LMFDB - Embedded newform 3150.2.g.s.2899.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3150,2,Mod(2899,3150)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3150, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3150.2899");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3150.g (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$25.1528766367$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 630)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2899.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	3150.2899
Dual form			3150.2.g.s.2899.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} +1.00000i q^{8} +O(q^{10})$	$q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} +1.00000i q^{8} +4.00000 q^{11} +6.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} +4.00000i q^{17} -6.00000 q^{19} -4.00000i q^{22} +6.00000 q^{26} -1.00000i q^{28} -6.00000 q^{29} -4.00000 q^{31} -1.00000i q^{32} +4.00000 q^{34} -8.00000i q^{37} +6.00000i q^{38} -10.0000 q^{41} -2.00000i q^{43} -4.00000 q^{44} +10.0000i q^{47} -1.00000 q^{49} -6.00000i q^{52} -14.0000i q^{53} -1.00000 q^{56} +6.00000i q^{58} -4.00000 q^{59} -8.00000 q^{61} +4.00000i q^{62} -1.00000 q^{64} -6.00000i q^{67} -4.00000i q^{68} +2.00000 q^{71} -10.0000i q^{73} -8.00000 q^{74} +6.00000 q^{76} +4.00000i q^{77} -16.0000 q^{79} +10.0000i q^{82} +8.00000i q^{83} -2.00000 q^{86} +4.00000i q^{88} +2.00000 q^{89} -6.00000 q^{91} +10.0000 q^{94} -2.00000i q^{97} +1.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4}+O(q^{10}) $ 2 * q - 2 * q^4	$ 2 q - 2 q^{4} + 8 q^{11} + 2 q^{14} + 2 q^{16} - 12 q^{19} + 12 q^{26} - 12 q^{29} - 8 q^{31} + 8 q^{34} - 20 q^{41} - 8 q^{44} - 2 q^{49} - 2 q^{56} - 8 q^{59} - 16 q^{61} - 2 q^{64} + 4 q^{71} - 16 q^{74} + 12 q^{76} - 32 q^{79} - 4 q^{86} + 4 q^{89} - 12 q^{91} + 20 q^{94}+O(q^{100}) $ 2 * q - 2 * q^4 + 8 * q^11 + 2 * q^14 + 2 * q^16 - 12 * q^19 + 12 * q^26 - 12 * q^29 - 8 * q^31 + 8 * q^34 - 20 * q^41 - 8 * q^44 - 2 * q^49 - 2 * q^56 - 8 * q^59 - 16 * q^61 - 2 * q^64 + 4 * q^71 - 16 * q^74 + 12 * q^76 - 32 * q^79 - 4 * q^86 + 4 * q^89 - 12 * q^91 + 20 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times$.

$n$	$127$	$451$	$2801$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$	− 1.00000i	− 0.707107i
$2$	− 1.00000i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.00000i	0.377964i
$7$	1.00000i	0.377964i
$8$	1.00000i	0.353553i
$9$	0	0
$10$	0	0
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	0	0
$13$	6.00000i	1.66410i	0.554700	+	0.832050i	$0.312833\pi$
$13$	6.00000i	1.66410i	−0.554700	+	0.832050i	$0.687167\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	4.00000i	0.970143i	0.874475	+	0.485071i	$0.161206\pi$
$17$	4.00000i	0.970143i	−0.874475	+	0.485071i	$0.838794\pi$
$18$	0	0
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	0	0
$21$	0	0
$22$	− 4.00000i	− 0.852803i
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	0	0
$26$	6.00000	1.17670
$27$	0	0
$28$	− 1.00000i	− 0.188982i
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	− 1.00000i	− 0.176777i
$33$	0	0
$34$	4.00000	0.685994
$35$	0	0
$36$	0	0
$37$	− 8.00000i	− 1.31519i	−0.753371	−	0.657596i	$-0.771573\pi$
$37$	− 8.00000i	− 1.31519i	0.753371	−	0.657596i	$-0.228427\pi$
$38$	6.00000i	0.973329i
$39$	0	0
$40$	0	0
$41$	−10.0000	−1.56174	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000	−1.56174	−0.780869	+	0.624695i	$0.785223\pi$
$42$	0	0
$43$	− 2.00000i	− 0.304997i	−0.988304	−	0.152499i	$-0.951268\pi$
$43$	− 2.00000i	− 0.304997i	0.988304	−	0.152499i	$-0.0487319\pi$
$44$	−4.00000	−0.603023
$45$	0	0
$46$	0	0
$47$	10.0000i	1.45865i	0.684167	+	0.729325i	$0.260166\pi$
$47$	10.0000i	1.45865i	−0.684167	+	0.729325i	$0.739834\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	− 6.00000i	− 0.832050i
$53$	− 14.0000i	− 1.92305i	−0.274721	−	0.961524i	$-0.588586\pi$
$53$	− 14.0000i	− 1.92305i	0.274721	−	0.961524i	$-0.411414\pi$
$54$	0	0
$55$	0	0
$56$	−1.00000	−0.133631
$57$	0	0
$58$	6.00000i	0.787839i
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	−8.00000	−1.02430	−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000	−1.02430	−0.512148	+	0.858898i	$0.671150\pi$
$62$	4.00000i	0.508001i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	− 6.00000i	− 0.733017i	−0.930415	−	0.366508i	$-0.880553\pi$
$67$	− 6.00000i	− 0.733017i	0.930415	−	0.366508i	$-0.119447\pi$
$68$	− 4.00000i	− 0.485071i
$69$	0	0
$70$	0	0
$71$	2.00000	0.237356	0.118678	−	0.992933i	$-0.462134\pi$
$71$	2.00000	0.237356	0.118678	+	0.992933i	$0.462134\pi$
$72$	0	0
$73$	− 10.0000i	− 1.17041i	−0.810885	−	0.585206i	$-0.801014\pi$
