LMFDB - Embedded newform 3071.1.d.c.3070.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3071,1,Mod(3070,3071)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3071, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("3071.3070");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3071 = 37 \cdot 83 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3071.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$1.53262865380$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\Q(\zeta_{38})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 8x^{7} + 7x^{6} + 21x^{5} - 15x^{4} - 20x^{3} + 10x^{2} + 5x - 1 $ x^9 - x^8 - 8x^7 + 7x^6 + 21x^5 - 15x^4 - 20x^3 + 10x^2 + 5*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{19}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{19} - \cdots)$

Embedding invariants

Embedding label			3070.9
Root			$-1.57828$ of defining polynomial
Character	$\chi$	$=$	3071.3070

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.89163 q^{2} -1.75895 q^{3} +2.57828 q^{4} -1.35456 q^{5} -3.32729 q^{6} +1.89163 q^{7} +2.98553 q^{8} +2.09390 q^{9} +O(q^{10})$	\(q+1.89163 q^{2} -1.75895 q^{3} +2.57828 q^{4} -1.35456 q^{5} -3.32729 q^{6} +1.89163 q^{7} +2.98553 q^{8} +2.09390 q^{9} -2.56234 q^{10} -1.35456 q^{11} -4.53506 q^{12} +1.09390 q^{13} +3.57828 q^{14} +2.38261 q^{15} +3.06925 q^{16} +3.96089 q^{18} -0.803391 q^{19} -3.49244 q^{20} -3.32729 q^{21} -2.56234 q^{22} -5.25139 q^{24} +0.834841 q^{25} +2.06925 q^{26} -1.92411 q^{27} +4.87717 q^{28} +4.50702 q^{30} +2.82037 q^{32} +2.38261 q^{33} -2.56234 q^{35} +5.39865 q^{36} +1.00000 q^{37} -1.51972 q^{38} -1.92411 q^{39} -4.04409 q^{40} +1.09390 q^{41} -6.29401 q^{42} +1.57828 q^{43} -3.49244 q^{44} -2.83631 q^{45} -5.39865 q^{48} +2.57828 q^{49} +1.57921 q^{50} +2.82037 q^{52} -3.63971 q^{54} +1.83484 q^{55} +5.64753 q^{56} +1.41312 q^{57} +6.14303 q^{60} +3.96089 q^{63} +2.26586 q^{64} -1.48175 q^{65} +4.50702 q^{66} -4.84701 q^{70} +6.25139 q^{72} +1.89163 q^{74} -1.46844 q^{75} -2.07137 q^{76} -2.56234 q^{77} -3.63971 q^{78} -1.97272 q^{79} -4.15750 q^{80} +1.29051 q^{81} +2.06925 q^{82} +1.00000 q^{83} -8.57868 q^{84} +2.98553 q^{86} -4.04409 q^{88} -1.75895 q^{89} -5.36527 q^{90} +2.06925 q^{91} +1.08824 q^{95} -4.96089 q^{96} -1.75895 q^{97} +4.87717 q^{98} -2.83631 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - q^{2} - q^{3} + 8 q^{4} - q^{5} - 2 q^{6} - q^{7} - 2 q^{8} + 8 q^{9}+O(q^{10}) $ 9 * q - q^2 - q^3 + 8 * q^4 - q^5 - 2 * q^6 - q^7 - 2 * q^8 + 8 * q^9	$ 9 q - q^{2} - q^{3} + 8 q^{4} - q^{5} - 2 q^{6} - q^{7} - 2 q^{8} + 8 q^{9} - 2 q^{10} - q^{11} - 3 q^{12} - q^{13} + 17 q^{14} - 2 q^{15} + 7 q^{16} - 3 q^{18} - q^{19} - 3 q^{20} - 2 q^{21} - 2 q^{22} - 4 q^{24} + 8 q^{25} - 2 q^{26} - 2 q^{27} - 3 q^{28} + 15 q^{30} - 3 q^{32} - 2 q^{33} - 2 q^{35} + 5 q^{36} + 9 q^{37} - 2 q^{38} - 2 q^{39} - 4 q^{40} - q^{41} - 4 q^{42} - q^{43} - 3 q^{44} - 3 q^{45} - 5 q^{48} + 8 q^{49} - 3 q^{50} - 3 q^{52} - 4 q^{54} + 17 q^{55} + 15 q^{56} - 2 q^{57} - 6 q^{60} - 3 q^{63} + 6 q^{64} - 2 q^{65} + 15 q^{66} - 4 q^{70} + 13 q^{72} - q^{74} - 3 q^{75} - 3 q^{76} - 2 q^{77} - 4 q^{78} - q^{79} - 5 q^{80} + 7 q^{81} - 2 q^{82} + 9 q^{83} - 6 q^{84} - 2 q^{86} - 4 q^{88} - q^{89} - 6 q^{90} - 2 q^{91} - 2 q^{95} - 6 q^{96} - q^{97} - 3 q^{98} - 3 q^{99}+O(q^{100}) $ 9 * q - q^2 - q^3 + 8 * q^4 - q^5 - 2 * q^6 - q^7 - 2 * q^8 + 8 * q^9 - 2 * q^10 - q^11 - 3 * q^12 - q^13 + 17 * q^14 - 2 * q^15 + 7 * q^16 - 3 * q^18 - q^19 - 3 * q^20 - 2 * q^21 - 2 * q^22 - 4 * q^24 + 8 * q^25 - 2 * q^26 - 2 * q^27 - 3 * q^28 + 15 * q^30 - 3 * q^32 - 2 * q^33 - 2 * q^35 + 5 * q^36 + 9 * q^37 - 2 * q^38 - 2 * q^39 - 4 * q^40 - q^41 - 4 * q^42 - q^43 - 3 * q^44 - 3 * q^45 - 5 * q^48 + 8 * q^49 - 3 * q^50 - 3 * q^52 - 4 * q^54 + 17 * q^55 + 15 * q^56 - 2 * q^57 - 6 * q^60 - 3 * q^63 + 6 * q^64 - 2 * q^65 + 15 * q^66 - 4 * q^70 + 13 * q^72 - q^74 - 3 * q^75 - 3 * q^76 - 2 * q^77 - 4 * q^78 - q^79 - 5 * q^80 + 7 * q^81 - 2 * q^82 + 9 * q^83 - 6 * q^84 - 2 * q^86 - 4 * q^88 - q^89 - 6 * q^90 - 2 * q^91 - 2 * q^95 - 6 * q^96 - q^97 - 3 * q^98 - 3 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3071\mathbb{Z}\right)^\times$.

$n$	$334$	$2740$
$\chi(n)$	$-1$	$-1$

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 $

$2$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$2$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$3$	−1.75895	−1.75895	−0.879474	−	0.475947i	$-0.842105\pi$
