LMFDB - Embedded newform 975.2.s.b.857.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [975,2,Mod(818,975)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(975, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 3, 2]))

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

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

chi := DirichletCharacter("975.818");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 975 = 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	975.s (of order $4$, degree $2$, 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:	$7.78541419707$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	8.0.151613669376.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 7x^{4} + 256 $ x^8 - 7*x^4 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			857.4
Root			$0.662382 - 1.88713i$ of defining polynomial
Character	$\chi$	$=$	975.857
Dual form			975.2.s.b.818.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.22474 + 1.22474i) q^{3} -2.00000i q^{4} +(2.54951 - 2.54951i) q^{7} +3.00000i q^{9} +6.24500 q^{11} +(2.44949 - 2.44949i) q^{12} +(-2.54951 - 2.54951i) q^{13} -4.00000 q^{16} +(-1.22474 + 1.22474i) q^{17} +6.24500 q^{21} +(-3.67423 - 3.67423i) q^{23} +(-3.67423 + 3.67423i) q^{27} +(-5.09902 - 5.09902i) q^{28} +(7.64853 + 7.64853i) q^{33} +6.00000 q^{36} +(2.54951 - 2.54951i) q^{37} -6.24500i q^{39} +6.24500 q^{41} -12.4900i q^{44} +(-4.89898 - 4.89898i) q^{48} -6.00000i q^{49} -3.00000 q^{51} +(-5.09902 + 5.09902i) q^{52} +(8.57321 + 8.57321i) q^{53} -12.4900i q^{59} +7.00000 q^{61} +(7.64853 + 7.64853i) q^{63} +8.00000i q^{64} +(-10.1980 + 10.1980i) q^{67} +(2.44949 + 2.44949i) q^{68} -9.00000i q^{69} +6.24500 q^{71} +(-5.09902 - 5.09902i) q^{73} +(15.9217 - 15.9217i) q^{77} +11.0000i q^{79} -9.00000 q^{81} -12.4900i q^{84} +18.7350i q^{89} -13.0000 q^{91} +(-7.34847 + 7.34847i) q^{92} +(2.54951 - 2.54951i) q^{97} +18.7350i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 32 q^{16} + 48 q^{36} - 24 q^{51} + 56 q^{61} - 72 q^{81} - 104 q^{91}+O(q^{100}) $ 8 * q - 32 * q^16 + 48 * q^36 - 24 * q^51 + 56 * q^61 - 72 * q^81 - 104 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/975\mathbb{Z}\right)^\times$.

$n$	$301$	$326$	$352$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{1}{4}\right)$

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 $

$2$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$2$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$3$	1.22474	+	1.22474i	0.707107	+	0.707107i
$3$	1.22474	+	1.22474i	0.707107	+	0.707107i
$4$		−	2.00000i		−	1.00000i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	2.54951	−	2.54951i	0.963624	−	0.963624i	−0.0357371	−	0.999361i	$-0.511378\pi$
$7$	2.54951	−	2.54951i	0.963624	−	0.963624i	0.999361	+	0.0357371i	$0.0113779\pi$
$8$		0			0
$9$			3.00000i			1.00000i
$10$		0			0
$11$	6.24500			1.88294			0.941469	−	0.337100i	$-0.109446\pi$
$11$	6.24500			1.88294			0.941469	+	0.337100i	$0.109446\pi$
$12$	2.44949	−	2.44949i	0.707107	−	0.707107i
$13$	−2.54951	−	2.54951i	−0.707107	−	0.707107i
$13$	−2.54951	−	2.54951i	−0.707107	−	0.707107i
$14$		0			0
$15$		0			0
$16$	−4.00000			−1.00000
$17$	−1.22474	+	1.22474i	−0.297044	+	0.297044i	−0.839855	−	0.542811i	$-0.817360\pi$
$17$	−1.22474	+	1.22474i	−0.297044	+	0.297044i	0.542811	+	0.839855i	$0.317360\pi$
$18$		0			0
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$		0			0
