.27448e8i	− 0.717018i
$683$	−8.21931e7	−0.257972	−0.128986	−	0.991646i	$-0.541172\pi$
$683$	−8.21931e7	−0.257972	−0.128986	+	0.991646i	$0.541172\pi$
$684$	0	0
$685$	2.58537e8i	0.804360i
$686$	0	0
$687$	0	0
$688$	4.90378e7	0.150580
$689$	2.93472e7i	0.0897242i
$690$	0	0
$691$	3.47176e7i	0.105224i	0.998615	+	0.0526122i	$0.0167547\pi$
$691$	3.47176e7i	0.105224i	−0.998615	+	0.0526122i	$0.983245\pi$
$692$	− 2.50866e8i	− 0.757050i
$693$	0	0
$694$	−4.07730e8	−1.21981
$695$	−4.43029e8	−1.31971
$696$	0	0
$697$	3.33819e8	0.985855
$698$	6.57654e8i	1.93389i
$699$	0	0
$700$	0	0
$701$	3.04543e8	0.884087	0.442044	−	0.896994i	$-0.354254\pi$
$701$	3.04543e8	0.884087	0.442044	+	0.896994i	$0.354254\pi$
$702$	0	0
$703$	− 4.23312e8i	− 1.21841i
$704$	5.51277e8	1.57998
$705$	0	0
$706$	− 5.71204e7i	− 0.162322i
$707$	0	0
$708$	0	0
$709$	−2.28892e8	−0.642231	−0.321116	−	0.947040i	$-0.604058\pi$
$709$	−2.28892e8	−0.642231	−0.321116	+	0.947040i	$0.604058\pi$
$710$	2.22433e8i	0.621476i
$711$	0	0
$712$	2.56236e8i	0.709906i
$713$	7.57777e7i	0.209061i
$714$	0	0
$715$	−1.30448e8	−0.356877
$716$	−8.13023e8	−2.21495
$717$	0	0
$718$	6.37403e7	0.172203
$719$	− 4.47898e8i	− 1.20501i	−0.798114	−	0.602507i	$-0.794169\pi$
$719$	− 4.47898e8i	− 1.20501i	0.798114	−	0.602507i	$-0.205831\pi$
$720$	0	0
$721$	0	0
$722$	−2.55602e6	−0.00679131
$723$	0	0
$724$	− 6.98014e8i	− 1.83928i
$725$	6.94161e7	0.182157
$726$	0	0
$727$	− 3.71752e7i	− 0.0967498i	−0.998829	−	0.0483749i	$-0.984596\pi$
$727$	− 3.71752e7i	− 0.0967498i	0.998829	−	0.0483749i	$-0.0154042\pi$
$728$	0	0
$729$	0	0
$730$	−6.93226e8	−1.78199
$731$	5.25722e7i	0.134587i
$732$	0	0
$733$	6.05169e8i	1.53661i	0.640081	+	0.768307i	$0.278901\pi$
$733$	6.05169e8i	1.53661i	−0.640081	+	0.768307i	$0.721099\pi$
$734$	− 9.58030e8i	− 2.42265i
$735$	0	0
$736$	−2.72471e8	−0.683419
$737$	3.97521e8	0.993020
$738$	0	0
$739$	−2.01787e8	−0.499989	−0.249994	−	0.968247i	$-0.580429\pi$
$739$	−2.01787e8	−0.499989	−0.249994	+	0.968247i	$0.580429\pi$
$740$	8.95898e8i	2.21087i
$741$	0	0
$742$	0	0
$743$	9.81196e7	0.239216	0.119608	−	0.992821i	$-0.461836\pi$
$743$	9.81196e7	0.239216	0.119608	+	0.992821i	$0.461836\pi$
$744$	0	0
$745$	5.71739e8i	1.38270i
$746$	5.07121e8	1.22150
$747$	0	0
$748$	3.57204e8i	0.853515i
$749$	0	0
$750$	0	0
$751$	−4.25827e8	−1.00534	−0.502670	−	0.864478i	$-0.667649\pi$
