LMFDB - Embedded newform 1950.2.e.e.1249.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1950,2,Mod(1249,1950)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1950, 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("1950.1249");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1950.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$15.5708283941$
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 390)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1950.1249
Dual form			1950.2.e.e.1249.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -1.00000i q^{12} -1.00000i q^{13} +4.00000 q^{14} +1.00000 q^{16} +2.00000i q^{17} -1.00000i q^{18} -4.00000 q^{19} +4.00000 q^{21} +8.00000i q^{23} +1.00000 q^{24} +1.00000 q^{26} -1.00000i q^{27} +4.00000i q^{28} -2.00000 q^{29} -8.00000 q^{31} +1.00000i q^{32} -2.00000 q^{34} +1.00000 q^{36} -2.00000i q^{37} -4.00000i q^{38} +1.00000 q^{39} -6.00000 q^{41} +4.00000i q^{42} +12.0000i q^{43} -8.00000 q^{46} +1.00000i q^{48} -9.00000 q^{49} -2.00000 q^{51} +1.00000i q^{52} +10.0000i q^{53} +1.00000 q^{54} -4.00000 q^{56} -4.00000i q^{57} -2.00000i q^{58} -10.0000 q^{61} -8.00000i q^{62} +4.00000i q^{63} -1.00000 q^{64} +4.00000i q^{67} -2.00000i q^{68} -8.00000 q^{69} -16.0000 q^{71} +1.00000i q^{72} -6.00000i q^{73} +2.00000 q^{74} +4.00000 q^{76} +1.00000i q^{78} +8.00000 q^{79} +1.00000 q^{81} -6.00000i q^{82} -4.00000i q^{83} -4.00000 q^{84} -12.0000 q^{86} -2.00000i q^{87} +14.0000 q^{89} -4.00000 q^{91} -8.00000i q^{92} -8.00000i q^{93} -1.00000 q^{96} +6.00000i q^{97} -9.00000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9	$ 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} + 8 q^{14} + 2 q^{16} - 8 q^{19} + 8 q^{21} + 2 q^{24} + 2 q^{26} - 4 q^{29} - 16 q^{31} - 4 q^{34} + 2 q^{36} + 2 q^{39} - 12 q^{41} - 16 q^{46} - 18 q^{49} - 4 q^{51} + 2 q^{54} - 8 q^{56} - 20 q^{61} - 2 q^{64} - 16 q^{69} - 32 q^{71} + 4 q^{74} + 8 q^{76} + 16 q^{79} + 2 q^{81} - 8 q^{84} - 24 q^{86} + 28 q^{89} - 8 q^{91} - 2 q^{96}+O(q^{100}) $ 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 + 8 * q^14 + 2 * q^16 - 8 * q^19 + 8 * q^21 + 2 * q^24 + 2 * q^26 - 4 * q^29 - 16 * q^31 - 4 * q^34 + 2 * q^36 + 2 * q^39 - 12 * q^41 - 16 * q^46 - 18 * q^49 - 4 * q^51 + 2 * q^54 - 8 * q^56 - 20 * q^61 - 2 * q^64 - 16 * q^69 - 32 * q^71 + 4 * q^74 + 8 * q^76 + 16 * q^79 + 2 * q^81 - 8 * q^84 - 24 * q^86 + 28 * q^89 - 8 * q^91 - 2 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1950\mathbb{Z}\right)^\times$.

$n$	$301$	$1301$	$1327$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	−1.00000	−0.408248
$7$	− 4.00000i	− 1.51186i	−0.654654	−	0.755929i	$-0.727186\pi$
$7$	− 4.00000i	− 1.51186i	0.654654	−	0.755929i	$-0.272814\pi$
$8$	− 1.00000i	− 0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	− 1.00000i	− 0.288675i
$13$	− 1.00000i	− 0.277350i
$13$	− 1.00000i	− 0.277350i
$14$	4.00000	1.06904
$15$	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	$0.922021\pi$
$18$	− 1.00000i	− 0.235702i
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	4.00000	0.872872
$22$	0	0
$23$	8.00000i	1.66812i	0.551677	+	0.834058i	$0.313988\pi$
$23$	8.00000i	1.66812i	−0.551677	+	0.834058i	$0.686012\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	1.00000	0.196116
$27$	− 1.00000i	− 0.192450i
$28$	4.00000i	0.755929i
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	1.00000i	0.176777i
$33$	0	0
$34$	−2.00000	−0.342997
$35$	0	0
$36$	1.00000	0.166667
$37$	− 2.00000i	− 0.328798i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$	− 2.00000i	− 0.328798i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$	− 4.00000i	− 0.648886i
$39$	1.00000	0.160128
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	4.00000i	0.617213i
$43$	12.0000i	1.82998i	0.403473	+	0.914991i	$0.367803\pi$
$43$	12.0000i	1.82998i	−0.403473	+	0.914991i	$0.632197\pi$
$44$	0	0
$45$	0	0
$46$	−8.00000	−1.17954
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	1.00000i	0.144338i
$49$	−9.00000	−1.28571
$50$	0	0
$51$	−2.00000	−0.280056
$52$	1.00000i	0.138675i
$53$	10.0000i	1.37361i	0.726844	+	0.686803i	$0.240986\pi$
$53$	10.0000i	1.37361i	−0.726844	+	0.686803i	$0.759014\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	−4.00000	−0.534522
$57$	− 4.00000i	− 0.529813i
$58$	− 2.00000i	− 0.262613i
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	− 8.00000i	− 1.01600i
$63$	4.00000i	0.503953i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	4.00000i	0.488678i	0.969690	+	0.244339i	$0.0785709\pi$
$67$	4.00000i	0.488678i	−0.969690	+	0.244339i	$0.921429\pi$
$68$	− 2.00000i	− 0.242536i
$69$	−8.00000	−0.963087
$70$	0	0
$71$	−16.0000	−1.89885	−0.949425	−	0.313993i	$-0.898333\pi$
$71$	−16.0000	−1.89885	−0.949425	+	0.313993i	$0.898333\pi$
$72$	1.00000i	0.117851i
$73$	− 6.00000i	− 0.702247i	−0.936329	−	0.351123i	$-0.885800\pi$
$73$	− 6.00000i	− 0.702247i	0.936329	−	0.351123i	$-0.114200\pi$
$74$	2.00000	0.232495
$75$	0	0
