LMFDB - Embedded newform 1350.3.d.d.701.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1350,3,Mod(701,1350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1350, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1350.701");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1350 = 2 \cdot 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1350.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:	$36.7848356886$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 54)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			701.1
Root			$-1.41421i$ of defining polynomial
Character	$\chi$	$=$	1350.701
Dual form			1350.3.d.d.701.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421i q^{2} -2.00000 q^{4} -5.00000 q^{7} +2.82843i q^{8} +O(q^{10})$	$q-1.41421i q^{2} -2.00000 q^{4} -5.00000 q^{7} +2.82843i q^{8} -8.48528i q^{11} +1.00000 q^{13} +7.07107i q^{14} +4.00000 q^{16} +25.4558i q^{17} +29.0000 q^{19} -12.0000 q^{22} -8.48528i q^{23} -1.41421i q^{26} +10.0000 q^{28} +16.9706i q^{29} -10.0000 q^{31} -5.65685i q^{32} +36.0000 q^{34} +25.0000 q^{37} -41.0122i q^{38} -16.9706i q^{41} -14.0000 q^{43} +16.9706i q^{44} -12.0000 q^{46} +8.48528i q^{47} -24.0000 q^{49} -2.00000 q^{52} -50.9117i q^{53} -14.1421i q^{56} +24.0000 q^{58} -93.3381i q^{59} +23.0000 q^{61} +14.1421i q^{62} -8.00000 q^{64} +19.0000 q^{67} -50.9117i q^{68} -101.823i q^{71} +97.0000 q^{73} -35.3553i q^{74} -58.0000 q^{76} +42.4264i q^{77} +77.0000 q^{79} -24.0000 q^{82} -118.794i q^{83} +19.7990i q^{86} +24.0000 q^{88} +76.3675i q^{89} -5.00000 q^{91} +16.9706i q^{92} +12.0000 q^{94} +49.0000 q^{97} +33.9411i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{4} - 10 q^{7}+O(q^{10}) $ 2 * q - 4 * q^4 - 10 * q^7	$ 2 q - 4 q^{4} - 10 q^{7} + 2 q^{13} + 8 q^{16} + 58 q^{19} - 24 q^{22} + 20 q^{28} - 20 q^{31} + 72 q^{34} + 50 q^{37} - 28 q^{43} - 24 q^{46} - 48 q^{49} - 4 q^{52} + 48 q^{58} + 46 q^{61} - 16 q^{64} + 38 q^{67} + 194 q^{73} - 116 q^{76} + 154 q^{79} - 48 q^{82} + 48 q^{88} - 10 q^{91} + 24 q^{94} + 98 q^{97}+O(q^{100}) $ 2 * q - 4 * q^4 - 10 * q^7 + 2 * q^13 + 8 * q^16 + 58 * q^19 - 24 * q^22 + 20 * q^28 - 20 * q^31 + 72 * q^34 + 50 * q^37 - 28 * q^43 - 24 * q^46 - 48 * q^49 - 4 * q^52 + 48 * q^58 + 46 * q^61 - 16 * q^64 + 38 * q^67 + 194 * q^73 - 116 * q^76 + 154 * q^79 - 48 * q^82 + 48 * q^88 - 10 * q^91 + 24 * q^94 + 98 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1350\mathbb{Z}\right)^\times$.

$n$	$1001$	$1027$
$\chi(n)$	$-1$	$1$

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 $

$2$	− 1.41421i	− 0.707107i
$2$	− 1.41421i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−2.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−5.00000	−0.714286	−0.357143	−	0.934050i	$-0.616249\pi$
$7$	−5.00000	−0.714286	−0.357143	+	0.934050i	$0.616249\pi$
$8$	2.82843i	0.353553i
$9$	0	0
$10$	0	0
$11$	− 8.48528i	− 0.771389i	−0.922627	−	0.385695i	$-0.873962\pi$
$11$	− 8.48528i	− 0.771389i	0.922627	−	0.385695i	$-0.126038\pi$
$12$	0	0
$13$	1.00000	0.0769231	0.0384615	−	0.999260i	$-0.487754\pi$
$13$	1.00000	0.0769231	0.0384615	+	0.999260i	$0.487754\pi$
$14$	7.07107i	0.505076i
$15$	0	0
$16$	4.00000	0.250000
$17$	25.4558i	1.49740i	0.662908	+	0.748701i	$0.269322\pi$
$17$	25.4558i	1.49740i	−0.662908	+	0.748701i	$0.730678\pi$
$18$	0	0
$19$	29.0000	1.52632	0.763158	−	0.646212i	$-0.223648\pi$
$19$	29.0000	1.52632	0.763158	+	0.646212i	$0.223648\pi$
$20$	0	0
$21$	0	0
$22$	−12.0000	−0.545455
$23$	− 8.48528i	− 0.368925i	−0.982840	−	0.184463i	$-0.940946\pi$
$23$	− 8.48528i	− 0.368925i	0.982840	−	0.184463i	$-0.0590545\pi$
$24$	0	0
$25$	0	0
$26$	− 1.41421i	− 0.0543928i
$27$	0	0
$28$	10.0000	0.357143
$29$	16.9706i	0.585192i	0.956236	+	0.292596i	$0.0945191\pi$
$29$	16.9706i	0.585192i	−0.956236	+	0.292596i	$0.905481\pi$
$30$	0	0
$31$	−10.0000	−0.322581	−0.161290	−	0.986907i	$-0.551566\pi$
$31$	−10.0000	−0.322581	−0.161290	+	0.986907i	$0.551566\pi$
$32$	− 5.65685i	− 0.176777i
$33$	0	0
$34$	36.0000	1.05882
$35$	0	0
$36$	0	0
$37$	25.0000	0.675676	0.337838	−	0.941204i	$-0.390304\pi$
$37$	25.0000	0.675676	0.337838	+	0.941204i	$0.390304\pi$
$38$	− 41.0122i	− 1.07927i
$39$	0	0
$40$	0	0
$41$	− 16.9706i	− 0.413916i	−0.978350	−	0.206958i	$-0.933644\pi$
$41$	− 16.9706i	− 0.413916i	0.978350	−	0.206958i	$-0.0663564\pi$
$42$	0	0
$43$	−14.0000	−0.325581	−0.162791	−	0.986661i	$-0.552050\pi$
$43$	−14.0000	−0.325581	−0.162791	+	0.986661i	$0.552050\pi$
$44$	16.9706i	0.385695i
$45$	0	0
$46$	−12.0000	−0.260870
$47$	8.48528i	0.180538i	0.995917	+	0.0902690i	$0.0287727\pi$
$47$	8.48528i	0.180538i	−0.995917	+	0.0902690i	$0.971227\pi$
$48$	0	0
$49$	−24.0000	−0.489796
$50$	0	0
$51$	0	0
$52$	−2.00000	−0.0384615
$53$	− 50.9117i	− 0.960598i	−0.877105	−	0.480299i	$-0.840528\pi$
$53$	− 50.9117i	− 0.960598i	0.877105	−	0.480299i	$-0.159472\pi$
$54$	0	0
$55$	0	0
$56$	− 14.1421i	− 0.252538i
$57$	0	0
$58$	24.0000	0.413793
$59$	− 93.3381i	− 1.58200i	−0.611815	−	0.791001i	$-0.709560\pi$
$59$	− 93.3381i	− 1.58200i	0.611815	−	0.791001i	$-0.290440\pi$
$60$	0	0
$61$	23.0000	0.377049	0.188525	−	0.982068i	$-0.439629\pi$