$751$	−4.25827e8	−1.00534	−0.502670	+	0.864478i	$0.667649\pi$
$752$	− 8.63456e7i	− 0.203042i
$753$	0	0
$754$	− 2.31507e7i	− 0.0540071i
$755$	4.15053e8i	0.964413i
$756$	0	0
$757$	3.07427e7	0.0708687	0.0354344	−	0.999372i	$-0.488719\pi$
$757$	3.07427e7	0.0708687	0.0354344	+	0.999372i	$0.488719\pi$
$758$	1.54608e8	0.354996
$759$	0	0
$760$	2.40045e8	0.546830
$761$	− 3.71229e8i	− 0.842340i	−0.906982	−	0.421170i	$-0.861620\pi$
$761$	− 3.71229e8i	− 0.842340i	0.906982	−	0.421170i	$-0.138380\pi$
$762$	0	0
$763$	0	0
$764$	−3.39075e8	−0.760354
$765$	0	0
$766$	3.21084e8i	0.714385i
$767$	1.04623e8	0.231868
$768$	0	0
$769$	4.59933e8i	1.01138i	0.862714	+	0.505691i	$0.168763\pi$
$769$	4.59933e8i	1.01138i	−0.862714	+	0.505691i	$0.831237\pi$
$770$	0	0
$771$	0	0
$772$	4.91857e8	1.06902
$773$	− 6.86231e8i	− 1.48570i	−0.669456	−	0.742852i	$-0.733473\pi$
$773$	− 6.86231e8i	− 1.48570i	0.669456	−	0.742852i	$-0.266527\pi$
$774$	0	0
$775$	− 2.23629e8i	− 0.480423i
$776$	2.90705e8i	0.622111i
$777$	0	0
$778$	1.29069e8	0.274083
$779$	−7.60143e8	−1.60799
$780$	0	0
$781$	1.50743e8	0.316434
$782$	− 2.14214e8i	− 0.447947i
$783$	0	0
$784$	0	0
$785$	2.70090e7	0.0558342
$786$	0	0
$787$	5.62391e8i	1.15376i	0.816830	+	0.576879i	$0.195729\pi$
$787$	5.62391e8i	1.15376i	−0.816830	+	0.576879i	$0.804271\pi$
$788$	−6.32443e7	−0.129254
$789$	0	0
$790$	− 7.90527e8i	− 1.60338i
$791$	0	0
$792$	0	0
$793$	−7.89272e7	−0.158273
$794$	1.14280e8i	0.228301i
$795$	0	0
$796$	− 6.50066e8i	− 1.28890i
$797$	9.95198e8i	1.96578i	0.184200	+	0.982889i	$0.441031\pi$
$797$	9.95198e8i	1.96578i	−0.184200	+	0.982889i	$0.558969\pi$
$798$	0	0
$799$	9.25690e7	0.181478
$800$	8.04096e8	1.57050
$801$	0	0
$802$	−9.21461e8	−1.78630
$803$	4.69798e8i	0.907329i
$804$	0	0
$805$	0	0
$806$	−7.45819e7	−0.142439
$807$	0	0
$808$	− 2.54337e8i	− 0.482142i
$809$	−7.15167e8	−1.35071	−0.675355	−	0.737493i	$-0.736010\pi$
$809$	−7.15167e8	−1.35071	−0.675355	+	0.737493i	$0.736010\pi$
$810$	0	0
$811$	3.33892e8i	0.625955i	0.949761	+	0.312977i	$0.101326\pi$
$811$	3.33892e8i	0.625955i	−0.949761	+	0.312977i	$0.898674\pi$
$812$	0	0
$813$	0	0
$814$	1.09287e9	2.02626
$815$	1.29078e9i	2.38441i
$816$	0	0
$817$	− 1.19713e8i	− 0.219520i
$818$	− 3.60705e8i	− 0.659011i
$819$	0	0
$820$	1.60877e9	2.91778
$821$	4.83959e8	0.874540	0.437270	−	0.899330i	$-0.355945\pi$