$21$	6.24500			1.36277
$22$		0			0
$23$	−3.67423	−	3.67423i	−0.766131	−	0.766131i	0.211292	−	0.977423i	$-0.432233\pi$
$23$	−3.67423	−	3.67423i	−0.766131	−	0.766131i	−0.977423	+	0.211292i	$0.932233\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−3.67423	+	3.67423i	−0.707107	+	0.707107i
$28$	−5.09902	−	5.09902i	−0.963624	−	0.963624i
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		0			0
$33$	7.64853	+	7.64853i	1.33144	+	1.33144i
$34$		0			0
$35$		0			0
$36$	6.00000			1.00000
$37$	2.54951	−	2.54951i	0.419137	−	0.419137i	−0.465769	−	0.884906i	$-0.654222\pi$
$37$	2.54951	−	2.54951i	0.419137	−	0.419137i	0.884906	+	0.465769i	$0.154222\pi$
$38$		0			0
$39$		−	6.24500i		−	1.00000i
$40$		0			0
$41$	6.24500			0.975305			0.487652	−	0.873038i	$-0.337853\pi$
$41$	6.24500			0.975305			0.487652	+	0.873038i	$0.337853\pi$
$42$		0			0
$43$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$43$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$44$		−	12.4900i		−	1.88294i
$45$		0			0
$46$		0			0
$47$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$47$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$48$	−4.89898	−	4.89898i	−0.707107	−	0.707107i
$49$		−	6.00000i		−	0.857143i
$50$		0			0
$51$	−3.00000			−0.420084
$52$	−5.09902	+	5.09902i	−0.707107	+	0.707107i
$53$	8.57321	+	8.57321i	1.17762	+	1.17762i	0.980349	+	0.197273i	$0.0632085\pi$
$53$	8.57321	+	8.57321i	1.17762	+	1.17762i	0.197273	+	0.980349i	$0.436791\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	12.4900i		−	1.62606i	−0.582223	−	0.813029i	$-0.697817\pi$
$59$		−	12.4900i		−	1.62606i	0.582223	−	0.813029i	$-0.302183\pi$
$60$		0			0
$61$	7.00000			0.896258			0.448129	−	0.893969i	$-0.352090\pi$
$61$	7.00000			0.896258			0.448129	+	0.893969i	$0.352090\pi$
$62$		0			0
$63$	7.64853	+	7.64853i	0.963624	+	0.963624i
$64$			8.00000i			1.00000i
$65$		0			0
$66$		0			0
$67$	−10.1980	+	10.1980i	−1.24589	+	1.24589i	−0.288370	+	0.957519i	$0.593113\pi$
$67$	−10.1980	+	10.1980i	−1.24589	+	1.24589i	−0.957519	+	0.288370i	$0.906887\pi$
$68$	2.44949	+	2.44949i	0.297044	+	0.297044i
$69$		−	9.00000i		−	1.08347i
$70$		0			0
$71$	6.24500			0.741145			0.370572	−	0.928804i	$-0.379162\pi$
$71$	6.24500			0.741145			0.370572	+	0.928804i	$0.379162\pi$
$72$		0			0
$73$	−5.09902	−	5.09902i	−0.596795	−	0.596795i	0.342663	−	0.939458i	$-0.388671\pi$
$73$	−5.09902	−	5.09902i	−0.596795	−	0.596795i	−0.939458	+	0.342663i	$0.888671\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	15.9217	−	15.9217i	1.81444	−	1.81444i
$78$		0			0
$79$			11.0000i			1.23760i	0.785550	+	0.618798i	$0.212380\pi$
$79$			11.0000i			1.23760i	−0.785550	+	0.618798i	$0.787620\pi$
$80$		0			0
$81$	−9.00000			−1.00000
$82$		0			0
$83$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$83$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$84$		−	12.4900i		−	1.36277i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$			18.7350i			1.98591i	0.118511	+	0.992953i	$0.462188\pi$
$89$			18.7350i			1.98591i	−0.118511	+	0.992953i	$0.537812\pi$
$90$		0			0
$91$	−13.0000			−1.36277
$92$	−7.34847	+	7.34847i	−0.766131	+	0.766131i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	2.54951	−	2.54951i	0.258863	−	0.258863i	−0.565728	−	0.824592i	$-0.691405\pi$
$97$	2.54951	−	2.54951i	0.258863	−	0.258863i	0.824592	+	0.565728i	$0.191405\pi$
$98$		0			0
$99$			18.7350i			1.88294i
$100$		0			0