$61$	23.0000	0.377049	0.188525	+	0.982068i	$0.439629\pi$
$62$	14.1421i	0.228099i
$63$	0	0
$64$	−8.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	19.0000	0.283582	0.141791	−	0.989897i	$-0.454714\pi$
$67$	19.0000	0.283582	0.141791	+	0.989897i	$0.454714\pi$
$68$	− 50.9117i	− 0.748701i
$69$	0	0
$70$	0	0
$71$	− 101.823i	− 1.43413i	−0.697005	−	0.717066i	$-0.745485\pi$
$71$	− 101.823i	− 1.43413i	0.697005	−	0.717066i	$-0.254515\pi$
$72$	0	0
$73$	97.0000	1.32877	0.664384	−	0.747392i	$-0.268694\pi$
$73$	97.0000	1.32877	0.664384	+	0.747392i	$0.268694\pi$
$74$	− 35.3553i	− 0.477775i
$75$	0	0
$76$	−58.0000	−0.763158
$77$	42.4264i	0.550992i
$78$	0	0
$79$	77.0000	0.974684	0.487342	−	0.873211i	$-0.337967\pi$
$79$	77.0000	0.974684	0.487342	+	0.873211i	$0.337967\pi$
$80$	0	0
$81$	0	0
$82$	−24.0000	−0.292683
$83$	− 118.794i	− 1.43125i	−0.698484	−	0.715626i	$-0.746141\pi$
$83$	− 118.794i	− 1.43125i	0.698484	−	0.715626i	$-0.253859\pi$
$84$	0	0
$85$	0	0
$86$	19.7990i	0.230221i
$87$	0	0
$88$	24.0000	0.272727
$89$	76.3675i	0.858062i	0.903290	+	0.429031i	$0.141145\pi$
$89$	76.3675i	0.858062i	−0.903290	+	0.429031i	$0.858855\pi$
$90$	0	0
$91$	−5.00000	−0.0549451
$92$	16.9706i	0.184463i
$93$	0	0
$94$	12.0000	0.127660
$95$	0	0
$96$	0	0
$97$	49.0000	0.505155	0.252577	−	0.967577i	$-0.418722\pi$
$97$	49.0000	0.505155	0.252577	+	0.967577i	$0.418722\pi$
$98$	33.9411i	0.346338i
$99$	0	0
$100$	0	0
$101$	− 135.765i	− 1.34420i	−0.740459	−	0.672101i	$-0.765392\pi$
$101$	− 135.765i	− 1.34420i	0.740459	−	0.672101i	$-0.234608\pi$
$102$	0	0
$103$	163.000	1.58252	0.791262	−	0.611477i	$-0.209424\pi$
$103$	163.000	1.58252	0.791262	+	0.611477i	$0.209424\pi$
$104$	2.82843i	0.0271964i
$105$	0	0
$106$	−72.0000	−0.679245
$107$	25.4558i	0.237905i	0.992900	+	0.118953i	$0.0379536\pi$
$107$	25.4558i	0.237905i	−0.992900	+	0.118953i	$0.962046\pi$
$108$	0	0
$109$	2.00000	0.0183486	0.00917431	−	0.999958i	$-0.497080\pi$
$109$	2.00000	0.0183486	0.00917431	+	0.999958i	$0.497080\pi$
$110$	0	0
$111$	0	0
$112$	−20.0000	−0.178571
$113$	− 110.309i	− 0.976183i	−0.872793	−	0.488091i	$-0.837693\pi$
$113$	− 110.309i	− 0.976183i	0.872793	−	0.488091i	$-0.162307\pi$
$114$	0	0
$115$	0	0
$116$	− 33.9411i	− 0.292596i
$117$	0	0
$118$	−132.000	−1.11864
$119$	− 127.279i	− 1.06957i
$120$	0	0
$121$	49.0000	0.404959
$122$	− 32.5269i	− 0.266614i
$123$	0	0
$124$	20.0000	0.161290
$125$	0	0
$126$	0	0
$127$	178.000	1.40157	0.700787	−	0.713370i	$-0.252832\pi$
$127$	178.000	1.40157	0.700787	+	0.713370i	$0.252832\pi$
$128$	11.3137i	0.0883883i
$129$	0	0
$130$	0	0
$131$	− 118.794i	− 0.906824i	−0.891301	−	0.453412i	$-0.850207\pi$
$131$	− 118.794i	− 0.906824i	0.891301	−	0.453412i	$-0.149793\pi$
$132$	0	0
$133$	−145.000	−1.09023
$134$	− 26.8701i	− 0.200523i
$135$	0	0
$136$	−72.0000	−0.529412
$137$	− 93.3381i	− 0.681300i	−0.940190	−	0.340650i	$-0.889353\pi$
$137$	− 93.3381i	− 0.681300i	0.940190	−	0.340650i	$-0.110647\pi$
$138$	0	0
$139$	−163.000	−1.17266	−0.586331	−	0.810072i	$-0.699428\pi$
$139$	−163.000	−1.17266	−0.586331	+	0.810072i	$0.699428\pi$
$140$	0	0
$141$	0	0
$142$	−144.000	−1.01408
$143$	− 8.48528i	− 0.0593376i
$144$	0	0
$145$	0	0
$146$	− 137.179i	− 0.939580i
$147$	0	0
$148$	−50.0000	−0.337838
$149$	33.9411i	0.227793i	0.993493	+	0.113896i	$0.0363332\pi$
$149$	33.9411i	0.227793i	−0.993493	+	0.113896i	$0.963667\pi$
$150$	0	0
$151$	149.000	0.986755	0.493377	−	0.869815i	$-0.335762\pi$
$151$	149.000	0.986755	0.493377	+	0.869815i	$0.335762\pi$
$152$	82.0244i	0.539634i
$153$	0	0
$154$	60.0000	0.389610
$155$	0	0
$156$	0	0
$157$	−242.000	−1.54140	−0.770701	−	0.637197i	$-0.780094\pi$
$157$	−242.000	−1.54140	−0.770701	+	0.637197i	$0.780094\pi$
$158$	− 108.894i	− 0.689205i
$159$	0	0
$160$	0	0
$161$	42.4264i	0.263518i
$162$	0	0
$163$	−173.000	−1.06135	−0.530675	−	0.847575i	$-0.678061\pi$
$163$	−173.000	−1.06135	−0.530675	+	0.847575i	$0.678061\pi$
$164$	33.9411i	0.206958i
$165$	0	0
$166$	−168.000	−1.01205
$167$	195.161i	1.16863i	0.811526	+	0.584316i	$0.198637\pi$
$167$	195.161i	1.16863i	−0.811526	+	0.584316i	$0.801363\pi$
$168$	0	0
$169$	−168.000	−0.994083
$170$	0	0
$171$	0	0
$172$	28.0000	0.162791
$173$	− 322.441i	− 1.86382i	−0.362691	−	0.931910i	$-0.618142\pi$
$173$	− 322.441i	− 1.86382i	0.362691	−	0.931910i	$-0.381858\pi$
$174$	0	0
$175$	0	0
$176$	− 33.9411i	− 0.192847i
$177$	0	0
$178$	108.000	0.606742
$179$	305.470i	1.70654i	0.521472	+	0.853269i	$0.325383\pi$
$179$	305.470i	1.70654i	−0.521472	+	0.853269i	$0.674617\pi$
$180$	0	0
$181$	263.000	1.45304	0.726519	−	0.687146i	$-0.241137\pi$
$181$	263.000	1.45304	0.726519	+	0.687146i	$0.241137\pi$
$182$	7.07107i	0.0388520i
$183$	0	0
$184$	24.0000	0.130435
$185$	0	0
$186$	0	0
$187$	216.000	1.15508
$188$	− 16.9706i	− 0.0902690i
$189$	0	0
$190$	0	0
$191$	144.250i	0.755234i	0.925962	+	0.377617i	$0.123256\pi$
$191$	144.250i	0.755234i	−0.925962	+	0.377617i	$0.876744\pi$
$192$	0	0
$193$	−71.0000	−0.367876	−0.183938	−	0.982938i	$-0.558885\pi$
$193$	−71.0000	−0.367876	−0.183938	+	0.982938i	$0.558885\pi$
$194$	− 69.2965i	− 0.357198i
$195$	0	0
$196$	48.0000	0.244898