$76$	4.00000	0.458831
$77$	0	0
$78$	1.00000i	0.113228i
$79$	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
$81$	1.00000	0.111111
$82$	− 6.00000i	− 0.662589i
$83$	− 4.00000i	− 0.439057i	−0.975606	−	0.219529i	$-0.929548\pi$
$83$	− 4.00000i	− 0.439057i	0.975606	−	0.219529i	$-0.0704519\pi$
$84$	−4.00000	−0.436436
$85$	0	0
$86$	−12.0000	−1.29399
$87$	− 2.00000i	− 0.214423i
$88$	0	0
$89$	14.0000	1.48400	0.741999	−	0.670402i	$-0.233878\pi$
$89$	14.0000	1.48400	0.741999	+	0.670402i	$0.233878\pi$
$90$	0	0
$91$	−4.00000	−0.419314
$92$	− 8.00000i	− 0.834058i
$93$	− 8.00000i	− 0.829561i
$94$	0	0
$95$	0	0
$96$	−1.00000	−0.102062
$97$	6.00000i	0.609208i	0.952479	+	0.304604i	$0.0985241\pi$
$97$	6.00000i	0.609208i	−0.952479	+	0.304604i	$0.901476\pi$
$98$	− 9.00000i	− 0.909137i
$99$	0	0
$100$	0	0
$101$	10.0000	0.995037	0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000	0.995037	0.497519	+	0.867453i	$0.334245\pi$
$102$	− 2.00000i	− 0.198030i
$103$	− 4.00000i	− 0.394132i	−0.980390	−	0.197066i	$-0.936859\pi$
$103$	− 4.00000i	− 0.394132i	0.980390	−	0.197066i	$-0.0631413\pi$
$104$	−1.00000	−0.0980581
$105$	0	0
$106$	−10.0000	−0.971286
$107$	12.0000i	1.16008i	0.814587	+	0.580042i	$0.196964\pi$
$107$	12.0000i	1.16008i	−0.814587	+	0.580042i	$0.803036\pi$
$108$	1.00000i	0.0962250i
$109$	−14.0000	−1.34096	−0.670478	−	0.741929i	$-0.733911\pi$
$109$	−14.0000	−1.34096	−0.670478	+	0.741929i	$0.733911\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	− 4.00000i	− 0.377964i
$113$	− 10.0000i	− 0.940721i	−0.882474	−	0.470360i	$-0.844124\pi$
$113$	− 10.0000i	− 0.940721i	0.882474	−	0.470360i	$-0.155876\pi$
$114$	4.00000	0.374634
$115$	0	0
$116$	2.00000	0.185695
$117$	1.00000i	0.0924500i
$118$	0	0
$119$	8.00000	0.733359
$120$	0	0
$121$	−11.0000	−1.00000
$122$	− 10.0000i	− 0.905357i
$123$	− 6.00000i	− 0.541002i
$124$	8.00000	0.718421
$125$	0	0
$126$	−4.00000	−0.356348
$127$	12.0000i	1.06483i	0.846484	+	0.532414i	$0.178715\pi$
$127$	12.0000i	1.06483i	−0.846484	+	0.532414i	$0.821285\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	−12.0000	−1.05654
$130$	0	0
$131$	8.00000	0.698963	0.349482	−	0.936943i	$-0.386358\pi$
$131$	8.00000	0.698963	0.349482	+	0.936943i	$0.386358\pi$
$132$	0	0
$133$	16.0000i	1.38738i
$134$	−4.00000	−0.345547
$135$	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	$-0.0825059\pi$
$138$	− 8.00000i	− 0.681005i
$139$	20.0000	1.69638	0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000	1.69638	0.848189	+	0.529694i	$0.177693\pi$
$140$	0	0
$141$	0	0
$142$	− 16.0000i	− 1.34269i
$143$	0	0
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	6.00000	0.496564
$147$	− 9.00000i	− 0.742307i
$148$	2.00000i	0.164399i
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	4.00000i	0.324443i
$153$	− 2.00000i	− 0.161690i
$154$	0	0
$155$	0	0
$156$	−1.00000	−0.0800641
$157$	− 18.0000i	− 1.43656i	−0.695756	−	0.718278i	$-0.744931\pi$
$157$	− 18.0000i	− 1.43656i	0.695756	−	0.718278i	$-0.255069\pi$
$158$	8.00000i	0.636446i
$159$	−10.0000	−0.793052
$160$	0	0
$161$	32.0000	2.52195
$162$	1.00000i	0.0785674i
$163$	− 20.0000i	− 1.56652i	−0.621694	−	0.783260i	$-0.713555\pi$
$163$	− 20.0000i	− 1.56652i	0.621694	−	0.783260i	$-0.286445\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	4.00000	0.310460
$167$	− 16.0000i	− 1.23812i	−0.785345	−	0.619059i	$-0.787514\pi$
$167$	− 16.0000i	− 1.23812i	0.785345	−	0.619059i	$-0.212486\pi$
$168$	− 4.00000i	− 0.308607i
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	4.00000	0.305888
$172$	− 12.0000i	− 0.914991i
$173$	2.00000i	0.152057i	0.997106	+	0.0760286i	$0.0242240\pi$
$173$	2.00000i	0.152057i	−0.997106	+	0.0760286i	$0.975776\pi$
$174$	2.00000	0.151620
$175$	0	0
$176$	0	0
$177$	0	0
$178$	14.0000i	1.04934i
$179$	−16.0000	−1.19590	−0.597948	−	0.801535i	$-0.704017\pi$
$179$	−16.0000	−1.19590	−0.597948	+	0.801535i	$0.704017\pi$
$180$	0	0
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	− 4.00000i	− 0.296500i
$183$	− 10.0000i	− 0.739221i
$184$	8.00000	0.589768
$185$	0	0
$186$	8.00000	0.586588
$187$	0	0
$188$	0	0
$189$	−4.00000	−0.290957
$190$	0	0
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	− 1.00000i	− 0.0721688i
$193$	18.0000i	1.29567i	0.761781	+	0.647834i	$0.224325\pi$
$193$	18.0000i	1.29567i	−0.761781	+	0.647834i	$0.775675\pi$
$194$	−6.00000	−0.430775
$195$	0	0
$196$	9.00000	0.642857
$197$	6.00000i	0.427482i	0.976890	+	0.213741i	$0.0685649\pi$
$197$	6.00000i	0.427482i	−0.976890	+	0.213741i	$0.931435\pi$
$198$	0	0
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	0	0
$201$	−4.00000	−0.282138
$202$	10.0000i	0.703598i
$203$	8.00000i	0.561490i
$204$	2.00000	0.140028
$205$	0	0
$206$	4.00000	0.278693
$207$	− 8.00000i	− 0.556038i
$208$	− 1.00000i	− 0.0693375i
$209$	0	0
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	− 10.0000i	− 0.686803i
$213$	− 16.0000i	− 1.09630i
$214$	−12.0000	−0.820303
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	32.0000i	2.17230i
