LMFDB - Embedded newform 700.6.e.g.449.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [700,6,Mod(449,700)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(700, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("700.449");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 700 = 2^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	700.e (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:	$112.268673869$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 998x^{4} + 249001x^{2} + 44100 $ x^6 + 998x^4 + 249001x^2 + 44100
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 140)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.5
Root			$22.1248i$ of defining polynomial
Character	$\chi$	$=$	700.449
Dual form			700.6.e.g.449.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+20.1248i q^{3} +49.0000i q^{7} -162.009 q^{9} +O(q^{10})$	$q+20.1248i q^{3} +49.0000i q^{7} -162.009 q^{9} -427.006 q^{11} +646.889i q^{13} -1029.14i q^{17} -2253.95 q^{19} -986.117 q^{21} -2595.11i q^{23} +1629.93i q^{27} -869.860 q^{29} +4407.19 q^{31} -8593.44i q^{33} +2555.64i q^{37} -13018.5 q^{39} +6222.90 q^{41} -7757.57i q^{43} +1887.76i q^{47} -2401.00 q^{49} +20711.2 q^{51} +11718.0i q^{53} -45360.4i q^{57} -33385.2 q^{59} -1212.23 q^{61} -7938.44i q^{63} -50421.1i q^{67} +52226.1 q^{69} +57251.5 q^{71} -84023.5i q^{73} -20923.3i q^{77} -98400.4 q^{79} -72170.3 q^{81} +32157.5i q^{83} -17505.8i q^{87} -54323.7 q^{89} -31697.6 q^{91} +88694.1i q^{93} -21700.3i q^{97} +69178.9 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 562 q^{9}+O(q^{10}) $ 6 * q - 562 * q^9	$ 6 q - 562 q^{9} - 28 q^{11} - 4656 q^{19} + 588 q^{21} - 8184 q^{29} + 11776 q^{31} - 27404 q^{39} + 22900 q^{41} - 14406 q^{49} + 63524 q^{51} - 24776 q^{59} + 54364 q^{61} + 124280 q^{69} + 163984 q^{71} + 31852 q^{79} - 64202 q^{81} + 191420 q^{89} - 392 q^{91} + 228736 q^{99}+O(q^{100}) $ 6 * q - 562 * q^9 - 28 * q^11 - 4656 * q^19 + 588 * q^21 - 8184 * q^29 + 11776 * q^31 - 27404 * q^39 + 22900 * q^41 - 14406 * q^49 + 63524 * q^51 - 24776 * q^59 + 54364 * q^61 + 124280 * q^69 + 163984 * q^71 + 31852 * q^79 - 64202 * q^81 + 191420 * q^89 - 392 * q^91 + 228736 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/700\mathbb{Z}\right)^\times$.

$n$	$101$	$351$	$477$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	20.1248i	1.29101i	0.763757	+	0.645504i	$0.223353\pi$
$3$	20.1248i	1.29101i	−0.763757	+	0.645504i	$0.776647\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	49.0000i	0.377964i
$7$	49.0000i	0.377964i
$8$	0	0
$9$	−162.009	−0.666704
$10$	0	0
$11$	−427.006	−1.06403	−0.532014	−	0.846736i	$-0.678564\pi$
$11$	−427.006	−1.06403	−0.532014	+	0.846736i	$0.678564\pi$
$12$	0	0
$13$	646.889i	1.06163i	0.847489	+	0.530813i	$0.178113\pi$
$13$	646.889i	1.06163i	−0.847489	+	0.530813i	$0.821887\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 1029.14i	− 0.863676i	−0.901951	−	0.431838i	$-0.857865\pi$
$17$	− 1029.14i	− 0.863676i	0.901951	−	0.431838i	$-0.142135\pi$
$18$	0	0
$19$	−2253.95	−1.43239	−0.716193	−	0.697902i	$-0.754117\pi$
$19$	−2253.95	−1.43239	−0.716193	+	0.697902i	$0.754117\pi$
$20$	0	0
$21$	−986.117	−0.487955
$22$	0	0
$23$	− 2595.11i	− 1.02291i	−0.859311	−	0.511453i	$-0.829108\pi$
$23$	− 2595.11i	− 1.02291i	0.859311	−	0.511453i	$-0.170892\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1629.93i	0.430288i
$28$	0	0
$29$	−869.860	−0.192068	−0.0960339	−	0.995378i	$-0.530616\pi$
$29$	−869.860	−0.192068	−0.0960339	+	0.995378i	$0.530616\pi$
$30$	0	0
$31$	4407.19	0.823679	0.411839	−	0.911256i	$-0.364886\pi$
$31$	4407.19	0.823679	0.411839	+	0.911256i	$0.364886\pi$
$32$	0	0
$33$	− 8593.44i	− 1.37367i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2555.64i	0.306899i	0.988156	+	0.153450i	$0.0490383\pi$
$37$	2555.64i	0.306899i	−0.988156	+	0.153450i	$0.950962\pi$
$38$	0	0
$39$	−13018.5	−1.37057
$40$	0	0
$41$	6222.90	0.578141	0.289070	−	0.957308i	$-0.406654\pi$
$41$	6222.90	0.578141	0.289070	+	0.957308i	$0.406654\pi$
$42$	0	0
$43$	− 7757.57i	− 0.639815i	−0.947449	−	0.319907i	$-0.896348\pi$
$43$	− 7757.57i	− 0.639815i	0.947449	−	0.319907i	$-0.103652\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1887.76i	0.124653i	0.998056	+	0.0623265i	$0.0198520\pi$
$47$	1887.76i	0.124653i	−0.998056	+	0.0623265i	$0.980148\pi$
$48$	0	0
$49$	−2401.00	−0.142857
$50$	0	0
$51$	20711.2	1.11501
$52$	0	0
$53$	11718.0i	0.573010i	0.958079	+	0.286505i	$0.0924935\pi$
$53$	11718.0i	0.573010i	−0.958079	+	0.286505i	$0.907507\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 45360.4i	− 1.84922i
$58$	0	0
$59$	−33385.2	−1.24860	−0.624300	−	0.781184i	$-0.714616\pi$
$59$	−33385.2	−1.24860	−0.624300	+	0.781184i	$0.714616\pi$
$60$	0	0
$61$	−1212.23	−0.0417119	−0.0208559	−	0.999782i	$-0.506639\pi$
$61$	−1212.23	−0.0417119	−0.0208559	+	0.999782i	$0.506639\pi$
$62$	0	0