$197$	− 76.3675i	− 0.387652i	−0.981036	−	0.193826i	$-0.937910\pi$
$197$	− 76.3675i	− 0.387652i	0.981036	−	0.193826i	$-0.0620898\pi$
$198$	0	0
$199$	173.000	0.869347	0.434673	−	0.900588i	$-0.356864\pi$
$199$	173.000	0.869347	0.434673	+	0.900588i	$0.356864\pi$
$200$	0	0
$201$	0	0
$202$	−192.000	−0.950495
$203$	− 84.8528i	− 0.417994i
$204$	0	0
$205$	0	0
$206$	− 230.517i	− 1.11901i
$207$	0	0
$208$	4.00000	0.0192308
$209$	− 246.073i	− 1.17738i
$210$	0	0
$211$	341.000	1.61611	0.808057	−	0.589104i	$-0.200519\pi$
$211$	341.000	1.61611	0.808057	+	0.589104i	$0.200519\pi$
$212$	101.823i	0.480299i
$213$	0	0
$214$	36.0000	0.168224
$215$	0	0
$216$	0	0
$217$	50.0000	0.230415
$218$	− 2.82843i	− 0.0129744i
$219$	0	0
$220$	0	0
$221$	25.4558i	0.115185i
$222$	0	0
$223$	58.0000	0.260090	0.130045	−	0.991508i	$-0.458488\pi$
$223$	58.0000	0.260090	0.130045	+	0.991508i	$0.458488\pi$
$224$	28.2843i	0.126269i
$225$	0	0
$226$	−156.000	−0.690265
$227$	135.765i	0.598082i	0.954240	+	0.299041i	$0.0966667\pi$
$227$	135.765i	0.598082i	−0.954240	+	0.299041i	$0.903333\pi$
$228$	0	0
$229$	−262.000	−1.14410	−0.572052	−	0.820217i	$-0.693853\pi$
$229$	−262.000	−1.14410	−0.572052	+	0.820217i	$0.693853\pi$
$230$	0	0
$231$	0	0
$232$	−48.0000	−0.206897
$233$	305.470i	1.31103i	0.755182	+	0.655515i	$0.227549\pi$
$233$	305.470i	1.31103i	−0.755182	+	0.655515i	$0.772451\pi$
$234$	0	0
$235$	0	0
$236$	186.676i	0.791001i
$237$	0	0
$238$	−180.000	−0.756303
$239$	− 169.706i	− 0.710065i	−0.934854	−	0.355033i	$-0.884470\pi$
$239$	− 169.706i	− 0.710065i	0.934854	−	0.355033i	$-0.115530\pi$
$240$	0	0
$241$	−121.000	−0.502075	−0.251037	−	0.967977i	$-0.580772\pi$
$241$	−121.000	−0.502075	−0.251037	+	0.967977i	$0.580772\pi$
$242$	− 69.2965i	− 0.286349i
$243$	0	0
$244$	−46.0000	−0.188525
$245$	0	0
$246$	0	0
$247$	29.0000	0.117409
$248$	− 28.2843i	− 0.114049i
$249$	0	0
$250$	0	0
$251$	− 356.382i	− 1.41985i	−0.704278	−	0.709924i	$-0.748729\pi$
$251$	− 356.382i	− 1.41985i	0.704278	−	0.709924i	$-0.251271\pi$
$252$	0	0
$253$	−72.0000	−0.284585
$254$	− 251.730i	− 0.991063i
$255$	0	0
$256$	16.0000	0.0625000
$257$	− 84.8528i	− 0.330167i	−0.986280	−	0.165083i	$-0.947211\pi$
$257$	− 84.8528i	− 0.330167i	0.986280	−	0.165083i	$-0.0527893\pi$
$258$	0	0
$259$	−125.000	−0.482625
$260$	0	0
$261$	0	0
$262$	−168.000	−0.641221
$263$	− 169.706i	− 0.645269i	−0.946524	−	0.322634i	$-0.895432\pi$
$263$	− 169.706i	− 0.645269i	0.946524	−	0.322634i	$-0.104568\pi$
$264$	0	0
$265$	0	0
$266$	205.061i	0.770906i
$267$	0	0
$268$	−38.0000	−0.141791
$269$	280.014i	1.04095i	0.853878	+	0.520473i	$0.174244\pi$
$269$	280.014i	1.04095i	−0.853878	+	0.520473i	$0.825756\pi$
$270$	0	0
$271$	29.0000	0.107011	0.0535055	−	0.998568i	$-0.482961\pi$
$271$	29.0000	0.107011	0.0535055	+	0.998568i	$0.482961\pi$
$272$	101.823i	0.374351i
$273$	0	0
$274$	−132.000	−0.481752
$275$	0	0
$276$	0	0
$277$	382.000	1.37906	0.689531	−	0.724256i	$-0.257817\pi$
$277$	382.000	1.37906	0.689531	+	0.724256i	$0.257817\pi$
$278$	230.517i	0.829197i
$279$	0	0
$280$	0	0
$281$	220.617i	0.785115i	0.919727	+	0.392558i	$0.128410\pi$
$281$	220.617i	0.785115i	−0.919727	+	0.392558i	$0.871590\pi$
$282$	0	0
$283$	−62.0000	−0.219081	−0.109541	−	0.993982i	$-0.534938\pi$
$283$	−62.0000	−0.219081	−0.109541	+	0.993982i	$0.534938\pi$
$284$	203.647i	0.717066i
$285$	0	0
$286$	−12.0000	−0.0419580
$287$	84.8528i	0.295654i
$288$	0	0
$289$	−359.000	−1.24221
$290$	0	0
$291$	0	0
$292$	−194.000	−0.664384
$293$	− 313.955i	− 1.07152i	−0.844370	−	0.535760i	$-0.820025\pi$
$293$	− 313.955i	− 1.07152i	0.844370	−	0.535760i	$-0.179975\pi$
$294$	0	0
$295$	0	0
$296$	70.7107i	0.238887i
$297$	0	0
$298$	48.0000	0.161074
$299$	− 8.48528i	− 0.0283789i
$300$	0	0
$301$	70.0000	0.232558
$302$	− 210.718i	− 0.697741i
$303$	0	0
$304$	116.000	0.381579
$305$	0	0
$306$	0	0
$307$	34.0000	0.110749	0.0553746	−	0.998466i	$-0.482365\pi$
$307$	34.0000	0.110749	0.0553746	+	0.998466i	$0.482365\pi$
$308$	− 84.8528i	− 0.275496i
$309$	0	0
$310$	0	0
$311$	59.3970i	0.190987i	0.995430	+	0.0954935i	$0.0304429\pi$
$311$	59.3970i	0.190987i	−0.995430	+	0.0954935i	$0.969557\pi$
$312$	0	0
$313$	−239.000	−0.763578	−0.381789	−	0.924249i	$-0.624692\pi$
$313$	−239.000	−0.763578	−0.381789	+	0.924249i	$0.624692\pi$
$314$	342.240i	1.08994i
$315$	0	0
$316$	−154.000	−0.487342
$317$	− 16.9706i	− 0.0535349i	−0.999642	−	0.0267674i	$-0.991479\pi$
$317$	− 16.9706i	− 0.0535349i	0.999642	−	0.0267674i	$-0.00852136\pi$
$318$	0	0
$319$	144.000	0.451411
$320$	0	0
$321$	0	0
$322$	60.0000	0.186335
$323$	738.219i	2.28551i
$324$	0	0
$325$	0	0
$326$	244.659i	0.750488i
$327$	0	0
$328$	48.0000	0.146341
$329$	− 42.4264i	− 0.128956i
$330$	0	0
$331$	−499.000	−1.50755	−0.753776	−	0.657131i	$-0.771770\pi$
$331$	−499.000	−1.50755	−0.753776	+	0.657131i	$0.771770\pi$
$332$	237.588i	0.715626i
$333$	0	0
$334$	276.000	0.826347
$335$	0	0
$336$	0	0
$337$	145.000	0.430267	0.215134	−	0.976585i	$-0.430981\pi$
$337$	145.000	0.430267	0.215134	+	0.976585i	$0.430981\pi$
$338$	237.588i	0.702923i
$339$	0	0
$340$	0	0
$341$	84.8528i	0.248835i
$342$	0	0
$343$	365.000	1.06414
$344$	− 39.5980i	− 0.115110i
$345$	0	0