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$103$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−13.4722	+	13.4722i	−1.30241	+	1.30241i	−0.375641	+	0.926765i	$0.622577\pi$
$107$	−13.4722	+	13.4722i	−1.30241	+	1.30241i	−0.926765	+	0.375641i	$0.877423\pi$
$108$	7.34847	+	7.34847i	0.707107	+	0.707107i
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$	6.24500			0.592749
$112$	−10.1980	+	10.1980i	−0.963624	+	0.963624i
$113$	−9.79796	−	9.79796i	−0.921714	−	0.921714i	0.0754362	−	0.997151i	$-0.475965\pi$
$113$	−9.79796	−	9.79796i	−0.921714	−	0.921714i	−0.997151	+	0.0754362i	$0.975965\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	7.64853	−	7.64853i	0.707107	−	0.707107i
$118$		0			0
$119$			6.24500i			0.572478i
$120$		0			0
$121$	28.0000			2.54545
$122$		0			0
$123$	7.64853	+	7.64853i	0.689645	+	0.689645i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$127$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$	15.2971	−	15.2971i	1.33144	−	1.33144i
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$137$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$138$		0			0
$139$		−	19.0000i		−	1.61156i	−0.592216	−	0.805779i	$-0.701747\pi$
$139$		−	19.0000i		−	1.61156i	0.592216	−	0.805779i	$-0.298253\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−15.9217	−	15.9217i	−1.33144	−	1.33144i
$144$		−	12.0000i		−	1.00000i
$145$		0			0
$146$		0			0
$147$	7.34847	−	7.34847i	0.606092	−	0.606092i
$148$	−5.09902	−	5.09902i	−0.419137	−	0.419137i
$149$			18.7350i			1.53483i	0.641150	+	0.767415i	$0.278457\pi$
$149$			18.7350i			1.53483i	−0.641150	+	0.767415i	$0.721543\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$	−3.67423	−	3.67423i	−0.297044	−	0.297044i
$154$		0			0
$155$		0			0
$156$	−12.4900			−1.00000
$157$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$157$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$158$		0			0
$159$			21.0000i			1.66541i
$160$		0			0
$161$	−18.7350			−1.47652
$162$		0			0
$163$	−17.8466	−	17.8466i	−1.39785	−	1.39785i	−0.806178	−	0.591673i	$-0.798468\pi$
$163$	−17.8466	−	17.8466i	−1.39785	−	1.39785i	−0.591673	−	0.806178i	$-0.701532\pi$
$164$		−	12.4900i		−	0.975305i
$165$		0			0
$166$		0			0
$167$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$167$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$168$		0			0
$169$			13.0000i			1.00000i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	14.6969	+	14.6969i	1.11739	+	1.11739i	0.992123	+	0.125264i	$0.0399778\pi$
$173$	14.6969	+	14.6969i	1.11739	+	1.11739i	0.125264	+	0.992123i	$0.460022\pi$
$174$		0			0
$175$		0			0
$176$	−24.9800			−1.88294
$177$	15.2971	−	15.2971i	1.14980	−	1.14980i
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$	−23.0000			−1.70958			−0.854788	−	0.518977i	$-0.826313\pi$
$181$	−23.0000			−1.70958			−0.854788	+	0.518977i	$0.826313\pi$
$182$		0			0
$183$	8.57321	+	8.57321i	0.633750	+	0.633750i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−7.64853	+	7.64853i	−0.559316	+	0.559316i
$188$		0			0
$189$			18.7350i			1.36277i
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$	−9.79796	+	9.79796i	−0.707107	+	0.707107i
$193$	−17.8466	−	17.8466i	−1.28462	−	1.28462i	−0.938007	−	0.346618i	$-0.887330\pi$
$193$	−17.8466	−	17.8466i	−1.28462	−	1.28462i	−0.346618	−	0.938007i	$-0.612670\pi$
$194$		0			0
$195$		0			0
$196$	−12.0000			−0.857143
$197$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$197$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$198$		0			0