$73$	− 10.0000i	− 1.17041i	0.810885	−	0.585206i	$-0.198986\pi$
$74$	−8.00000	−0.929981
$75$	0	0
$76$	6.00000	0.688247
$77$	4.00000i	0.455842i
$78$	0	0
$79$	−16.0000	−1.80014	−0.900070	−	0.435745i	$-0.856485\pi$
$79$	−16.0000	−1.80014	−0.900070	+	0.435745i	$0.856485\pi$
$80$	0	0
$81$	0	0
$82$	10.0000i	1.10432i
$83$	8.00000i	0.878114i	0.898459	+	0.439057i	$0.144687\pi$
$83$	8.00000i	0.878114i	−0.898459	+	0.439057i	$0.855313\pi$
$84$	0	0
$85$	0	0
$86$	−2.00000	−0.215666
$87$	0	0
$88$	4.00000i	0.426401i
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	0	0
$91$	−6.00000	−0.628971
$92$	0	0
$93$	0	0
$94$	10.0000	1.03142
$95$	0	0
$96$	0	0
$97$	− 2.00000i	− 0.203069i	−0.994832	−	0.101535i	$-0.967625\pi$
$97$	− 2.00000i	− 0.203069i	0.994832	−	0.101535i	$-0.0323753\pi$
$98$	1.00000i	0.101015i
$99$	0	0
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	− 8.00000i	− 0.788263i	−0.919054	−	0.394132i	$-0.871045\pi$
$103$	− 8.00000i	− 0.788263i	0.919054	−	0.394132i	$-0.128955\pi$
$104$	−6.00000	−0.588348
$105$	0	0
$106$	−14.0000	−1.35980
$107$	12.0000i	1.16008i	0.814587	+	0.580042i	$0.196964\pi$
$107$	12.0000i	1.16008i	−0.814587	+	0.580042i	$0.803036\pi$
$108$	0	0
$109$	18.0000	1.72409	0.862044	−	0.506834i	$-0.169184\pi$
$109$	18.0000	1.72409	0.862044	+	0.506834i	$0.169184\pi$
$110$	0	0
$111$	0	0
$112$	1.00000i	0.0944911i
$113$	14.0000i	1.31701i	0.752577	+	0.658505i	$0.228811\pi$
$113$	14.0000i	1.31701i	−0.752577	+	0.658505i	$0.771189\pi$
$114$	0	0
$115$	0	0
$116$	6.00000	0.557086
$117$	0	0
$118$	4.00000i	0.368230i
$119$	−4.00000	−0.366679
$120$	0	0
$121$	5.00000	0.454545
$122$	8.00000i	0.724286i
$123$	0	0
$124$	4.00000	0.359211
$125$	0	0
$126$	0	0
$127$	8.00000i	0.709885i	0.934888	+	0.354943i	$0.115500\pi$
$127$	8.00000i	0.709885i	−0.934888	+	0.354943i	$0.884500\pi$
$128$	1.00000i	0.0883883i
$129$	0	0
$130$	0	0
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	0	0
$133$	− 6.00000i	− 0.520266i
$134$	−6.00000	−0.518321
$135$	0	0
$136$	−4.00000	−0.342997
$137$	6.00000i	0.512615i	0.966595	+	0.256307i	$0.0825059\pi$
$137$	6.00000i	0.512615i	−0.966595	+	0.256307i	$0.917494\pi$
$138$	0	0
$139$	−10.0000	−0.848189	−0.424094	−	0.905618i	$-0.639408\pi$
$139$	−10.0000	−0.848189	−0.424094	+	0.905618i	$0.639408\pi$
$140$	0	0
$141$	0	0
$142$	− 2.00000i	− 0.167836i
$143$	24.0000i	2.00698i
$144$	0	0
$145$	0	0
$146$	−10.0000	−0.827606
$147$	0	0
$148$	8.00000i	0.657596i
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	− 6.00000i	− 0.486664i
$153$	0	0
$154$	4.00000	0.322329
$155$	0	0
$156$	0	0
$157$	14.0000i	1.11732i	0.829396	+	0.558661i	$0.188685\pi$
$157$	14.0000i	1.11732i	−0.829396	+	0.558661i	$0.811315\pi$
$158$	16.0000i	1.27289i
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 10.0000i	− 0.783260i	−0.920123	−	0.391630i	$-0.871911\pi$
$163$	− 10.0000i	− 0.783260i	0.920123	−	0.391630i	$-0.128089\pi$
$164$	10.0000	0.780869
$165$	0	0
$166$	8.00000	0.620920
$167$	6.00000i	0.464294i	0.972681	+	0.232147i	$0.0745750\pi$
$167$	6.00000i	0.464294i	−0.972681	+	0.232147i	$0.925425\pi$
$168$	0	0
$169$	−23.0000	−1.76923
$170$	0	0
$171$	0	0
$172$	2.00000i	0.152499i
$173$	2.00000i	0.152057i	0.997106	+	0.0760286i	$0.0242240\pi$
$173$	2.00000i	0.152057i	−0.997106	+	0.0760286i	$0.975776\pi$
$174$	0	0
$175$	0	0
$176$	4.00000	0.301511
$177$	0	0
$178$	− 2.00000i	− 0.149906i
$179$	−24.0000	−1.79384	−0.896922	−	0.442189i	$-0.854202\pi$
$179$	−24.0000	−1.79384	−0.896922	+	0.442189i	$0.854202\pi$
$180$	0	0
$181$	−8.00000	−0.594635	−0.297318	−	0.954779i	$-0.596092\pi$
$181$	−8.00000	−0.594635	−0.297318	+	0.954779i	$0.596092\pi$
$182$	6.00000i	0.444750i
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	16.0000i	1.17004i
$188$	− 10.0000i	− 0.729325i
$189$	0	0
$190$	0	0
$191$	18.0000	1.30243	0.651217	−	0.758891i	$-0.274259\pi$
$191$	18.0000	1.30243	0.651217	+	0.758891i	$0.274259\pi$
$192$	0	0
$193$	− 10.0000i	− 0.719816i	−0.932988	−	0.359908i	$-0.882808\pi$
$193$	− 10.0000i	− 0.719816i	0.932988	−	0.359908i	$-0.117192\pi$
$194$	−2.00000	−0.143592
$195$	0	0
$196$	1.00000	0.0714286
$197$	22.0000i	1.56744i	0.621117	+	0.783718i	$0.286679\pi$
$197$	22.0000i	1.56744i	−0.621117	+	0.783718i	$0.713321\pi$
$198$	0	0
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	0	0
$201$	0	0
$202$	− 6.00000i	− 0.422159i
$203$	− 6.00000i	− 0.421117i
$204$	0	0
$205$	0	0
$206$	−8.00000	−0.557386
$207$	0	0
$208$	6.00000i	0.416025i
$209$	−24.0000	−1.66011
$210$	0	0
$211$	16.0000	1.10149	0.550743	−	0.834675i	$-0.314345\pi$
$211$	16.0000	1.10149	0.550743	+	0.834675i	$0.314345\pi$
$212$	14.0000i	0.961524i
$213$	0	0
$214$	12.0000	0.820303
$215$	0	0
$216$	0	0
$217$	− 4.00000i	− 0.271538i
$218$	− 18.0000i	− 1.21911i
