LMFDB - Embedded newform 350.2.c.b.99.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [350,2,Mod(99,350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(350, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("350.99");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 350 = 2 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	350.c (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:	$2.79476407074$
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 70)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			99.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	350.99
Dual form			350.2.c.b.99.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{7} -1.00000i q^{8} +3.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{7} -1.00000i q^{8} +3.00000 q^{9} +4.00000 q^{11} +6.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} +2.00000i q^{17} +3.00000i q^{18} +4.00000i q^{22} -6.00000 q^{26} +1.00000i q^{28} -6.00000 q^{29} +8.00000 q^{31} +1.00000i q^{32} -2.00000 q^{34} -3.00000 q^{36} -10.0000i q^{37} +2.00000 q^{41} -4.00000i q^{43} -4.00000 q^{44} +8.00000i q^{47} -1.00000 q^{49} -6.00000i q^{52} +2.00000i q^{53} -1.00000 q^{56} -6.00000i q^{58} +8.00000 q^{59} -14.0000 q^{61} +8.00000i q^{62} -3.00000i q^{63} -1.00000 q^{64} -12.0000i q^{67} -2.00000i q^{68} -16.0000 q^{71} -3.00000i q^{72} -2.00000i q^{73} +10.0000 q^{74} -4.00000i q^{77} +8.00000 q^{79} +9.00000 q^{81} +2.00000i q^{82} -8.00000i q^{83} +4.00000 q^{86} -4.00000i q^{88} -10.0000 q^{89} +6.00000 q^{91} -8.00000 q^{94} +2.00000i q^{97} -1.00000i q^{98} +12.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} + 6 q^{9}+O(q^{10}) $ 2 * q - 2 * q^4 + 6 * q^9	$ 2 q - 2 q^{4} + 6 q^{9} + 8 q^{11} + 2 q^{14} + 2 q^{16} - 12 q^{26} - 12 q^{29} + 16 q^{31} - 4 q^{34} - 6 q^{36} + 4 q^{41} - 8 q^{44} - 2 q^{49} - 2 q^{56} + 16 q^{59} - 28 q^{61} - 2 q^{64} - 32 q^{71} + 20 q^{74} + 16 q^{79} + 18 q^{81} + 8 q^{86} - 20 q^{89} + 12 q^{91} - 16 q^{94} + 24 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 + 6 * q^9 + 8 * q^11 + 2 * q^14 + 2 * q^16 - 12 * q^26 - 12 * q^29 + 16 * q^31 - 4 * q^34 - 6 * q^36 + 4 * q^41 - 8 * q^44 - 2 * q^49 - 2 * q^56 + 16 * q^59 - 28 * q^61 - 2 * q^64 - 32 * q^71 + 20 * q^74 + 16 * q^79 + 18 * q^81 + 8 * q^86 - 20 * q^89 + 12 * q^91 - 16 * q^94 + 24 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/350\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$1$	$-1$

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 $

$2$	1.00000i	0.707107i
$2$	1.00000i	0.707107i
$3$	0	0	1.00000			$0$
$3$	0	0	−1.00000			$\pi$
$4$	−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
$9$	3.00000	1.00000
$10$	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
$17$	2.00000i	0.485071i	0.970143	+	0.242536i	$0.0779791\pi$
$17$	2.00000i	0.485071i	−0.970143	+	0.242536i	$0.922021\pi$
$18$	3.00000i	0.707107i
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	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
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	1.00000i	0.176777i
$33$	0	0
$34$	−2.00000	−0.342997
$35$	0	0
$36$	−3.00000	−0.500000
$37$	− 10.0000i	− 1.64399i	−0.569495	−	0.821995i	$-0.692861\pi$
$37$	− 10.0000i	− 1.64399i	0.569495	−	0.821995i	$-0.307139\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	− 4.00000i	− 0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$	− 4.00000i	− 0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$	−4.00000	−0.603023
$45$	0	0
$46$	0	0
$47$	8.00000i	1.16692i	0.812142	+	0.583460i	$0.198301\pi$
$47$	8.00000i	1.16692i	−0.812142	+	0.583460i	$0.801699\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	− 6.00000i	− 0.832050i
$53$	2.00000i	0.274721i	0.990521	+	0.137361i	$0.0438619\pi$
$53$	2.00000i	0.274721i	−0.990521	+	0.137361i	$0.956138\pi$
$54$	0	0
$55$	0	0
$56$	−1.00000	−0.133631
$57$	0	0
$58$	− 6.00000i	− 0.787839i
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	0	0
$61$	−14.0000	−1.79252	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−14.0000	−1.79252	−0.896258	+	0.443533i	$0.853725\pi$
$62$	8.00000i	1.01600i
$63$	− 3.00000i	− 0.377964i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	− 12.0000i	− 1.46603i	−0.680211	−	0.733017i	$-0.738112\pi$
$67$	− 12.0000i	− 1.46603i	0.680211	−	0.733017i	$-0.261888\pi$
$68$	− 2.00000i	− 0.242536i
$69$	0	0
$70$	0	0
$71$	−16.0000	−1.89885	−0.949425	−	0.313993i	$-0.898333\pi$
$71$	−16.0000	−1.89885	−0.949425	+	0.313993i	$0.898333\pi$
$72$	− 3.00000i	− 0.353553i
$73$	− 2.00000i	− 0.234082i	−0.993127	−	0.117041i	$-0.962659\pi$
$73$	− 2.00000i	− 0.234082i	0.993127	−	0.117041i	$-0.0373409\pi$
$74$	10.0000	1.16248
$75$	0	0
$76$	0	0
$77$	− 4.00000i	− 0.455842i