$199$		−	4.00000i		−	0.283552i	−0.989899	−	0.141776i	$-0.954719\pi$
$199$		−	4.00000i		−	0.283552i	0.989899	−	0.141776i	$-0.0452813\pi$
$200$		0			0
$201$	−24.9800			−1.76195
$202$		0			0
$203$		0			0
$204$			6.00000i			0.420084i
$205$		0			0
$206$		0			0
$207$	11.0227	−	11.0227i	0.766131	−	0.766131i
$208$	10.1980	+	10.1980i	0.707107	+	0.707107i
$209$		0			0
$210$		0			0
$211$	−8.00000			−0.550743			−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000			−0.550743			−0.275371	+	0.961338i	$0.588801\pi$
$212$	17.1464	−	17.1464i	1.17762	−	1.17762i
$213$	7.64853	+	7.64853i	0.524069	+	0.524069i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		−	12.4900i		−	0.843996i
$220$		0			0
$221$	6.24500			0.420084
$222$		0			0
$223$	20.3961	+	20.3961i	1.36582	+	1.36582i	0.866302	+	0.499520i	$0.166490\pi$
$223$	20.3961	+	20.3961i	1.36582	+	1.36582i	0.499520	+	0.866302i	$0.333510\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$227$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$228$		0			0
$229$		0			0			−	1.00000i	$-0.5\pi$
$229$		0			0				1.00000i	$0.5\pi$
$230$		0			0
$231$	39.0000			2.56601
$232$		0			0
$233$	20.8207	+	20.8207i	1.36401	+	1.36401i	0.868742	+	0.495265i	$0.164929\pi$
$233$	20.8207	+	20.8207i	1.36401	+	1.36401i	0.495265	+	0.868742i	$0.335071\pi$
$234$		0			0
$235$		0			0
$236$	−24.9800			−1.62606
$237$	−13.4722	+	13.4722i	−0.875113	+	0.875113i
$238$		0			0
$239$			18.7350i			1.21187i	0.795516	+	0.605933i	$0.207200\pi$
$239$			18.7350i			1.21187i	−0.795516	+	0.605933i	$0.792800\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		0			0
$243$	−11.0227	−	11.0227i	−0.707107	−	0.707107i
$244$		−	14.0000i		−	0.896258i
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$	15.2971	−	15.2971i	0.963624	−	0.963624i
$253$	−22.9456	−	22.9456i	−1.44258	−	1.44258i
$254$		0			0
$255$		0			0
$256$	16.0000			1.00000
$257$	−19.5959	+	19.5959i	−1.22236	+	1.22236i	−0.255569	+	0.966791i	$0.582263\pi$
$257$	−19.5959	+	19.5959i	−1.22236	+	1.22236i	−0.966791	+	0.255569i	$0.917737\pi$
$258$		0			0
$259$		−	13.0000i		−	0.807781i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	2.44949	+	2.44949i	0.151042	+	0.151042i	0.778583	−	0.627541i	$-0.215939\pi$
$263$	2.44949	+	2.44949i	0.151042	+	0.151042i	−0.627541	+	0.778583i	$0.715939\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−22.9456	+	22.9456i	−1.40425	+	1.40425i
$268$	20.3961	+	20.3961i	1.24589	+	1.24589i
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$	4.89898	−	4.89898i	0.297044	−	0.297044i
$273$	−15.9217	−	15.9217i	−0.963624	−	0.963624i
$274$		0			0
$275$		0			0
$276$	−18.0000			−1.08347
$277$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$277$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−24.9800			−1.49018			−0.745091	−	0.666963i	$-0.767594\pi$
$281$	−24.9800			−1.49018			−0.745091	+	0.666963i	$0.767594\pi$
$282$		0			0
$283$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$283$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$284$		−	12.4900i		−	0.741145i
$285$		0			0
$286$		0			0
$287$	15.9217	−	15.9217i	0.939827	−	0.939827i
$288$		0			0
$289$			14.0000i			0.823529i
$290$		0			0
$291$	6.24500			0.366088
$292$	−10.1980	+	10.1980i	−0.596795	+	0.596795i
$293$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$293$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−22.9456	+	22.9456i	−1.33144	+	1.33144i