$218$	− 14.0000i	− 0.948200i
$219$	6.00000	0.405442
$220$	0	0
$221$	2.00000	0.134535
$222$	2.00000i	0.134231i
$223$	12.0000i	0.803579i	0.915732	+	0.401790i	$0.131612\pi$
$223$	12.0000i	0.803579i	−0.915732	+	0.401790i	$0.868388\pi$
$224$	4.00000	0.267261
$225$	0	0
$226$	10.0000	0.665190
$227$	12.0000i	0.796468i	0.917284	+	0.398234i	$0.130377\pi$
$227$	12.0000i	0.796468i	−0.917284	+	0.398234i	$0.869623\pi$
$228$	4.00000i	0.264906i
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	0	0
$231$	0	0
$232$	2.00000i	0.131306i
$233$	− 26.0000i	− 1.70332i	−0.524097	−	0.851658i	$-0.675597\pi$
$233$	− 26.0000i	− 1.70332i	0.524097	−	0.851658i	$-0.324403\pi$
$234$	−1.00000	−0.0653720
$235$	0	0
$236$	0	0
$237$	8.00000i	0.519656i
$238$	8.00000i	0.518563i
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	− 11.0000i	− 0.707107i
$243$	1.00000i	0.0641500i
$244$	10.0000	0.640184
$245$	0	0
$246$	6.00000	0.382546
$247$	4.00000i	0.254514i
$248$	8.00000i	0.508001i
$249$	4.00000	0.253490
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	− 4.00000i	− 0.251976i
$253$	0	0
$254$	−12.0000	−0.752947
$255$	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	$-0.240706\pi$
$258$	− 12.0000i	− 0.747087i
$259$	−8.00000	−0.497096
$260$	0	0
$261$	2.00000	0.123797
$262$	8.00000i	0.494242i
$263$	24.0000i	1.47990i	0.672660	+	0.739952i	$0.265152\pi$
$263$	24.0000i	1.47990i	−0.672660	+	0.739952i	$0.734848\pi$
$264$	0	0
$265$	0	0
$266$	−16.0000	−0.981023
$267$	14.0000i	0.856786i
$268$	− 4.00000i	− 0.244339i
$269$	14.0000	0.853595	0.426798	−	0.904347i	$-0.359642\pi$
$269$	14.0000	0.853595	0.426798	+	0.904347i	$0.359642\pi$
$270$	0	0
$271$	−32.0000	−1.94386	−0.971931	−	0.235267i	$-0.924404\pi$
$271$	−32.0000	−1.94386	−0.971931	+	0.235267i	$0.924404\pi$
$272$	2.00000i	0.121268i
$273$	− 4.00000i	− 0.242091i
$274$	6.00000	0.362473
$275$	0	0
$276$	8.00000	0.481543
$277$	− 18.0000i	− 1.08152i	−0.841178	−	0.540758i	$-0.818138\pi$
$277$	− 18.0000i	− 1.08152i	0.841178	−	0.540758i	$-0.181862\pi$
$278$	20.0000i	1.19952i
$279$	8.00000	0.478947
$280$	0	0
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	0	0
$283$	20.0000i	1.18888i	0.804141	+	0.594438i	$0.202626\pi$
$283$	20.0000i	1.18888i	−0.804141	+	0.594438i	$0.797374\pi$
$284$	16.0000	0.949425
$285$	0	0
$286$	0	0
$287$	24.0000i	1.41668i
$288$	− 1.00000i	− 0.0589256i
$289$	13.0000	0.764706
$290$	0	0
$291$	−6.00000	−0.351726
$292$	6.00000i	0.351123i
$293$	− 6.00000i	− 0.350524i	−0.984522	−	0.175262i	$-0.943923\pi$
$293$	− 6.00000i	− 0.350524i	0.984522	−	0.175262i	$-0.0560772\pi$
$294$	9.00000	0.524891
$295$	0	0
$296$	−2.00000	−0.116248
$297$	0	0
$298$	− 10.0000i	− 0.579284i
$299$	8.00000	0.462652
$300$	0	0
$301$	48.0000	2.76667
$302$	− 8.00000i	− 0.460348i
$303$	10.0000i	0.574485i
$304$	−4.00000	−0.229416
$305$	0	0
$306$	2.00000	0.114332
$307$	12.0000i	0.684876i	0.939540	+	0.342438i	$0.111253\pi$
$307$	12.0000i	0.684876i	−0.939540	+	0.342438i	$0.888747\pi$
$308$	0	0
$309$	4.00000	0.227552
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	− 1.00000i	− 0.0566139i
$313$	18.0000i	1.01742i	0.860938	+	0.508710i	$0.169877\pi$
$313$	18.0000i	1.01742i	−0.860938	+	0.508710i	$0.830123\pi$
$314$	18.0000	1.01580
$315$	0	0
$316$	−8.00000	−0.450035
$317$	14.0000i	0.786318i	0.919470	+	0.393159i	$0.128618\pi$
$317$	14.0000i	0.786318i	−0.919470	+	0.393159i	$0.871382\pi$
$318$	− 10.0000i	− 0.560772i
$319$	0	0
$320$	0	0
$321$	−12.0000	−0.669775
$322$	32.0000i	1.78329i
$323$	− 8.00000i	− 0.445132i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	20.0000	1.10770
$327$	− 14.0000i	− 0.774202i
$328$	6.00000i	0.331295i
$329$	0	0
$330$	0	0
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	4.00000i	0.219529i
$333$	2.00000i	0.109599i
$334$	16.0000	0.875481
$335$	0	0
$336$	4.00000	0.218218
$337$	22.0000i	1.19842i	0.800593	+	0.599208i	$0.204518\pi$
$337$	22.0000i	1.19842i	−0.800593	+	0.599208i	$0.795482\pi$
$338$	− 1.00000i	− 0.0543928i
$339$	10.0000	0.543125
$340$	0	0
$341$	0	0
$342$	4.00000i	0.216295i
$343$	8.00000i	0.431959i
$344$	12.0000	0.646997
$345$	0	0
$346$	−2.00000	−0.107521
$347$	4.00000i	0.214731i	0.994220	+	0.107366i	$0.0342415\pi$
$347$	4.00000i	0.214731i	−0.994220	+	0.107366i	$0.965758\pi$
$348$	2.00000i	0.107211i
$349$	−14.0000	−0.749403	−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000	−0.749403	−0.374701	+	0.927146i	$0.622255\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	− 26.0000i	− 1.38384i	−0.721974	−	0.691920i	$-0.756765\pi$
$353$	− 26.0000i	− 1.38384i	0.721974	−	0.691920i	$-0.243235\pi$
$354$	0	0
$355$	0	0
$356$	−14.0000	−0.741999
$357$	8.00000i	0.423405i
$358$	− 16.0000i	− 0.845626i
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	− 2.00000i	− 0.105118i
$363$	− 11.0000i	− 0.577350i
$364$	4.00000	0.209657
$365$	0	0
$366$	10.0000	0.522708
$367$	− 4.00000i	− 0.208798i	−0.994535	−	0.104399i	$-0.966708\pi$
$367$	− 4.00000i	− 0.208798i	0.994535	−	0.104399i	$-0.0332919\pi$
$368$	8.00000i	0.417029i