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	2.00000i	0.220863i
$83$	− 8.00000i	− 0.878114i	−0.898459	−	0.439057i	$-0.855313\pi$
$83$	− 8.00000i	− 0.878114i	0.898459	−	0.439057i	$-0.144687\pi$
$84$	0	0
$85$	0	0
$86$	4.00000	0.431331
$87$	0	0
$88$	− 4.00000i	− 0.426401i
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	6.00000	0.628971
$92$	0	0
$93$	0	0
$94$	−8.00000	−0.825137
$95$	0	0
$96$	0	0
$97$	2.00000i	0.203069i	0.994832	+	0.101535i	$0.0323753\pi$
$97$	2.00000i	0.203069i	−0.994832	+	0.101535i	$0.967625\pi$
$98$	− 1.00000i	− 0.101015i
$99$	12.0000	1.20605
$100$	0	0
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	− 16.0000i	− 1.57653i	−0.615338	−	0.788263i	$-0.710980\pi$
$103$	− 16.0000i	− 1.57653i	0.615338	−	0.788263i	$-0.289020\pi$
$104$	6.00000	0.588348
$105$	0	0
$106$	−2.00000	−0.194257
$107$	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
$109$	−6.00000	−0.574696	−0.287348	−	0.957826i	$-0.592774\pi$
$109$	−6.00000	−0.574696	−0.287348	+	0.957826i	$0.592774\pi$
$110$	0	0
$111$	0	0
$112$	− 1.00000i	− 0.0944911i
$113$	− 2.00000i	− 0.188144i	−0.995565	−	0.0940721i	$-0.970012\pi$
$113$	− 2.00000i	− 0.188144i	0.995565	−	0.0940721i	$-0.0299884\pi$
$114$	0	0
$115$	0	0
$116$	6.00000	0.557086
$117$	18.0000i	1.66410i
$118$	8.00000i	0.736460i
$119$	2.00000	0.183340
$120$	0	0
$121$	5.00000	0.454545
$122$	− 14.0000i	− 1.26750i
$123$	0	0
$124$	−8.00000	−0.718421
$125$	0	0
$126$	3.00000	0.267261
$127$	− 8.00000i	− 0.709885i	−0.934888	−	0.354943i	$-0.884500\pi$
$127$	− 8.00000i	− 0.709885i	0.934888	−	0.354943i	$-0.115500\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	0	0
$130$	0	0
$131$	−16.0000	−1.39793	−0.698963	−	0.715158i	$-0.746355\pi$
$131$	−16.0000	−1.39793	−0.698963	+	0.715158i	$0.746355\pi$
$132$	0	0
$133$	0	0
$134$	12.0000	1.03664
$135$	0	0
$136$	2.00000	0.171499
$137$	− 6.00000i	− 0.512615i	−0.966595	−	0.256307i	$-0.917494\pi$
$137$	− 6.00000i	− 0.512615i	0.966595	−	0.256307i	$-0.0825059\pi$
$138$	0	0
$139$	−16.0000	−1.35710	−0.678551	−	0.734553i	$-0.737392\pi$
$139$	−16.0000	−1.35710	−0.678551	+	0.734553i	$0.737392\pi$
$140$	0	0
$141$	0	0
$142$	− 16.0000i	− 1.34269i
$143$	24.0000i	2.00698i
$144$	3.00000	0.250000
$145$	0	0
$146$	2.00000	0.165521
$147$	0	0
$148$	10.0000i	0.821995i
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	0	0
$153$	6.00000i	0.485071i
$154$	4.00000	0.322329
$155$	0	0
$156$	0	0
$157$	10.0000i	0.798087i	0.916932	+	0.399043i	$0.130658\pi$
$157$	10.0000i	0.798087i	−0.916932	+	0.399043i	$0.869342\pi$
$158$	8.00000i	0.636446i
$159$	0	0
$160$	0	0
$161$	0	0
$162$	9.00000i	0.707107i
$163$	4.00000i	0.313304i	0.987654	+	0.156652i	$0.0500701\pi$
$163$	4.00000i	0.313304i	−0.987654	+	0.156652i	$0.949930\pi$
$164$	−2.00000	−0.156174
$165$	0	0
$166$	8.00000	0.620920
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	−23.0000	−1.76923
$170$	0	0
$171$	0	0
$172$	4.00000i	0.304997i
$173$	22.0000i	1.67263i	0.548250	+	0.836315i	$0.315294\pi$
$173$	22.0000i	1.67263i	−0.548250	+	0.836315i	$0.684706\pi$
$174$	0	0
$175$	0	0
$176$	4.00000	0.301511
$177$	0	0
$178$	− 10.0000i	− 0.749532i
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	−14.0000	−1.04061	−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000	−1.04061	−0.520306	+	0.853980i	$0.674182\pi$
$182$	6.00000i	0.444750i
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	8.00000i	0.585018i
$188$	− 8.00000i	− 0.583460i
$189$	0	0
$190$	0	0
$191$	24.0000	1.73658	0.868290	−	0.496058i	$-0.165220\pi$
$191$	24.0000	1.73658	0.868290	+	0.496058i	$0.165220\pi$
$192$	0	0
$193$	− 2.00000i	− 0.143963i	−0.997406	−	0.0719816i	$-0.977068\pi$
$193$	− 2.00000i	− 0.143963i	0.997406	−	0.0719816i	$-0.0229323\pi$
$194$	−2.00000	−0.143592
$195$	0	0
$196$	1.00000	0.0714286
$197$	14.0000i	0.997459i	0.866758	+	0.498729i	$0.166200\pi$
$197$	14.0000i	0.997459i	−0.866758	+	0.498729i	$0.833800\pi$
$198$	12.0000i	0.852803i
$199$	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
$206$	16.0000	1.11477
$207$	0	0
$208$	6.00000i	0.416025i
$209$	0	0
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	− 2.00000i	− 0.137361i
$213$	0	0
$214$	−12.0000	−0.820303
$215$	0	0
$216$	0	0
$217$	− 8.00000i	− 0.543075i
$218$	− 6.00000i	− 0.406371i
$219$	0	0
$220$	0	0
$221$	−12.0000	−0.807207
$222$	0	0
$223$	− 16.0000i	− 1.07144i	−0.844396	−	0.535720i	$-0.820040\pi$
$223$	− 16.0000i	− 1.07144i	0.844396	−	0.535720i	$-0.179960\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	2.00000	0.133038
$227$	8.00000i	0.530979i	0.964114	+	0.265489i	$0.0855335\pi$
$227$	8.00000i	0.530979i	−0.964114	+	0.265489i	$0.914466\pi$
$228$	0	0
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	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	$0.937031\pi$