$3$	−1.75895	−1.75895	−0.879474	+	0.475947i	$0.842105\pi$
$4$	2.57828	2.57828
$5$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$5$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$6$	−3.32729	−3.32729
$7$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$7$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$8$	2.98553	2.98553
$9$	2.09390	2.09390
$10$	−2.56234	−2.56234
$11$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$11$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$12$	−4.53506	−4.53506
$13$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$13$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$14$	3.57828	3.57828
$15$	2.38261	2.38261
$16$	3.06925	3.06925
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	3.96089	3.96089
$19$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$19$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$20$	−3.49244	−3.49244
$21$	−3.32729	−3.32729
$22$	−2.56234	−2.56234
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	−5.25139	−5.25139
$25$	0.834841	0.834841
$26$	2.06925	2.06925
$27$	−1.92411	−1.92411
$28$	4.87717	4.87717
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	4.50702	4.50702
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	2.82037	2.82037
$33$	2.38261	2.38261
$34$	0	0
$35$	−2.56234	−2.56234
$36$	5.39865	5.39865
$37$	1.00000	1.00000
$37$	1.00000	1.00000
$38$	−1.51972	−1.51972
$39$	−1.92411	−1.92411
$40$	−4.04409	−4.04409
$41$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$41$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$42$	−6.29401	−6.29401
$43$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$43$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$44$	−3.49244	−3.49244
$45$	−2.83631	−2.83631
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	−5.39865	−5.39865
$49$	2.57828	2.57828
$50$	1.57921	1.57921
$51$	0	0
$52$	2.82037	2.82037
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	−3.63971	−3.63971
$55$	1.83484	1.83484
$56$	5.64753	5.64753
$57$	1.41312	1.41312
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	6.14303	6.14303
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	3.96089	3.96089
$64$	2.26586	2.26586
$65$	−1.48175	−1.48175
$66$	4.50702	4.50702
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	0	0
$70$	−4.84701	−4.84701
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	6.25139	6.25139
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	1.89163	1.89163
$75$	−1.46844	−1.46844
$76$	−2.07137	−2.07137
$77$	−2.56234	−2.56234
$78$	−3.63971	−3.63971
$79$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$79$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$80$	−4.15750	−4.15750
$81$	1.29051	1.29051
$82$	2.06925	2.06925
$83$	1.00000	1.00000
$83$	1.00000	1.00000
$84$	−8.57868	−8.57868
$85$	0	0
$86$	2.98553	2.98553
$87$	0	0
$88$	−4.04409	−4.04409
$89$	−1.75895	−1.75895	−0.879474	−	0.475947i	$-0.842105\pi$
$89$	−1.75895	−1.75895	−0.879474	+	0.475947i	$0.842105\pi$
$90$	−5.36527	−5.36527
$91$	2.06925	2.06925
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.08824	1.08824
$96$	−4.96089	−4.96089
$97$	−1.75895	−1.75895	−0.879474	−	0.475947i	$-0.842105\pi$
$97$	−1.75895	−1.75895	−0.879474	+	0.475947i	$0.842105\pi$
$98$	4.87717	4.87717
$99$	−2.83631	−2.83631
$100$	2.15246	2.15246
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$103$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$104$	3.26586	3.26586
$105$	4.50702	4.50702
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	−4.96089	−4.96089
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	3.47085	3.47085
$111$	−1.75895	−1.75895
$112$	5.80590	5.80590
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	2.67311	2.67311
$115$	0	0
$116$	0	0
$117$	2.29051	2.29051
$118$	0	0
$119$	0	0
$120$	7.11334	7.11334
$121$	0.834841	0.834841
$122$	0	0
$123$	−1.92411	−1.92411
$124$	0	0
$125$	0.223718	0.223718
$126$	7.49255	7.49255
$127$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$127$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$128$	1.46581	1.46581
$129$	−2.77611	−2.77611
$130$	−2.80293	−2.80293
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	6.14303	6.14303
$133$	−1.51972	−1.51972
$134$	0	0
$135$	2.60632	2.60632
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	−6.60643	−6.60643
$141$	0	0
$142$	0	0