$219$	0	0
$220$	0	0
$221$	−24.0000	−1.61441
$222$	0	0
$223$	16.0000i	1.07144i	0.844396	+	0.535720i	$0.179960\pi$
$223$	16.0000i	1.07144i	−0.844396	+	0.535720i	$0.820040\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	14.0000	0.931266
$227$	− 20.0000i	− 1.32745i	−0.747978	−	0.663723i	$-0.768975\pi$
$227$	− 20.0000i	− 1.32745i	0.747978	−	0.663723i	$-0.231025\pi$
$228$	0	0
$229$	−4.00000	−0.264327	−0.132164	−	0.991228i	$-0.542192\pi$
$229$	−4.00000	−0.264327	−0.132164	+	0.991228i	$0.542192\pi$
$230$	0	0
$231$	0	0
$232$	− 6.00000i	− 0.393919i
$233$	6.00000i	0.393073i	0.980497	+	0.196537i	$0.0629694\pi$
$233$	6.00000i	0.393073i	−0.980497	+	0.196537i	$0.937031\pi$
$234$	0	0
$235$	0	0
$236$	4.00000	0.260378
$237$	0	0
$238$	4.00000i	0.259281i
$239$	−22.0000	−1.42306	−0.711531	−	0.702655i	$-0.751998\pi$
$239$	−22.0000	−1.42306	−0.711531	+	0.702655i	$0.751998\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	− 5.00000i	− 0.321412i
$243$	0	0
$244$	8.00000	0.512148
$245$	0	0
$246$	0	0
$247$	− 36.0000i	− 2.29063i
$248$	− 4.00000i	− 0.254000i
$249$	0	0
$250$	0	0
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	0	0
$253$	0	0
$254$	8.00000	0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	16.0000i	0.998053i	0.866587	+	0.499026i	$0.166309\pi$
$257$	16.0000i	0.998053i	−0.866587	+	0.499026i	$0.833691\pi$
$258$	0	0
$259$	8.00000	0.497096
$260$	0	0
$261$	0	0
$262$	4.00000i	0.247121i
$263$	− 4.00000i	− 0.246651i	−0.992366	−	0.123325i	$-0.960644\pi$
$263$	− 4.00000i	− 0.246651i	0.992366	−	0.123325i	$-0.0393559\pi$
$264$	0	0
$265$	0	0
$266$	−6.00000	−0.367884
$267$	0	0
$268$	6.00000i	0.366508i
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	0	0
$271$	−12.0000	−0.728948	−0.364474	−	0.931214i	$-0.618751\pi$
$271$	−12.0000	−0.728948	−0.364474	+	0.931214i	$0.618751\pi$
$272$	4.00000i	0.242536i
$273$	0	0
$274$	6.00000	0.362473
$275$	0	0
$276$	0	0
$277$	12.0000i	0.721010i	0.932757	+	0.360505i	$0.117396\pi$
$277$	12.0000i	0.721010i	−0.932757	+	0.360505i	$0.882604\pi$
$278$	10.0000i	0.599760i
$279$	0	0
$280$	0	0
$281$	8.00000	0.477240	0.238620	−	0.971113i	$-0.423305\pi$
$281$	8.00000	0.477240	0.238620	+	0.971113i	$0.423305\pi$
$282$	0	0
$283$	− 4.00000i	− 0.237775i	−0.992908	−	0.118888i	$-0.962067\pi$
$283$	− 4.00000i	− 0.237775i	0.992908	−	0.118888i	$-0.0379328\pi$
$284$	−2.00000	−0.118678
$285$	0	0
$286$	24.0000	1.41915
$287$	− 10.0000i	− 0.590281i
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	0	0
$292$	10.0000i	0.585206i
$293$	− 2.00000i	− 0.116841i	−0.998292	−	0.0584206i	$-0.981394\pi$
$293$	− 2.00000i	− 0.116841i	0.998292	−	0.0584206i	$-0.0186065\pi$
$294$	0	0
$295$	0	0
$296$	8.00000	0.464991
$297$	0	0
$298$	− 6.00000i	− 0.347571i
$299$	0	0
$300$	0	0
$301$	2.00000	0.115278
$302$	16.0000i	0.920697i
$303$	0	0
$304$	−6.00000	−0.344124
$305$	0	0
$306$	0	0
$307$	− 4.00000i	− 0.228292i	−0.993464	−	0.114146i	$-0.963587\pi$
$307$	− 4.00000i	− 0.228292i	0.993464	−	0.114146i	$-0.0364132\pi$
$308$	− 4.00000i	− 0.227921i
$309$	0	0
$310$	0	0
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	0	0
$313$	− 10.0000i	− 0.565233i	−0.959233	−	0.282617i	$-0.908798\pi$
$313$	− 10.0000i	− 0.565233i	0.959233	−	0.282617i	$-0.0912024\pi$
$314$	14.0000	0.790066
$315$	0	0
$316$	16.0000	0.900070
$317$	2.00000i	0.112331i	0.998421	+	0.0561656i	$0.0178875\pi$
$317$	2.00000i	0.112331i	−0.998421	+	0.0561656i	$0.982113\pi$
$318$	0	0
$319$	−24.0000	−1.34374
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 24.0000i	− 1.33540i
$324$	0	0
$325$	0	0
$326$	−10.0000	−0.553849
$327$	0	0
$328$	− 10.0000i	− 0.552158i
$329$	−10.0000	−0.551318
$330$	0	0
$331$	16.0000	0.879440	0.439720	−	0.898135i	$-0.355078\pi$
$331$	16.0000	0.879440	0.439720	+	0.898135i	$0.355078\pi$
$332$	− 8.00000i	− 0.439057i
$333$	0	0
$334$	6.00000	0.328305
$335$	0	0
$336$	0	0
$337$	26.0000i	1.41631i	0.706057	+	0.708155i	$0.250472\pi$
$337$	26.0000i	1.41631i	−0.706057	+	0.708155i	$0.749528\pi$
$338$	23.0000i	1.25104i
$339$	0	0
$340$	0	0
$341$	−16.0000	−0.866449
$342$	0	0
$343$	− 1.00000i	− 0.0539949i
$344$	2.00000	0.107833
$345$	0	0
$346$	2.00000	0.107521
$347$	20.0000i	1.07366i	0.843692	+	0.536828i	$0.180378\pi$
$347$	20.0000i	1.07366i	−0.843692	+	0.536828i	$0.819622\pi$
$348$	0	0
$349$	−28.0000	−1.49881	−0.749403	−	0.662114i	$-0.769659\pi$
$349$	−28.0000	−1.49881	−0.749403	+	0.662114i	$0.769659\pi$
$350$	0	0
$351$	0	0
$352$	− 4.00000i	− 0.213201i
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	0	0
$356$	−2.00000	−0.106000
$357$	0	0
$358$	24.0000i	1.26844i
$359$	34.0000	1.79445	0.897226	−	0.441572i	$-0.145579\pi$
$359$	34.0000	1.79445	0.897226	+	0.441572i	$0.145579\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	8.00000i	0.420471i