$234$	−18.0000	−1.17670
$235$	0	0
$236$	−8.00000	−0.520756
$237$	0	0
$238$	2.00000i	0.129641i
$239$	−16.0000	−1.03495	−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000	−1.03495	−0.517477	+	0.855697i	$0.673129\pi$
$240$	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
$244$	14.0000	0.896258
$245$	0	0
$246$	0	0
$247$	0	0
$248$	− 8.00000i	− 0.508001i
$249$	0	0
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	3.00000i	0.188982i
$253$	0	0
$254$	8.00000	0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 22.0000i	− 1.37232i	−0.727450	−	0.686161i	$-0.759294\pi$
$257$	− 22.0000i	− 1.37232i	0.727450	−	0.686161i	$-0.240706\pi$
$258$	0	0
$259$	−10.0000	−0.621370
$260$	0	0
$261$	−18.0000	−1.11417
$262$	− 16.0000i	− 0.988483i
$263$	− 8.00000i	− 0.493301i	−0.969104	−	0.246651i	$-0.920670\pi$
$263$	− 8.00000i	− 0.493301i	0.969104	−	0.246651i	$-0.0793300\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	12.0000i	0.733017i
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	2.00000i	0.121268i
$273$	0	0
$274$	6.00000	0.362473
$275$	0	0
$276$	0	0
$277$	− 18.0000i	− 1.08152i	−0.841178	−	0.540758i	$-0.818138\pi$
$277$	− 18.0000i	− 1.08152i	0.841178	−	0.540758i	$-0.181862\pi$
$278$	− 16.0000i	− 0.959616i
$279$	24.0000	1.43684
$280$	0	0
$281$	26.0000	1.55103	0.775515	−	0.631329i	$-0.217490\pi$
$281$	26.0000	1.55103	0.775515	+	0.631329i	$0.217490\pi$
$282$	0	0
$283$	− 32.0000i	− 1.90220i	−0.308879	−	0.951101i	$-0.599954\pi$
$283$	− 32.0000i	− 1.90220i	0.308879	−	0.951101i	$-0.400046\pi$
$284$	16.0000	0.949425
$285$	0	0
$286$	−24.0000	−1.41915
$287$	− 2.00000i	− 0.118056i
$288$	3.00000i	0.176777i
$289$	13.0000	0.764706
$290$	0	0
$291$	0	0
$292$	2.00000i	0.117041i
$293$	− 10.0000i	− 0.584206i	−0.956387	−	0.292103i	$-0.905645\pi$
$293$	− 10.0000i	− 0.584206i	0.956387	−	0.292103i	$-0.0943550\pi$
$294$	0	0
$295$	0	0
$296$	−10.0000	−0.581238
$297$	0	0
$298$	− 6.00000i	− 0.347571i
$299$	0	0
$300$	0	0
$301$	−4.00000	−0.230556
$302$	8.00000i	0.460348i
$303$	0	0
$304$	0	0
$305$	0	0
$306$	−6.00000	−0.342997
$307$	− 8.00000i	− 0.456584i	−0.973593	−	0.228292i	$-0.926686\pi$
$307$	− 8.00000i	− 0.456584i	0.973593	−	0.228292i	$-0.0733141\pi$
$308$	4.00000i	0.227921i
$309$	0	0
$310$	0	0
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	0	0
$313$	22.0000i	1.24351i	0.783210	+	0.621757i	$0.213581\pi$
$313$	22.0000i	1.24351i	−0.783210	+	0.621757i	$0.786419\pi$
$314$	−10.0000	−0.564333
$315$	0	0
$316$	−8.00000	−0.450035
$317$	22.0000i	1.23564i	0.786318	+	0.617822i	$0.211985\pi$
$317$	22.0000i	1.23564i	−0.786318	+	0.617822i	$0.788015\pi$
$318$	0	0
$319$	−24.0000	−1.34374
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	−9.00000	−0.500000
$325$	0	0
$326$	−4.00000	−0.221540
$327$	0	0
$328$	− 2.00000i	− 0.110432i
$329$	8.00000	0.441054
$330$	0	0
$331$	4.00000	0.219860	0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000	0.219860	0.109930	+	0.993939i	$0.464937\pi$
$332$	8.00000i	0.439057i
$333$	− 30.0000i	− 1.64399i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 14.0000i	− 0.762629i	−0.924445	−	0.381314i	$-0.875472\pi$
$337$	− 14.0000i	− 0.762629i	0.924445	−	0.381314i	$-0.124528\pi$
$338$	− 23.0000i	− 1.25104i
$339$	0	0
$340$	0	0
$341$	32.0000	1.73290
$342$	0	0
$343$	1.00000i	0.0539949i
$344$	−4.00000	−0.215666
$345$	0	0
$346$	−22.0000	−1.18273
$347$	4.00000i	0.214731i	0.994220	+	0.107366i	$0.0342415\pi$
$347$	4.00000i	0.214731i	−0.994220	+	0.107366i	$0.965758\pi$
$348$	0	0
$349$	−10.0000	−0.535288	−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000	−0.535288	−0.267644	+	0.963518i	$0.586245\pi$
$350$	0	0
$351$	0	0
$352$	4.00000i	0.213201i
$353$	6.00000i	0.319348i	0.987170	+	0.159674i	$0.0510443\pi$
$353$	6.00000i	0.319348i	−0.987170	+	0.159674i	$0.948956\pi$
$354$	0	0
$355$	0	0
$356$	10.0000	0.529999
$357$	0	0
$358$	12.0000i	0.634220i
$359$	−8.00000	−0.422224	−0.211112	−	0.977462i	$-0.567708\pi$
$359$	−8.00000	−0.422224	−0.211112	+	0.977462i	$0.567708\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	− 14.0000i	− 0.735824i
$363$	0	0
$364$	−6.00000	−0.314485
$365$	0	0
$366$	0	0
$367$	16.0000i	0.835193i	0.908633	+	0.417597i	$0.137127\pi$
$367$	16.0000i	0.835193i	−0.908633	+	0.417597i	$0.862873\pi$
$368$	0	0
$369$	6.00000	0.312348
$370$	0	0
$371$	2.00000	0.103835
$372$	0	0
$373$	− 14.0000i	− 0.724893i	−0.932005	−	0.362446i	$-0.881942\pi$
$373$	− 14.0000i	− 0.724893i	0.932005	−	0.362446i	$-0.118058\pi$
$374$	−8.00000	−0.413670
$375$	0	0
$376$	8.00000	0.412568
$377$	− 36.0000i	− 1.85409i
$378$	0	0
$379$	12.0000	0.616399	0.308199	−	0.951322i	$-0.400274\pi$
$379$	12.0000	0.616399	0.308199	+	0.951322i	$0.400274\pi$
$380$	0	0
$381$	0	0