$369$	6.00000	0.312348
$370$	0	0
$371$	40.0000	2.07670
$372$	8.00000i	0.414781i
$373$	− 22.0000i	− 1.13912i	−0.821951	−	0.569558i	$-0.807114\pi$
$373$	− 22.0000i	− 1.13912i	0.821951	−	0.569558i	$-0.192886\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.00000i	0.103005i
$378$	− 4.00000i	− 0.205738i
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	0	0
$381$	−12.0000	−0.614779
$382$	8.00000i	0.409316i
$383$	− 16.0000i	− 0.817562i	−0.912633	−	0.408781i	$-0.865954\pi$
$383$	− 16.0000i	− 0.817562i	0.912633	−	0.408781i	$-0.134046\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−18.0000	−0.916176
$387$	− 12.0000i	− 0.609994i
$388$	− 6.00000i	− 0.304604i
$389$	14.0000	0.709828	0.354914	−	0.934899i	$-0.384510\pi$
$389$	14.0000	0.709828	0.354914	+	0.934899i	$0.384510\pi$
$390$	0	0
$391$	−16.0000	−0.809155
$392$	9.00000i	0.454569i
$393$	8.00000i	0.403547i
$394$	−6.00000	−0.302276
$395$	0	0
$396$	0	0
$397$	− 10.0000i	− 0.501886i	−0.968002	−	0.250943i	$-0.919259\pi$
$397$	− 10.0000i	− 0.501886i	0.968002	−	0.250943i	$-0.0807406\pi$
$398$	8.00000i	0.401004i
$399$	−16.0000	−0.801002
$400$	0	0
$401$	26.0000	1.29838	0.649189	−	0.760627i	$-0.275108\pi$
$401$	26.0000	1.29838	0.649189	+	0.760627i	$0.275108\pi$
$402$	− 4.00000i	− 0.199502i
$403$	8.00000i	0.398508i
$404$	−10.0000	−0.497519
$405$	0	0
$406$	−8.00000	−0.397033
$407$	0	0
$408$	2.00000i	0.0990148i
$409$	−10.0000	−0.494468	−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000	−0.494468	−0.247234	+	0.968956i	$0.579522\pi$
$410$	0	0
$411$	6.00000	0.295958
$412$	4.00000i	0.197066i
$413$	0	0
$414$	8.00000	0.393179
$415$	0	0
$416$	1.00000	0.0490290
$417$	20.0000i	0.979404i
$418$	0	0
$419$	24.0000	1.17248	0.586238	−	0.810139i	$-0.300608\pi$
$419$	24.0000	1.17248	0.586238	+	0.810139i	$0.300608\pi$
$420$	0	0
$421$	30.0000	1.46211	0.731055	−	0.682318i	$-0.239028\pi$
$421$	30.0000	1.46211	0.731055	+	0.682318i	$0.239028\pi$
$422$	− 12.0000i	− 0.584151i
$423$	0	0
$424$	10.0000	0.485643
$425$	0	0
$426$	16.0000	0.775203
$427$	40.0000i	1.93574i
$428$	− 12.0000i	− 0.580042i
$429$	0	0
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	− 1.00000i	− 0.0481125i
$433$	10.0000i	0.480569i	0.970702	+	0.240285i	$0.0772408\pi$
$433$	10.0000i	0.480569i	−0.970702	+	0.240285i	$0.922759\pi$
$434$	−32.0000	−1.53605
$435$	0	0
$436$	14.0000	0.670478
$437$	− 32.0000i	− 1.53077i
$438$	6.00000i	0.286691i
$439$	−40.0000	−1.90910	−0.954548	−	0.298057i	$-0.903661\pi$
$439$	−40.0000	−1.90910	−0.954548	+	0.298057i	$0.903661\pi$
$440$	0	0
$441$	9.00000	0.428571
$442$	2.00000i	0.0951303i
$443$	− 36.0000i	− 1.71041i	−0.518289	−	0.855206i	$-0.673431\pi$
$443$	− 36.0000i	− 1.71041i	0.518289	−	0.855206i	$-0.326569\pi$
$444$	−2.00000	−0.0949158
$445$	0	0
$446$	−12.0000	−0.568216
$447$	− 10.0000i	− 0.472984i
$448$	4.00000i	0.188982i
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	0	0
$451$	0	0
$452$	10.0000i	0.470360i
$453$	− 8.00000i	− 0.375873i
$454$	−12.0000	−0.563188
$455$	0	0
$456$	−4.00000	−0.187317
$457$	− 18.0000i	− 0.842004i	−0.907060	−	0.421002i	$-0.861678\pi$
$457$	− 18.0000i	− 0.842004i	0.907060	−	0.421002i	$-0.138322\pi$
$458$	− 14.0000i	− 0.654177i
$459$	2.00000	0.0933520
$460$	0	0
$461$	2.00000	0.0931493	0.0465746	−	0.998915i	$-0.485169\pi$
$461$	2.00000	0.0931493	0.0465746	+	0.998915i	$0.485169\pi$
$462$	0	0
$463$	20.0000i	0.929479i	0.885448	+	0.464739i	$0.153852\pi$
$463$	20.0000i	0.929479i	−0.885448	+	0.464739i	$0.846148\pi$
$464$	−2.00000	−0.0928477
$465$	0	0
$466$	26.0000	1.20443
$467$	36.0000i	1.66588i	0.553362	+	0.832941i	$0.313345\pi$
$467$	36.0000i	1.66588i	−0.553362	+	0.832941i	$0.686655\pi$
$468$	− 1.00000i	− 0.0462250i
$469$	16.0000	0.738811
$470$	0	0
$471$	18.0000	0.829396
$472$	0	0
$473$	0	0
$474$	−8.00000	−0.367452
$475$	0	0
$476$	−8.00000	−0.366679
$477$	− 10.0000i	− 0.457869i
$478$	− 24.0000i	− 1.09773i
$479$	−8.00000	−0.365529	−0.182765	−	0.983157i	$-0.558505\pi$
$479$	−8.00000	−0.365529	−0.182765	+	0.983157i	$0.558505\pi$
$480$	0	0
$481$	−2.00000	−0.0911922
$482$	2.00000i	0.0910975i
$483$	32.0000i	1.45605i
$484$	11.0000	0.500000
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	20.0000i	0.906287i	0.891438	+	0.453143i	$0.149697\pi$
$487$	20.0000i	0.906287i	−0.891438	+	0.453143i	$0.850303\pi$
$488$	10.0000i	0.452679i
$489$	20.0000	0.904431
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	6.00000i	0.270501i
$493$	− 4.00000i	− 0.180151i
$494$	−4.00000	−0.179969
$495$	0	0
$496$	−8.00000	−0.359211
$497$	64.0000i	2.87079i
$498$	4.00000i	0.179244i
$499$	44.0000	1.96971	0.984855	−	0.173379i	$-0.0554684\pi$
$499$	44.0000	1.96971	0.984855	+	0.173379i	$0.0554684\pi$
$500$	0	0
$501$	16.0000	0.714827
$502$	0	0
$503$	16.0000i	0.713405i	0.934218	+	0.356702i	$0.116099\pi$
$503$	16.0000i	0.713405i	−0.934218	+	0.356702i	$0.883901\pi$
$504$	4.00000	0.178174
$505$	0	0
$506$	0	0
$507$	− 1.00000i	− 0.0444116i
$508$	− 12.0000i	− 0.532414i
$509$	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