$298$		0			0
$299$			18.7350i			1.08347i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	2.54951	−	2.54951i	0.145508	−	0.145508i	−0.630600	−	0.776108i	$-0.717191\pi$
$307$	2.54951	−	2.54951i	0.145508	−	0.145508i	0.776108	+	0.630600i	$0.217191\pi$
$308$	−31.8434	−	31.8434i	−1.81444	−	1.81444i
$309$		0			0
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$313$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$314$		0			0
$315$		0			0
$316$	22.0000			1.23760
$317$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$317$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−33.0000			−1.84188
$322$		0			0
$323$		0			0
$324$			18.0000i			1.00000i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$		0			0
$333$	7.64853	+	7.64853i	0.419137	+	0.419137i
$334$		0			0
$335$		0			0
$336$	−24.9800			−1.36277
$337$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$337$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$338$		0			0
$339$		−	24.0000i		−	1.30350i
$340$		0			0
$341$		0			0
$342$		0			0
$343$	2.54951	+	2.54951i	0.137661	+	0.137661i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−25.7196	+	25.7196i	−1.38070	+	1.38070i	−0.537332	+	0.843371i	$0.680568\pi$
$347$	−25.7196	+	25.7196i	−1.38070	+	1.38070i	−0.843371	+	0.537332i	$0.819432\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$		0			0
$351$	18.7350			1.00000
$352$		0			0
$353$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$353$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$354$		0			0
$355$		0			0
$356$	37.4700			1.98591
$357$	−7.64853	+	7.64853i	−0.404803	+	0.404803i
$358$		0			0
$359$		−	12.4900i		−	0.659197i	−0.944121	−	0.329598i	$-0.893087\pi$
$359$		−	12.4900i		−	0.659197i	0.944121	−	0.329598i	$-0.106913\pi$
$360$		0			0
$361$	−19.0000			−1.00000
$362$		0			0
$363$	34.2929	+	34.2929i	1.79991	+	1.79991i
$364$			26.0000i			1.36277i
$365$		0			0
$366$		0			0
$367$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$367$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$368$	14.6969	+	14.6969i	0.766131	+	0.766131i
$369$			18.7350i			0.975305i
$370$		0			0
$371$	43.7150			2.26957
$372$		0			0
$373$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$373$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$383$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	−5.09902	−	5.09902i	−0.258863	−	0.258863i
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$		0			0
$391$	9.00000			0.455150
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$	37.4700			1.88294
$397$	28.0446	−	28.0446i	1.40752	−	1.40752i	0.635036	−	0.772483i	$-0.280985\pi$
$397$	28.0446	−	28.0446i	1.40752	−	1.40752i	0.772483	−	0.635036i	$-0.219015\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−24.9800			−1.24744			−0.623721	−	0.781647i	$-0.714380\pi$
$401$	−24.9800			−1.24744			−0.623721	+	0.781647i	$0.714380\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$	15.9217	−	15.9217i	0.789209	−	0.789209i
$408$		0			0
$409$		0			0			−	1.00000i	$-0.5\pi$
$409$		0			0				1.00000i	$0.5\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−31.8434	−	31.8434i	−1.56691	−	1.56691i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	23.2702	−	23.2702i	1.13954	−	1.13954i
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	17.8466	−	17.8466i	0.863656	−	0.863656i
$428$	26.9444	+	26.9444i	1.30241	+	1.30241i