$382$	24.0000i	1.22795i
$383$	− 24.0000i	− 1.22634i	−0.789950	−	0.613171i	$-0.789894\pi$
$383$	− 24.0000i	− 1.22634i	0.789950	−	0.613171i	$-0.210106\pi$
$384$	0	0
$385$	0	0
$386$	2.00000	0.101797
$387$	− 12.0000i	− 0.609994i
$388$	− 2.00000i	− 0.101535i
$389$	−14.0000	−0.709828	−0.354914	−	0.934899i	$-0.615490\pi$
$389$	−14.0000	−0.709828	−0.354914	+	0.934899i	$0.615490\pi$
$390$	0	0
$391$	0	0
$392$	1.00000i	0.0505076i
$393$	0	0
$394$	−14.0000	−0.705310
$395$	0	0
$396$	−12.0000	−0.603023
$397$	10.0000i	0.501886i	0.968002	+	0.250943i	$0.0807406\pi$
$397$	10.0000i	0.501886i	−0.968002	+	0.250943i	$0.919259\pi$
$398$	16.0000i	0.802008i
$399$	0	0
$400$	0	0
$401$	−14.0000	−0.699127	−0.349563	−	0.936913i	$-0.613670\pi$
$401$	−14.0000	−0.699127	−0.349563	+	0.936913i	$0.613670\pi$
$402$	0	0
$403$	48.0000i	2.39105i
$404$	6.00000	0.298511
$405$	0	0
$406$	−6.00000	−0.297775
$407$	− 40.0000i	− 1.98273i
$408$	0	0
$409$	30.0000	1.48340	0.741702	−	0.670729i	$-0.234019\pi$
$409$	30.0000	1.48340	0.741702	+	0.670729i	$0.234019\pi$
$410$	0	0
$411$	0	0
$412$	16.0000i	0.788263i
$413$	− 8.00000i	− 0.393654i
$414$	0	0
$415$	0	0
$416$	−6.00000	−0.294174
$417$	0	0
$418$	0	0
$419$	−24.0000	−1.17248	−0.586238	−	0.810139i	$-0.699392\pi$
$419$	−24.0000	−1.17248	−0.586238	+	0.810139i	$0.699392\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	4.00000i	0.194717i
$423$	24.0000i	1.16692i
$424$	2.00000	0.0971286
$425$	0	0
$426$	0	0
$427$	14.0000i	0.677507i
$428$	− 12.0000i	− 0.580042i
$429$	0	0
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	− 2.00000i	− 0.0961139i	−0.998845	−	0.0480569i	$-0.984697\pi$
$433$	− 2.00000i	− 0.0961139i	0.998845	−	0.0480569i	$-0.0153029\pi$
$434$	8.00000	0.384012
$435$	0	0
$436$	6.00000	0.287348
$437$	0	0
$438$	0	0
$439$	8.00000	0.381819	0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000	0.381819	0.190910	+	0.981608i	$0.438856\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	− 12.0000i	− 0.570782i
$443$	20.0000i	0.950229i	0.879924	+	0.475114i	$0.157593\pi$
$443$	20.0000i	0.950229i	−0.879924	+	0.475114i	$0.842407\pi$
$444$	0	0
$445$	0	0
$446$	16.0000	0.757622
$447$	0	0
$448$	1.00000i	0.0472456i
$449$	30.0000	1.41579	0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000	1.41579	0.707894	+	0.706319i	$0.249646\pi$
$450$	0	0
$451$	8.00000	0.376705
$452$	2.00000i	0.0940721i
$453$	0	0
$454$	−8.00000	−0.375459
$455$	0	0
$456$	0	0
$457$	10.0000i	0.467780i	0.972263	+	0.233890i	$0.0751456\pi$
$457$	10.0000i	0.467780i	−0.972263	+	0.233890i	$0.924854\pi$
$458$	14.0000i	0.654177i
$459$	0	0
$460$	0	0
$461$	10.0000	0.465746	0.232873	−	0.972507i	$-0.425187\pi$
$461$	10.0000	0.465746	0.232873	+	0.972507i	$0.425187\pi$
$462$	0	0
$463$	16.0000i	0.743583i	0.928316	+	0.371792i	$0.121256\pi$
$463$	16.0000i	0.743583i	−0.928316	+	0.371792i	$0.878744\pi$
$464$	−6.00000	−0.278543
$465$	0	0
$466$	−6.00000	−0.277945
$467$	− 40.0000i	− 1.85098i	−0.378773	−	0.925490i	$-0.623654\pi$
$467$	− 40.0000i	− 1.85098i	0.378773	−	0.925490i	$-0.376346\pi$
$468$	− 18.0000i	− 0.832050i
$469$	−12.0000	−0.554109
$470$	0	0
$471$	0	0
$472$	− 8.00000i	− 0.368230i
$473$	− 16.0000i	− 0.735681i
$474$	0	0
$475$	0	0
$476$	−2.00000	−0.0916698
$477$	6.00000i	0.274721i
$478$	− 16.0000i	− 0.731823i
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	0	0
$481$	60.0000	2.73576
$482$	10.0000i	0.455488i
$483$	0	0
$484$	−5.00000	−0.227273
$485$	0	0
$486$	0	0
$487$	32.0000i	1.45006i	0.688718	+	0.725029i	$0.258174\pi$
$487$	32.0000i	1.45006i	−0.688718	+	0.725029i	$0.741826\pi$
$488$	14.0000i	0.633750i
$489$	0	0
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	0	0
$493$	− 12.0000i	− 0.540453i
$494$	0	0
$495$	0	0
$496$	8.00000	0.359211
$497$	16.0000i	0.717698i
$498$	0	0
$499$	12.0000	0.537194	0.268597	−	0.963253i	$-0.413440\pi$
$499$	12.0000	0.537194	0.268597	+	0.963253i	$0.413440\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	24.0000i	1.07011i	0.844818	+	0.535054i	$0.179709\pi$
$503$	24.0000i	1.07011i	−0.844818	+	0.535054i	$0.820291\pi$
$504$	−3.00000	−0.133631
$505$	0	0
$506$	0	0
$507$	0	0
$508$	8.00000i	0.354943i
$509$	−10.0000	−0.443242	−0.221621	−	0.975133i	$-0.571135\pi$
$509$	−10.0000	−0.443242	−0.221621	+	0.975133i	$0.571135\pi$
$510$	0	0
$511$	−2.00000	−0.0884748
$512$	1.00000i	0.0441942i
$513$	0	0
$514$	22.0000	0.970378
$515$	0	0
$516$	0	0
$517$	32.0000i	1.40736i
$518$	− 10.0000i	− 0.439375i
$519$	0	0
$520$	0	0
$521$	2.00000	0.0876216	0.0438108	−	0.999040i	$-0.486050\pi$
$521$	2.00000	0.0876216	0.0438108	+	0.999040i	$0.486050\pi$
$522$	− 18.0000i	− 0.787839i
$523$	− 16.0000i	− 0.699631i	−0.936819	−	0.349816i	$-0.886244\pi$
$523$	− 16.0000i	− 0.699631i	0.936819	−	0.349816i	$-0.113756\pi$
$524$	16.0000	0.698963