$363$	0	0
$364$	6.00000	0.314485
$365$	0	0
$366$	0	0
$367$	− 16.0000i	− 0.835193i	−0.908633	−	0.417597i	$-0.862873\pi$
$367$	− 16.0000i	− 0.835193i	0.908633	−	0.417597i	$-0.137127\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	14.0000	0.726844
$372$	0	0
$373$	32.0000i	1.65690i	0.560065	+	0.828449i	$0.310776\pi$
$373$	32.0000i	1.65690i	−0.560065	+	0.828449i	$0.689224\pi$
$374$	16.0000	0.827340
$375$	0	0
$376$	−10.0000	−0.515711
$377$	− 36.0000i	− 1.85409i
$378$	0	0
$379$	0	0		−	1.00000i	$-0.5\pi$
$379$	0	0			1.00000i	$0.5\pi$
$380$	0	0
$381$	0	0
$382$	− 18.0000i	− 0.920960i
$383$	30.0000i	1.53293i	0.642287	+	0.766464i	$0.277986\pi$
$383$	30.0000i	1.53293i	−0.642287	+	0.766464i	$0.722014\pi$
$384$	0	0
$385$	0	0
$386$	−10.0000	−0.508987
$387$	0	0
$388$	2.00000i	0.101535i
$389$	22.0000	1.11544	0.557722	−	0.830028i	$-0.311675\pi$
$389$	22.0000	1.11544	0.557722	+	0.830028i	$0.311675\pi$
$390$	0	0
$391$	0	0
$392$	− 1.00000i	− 0.0505076i
$393$	0	0
$394$	22.0000	1.10834
$395$	0	0
$396$	0	0
$397$	26.0000i	1.30490i	0.757831	+	0.652451i	$0.226259\pi$
$397$	26.0000i	1.30490i	−0.757831	+	0.652451i	$0.773741\pi$
$398$	− 16.0000i	− 0.802008i
$399$	0	0
$400$	0	0
$401$	16.0000	0.799002	0.399501	−	0.916733i	$-0.369183\pi$
$401$	16.0000	0.799002	0.399501	+	0.916733i	$0.369183\pi$
$402$	0	0
$403$	− 24.0000i	− 1.19553i
$404$	−6.00000	−0.298511
$405$	0	0
$406$	−6.00000	−0.297775
$407$	− 32.0000i	− 1.58618i
$408$	0	0
$409$	−18.0000	−0.890043	−0.445021	−	0.895520i	$-0.646804\pi$
$409$	−18.0000	−0.890043	−0.445021	+	0.895520i	$0.646804\pi$
$410$	0	0
$411$	0	0
$412$	8.00000i	0.394132i
$413$	− 4.00000i	− 0.196827i
$414$	0	0
$415$	0	0
$416$	6.00000	0.294174
$417$	0	0
$418$	24.0000i	1.17388i
$419$	−36.0000	−1.75872	−0.879358	−	0.476162i	$-0.842028\pi$
$419$	−36.0000	−1.75872	−0.879358	+	0.476162i	$0.842028\pi$
$420$	0	0
$421$	14.0000	0.682318	0.341159	−	0.940006i	$-0.389181\pi$
$421$	14.0000	0.682318	0.341159	+	0.940006i	$0.389181\pi$
$422$	− 16.0000i	− 0.778868i
$423$	0	0
$424$	14.0000	0.679900
$425$	0	0
$426$	0	0
$427$	− 8.00000i	− 0.387147i
$428$	− 12.0000i	− 0.580042i
$429$	0	0
$430$	0	0
$431$	−30.0000	−1.44505	−0.722525	−	0.691345i	$-0.757018\pi$
$431$	−30.0000	−1.44505	−0.722525	+	0.691345i	$0.757018\pi$
$432$	0	0
$433$	− 22.0000i	− 1.05725i	−0.848855	−	0.528626i	$-0.822707\pi$
$433$	− 22.0000i	− 1.05725i	0.848855	−	0.528626i	$-0.177293\pi$
$434$	−4.00000	−0.192006
$435$	0	0
$436$	−18.0000	−0.862044
$437$	0	0
$438$	0	0
$439$	−28.0000	−1.33637	−0.668184	−	0.743996i	$-0.732928\pi$
$439$	−28.0000	−1.33637	−0.668184	+	0.743996i	$0.732928\pi$
$440$	0	0
$441$	0	0
$442$	24.0000i	1.14156i
$443$	4.00000i	0.190046i	0.995475	+	0.0950229i	$0.0302924\pi$
$443$	4.00000i	0.190046i	−0.995475	+	0.0950229i	$0.969708\pi$
$444$	0	0
$445$	0	0
$446$	16.0000	0.757622
$447$	0	0
$448$	− 1.00000i	− 0.0472456i
$449$	−12.0000	−0.566315	−0.283158	−	0.959073i	$-0.591382\pi$
$449$	−12.0000	−0.566315	−0.283158	+	0.959073i	$0.591382\pi$
$450$	0	0
$451$	−40.0000	−1.88353
$452$	− 14.0000i	− 0.658505i
$453$	0	0
$454$	−20.0000	−0.938647
$455$	0	0
$456$	0	0
$457$	14.0000i	0.654892i	0.944870	+	0.327446i	$0.106188\pi$
$457$	14.0000i	0.654892i	−0.944870	+	0.327446i	$0.893812\pi$
$458$	4.00000i	0.186908i
$459$	0	0
$460$	0	0
$461$	−2.00000	−0.0931493	−0.0465746	−	0.998915i	$-0.514831\pi$
$461$	−2.00000	−0.0931493	−0.0465746	+	0.998915i	$0.514831\pi$
$462$	0	0
$463$	− 4.00000i	− 0.185896i	−0.995671	−	0.0929479i	$-0.970371\pi$
$463$	− 4.00000i	− 0.185896i	0.995671	−	0.0929479i	$-0.0296290\pi$
$464$	−6.00000	−0.278543
$465$	0	0
$466$	6.00000	0.277945
$467$	− 32.0000i	− 1.48078i	−0.672176	−	0.740392i	$-0.734640\pi$
$467$	− 32.0000i	− 1.48078i	0.672176	−	0.740392i	$-0.265360\pi$
$468$	0	0
$469$	6.00000	0.277054
$470$	0	0
$471$	0	0
$472$	− 4.00000i	− 0.184115i
$473$	− 8.00000i	− 0.367840i
$474$	0	0
$475$	0	0
$476$	4.00000	0.183340
$477$	0	0
$478$	22.0000i	1.00626i
$479$	36.0000	1.64488	0.822441	−	0.568850i	$-0.192612\pi$
$479$	36.0000	1.64488	0.822441	+	0.568850i	$0.192612\pi$
$480$	0	0
$481$	48.0000	2.18861
$482$	− 10.0000i	− 0.455488i
$483$	0	0
$484$	−5.00000	−0.227273
$485$	0	0
$486$	0	0
$487$	− 8.00000i	− 0.362515i	−0.983436	−	0.181257i	$-0.941983\pi$
$487$	− 8.00000i	− 0.362515i	0.983436	−	0.181257i	$-0.0580167\pi$
$488$	− 8.00000i	− 0.362143i
$489$	0	0
$490$	0	0
$491$	−24.0000	−1.08310	−0.541552	−	0.840667i	$-0.682163\pi$
$491$	−24.0000	−1.08310	−0.541552	+	0.840667i	$0.682163\pi$
$492$	0	0
$493$	− 24.0000i	− 1.08091i
$494$	−36.0000	−1.61972
$495$	0	0
$496$	−4.00000	−0.179605
$497$	2.00000i	0.0897123i
$498$	0	0