$429$		−	39.0000i		−	1.88294i
$430$		0			0
$431$	37.4700			1.80487			0.902433	−	0.430830i	$-0.141779\pi$
$431$	37.4700			1.80487			0.902433	+	0.430830i	$0.141779\pi$
$432$	14.6969	−	14.6969i	0.707107	−	0.707107i
$433$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$433$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$			1.00000i			0.0477274i	0.999715	+	0.0238637i	$0.00759677\pi$
$439$			1.00000i			0.0477274i	−0.999715	+	0.0238637i	$0.992403\pi$
$440$		0			0
$441$	18.0000			0.857143
$442$		0			0
$443$	8.57321	+	8.57321i	0.407326	+	0.407326i	0.880805	−	0.473479i	$-0.157002\pi$
$443$	8.57321	+	8.57321i	0.407326	+	0.407326i	−0.473479	+	0.880805i	$0.657002\pi$
$444$		−	12.4900i		−	0.592749i
$445$		0			0
$446$		0			0
$447$	−22.9456	+	22.9456i	−1.08529	+	1.08529i
$448$	20.3961	+	20.3961i	0.963624	+	0.963624i
$449$			18.7350i			0.884159i	0.896976	+	0.442080i	$0.145759\pi$
$449$			18.7350i			0.884159i	−0.896976	+	0.442080i	$0.854241\pi$
$450$		0			0
$451$	39.0000			1.83644
$452$	−19.5959	+	19.5959i	−0.921714	+	0.921714i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	2.54951	−	2.54951i	0.119261	−	0.119261i	−0.644957	−	0.764218i	$-0.723125\pi$
$457$	2.54951	−	2.54951i	0.119261	−	0.119261i	0.764218	+	0.644957i	$0.223125\pi$
$458$		0			0
$459$		−	9.00000i		−	0.420084i
$460$		0			0
$461$	6.24500			0.290859			0.145429	−	0.989369i	$-0.453544\pi$
$461$	6.24500			0.290859			0.145429	+	0.989369i	$0.453544\pi$
$462$		0			0
$463$	−17.8466	−	17.8466i	−0.829400	−	0.829400i	0.158033	−	0.987434i	$-0.449485\pi$
$463$	−17.8466	−	17.8466i	−0.829400	−	0.829400i	−0.987434	+	0.158033i	$0.949485\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	11.0227	−	11.0227i	0.510070	−	0.510070i	−0.404478	−	0.914548i	$-0.632547\pi$
$467$	11.0227	−	11.0227i	0.510070	−	0.510070i	0.914548	+	0.404478i	$0.132547\pi$
$468$	−15.2971	−	15.2971i	−0.707107	−	0.707107i
$469$			52.0000i			2.40114i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$	12.4900			0.572478
$477$	−25.7196	+	25.7196i	−1.17762	+	1.17762i
$478$		0			0
$479$		−	43.7150i		−	1.99739i	−0.0510843	−	0.998694i	$-0.516268\pi$
$479$		−	43.7150i		−	1.99739i	0.0510843	−	0.998694i	$-0.483732\pi$
$480$		0			0
$481$	−13.0000			−0.592749
$482$		0			0
$483$	−22.9456	−	22.9456i	−1.04406	−	1.04406i
$484$		−	56.0000i		−	2.54545i
$485$		0			0
$486$		0			0
$487$	28.0446	−	28.0446i	1.27082	−	1.27082i	0.325165	−	0.945657i	$-0.394580\pi$
$487$	28.0446	−	28.0446i	1.27082	−	1.27082i	0.945657	−	0.325165i	$-0.105420\pi$
$488$		0			0
$489$		−	43.7150i		−	1.97686i
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$	15.2971	−	15.2971i	0.689645	−	0.689645i
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	15.9217	−	15.9217i	0.714185	−	0.714185i
$498$		0			0
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−22.0454	−	22.0454i	−0.982956	−	0.982956i	0.0169010	−	0.999857i	$-0.494620\pi$
$503$	−22.0454	−	22.0454i	−0.982956	−	0.982956i	−0.999857	+	0.0169010i	$0.994620\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−15.9217	+	15.9217i	−0.707107	+	0.707107i
$508$		0			0
$509$		−	43.7150i		−	1.93763i	−0.247780	−	0.968816i	$-0.579701\pi$
$509$		−	43.7150i		−	1.93763i	0.247780	−	0.968816i	$-0.420299\pi$
$510$		0			0
$511$	−26.0000			−1.15017
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$			36.0000i			1.58022i
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$523$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$	−30.5941	−	30.5941i	−1.33144	−	1.33144i