$143$	−1.48175	−1.48175
$144$	6.42670	6.42670
$145$	0	0
$146$	0	0
$147$	−4.53506	−4.53506
$148$	2.57828	2.57828
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	−2.77776	−2.77776
$151$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$151$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$152$	−2.39855	−2.39855
$153$	0	0
$154$	−4.84701	−4.84701
$155$	0	0
$156$	−4.96089	−4.96089
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	−3.73167	−3.73167
$159$	0	0
$160$	−3.82037	−3.82037
$161$	0	0
$162$	2.44116	2.44116
$163$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$163$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$164$	2.82037	2.82037
$165$	−3.22739	−3.22739
$166$	1.89163	1.89163
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	−9.93371	−9.93371
$169$	0.196609	0.196609
$170$	0	0
$171$	−1.68222	−1.68222
$172$	4.06925	4.06925
$173$	−0.165159	−0.165159	−0.0825793	−	0.996584i	$-0.526316\pi$
$173$	−0.165159	−0.165159	−0.0825793	+	0.996584i	$0.526316\pi$
$174$	0	0
$175$	1.57921	1.57921
$176$	−4.15750	−4.15750
$177$	0	0
$178$	−3.32729	−3.32729
$179$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$179$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$180$	−7.31282	−7.31282
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	3.91427	3.91427
$183$	0	0
$184$	0	0
$185$	−1.35456	−1.35456
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−3.63971	−3.63971
$190$	2.05856	2.05856
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	−3.98553	−3.98553
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	−3.32729	−3.32729
$195$	2.60632	2.60632
$196$	6.64753	6.64753
$197$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$197$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$198$	−5.36527	−5.36527
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	2.49244	2.49244
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−1.48175	−1.48175
$206$	2.06925	2.06925
$207$	0	0
$208$	3.35744	3.35744
$209$	1.08824	1.08824
$210$	8.52563	8.52563
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−2.13788	−2.13788
$216$	−5.74448	−5.74448
$217$	0	0
$218$	0	0
$219$	0	0
$220$	4.73074	4.73074
$221$	0	0
$222$	−3.32729	−3.32729
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	5.33511	5.33511
$225$	1.74807	1.74807
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	3.64343	3.64343
$229$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$229$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$230$	0	0
$231$	4.50702	4.50702
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	4.33280	4.33280
$235$	0	0
$236$	0	0
$237$	3.46992	3.46992
$238$	0	0
$239$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$239$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$240$	7.31282	7.31282
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	1.57921	1.57921
$243$	−0.345825	−0.345825
$244$	0	0
$245$	−3.49244	−3.49244
$246$	−3.63971	−3.63971
$247$	−0.878826	−0.878826
$248$	0	0
$249$	−1.75895	−1.75895
$250$	0.423192	0.423192
$251$	−0.165159	−0.165159	−0.0825793	−	0.996584i	$-0.526316\pi$
$251$	−0.165159	−0.165159	−0.0825793	+	0.996584i	$0.526316\pi$
$252$	10.2123	10.2123
$253$	0	0
$254$	−3.73167	−3.73167
$255$	0	0
$256$	0.506914	0.506914
$257$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$257$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$258$	−5.25139	−5.25139
$259$	1.89163	1.89163
$260$	−3.82037	−3.82037
$261$	0	0
$262$	0	0
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	7.11334	7.11334
$265$	0	0
$266$	−2.87476	−2.87476
$267$	3.09390	3.09390
$268$	0	0
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	4.93021	4.93021
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	−3.63971	−3.63971
$274$	0	0
$275$	−1.13085	−1.13085
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	0	0
$280$	−7.64994	−7.64994
$281$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$281$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$282$	0	0
$283$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$283$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$284$	0	0
$285$	−1.91416	−1.91416
$286$	−2.80293	−2.80293
$287$	2.06925	2.06925
$288$	5.90557	5.90557
$289$	1.00000	1.00000
$290$	0	0
$291$	3.09390	3.09390
$292$	0	0
$293$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$293$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$294$	−8.57868	−8.57868
$295$	0	0
$296$	2.98553	2.98553