$821$	4.83959e8	0.874540	0.437270	+	0.899330i	$0.355945\pi$
$822$	0	0
$823$	1.57007e8	0.281657	0.140829	−	0.990034i	$-0.455023\pi$
$823$	1.57007e8	0.281657	0.140829	+	0.990034i	$0.455023\pi$
$824$	1.22889e7i	0.0219649i
$825$	0	0
$826$	0	0
$827$	−8.84716e8	−1.56418	−0.782091	−	0.623164i	$-0.785847\pi$
$827$	−8.84716e8	−1.56418	−0.782091	+	0.623164i	$0.785847\pi$
$828$	0	0
$829$	8.14222e7i	0.142916i	0.997444	+	0.0714578i	$0.0227651\pi$
$829$	8.14222e7i	0.142916i	−0.997444	+	0.0714578i	$0.977235\pi$
$830$	−4.73105e8	−0.827414
$831$	0	0
$832$	− 1.80767e8i	− 0.313870i
$833$	0	0
$834$	0	0
$835$	1.17482e8	0.201796
$836$	− 8.13392e8i	− 1.39213i
$837$	0	0
$838$	5.71662e8i	0.971421i
$839$	4.66806e8i	0.790407i	0.918594	+	0.395203i	$0.129326\pi$
$839$	4.66806e8i	0.790407i	−0.918594	+	0.395203i	$0.870674\pi$
$840$	0	0
$841$	−5.78999e8	−0.973396
$842$	−3.35029e8	−0.561237
$843$	0	0
$844$	9.33789e8	1.55318
$845$	− 8.35055e8i	− 1.38403i
$846$	0	0
$847$	0	0
$848$	−1.70405e8	−0.279443
$849$	0	0
$850$	6.32171e8i	1.02939i
$851$	−3.64105e8	−0.590796
$852$	0	0
$853$	− 8.80430e8i	− 1.41856i	−0.704926	−	0.709280i	$-0.749020\pi$
$853$	− 8.80430e8i	− 1.41856i	0.704926	−	0.709280i	$-0.250980\pi$
$854$	0	0
$855$	0	0
$856$	−1.26824e8	−0.202200
$857$	1.79898e8i	0.285815i	0.989736	+	0.142907i	$0.0456451\pi$
$857$	1.79898e8i	0.285815i	−0.989736	+	0.142907i	$0.954355\pi$
$858$	0	0
$859$	− 9.06857e8i	− 1.43074i	−0.698748	−	0.715368i	$-0.746259\pi$
$859$	− 9.06857e8i	− 1.43074i	0.698748	−	0.715368i	$-0.253741\pi$
$860$	2.53360e8i	0.398330i
$861$	0	0
$862$	−3.97733e7	−0.0620968
$863$	−5.44143e7	−0.0846605	−0.0423303	−	0.999104i	$-0.513478\pi$
$863$	−5.44143e7	−0.0846605	−0.0423303	+	0.999104i	$0.513478\pi$
$864$	0	0
$865$	−5.70299e8	−0.881159
$866$	− 5.51695e8i	− 0.849465i
$867$	0	0
$868$	0	0
$869$	−5.35740e8	−0.816384
$870$	0	0
$871$	− 1.30350e8i	− 0.197268i
$872$	−1.40484e7	−0.0211875
$873$	0	0
$874$	4.87788e8i	0.730628i
$875$	0	0
$876$	0	0
$877$	−1.26037e8	−0.186853	−0.0934263	−	0.995626i	$-0.529782\pi$
$877$	−1.26037e8	−0.186853	−0.0934263	+	0.995626i	$0.529782\pi$
$878$	6.53279e8i	0.965195i
$879$	0	0
$880$	− 7.57444e8i	− 1.11148i
$881$	1.37857e8i	0.201605i	0.994906	+	0.100802i	$0.0321410\pi$
$881$	1.37857e8i	0.201605i	−0.994906	+	0.100802i	$0.967859\pi$
$882$	0	0