$63$	− 7938.44i	− 0.251990i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 50421.1i	− 1.37222i	−0.727496	−	0.686112i	$-0.759316\pi$
$67$	− 50421.1i	− 1.37222i	0.727496	−	0.686112i	$-0.240684\pi$
$68$	0	0
$69$	52226.1	1.32058
$70$	0	0
$71$	57251.5	1.34785	0.673925	−	0.738800i	$-0.264607\pi$
$71$	57251.5	1.34785	0.673925	+	0.738800i	$0.264607\pi$
$72$	0	0
$73$	− 84023.5i	− 1.84541i	−0.385502	−	0.922707i	$-0.625972\pi$
$73$	− 84023.5i	− 1.84541i	0.385502	−	0.922707i	$-0.374028\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 20923.3i	− 0.402164i
$78$	0	0
$79$	−98400.4	−1.77390	−0.886950	−	0.461866i	$-0.847180\pi$
$79$	−98400.4	−1.77390	−0.886950	+	0.461866i	$0.847180\pi$
$80$	0	0
$81$	−72170.3	−1.22221
$82$	0	0
$83$	32157.5i	0.512375i	0.966627	+	0.256187i	$0.0824664\pi$
$83$	32157.5i	0.512375i	−0.966627	+	0.256187i	$0.917534\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 17505.8i	− 0.247961i
$88$	0	0
$89$	−54323.7	−0.726966	−0.363483	−	0.931601i	$-0.618413\pi$
$89$	−54323.7	−0.726966	−0.363483	+	0.931601i	$0.618413\pi$
$90$	0	0
$91$	−31697.6	−0.401257
$92$	0	0
$93$	88694.1i	1.06338i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 21700.3i	− 0.234173i	−0.993122	−	0.117086i	$-0.962645\pi$
$97$	− 21700.3i	− 0.234173i	0.993122	−	0.117086i	$-0.0373555\pi$
$98$	0	0
$99$	69178.9	0.709391
$100$	0	0
$101$	127360.	1.24231	0.621154	−	0.783689i	$-0.286664\pi$
$101$	127360.	1.24231	0.621154	+	0.783689i	$0.286664\pi$
$102$	0	0
$103$	− 44694.5i	− 0.415108i	−0.978224	−	0.207554i	$-0.933450\pi$
$103$	− 44694.5i	− 0.415108i	0.978224	−	0.207554i	$-0.0665503\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 125360.i	− 1.05852i	−0.848461	−	0.529258i	$-0.822470\pi$
$107$	− 125360.i	− 1.05852i	0.848461	−	0.529258i	$-0.177530\pi$
$108$	0	0
$109$	22940.1	0.184939	0.0924696	−	0.995716i	$-0.470524\pi$
$109$	22940.1	0.184939	0.0924696	+	0.995716i	$0.470524\pi$
$110$	0	0
$111$	−51431.9	−0.396210
$112$	0	0
$113$	191652.i	1.41194i	0.708240	+	0.705971i	$0.249489\pi$
$113$	191652.i	1.41194i	−0.708240	+	0.705971i	$0.750511\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 104802.i	− 0.707790i
$118$	0	0
$119$	50427.7	0.326439
$120$	0	0
$121$	21283.5	0.132154
$122$	0	0
$123$	125235.i	0.746385i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 280121.i	− 1.54112i	−0.637368	−	0.770559i	$-0.719977\pi$
$127$	− 280121.i	− 1.54112i	0.637368	−	0.770559i	$-0.280023\pi$
$128$	0	0
$129$	156120.	0.826007
$130$	0	0
$131$	245780.	1.25132	0.625660	−	0.780096i	$-0.284830\pi$
$131$	245780.	1.25132	0.625660	+	0.780096i	$0.284830\pi$
$132$	0	0
$133$	− 110444.i	− 0.541391i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 13417.8i	− 0.0610773i	−0.999534	−	0.0305386i	$-0.990278\pi$
$137$	− 13417.8i	− 0.0610773i	0.999534	−	0.0305386i	$-0.00972226\pi$
$138$	0	0
$139$	363369.	1.59519	0.797593	−	0.603196i	$-0.206106\pi$
$139$	363369.	1.59519	0.797593	+	0.603196i	$0.206106\pi$
$140$	0	0
$141$	−37990.9	−0.160928
$142$	0	0
$143$	− 276226.i	− 1.12960i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 48319.7i	− 0.184430i
$148$	0	0
$149$	260049.	0.959597	0.479798	−	0.877379i	$-0.340710\pi$
$149$	260049.	0.959597	0.479798	+	0.877379i	$0.340710\pi$
$150$	0	0
$151$	219894.	0.784821	0.392410	−	0.919790i	$-0.371641\pi$
$151$	219894.	0.784821	0.392410	+	0.919790i	$0.371641\pi$
$152$	0	0
$153$	166729.i	0.575816i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	430845.i	1.39499i	0.716589	+	0.697496i	$0.245702\pi$
$157$	430845.i	1.39499i	−0.716589	+	0.697496i	$0.754298\pi$
$158$	0	0
$159$	−235822.	−0.739761
$160$	0	0
$161$	127160.	0.386622
$162$	0	0
$163$	− 462115.i	− 1.36233i	−0.732132	−	0.681163i	$-0.761474\pi$
$163$	− 462115.i	− 1.36233i	0.732132	−	0.681163i	$-0.238526\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	493069.i	1.36809i	0.729438	+	0.684047i	$0.239782\pi$
$167$	493069.i	1.36809i	−0.729438	+	0.684047i	$0.760218\pi$
$168$	0	0
$169$	−47172.8	−0.127050
$170$	0	0
$171$	365160.	0.954978
$172$	0	0
$173$	614223.i	1.56031i	0.625586	+	0.780155i	$0.284860\pi$
$173$	614223.i	1.56031i	−0.625586	+	0.780155i	$0.715140\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 671871.i	− 1.61195i
$178$	0	0
$179$	92331.1	0.215385	0.107692	−	0.994184i	$-0.465654\pi$
$179$	92331.1	0.215385	0.107692	+	0.994184i	$0.465654\pi$
$180$	0	0
$181$	694842.	1.57649	0.788243	−	0.615365i	$-0.210991\pi$
$181$	694842.	1.57649	0.788243	+	0.615365i	$0.210991\pi$
$182$	0	0
$183$	− 24395.9i	− 0.0538504i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	439448.i	0.918974i
$188$	0	0
$189$	−79866.5	−0.162634
$190$	0	0
$191$	−459214.	−0.910818	−0.455409	−	0.890282i	$-0.650507\pi$
$191$	−459214.	−0.910818	−0.455409	+	0.890282i	$0.650507\pi$
$192$	0	0
$193$	− 182792.i	− 0.353236i	−0.984279	−	0.176618i	$-0.943484\pi$
$193$	− 182792.i	− 0.353236i	0.984279	−	0.176618i	$-0.0565157\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	345493.i	0.634270i	0.948380	+	0.317135i	$0.102721\pi$