$499$	24.0000	1.07439	0.537194	−	0.843459i	$-0.319484\pi$
$499$	24.0000	1.07439	0.537194	+	0.843459i	$0.319484\pi$
$500$	0	0
$501$	0	0
$502$	− 12.0000i	− 0.535586i
$503$	− 30.0000i	− 1.33763i	−0.743427	−	0.668817i	$-0.766801\pi$
$503$	− 30.0000i	− 1.33763i	0.743427	−	0.668817i	$-0.233199\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	− 8.00000i	− 0.354943i
$509$	38.0000	1.68432	0.842160	−	0.539227i	$-0.181284\pi$
$509$	38.0000	1.68432	0.842160	+	0.539227i	$0.181284\pi$
$510$	0	0
$511$	10.0000	0.442374
$512$	− 1.00000i	− 0.0441942i
$513$	0	0
$514$	16.0000	0.705730
$515$	0	0
$516$	0	0
$517$	40.0000i	1.75920i
$518$	− 8.00000i	− 0.351500i
$519$	0	0
$520$	0	0
$521$	38.0000	1.66481	0.832405	−	0.554168i	$-0.186963\pi$
$521$	38.0000	1.66481	0.832405	+	0.554168i	$0.186963\pi$
$522$	0	0
$523$	16.0000i	0.699631i	0.936819	+	0.349816i	$0.113756\pi$
$523$	16.0000i	0.699631i	−0.936819	+	0.349816i	$0.886244\pi$
$524$	4.00000	0.174741
$525$	0	0
$526$	−4.00000	−0.174408
$527$	− 16.0000i	− 0.696971i
$528$	0	0
$529$	23.0000	1.00000
$530$	0	0
$531$	0	0
$532$	6.00000i	0.260133i
$533$	− 60.0000i	− 2.59889i
$534$	0	0
$535$	0	0
$536$	6.00000	0.259161
$537$	0	0
$538$	− 18.0000i	− 0.776035i
$539$	−4.00000	−0.172292
$540$	0	0
$541$	18.0000	0.773880	0.386940	−	0.922105i	$-0.373532\pi$
$541$	18.0000	0.773880	0.386940	+	0.922105i	$0.373532\pi$
$542$	12.0000i	0.515444i
$543$	0	0
$544$	4.00000	0.171499
$545$	0	0
$546$	0	0
$547$	− 30.0000i	− 1.28271i	−0.767245	−	0.641354i	$-0.778373\pi$
$547$	− 30.0000i	− 1.28271i	0.767245	−	0.641354i	$-0.221627\pi$
$548$	− 6.00000i	− 0.256307i
$549$	0	0
$550$	0	0
$551$	36.0000	1.53365
$552$	0	0
$553$	− 16.0000i	− 0.680389i
$554$	12.0000	0.509831
$555$	0	0
$556$	10.0000	0.424094
$557$	2.00000i	0.0847427i	0.999102	+	0.0423714i	$0.0134913\pi$
$557$	2.00000i	0.0847427i	−0.999102	+	0.0423714i	$0.986509\pi$
$558$	0	0
$559$	12.0000	0.507546
$560$	0	0
$561$	0	0
$562$	− 8.00000i	− 0.337460i
$563$	44.0000i	1.85438i	0.374593	+	0.927189i	$0.377783\pi$
$563$	44.0000i	1.85438i	−0.374593	+	0.927189i	$0.622217\pi$
$564$	0	0
$565$	0	0
$566$	−4.00000	−0.168133
$567$	0	0
$568$	2.00000i	0.0839181i
$569$	−24.0000	−1.00613	−0.503066	−	0.864248i	$-0.667795\pi$
$569$	−24.0000	−1.00613	−0.503066	+	0.864248i	$0.667795\pi$
$570$	0	0
$571$	−36.0000	−1.50655	−0.753277	−	0.657704i	$-0.771528\pi$
$571$	−36.0000	−1.50655	−0.753277	+	0.657704i	$0.771528\pi$
$572$	− 24.0000i	− 1.00349i
$573$	0	0
$574$	−10.0000	−0.417392
$575$	0	0
$576$	0	0
$577$	2.00000i	0.0832611i	0.999133	+	0.0416305i	$0.0132552\pi$
$577$	2.00000i	0.0832611i	−0.999133	+	0.0416305i	$0.986745\pi$
$578$	− 1.00000i	− 0.0415945i
$579$	0	0
$580$	0	0
$581$	−8.00000	−0.331896
$582$	0	0
$583$	− 56.0000i	− 2.31928i
$584$	10.0000	0.413803
$585$	0	0
$586$	−2.00000	−0.0826192
$587$	4.00000i	0.165098i	0.996587	+	0.0825488i	$0.0263060\pi$
$587$	4.00000i	0.165098i	−0.996587	+	0.0825488i	$0.973694\pi$
$588$	0	0
$589$	24.0000	0.988903
$590$	0	0
$591$	0	0
$592$	− 8.00000i	− 0.328798i
$593$	− 12.0000i	− 0.492781i	−0.969171	−	0.246390i	$-0.920755\pi$
$593$	− 12.0000i	− 0.492781i	0.969171	−	0.246390i	$-0.0792446\pi$
$594$	0	0
$595$	0	0
$596$	−6.00000	−0.245770
$597$	0	0
$598$	0	0
$599$	6.00000	0.245153	0.122577	−	0.992459i	$-0.460884\pi$
$599$	6.00000	0.245153	0.122577	+	0.992459i	$0.460884\pi$
$600$	0	0
$601$	−30.0000	−1.22373	−0.611863	−	0.790964i	$-0.709580\pi$
$601$	−30.0000	−1.22373	−0.611863	+	0.790964i	$0.709580\pi$
$602$	− 2.00000i	− 0.0815139i
$603$	0	0
$604$	16.0000	0.651031
$605$	0	0
$606$	0	0
$607$	32.0000i	1.29884i	0.760430	+	0.649420i	$0.224988\pi$
$607$	32.0000i	1.29884i	−0.760430	+	0.649420i	$0.775012\pi$
$608$	6.00000i	0.243332i
$609$	0	0
$610$	0	0
$611$	−60.0000	−2.42734
$612$	0	0
$613$	48.0000i	1.93870i	0.245680	+	0.969351i	$0.420989\pi$
$613$	48.0000i	1.93870i	−0.245680	+	0.969351i	$0.579011\pi$
$614$	−4.00000	−0.161427
$615$	0	0
$616$	−4.00000	−0.161165
$617$	10.0000i	0.402585i	0.979531	+	0.201292i	$0.0645141\pi$
$617$	10.0000i	0.402585i	−0.979531	+	0.201292i	$0.935486\pi$
$618$	0	0
$619$	10.0000	0.401934	0.200967	−	0.979598i	$-0.435592\pi$
$619$	10.0000	0.401934	0.200967	+	0.979598i	$0.435592\pi$
$620$	0	0
$621$	0	0
$622$	− 12.0000i	− 0.481156i
$623$	2.00000i	0.0801283i
$624$	0	0
$625$	0	0
$626$	−10.0000	−0.399680
$627$	0	0
$628$	− 14.0000i	− 0.558661i
$629$	32.0000	1.27592
$630$	0	0
$631$	−40.0000	−1.59237	−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000	−1.59237	−0.796187	+	0.605050i	$0.793153\pi$
$632$	− 16.0000i	− 0.636446i
$633$	0	0
$634$	2.00000	0.0794301
$635$	0	0
$636$	0	0
$637$	− 6.00000i	− 0.237729i
$638$	24.0000i	0.950169i
$639$	0	0
$640$	0	0
$641$	−4.00000	−0.157991	−0.0789953	−	0.996875i	$-0.525171\pi$