$511$	−24.0000	−1.06170
$512$	1.00000i	0.0441942i
$513$	4.00000i	0.176604i
$514$	22.0000	0.970378
$515$	0	0
$516$	12.0000	0.528271
$517$	0	0
$518$	− 8.00000i	− 0.351500i
$519$	−2.00000	−0.0877903
$520$	0	0
$521$	2.00000	0.0876216	0.0438108	−	0.999040i	$-0.486050\pi$
$521$	2.00000	0.0876216	0.0438108	+	0.999040i	$0.486050\pi$
$522$	2.00000i	0.0875376i
$523$	20.0000i	0.874539i	0.899331	+	0.437269i	$0.144054\pi$
$523$	20.0000i	0.874539i	−0.899331	+	0.437269i	$0.855946\pi$
$524$	−8.00000	−0.349482
$525$	0	0
$526$	−24.0000	−1.04645
$527$	− 16.0000i	− 0.696971i
$528$	0	0
$529$	−41.0000	−1.78261
$530$	0	0
$531$	0	0
$532$	− 16.0000i	− 0.693688i
$533$	6.00000i	0.259889i
$534$	−14.0000	−0.605839
$535$	0	0
$536$	4.00000	0.172774
$537$	− 16.0000i	− 0.690451i
$538$	14.0000i	0.603583i
$539$	0	0
$540$	0	0
$541$	−2.00000	−0.0859867	−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000	−0.0859867	−0.0429934	+	0.999075i	$0.513689\pi$
$542$	− 32.0000i	− 1.37452i
$543$	− 2.00000i	− 0.0858282i
$544$	−2.00000	−0.0857493
$545$	0	0
$546$	4.00000	0.171184
$547$	− 36.0000i	− 1.53925i	−0.638497	−	0.769624i	$-0.720443\pi$
$547$	− 36.0000i	− 1.53925i	0.638497	−	0.769624i	$-0.279557\pi$
$548$	6.00000i	0.256307i
$549$	10.0000	0.426790
$550$	0	0
$551$	8.00000	0.340811
$552$	8.00000i	0.340503i
$553$	− 32.0000i	− 1.36078i
$554$	18.0000	0.764747
$555$	0	0
$556$	−20.0000	−0.848189
$557$	− 18.0000i	− 0.762684i	−0.924434	−	0.381342i	$-0.875462\pi$
$557$	− 18.0000i	− 0.762684i	0.924434	−	0.381342i	$-0.124538\pi$
$558$	8.00000i	0.338667i
$559$	12.0000	0.507546
$560$	0	0
$561$	0	0
$562$	18.0000i	0.759284i
$563$	36.0000i	1.51722i	0.651546	+	0.758610i	$0.274121\pi$
$563$	36.0000i	1.51722i	−0.651546	+	0.758610i	$0.725879\pi$
$564$	0	0
$565$	0	0
$566$	−20.0000	−0.840663
$567$	− 4.00000i	− 0.167984i
$568$	16.0000i	0.671345i
$569$	30.0000	1.25767	0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000	1.25767	0.628833	+	0.777541i	$0.283533\pi$
$570$	0	0
$571$	36.0000	1.50655	0.753277	−	0.657704i	$-0.228472\pi$
$571$	36.0000	1.50655	0.753277	+	0.657704i	$0.228472\pi$
$572$	0	0
$573$	8.00000i	0.334205i
$574$	−24.0000	−1.00174
$575$	0	0
$576$	1.00000	0.0416667
$577$	22.0000i	0.915872i	0.888985	+	0.457936i	$0.151411\pi$
$577$	22.0000i	0.915872i	−0.888985	+	0.457936i	$0.848589\pi$
$578$	13.0000i	0.540729i
$579$	−18.0000	−0.748054
$580$	0	0
$581$	−16.0000	−0.663792
$582$	− 6.00000i	− 0.248708i
$583$	0	0
$584$	−6.00000	−0.248282
$585$	0	0
$586$	6.00000	0.247858
$587$	12.0000i	0.495293i	0.968850	+	0.247647i	$0.0796572\pi$
$587$	12.0000i	0.495293i	−0.968850	+	0.247647i	$0.920343\pi$
$588$	9.00000i	0.371154i
$589$	32.0000	1.31854
$590$	0	0
$591$	−6.00000	−0.246807
$592$	− 2.00000i	− 0.0821995i
$593$	14.0000i	0.574911i	0.957794	+	0.287456i	$0.0928094\pi$
$593$	14.0000i	0.574911i	−0.957794	+	0.287456i	$0.907191\pi$
$594$	0	0
$595$	0	0
$596$	10.0000	0.409616
$597$	8.00000i	0.327418i
$598$	8.00000i	0.327144i
$599$	−40.0000	−1.63436	−0.817178	−	0.576386i	$-0.804463\pi$
$599$	−40.0000	−1.63436	−0.817178	+	0.576386i	$0.804463\pi$
$600$	0	0
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	48.0000i	1.95633i
$603$	− 4.00000i	− 0.162893i
$604$	8.00000	0.325515
$605$	0	0
$606$	−10.0000	−0.406222
$607$	− 28.0000i	− 1.13648i	−0.822861	−	0.568242i	$-0.807624\pi$
$607$	− 28.0000i	− 1.13648i	0.822861	−	0.568242i	$-0.192376\pi$
$608$	− 4.00000i	− 0.162221i
$609$	−8.00000	−0.324176
$610$	0	0
$611$	0	0
$612$	2.00000i	0.0808452i
$613$	34.0000i	1.37325i	0.727013	+	0.686624i	$0.240908\pi$
$613$	34.0000i	1.37325i	−0.727013	+	0.686624i	$0.759092\pi$
$614$	−12.0000	−0.484281
$615$	0	0
$616$	0	0
$617$	2.00000i	0.0805170i	0.999189	+	0.0402585i	$0.0128181\pi$
$617$	2.00000i	0.0805170i	−0.999189	+	0.0402585i	$0.987182\pi$
$618$	4.00000i	0.160904i
$619$	28.0000	1.12542	0.562708	−	0.826656i	$-0.309760\pi$
$619$	28.0000	1.12542	0.562708	+	0.826656i	$0.309760\pi$
$620$	0	0
$621$	8.00000	0.321029
$622$	0	0
$623$	− 56.0000i	− 2.24359i
$624$	1.00000	0.0400320
$625$	0	0
$626$	−18.0000	−0.719425
$627$	0	0
$628$	18.0000i	0.718278i
$629$	4.00000	0.159490
$630$	0	0
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	− 8.00000i	− 0.318223i
$633$	− 12.0000i	− 0.476957i
$634$	−14.0000	−0.556011
$635$	0	0
$636$	10.0000	0.396526
$637$	9.00000i	0.356593i
$638$	0	0
$639$	16.0000	0.632950
$640$	0	0
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	− 12.0000i	− 0.473602i
$643$	28.0000i	1.10421i	0.833774	+	0.552106i	$0.186176\pi$
$643$	28.0000i	1.10421i	−0.833774	+	0.552106i	$0.813824\pi$
$644$	−32.0000	−1.26098
$645$	0	0
$646$	8.00000	0.314756
$647$	48.0000i	1.88707i	0.331266	+	0.943537i	$0.392524\pi$
$647$	48.0000i	1.88707i	−0.331266	+	0.943537i	$0.607476\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	0	0
$650$	0	0
$651$	−32.0000	−1.25418
$652$	20.0000i	0.783260i
$653$	18.0000i	0.704394i	0.935926	+	0.352197i	$0.114565\pi$
$653$	18.0000i	0.704394i	−0.935926	+	0.352197i	$0.885435\pi$