$197$	345493.i	0.634270i	−0.948380	+	0.317135i	$0.897279\pi$
$198$	0	0
$199$	−455123.	−0.814697	−0.407348	−	0.913273i	$-0.633547\pi$
$199$	−455123.	−0.814697	−0.407348	+	0.913273i	$0.633547\pi$
$200$	0	0
$201$	1.01472e6	1.77155
$202$	0	0
$203$	− 42623.2i	− 0.0725948i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	420431.i	0.681975i
$208$	0	0
$209$	962451.	1.52410
$210$	0	0
$211$	364003.	0.562858	0.281429	−	0.959582i	$-0.409192\pi$
$211$	364003.	0.562858	0.281429	+	0.959582i	$0.409192\pi$
$212$	0	0
$213$	1.15218e6i	1.74009i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	215953.i	0.311321i
$218$	0	0
$219$	1.69096e6	2.38245
$220$	0	0
$221$	665737.	0.916901
$222$	0	0
$223$	− 275681.i	− 0.371232i	−0.982622	−	0.185616i	$-0.940572\pi$
$223$	− 275681.i	− 0.371232i	0.982622	−	0.185616i	$-0.0594281\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 1.14571e6i	− 1.47574i	−0.674943	−	0.737869i	$-0.735832\pi$
$227$	− 1.14571e6i	− 1.47574i	0.674943	−	0.737869i	$-0.264168\pi$
$228$	0	0
$229$	908996.	1.14544	0.572721	−	0.819750i	$-0.305888\pi$
$229$	908996.	1.14544	0.572721	+	0.819750i	$0.305888\pi$
$230$	0	0
$231$	421078.	0.519198
$232$	0	0
$233$	788658.i	0.951698i	0.879527	+	0.475849i	$0.157859\pi$
$233$	788658.i	0.951698i	−0.879527	+	0.475849i	$0.842141\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 1.98029e6i	− 2.29012i
$238$	0	0
$239$	−1.50376e6	−1.70288	−0.851441	−	0.524451i	$-0.824271\pi$
$239$	−1.50376e6	−1.70288	−0.851441	+	0.524451i	$0.824271\pi$
$240$	0	0
$241$	114870.	0.127399	0.0636993	−	0.997969i	$-0.479710\pi$
$241$	114870.	0.127399	0.0636993	+	0.997969i	$0.479710\pi$
$242$	0	0
$243$	− 1.05634e6i	− 1.14760i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 1.45806e6i	− 1.52066i
$248$	0	0
$249$	−647165.	−0.661480
$250$	0	0
$251$	−893691.	−0.895371	−0.447686	−	0.894191i	$-0.647752\pi$
$251$	−893691.	−0.895371	−0.447686	+	0.894191i	$0.647752\pi$
$252$	0	0
$253$	1.10813e6i	1.08840i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 839193.i	− 0.792555i	−0.918131	−	0.396277i	$-0.870302\pi$
$257$	− 839193.i	− 0.792555i	0.918131	−	0.396277i	$-0.129698\pi$
$258$	0	0
$259$	−125227.	−0.115997
$260$	0	0
$261$	140925.	0.128052
$262$	0	0
$263$	− 1.19862e6i	− 1.06855i	−0.845312	−	0.534273i	$-0.820585\pi$
$263$	− 1.19862e6i	− 1.06855i	0.845312	−	0.534273i	$-0.179415\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 1.09326e6i	− 0.938520i
$268$	0	0
$269$	−333286.	−0.280825	−0.140413	−	0.990093i	$-0.544843\pi$
$269$	−333286.	−0.280825	−0.140413	+	0.990093i	$0.544843\pi$
$270$	0	0
$271$	1.99242e6	1.64800	0.824000	−	0.566589i	$-0.191737\pi$
$271$	1.99242e6	1.64800	0.824000	+	0.566589i	$0.191737\pi$
$272$	0	0
$273$	− 637909.i	− 0.518026i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 201374.i	− 0.157690i	−0.996887	−	0.0788449i	$-0.974877\pi$
$277$	− 201374.i	− 0.157690i	0.996887	−	0.0788449i	$-0.0251232\pi$
$278$	0	0
$279$	−714005.	−0.549150
$280$	0	0
$281$	−456782.	−0.345099	−0.172549	−	0.985001i	$-0.555200\pi$
$281$	−456782.	−0.345099	−0.172549	+	0.985001i	$0.555200\pi$
$282$	0	0
$283$	− 590693.i	− 0.438425i	−0.975677	−	0.219213i	$-0.929651\pi$
$283$	− 590693.i	− 0.438425i	0.975677	−	0.219213i	$-0.0703488\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	304922.i	0.218517i
$288$	0	0
$289$	360735.	0.254064
$290$	0	0
$291$	436715.	0.302319
$292$	0	0
$293$	416363.i	0.283337i	0.989914	+	0.141668i	$0.0452466\pi$
$293$	416363.i	0.283337i	−0.989914	+	0.141668i	$0.954753\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 695990.i	− 0.457838i
$298$	0	0
$299$	1.67875e6	1.08594
$300$	0	0
$301$	380121.	0.241827
$302$	0	0
$303$	2.56310e6i	1.60383i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 1.25575e6i	− 0.760427i	−0.924899	−	0.380213i	$-0.875851\pi$
$307$	− 1.25575e6i	− 0.760427i	0.924899	−	0.380213i	$-0.124149\pi$
$308$	0	0
$309$	899470.	0.535909
$310$	0	0
$311$	−1.16270e6	−0.681659	−0.340830	−	0.940125i	$-0.610708\pi$
$311$	−1.16270e6	−0.681659	−0.340830	+	0.940125i	$0.610708\pi$
$312$	0	0
$313$	− 3.45256e6i	− 1.99196i	−0.0895747	−	0.995980i	$-0.528551\pi$
$313$	− 3.45256e6i	− 1.99196i	0.0895747	−	0.995980i	$-0.471449\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 1.06034e6i	− 0.592651i	−0.955087	−	0.296325i	$-0.904239\pi$
$317$	− 1.06034e6i	− 0.592651i	0.955087	−	0.296325i	$-0.0957613\pi$
$318$	0	0
$319$	371436.	0.204365
$320$	0	0
$321$	2.52284e6	1.36655
$322$	0	0
$323$	2.31962e6i	1.23712i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	461666.i	0.238758i
$328$	0	0
$329$	−92500.3	−0.0471144
$330$	0	0
$331$	−1.80273e6	−0.904402	−0.452201	−	0.891916i	$-0.649361\pi$
$331$	−1.80273e6	−0.904402	−0.452201	+	0.891916i	$0.649361\pi$
$332$	0	0
$333$	− 414037.i	− 0.204611i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	390541.i	0.187323i	0.995604	+	0.0936616i	$0.0298572\pi$