$346$	−456.000	−1.31792
$347$	− 441.235i	− 1.27157i	−0.771866	−	0.635785i	$-0.780677\pi$
$347$	− 441.235i	− 1.27157i	0.771866	−	0.635785i	$-0.219323\pi$
$348$	0	0
$349$	−121.000	−0.346705	−0.173352	−	0.984860i	$-0.555460\pi$
$349$	−121.000	−0.346705	−0.173352	+	0.984860i	$0.555460\pi$
$350$	0	0
$351$	0	0
$352$	−48.0000	−0.136364
$353$	441.235i	1.24996i	0.780642	+	0.624978i	$0.214892\pi$
$353$	441.235i	1.24996i	−0.780642	+	0.624978i	$0.785108\pi$
$354$	0	0
$355$	0	0
$356$	− 152.735i	− 0.429031i
$357$	0	0
$358$	432.000	1.20670
$359$	432.749i	1.20543i	0.797957	+	0.602715i	$0.205914\pi$
$359$	432.749i	1.20543i	−0.797957	+	0.602715i	$0.794086\pi$
$360$	0	0
$361$	480.000	1.32964
$362$	− 371.938i	− 1.02745i
$363$	0	0
$364$	10.0000	0.0274725
$365$	0	0
$366$	0	0
$367$	−149.000	−0.405995	−0.202997	−	0.979179i	$-0.565068\pi$
$367$	−149.000	−0.405995	−0.202997	+	0.979179i	$0.565068\pi$
$368$	− 33.9411i	− 0.0922313i
$369$	0	0
$370$	0	0
$371$	254.558i	0.686141i
$372$	0	0
$373$	361.000	0.967828	0.483914	−	0.875115i	$-0.339215\pi$
$373$	361.000	0.967828	0.483914	+	0.875115i	$0.339215\pi$
$374$	− 305.470i	− 0.816765i
$375$	0	0
$376$	−24.0000	−0.0638298
$377$	16.9706i	0.0450148i
$378$	0	0
$379$	173.000	0.456464	0.228232	−	0.973607i	$-0.426705\pi$
$379$	173.000	0.456464	0.228232	+	0.973607i	$0.426705\pi$
$380$	0	0
$381$	0	0
$382$	204.000	0.534031
$383$	678.823i	1.77238i	0.463320	+	0.886191i	$0.346658\pi$
$383$	678.823i	1.77238i	−0.463320	+	0.886191i	$0.653342\pi$
$384$	0	0
$385$	0	0
$386$	100.409i	0.260127i
$387$	0	0
$388$	−98.0000	−0.252577
$389$	− 415.779i	− 1.06884i	−0.845219	−	0.534420i	$-0.820530\pi$
$389$	− 415.779i	− 1.06884i	0.845219	−	0.534420i	$-0.179470\pi$
$390$	0	0
$391$	216.000	0.552430
$392$	− 67.8823i	− 0.173169i
$393$	0	0
$394$	−108.000	−0.274112
$395$	0	0
$396$	0	0
$397$	286.000	0.720403	0.360202	−	0.932875i	$-0.382708\pi$
$397$	286.000	0.720403	0.360202	+	0.932875i	$0.382708\pi$
$398$	− 244.659i	− 0.614721i
$399$	0	0
$400$	0	0
$401$	339.411i	0.846412i	0.906033	+	0.423206i	$0.139095\pi$
$401$	339.411i	0.846412i	−0.906033	+	0.423206i	$0.860905\pi$
$402$	0	0
$403$	−10.0000	−0.0248139
$404$	271.529i	0.672101i
$405$	0	0
$406$	−120.000	−0.295567
$407$	− 212.132i	− 0.521209i
$408$	0	0
$409$	215.000	0.525672	0.262836	−	0.964840i	$-0.415342\pi$
$409$	215.000	0.525672	0.262836	+	0.964840i	$0.415342\pi$
$410$	0	0
$411$	0	0
$412$	−326.000	−0.791262
$413$	466.690i	1.13000i
$414$	0	0
$415$	0	0
$416$	− 5.65685i	− 0.0135982i
$417$	0	0
$418$	−348.000	−0.832536
$419$	− 602.455i	− 1.43784i	−0.695093	−	0.718920i	$-0.744637\pi$
$419$	− 602.455i	− 1.43784i	0.695093	−	0.718920i	$-0.255363\pi$
$420$	0	0
$421$	455.000	1.08076	0.540380	−	0.841421i	$-0.318280\pi$
$421$	455.000	1.08076	0.540380	+	0.841421i	$0.318280\pi$
$422$	− 482.247i	− 1.14276i
$423$	0	0
$424$	144.000	0.339623
$425$	0	0
$426$	0	0
$427$	−115.000	−0.269321
$428$	− 50.9117i	− 0.118953i
$429$	0	0
$430$	0	0
$431$	330.926i	0.767810i	0.923373	+	0.383905i	$0.125421\pi$
$431$	330.926i	0.767810i	−0.923373	+	0.383905i	$0.874579\pi$
$432$	0	0
$433$	−218.000	−0.503464	−0.251732	−	0.967797i	$-0.581000\pi$
$433$	−218.000	−0.503464	−0.251732	+	0.967797i	$0.581000\pi$
$434$	− 70.7107i	− 0.162928i
$435$	0	0
$436$	−4.00000	−0.00917431
$437$	− 246.073i	− 0.563096i
$438$	0	0
$439$	374.000	0.851936	0.425968	−	0.904738i	$-0.359934\pi$
$439$	374.000	0.851936	0.425968	+	0.904738i	$0.359934\pi$
$440$	0	0
$441$	0	0
$442$	36.0000	0.0814480
$443$	− 16.9706i	− 0.0383083i	−0.999817	−	0.0191541i	$-0.993903\pi$
$443$	− 16.9706i	− 0.0383083i	0.999817	−	0.0191541i	$-0.00609732\pi$
$444$	0	0
$445$	0	0
$446$	− 82.0244i	− 0.183911i
$447$	0	0
$448$	40.0000	0.0892857
$449$	483.661i	1.07720i	0.842563	+	0.538598i	$0.181046\pi$
$449$	483.661i	1.07720i	−0.842563	+	0.538598i	$0.818954\pi$
$450$	0	0
$451$	−144.000	−0.319290
$452$	220.617i	0.488091i
$453$	0	0
$454$	192.000	0.422907
$455$	0	0
$456$	0	0
$457$	22.0000	0.0481400	0.0240700	−	0.999710i	$-0.492338\pi$
$457$	22.0000	0.0481400	0.0240700	+	0.999710i	$0.492338\pi$
$458$	370.524i	0.809004i
$459$	0	0
$460$	0	0
$461$	398.808i	0.865094i	0.901611	+	0.432547i	$0.142385\pi$
$461$	398.808i	0.865094i	−0.901611	+	0.432547i	$0.857615\pi$
$462$	0	0
$463$	883.000	1.90713	0.953564	−	0.301191i	$-0.0973843\pi$
$463$	883.000	1.90713	0.953564	+	0.301191i	$0.0973843\pi$
$464$	67.8823i	0.146298i
$465$	0	0
$466$	432.000	0.927039
$467$	127.279i	0.272547i	0.990671	+	0.136273i	$0.0435125\pi$
$467$	127.279i	0.272547i	−0.990671	+	0.136273i	$0.956487\pi$
$468$	0	0
$469$	−95.0000	−0.202559
$470$	0	0
$471$	0	0
$472$	264.000	0.559322
$473$	118.794i	0.251150i
$474$	0	0
$475$	0	0
$476$	254.558i	0.534787i
$477$	0	0
$478$	−240.000	−0.502092
$479$	− 390.323i	− 0.814870i	−0.913234	−	0.407435i	$-0.866423\pi$
$479$	− 390.323i	− 0.814870i	0.913234	−	0.407435i	$-0.133577\pi$
$480$	0	0
$481$	25.0000	0.0519751
$482$	171.120i	0.355020i
$483$	0	0
$484$	−98.0000	−0.202479
$485$	0	0
$486$	0	0
$487$	−317.000	−0.650924	−0.325462	−	0.945555i	$-0.605520\pi$
$487$	−317.000	−0.650924	−0.325462	+	0.945555i	$0.605520\pi$
$488$	65.0538i	0.133307i