$654$	14.0000	0.547443
$655$	0	0
$656$	−6.00000	−0.234261
$657$	6.00000i	0.234082i
$658$	0	0
$659$	−24.0000	−0.934907	−0.467454	−	0.884018i	$-0.654829\pi$
$659$	−24.0000	−0.934907	−0.467454	+	0.884018i	$0.654829\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	− 20.0000i	− 0.777322i
$663$	2.00000i	0.0776736i
$664$	−4.00000	−0.155230
$665$	0	0
$666$	−2.00000	−0.0774984
$667$	− 16.0000i	− 0.619522i
$668$	16.0000i	0.619059i
$669$	−12.0000	−0.463947
$670$	0	0
$671$	0	0
$672$	4.00000i	0.154303i
$673$	− 38.0000i	− 1.46479i	−0.680879	−	0.732396i	$-0.738402\pi$
$673$	− 38.0000i	− 1.46479i	0.680879	−	0.732396i	$-0.261598\pi$
$674$	−22.0000	−0.847408
$675$	0	0
$676$	1.00000	0.0384615
$677$	− 26.0000i	− 0.999261i	−0.866239	−	0.499631i	$-0.833469\pi$
$677$	− 26.0000i	− 0.999261i	0.866239	−	0.499631i	$-0.166531\pi$
$678$	10.0000i	0.384048i
$679$	24.0000	0.921035
$680$	0	0
$681$	−12.0000	−0.459841
$682$	0	0
$683$	36.0000i	1.37750i	0.724998	+	0.688751i	$0.241841\pi$
$683$	36.0000i	1.37750i	−0.724998	+	0.688751i	$0.758159\pi$
$684$	−4.00000	−0.152944
$685$	0	0
$686$	−8.00000	−0.305441
$687$	− 14.0000i	− 0.534133i
$688$	12.0000i	0.457496i
$689$	10.0000	0.380970
$690$	0	0
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	− 2.00000i	− 0.0760286i
$693$	0	0
$694$	−4.00000	−0.151838
$695$	0	0
$696$	−2.00000	−0.0758098
$697$	− 12.0000i	− 0.454532i
$698$	− 14.0000i	− 0.529908i
$699$	26.0000	0.983410
$700$	0	0
$701$	18.0000	0.679851	0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000	0.679851	0.339925	+	0.940452i	$0.389598\pi$
$702$	− 1.00000i	− 0.0377426i
$703$	8.00000i	0.301726i
$704$	0	0
$705$	0	0
$706$	26.0000	0.978523
$707$	− 40.0000i	− 1.50435i
$708$	0	0
$709$	34.0000	1.27690	0.638448	−	0.769665i	$-0.279577\pi$
$709$	34.0000	1.27690	0.638448	+	0.769665i	$0.279577\pi$
$710$	0	0
$711$	−8.00000	−0.300023
$712$	− 14.0000i	− 0.524672i
$713$	− 64.0000i	− 2.39682i
$714$	−8.00000	−0.299392
$715$	0	0
$716$	16.0000	0.597948
$717$	− 24.0000i	− 0.896296i
$718$	− 24.0000i	− 0.895672i
$719$	−16.0000	−0.596699	−0.298350	−	0.954457i	$-0.596436\pi$
$719$	−16.0000	−0.596699	−0.298350	+	0.954457i	$0.596436\pi$
$720$	0	0
$721$	−16.0000	−0.595871
$722$	− 3.00000i	− 0.111648i
$723$	2.00000i	0.0743808i
$724$	2.00000	0.0743294
$725$	0	0
$726$	11.0000	0.408248
$727$	− 52.0000i	− 1.92857i	−0.264861	−	0.964287i	$-0.585326\pi$
$727$	− 52.0000i	− 1.92857i	0.264861	−	0.964287i	$-0.414674\pi$
$728$	4.00000i	0.148250i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−24.0000	−0.887672
$732$	10.0000i	0.369611i
$733$	− 22.0000i	− 0.812589i	−0.913742	−	0.406294i	$-0.866821\pi$
$733$	− 22.0000i	− 0.812589i	0.913742	−	0.406294i	$-0.133179\pi$
$734$	4.00000	0.147643
$735$	0	0
$736$	−8.00000	−0.294884
$737$	0	0
$738$	6.00000i	0.220863i
$739$	−4.00000	−0.147142	−0.0735712	−	0.997290i	$-0.523440\pi$
$739$	−4.00000	−0.147142	−0.0735712	+	0.997290i	$0.523440\pi$
$740$	0	0
$741$	−4.00000	−0.146944
$742$	40.0000i	1.46845i
$743$	− 40.0000i	− 1.46746i	−0.679442	−	0.733729i	$-0.737778\pi$
$743$	− 40.0000i	− 1.46746i	0.679442	−	0.733729i	$-0.262222\pi$
$744$	−8.00000	−0.293294
$745$	0	0
$746$	22.0000	0.805477
$747$	4.00000i	0.146352i
$748$	0	0
$749$	48.0000	1.75388
$750$	0	0
$751$	−8.00000	−0.291924	−0.145962	−	0.989290i	$-0.546628\pi$
$751$	−8.00000	−0.291924	−0.145962	+	0.989290i	$0.546628\pi$
$752$	0	0
$753$	0	0
$754$	−2.00000	−0.0728357
$755$	0	0
$756$	4.00000	0.145479
$757$	− 50.0000i	− 1.81728i	−0.417579	−	0.908640i	$-0.637121\pi$
$757$	− 50.0000i	− 1.81728i	0.417579	−	0.908640i	$-0.362879\pi$
$758$	− 20.0000i	− 0.726433i
$759$	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	$0.394215\pi$
$762$	− 12.0000i	− 0.434714i
$763$	56.0000i	2.02734i
$764$	−8.00000	−0.289430
$765$	0	0
$766$	16.0000	0.578103
$767$	0	0
$768$	1.00000i	0.0360844i
$769$	−2.00000	−0.0721218	−0.0360609	−	0.999350i	$-0.511481\pi$
$769$	−2.00000	−0.0721218	−0.0360609	+	0.999350i	$0.511481\pi$
$770$	0	0
$771$	22.0000	0.792311
$772$	− 18.0000i	− 0.647834i
$773$	10.0000i	0.359675i	0.983696	+	0.179838i	$0.0575572\pi$
$773$	10.0000i	0.359675i	−0.983696	+	0.179838i	$0.942443\pi$
$774$	12.0000	0.431331
$775$	0	0
$776$	6.00000	0.215387
$777$	− 8.00000i	− 0.286998i
$778$	14.0000i	0.501924i
$779$	24.0000	0.859889
$780$	0	0
$781$	0	0
$782$	− 16.0000i	− 0.572159i
$783$	2.00000i	0.0714742i
$784$	−9.00000	−0.321429
$785$	0	0
$786$	−8.00000	−0.285351
$787$	− 44.0000i	− 1.56843i	−0.620489	−	0.784215i	$-0.713066\pi$
$787$	− 44.0000i	− 1.56843i	0.620489	−	0.784215i	$-0.286934\pi$
$788$	− 6.00000i	− 0.213741i
$789$	−24.0000	−0.854423
$790$	0	0
$791$	−40.0000	−1.42224
$792$	0	0
$793$	10.0000i	0.355110i
$794$	10.0000	0.354887
$795$	0	0
$796$	−8.00000	−0.283552
$797$	46.0000i	1.62940i	0.579880	+	0.814702i	$0.303099\pi$
$797$	46.0000i	1.62940i	−0.579880	+	0.814702i	$0.696901\pi$
$798$	− 16.0000i	− 0.566394i
$799$	0	0
$800$	0	0
$801$	−14.0000	−0.494666
$802$	26.0000i	0.918092i
$803$	0	0
$804$	4.00000	0.141069
$805$	0	0