$337$	390541.i	0.187323i	−0.995604	+	0.0936616i	$0.970143\pi$
$338$	0	0
$339$	−3.85696e6	−1.82283
$340$	0	0
$341$	−1.88190e6	−0.876417
$342$	0	0
$343$	− 117649.i	− 0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 3.38976e6i	− 1.51128i	−0.654988	−	0.755640i	$-0.727326\pi$
$347$	− 3.38976e6i	− 1.51128i	0.654988	−	0.755640i	$-0.272674\pi$
$348$	0	0
$349$	−312893.	−0.137510	−0.0687548	−	0.997634i	$-0.521903\pi$
$349$	−312893.	−0.137510	−0.0687548	+	0.997634i	$0.521903\pi$
$350$	0	0
$351$	−1.05438e6	−0.456805
$352$	0	0
$353$	− 1.24260e6i	− 0.530756i	−0.964144	−	0.265378i	$-0.914503\pi$
$353$	− 1.24260e6i	− 0.530756i	0.964144	−	0.265378i	$-0.0854968\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	1.01485e6i	0.421435i
$358$	0	0
$359$	2.84568e6	1.16533	0.582666	−	0.812712i	$-0.302010\pi$
$359$	2.84568e6	1.16533	0.582666	+	0.812712i	$0.302010\pi$
$360$	0	0
$361$	2.60419e6	1.05173
$362$	0	0
$363$	428327.i	0.170612i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	384128.i	0.148871i	0.997226	+	0.0744357i	$0.0237155\pi$
$367$	384128.i	0.148871i	−0.997226	+	0.0744357i	$0.976284\pi$
$368$	0	0
$369$	−1.00817e6	−0.385449
$370$	0	0
$371$	−574180.	−0.216577
$372$	0	0
$373$	76006.4i	0.0282864i	0.999900	+	0.0141432i	$0.00450207\pi$
$373$	76006.4i	0.0282864i	−0.999900	+	0.0141432i	$0.995498\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 562703.i	− 0.203904i
$378$	0	0
$379$	−5.54214e6	−1.98189	−0.990944	−	0.134273i	$-0.957130\pi$
$379$	−5.54214e6	−1.98189	−0.990944	+	0.134273i	$0.957130\pi$
$380$	0	0
$381$	5.63739e6	1.98960
$382$	0	0
$383$	− 4.85688e6i	− 1.69185i	−0.533305	−	0.845923i	$-0.679050\pi$
$383$	− 4.85688e6i	− 1.69185i	0.533305	−	0.845923i	$-0.320950\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1.25680e6i	0.426567i
$388$	0	0
$389$	−3.98689e6	−1.33586	−0.667928	−	0.744226i	$-0.732819\pi$
$389$	−3.98689e6	−1.33586	−0.667928	+	0.744226i	$0.732819\pi$
$390$	0	0
$391$	−2.67072e6	−0.883458
$392$	0	0
$393$	4.94629e6i	1.61547i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	3.96757e6i	1.26342i	0.775205	+	0.631710i	$0.217647\pi$
$397$	3.96757e6i	1.26342i	−0.775205	+	0.631710i	$0.782353\pi$
$398$	0	0
$399$	2.22266e6	0.698941
$400$	0	0
$401$	4.93573e6	1.53282	0.766409	−	0.642353i	$-0.222042\pi$
$401$	4.93573e6	1.53282	0.766409	+	0.642353i	$0.222042\pi$
$402$	0	0
$403$	2.85097e6i	0.874439i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 1.09128e6i	− 0.326549i
$408$	0	0
$409$	−6.35318e6	−1.87794	−0.938972	−	0.343994i	$-0.888220\pi$
$409$	−6.35318e6	−1.87794	−0.938972	+	0.343994i	$0.888220\pi$
$410$	0	0
$411$	270031.	0.0788513
$412$	0	0
$413$	− 1.63587e6i	− 0.471927i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	7.31275e6i	2.05940i
$418$	0	0
$419$	5.86656e6	1.63248	0.816241	−	0.577711i	$-0.196054\pi$
$419$	5.86656e6	1.63248	0.816241	+	0.577711i	$0.196054\pi$
$420$	0	0
$421$	−4.13207e6	−1.13622	−0.568109	−	0.822953i	$-0.692325\pi$
$421$	−4.13207e6	−1.13622	−0.568109	+	0.822953i	$0.692325\pi$
$422$	0	0
$423$	− 305834.i	− 0.0831066i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 59399.2i	− 0.0157656i
$428$	0	0
$429$	5.55900e6	1.45832
$430$	0	0
$431$	−3.28044e6	−0.850627	−0.425314	−	0.905046i	$-0.639836\pi$
$431$	−3.28044e6	−0.850627	−0.425314	+	0.905046i	$0.639836\pi$
$432$	0	0
$433$	− 7.35755e6i	− 1.88588i	−0.332968	−	0.942938i	$-0.608050\pi$
$433$	− 7.35755e6i	− 1.88588i	0.332968	−	0.942938i	$-0.391950\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	5.84924e6i	1.46520i
$438$	0	0
$439$	296949.	0.0735395	0.0367698	−	0.999324i	$-0.488293\pi$
$439$	296949.	0.0735395	0.0367698	+	0.999324i	$0.488293\pi$
$440$	0	0
$441$	388984.	0.0952434
$442$	0	0
$443$	− 3.07626e6i	− 0.744756i	−0.928081	−	0.372378i	$-0.878542\pi$
$443$	− 3.07626e6i	− 0.744756i	0.928081	−	0.372378i	$-0.121458\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	5.23343e6i	1.23885i
$448$	0	0
$449$	2.95674e6	0.692145	0.346073	−	0.938208i	$-0.387515\pi$
$449$	2.95674e6	0.692145	0.346073	+	0.938208i	$0.387515\pi$
$450$	0	0
$451$	−2.65722e6	−0.615157
$452$	0	0
$453$	4.42532e6i	1.01321i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	5.68155e6i	1.27255i	0.771461	+	0.636277i	$0.219526\pi$
$457$	5.68155e6i	1.27255i	−0.771461	+	0.636277i	$0.780474\pi$
$458$	0	0
$459$	1.67742e6	0.371629
$460$	0	0
$461$	−5.75693e6	−1.26165	−0.630825	−	0.775925i	$-0.717283\pi$
$461$	−5.75693e6	−1.26165	−0.630825	+	0.775925i	$0.717283\pi$
$462$	0	0
$463$	2.82533e6i	0.612514i	0.951949	+	0.306257i	$0.0990768\pi$
$463$	2.82533e6i	0.612514i	−0.951949	+	0.306257i	$0.900923\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	1.29610e6i	0.275008i	0.990501	+	0.137504i	$0.0439080\pi$
$467$	1.29610e6i	0.275008i	−0.990501	+	0.137504i	$0.956092\pi$
$468$	0	0
$469$	2.47063e6	0.518652
$470$	0	0
$471$	−8.67068e6	−1.80095
$472$	0	0
$473$	3.31253e6i	0.680781i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 1.89841e6i	− 0.382028i