$883$	7.41288e8	1.07672	0.538362	−	0.842713i	$-0.319043\pi$
$883$	7.41288e8	1.07672	0.538362	+	0.842713i	$0.319043\pi$
$884$	1.17130e8	0.169555
$885$	0	0
$886$	−1.27064e9	−1.82693
$887$	3.49873e8i	0.501348i	0.968072	+	0.250674i	$0.0806522\pi$
$887$	3.49873e8i	0.501348i	−0.968072	+	0.250674i	$0.919348\pi$
$888$	0	0
$889$	0	0
$890$	2.91253e9	4.13143
$891$	0	0
$892$	− 2.31856e8i	− 0.326682i
$893$	−2.10790e8	−0.296002
$894$	0	0
$895$	1.84826e9i	2.57806i
$896$	0	0
$897$	0	0
$898$	−1.36864e9	−1.88999
$899$	− 5.09798e7i	− 0.0701648i
$900$	0	0
$901$	− 1.82687e8i	− 0.249765i
$902$	− 1.96247e9i	− 2.67413i
$903$	0	0
$904$	−3.07873e8	−0.416741
$905$	−1.58681e9	−2.14081
$906$	0	0
$907$	7.32341e8	0.981503	0.490751	−	0.871300i	$-0.336722\pi$
$907$	7.32341e8	0.981503	0.490751	+	0.871300i	$0.336722\pi$
$908$	− 1.34168e9i	− 1.79222i
$909$	0	0
$910$	0	0
$911$	8.09031e8	1.07007	0.535033	−	0.844831i	$-0.320299\pi$
$911$	8.09031e8	1.07007	0.535033	+	0.844831i	$0.320299\pi$
$912$	0	0
$913$	3.20623e8i	0.421291i
$914$	1.18168e9	1.54760
$915$	0	0
$916$	− 1.39823e9i	− 1.81925i
$917$	0	0
$918$	0	0
$919$	1.23948e9	1.59696	0.798479	−	0.602022i	$-0.205638\pi$
$919$	1.23948e9	1.59696	0.798479	+	0.602022i	$0.205638\pi$
$920$	− 2.06471e8i	− 0.265152i
$921$	0	0
$922$	4.29999e8i	0.548624i
$923$	− 4.94295e7i	− 0.0628610i
$924$	0	0
$925$	1.07452e9	1.35765
$926$	2.17199e9	2.73543
$927$	0	0
$928$	1.83306e8	0.229368
$929$	1.21156e9i	1.51112i	0.655082	+	0.755558i	$0.272634\pi$
$929$	1.21156e9i	1.51112i	−0.655082	+	0.755558i	$0.727366\pi$
$930$	0	0
$931$	0	0
$932$	8.16348e8	1.00839
$933$	0	0
$934$	− 1.01825e9i	− 1.24972i
$935$	8.12038e8	0.993439
$936$	0	0
$937$	− 1.26862e8i	− 0.154211i	−0.997023	−	0.0771053i	$-0.975432\pi$
$937$	− 1.26862e8i	− 0.154211i	0.997023	−	0.0771053i	$-0.0245678\pi$
$938$	0	0
$939$	0	0
$940$	4.46116e8	0.537111
$941$	− 4.23325e8i	− 0.508048i	−0.967198	−	0.254024i	$-0.918246\pi$
$941$	− 4.23325e8i	− 0.508048i	0.967198	−	0.254024i	$-0.0817543\pi$
$942$	0	0
$943$	6.53825e8i	0.779698i
$944$	6.07492e8i	0.722145i
$945$	0	0
$946$	3.09064e8	0.365069
$947$	1.59549e9	1.87865	0.939324	−	0.343032i	$-0.111454\pi$
$947$	1.59549e9	1.87865	0.939324	+	0.343032i	$0.111454\pi$
$948$	0	0
$949$	1.54050e8	0.180245
$950$	− 1.43952e9i	− 1.67899i
$951$	0	0
$952$	0	0
$953$	3.42254e8	0.395430	0.197715	−	0.980260i	$-0.436648\pi$