$806$	−8.00000	−0.281788
$807$	14.0000i	0.492823i
$808$	− 10.0000i	− 0.351799i
$809$	−10.0000	−0.351581	−0.175791	−	0.984428i	$-0.556248\pi$
$809$	−10.0000	−0.351581	−0.175791	+	0.984428i	$0.556248\pi$
$810$	0	0
$811$	4.00000	0.140459	0.0702295	−	0.997531i	$-0.477627\pi$
$811$	4.00000	0.140459	0.0702295	+	0.997531i	$0.477627\pi$
$812$	− 8.00000i	− 0.280745i
$813$	− 32.0000i	− 1.12229i
$814$	0	0
$815$	0	0
$816$	−2.00000	−0.0700140
$817$	− 48.0000i	− 1.67931i
$818$	− 10.0000i	− 0.349642i
$819$	4.00000	0.139771
$820$	0	0
$821$	2.00000	0.0698005	0.0349002	−	0.999391i	$-0.488889\pi$
$821$	2.00000	0.0698005	0.0349002	+	0.999391i	$0.488889\pi$
$822$	6.00000i	0.209274i
$823$	− 44.0000i	− 1.53374i	−0.641800	−	0.766872i	$-0.721812\pi$
$823$	− 44.0000i	− 1.53374i	0.641800	−	0.766872i	$-0.278188\pi$
$824$	−4.00000	−0.139347
$825$	0	0
$826$	0	0
$827$	36.0000i	1.25184i	0.779886	+	0.625921i	$0.215277\pi$
$827$	36.0000i	1.25184i	−0.779886	+	0.625921i	$0.784723\pi$
$828$	8.00000i	0.278019i
$829$	−38.0000	−1.31979	−0.659897	−	0.751356i	$-0.729400\pi$
$829$	−38.0000	−1.31979	−0.659897	+	0.751356i	$0.729400\pi$
$830$	0	0
$831$	18.0000	0.624413
$832$	1.00000i	0.0346688i
$833$	− 18.0000i	− 0.623663i
$834$	−20.0000	−0.692543
$835$	0	0
$836$	0	0
$837$	8.00000i	0.276520i
$838$	24.0000i	0.829066i
$839$	8.00000	0.276191	0.138095	−	0.990419i	$-0.455902\pi$
$839$	8.00000	0.276191	0.138095	+	0.990419i	$0.455902\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	30.0000i	1.03387i
$843$	18.0000i	0.619953i
$844$	12.0000	0.413057
$845$	0	0
$846$	0	0
$847$	44.0000i	1.51186i
$848$	10.0000i	0.343401i
$849$	−20.0000	−0.686398
$850$	0	0
$851$	16.0000	0.548473
$852$	16.0000i	0.548151i
$853$	− 14.0000i	− 0.479351i	−0.970853	−	0.239675i	$-0.922959\pi$
$853$	− 14.0000i	− 0.479351i	0.970853	−	0.239675i	$-0.0770410\pi$
$854$	−40.0000	−1.36877
$855$	0	0
$856$	12.0000	0.410152
$857$	10.0000i	0.341593i	0.985306	+	0.170797i	$0.0546341\pi$
$857$	10.0000i	0.341593i	−0.985306	+	0.170797i	$0.945366\pi$
$858$	0	0
$859$	36.0000	1.22830	0.614152	−	0.789188i	$-0.289498\pi$
$859$	36.0000	1.22830	0.614152	+	0.789188i	$0.289498\pi$
$860$	0	0
$861$	−24.0000	−0.817918
$862$	0	0
$863$	− 24.0000i	− 0.816970i	−0.912765	−	0.408485i	$-0.866057\pi$
$863$	− 24.0000i	− 0.816970i	0.912765	−	0.408485i	$-0.133943\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	−10.0000	−0.339814
$867$	13.0000i	0.441503i
$868$	− 32.0000i	− 1.08615i
$869$	0	0
$870$	0	0
$871$	4.00000	0.135535
$872$	14.0000i	0.474100i
$873$	− 6.00000i	− 0.203069i
$874$	32.0000	1.08242
$875$	0	0
$876$	−6.00000	−0.202721
$877$	− 18.0000i	− 0.607817i	−0.952701	−	0.303908i	$-0.901708\pi$
$877$	− 18.0000i	− 0.607817i	0.952701	−	0.303908i	$-0.0982917\pi$
$878$	− 40.0000i	− 1.34993i
$879$	6.00000	0.202375
$880$	0	0
$881$	−6.00000	−0.202145	−0.101073	−	0.994879i	$-0.532227\pi$
$881$	−6.00000	−0.202145	−0.101073	+	0.994879i	$0.532227\pi$
$882$	9.00000i	0.303046i
$883$	− 36.0000i	− 1.21150i	−0.795656	−	0.605748i	$-0.792874\pi$
$883$	− 36.0000i	− 1.21150i	0.795656	−	0.605748i	$-0.207126\pi$
$884$	−2.00000	−0.0672673
$885$	0	0
$886$	36.0000	1.20944
$887$	− 8.00000i	− 0.268614i	−0.990940	−	0.134307i	$-0.957119\pi$
$887$	− 8.00000i	− 0.268614i	0.990940	−	0.134307i	$-0.0428808\pi$
$888$	− 2.00000i	− 0.0671156i
$889$	48.0000	1.60987
$890$	0	0
$891$	0	0
$892$	− 12.0000i	− 0.401790i
$893$	0	0
$894$	10.0000	0.334450
$895$	0	0
$896$	−4.00000	−0.133631
$897$	8.00000i	0.267112i
$898$	6.00000i	0.200223i
$899$	16.0000	0.533630
$900$	0	0
$901$	−20.0000	−0.666297
$902$	0	0
$903$	48.0000i	1.59734i
$904$	−10.0000	−0.332595
$905$	0	0
$906$	8.00000	0.265782
$907$	− 28.0000i	− 0.929725i	−0.885383	−	0.464862i	$-0.846104\pi$
$907$	− 28.0000i	− 0.929725i	0.885383	−	0.464862i	$-0.153896\pi$
$908$	− 12.0000i	− 0.398234i
$909$	−10.0000	−0.331679
$910$	0	0
$911$	−32.0000	−1.06021	−0.530104	−	0.847933i	$-0.677847\pi$
$911$	−32.0000	−1.06021	−0.530104	+	0.847933i	$0.677847\pi$
$912$	− 4.00000i	− 0.132453i
$913$	0	0
$914$	18.0000	0.595387
$915$	0	0
$916$	14.0000	0.462573
$917$	− 32.0000i	− 1.05673i
$918$	2.00000i	0.0660098i
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	0	0
$921$	−12.0000	−0.395413
$922$	2.00000i	0.0658665i
$923$	16.0000i	0.526646i
$924$	0	0
$925$	0	0
$926$	−20.0000	−0.657241
$927$	4.00000i	0.131377i
$928$	− 2.00000i	− 0.0656532i
$929$	−26.0000	−0.853032	−0.426516	−	0.904480i	$-0.640259\pi$
$929$	−26.0000	−0.853032	−0.426516	+	0.904480i	$0.640259\pi$
$930$	0	0
$931$	36.0000	1.17985
$932$	26.0000i	0.851658i
$933$	0	0
$934$	−36.0000	−1.17796
$935$	0	0
$936$	1.00000	0.0326860
$937$	6.00000i	0.196011i	0.995186	+	0.0980057i	$0.0312463\pi$
$937$	6.00000i	0.196011i	−0.995186	+	0.0980057i	$0.968754\pi$
$938$	16.0000i	0.522419i
$939$	−18.0000	−0.587408
$940$	0	0
$941$	−30.0000	−0.977972	−0.488986	−	0.872292i	$-0.662633\pi$
$941$	−30.0000	−0.977972	−0.488986	+	0.872292i	$0.662633\pi$
$942$	18.0000i	0.586472i