$478$	0	0
$479$	−854121.	−0.170091	−0.0850453	−	0.996377i	$-0.527104\pi$
$479$	−854121.	−0.170091	−0.0850453	+	0.996377i	$0.527104\pi$
$480$	0	0
$481$	−1.65322e6	−0.325812
$482$	0	0
$483$	2.55908e6i	0.499132i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	2.05453e6i	0.392545i	0.980549	+	0.196273i	$0.0628838\pi$
$487$	2.05453e6i	0.392545i	−0.980549	+	0.196273i	$0.937116\pi$
$488$	0	0
$489$	9.29999e6	1.75877
$490$	0	0
$491$	−7.21316e6	−1.35027	−0.675137	−	0.737692i	$-0.735916\pi$
$491$	−7.21316e6	−1.35027	−0.675137	+	0.737692i	$0.735916\pi$
$492$	0	0
$493$	895205.i	0.165884i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2.80533e6i	0.509439i
$498$	0	0
$499$	−4.85347e6	−0.872571	−0.436286	−	0.899808i	$-0.643706\pi$
$499$	−4.85347e6	−0.872571	−0.436286	+	0.899808i	$0.643706\pi$
$500$	0	0
$501$	−9.92292e6	−1.76622
$502$	0	0
$503$	2.60704e6i	0.459439i	0.973257	+	0.229720i	$0.0737809\pi$
$503$	2.60704e6i	0.459439i	−0.973257	+	0.229720i	$0.926219\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 949346.i	− 0.164023i
$508$	0	0
$509$	4.56269e6	0.780596	0.390298	−	0.920689i	$-0.372372\pi$
$509$	4.56269e6	0.780596	0.390298	+	0.920689i	$0.372372\pi$
$510$	0	0
$511$	4.11715e6	0.697501
$512$	0	0
$513$	− 3.67378e6i	− 0.616339i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 806086.i	− 0.132634i
$518$	0	0
$519$	−1.23611e7	−2.01437
$520$	0	0
$521$	−3.14831e6	−0.508140	−0.254070	−	0.967186i	$-0.581769\pi$
$521$	−3.14831e6	−0.508140	−0.254070	+	0.967186i	$0.581769\pi$
$522$	0	0
$523$	− 7.28996e6i	− 1.16539i	−0.812692	−	0.582694i	$-0.801999\pi$
$523$	− 7.28996e6i	− 1.16539i	0.812692	−	0.582694i	$-0.198001\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 4.53560e6i	− 0.711391i
$528$	0	0
$529$	−298229.	−0.0463351
$530$	0	0
$531$	5.40870e6	0.832447
$532$	0	0
$533$	4.02553e6i	0.613769i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	1.85815e6i	0.278064i
$538$	0	0
$539$	1.02524e6	0.152004
$540$	0	0
$541$	−4131.68	−0.000606923 0	−0.000303462	−	1.00000i	$-0.500097\pi$
$541$	−4131.68	−0.000606923 0	−0.000303462		1.00000i	$0.500097\pi$
$542$	0	0
$543$	1.39836e7i	2.03526i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	1.13419e7i	1.62075i	0.585910	+	0.810376i	$0.300737\pi$
$547$	1.13419e7i	1.62075i	−0.585910	+	0.810376i	$0.699263\pi$
$548$	0	0
$549$	196392.	0.0278095
$550$	0	0
$551$	1.96062e6	0.275115
$552$	0	0
$553$	− 4.82162e6i	− 0.670471i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 3.06648e6i	− 0.418796i	−0.977831	−	0.209398i	$-0.932850\pi$
$557$	− 3.06648e6i	− 0.418796i	0.977831	−	0.209398i	$-0.0671504\pi$
$558$	0	0
$559$	5.01829e6	0.679244
$560$	0	0
$561$	−8.84382e6	−1.18640
$562$	0	0
$563$	8.73658e6i	1.16164i	0.814033	+	0.580819i	$0.197268\pi$
$563$	8.73658e6i	1.16164i	−0.814033	+	0.580819i	$0.802732\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 3.53634e6i	− 0.461952i
$568$	0	0
$569$	−8.05550e6	−1.04307	−0.521533	−	0.853231i	$-0.674640\pi$
$569$	−8.05550e6	−1.04307	−0.521533	+	0.853231i	$0.674640\pi$
$570$	0	0
$571$	2.25192e6	0.289043	0.144522	−	0.989502i	$-0.453836\pi$
$571$	2.25192e6	0.289043	0.144522	+	0.989502i	$0.453836\pi$
$572$	0	0
$573$	− 9.24161e6i	− 1.17587i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 1.56246e6i	− 0.195375i	−0.995217	−	0.0976874i	$-0.968855\pi$
$577$	− 1.56246e6i	− 0.195375i	0.995217	−	0.0976874i	$-0.0311445\pi$
$578$	0	0
$579$	3.67867e6	0.456031
$580$	0	0
$581$	−1.57572e6	−0.193659
$582$	0	0
$583$	− 5.00364e6i	− 0.609698i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.31305e7i	1.57284i	0.617690	+	0.786421i	$0.288069\pi$
$587$	1.31305e7i	1.57284i	−0.617690	+	0.786421i	$0.711931\pi$
$588$	0	0
$589$	−9.93360e6	−1.17983
$590$	0	0
$591$	−6.95299e6	−0.818848
$592$	0	0
$593$	− 8.96601e6i	− 1.04704i	−0.852014	−	0.523519i	$-0.824619\pi$
$593$	− 8.96601e6i	− 1.04704i	0.852014	−	0.523519i	$-0.175381\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 9.15927e6i	− 1.05178i
$598$	0	0
$599$	−1.29511e7	−1.47483	−0.737414	−	0.675441i	$-0.763953\pi$
$599$	−1.29511e7	−1.47483	−0.737414	+	0.675441i	$0.763953\pi$
$600$	0	0
$601$	1.96489e6	0.221897	0.110949	−	0.993826i	$-0.464611\pi$
$601$	1.96489e6	0.221897	0.110949	+	0.993826i	$0.464611\pi$
$602$	0	0
$603$	8.16867e6i	0.914867i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 1.13138e7i	− 1.24634i	−0.782085	−	0.623172i	$-0.785844\pi$
$607$	− 1.13138e7i	− 1.24634i	0.782085	−	0.623172i	$-0.214156\pi$
$608$	0	0
$609$	857784.	0.0937205
$610$	0	0
$611$	−1.22117e6	−0.132335
$612$	0	0
$613$	− 7.29872e6i	− 0.784505i	−0.919858	−	0.392252i	$-0.871696\pi$
$613$	− 7.29872e6i	− 0.784505i	0.919858	−	0.392252i	$-0.128304\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	19580.9i	0.00207071i	0.999999	+	0.00103536i	$0.000329564\pi$
$617$	19580.9i	0.00207071i	−0.999999	+	0.00103536i	$0.999670\pi$
$618$	0	0
$619$	−6.49314e6	−0.681127	−0.340563	−	0.940222i	$-0.610618\pi$