$953$	3.42254e8	0.395430	0.197715	+	0.980260i	$0.436648\pi$
$954$	0	0
$955$	7.70825e8i	0.885005i
$956$	−1.14491e8	−0.131038
$957$	0	0
$958$	− 1.91069e9i	− 2.17317i
$959$	0	0
$960$	0	0
$961$	7.23268e8	0.814947
$962$	− 3.58359e8i	− 0.402525i
$963$	0	0
$964$	− 1.24031e9i	− 1.38453i
$965$	− 1.11815e9i	− 1.24428i
$966$	0	0
$967$	−1.56835e9	−1.73445	−0.867227	−	0.497914i	$-0.834100\pi$
$967$	−1.56835e9	−1.73445	−0.867227	+	0.497914i	$0.834100\pi$
$968$	7.98490e7	0.0880325
$969$	0	0
$970$	3.30432e9	3.62049
$971$	− 6.00734e8i	− 0.656182i	−0.944646	−	0.328091i	$-0.893595\pi$
$971$	− 6.00734e8i	− 0.656182i	0.944646	−	0.328091i	$-0.106405\pi$
$972$	0	0
$973$	0	0
$974$	−9.67371e8	−1.04693
$975$	0	0
$976$	− 4.58291e8i	− 0.492937i
$977$	1.17902e8	0.126426	0.0632132	−	0.998000i	$-0.479865\pi$
$977$	1.17902e8	0.126426	0.0632132	+	0.998000i	$0.479865\pi$
$978$	0	0
$979$	− 1.97382e9i	− 2.10358i
$980$	0	0
$981$	0	0
$982$	1.75852e8	0.185701
$983$	1.04078e9i	1.09572i	0.836570	+	0.547860i	$0.184558\pi$
$983$	1.04078e9i	1.09572i	−0.836570	+	0.547860i	$0.815442\pi$
$984$	0	0
$985$	1.43774e8i	0.150443i
$986$	1.44113e8i	0.150340i
$987$	0	0
$988$	−2.66717e8	−0.276554
$989$	−1.02969e8	−0.106443
$990$	0	0
$991$	−1.41876e9	−1.45777	−0.728883	−	0.684639i	$-0.759960\pi$
$991$	−1.41876e9	−1.45777	−0.728883	+	0.684639i	$0.759960\pi$
$992$	− 5.90536e8i	− 0.604938i
$993$	0	0
$994$	0	0
$995$	−1.47781e9	−1.50020
$996$	0	0
$997$	− 8.14278e8i	− 0.821651i	−0.911714	−	0.410826i	$-0.865241\pi$
$997$	− 8.14278e8i	− 0.821651i	0.911714	−	0.410826i	$-0.134759\pi$
$998$	−3.75068e8	−0.377327
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.7.d.a.244.1		2
3.2	odd	2		49.7.b.a.48.2		2
7.2	even	3		63.7.m.a.10.1		2
7.3	odd	6		63.7.m.a.19.1		2
7.6	odd	2	inner	441.7.d.a.244.2		2
21.2	odd	6		7.7.d.a.3.1	✓	2
21.5	even	6		49.7.d.b.31.1		2
21.11	odd	6		49.7.d.b.19.1		2
21.17	even	6		7.7.d.a.5.1	yes	2
21.20	even	2		49.7.b.a.48.1		2
84.23	even	6		112.7.s.a.17.1		2
84.59	odd	6		112.7.s.a.33.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.7.d.a.3.1	✓	2	21.2	odd	6
7.7.d.a.5.1	yes	2	21.17	even	6
49.7.b.a.48.1		2	21.20	even	2
49.7.b.a.48.2		2	3.2	odd	2
49.7.d.b.19.1		2	21.11	odd	6
49.7.d.b.31.1		2	21.5	even	6
63.7.m.a.10.1		2	7.2	even	3
63.7.m.a.19.1		2	7.3	odd	6
112.7.s.a.17.1		2	84.23	even	6
112.7.s.a.33.1		2	84.59	odd	6