$641$	−4.00000	−0.157991	−0.0789953	+	0.996875i	$0.525171\pi$
$642$	0	0
$643$	16.0000i	0.630978i	0.948929	+	0.315489i	$0.102169\pi$
$643$	16.0000i	0.630978i	−0.948929	+	0.315489i	$0.897831\pi$
$644$	0	0
$645$	0	0
$646$	−24.0000	−0.944267
$647$	2.00000i	0.0786281i	0.999227	+	0.0393141i	$0.0125173\pi$
$647$	2.00000i	0.0786281i	−0.999227	+	0.0393141i	$0.987483\pi$
$648$	0	0
$649$	−16.0000	−0.628055
$650$	0	0
$651$	0	0
$652$	10.0000i	0.391630i
$653$	38.0000i	1.48705i	0.668705	+	0.743527i	$0.266849\pi$
$653$	38.0000i	1.48705i	−0.668705	+	0.743527i	$0.733151\pi$
$654$	0	0
$655$	0	0
$656$	−10.0000	−0.390434
$657$	0	0
$658$	10.0000i	0.389841i
$659$	20.0000	0.779089	0.389545	−	0.921008i	$-0.372632\pi$
$659$	20.0000	0.779089	0.389545	+	0.921008i	$0.372632\pi$
$660$	0	0
$661$	−8.00000	−0.311164	−0.155582	−	0.987823i	$-0.549725\pi$
$661$	−8.00000	−0.311164	−0.155582	+	0.987823i	$0.549725\pi$
$662$	− 16.0000i	− 0.621858i
$663$	0	0
$664$	−8.00000	−0.310460
$665$	0	0
$666$	0	0
$667$	0	0
$668$	− 6.00000i	− 0.232147i
$669$	0	0
$670$	0	0
$671$	−32.0000	−1.23535
$672$	0	0
$673$	− 26.0000i	− 1.00223i	−0.865382	−	0.501113i	$-0.832924\pi$
$673$	− 26.0000i	− 1.00223i	0.865382	−	0.501113i	$-0.167076\pi$
$674$	26.0000	1.00148
$675$	0	0
$676$	23.0000	0.884615
$677$	6.00000i	0.230599i	0.993331	+	0.115299i	$0.0367827\pi$
$677$	6.00000i	0.230599i	−0.993331	+	0.115299i	$0.963217\pi$
$678$	0	0
$679$	2.00000	0.0767530
$680$	0	0
$681$	0	0
$682$	16.0000i	0.612672i
$683$	− 52.0000i	− 1.98972i	−0.101237	−	0.994862i	$-0.532280\pi$
$683$	− 52.0000i	− 1.98972i	0.101237	−	0.994862i	$-0.467720\pi$
$684$	0	0
$685$	0	0
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	− 2.00000i	− 0.0762493i
$689$	84.0000	3.20015
$690$	0	0
$691$	−26.0000	−0.989087	−0.494543	−	0.869153i	$-0.664665\pi$
$691$	−26.0000	−0.989087	−0.494543	+	0.869153i	$0.664665\pi$
$692$	− 2.00000i	− 0.0760286i
$693$	0	0
$694$	20.0000	0.759190
$695$	0	0
$696$	0	0
$697$	− 40.0000i	− 1.51511i
$698$	28.0000i	1.05982i
$699$	0	0
$700$	0	0
$701$	−2.00000	−0.0755390	−0.0377695	−	0.999286i	$-0.512025\pi$
$701$	−2.00000	−0.0755390	−0.0377695	+	0.999286i	$0.512025\pi$
$702$	0	0
$703$	48.0000i	1.81035i
$704$	−4.00000	−0.150756
$705$	0	0
$706$	0	0
$707$	6.00000i	0.225653i
$708$	0	0
$709$	14.0000	0.525781	0.262891	−	0.964826i	$-0.415324\pi$
$709$	14.0000	0.525781	0.262891	+	0.964826i	$0.415324\pi$
$710$	0	0
$711$	0	0
$712$	2.00000i	0.0749532i
$713$	0	0
$714$	0	0
$715$	0	0
$716$	24.0000	0.896922
$717$	0	0
$718$	− 34.0000i	− 1.26887i
$719$	36.0000	1.34257	0.671287	−	0.741198i	$-0.265742\pi$
$719$	36.0000	1.34257	0.671287	+	0.741198i	$0.265742\pi$
$720$	0	0
$721$	8.00000	0.297936
$722$	− 17.0000i	− 0.632674i
$723$	0	0
$724$	8.00000	0.297318
$725$	0	0
$726$	0	0
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	− 6.00000i	− 0.222375i
$729$	0	0
$730$	0	0
$731$	8.00000	0.295891
$732$	0	0
$733$	22.0000i	0.812589i	0.913742	+	0.406294i	$0.133179\pi$
$733$	22.0000i	0.812589i	−0.913742	+	0.406294i	$0.866821\pi$
$734$	−16.0000	−0.590571
$735$	0	0
$736$	0	0
$737$	− 24.0000i	− 0.884051i
$738$	0	0
$739$	−40.0000	−1.47142	−0.735712	−	0.677295i	$-0.763152\pi$
$739$	−40.0000	−1.47142	−0.735712	+	0.677295i	$0.763152\pi$
$740$	0	0
$741$	0	0
$742$	− 14.0000i	− 0.513956i
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	0	0
$746$	32.0000	1.17160
$747$	0	0
$748$	− 16.0000i	− 0.585018i
$749$	−12.0000	−0.438470
$750$	0	0
$751$	−40.0000	−1.45962	−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000	−1.45962	−0.729810	+	0.683650i	$0.760392\pi$
$752$	10.0000i	0.364662i
$753$	0	0
$754$	−36.0000	−1.31104
$755$	0	0
$756$	0	0
$757$	− 40.0000i	− 1.45382i	−0.686730	−	0.726912i	$-0.740955\pi$
$757$	− 40.0000i	− 1.45382i	0.686730	−	0.726912i	$-0.259045\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−6.00000	−0.217500	−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000	−0.217500	−0.108750	+	0.994069i	$0.534685\pi$
$762$	0	0
$763$	18.0000i	0.651644i
$764$	−18.0000	−0.651217
$765$	0	0
$766$	30.0000	1.08394
$767$	− 24.0000i	− 0.866590i
$768$	0	0
$769$	26.0000	0.937584	0.468792	−	0.883309i	$-0.344689\pi$
$769$	26.0000	0.937584	0.468792	+	0.883309i	$0.344689\pi$
$770$	0	0
$771$	0	0
$772$	10.0000i	0.359908i
$773$	54.0000i	1.94225i	0.238581	+	0.971123i	$0.423318\pi$
$773$	54.0000i	1.94225i	−0.238581	+	0.971123i	$0.576682\pi$
$774$	0	0
$775$	0	0
$776$	2.00000	0.0717958
$777$	0	0
$778$	− 22.0000i	− 0.788738i
$779$	60.0000	2.14972
$780$	0	0
$781$	8.00000	0.286263
$782$	0	0
$783$	0	0
$784$	−1.00000	−0.0357143
$785$	0	0
$786$	0	0
$787$	− 52.0000i	− 1.85360i	−0.375555	−	0.926800i	$-0.622548\pi$