$489$	0	0
$490$	0	0
$491$	364.867i	0.743110i	0.928411	+	0.371555i	$0.121175\pi$
$491$	364.867i	0.743110i	−0.928411	+	0.371555i	$0.878825\pi$
$492$	0	0
$493$	−432.000	−0.876268
$494$	− 41.0122i	− 0.0830206i
$495$	0	0
$496$	−40.0000	−0.0806452
$497$	509.117i	1.02438i
$498$	0	0
$499$	710.000	1.42285	0.711423	−	0.702764i	$-0.248051\pi$
$499$	710.000	1.42285	0.711423	+	0.702764i	$0.248051\pi$
$500$	0	0
$501$	0	0
$502$	−504.000	−1.00398
$503$	280.014i	0.556688i	0.960481	+	0.278344i	$0.0897856\pi$
$503$	280.014i	0.556688i	−0.960481	+	0.278344i	$0.910214\pi$
$504$	0	0
$505$	0	0
$506$	101.823i	0.201232i
$507$	0	0
$508$	−356.000	−0.700787
$509$	263.044i	0.516785i	0.966040	+	0.258393i	$0.0831928\pi$
$509$	263.044i	0.516785i	−0.966040	+	0.258393i	$0.916807\pi$
$510$	0	0
$511$	−485.000	−0.949119
$512$	− 22.6274i	− 0.0441942i
$513$	0	0
$514$	−120.000	−0.233463
$515$	0	0
$516$	0	0
$517$	72.0000	0.139265
$518$	176.777i	0.341268i
$519$	0	0
$520$	0	0
$521$	− 687.308i	− 1.31921i	−0.751613	−	0.659604i	$-0.770724\pi$
$521$	− 687.308i	− 1.31921i	0.751613	−	0.659604i	$-0.229276\pi$
$522$	0	0
$523$	763.000	1.45889	0.729446	−	0.684039i	$-0.239778\pi$
$523$	763.000	1.45889	0.729446	+	0.684039i	$0.239778\pi$
$524$	237.588i	0.453412i
$525$	0	0
$526$	−240.000	−0.456274
$527$	− 254.558i	− 0.483033i
$528$	0	0
$529$	457.000	0.863894
$530$	0	0
$531$	0	0
$532$	290.000	0.545113
$533$	− 16.9706i	− 0.0318397i
$534$	0	0
$535$	0	0
$536$	53.7401i	0.100261i
$537$	0	0
$538$	396.000	0.736059
$539$	203.647i	0.377823i
$540$	0	0
$541$	−313.000	−0.578558	−0.289279	−	0.957245i	$-0.593416\pi$
$541$	−313.000	−0.578558	−0.289279	+	0.957245i	$0.593416\pi$
$542$	− 41.0122i	− 0.0756683i
$543$	0	0
$544$	144.000	0.264706
$545$	0	0
$546$	0	0
$547$	139.000	0.254113	0.127057	−	0.991895i	$-0.459447\pi$
$547$	139.000	0.254113	0.127057	+	0.991895i	$0.459447\pi$
$548$	186.676i	0.340650i
$549$	0	0
$550$	0	0
$551$	492.146i	0.893188i
$552$	0	0
$553$	−385.000	−0.696203
$554$	− 540.230i	− 0.975144i
$555$	0	0
$556$	326.000	0.586331
$557$	− 280.014i	− 0.502719i	−0.967894	−	0.251359i	$-0.919122\pi$
$557$	− 280.014i	− 0.502719i	0.967894	−	0.251359i	$-0.0808776\pi$
$558$	0	0
$559$	−14.0000	−0.0250447
$560$	0	0
$561$	0	0
$562$	312.000	0.555160
$563$	− 593.970i	− 1.05501i	−0.849552	−	0.527504i	$-0.823128\pi$
$563$	− 593.970i	− 1.05501i	0.849552	−	0.527504i	$-0.176872\pi$
$564$	0	0
$565$	0	0
$566$	87.6812i	0.154914i
$567$	0	0
$568$	288.000	0.507042
$569$	− 619.426i	− 1.08862i	−0.838884	−	0.544311i	$-0.816791\pi$
$569$	− 619.426i	− 1.08862i	0.838884	−	0.544311i	$-0.183209\pi$
$570$	0	0
$571$	−163.000	−0.285464	−0.142732	−	0.989761i	$-0.545589\pi$
$571$	−163.000	−0.285464	−0.142732	+	0.989761i	$0.545589\pi$
$572$	16.9706i	0.0296688i
$573$	0	0
$574$	120.000	0.209059
$575$	0	0
$576$	0	0
$577$	−1127.00	−1.95321	−0.976603	−	0.215050i	$-0.931009\pi$
$577$	−1127.00	−1.95321	−0.976603	+	0.215050i	$0.931009\pi$
$578$	507.703i	0.878378i
$579$	0	0
$580$	0	0
$581$	593.970i	1.02232i
$582$	0	0
$583$	−432.000	−0.740995
$584$	274.357i	0.469790i
$585$	0	0
$586$	−444.000	−0.757679
$587$	− 1009.75i	− 1.72018i	−0.510138	−	0.860092i	$-0.670406\pi$
$587$	− 1009.75i	− 1.72018i	0.510138	−	0.860092i	$-0.329594\pi$
$588$	0	0
$589$	−290.000	−0.492360
$590$	0	0
$591$	0	0
$592$	100.000	0.168919
$593$	356.382i	0.600981i	0.953785	+	0.300491i	$0.0971504\pi$
$593$	356.382i	0.600981i	−0.953785	+	0.300491i	$0.902850\pi$
$594$	0	0
$595$	0	0
$596$	− 67.8823i	− 0.113896i
$597$	0	0
$598$	−12.0000	−0.0200669
$599$	899.440i	1.50157i	0.660547	+	0.750784i	$0.270324\pi$
$599$	899.440i	1.50157i	−0.660547	+	0.750784i	$0.729676\pi$
$600$	0	0
$601$	−166.000	−0.276206	−0.138103	−	0.990418i	$-0.544101\pi$
$601$	−166.000	−0.276206	−0.138103	+	0.990418i	$0.544101\pi$
$602$	− 98.9949i	− 0.164443i
$603$	0	0
$604$	−298.000	−0.493377
$605$	0	0
$606$	0	0
$607$	523.000	0.861614	0.430807	−	0.902444i	$-0.358229\pi$
$607$	523.000	0.861614	0.430807	+	0.902444i	$0.358229\pi$
$608$	− 164.049i	− 0.269817i
$609$	0	0
$610$	0	0
$611$	8.48528i	0.0138875i
$612$	0	0
$613$	−335.000	−0.546493	−0.273246	−	0.961944i	$-0.588097\pi$
$613$	−335.000	−0.546493	−0.273246	+	0.961944i	$0.588097\pi$
$614$	− 48.0833i	− 0.0783115i
$615$	0	0
$616$	−120.000	−0.194805
$617$	500.632i	0.811396i	0.914007	+	0.405698i	$0.132972\pi$
$617$	500.632i	0.811396i	−0.914007	+	0.405698i	$0.867028\pi$
$618$	0	0
$619$	5.00000	0.00807754	0.00403877	−	0.999992i	$-0.498714\pi$
$619$	5.00000	0.00807754	0.00403877	+	0.999992i	$0.498714\pi$
$620$	0	0
$621$	0	0
$622$	84.0000	0.135048
$623$	− 381.838i	− 0.612902i
$624$	0	0
$625$	0	0
$626$	337.997i	0.539931i
$627$	0	0
$628$	484.000	0.770701
$629$	636.396i	1.01176i
$630$	0	0
$631$	245.000	0.388273	0.194136	−	0.980975i	$-0.437810\pi$
$631$	245.000	0.388273	0.194136	+	0.980975i	$0.437810\pi$
$632$	217.789i	0.344603i
$633$	0	0
$634$	−24.0000	−0.0378549
$635$	0	0
$636$	0	0
$637$	−24.0000	−0.0376766
$638$	− 203.647i	− 0.319196i
$639$	0	0
$640$	0	0
$641$	− 644.881i	− 1.00606i	−0.864270	−	0.503028i	$-0.832219\pi$
$641$	− 644.881i	− 1.00606i	0.864270	−	0.503028i	$-0.167781\pi$
$642$	0	0