$297$	2.60632	2.60632
$298$	0	0
$299$	0	0
$300$	−3.78606	−3.78606
$301$	2.98553	2.98553
$302$	−2.56234	−2.56234
$303$	0	0
$304$	−2.46581	−2.46581
$305$	0	0
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	−6.60643	−6.60643
$309$	−1.92411	−1.92411
$310$	0	0
$311$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$311$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$312$	−5.74448	−5.74448
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	0	0
$315$	−5.36527	−5.36527
$316$	−5.08623	−5.08623
$317$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$317$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$318$	0	0
$319$	0	0
$320$	−3.06925	−3.06925
$321$	0	0
$322$	0	0
$323$	0	0
$324$	3.32729	3.32729
$325$	0.913230	0.913230
$326$	−3.73167	−3.73167
$327$	0	0
$328$	3.26586	3.26586
$329$	0	0
$330$	−6.10504	−6.10504
$331$	−0.165159	−0.165159	−0.0825793	−	0.996584i	$-0.526316\pi$
$331$	−0.165159	−0.165159	−0.0825793	+	0.996584i	$0.526316\pi$
$332$	2.57828	2.57828
$333$	2.09390	2.09390
$334$	0	0
$335$	0	0
$336$	−10.2123	−10.2123
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	0.371913	0.371913
$339$	0	0
$340$	0	0
$341$	0	0
$342$	−3.18214	−3.18214
$343$	2.98553	2.98553
$344$	4.71201	4.71201
$345$	0	0
$346$	−0.312420	−0.312420
$347$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$347$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$348$	0	0
$349$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$349$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$350$	2.98730	2.98730
$351$	−2.10477	−2.10477
$352$	−3.82037	−3.82037
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	0	0
$356$	−4.53506	−4.53506
$357$	0	0
$358$	−2.56234	−2.56234
$359$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$359$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$360$	−8.46791	−8.46791
$361$	−0.354563	−0.354563
$362$	0	0
$363$	−1.46844	−1.46844
$364$	5.33511	5.33511
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	2.29051	2.29051
$370$	−2.56234	−2.56234
$371$	0	0
$372$	0	0
$373$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$373$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$374$	0	0
$375$	−0.393508	−0.393508
$376$	0	0
$377$	0	0
$378$	−6.88499	−6.88499
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	2.80580	2.80580
$381$	3.46992	3.46992
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	−2.57828	−2.57828
$385$	3.47085	3.47085
$386$	0	0
$387$	3.30476	3.30476
$388$	−4.53506	−4.53506
$389$	−0.165159	−0.165159	−0.0825793	−	0.996584i	$-0.526316\pi$
$389$	−0.165159	−0.165159	−0.0825793	+	0.996584i	$0.526316\pi$
$390$	4.93021	4.93021
$391$	0	0
$392$	7.69754	7.69754
$393$	0	0
$394$	2.98553	2.98553
$395$	2.67218	2.67218
$396$	−7.31282	−7.31282
$397$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$397$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$398$	0	0
$399$	2.67311	2.67311
$400$	2.56234	2.56234
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−1.74807	−1.74807
$406$	0	0
$407$	−1.35456	−1.35456
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	−2.80293	−2.80293
$411$	0	0
$412$	2.82037	2.82037
$413$	0	0
$414$	0	0
$415$	−1.35456	−1.35456
$416$	3.08519	3.08519
$417$	0	0
$418$	2.05856	2.05856
$419$	−1.75895	−1.75895	−0.879474	−	0.475947i	$-0.842105\pi$
$419$	−1.75895	−1.75895	−0.879474	+	0.475947i	$0.842105\pi$
$420$	11.6204	11.6204
$421$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$421$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	2.60632	2.60632
$430$	−4.04409	−4.04409
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	−5.90557	−5.90557
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$439$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$440$	5.47798	5.47798
$441$	5.39865	5.39865
$442$	0	0
$443$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$443$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$444$	−4.53506	−4.53506
$445$	2.38261	2.38261
$446$	0	0
$447$	0	0
$448$	4.28618	4.28618
$449$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$449$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$450$	3.30671	3.30671
$451$	−1.48175	−1.48175
$452$	0	0
$453$	2.38261	2.38261
$454$	0	0
$455$	−2.80293	−2.80293
$456$	4.21892	4.21892
$457$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$457$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$458$	−3.73167	−3.73167
$459$	0	0
$460$	0	0