$525$	0	0
$526$	8.00000	0.348817
$527$	16.0000i	0.696971i
$528$	0	0
$529$	23.0000	1.00000
$530$	0	0
$531$	24.0000	1.04151
$532$	0	0
$533$	12.0000i	0.519778i
$534$	0	0
$535$	0	0
$536$	−12.0000	−0.518321
$537$	0	0
$538$	6.00000i	0.258678i
$539$	−4.00000	−0.172292
$540$	0	0
$541$	−18.0000	−0.773880	−0.386940	−	0.922105i	$-0.626468\pi$
$541$	−18.0000	−0.773880	−0.386940	+	0.922105i	$0.626468\pi$
$542$	0	0
$543$	0	0
$544$	−2.00000	−0.0857493
$545$	0	0
$546$	0	0
$547$	12.0000i	0.513083i	0.966533	+	0.256541i	$0.0825830\pi$
$547$	12.0000i	0.513083i	−0.966533	+	0.256541i	$0.917417\pi$
$548$	6.00000i	0.256307i
$549$	−42.0000	−1.79252
$550$	0	0
$551$	0	0
$552$	0	0
$553$	− 8.00000i	− 0.340195i
$554$	18.0000	0.764747
$555$	0	0
$556$	16.0000	0.678551
$557$	22.0000i	0.932170i	0.884740	+	0.466085i	$0.154336\pi$
$557$	22.0000i	0.932170i	−0.884740	+	0.466085i	$0.845664\pi$
$558$	24.0000i	1.01600i
$559$	24.0000	1.01509
$560$	0	0
$561$	0	0
$562$	26.0000i	1.09674i
$563$	16.0000i	0.674320i	0.941447	+	0.337160i	$0.109466\pi$
$563$	16.0000i	0.674320i	−0.941447	+	0.337160i	$0.890534\pi$
$564$	0	0
$565$	0	0
$566$	32.0000	1.34506
$567$	− 9.00000i	− 0.377964i
$568$	16.0000i	0.671345i
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	−12.0000	−0.502184	−0.251092	−	0.967963i	$-0.580790\pi$
$571$	−12.0000	−0.502184	−0.251092	+	0.967963i	$0.580790\pi$
$572$	− 24.0000i	− 1.00349i
$573$	0	0
$574$	2.00000	0.0834784
$575$	0	0
$576$	−3.00000	−0.125000
$577$	10.0000i	0.416305i	0.978096	+	0.208153i	$0.0667451\pi$
$577$	10.0000i	0.416305i	−0.978096	+	0.208153i	$0.933255\pi$
$578$	13.0000i	0.540729i
$579$	0	0
$580$	0	0
$581$	−8.00000	−0.331896
$582$	0	0
$583$	8.00000i	0.331326i
$584$	−2.00000	−0.0827606
$585$	0	0
$586$	10.0000	0.413096
$587$	8.00000i	0.330195i	0.986277	+	0.165098i	$0.0527939\pi$
$587$	8.00000i	0.330195i	−0.986277	+	0.165098i	$0.947206\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	− 10.0000i	− 0.410997i
$593$	6.00000i	0.246390i	0.992382	+	0.123195i	$0.0393141\pi$
$593$	6.00000i	0.246390i	−0.992382	+	0.123195i	$0.960686\pi$
$594$	0	0
$595$	0	0
$596$	6.00000	0.245770
$597$	0	0
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	42.0000	1.71322	0.856608	−	0.515968i	$-0.172568\pi$
$601$	42.0000	1.71322	0.856608	+	0.515968i	$0.172568\pi$
$602$	− 4.00000i	− 0.163028i
$603$	− 36.0000i	− 1.46603i
$604$	−8.00000	−0.325515
$605$	0	0
$606$	0	0
$607$	− 32.0000i	− 1.29884i	−0.760430	−	0.649420i	$-0.775012\pi$
$607$	− 32.0000i	− 1.29884i	0.760430	−	0.649420i	$-0.224988\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−48.0000	−1.94187
$612$	− 6.00000i	− 0.242536i
$613$	− 6.00000i	− 0.242338i	−0.992632	−	0.121169i	$-0.961336\pi$
$613$	− 6.00000i	− 0.242338i	0.992632	−	0.121169i	$-0.0386643\pi$
$614$	8.00000	0.322854
$615$	0	0
$616$	−4.00000	−0.161165
$617$	− 22.0000i	− 0.885687i	−0.896599	−	0.442843i	$-0.853970\pi$
$617$	− 22.0000i	− 0.885687i	0.896599	−	0.442843i	$-0.146030\pi$
$618$	0	0
$619$	−8.00000	−0.321547	−0.160774	−	0.986991i	$-0.551399\pi$
$619$	−8.00000	−0.321547	−0.160774	+	0.986991i	$0.551399\pi$
$620$	0	0
$621$	0	0
$622$	− 24.0000i	− 0.962312i
$623$	10.0000i	0.400642i
$624$	0	0
$625$	0	0
$626$	−22.0000	−0.879297
$627$	0	0
$628$	− 10.0000i	− 0.399043i
$629$	20.0000	0.797452
$630$	0	0
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	− 8.00000i	− 0.318223i
$633$	0	0
$634$	−22.0000	−0.873732
$635$	0	0
$636$	0	0
$637$	− 6.00000i	− 0.237729i
$638$	− 24.0000i	− 0.950169i
$639$	−48.0000	−1.89885
$640$	0	0
$641$	2.00000	0.0789953	0.0394976	−	0.999220i	$-0.487424\pi$
$641$	2.00000	0.0789953	0.0394976	+	0.999220i	$0.487424\pi$
$642$	0	0
$643$	32.0000i	1.26196i	0.775800	+	0.630978i	$0.217346\pi$
$643$	32.0000i	1.26196i	−0.775800	+	0.630978i	$0.782654\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	16.0000i	0.629025i	0.949253	+	0.314512i	$0.101841\pi$
$647$	16.0000i	0.629025i	−0.949253	+	0.314512i	$0.898159\pi$
$648$	− 9.00000i	− 0.353553i
$649$	32.0000	1.25611
$650$	0	0
$651$	0	0
$652$	− 4.00000i	− 0.156652i
$653$	− 14.0000i	− 0.547862i	−0.961749	−	0.273931i	$-0.911676\pi$
$653$	− 14.0000i	− 0.547862i	0.961749	−	0.273931i	$-0.0883240\pi$
$654$	0	0
$655$	0	0
$656$	2.00000	0.0780869
$657$	− 6.00000i	− 0.234082i
$658$	8.00000i	0.311872i
$659$	−28.0000	−1.09073	−0.545363	−	0.838200i	$-0.683608\pi$
$659$	−28.0000	−1.09073	−0.545363	+	0.838200i	$0.683608\pi$
$660$	0	0
$661$	−38.0000	−1.47803	−0.739014	−	0.673690i	$-0.764708\pi$
$661$	−38.0000	−1.47803	−0.739014	+	0.673690i	$0.764708\pi$
$662$	4.00000i	0.155464i
$663$	0	0
$664$	−8.00000	−0.310460
$665$	0	0
$666$	30.0000	1.16248
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−56.0000	−2.16186
$672$	0	0