$619$	−6.49314e6	−0.681127	−0.340563	+	0.940222i	$0.610618\pi$
$620$	0	0
$621$	4.22984e6	0.440144
$622$	0	0
$623$	− 2.66186e6i	− 0.274767i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	1.93692e7i	1.96762i
$628$	0	0
$629$	2.63011e6	0.265061
$630$	0	0
$631$	4.09998e6	0.409929	0.204964	−	0.978769i	$-0.434292\pi$
$631$	4.09998e6	0.409929	0.204964	+	0.978769i	$0.434292\pi$
$632$	0	0
$633$	7.32550e6i	0.726655i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 1.55318e6i	− 0.151661i
$638$	0	0
$639$	−9.27527e6	−0.898616
$640$	0	0
$641$	−8.51977e6	−0.818998	−0.409499	−	0.912311i	$-0.634296\pi$
$641$	−8.51977e6	−0.818998	−0.409499	+	0.912311i	$0.634296\pi$
$642$	0	0
$643$	4.17027e6i	0.397774i	0.980022	+	0.198887i	$0.0637327\pi$
$643$	4.17027e6i	0.397774i	−0.980022	+	0.198887i	$0.936267\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 2.00875e7i	− 1.88654i	−0.332029	−	0.943269i	$-0.607733\pi$
$647$	− 2.00875e7i	− 1.88654i	0.332029	−	0.943269i	$-0.392267\pi$
$648$	0	0
$649$	1.42557e7	1.32855
$650$	0	0
$651$	−4.34601e6	−0.401919
$652$	0	0
$653$	− 1.88656e7i	− 1.73136i	−0.500595	−	0.865682i	$-0.666885\pi$
$653$	− 1.88656e7i	− 1.73136i	0.500595	−	0.865682i	$-0.333115\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	1.36126e7i	1.23034i
$658$	0	0
$659$	−1.99030e7	−1.78527	−0.892636	−	0.450777i	$-0.851147\pi$
$659$	−1.99030e7	−1.78527	−0.892636	+	0.450777i	$0.851147\pi$
$660$	0	0
$661$	−2.07323e7	−1.84563	−0.922815	−	0.385244i	$-0.874117\pi$
$661$	−2.07323e7	−1.84563	−0.922815	+	0.385244i	$0.874117\pi$
$662$	0	0
$663$	1.33979e7i	1.18373i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	2.25738e6i	0.196467i
$668$	0	0
$669$	5.54805e6	0.479264
$670$	0	0
$671$	517629.	0.0443826
$672$	0	0
$673$	− 1.43855e7i	− 1.22430i	−0.790741	−	0.612151i	$-0.790304\pi$
$673$	− 1.43855e7i	− 1.22430i	0.790741	−	0.612151i	$-0.209696\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 7.15539e6i	− 0.600014i	−0.953937	−	0.300007i	$-0.903011\pi$
$677$	− 7.15539e6i	− 0.600014i	0.953937	−	0.300007i	$-0.0969890\pi$
$678$	0	0
$679$	1.06332e6	0.0885091
$680$	0	0
$681$	2.30572e7	1.90519
$682$	0	0
$683$	− 9.42223e6i	− 0.772862i	−0.922318	−	0.386431i	$-0.873708\pi$
$683$	− 9.42223e6i	− 0.772862i	0.922318	−	0.386431i	$-0.126292\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	1.82934e7i	1.47878i
$688$	0	0
$689$	−7.58022e6	−0.608322
$690$	0	0
$691$	−7.56428e6	−0.602660	−0.301330	−	0.953520i	$-0.597431\pi$
$691$	−7.56428e6	−0.602660	−0.301330	+	0.953520i	$0.597431\pi$
$692$	0	0
$693$	3.38977e6i	0.268125i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 6.40422e6i	− 0.499326i
$698$	0	0
$699$	−1.58716e7	−1.22865
$700$	0	0
$701$	7.75589e6	0.596124	0.298062	−	0.954547i	$-0.403660\pi$
$701$	7.75589e6	0.596124	0.298062	+	0.954547i	$0.403660\pi$
$702$	0	0
$703$	− 5.76029e6i	− 0.439599i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	6.24064e6i	0.469548i
$708$	0	0
$709$	1.08957e7	0.814030	0.407015	−	0.913422i	$-0.366570\pi$
$709$	1.08957e7	0.814030	0.407015	+	0.913422i	$0.366570\pi$
$710$	0	0
$711$	1.59418e7	1.18267
$712$	0	0
$713$	− 1.14371e7i	− 0.842546i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 3.02630e7i	− 2.19843i
$718$	0	0
$719$	−1.83953e7	−1.32704	−0.663519	−	0.748159i	$-0.730938\pi$
$719$	−1.83953e7	−1.32704	−0.663519	+	0.748159i	$0.730938\pi$
$720$	0	0
$721$	2.19003e6	0.156896
$722$	0	0
$723$	2.31174e6i	0.164473i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 3.74257e6i	− 0.262623i	−0.991341	−	0.131312i	$-0.958081\pi$
$727$	− 3.74257e6i	− 0.262623i	0.991341	−	0.131312i	$-0.0419189\pi$
$728$	0	0
$729$	3.72133e6	0.259346
$730$	0	0
$731$	−7.98359e6	−0.552593
$732$	0	0
$733$	− 1.63705e7i	− 1.12539i	−0.826665	−	0.562695i	$-0.809765\pi$
$733$	− 1.63705e7i	− 1.12539i	0.826665	−	0.562695i	$-0.190235\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	2.15301e7i	1.46008i
$738$	0	0
$739$	2.72667e7	1.83663	0.918315	−	0.395851i	$-0.129550\pi$
$739$	2.72667e7	1.83663	0.918315	+	0.395851i	$0.129550\pi$
$740$	0	0
$741$	2.93431e7	1.96318
$742$	0	0
$743$	2.19134e7i	1.45625i	0.685443	+	0.728127i	$0.259609\pi$
$743$	2.19134e7i	1.45625i	−0.685443	+	0.728127i	$0.740391\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 5.20981e6i	− 0.341602i
$748$	0	0
$749$	6.14262e6	0.400082
$750$	0	0
$751$	1.19293e7	0.771820	0.385910	−	0.922536i	$-0.373887\pi$
$751$	1.19293e7	0.771820	0.385910	+	0.922536i	$0.373887\pi$
$752$	0	0
$753$	− 1.79854e7i	− 1.15593i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 2.46216e7i	− 1.56162i	−0.624767	−	0.780811i	$-0.714806\pi$
$757$	− 2.46216e7i	− 1.56162i	0.624767	−	0.780811i	$-0.285194\pi$
$758$	0	0
$759$	−2.23009e7	−1.40513
$760$	0	0
$761$	−1.11065e6	−0.0695208	−0.0347604	−	0.999396i	$-0.511067\pi$
$761$	−1.11065e6	−0.0695208	−0.0347604	+	0.999396i	$0.511067\pi$
$762$	0	0