$461$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$461$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$462$	8.52563	8.52563
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−1.75895	−1.75895	−0.879474	−	0.475947i	$-0.842105\pi$
$467$	−1.75895	−1.75895	−0.879474	+	0.475947i	$0.842105\pi$
$468$	5.90557	5.90557
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−2.13788	−2.13788
$474$	6.56381	6.56381
$475$	−0.670704	−0.670704
$476$	0	0
$477$	0	0
$478$	2.06925	2.06925
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	6.71983	6.71983
$481$	1.09390	1.09390
$482$	0	0
$483$	0	0
$484$	2.15246	2.15246
$485$	2.38261	2.38261
$486$	−0.654175	−0.654175
$487$	−1.75895	−1.75895	−0.879474	−	0.475947i	$-0.842105\pi$
$487$	−1.75895	−1.75895	−0.879474	+	0.475947i	$0.842105\pi$
$488$	0	0
$489$	3.46992	3.46992
$490$	−6.60643	−6.60643
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	−4.96089	−4.96089
$493$	0	0
$494$	−1.66242	−1.66242
$495$	3.84197	3.84197
$496$	0	0
$497$	0	0
$498$	−3.32729	−3.32729
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	0.576808	0.576808
$501$	0	0
$502$	−0.312420	−0.312420
$503$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$503$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$504$	11.8253	11.8253
$505$	0	0
$506$	0	0
$507$	−0.345825	−0.345825
$508$	−5.08623	−5.08623
$509$	−0.165159	−0.165159	−0.0825793	−	0.996584i	$-0.526316\pi$
$509$	−0.165159	−0.165159	−0.0825793	+	0.996584i	$0.526316\pi$
$510$	0	0
$511$	0	0
$512$	−0.506914	−0.506914
$513$	1.54581	1.54581
$514$	0.928738	0.928738
$515$	−1.48175	−1.48175
$516$	−7.15760	−7.15760
$517$	0	0
$518$	3.57828	3.57828
$519$	0.290505	0.290505
$520$	−4.42382	−4.42382
$521$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$521$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$522$	0	0
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	0	0
$525$	−2.77776	−2.77776
$526$	0	0
$527$	0	0
$528$	7.31282	7.31282
$529$	1.00000	1.00000
$530$	0	0
$531$	0	0
$532$	−3.91827	−3.91827
$533$	1.19661	1.19661
$534$	5.85252	5.85252
$535$	0	0
$536$	0	0
$537$	2.38261	2.38261
$538$	0	0
$539$	−3.49244	−3.49244
$540$	6.71983	6.71983
$541$	2.00000	2.00000	1.00000			$0$
$541$	2.00000	2.00000	1.00000			$0$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	−6.88499	−6.88499
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0	0
$549$	0	0
$550$	−2.13915	−2.13915
$551$	0	0
$552$	0	0
$553$	−3.73167	−3.73167
$554$	0	0
$555$	2.38261	2.38261
$556$	0	0
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	1.72648	1.72648
$560$	−7.86446	−7.86446
$561$	0	0
$562$	0.928738	0.928738
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	0	0
$565$	0	0
$566$	−2.56234	−2.56234
$567$	2.44116	2.44116
$568$	0	0
$569$	2.00000	2.00000	1.00000			$0$
$569$	2.00000	2.00000	1.00000			$0$
$570$	−3.62090	−3.62090
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	−3.82037	−3.82037
$573$	0	0
$574$	3.91427	3.91427
$575$	0	0
$576$	4.74448	4.74448
$577$	2.00000	2.00000	1.00000			$0$
$577$	2.00000	2.00000	1.00000			$0$
$578$	1.89163	1.89163
$579$	0	0
$580$	0	0
$581$	1.89163	1.89163
$582$	5.85252	5.85252
$583$	0	0
$584$	0	0
$585$	−3.10263	−3.10263
$586$	−1.51972	−1.51972
$587$	−0.165159	−0.165159	−0.0825793	−	0.996584i	$-0.526316\pi$
$587$	−0.165159	−0.165159	−0.0825793	+	0.996584i	$0.526316\pi$
$588$	−11.6927	−11.6927
$589$	0	0
$590$	0	0
$591$	−2.77611	−2.77611
$592$	3.06925	3.06925
$593$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$593$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$594$	4.93021	4.93021
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	−4.38408	−4.38408
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	5.64753	5.64753
$603$	0	0
$604$	−3.49244	−3.49244
$605$	−1.13085	−1.13085
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	−2.26586	−2.26586
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0	0
$615$	2.60632	2.60632
$616$	−7.64994	−7.64994
$617$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$617$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$618$	−3.63971	−3.63971
$619$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$619$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$620$	0	0
$621$	0	0
$622$	2.98553	2.98553
$623$	−3.32729	−3.32729
$624$	−5.90557	−5.90557
$625$	−1.13788	−1.13788
$626$	0	0
$627$	−1.91416	−1.91416
$628$	0	0
$629$	0	0
$630$	−10.1491	−10.1491