$787$	− 52.0000i	− 1.85360i	0.375555	−	0.926800i	$-0.377452\pi$
$788$	− 22.0000i	− 0.783718i
$789$	0	0
$790$	0	0
$791$	−14.0000	−0.497783
$792$	0	0
$793$	− 48.0000i	− 1.70453i
$794$	26.0000	0.922705
$795$	0	0
$796$	−16.0000	−0.567105
$797$	− 42.0000i	− 1.48772i	−0.668338	−	0.743858i	$-0.732994\pi$
$797$	− 42.0000i	− 1.48772i	0.668338	−	0.743858i	$-0.267006\pi$
$798$	0	0
$799$	−40.0000	−1.41510
$800$	0	0
$801$	0	0
$802$	− 16.0000i	− 0.564980i
$803$	− 40.0000i	− 1.41157i
$804$	0	0
$805$	0	0
$806$	−24.0000	−0.845364
$807$	0	0
$808$	6.00000i	0.211079i
$809$	−24.0000	−0.843795	−0.421898	−	0.906644i	$-0.638636\pi$
$809$	−24.0000	−0.843795	−0.421898	+	0.906644i	$0.638636\pi$
$810$	0	0
$811$	−22.0000	−0.772524	−0.386262	−	0.922389i	$-0.626234\pi$
$811$	−22.0000	−0.772524	−0.386262	+	0.922389i	$0.626234\pi$
$812$	6.00000i	0.210559i
$813$	0	0
$814$	−32.0000	−1.12160
$815$	0	0
$816$	0	0
$817$	12.0000i	0.419827i
$818$	18.0000i	0.629355i
$819$	0	0
$820$	0	0
$821$	30.0000	1.04701	0.523504	−	0.852023i	$-0.324625\pi$
$821$	30.0000	1.04701	0.523504	+	0.852023i	$0.324625\pi$
$822$	0	0
$823$	4.00000i	0.139431i	0.997567	+	0.0697156i	$0.0222092\pi$
$823$	4.00000i	0.139431i	−0.997567	+	0.0697156i	$0.977791\pi$
$824$	8.00000	0.278693
$825$	0	0
$826$	−4.00000	−0.139178
$827$	4.00000i	0.139094i	0.997579	+	0.0695468i	$0.0221553\pi$
$827$	4.00000i	0.139094i	−0.997579	+	0.0695468i	$0.977845\pi$
$828$	0	0
$829$	4.00000	0.138926	0.0694629	−	0.997585i	$-0.477871\pi$
$829$	4.00000	0.138926	0.0694629	+	0.997585i	$0.477871\pi$
$830$	0	0
$831$	0	0
$832$	− 6.00000i	− 0.208013i
$833$	− 4.00000i	− 0.138592i
$834$	0	0
$835$	0	0
$836$	24.0000	0.830057
$837$	0	0
$838$	36.0000i	1.24360i
$839$	−32.0000	−1.10476	−0.552381	−	0.833592i	$-0.686281\pi$
$839$	−32.0000	−1.10476	−0.552381	+	0.833592i	$0.686281\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	− 14.0000i	− 0.482472i
$843$	0	0
$844$	−16.0000	−0.550743
$845$	0	0
$846$	0	0
$847$	5.00000i	0.171802i
$848$	− 14.0000i	− 0.480762i
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	38.0000i	1.30110i	0.759465	+	0.650548i	$0.225461\pi$
$853$	38.0000i	1.30110i	−0.759465	+	0.650548i	$0.774539\pi$
$854$	−8.00000	−0.273754
$855$	0	0
$856$	−12.0000	−0.410152
$857$	− 48.0000i	− 1.63965i	−0.572615	−	0.819824i	$-0.694071\pi$
$857$	− 48.0000i	− 1.63965i	0.572615	−	0.819824i	$-0.305929\pi$
$858$	0	0
$859$	−30.0000	−1.02359	−0.511793	−	0.859109i	$-0.671019\pi$
$859$	−30.0000	−1.02359	−0.511793	+	0.859109i	$0.671019\pi$
$860$	0	0
$861$	0	0
$862$	30.0000i	1.02180i
$863$	− 44.0000i	− 1.49778i	−0.662696	−	0.748889i	$-0.730588\pi$
$863$	− 44.0000i	− 1.49778i	0.662696	−	0.748889i	$-0.269412\pi$
$864$	0	0
$865$	0	0
$866$	−22.0000	−0.747590
$867$	0	0
$868$	4.00000i	0.135769i
$869$	−64.0000	−2.17105
$870$	0	0
$871$	36.0000	1.21981
$872$	18.0000i	0.609557i
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	28.0000i	0.945493i	0.881199	+	0.472746i	$0.156737\pi$
$877$	28.0000i	0.945493i	−0.881199	+	0.472746i	$0.843263\pi$
$878$	28.0000i	0.944954i
$879$	0	0
$880$	0	0
$881$	26.0000	0.875962	0.437981	−	0.898984i	$-0.355694\pi$
$881$	26.0000	0.875962	0.437981	+	0.898984i	$0.355694\pi$
$882$	0	0
$883$	6.00000i	0.201916i	0.994891	+	0.100958i	$0.0321908\pi$
$883$	6.00000i	0.201916i	−0.994891	+	0.100958i	$0.967809\pi$
$884$	24.0000	0.807207
$885$	0	0
$886$	4.00000	0.134383
$887$	− 2.00000i	− 0.0671534i	−0.999436	−	0.0335767i	$-0.989310\pi$
$887$	− 2.00000i	− 0.0671534i	0.999436	−	0.0335767i	$-0.0106898\pi$
$888$	0	0
$889$	−8.00000	−0.268311
$890$	0	0
$891$	0	0
$892$	− 16.0000i	− 0.535720i
$893$	− 60.0000i	− 2.00782i
$894$	0	0
$895$	0	0
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	12.0000i	0.400445i
$899$	24.0000	0.800445
$900$	0	0
$901$	56.0000	1.86563
$902$	40.0000i	1.33185i
$903$	0	0
$904$	−14.0000	−0.465633
$905$	0	0
$906$	0	0
$907$	10.0000i	0.332045i	0.986122	+	0.166022i	$0.0530924\pi$
$907$	10.0000i	0.332045i	−0.986122	+	0.166022i	$0.946908\pi$
$908$	20.0000i	0.663723i
$909$	0	0
$910$	0	0
$911$	−30.0000	−0.993944	−0.496972	−	0.867766i	$-0.665555\pi$
$911$	−30.0000	−0.993944	−0.496972	+	0.867766i	$0.665555\pi$
$912$	0	0
$913$	32.0000i	1.05905i
$914$	14.0000	0.463079
$915$	0	0
$916$	4.00000	0.132164
$917$	− 4.00000i	− 0.132092i
$918$	0	0
$919$	−8.00000	−0.263896	−0.131948	−	0.991257i	$-0.542123\pi$
$919$	−8.00000	−0.263896	−0.131948	+	0.991257i	$0.542123\pi$
$920$	0	0
$921$	0	0
$922$	2.00000i	0.0658665i
$923$	12.0000i	0.394985i
$924$	0	0
$925$	0	0
$926$	−4.00000	−0.131448
$927$	0	0
$928$	6.00000i	0.196960i
$929$	−34.0000	−1.11550	−0.557752	−	0.830008i	$-0.688336\pi$
$929$	−34.0000	−1.11550	−0.557752	+	0.830008i	$0.688336\pi$
$930$	0	0
$931$	6.00000	0.196642