$943$	− 48.0000i	− 1.56310i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 52.0000i	− 1.68977i	−0.534946	−	0.844886i	$-0.679668\pi$
$947$	− 52.0000i	− 1.68977i	0.534946	−	0.844886i	$-0.320332\pi$
$948$	− 8.00000i	− 0.259828i
$949$	−6.00000	−0.194768
$950$	0	0
$951$	−14.0000	−0.453981
$952$	− 8.00000i	− 0.259281i
$953$	30.0000i	0.971795i	0.874016	+	0.485898i	$0.161507\pi$
$953$	30.0000i	0.971795i	−0.874016	+	0.485898i	$0.838493\pi$
$954$	10.0000	0.323762
$955$	0	0
$956$	24.0000	0.776215
$957$	0	0
$958$	− 8.00000i	− 0.258468i
$959$	−24.0000	−0.775000
$960$	0	0
$961$	33.0000	1.06452
$962$	− 2.00000i	− 0.0644826i
$963$	− 12.0000i	− 0.386695i
$964$	−2.00000	−0.0644157
$965$	0	0
$966$	−32.0000	−1.02958
$967$	28.0000i	0.900419i	0.892923	+	0.450210i	$0.148651\pi$
$967$	28.0000i	0.900419i	−0.892923	+	0.450210i	$0.851349\pi$
$968$	11.0000i	0.353553i
$969$	8.00000	0.256997
$970$	0	0
$971$	−48.0000	−1.54039	−0.770197	−	0.637806i	$-0.779842\pi$
$971$	−48.0000	−1.54039	−0.770197	+	0.637806i	$0.779842\pi$
$972$	− 1.00000i	− 0.0320750i
$973$	− 80.0000i	− 2.56468i
$974$	−20.0000	−0.640841
$975$	0	0
$976$	−10.0000	−0.320092
$977$	− 38.0000i	− 1.21573i	−0.794041	−	0.607864i	$-0.792027\pi$
$977$	− 38.0000i	− 1.21573i	0.794041	−	0.607864i	$-0.207973\pi$
$978$	20.0000i	0.639529i
$979$	0	0
$980$	0	0
$981$	14.0000	0.446986
$982$	0	0
$983$	56.0000i	1.78612i	0.449935	+	0.893061i	$0.351447\pi$
$983$	56.0000i	1.78612i	−0.449935	+	0.893061i	$0.648553\pi$
$984$	−6.00000	−0.191273
$985$	0	0
$986$	4.00000	0.127386
$987$	0	0
$988$	− 4.00000i	− 0.127257i
$989$	−96.0000	−3.05262
$990$	0	0
$991$	−32.0000	−1.01651	−0.508257	−	0.861206i	$-0.669710\pi$
$991$	−32.0000	−1.01651	−0.508257	+	0.861206i	$0.669710\pi$
$992$	− 8.00000i	− 0.254000i
$993$	− 20.0000i	− 0.634681i
$994$	−64.0000	−2.02996
$995$	0	0
$996$	−4.00000	−0.126745
$997$	22.0000i	0.696747i	0.937356	+	0.348373i	$0.113266\pi$
$997$	22.0000i	0.696747i	−0.937356	+	0.348373i	$0.886734\pi$
$998$	44.0000i	1.39280i
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1950.2.e.e.1249.2		2
3.2	odd	2		5850.2.e.m.5149.1		2
5.2	odd	4		390.2.a.c.1.1	✓	1
5.3	odd	4		1950.2.a.n.1.1		1
5.4	even	2	inner	1950.2.e.e.1249.1		2
15.2	even	4		1170.2.a.n.1.1		1
15.8	even	4		5850.2.a.c.1.1		1
15.14	odd	2		5850.2.e.m.5149.2		2
20.7	even	4		3120.2.a.a.1.1		1
60.47	odd	4		9360.2.a.bc.1.1		1
65.12	odd	4		5070.2.a.u.1.1		1
65.47	even	4		5070.2.b.i.1351.1		2
65.57	even	4		5070.2.b.i.1351.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.a.c.1.1	✓	1	5.2	odd	4
1170.2.a.n.1.1		1	15.2	even	4
1950.2.a.n.1.1		1	5.3	odd	4
1950.2.e.e.1249.1		2	5.4	even	2	inner
1950.2.e.e.1249.2		2	1.1	even	1	trivial
3120.2.a.a.1.1		1	20.7	even	4
5070.2.a.u.1.1		1	65.12	odd	4
5070.2.b.i.1351.1		2	65.47	even	4
5070.2.b.i.1351.2		2	65.57	even	4
5850.2.a.c.1.1		1	15.8	even	4
5850.2.e.m.5149.1		2	3.2	odd	2
5850.2.e.m.5149.2		2	15.14	odd	2
9360.2.a.bc.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:	\( N \)	\(=\)	\( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1950.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -1.00000i q^{12} -1.00000i q^{13} +4.00000 q^{14} +1.00000 q^{16} +2.00000i q^{17} -1.00000i q^{18} -4.00000 q^{19} +4.00000 q^{21} +8.00000i q^{23} +1.00000 q^{24} +1.00000 q^{26} -1.00000i q^{27} +4.00000i q^{28} -2.00000 q^{29} -8.00000 q^{31} +1.00000i q^{32} -2.00000 q^{34} +1.00000 q^{36} -2.00000i q^{37} -4.00000i q^{38} +1.00000 q^{39} -6.00000 q^{41} +4.00000i q^{42} +12.0000i q^{43} -8.00000 q^{46} +1.00000i q^{48} -9.00000 q^{49} -2.00000 q^{51} +1.00000i q^{52} +10.0000i q^{53} +1.00000 q^{54} -4.00000 q^{56} -4.00000i q^{57} -2.00000i q^{58} -10.0000 q^{61} -8.00000i q^{62} +4.00000i q^{63} -1.00000 q^{64} +4.00000i q^{67} -2.00000i q^{68} -8.00000 q^{69} -16.0000 q^{71} +1.00000i q^{72} -6.00000i q^{73} +2.00000 q^{74} +4.00000 q^{76} +1.00000i q^{78} +8.00000 q^{79} +1.00000 q^{81} -6.00000i q^{82} -4.00000i q^{83} -4.00000 q^{84} -12.0000 q^{86} -2.00000i q^{87} +14.0000 q^{89} -4.00000 q^{91} -8.00000i q^{92} -8.00000i q^{93} -1.00000 q^{96} +6.00000i q^{97} -9.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} + 8 q^{14} + 2 q^{16} - 8 q^{19} + 8 q^{21} + 2 q^{24} + 2 q^{26} - 4 q^{29} - 16 q^{31} - 4 q^{34} + 2 q^{36} + 2 q^{39} - 12 q^{41} - 16 q^{46} - 18 q^{49} - 4 q^{51} + 2 q^{54} - 8 q^{56} - 20 q^{61} - 2 q^{64} - 16 q^{69} - 32 q^{71} + 4 q^{74} + 8 q^{76} + 16 q^{79} + 2 q^{81} - 8 q^{84} - 24 q^{86} + 28 q^{89} - 8 q^{91} - 2 q^{96}+O(q^{100}) \) 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 + 8 * q^14 + 2 * q^16 - 8 * q^19 + 8 * q^21 + 2 * q^24 + 2 * q^26 - 4 * q^29 - 16 * q^31 - 4 * q^34 + 2 * q^36 + 2 * q^39 - 12 * q^41 - 16 * q^46 - 18 * q^49 - 4 * q^51 + 2 * q^54 - 8 * q^56 - 20 * q^61 - 2 * q^64 - 16 * q^69 - 32 * q^71 + 4 * q^74 + 8 * q^76 + 16 * q^79 + 2 * q^81 - 8 * q^84 - 24 * q^86 + 28 * q^89 - 8 * q^91 - 2 * q^96

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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