$631$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$631$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$632$	−5.88962	−5.88962
$633$	0	0
$634$	2.98553	2.98553
$635$	2.67218	2.67218
$636$	0	0
$637$	2.82037	2.82037
$638$	0	0
$639$	0	0
$640$	−1.98553	−1.98553
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$643$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$644$	0	0
$645$	3.76042	3.76042
$646$	0	0
$647$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$647$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$648$	3.85284	3.85284
$649$	0	0
$650$	1.72750	1.72750
$651$	0	0
$652$	−5.08623	−5.08623
$653$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$653$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$654$	0	0
$655$	0	0
$656$	3.35744	3.35744
$657$	0	0
$658$	0	0
$659$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$659$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$660$	−8.32112	−8.32112
$661$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$661$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$662$	−0.312420	−0.312420
$663$	0	0
$664$	2.98553	2.98553
$665$	2.05856	2.05856
$666$	3.96089	3.96089
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	−9.38418	−9.38418
$673$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$673$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$674$	0	0
$675$	−1.60632	−1.60632
$676$	0.506914	0.506914
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	−3.32729	−3.32729
$680$	0	0
$681$	0	0
$682$	0	0
$683$	2.00000	2.00000	1.00000			$0$
$683$	2.00000	2.00000	1.00000			$0$
$684$	−4.33723	−4.33723
$685$	0	0
$686$	5.64753	5.64753
$687$	3.46992	3.46992
$688$	4.84414	4.84414
$689$	0	0
$690$	0	0
$691$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$691$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$692$	−0.425826	−0.425826
$693$	−5.36527	−5.36527
$694$	−3.73167	−3.73167
$695$	0	0
$696$	0	0
$697$	0	0
$698$	2.06925	2.06925
$699$	0	0
$700$	4.07166	4.07166
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	−3.98146	−3.98146
$703$	−0.803391	−0.803391
$704$	−3.06925	−3.06925
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$709$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$710$	0	0
$711$	−4.13068	−4.13068
$712$	−5.25139	−5.25139
$713$	0	0
$714$	0	0
$715$	2.00713	2.00713
$716$	−3.49244	−3.49244
$717$	−1.92411	−1.92411
$718$	−1.51972	−1.51972
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	−8.70536	−8.70536
$721$	2.06925	2.06925
$722$	−0.670704	−0.670704
$723$	0	0
$724$	0	0
$725$	0	0
$726$	−2.77776	−2.77776
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	6.17782	6.17782
$729$	−0.682217	−0.682217
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$733$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$734$	0	0
$735$	6.14303	6.14303
$736$	0	0
$737$	0	0
$738$	4.33280	4.33280
$739$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$739$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$740$	−3.49244	−3.49244
$741$	1.54581	1.54581
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	0	0
$746$	0.928738	0.928738
$747$	2.09390	2.09390
$748$	0	0
$749$	0	0
$750$	−0.744373	−0.744373
$751$	−1.75895	−1.75895	−0.879474	−	0.475947i	$-0.842105\pi$
$751$	−1.75895	−1.75895	−0.879474	+	0.475947i	$0.842105\pi$
$752$	0	0
$753$	0.290505	0.290505
$754$	0	0
$755$	1.83484	1.83484
$756$	−9.38418	−9.38418
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	0	0
$760$	3.24898	3.24898
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	6.56381	6.56381
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	−0.891634	−0.891634
$769$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$769$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$770$	6.56558	6.56558
$771$	−0.863592	−0.863592
$772$	0	0
$773$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$773$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$774$	6.25139	6.25139
$775$	0	0
$776$	−5.25139	−5.25139
$777$	−3.32729	−3.32729
$778$	−0.312420	−0.312420
$779$	−0.878826	−0.878826
$780$	6.71983	6.71983
$781$	0	0
$782$	0	0
$783$	0	0
$784$	7.91339	7.91339
$785$	0	0
$786$	0	0
$787$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$787$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$788$	4.06925	4.06925
$789$	0	0
$790$	5.05478	5.05478
$791$	0	0
$792$	−8.46791	−8.46791
$793$	0	0
$794$	−2.56234	−2.56234
$795$	0	0
$796$	0	0
$797$	−1.75895	−1.75895	−0.879474	−	0.475947i	$-0.842105\pi$