$932$	− 6.00000i	− 0.196537i
$933$	0	0
$934$	−32.0000	−1.04707
$935$	0	0
$936$	0	0
$937$	22.0000i	0.718709i	0.933201	+	0.359354i	$0.117003\pi$
$937$	22.0000i	0.718709i	−0.933201	+	0.359354i	$0.882997\pi$
$938$	− 6.00000i	− 0.195907i
$939$	0	0
$940$	0	0
$941$	−58.0000	−1.89075	−0.945373	−	0.325991i	$-0.894302\pi$
$941$	−58.0000	−1.89075	−0.945373	+	0.325991i	$0.894302\pi$
$942$	0	0
$943$	0	0
$944$	−4.00000	−0.130189
$945$	0	0
$946$	−8.00000	−0.260102
$947$	4.00000i	0.129983i	0.997886	+	0.0649913i	$0.0207020\pi$
$947$	4.00000i	0.129983i	−0.997886	+	0.0649913i	$0.979298\pi$
$948$	0	0
$949$	60.0000	1.94768
$950$	0	0
$951$	0	0
$952$	− 4.00000i	− 0.129641i
$953$	30.0000i	0.971795i	0.874016	+	0.485898i	$0.161507\pi$
$953$	30.0000i	0.971795i	−0.874016	+	0.485898i	$0.838493\pi$
$954$	0	0
$955$	0	0
$956$	22.0000	0.711531
$957$	0	0
$958$	− 36.0000i	− 1.16311i
$959$	−6.00000	−0.193750
$960$	0	0
$961$	−15.0000	−0.483871
$962$	− 48.0000i	− 1.54758i
$963$	0	0
$964$	−10.0000	−0.322078
$965$	0	0
$966$	0	0
$967$	8.00000i	0.257263i	0.991692	+	0.128631i	$0.0410584\pi$
$967$	8.00000i	0.257263i	−0.991692	+	0.128631i	$0.958942\pi$
$968$	5.00000i	0.160706i
$969$	0	0
$970$	0	0
$971$	36.0000	1.15529	0.577647	−	0.816286i	$-0.303971\pi$
$971$	36.0000	1.15529	0.577647	+	0.816286i	$0.303971\pi$
$972$	0	0
$973$	− 10.0000i	− 0.320585i
$974$	−8.00000	−0.256337
$975$	0	0
$976$	−8.00000	−0.256074
$977$	− 6.00000i	− 0.191957i	−0.995383	−	0.0959785i	$-0.969402\pi$
$977$	− 6.00000i	− 0.191957i	0.995383	−	0.0959785i	$-0.0305980\pi$
$978$	0	0
$979$	8.00000	0.255681
$980$	0	0
$981$	0	0
$982$	24.0000i	0.765871i
$983$	38.0000i	1.21201i	0.795460	+	0.606006i	$0.207229\pi$
$983$	38.0000i	1.21201i	−0.795460	+	0.606006i	$0.792771\pi$
$984$	0	0
$985$	0	0
$986$	−24.0000	−0.764316
$987$	0	0
$988$	36.0000i	1.14531i
$989$	0	0
$990$	0	0
$991$	−8.00000	−0.254128	−0.127064	−	0.991894i	$-0.540555\pi$
$991$	−8.00000	−0.254128	−0.127064	+	0.991894i	$0.540555\pi$
$992$	4.00000i	0.127000i
$993$	0	0
$994$	2.00000	0.0634361
$995$	0	0
$996$	0	0
$997$	− 14.0000i	− 0.443384i	−0.975117	−	0.221692i	$-0.928842\pi$
$997$	− 14.0000i	− 0.443384i	0.975117	−	0.221692i	$-0.0711580\pi$
$998$	− 24.0000i	− 0.759707i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3150.2.g.s.2899.1		2
3.2	odd	2		3150.2.g.b.2899.2		2
5.2	odd	4		630.2.a.g.1.1	yes	1
5.3	odd	4		3150.2.a.s.1.1		1
5.4	even	2	inner	3150.2.g.s.2899.2		2
15.2	even	4		630.2.a.e.1.1	✓	1
15.8	even	4		3150.2.a.bh.1.1		1
15.14	odd	2		3150.2.g.b.2899.1		2
20.7	even	4		5040.2.a.n.1.1		1
35.27	even	4		4410.2.a.bl.1.1		1
60.47	odd	4		5040.2.a.bp.1.1		1
105.62	odd	4		4410.2.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
630.2.a.e.1.1	✓	1	15.2	even	4
630.2.a.g.1.1	yes	1	5.2	odd	4
3150.2.a.s.1.1		1	5.3	odd	4
3150.2.a.bh.1.1		1	15.8	even	4
3150.2.g.b.2899.1		2	15.14	odd	2
3150.2.g.b.2899.2		2	3.2	odd	2
3150.2.g.s.2899.1		2	1.1	even	1	trivial
3150.2.g.s.2899.2		2	5.4	even	2	inner
4410.2.a.a.1.1		1	105.62	odd	4
4410.2.a.bl.1.1		1	35.27	even	4
5040.2.a.n.1.1		1	20.7	even	4
5040.2.a.bp.1.1		1	60.47	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3150.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} +1.00000i q^{8} +4.00000 q^{11} +6.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} +4.00000i q^{17} -6.00000 q^{19} -4.00000i q^{22} +6.00000 q^{26} -1.00000i q^{28} -6.00000 q^{29} -4.00000 q^{31} -1.00000i q^{32} +4.00000 q^{34} -8.00000i q^{37} +6.00000i q^{38} -10.0000 q^{41} -2.00000i q^{43} -4.00000 q^{44} +10.0000i q^{47} -1.00000 q^{49} -6.00000i q^{52} -14.0000i q^{53} -1.00000 q^{56} +6.00000i q^{58} -4.00000 q^{59} -8.00000 q^{61} +4.00000i q^{62} -1.00000 q^{64} -6.00000i q^{67} -4.00000i q^{68} +2.00000 q^{71} -10.0000i q^{73} -8.00000 q^{74} +6.00000 q^{76} +4.00000i q^{77} -16.0000 q^{79} +10.0000i q^{82} +8.00000i q^{83} -2.00000 q^{86} +4.00000i q^{88} +2.00000 q^{89} -6.00000 q^{91} +10.0000 q^{94} -2.00000i q^{97} +1.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4}+O(q^{10}) \) 2 * q - 2 * q^4	\( 2 q - 2 q^{4} + 8 q^{11} + 2 q^{14} + 2 q^{16} - 12 q^{19} + 12 q^{26} - 12 q^{29} - 8 q^{31} + 8 q^{34} - 20 q^{41} - 8 q^{44} - 2 q^{49} - 2 q^{56} - 8 q^{59} - 16 q^{61} - 2 q^{64} + 4 q^{71} - 16 q^{74} + 12 q^{76} - 32 q^{79} - 4 q^{86} + 4 q^{89} - 12 q^{91} + 20 q^{94}+O(q^{100}) \) 2 * q - 2 * q^4 + 8 * q^11 + 2 * q^14 + 2 * q^16 - 12 * q^19 + 12 * q^26 - 12 * q^29 - 8 * q^31 + 8 * q^34 - 20 * q^41 - 8 * q^44 - 2 * q^49 - 2 * q^56 - 8 * q^59 - 16 * q^61 - 2 * q^64 + 4 * q^71 - 16 * q^74 + 12 * q^76 - 32 * q^79 - 4 * q^86 + 4 * q^89 - 12 * q^91 + 20 * q^94

Introduction

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