$673$	14.0000i	0.539660i	0.962908	+	0.269830i	$0.0869676\pi$
$673$	14.0000i	0.539660i	−0.962908	+	0.269830i	$0.913032\pi$
$674$	14.0000	0.539260
$675$	0	0
$676$	23.0000	0.884615
$677$	18.0000i	0.691796i	0.938272	+	0.345898i	$0.112426\pi$
$677$	18.0000i	0.691796i	−0.938272	+	0.345898i	$0.887574\pi$
$678$	0	0
$679$	2.00000	0.0767530
$680$	0	0
$681$	0	0
$682$	32.0000i	1.22534i
$683$	4.00000i	0.153056i	0.997067	+	0.0765279i	$0.0243834\pi$
$683$	4.00000i	0.153056i	−0.997067	+	0.0765279i	$0.975617\pi$
$684$	0	0
$685$	0	0
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	− 4.00000i	− 0.152499i
$689$	−12.0000	−0.457164
$690$	0	0
$691$	−32.0000	−1.21734	−0.608669	−	0.793424i	$-0.708296\pi$
$691$	−32.0000	−1.21734	−0.608669	+	0.793424i	$0.708296\pi$
$692$	− 22.0000i	− 0.836315i
$693$	− 12.0000i	− 0.455842i
$694$	−4.00000	−0.151838
$695$	0	0
$696$	0	0
$697$	4.00000i	0.151511i
$698$	− 10.0000i	− 0.378506i
$699$	0	0
$700$	0	0
$701$	22.0000	0.830929	0.415464	−	0.909610i	$-0.363619\pi$
$701$	22.0000	0.830929	0.415464	+	0.909610i	$0.363619\pi$
$702$	0	0
$703$	0	0
$704$	−4.00000	−0.150756
$705$	0	0
$706$	−6.00000	−0.225813
$707$	6.00000i	0.225653i
$708$	0	0
$709$	2.00000	0.0751116	0.0375558	−	0.999295i	$-0.488043\pi$
$709$	2.00000	0.0751116	0.0375558	+	0.999295i	$0.488043\pi$
$710$	0	0
$711$	24.0000	0.900070
$712$	10.0000i	0.374766i
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−12.0000	−0.448461
$717$	0	0
$718$	− 8.00000i	− 0.298557i
$719$	−24.0000	−0.895049	−0.447524	−	0.894272i	$-0.647694\pi$
$719$	−24.0000	−0.895049	−0.447524	+	0.894272i	$0.647694\pi$
$720$	0	0
$721$	−16.0000	−0.595871
$722$	− 19.0000i	− 0.707107i
$723$	0	0
$724$	14.0000	0.520306
$725$	0	0
$726$	0	0
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	− 6.00000i	− 0.222375i
$729$	27.0000	1.00000
$730$	0	0
$731$	8.00000	0.295891
$732$	0	0
$733$	14.0000i	0.517102i	0.965998	+	0.258551i	$0.0832450\pi$
$733$	14.0000i	0.517102i	−0.965998	+	0.258551i	$0.916755\pi$
$734$	−16.0000	−0.590571
$735$	0	0
$736$	0	0
$737$	− 48.0000i	− 1.76810i
$738$	6.00000i	0.220863i
$739$	−28.0000	−1.03000	−0.514998	−	0.857191i	$-0.672207\pi$
$739$	−28.0000	−1.03000	−0.514998	+	0.857191i	$0.672207\pi$
$740$	0	0
$741$	0	0
$742$	2.00000i	0.0734223i
$743$	− 48.0000i	− 1.76095i	−0.474093	−	0.880475i	$-0.657224\pi$
$743$	− 48.0000i	− 1.76095i	0.474093	−	0.880475i	$-0.342776\pi$
$744$	0	0
$745$	0	0
$746$	14.0000	0.512576
$747$	− 24.0000i	− 0.878114i
$748$	− 8.00000i	− 0.292509i
$749$	12.0000	0.438470
$750$	0	0
$751$	−16.0000	−0.583848	−0.291924	−	0.956441i	$-0.594295\pi$
$751$	−16.0000	−0.583848	−0.291924	+	0.956441i	$0.594295\pi$
$752$	8.00000i	0.291730i
$753$	0	0
$754$	36.0000	1.31104
$755$	0	0
$756$	0	0
$757$	22.0000i	0.799604i	0.916602	+	0.399802i	$0.130921\pi$
$757$	22.0000i	0.799604i	−0.916602	+	0.399802i	$0.869079\pi$
$758$	12.0000i	0.435860i
$759$	0	0
$760$	0	0
$761$	18.0000	0.652499	0.326250	−	0.945284i	$-0.394215\pi$
$761$	18.0000	0.652499	0.326250	+	0.945284i	$0.394215\pi$
$762$	0	0
$763$	6.00000i	0.217215i
$764$	−24.0000	−0.868290
$765$	0	0
$766$	24.0000	0.867155
$767$	48.0000i	1.73318i
$768$	0	0
$769$	−10.0000	−0.360609	−0.180305	−	0.983611i	$-0.557708\pi$
$769$	−10.0000	−0.360609	−0.180305	+	0.983611i	$0.557708\pi$
$770$	0	0
$771$	0	0
$772$	2.00000i	0.0719816i
$773$	− 42.0000i	− 1.51064i	−0.655359	−	0.755318i	$-0.727483\pi$
$773$	− 42.0000i	− 1.51064i	0.655359	−	0.755318i	$-0.272517\pi$
$774$	12.0000	0.431331
$775$	0	0
$776$	2.00000	0.0717958
$777$	0	0
$778$	− 14.0000i	− 0.501924i
$779$	0	0
$780$	0	0
$781$	−64.0000	−2.29010
$782$	0	0
$783$	0	0
$784$	−1.00000	−0.0357143
$785$	0	0
$786$	0	0
$787$	− 8.00000i	− 0.285169i	−0.989783	−	0.142585i	$-0.954459\pi$
$787$	− 8.00000i	− 0.285169i	0.989783	−	0.142585i	$-0.0455413\pi$
$788$	− 14.0000i	− 0.498729i
$789$	0	0
$790$	0	0
$791$	−2.00000	−0.0711118
$792$	− 12.0000i	− 0.426401i
$793$	− 84.0000i	− 2.98293i
$794$	−10.0000	−0.354887
$795$	0	0
$796$	−16.0000	−0.567105
$797$	− 6.00000i	− 0.212531i	−0.994338	−	0.106265i	$-0.966111\pi$
$797$	− 6.00000i	− 0.212531i	0.994338	−	0.106265i	$-0.0338893\pi$
$798$	0	0
$799$	−16.0000	−0.566039
$800$	0	0
$801$	−30.0000	−1.06000
$802$	− 14.0000i	− 0.494357i
$803$	− 8.00000i	− 0.282314i
$804$	0	0
$805$	0	0
$806$	−48.0000	−1.69073
$807$	0	0
$808$	6.00000i	0.211079i
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	0	0
$811$	8.00000	0.280918	0.140459	−	0.990086i	$-0.455142\pi$
$811$	8.00000	0.280918	0.140459	+	0.990086i	$0.455142\pi$
$812$	− 6.00000i	− 0.210559i
$813$	0	0
$814$	40.0000	1.40200
$815$	0	0
$816$	0	0
$817$	0	0
$818$	30.0000i	1.04893i
$819$	18.0000	0.628971
$820$	0	0
$821$	6.00000	0.209401	0.104701	−	0.994504i	$-0.466612\pi$
$821$	6.00000	0.209401	0.104701	+	0.994504i	$0.466612\pi$