$643$	82.0000	0.127527	0.0637636	−	0.997965i	$-0.479690\pi$
$643$	82.0000	0.127527	0.0637636	+	0.997965i	$0.479690\pi$
$644$	− 84.8528i	− 0.131759i
$645$	0	0
$646$	1044.00	1.61610
$647$	458.205i	0.708200i	0.935208	+	0.354100i	$0.115213\pi$
$647$	458.205i	0.708200i	−0.935208	+	0.354100i	$0.884787\pi$
$648$	0	0
$649$	−792.000	−1.22034
$650$	0	0
$651$	0	0
$652$	346.000	0.530675
$653$	322.441i	0.493784i	0.969043	+	0.246892i	$0.0794092\pi$
$653$	322.441i	0.493784i	−0.969043	+	0.246892i	$0.920591\pi$
$654$	0	0
$655$	0	0
$656$	− 67.8823i	− 0.103479i
$657$	0	0
$658$	−60.0000	−0.0911854
$659$	1035.20i	1.57087i	0.618943	+	0.785436i	$0.287561\pi$
$659$	1035.20i	1.57087i	−0.618943	+	0.785436i	$0.712439\pi$
$660$	0	0
$661$	143.000	0.216339	0.108169	−	0.994132i	$-0.465501\pi$
$661$	143.000	0.216339	0.108169	+	0.994132i	$0.465501\pi$
$662$	705.693i	1.06600i
$663$	0	0
$664$	336.000	0.506024
$665$	0	0
$666$	0	0
$667$	144.000	0.215892
$668$	− 390.323i	− 0.584316i
$669$	0	0
$670$	0	0
$671$	− 195.161i	− 0.290852i
$672$	0	0
$673$	−671.000	−0.997028	−0.498514	−	0.866882i	$-0.666121\pi$
$673$	−671.000	−0.997028	−0.498514	+	0.866882i	$0.666121\pi$
$674$	− 205.061i	− 0.304245i
$675$	0	0
$676$	336.000	0.497041
$677$	8.48528i	0.0125337i	0.999980	+	0.00626683i	$0.00199481\pi$
$677$	8.48528i	0.0125337i	−0.999980	+	0.00626683i	$0.998005\pi$
$678$	0	0
$679$	−245.000	−0.360825
$680$	0	0
$681$	0	0
$682$	120.000	0.175953
$683$	101.823i	0.149083i	0.997218	+	0.0745413i	$0.0237493\pi$
$683$	101.823i	0.149083i	−0.997218	+	0.0745413i	$0.976251\pi$
$684$	0	0
$685$	0	0
$686$	− 516.188i	− 0.752461i
$687$	0	0
$688$	−56.0000	−0.0813953
$689$	− 50.9117i	− 0.0738921i
$690$	0	0
$691$	−850.000	−1.23010	−0.615051	−	0.788488i	$-0.710864\pi$
$691$	−850.000	−1.23010	−0.615051	+	0.788488i	$0.710864\pi$
$692$	644.881i	0.931910i
$693$	0	0
$694$	−624.000	−0.899135
$695$	0	0
$696$	0	0
$697$	432.000	0.619799
$698$	171.120i	0.245157i
$699$	0	0
$700$	0	0
$701$	280.014i	0.399450i	0.979852	+	0.199725i	$0.0640048\pi$
$701$	280.014i	0.399450i	−0.979852	+	0.199725i	$0.935995\pi$
$702$	0	0
$703$	725.000	1.03129
$704$	67.8823i	0.0964237i
$705$	0	0
$706$	624.000	0.883853
$707$	678.823i	0.960145i
$708$	0	0
$709$	−1201.00	−1.69394	−0.846968	−	0.531645i	$-0.821574\pi$
$709$	−1201.00	−1.69394	−0.846968	+	0.531645i	$0.821574\pi$
$710$	0	0
$711$	0	0
$712$	−216.000	−0.303371
$713$	84.8528i	0.119008i
$714$	0	0
$715$	0	0
$716$	− 610.940i	− 0.853269i
$717$	0	0
$718$	612.000	0.852368
$719$	− 967.322i	− 1.34537i	−0.739928	−	0.672686i	$-0.765141\pi$
$719$	− 967.322i	− 1.34537i	0.739928	−	0.672686i	$-0.234859\pi$
$720$	0	0
$721$	−815.000	−1.13037
$722$	− 678.823i	− 0.940197i
$723$	0	0
$724$	−526.000	−0.726519
$725$	0	0
$726$	0	0
$727$	−950.000	−1.30674	−0.653370	−	0.757039i	$-0.726645\pi$
$727$	−950.000	−1.30674	−0.653370	+	0.757039i	$0.726645\pi$
$728$	− 14.1421i	− 0.0194260i
$729$	0	0
$730$	0	0
$731$	− 356.382i	− 0.487526i
$732$	0	0
$733$	−98.0000	−0.133697	−0.0668486	−	0.997763i	$-0.521294\pi$
$733$	−98.0000	−0.133697	−0.0668486	+	0.997763i	$0.521294\pi$
$734$	210.718i	0.287081i
$735$	0	0
$736$	−48.0000	−0.0652174
$737$	− 161.220i	− 0.218752i
$738$	0	0
$739$	1262.00	1.70771	0.853857	−	0.520508i	$-0.174258\pi$
$739$	1262.00	1.70771	0.853857	+	0.520508i	$0.174258\pi$
$740$	0	0
$741$	0	0
$742$	360.000	0.485175
$743$	− 772.161i	− 1.03925i	−0.854395	−	0.519624i	$-0.826072\pi$
$743$	− 772.161i	− 1.03925i	0.854395	−	0.519624i	$-0.173928\pi$
$744$	0	0
$745$	0	0
$746$	− 510.531i	− 0.684358i
$747$	0	0
$748$	−432.000	−0.577540
$749$	− 127.279i	− 0.169932i
$750$	0	0
$751$	197.000	0.262317	0.131158	−	0.991361i	$-0.458130\pi$
$751$	197.000	0.262317	0.131158	+	0.991361i	$0.458130\pi$
$752$	33.9411i	0.0451345i
$753$	0	0
$754$	24.0000	0.0318302
$755$	0	0
$756$	0	0
$757$	241.000	0.318362	0.159181	−	0.987249i	$-0.449115\pi$
$757$	241.000	0.318362	0.159181	+	0.987249i	$0.449115\pi$
$758$	− 244.659i	− 0.322769i
$759$	0	0
$760$	0	0
$761$	− 500.632i	− 0.657860i	−0.944354	−	0.328930i	$-0.893312\pi$
$761$	− 500.632i	− 0.657860i	0.944354	−	0.328930i	$-0.106688\pi$
$762$	0	0
$763$	−10.0000	−0.0131062
$764$	− 288.500i	− 0.377617i
$765$	0	0
$766$	960.000	1.25326
$767$	− 93.3381i	− 0.121692i
$768$	0	0
$769$	431.000	0.560468	0.280234	−	0.959932i	$-0.409588\pi$
$769$	431.000	0.560468	0.280234	+	0.959932i	$0.409588\pi$
$770$	0	0
$771$	0	0
$772$	142.000	0.183938
$773$	− 254.558i	− 0.329312i	−0.986351	−	0.164656i	$-0.947349\pi$
$773$	− 254.558i	− 0.329312i	0.986351	−	0.164656i	$-0.0526515\pi$
$774$	0	0
$775$	0	0
$776$	138.593i	0.178599i
$777$	0	0
$778$	−588.000	−0.755784
$779$	− 492.146i	− 0.631767i
$780$	0	0
$781$	−864.000	−1.10627
$782$	− 305.470i	− 0.390627i
$783$	0	0
$784$	−96.0000	−0.122449
$785$	0	0
$786$	0	0
$787$	−125.000	−0.158831	−0.0794155	−	0.996842i	$-0.525305\pi$
$787$	−125.000	−0.158831	−0.0794155	+	0.996842i	$0.525305\pi$
$788$	152.735i	0.193826i
$789$	0	0
$790$	0	0
$791$	551.543i	0.697273i
$792$	0	0
$793$	23.0000	0.0290038
$794$	− 404.465i	− 0.509402i
$795$	0	0
$796$	−346.000	−0.434673
$797$	− 848.528i	− 1.06465i	−0.846539	−	0.532326i	$-0.821318\pi$