441.7.d.a.244.1		2	1.1	even	1	trivial
441.7.d.a.244.2		2	7.6	odd	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:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	441.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+12.0000 q^{2} +80.0000 q^{4} -181.865i q^{5} +192.000 q^{8} +O(q^{10})\)	\(q+12.0000 q^{2} +80.0000 q^{4} -181.865i q^{5} +192.000 q^{8} -2182.38i q^{10} -1479.00 q^{11} +484.974i q^{13} -2816.00 q^{16} -3018.96i q^{17} +6874.51i q^{19} -14549.2i q^{20} -17748.0 q^{22} +5913.00 q^{23} -17450.0 q^{25} +5819.69i q^{26} -3978.00 q^{29} +12815.4i q^{31} -46080.0 q^{32} -36227.6i q^{34} -61577.0 q^{37} +82494.1i q^{38} -34918.1i q^{40} +110574. i q^{41} -17414.0 q^{43} -118320. q^{44} +70956.0 q^{46} +30662.5i q^{47} -209400. q^{50} +38797.9i q^{52} +60513.0 q^{53} +268979. i q^{55} -47736.0 q^{58} -215729. i q^{59} +162745. i q^{61} +153785. i q^{62} -372736. q^{64} +88200.0 q^{65} -268777. q^{67} -241517. i q^{68} -101922. q^{71} -317646. i q^{73} -738924. q^{74} +549961. i q^{76} +362231. q^{79} +512133. i q^{80} +1.32689e6i q^{82} -216783. i q^{83} -549045. q^{85} -208968. q^{86} -283968. q^{88} +1.33456e6i q^{89} +473040. q^{92} +367950. i q^{94} +1.25024e6 q^{95} +1.51409e6i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 24 q^{2} + 160 q^{4} + 384 q^{8}+O(q^{10}) \) 2 * q + 24 * q^2 + 160 * q^4 + 384 * q^8	\( 2 q + 24 q^{2} + 160 q^{4} + 384 q^{8} - 2958 q^{11} - 5632 q^{16} - 35496 q^{22} + 11826 q^{23} - 34900 q^{25} - 7956 q^{29} - 92160 q^{32} - 123154 q^{37} - 34828 q^{43} - 236640 q^{44} + 141912 q^{46} - 418800 q^{50} + 121026 q^{53} - 95472 q^{58} - 745472 q^{64} + 176400 q^{65} - 537554 q^{67} - 203844 q^{71} - 1477848 q^{74} + 724462 q^{79} - 1098090 q^{85} - 417936 q^{86} - 567936 q^{88} + 946080 q^{92} + 2500470 q^{95}+O(q^{100}) \) 2 * q + 24 * q^2 + 160 * q^4 + 384 * q^8 - 2958 * q^11 - 5632 * q^16 - 35496 * q^22 + 11826 * q^23 - 34900 * q^25 - 7956 * q^29 - 92160 * q^32 - 123154 * q^37 - 34828 * q^43 - 236640 * q^44 + 141912 * q^46 - 418800 * q^50 + 121026 * q^53 - 95472 * q^58 - 745472 * q^64 + 176400 * q^65 - 537554 * q^67 - 203844 * q^71 - 1477848 * q^74 + 724462 * q^79 - 1098090 * q^85 - 417936 * q^86 - 567936 * q^88 + 946080 * q^92 + 2500470 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more