$763$	1.12407e6i	0.0699005i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 2.15965e7i	− 1.32555i
$768$	0	0
$769$	516268.	0.0314818	0.0157409	−	0.999876i	$-0.494989\pi$
$769$	516268.	0.0314818	0.0157409	+	0.999876i	$0.494989\pi$
$770$	0	0
$771$	1.68886e7	1.02319
$772$	0	0
$773$	5.63544e6i	0.339218i	0.985511	+	0.169609i	$0.0542505\pi$
$773$	5.63544e6i	0.339218i	−0.985511	+	0.169609i	$0.945749\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 2.52016e6i	− 0.149753i
$778$	0	0
$779$	−1.40261e7	−0.828121
$780$	0	0
$781$	−2.44468e7	−1.43415
$782$	0	0
$783$	− 1.41781e6i	− 0.0826445i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 1.77616e6i	− 0.102222i	−0.998693	−	0.0511110i	$-0.983724\pi$
$787$	− 1.77616e6i	− 0.102222i	0.998693	−	0.0511110i	$-0.0162762\pi$
$788$	0	0
$789$	2.41221e7	1.37950
$790$	0	0
$791$	−9.39094e6	−0.533664
$792$	0	0
$793$	− 784177.i	− 0.0442824i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	4.79357e6i	0.267309i	0.991028	+	0.133654i	$0.0426712\pi$
$797$	4.79357e6i	0.267309i	−0.991028	+	0.133654i	$0.957329\pi$
$798$	0	0
$799$	1.94276e6	0.107660
$800$	0	0
$801$	8.80093e6	0.484671
$802$	0	0
$803$	3.58786e7i	1.96357i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 6.70733e6i	− 0.362548i
$808$	0	0
$809$	2.32430e7	1.24860	0.624298	−	0.781187i	$-0.285385\pi$
$809$	2.32430e7	1.24860	0.624298	+	0.781187i	$0.285385\pi$
$810$	0	0
$811$	−1.55958e7	−0.832637	−0.416318	−	0.909219i	$-0.636680\pi$
$811$	−1.55958e7	−0.832637	−0.416318	+	0.909219i	$0.636680\pi$
$812$	0	0
$813$	4.00971e7i	2.12758i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	1.74852e7i	0.916463i
$818$	0	0
$819$	5.13529e6	0.267520
$820$	0	0
$821$	−7.07206e6	−0.366174	−0.183087	−	0.983097i	$-0.558609\pi$
$821$	−7.07206e6	−0.366174	−0.183087	+	0.983097i	$0.558609\pi$
$822$	0	0
$823$	3.51052e7i	1.80664i	0.428965	+	0.903321i	$0.358878\pi$
$823$	3.51052e7i	1.80664i	−0.428965	+	0.903321i	$0.641122\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.26303e7i	0.642168i	0.947051	+	0.321084i	$0.104047\pi$
$827$	1.26303e7i	0.642168i	−0.947051	+	0.321084i	$0.895953\pi$
$828$	0	0
$829$	2.71570e7	1.37245	0.686223	−	0.727391i	$-0.259267\pi$
$829$	2.71570e7	1.37245	0.686223	+	0.727391i	$0.259267\pi$
$830$	0	0
$831$	4.05262e6	0.203579
$832$	0	0
$833$	2.47096e6i	0.123382i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	7.18342e6i	0.354419i
$838$	0	0
$839$	1.26528e7	0.620555	0.310278	−	0.950646i	$-0.399578\pi$
$839$	1.26528e7	0.620555	0.310278	+	0.950646i	$0.399578\pi$
$840$	0	0
$841$	−1.97545e7	−0.963110
$842$	0	0
$843$	− 9.19267e6i	− 0.445526i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	1.04289e6i	0.0499495i
$848$	0	0
$849$	1.18876e7	0.566011
$850$	0	0
$851$	6.63216e6	0.313929
$852$	0	0
$853$	1.52399e7i	0.717151i	0.933501	+	0.358576i	$0.116737\pi$
$853$	1.52399e7i	0.717151i	−0.933501	+	0.358576i	$0.883263\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 3.63637e7i	− 1.69128i	−0.533753	−	0.845641i	$-0.679219\pi$
$857$	− 3.63637e7i	− 1.69128i	0.533753	−	0.845641i	$-0.320781\pi$
$858$	0	0
$859$	2.51710e7	1.16391	0.581953	−	0.813222i	$-0.302289\pi$
$859$	2.51710e7	1.16391	0.581953	+	0.813222i	$0.302289\pi$
$860$	0	0
$861$	−6.13651e6	−0.282107
$862$	0	0
$863$	− 2.21176e7i	− 1.01091i	−0.862853	−	0.505454i	$-0.831325\pi$
$863$	− 2.21176e7i	− 1.01091i	0.862853	−	0.505454i	$-0.168675\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	7.25974e6i	0.327999i
$868$	0	0
$869$	4.20176e7	1.88748
$870$	0	0
$871$	3.26169e7	1.45679
$872$	0	0
$873$	3.51565e6i	0.156124i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	4.77311e6i	0.209557i	0.994496	+	0.104779i	$0.0334134\pi$
$877$	4.77311e6i	0.209557i	−0.994496	+	0.104779i	$0.966587\pi$
$878$	0	0
$879$	−8.37923e6	−0.365790
$880$	0	0
$881$	−1.79025e7	−0.777095	−0.388548	−	0.921429i	$-0.627023\pi$
$881$	−1.79025e7	−0.777095	−0.388548	+	0.921429i	$0.627023\pi$
$882$	0	0
$883$	− 1.23636e7i	− 0.533634i	−0.963747	−	0.266817i	$-0.914028\pi$
$883$	− 1.23636e7i	− 0.533634i	0.963747	−	0.266817i	$-0.0859719\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 2.33951e7i	− 0.998427i	−0.866479	−	0.499214i	$-0.833622\pi$
$887$	− 2.33951e7i	− 0.998427i	0.866479	−	0.499214i	$-0.166378\pi$
$888$	0	0
$889$	1.37259e7	0.582488
$890$	0	0
$891$	3.08172e7	1.30046
$892$	0	0
$893$	− 4.25492e6i	− 0.178551i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	3.37845e7i	1.40196i
$898$	0	0
$899$	−3.83364e6	−0.158202
$900$	0	0
$901$	1.20594e7	0.494895
$902$	0	0
$903$	7.64987e6i	0.312201i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	1.76517e7i	0.712474i	0.934396	+	0.356237i	$0.115940\pi$
$907$	1.76517e7i	0.712474i	−0.934396	+	0.356237i	$0.884060\pi$
$908$	0	0
$909$	−2.06335e7	−0.828252
$910$	0	0
$911$	2.50785e7	1.00117	0.500583	−	0.865688i	$-0.333119\pi$
$911$	2.50785e7	1.00117	0.500583	+	0.865688i	$0.333119\pi$
$912$	0	0
$913$	− 1.37315e7i	− 0.545181i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.20432e7i	0.472955i