$797$	−1.75895	−1.75895	−0.879474	+	0.475947i	$0.842105\pi$
$798$	5.05655	5.05655
$799$	0	0
$800$	2.35456	2.35456
$801$	−3.68305	−3.68305
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$809$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$810$	−3.30671	−3.30671
$811$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$811$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$812$	0	0
$813$	0	0
$814$	−2.56234	−2.56234
$815$	2.67218	2.67218
$816$	0	0
$817$	−1.26798	−1.26798
$818$	0	0
$819$	4.33280	4.33280
$820$	−3.82037	−3.82037
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	3.26586	3.26586
$825$	1.98910	1.98910
$826$	0	0
$827$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$827$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$828$	0	0
$829$	−0.165159	−0.165159	−0.0825793	−	0.996584i	$-0.526316\pi$
$829$	−0.165159	−0.165159	−0.0825793	+	0.996584i	$0.526316\pi$
$830$	−2.56234	−2.56234
$831$	0	0
$832$	2.47862	2.47862
$833$	0	0
$834$	0	0
$835$	0	0
$836$	2.80580	2.80580
$837$	0	0
$838$	−3.32729	−3.32729
$839$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$839$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$840$	13.4558	13.4558
$841$	1.00000	1.00000
$842$	−1.51972	−1.51972
$843$	−0.863592	−0.863592
$844$	0	0
$845$	−0.266320	−0.266320
$846$	0	0
$847$	1.57921	1.57921
$848$	0	0
$849$	2.38261	2.38261
$850$	0	0
$851$	0	0
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	2.27867	2.27867
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	4.93021	4.93021
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	−5.51206	−5.51206
$861$	−3.63971	−3.63971
$862$	0	0
$863$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$863$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$864$	−5.42670	−5.42670
$865$	0.223718	0.223718
$866$	0	0
$867$	−1.75895	−1.75895
$868$	0	0
$869$	2.67218	2.67218
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−3.68305	−3.68305
$874$	0	0
$875$	0.423192	0.423192
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	3.57828	3.57828
$879$	1.41312	1.41312
$880$	5.63159	5.63159
$881$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$881$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$882$	10.2123	10.2123
$883$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$883$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$884$	0	0
$885$	0	0
$886$	0.928738	0.928738
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	−5.25139	−5.25139
$889$	−3.73167	−3.73167
$890$	4.50702	4.50702
$891$	−1.74807	−1.74807
$892$	0	0
$893$	0	0
$894$	0	0
$895$	1.83484	1.83484
$896$	2.77277	2.77277
$897$	0	0
$898$	−3.73167	−3.73167
$899$	0	0
$900$	4.50702	4.50702
$901$	0	0
$902$	−2.80293	−2.80293
$903$	−5.25139	−5.25139
$904$	0	0
$905$	0	0
$906$	4.50702	4.50702
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	0	0
$910$	−5.30212	−5.30212
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	4.33723	4.33723
$913$	−1.35456	−1.35456
$914$	−3.73167	−3.73167
$915$	0	0
$916$	−5.08623	−5.08623
$917$	0	0
$918$	0	0
$919$	−1.75895	−1.75895	−0.879474	−	0.475947i	$-0.842105\pi$
$919$	−1.75895	−1.75895	−0.879474	+	0.475947i	$0.842105\pi$
$920$	0	0
$921$	0	0
$922$	0.928738	0.928738
$923$	0	0
$924$	11.6204	11.6204
$925$	0.834841	0.834841
$926$	0	0
$927$	2.29051	2.29051
$928$	0	0
$929$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$929$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$930$	0	0
$931$	−2.07137	−2.07137
$932$	0	0
$933$	−2.77611	−2.77611
$934$	−3.32729	−3.32729
$935$	0	0
$936$	6.83837	6.83837
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$941$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	4.93021	4.93021
$946$	−4.04409	−4.04409
$947$	−1.75895	−1.75895	−0.879474	−	0.475947i	$-0.842105\pi$
$947$	−1.75895	−1.75895	−0.879474	+	0.475947i	$0.842105\pi$
$948$	8.94642	8.94642
$949$	0	0
$950$	−1.26873	−1.26873
$951$	−2.77611	−2.77611
$952$	0	0
$953$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$953$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$954$	0	0
$955$	0	0
$956$	2.82037	2.82037
$957$	0	0
$958$	0	0
$959$	0	0
$960$	5.39865	5.39865
$961$	1.00000	1.00000
$962$	2.06925	2.06925
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$967$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$968$	2.49244	2.49244
$969$	0	0
$970$	4.50702	4.50702
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	−0.891634	−0.891634
$973$	0	0