$822$	0	0
$823$	8.00000i	0.278862i	0.990232	+	0.139431i	$0.0445274\pi$
$823$	8.00000i	0.278862i	−0.990232	+	0.139431i	$0.955473\pi$
$824$	−16.0000	−0.557386
$825$	0	0
$826$	8.00000	0.278356
$827$	44.0000i	1.53003i	0.644013	+	0.765015i	$0.277268\pi$
$827$	44.0000i	1.53003i	−0.644013	+	0.765015i	$0.722732\pi$
$828$	0	0
$829$	46.0000	1.59765	0.798823	−	0.601566i	$-0.205456\pi$
$829$	46.0000	1.59765	0.798823	+	0.601566i	$0.205456\pi$
$830$	0	0
$831$	0	0
$832$	− 6.00000i	− 0.208013i
$833$	− 2.00000i	− 0.0692959i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	− 24.0000i	− 0.829066i
$839$	16.0000	0.552381	0.276191	−	0.961103i	$-0.410928\pi$
$839$	16.0000	0.552381	0.276191	+	0.961103i	$0.410928\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	− 10.0000i	− 0.344623i
$843$	0	0
$844$	−4.00000	−0.137686
$845$	0	0
$846$	−24.0000	−0.825137
$847$	− 5.00000i	− 0.171802i
$848$	2.00000i	0.0686803i
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	46.0000i	1.57501i	0.616308	+	0.787505i	$0.288628\pi$
$853$	46.0000i	1.57501i	−0.616308	+	0.787505i	$0.711372\pi$
$854$	−14.0000	−0.479070
$855$	0	0
$856$	12.0000	0.410152
$857$	− 54.0000i	− 1.84460i	−0.386469	−	0.922302i	$-0.626305\pi$
$857$	− 54.0000i	− 1.84460i	0.386469	−	0.922302i	$-0.373695\pi$
$858$	0	0
$859$	48.0000	1.63774	0.818869	−	0.573980i	$-0.194601\pi$
$859$	48.0000	1.63774	0.818869	+	0.573980i	$0.194601\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	32.0000i	1.08929i	0.838666	+	0.544646i	$0.183336\pi$
$863$	32.0000i	1.08929i	−0.838666	+	0.544646i	$0.816664\pi$
$864$	0	0
$865$	0	0
$866$	2.00000	0.0679628
$867$	0	0
$868$	8.00000i	0.271538i
$869$	32.0000	1.08553
$870$	0	0
$871$	72.0000	2.43963
$872$	6.00000i	0.203186i
$873$	6.00000i	0.203069i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	14.0000i	0.472746i	0.971662	+	0.236373i	$0.0759588\pi$
$877$	14.0000i	0.472746i	−0.971662	+	0.236373i	$0.924041\pi$
$878$	8.00000i	0.269987i
$879$	0	0
$880$	0	0
$881$	2.00000	0.0673817	0.0336909	−	0.999432i	$-0.489274\pi$
$881$	2.00000	0.0673817	0.0336909	+	0.999432i	$0.489274\pi$
$882$	− 3.00000i	− 0.101015i
$883$	12.0000i	0.403832i	0.979403	+	0.201916i	$0.0647168\pi$
$883$	12.0000i	0.403832i	−0.979403	+	0.201916i	$0.935283\pi$
$884$	12.0000	0.403604
$885$	0	0
$886$	−20.0000	−0.671913
$887$	− 16.0000i	− 0.537227i	−0.963248	−	0.268614i	$-0.913434\pi$
$887$	− 16.0000i	− 0.537227i	0.963248	−	0.268614i	$-0.0865655\pi$
$888$	0	0
$889$	−8.00000	−0.268311
$890$	0	0
$891$	36.0000	1.20605
$892$	16.0000i	0.535720i
$893$	0	0
$894$	0	0
$895$	0	0
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	30.0000i	1.00111i
$899$	−48.0000	−1.60089
$900$	0	0
$901$	−4.00000	−0.133259
$902$	8.00000i	0.266371i
$903$	0	0
$904$	−2.00000	−0.0665190
$905$	0	0
$906$	0	0
$907$	− 52.0000i	− 1.72663i	−0.504664	−	0.863316i	$-0.668384\pi$
$907$	− 52.0000i	− 1.72663i	0.504664	−	0.863316i	$-0.331616\pi$
$908$	− 8.00000i	− 0.265489i
$909$	−18.0000	−0.597022
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	− 32.0000i	− 1.05905i
$914$	−10.0000	−0.330771
$915$	0	0
$916$	−14.0000	−0.462573
$917$	16.0000i	0.528367i
$918$	0	0
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	0	0
$921$	0	0
$922$	10.0000i	0.329332i
$923$	− 96.0000i	− 3.15988i
$924$	0	0
$925$	0	0
$926$	−16.0000	−0.525793
$927$	− 48.0000i	− 1.57653i
$928$	− 6.00000i	− 0.196960i
$929$	−58.0000	−1.90292	−0.951459	−	0.307775i	$-0.900416\pi$
$929$	−58.0000	−1.90292	−0.951459	+	0.307775i	$0.900416\pi$
$930$	0	0
$931$	0	0
$932$	− 6.00000i	− 0.196537i
$933$	0	0
$934$	40.0000	1.30884
$935$	0	0
$936$	18.0000	0.588348
$937$	50.0000i	1.63343i	0.577042	+	0.816714i	$0.304207\pi$
$937$	50.0000i	1.63343i	−0.577042	+	0.816714i	$0.695793\pi$
$938$	− 12.0000i	− 0.391814i
$939$	0	0
$940$	0	0
$941$	2.00000	0.0651981	0.0325991	−	0.999469i	$-0.489622\pi$
$941$	2.00000	0.0651981	0.0325991	+	0.999469i	$0.489622\pi$
$942$	0	0
$943$	0	0
$944$	8.00000	0.260378
$945$	0	0
$946$	16.0000	0.520205
$947$	44.0000i	1.42981i	0.699223	+	0.714904i	$0.253530\pi$
$947$	44.0000i	1.42981i	−0.699223	+	0.714904i	$0.746470\pi$
$948$	0	0
$949$	12.0000	0.389536
$950$	0	0
$951$	0	0
$952$	− 2.00000i	− 0.0648204i
$953$	54.0000i	1.74923i	0.484817	+	0.874616i	$0.338886\pi$
$953$	54.0000i	1.74923i	−0.484817	+	0.874616i	$0.661114\pi$
$954$	−6.00000	−0.194257
$955$	0	0
$956$	16.0000	0.517477
$957$	0	0
$958$	− 24.0000i	− 0.775405i
$959$	−6.00000	−0.193750
$960$	0	0
$961$	33.0000	1.06452
$962$	60.0000i	1.93448i
$963$	36.0000i	1.16008i
$964$	−10.0000	−0.322078
$965$	0	0
$966$	0	0
$967$	16.0000i	0.514525i	0.966342	+	0.257263i	$0.0828206\pi$
$967$	16.0000i	0.514525i	−0.966342	+	0.257263i	$0.917179\pi$
$968$	− 5.00000i	− 0.160706i