$797$	− 848.528i	− 1.06465i	0.846539	−	0.532326i	$-0.178682\pi$
$798$	0	0
$799$	−216.000	−0.270338
$800$	0	0
$801$	0	0
$802$	480.000	0.598504
$803$	− 823.072i	− 1.02500i
$804$	0	0
$805$	0	0
$806$	14.1421i	0.0175461i
$807$	0	0
$808$	384.000	0.475248
$809$	− 1120.06i	− 1.38450i	−0.721660	−	0.692248i	$-0.756620\pi$
$809$	− 1120.06i	− 1.38450i	0.721660	−	0.692248i	$-0.243380\pi$
$810$	0	0
$811$	−322.000	−0.397041	−0.198520	−	0.980097i	$-0.563614\pi$
$811$	−322.000	−0.397041	−0.198520	+	0.980097i	$0.563614\pi$
$812$	169.706i	0.208997i
$813$	0	0
$814$	−300.000	−0.368550
$815$	0	0
$816$	0	0
$817$	−406.000	−0.496940
$818$	− 304.056i	− 0.371706i
$819$	0	0
$820$	0	0
$821$	− 873.984i	− 1.06454i	−0.846576	−	0.532268i	$-0.821340\pi$
$821$	− 873.984i	− 1.06454i	0.846576	−	0.532268i	$-0.178660\pi$
$822$	0	0
$823$	−269.000	−0.326853	−0.163426	−	0.986556i	$-0.552255\pi$
$823$	−269.000	−0.326853	−0.163426	+	0.986556i	$0.552255\pi$
$824$	461.034i	0.559507i
$825$	0	0
$826$	660.000	0.799031
$827$	25.4558i	0.0307809i	0.999882	+	0.0153905i	$0.00489913\pi$
$827$	25.4558i	0.0307809i	−0.999882	+	0.0153905i	$0.995101\pi$
$828$	0	0
$829$	−1105.00	−1.33293	−0.666466	−	0.745536i	$-0.732194\pi$
$829$	−1105.00	−1.33293	−0.666466	+	0.745536i	$0.732194\pi$
$830$	0	0
$831$	0	0
$832$	−8.00000	−0.00961538
$833$	− 610.940i	− 0.733422i
$834$	0	0
$835$	0	0
$836$	492.146i	0.588692i
$837$	0	0
$838$	−852.000	−1.01671
$839$	118.794i	0.141590i	0.997491	+	0.0707950i	$0.0225536\pi$
$839$	118.794i	0.141590i	−0.997491	+	0.0707950i	$0.977446\pi$
$840$	0	0
$841$	553.000	0.657551
$842$	− 643.467i	− 0.764213i
$843$	0	0
$844$	−682.000	−0.808057
$845$	0	0
$846$	0	0
$847$	−245.000	−0.289256
$848$	− 203.647i	− 0.240149i
$849$	0	0
$850$	0	0
$851$	− 212.132i	− 0.249274i
$852$	0	0
$853$	−1391.00	−1.63072	−0.815358	−	0.578958i	$-0.803460\pi$
$853$	−1391.00	−1.63072	−0.815358	+	0.578958i	$0.803460\pi$
$854$	162.635i	0.190439i
$855$	0	0
$856$	−72.0000	−0.0841121
$857$	543.058i	0.633673i	0.948480	+	0.316837i	$0.102621\pi$
$857$	543.058i	0.633673i	−0.948480	+	0.316837i	$0.897379\pi$
$858$	0	0
$859$	845.000	0.983702	0.491851	−	0.870679i	$-0.336320\pi$
$859$	845.000	0.983702	0.491851	+	0.870679i	$0.336320\pi$
$860$	0	0
$861$	0	0
$862$	468.000	0.542923
$863$	− 1247.34i	− 1.44535i	−0.691188	−	0.722675i	$-0.742913\pi$
$863$	− 1247.34i	− 1.44535i	0.691188	−	0.722675i	$-0.257087\pi$
$864$	0	0
$865$	0	0
$866$	308.299i	0.356003i
$867$	0	0
$868$	−100.000	−0.115207
$869$	− 653.367i	− 0.751860i
$870$	0	0
$871$	19.0000	0.0218140
$872$	5.65685i	0.00648722i
$873$	0	0
$874$	−348.000	−0.398169
$875$	0	0
$876$	0	0
$877$	−1151.00	−1.31243	−0.656214	−	0.754575i	$-0.727843\pi$
$877$	−1151.00	−1.31243	−0.656214	+	0.754575i	$0.727843\pi$
$878$	− 528.916i	− 0.602410i
$879$	0	0
$880$	0	0
$881$	− 1298.25i	− 1.47361i	−0.676107	−	0.736804i	$-0.736334\pi$
$881$	− 1298.25i	− 1.47361i	0.676107	−	0.736804i	$-0.263666\pi$
$882$	0	0
$883$	−677.000	−0.766704	−0.383352	−	0.923602i	$-0.625230\pi$
$883$	−677.000	−0.766704	−0.383352	+	0.923602i	$0.625230\pi$
$884$	− 50.9117i	− 0.0575924i
$885$	0	0
$886$	−24.0000	−0.0270880
$887$	1035.20i	1.16708i	0.812083	+	0.583542i	$0.198334\pi$
$887$	1035.20i	1.16708i	−0.812083	+	0.583542i	$0.801666\pi$
$888$	0	0
$889$	−890.000	−1.00112
$890$	0	0
$891$	0	0
$892$	−116.000	−0.130045
$893$	246.073i	0.275558i
$894$	0	0
$895$	0	0
$896$	− 56.5685i	− 0.0631345i
$897$	0	0
$898$	684.000	0.761693
$899$	− 169.706i	− 0.188772i
$900$	0	0
$901$	1296.00	1.43840
$902$	203.647i	0.225772i
$903$	0	0
$904$	312.000	0.345133
$905$	0	0
$906$	0	0
$907$	−5.00000	−0.00551268	−0.00275634	−	0.999996i	$-0.500877\pi$
$907$	−5.00000	−0.00551268	−0.00275634	+	0.999996i	$0.500877\pi$
$908$	− 271.529i	− 0.299041i
$909$	0	0
$910$	0	0
$911$	− 1001.26i	− 1.09908i	−0.835467	−	0.549541i	$-0.814803\pi$
$911$	− 1001.26i	− 1.09908i	0.835467	−	0.549541i	$-0.185197\pi$
$912$	0	0
$913$	−1008.00	−1.10405
$914$	− 31.1127i	− 0.0340402i
$915$	0	0
$916$	524.000	0.572052
$917$	593.970i	0.647731i
$918$	0	0
$919$	1190.00	1.29489	0.647443	−	0.762114i	$-0.275838\pi$
$919$	1190.00	1.29489	0.647443	+	0.762114i	$0.275838\pi$
$920$	0	0
$921$	0	0
$922$	564.000	0.611714
$923$	− 101.823i	− 0.110318i
$924$	0	0
$925$	0	0
$926$	− 1248.75i	− 1.34854i
$927$	0	0
$928$	96.0000	0.103448
$929$	475.176i	0.511492i	0.966744	+	0.255746i	$0.0823210\pi$
$929$	475.176i	0.511492i	−0.966744	+	0.255746i	$0.917679\pi$
$930$	0	0
$931$	−696.000	−0.747583
$932$	− 610.940i	− 0.655515i
$933$	0	0
$934$	180.000	0.192719
$935$	0	0
$936$	0	0
$937$	−1775.00	−1.89434	−0.947172	−	0.320727i	$-0.896073\pi$
$937$	−1775.00	−1.89434	−0.947172	+	0.320727i	$0.896073\pi$
$938$	134.350i	0.143231i
$939$	0	0
$940$	0	0
$941$	161.220i	0.171329i	0.996324	+	0.0856644i	$0.0273013\pi$
$941$	161.220i	0.171329i	−0.996324	+	0.0856644i	$0.972699\pi$
$942$	0	0
$943$	−144.000	−0.152704
$944$	− 373.352i	− 0.395500i
$945$	0	0
$946$	168.000	0.177590
$947$	568.514i	0.600331i	0.953887	+	0.300166i	$0.0970420\pi$
$947$	568.514i	0.600331i	−0.953887	+	0.300166i	$0.902958\pi$
$948$	0	0
$949$	97.0000	0.102213
$950$	0	0