$974$	−3.32729	−3.32729
$975$	−1.60632	−1.60632
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	6.56381	6.56381
$979$	2.38261	2.38261
$980$	−9.00450	−9.00450
$981$	0	0
$982$	0	0
$983$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$983$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$984$	−5.74448	−5.74448
$985$	−2.13788	−2.13788
$986$	0	0
$987$	0	0
$988$	−2.26586	−2.26586
$989$	0	0
$990$	7.26760	7.26760
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	0.290505	0.290505
$994$	0	0
$995$	0	0
$996$	−4.53506	−4.53506
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	0	0
$999$	−1.92411	−1.92411

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3071.1.d.c.3070.9	✓	9
37.36	even	2		3071.1.d.d.3070.1	yes	9
83.82	odd	2		3071.1.d.d.3070.1	yes	9
3071.3070	odd	2	CM	3071.1.d.c.3070.9	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3071.1.d.c.3070.9	✓	9	1.1	even	1	trivial
3071.1.d.c.3070.9	✓	9	3071.3070	odd	2	CM
3071.1.d.d.3070.1	yes	9	37.36	even	2
3071.1.d.d.3070.1	yes	9	83.82	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3071 = 37 \cdot 83 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3071.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.89163 q^{2} -1.75895 q^{3} +2.57828 q^{4} -1.35456 q^{5} -3.32729 q^{6} +1.89163 q^{7} +2.98553 q^{8} +2.09390 q^{9} +O(q^{10})\)	\(q+1.89163 q^{2} -1.75895 q^{3} +2.57828 q^{4} -1.35456 q^{5} -3.32729 q^{6} +1.89163 q^{7} +2.98553 q^{8} +2.09390 q^{9} -2.56234 q^{10} -1.35456 q^{11} -4.53506 q^{12} +1.09390 q^{13} +3.57828 q^{14} +2.38261 q^{15} +3.06925 q^{16} +3.96089 q^{18} -0.803391 q^{19} -3.49244 q^{20} -3.32729 q^{21} -2.56234 q^{22} -5.25139 q^{24} +0.834841 q^{25} +2.06925 q^{26} -1.92411 q^{27} +4.87717 q^{28} +4.50702 q^{30} +2.82037 q^{32} +2.38261 q^{33} -2.56234 q^{35} +5.39865 q^{36} +1.00000 q^{37} -1.51972 q^{38} -1.92411 q^{39} -4.04409 q^{40} +1.09390 q^{41} -6.29401 q^{42} +1.57828 q^{43} -3.49244 q^{44} -2.83631 q^{45} -5.39865 q^{48} +2.57828 q^{49} +1.57921 q^{50} +2.82037 q^{52} -3.63971 q^{54} +1.83484 q^{55} +5.64753 q^{56} +1.41312 q^{57} +6.14303 q^{60} +3.96089 q^{63} +2.26586 q^{64} -1.48175 q^{65} +4.50702 q^{66} -4.84701 q^{70} +6.25139 q^{72} +1.89163 q^{74} -1.46844 q^{75} -2.07137 q^{76} -2.56234 q^{77} -3.63971 q^{78} -1.97272 q^{79} -4.15750 q^{80} +1.29051 q^{81} +2.06925 q^{82} +1.00000 q^{83} -8.57868 q^{84} +2.98553 q^{86} -4.04409 q^{88} -1.75895 q^{89} -5.36527 q^{90} +2.06925 q^{91} +1.08824 q^{95} -4.96089 q^{96} -1.75895 q^{97} +4.87717 q^{98} -2.83631 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - q^{2} - q^{3} + 8 q^{4} - q^{5} - 2 q^{6} - q^{7} - 2 q^{8} + 8 q^{9}+O(q^{10}) \) 9 * q - q^2 - q^3 + 8 * q^4 - q^5 - 2 * q^6 - q^7 - 2 * q^8 + 8 * q^9	\( 9 q - q^{2} - q^{3} + 8 q^{4} - q^{5} - 2 q^{6} - q^{7} - 2 q^{8} + 8 q^{9} - 2 q^{10} - q^{11} - 3 q^{12} - q^{13} + 17 q^{14} - 2 q^{15} + 7 q^{16} - 3 q^{18} - q^{19} - 3 q^{20} - 2 q^{21} - 2 q^{22} - 4 q^{24} + 8 q^{25} - 2 q^{26} - 2 q^{27} - 3 q^{28} + 15 q^{30} - 3 q^{32} - 2 q^{33} - 2 q^{35} + 5 q^{36} + 9 q^{37} - 2 q^{38} - 2 q^{39} - 4 q^{40} - q^{41} - 4 q^{42} - q^{43} - 3 q^{44} - 3 q^{45} - 5 q^{48} + 8 q^{49} - 3 q^{50} - 3 q^{52} - 4 q^{54} + 17 q^{55} + 15 q^{56} - 2 q^{57} - 6 q^{60} - 3 q^{63} + 6 q^{64} - 2 q^{65} + 15 q^{66} - 4 q^{70} + 13 q^{72} - q^{74} - 3 q^{75} - 3 q^{76} - 2 q^{77} - 4 q^{78} - q^{79} - 5 q^{80} + 7 q^{81} - 2 q^{82} + 9 q^{83} - 6 q^{84} - 2 q^{86} - 4 q^{88} - q^{89} - 6 q^{90} - 2 q^{91} - 2 q^{95} - 6 q^{96} - q^{97} - 3 q^{98} - 3 q^{99}+O(q^{100}) \) 9 * q - q^2 - q^3 + 8 * q^4 - q^5 - 2 * q^6 - q^7 - 2 * q^8 + 8 * q^9 - 2 * q^10 - q^11 - 3 * q^12 - q^13 + 17 * q^14 - 2 * q^15 + 7 * q^16 - 3 * q^18 - q^19 - 3 * q^20 - 2 * q^21 - 2 * q^22 - 4 * q^24 + 8 * q^25 - 2 * q^26 - 2 * q^27 - 3 * q^28 + 15 * q^30 - 3 * q^32 - 2 * q^33 - 2 * q^35 + 5 * q^36 + 9 * q^37 - 2 * q^38 - 2 * q^39 - 4 * q^40 - q^41 - 4 * q^42 - q^43 - 3 * q^44 - 3 * q^45 - 5 * q^48 + 8 * q^49 - 3 * q^50 - 3 * q^52 - 4 * q^54 + 17 * q^55 + 15 * q^56 - 2 * q^57 - 6 * q^60 - 3 * q^63 + 6 * q^64 - 2 * q^65 + 15 * q^66 - 4 * q^70 + 13 * q^72 - q^74 - 3 * q^75 - 3 * q^76 - 2 * q^77 - 4 * q^78 - q^79 - 5 * q^80 + 7 * q^81 - 2 * q^82 + 9 * q^83 - 6 * q^84 - 2 * q^86 - 4 * q^88 - q^89 - 6 * q^90 - 2 * q^91 - 2 * q^95 - 6 * q^96 - q^97 - 3 * q^98 - 3 * q^99

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