$969$	0	0
$970$	0	0
$971$	−24.0000	−0.770197	−0.385098	−	0.922876i	$-0.625832\pi$
$971$	−24.0000	−0.770197	−0.385098	+	0.922876i	$0.625832\pi$
$972$	0	0
$973$	16.0000i	0.512936i
$974$	−32.0000	−1.02535
$975$	0	0
$976$	−14.0000	−0.448129
$977$	− 30.0000i	− 0.959785i	−0.877327	−	0.479893i	$-0.840676\pi$
$977$	− 30.0000i	− 0.959785i	0.877327	−	0.479893i	$-0.159324\pi$
$978$	0	0
$979$	−40.0000	−1.27841
$980$	0	0
$981$	−18.0000	−0.574696
$982$	12.0000i	0.382935i
$983$	16.0000i	0.510321i	0.966899	+	0.255160i	$0.0821283\pi$
$983$	16.0000i	0.510321i	−0.966899	+	0.255160i	$0.917872\pi$
$984$	0	0
$985$	0	0
$986$	12.0000	0.382158
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	40.0000	1.27064	0.635321	−	0.772248i	$-0.280868\pi$
$991$	40.0000	1.27064	0.635321	+	0.772248i	$0.280868\pi$
$992$	8.00000i	0.254000i
$993$	0	0
$994$	−16.0000	−0.507489
$995$	0	0
$996$	0	0
$997$	− 22.0000i	− 0.696747i	−0.937356	−	0.348373i	$-0.886734\pi$
$997$	− 22.0000i	− 0.696747i	0.937356	−	0.348373i	$-0.113266\pi$
$998$	12.0000i	0.379853i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.2.c.b.99.2		2
3.2	odd	2		3150.2.g.c.2899.1		2
4.3	odd	2		2800.2.g.n.449.2		2
5.2	odd	4		350.2.a.b.1.1		1
5.3	odd	4		70.2.a.a.1.1	✓	1
5.4	even	2	inner	350.2.c.b.99.1		2
7.6	odd	2		2450.2.c.k.99.2		2
15.2	even	4		3150.2.a.bj.1.1		1
15.8	even	4		630.2.a.d.1.1		1
15.14	odd	2		3150.2.g.c.2899.2		2
20.3	even	4		560.2.a.d.1.1		1
20.7	even	4		2800.2.a.m.1.1		1
20.19	odd	2		2800.2.g.n.449.1		2
35.3	even	12		490.2.e.c.471.1		2
35.13	even	4		490.2.a.h.1.1		1
35.18	odd	12		490.2.e.d.471.1		2
35.23	odd	12		490.2.e.d.361.1		2
35.27	even	4		2450.2.a.l.1.1		1
35.33	even	12		490.2.e.c.361.1		2
35.34	odd	2		2450.2.c.k.99.1		2
40.3	even	4		2240.2.a.q.1.1		1
40.13	odd	4		2240.2.a.n.1.1		1
55.43	even	4		8470.2.a.j.1.1		1
60.23	odd	4		5040.2.a.bm.1.1		1
105.83	odd	4		4410.2.a.b.1.1		1
140.83	odd	4		3920.2.a.t.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
70.2.a.a.1.1	✓	1	5.3	odd	4
350.2.a.b.1.1		1	5.2	odd	4
350.2.c.b.99.1		2	5.4	even	2	inner
350.2.c.b.99.2		2	1.1	even	1	trivial
490.2.a.h.1.1		1	35.13	even	4
490.2.e.c.361.1		2	35.33	even	12
490.2.e.c.471.1		2	35.3	even	12
490.2.e.d.361.1		2	35.23	odd	12
490.2.e.d.471.1		2	35.18	odd	12
560.2.a.d.1.1		1	20.3	even	4
630.2.a.d.1.1		1	15.8	even	4
2240.2.a.n.1.1		1	40.13	odd	4
2240.2.a.q.1.1		1	40.3	even	4
2450.2.a.l.1.1		1	35.27	even	4
2450.2.c.k.99.1		2	35.34	odd	2
2450.2.c.k.99.2		2	7.6	odd	2
2800.2.a.m.1.1		1	20.7	even	4
2800.2.g.n.449.1		2	20.19	odd	2
2800.2.g.n.449.2		2	4.3	odd	2
3150.2.a.bj.1.1		1	15.2	even	4
3150.2.g.c.2899.1		2	3.2	odd	2
3150.2.g.c.2899.2		2	15.14	odd	2
3920.2.a.t.1.1		1	140.83	odd	4
4410.2.a.b.1.1		1	105.83	odd	4
5040.2.a.bm.1.1		1	60.23	odd	4
8470.2.a.j.1.1		1	55.43	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:	\( N \)	\(=\)	\( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	350.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{7} -1.00000i q^{8} +3.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{7} -1.00000i q^{8} +3.00000 q^{9} +4.00000 q^{11} +6.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} +2.00000i q^{17} +3.00000i q^{18} +4.00000i q^{22} -6.00000 q^{26} +1.00000i q^{28} -6.00000 q^{29} +8.00000 q^{31} +1.00000i q^{32} -2.00000 q^{34} -3.00000 q^{36} -10.0000i q^{37} +2.00000 q^{41} -4.00000i q^{43} -4.00000 q^{44} +8.00000i q^{47} -1.00000 q^{49} -6.00000i q^{52} +2.00000i q^{53} -1.00000 q^{56} -6.00000i q^{58} +8.00000 q^{59} -14.0000 q^{61} +8.00000i q^{62} -3.00000i q^{63} -1.00000 q^{64} -12.0000i q^{67} -2.00000i q^{68} -16.0000 q^{71} -3.00000i q^{72} -2.00000i q^{73} +10.0000 q^{74} -4.00000i q^{77} +8.00000 q^{79} +9.00000 q^{81} +2.00000i q^{82} -8.00000i q^{83} +4.00000 q^{86} -4.00000i q^{88} -10.0000 q^{89} +6.00000 q^{91} -8.00000 q^{94} +2.00000i q^{97} -1.00000i q^{98} +12.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 6 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 + 6 * q^9	\( 2 q - 2 q^{4} + 6 q^{9} + 8 q^{11} + 2 q^{14} + 2 q^{16} - 12 q^{26} - 12 q^{29} + 16 q^{31} - 4 q^{34} - 6 q^{36} + 4 q^{41} - 8 q^{44} - 2 q^{49} - 2 q^{56} + 16 q^{59} - 28 q^{61} - 2 q^{64} - 32 q^{71} + 20 q^{74} + 16 q^{79} + 18 q^{81} + 8 q^{86} - 20 q^{89} + 12 q^{91} - 16 q^{94} + 24 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 + 6 * q^9 + 8 * q^11 + 2 * q^14 + 2 * q^16 - 12 * q^26 - 12 * q^29 + 16 * q^31 - 4 * q^34 - 6 * q^36 + 4 * q^41 - 8 * q^44 - 2 * q^49 - 2 * q^56 + 16 * q^59 - 28 * q^61 - 2 * q^64 - 32 * q^71 + 20 * q^74 + 16 * q^79 + 18 * q^81 + 8 * q^86 - 20 * q^89 + 12 * q^91 - 16 * q^94 + 24 * q^99

Introduction

L-functions

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