$951$	0	0
$952$	360.000	0.378151
$953$	− 1807.36i	− 1.89650i	−0.317524	−	0.948250i	$-0.602851\pi$
$953$	− 1807.36i	− 1.89650i	0.317524	−	0.948250i	$-0.397149\pi$
$954$	0	0
$955$	0	0
$956$	339.411i	0.355033i
$957$	0	0
$958$	−552.000	−0.576200
$959$	466.690i	0.486643i
$960$	0	0
$961$	−861.000	−0.895942
$962$	− 35.3553i	− 0.0367519i
$963$	0	0
$964$	242.000	0.251037
$965$	0	0
$966$	0	0
$967$	−701.000	−0.724922	−0.362461	−	0.931999i	$-0.618063\pi$
$967$	−701.000	−0.724922	−0.362461	+	0.931999i	$0.618063\pi$
$968$	138.593i	0.143175i
$969$	0	0
$970$	0	0
$971$	− 381.838i	− 0.393242i	−0.980480	−	0.196621i	$-0.937003\pi$
$971$	− 381.838i	− 0.393242i	0.980480	−	0.196621i	$-0.0629968\pi$
$972$	0	0
$973$	815.000	0.837616
$974$	448.306i	0.460273i
$975$	0	0
$976$	92.0000	0.0942623
$977$	322.441i	0.330031i	0.986291	+	0.165016i	$0.0527675\pi$
$977$	322.441i	0.330031i	−0.986291	+	0.165016i	$0.947233\pi$
$978$	0	0
$979$	648.000	0.661900
$980$	0	0
$981$	0	0
$982$	516.000	0.525458
$983$	− 42.4264i	− 0.0431601i	−0.999767	−	0.0215801i	$-0.993130\pi$
$983$	− 42.4264i	− 0.0431601i	0.999767	−	0.0215801i	$-0.00686968\pi$
$984$	0	0
$985$	0	0
$986$	610.940i	0.619615i
$987$	0	0
$988$	−58.0000	−0.0587045
$989$	118.794i	0.120115i
$990$	0	0
$991$	−475.000	−0.479314	−0.239657	−	0.970858i	$-0.577035\pi$
$991$	−475.000	−0.479314	−0.239657	+	0.970858i	$0.577035\pi$
$992$	56.5685i	0.0570247i
$993$	0	0
$994$	720.000	0.724346
$995$	0	0
$996$	0	0
$997$	1534.00	1.53862	0.769308	−	0.638878i	$-0.220601\pi$
$997$	1534.00	1.53862	0.769308	+	0.638878i	$0.220601\pi$
$998$	− 1004.09i	− 1.00610i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1350.3.d.d.701.1		2
3.2	odd	2	inner	1350.3.d.d.701.2		2
5.2	odd	4		1350.3.b.b.1349.3		4
5.3	odd	4		1350.3.b.b.1349.2		4
5.4	even	2		54.3.b.a.53.2	yes	2
15.2	even	4		1350.3.b.b.1349.1		4
15.8	even	4		1350.3.b.b.1349.4		4
15.14	odd	2		54.3.b.a.53.1	✓	2
20.19	odd	2		432.3.e.d.161.2		2
40.19	odd	2		1728.3.e.f.1025.1		2
40.29	even	2		1728.3.e.l.1025.1		2
45.4	even	6		162.3.d.a.107.1		4
45.14	odd	6		162.3.d.a.107.2		4
45.29	odd	6		162.3.d.a.53.1		4
45.34	even	6		162.3.d.a.53.2		4
60.59	even	2		432.3.e.d.161.1		2
120.29	odd	2		1728.3.e.l.1025.2		2
120.59	even	2		1728.3.e.f.1025.2		2
180.59	even	6		1296.3.q.i.593.1		4
180.79	odd	6		1296.3.q.i.1025.1		4
180.119	even	6		1296.3.q.i.1025.2		4
180.139	odd	6		1296.3.q.i.593.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
54.3.b.a.53.1	✓	2	15.14	odd	2
54.3.b.a.53.2	yes	2	5.4	even	2
162.3.d.a.53.1		4	45.29	odd	6
162.3.d.a.53.2		4	45.34	even	6
162.3.d.a.107.1		4	45.4	even	6
162.3.d.a.107.2		4	45.14	odd	6
432.3.e.d.161.1		2	60.59	even	2
432.3.e.d.161.2		2	20.19	odd	2
1296.3.q.i.593.1		4	180.59	even	6
1296.3.q.i.593.2		4	180.139	odd	6
1296.3.q.i.1025.1		4	180.79	odd	6
1296.3.q.i.1025.2		4	180.119	even	6
1350.3.b.b.1349.1		4	15.2	even	4
1350.3.b.b.1349.2		4	5.3	odd	4
1350.3.b.b.1349.3		4	5.2	odd	4
1350.3.b.b.1349.4		4	15.8	even	4
1350.3.d.d.701.1		2	1.1	even	1	trivial
1350.3.d.d.701.2		2	3.2	odd	2	inner
1728.3.e.f.1025.1		2	40.19	odd	2
1728.3.e.f.1025.2		2	120.59	even	2
1728.3.e.l.1025.1		2	40.29	even	2
1728.3.e.l.1025.2		2	120.29	odd	2

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:	\( N \)	\(=\)	\( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1350.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} -2.00000 q^{4} -5.00000 q^{7} +2.82843i q^{8} +O(q^{10})\)	\(q-1.41421i q^{2} -2.00000 q^{4} -5.00000 q^{7} +2.82843i q^{8} -8.48528i q^{11} +1.00000 q^{13} +7.07107i q^{14} +4.00000 q^{16} +25.4558i q^{17} +29.0000 q^{19} -12.0000 q^{22} -8.48528i q^{23} -1.41421i q^{26} +10.0000 q^{28} +16.9706i q^{29} -10.0000 q^{31} -5.65685i q^{32} +36.0000 q^{34} +25.0000 q^{37} -41.0122i q^{38} -16.9706i q^{41} -14.0000 q^{43} +16.9706i q^{44} -12.0000 q^{46} +8.48528i q^{47} -24.0000 q^{49} -2.00000 q^{52} -50.9117i q^{53} -14.1421i q^{56} +24.0000 q^{58} -93.3381i q^{59} +23.0000 q^{61} +14.1421i q^{62} -8.00000 q^{64} +19.0000 q^{67} -50.9117i q^{68} -101.823i q^{71} +97.0000 q^{73} -35.3553i q^{74} -58.0000 q^{76} +42.4264i q^{77} +77.0000 q^{79} -24.0000 q^{82} -118.794i q^{83} +19.7990i q^{86} +24.0000 q^{88} +76.3675i q^{89} -5.00000 q^{91} +16.9706i q^{92} +12.0000 q^{94} +49.0000 q^{97} +33.9411i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{4} - 10 q^{7}+O(q^{10}) \) 2 * q - 4 * q^4 - 10 * q^7	\( 2 q - 4 q^{4} - 10 q^{7} + 2 q^{13} + 8 q^{16} + 58 q^{19} - 24 q^{22} + 20 q^{28} - 20 q^{31} + 72 q^{34} + 50 q^{37} - 28 q^{43} - 24 q^{46} - 48 q^{49} - 4 q^{52} + 48 q^{58} + 46 q^{61} - 16 q^{64} + 38 q^{67} + 194 q^{73} - 116 q^{76} + 154 q^{79} - 48 q^{82} + 48 q^{88} - 10 q^{91} + 24 q^{94} + 98 q^{97}+O(q^{100}) \) 2 * q - 4 * q^4 - 10 * q^7 + 2 * q^13 + 8 * q^16 + 58 * q^19 - 24 * q^22 + 20 * q^28 - 20 * q^31 + 72 * q^34 + 50 * q^37 - 28 * q^43 - 24 * q^46 - 48 * q^49 - 4 * q^52 + 48 * q^58 + 46 * q^61 - 16 * q^64 + 38 * q^67 + 194 * q^73 - 116 * q^76 + 154 * q^79 - 48 * q^82 + 48 * q^88 - 10 * q^91 + 24 * q^94 + 98 * q^97

Introduction

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