$918$	0	0
$919$	−4.36173e7	−1.70361	−0.851805	−	0.523859i	$-0.824492\pi$
$919$	−4.36173e7	−1.70361	−0.851805	+	0.523859i	$0.824492\pi$
$920$	0	0
$921$	2.52718e7	0.981718
$922$	0	0
$923$	3.70354e7i	1.43091i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	7.24092e6i	0.276754i
$928$	0	0
$929$	−4.60105e7	−1.74911	−0.874556	−	0.484924i	$-0.838847\pi$
$929$	−4.60105e7	−1.74911	−0.874556	+	0.484924i	$0.838847\pi$
$930$	0	0
$931$	5.41173e6	0.204627
$932$	0	0
$933$	− 2.33992e7i	− 0.880028i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 4.54956e7i	− 1.69286i	−0.532503	−	0.846428i	$-0.678748\pi$
$937$	− 4.54956e7i	− 1.69286i	0.532503	−	0.846428i	$-0.321252\pi$
$938$	0	0
$939$	6.94822e7	2.57164
$940$	0	0
$941$	5.13895e7	1.89191	0.945955	−	0.324299i	$-0.105128\pi$
$941$	5.13895e7	1.89191	0.945955	+	0.324299i	$0.105128\pi$
$942$	0	0
$943$	− 1.61491e7i	− 0.591383i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 8.35685e6i	− 0.302808i	−0.988472	−	0.151404i	$-0.951621\pi$
$947$	− 8.35685e6i	− 0.302808i	0.988472	−	0.151404i	$-0.0483795\pi$
$948$	0	0
$949$	5.43539e7	1.95914
$950$	0	0
$951$	2.13393e7	0.765118
$952$	0	0
$953$	− 3.47459e7i	− 1.23929i	−0.784884	−	0.619643i	$-0.787277\pi$
$953$	− 3.47459e7i	− 1.23929i	0.784884	−	0.619643i	$-0.212723\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	7.47509e6i	0.263837i
$958$	0	0
$959$	657472.	0.0230850
$960$	0	0
$961$	−9.20579e6	−0.321553
$962$	0	0
$963$	2.03094e7i	0.705717i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	1.15760e7i	0.398101i	0.979989	+	0.199051i	$0.0637858\pi$
$967$	1.15760e7i	0.398101i	−0.979989	+	0.199051i	$0.936214\pi$
$968$	0	0
$969$	−4.66820e7	−1.59713
$970$	0	0
$971$	−1.65286e7	−0.562583	−0.281292	−	0.959622i	$-0.590763\pi$
$971$	−1.65286e7	−0.562583	−0.281292	+	0.959622i	$0.590763\pi$
$972$	0	0
$973$	1.78051e7i	0.602924i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 4.85257e6i	− 0.162643i	−0.996688	−	0.0813215i	$-0.974086\pi$
$977$	− 4.85257e6i	− 0.162643i	0.996688	−	0.0813215i	$-0.0259140\pi$
$978$	0	0
$979$	2.31966e7	0.773512
$980$	0	0
$981$	−3.71650e6	−0.123300
$982$	0	0
$983$	− 8.52706e6i	− 0.281459i	−0.990048	−	0.140730i	$-0.955055\pi$
$983$	− 8.52706e6i	− 0.281459i	0.990048	−	0.140730i	$-0.0449448\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 1.86155e6i	− 0.0608251i
$988$	0	0
$989$	−2.01317e7	−0.654470
$990$	0	0
$991$	6271.13	0.000202844 0	0.000101422	−	1.00000i	$-0.499968\pi$
$991$	6271.13	0.000202844 0	0.000101422		1.00000i	$0.499968\pi$
$992$	0	0
$993$	− 3.62797e7i	− 1.16759i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 3.44729e7i	− 1.09835i	−0.835708	−	0.549174i	$-0.814942\pi$
$997$	− 3.44729e7i	− 1.09835i	0.835708	−	0.549174i	$-0.185058\pi$
$998$	0	0
$999$	−4.16552e6	−0.132055

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	700.6.e.g.449.5		6
5.2	odd	4		700.6.a.i.1.3		3
5.3	odd	4		140.6.a.d.1.1	✓	3
5.4	even	2	inner	700.6.e.g.449.2		6
20.3	even	4		560.6.a.t.1.3		3
35.13	even	4		980.6.a.h.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.6.a.d.1.1	✓	3	5.3	odd	4
560.6.a.t.1.3		3	20.3	even	4
700.6.a.i.1.3		3	5.2	odd	4
700.6.e.g.449.2		6	5.4	even	2	inner
700.6.e.g.449.5		6	1.1	even	1	trivial
980.6.a.h.1.3		3	35.13	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:	\( N \)	\(=\)	\( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	700.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+20.1248i q^{3} +49.0000i q^{7} -162.009 q^{9} +O(q^{10})\)	\(q+20.1248i q^{3} +49.0000i q^{7} -162.009 q^{9} -427.006 q^{11} +646.889i q^{13} -1029.14i q^{17} -2253.95 q^{19} -986.117 q^{21} -2595.11i q^{23} +1629.93i q^{27} -869.860 q^{29} +4407.19 q^{31} -8593.44i q^{33} +2555.64i q^{37} -13018.5 q^{39} +6222.90 q^{41} -7757.57i q^{43} +1887.76i q^{47} -2401.00 q^{49} +20711.2 q^{51} +11718.0i q^{53} -45360.4i q^{57} -33385.2 q^{59} -1212.23 q^{61} -7938.44i q^{63} -50421.1i q^{67} +52226.1 q^{69} +57251.5 q^{71} -84023.5i q^{73} -20923.3i q^{77} -98400.4 q^{79} -72170.3 q^{81} +32157.5i q^{83} -17505.8i q^{87} -54323.7 q^{89} -31697.6 q^{91} +88694.1i q^{93} -21700.3i q^{97} +69178.9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 562 q^{9}+O(q^{10}) \) 6 * q - 562 * q^9	\( 6 q - 562 q^{9} - 28 q^{11} - 4656 q^{19} + 588 q^{21} - 8184 q^{29} + 11776 q^{31} - 27404 q^{39} + 22900 q^{41} - 14406 q^{49} + 63524 q^{51} - 24776 q^{59} + 54364 q^{61} + 124280 q^{69} + 163984 q^{71} + 31852 q^{79} - 64202 q^{81} + 191420 q^{89} - 392 q^{91} + 228736 q^{99}+O(q^{100}) \) 6 * q - 562 * q^9 - 28 * q^11 - 4656 * q^19 + 588 * q^21 - 8184 * q^29 + 11776 * q^31 - 27404 * q^39 + 22900 * q^41 - 14406 * q^49 + 63524 * q^51 - 24776 * q^59 + 54364 * q^61 + 124280 * q^69 + 163984 * q^71 + 31852 * q^79 - 64202 * q^81 + 191420 * q^89 - 392 * q^91 + 228736 * q^99

Introduction

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