$529$			4.00000i			0.173913i
$530$		0			0
$531$	37.4700			1.62606
$532$		0			0
$533$	−15.9217	−	15.9217i	−0.689645	−	0.689645i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	37.4700i		−	1.61395i
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$	−28.1691	−	28.1691i	−1.20885	−	1.20885i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$547$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$548$		0			0
$549$			21.0000i			0.896258i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	28.0446	+	28.0446i	1.19258	+	1.19258i
$554$		0			0
$555$		0			0
$556$	−38.0000			−1.61156
$557$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$557$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	−18.7350			−0.790992
$562$		0			0
$563$	33.0681	+	33.0681i	1.39365	+	1.39365i	0.816959	+	0.576696i	$0.195658\pi$
$563$	33.0681	+	33.0681i	1.39365	+	1.39365i	0.576696	+	0.816959i	$0.304342\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−22.9456	+	22.9456i	−0.963624	+	0.963624i
$568$		0			0
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$	−23.0000			−0.962520			−0.481260	−	0.876578i	$-0.659821\pi$
$571$	−23.0000			−0.962520			−0.481260	+	0.876578i	$0.659821\pi$
$572$	−31.8434	+	31.8434i	−1.33144	+	1.33144i
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−24.0000			−1.00000
$577$	28.0446	−	28.0446i	1.16751	−	1.16751i	0.184721	−	0.982791i	$-0.440862\pi$
$577$	28.0446	−	28.0446i	1.16751	−	1.16751i	0.982791	−	0.184721i	$-0.0591383\pi$
$578$		0			0
$579$		−	43.7150i		−	1.81673i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	53.5397	+	53.5397i	2.21739	+	2.21739i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$587$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$588$	−14.6969	−	14.6969i	−0.606092	−	0.606092i
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−10.1980	+	10.1980i	−0.419137	+	0.419137i
$593$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$593$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$594$		0			0
$595$		0			0
$596$	37.4700			1.53483
$597$	4.89898	−	4.89898i	0.200502	−	0.200502i
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	47.0000			1.91717			0.958585	−	0.284807i	$-0.0919294\pi$
$601$	47.0000			1.91717			0.958585	+	0.284807i	$0.0919294\pi$
$602$		0			0
$603$	−30.5941	−	30.5941i	−1.24589	−	1.24589i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$607$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$	−7.34847	+	7.34847i	−0.297044	+	0.297044i
$613$	−17.8466	−	17.8466i	−0.720816	−	0.720816i	0.247955	−	0.968771i	$-0.420241\pi$
$613$	−17.8466	−	17.8466i	−0.720816	−	0.720816i	−0.968771	+	0.247955i	$0.920241\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$617$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$618$		0			0
$619$		0			0			−	1.00000i	$-0.5\pi$
$619$		0			0				1.00000i	$0.5\pi$
$620$		0			0
$621$	27.0000			1.08347
$622$		0			0
$623$	47.7650	+	47.7650i	1.91367	+	1.91367i
$624$			24.9800i			1.00000i
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$			6.24500i			0.249004i
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$	−9.79796	−	9.79796i	−0.389434	−	0.389434i
$634$		0			0
$635$		0			0
$636$	42.0000			1.66541
$637$	−15.2971	+	15.2971i	−0.606092	+	0.606092i
$638$		0			0
$639$			18.7350i			0.741145i
$640$		0			0
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$		0			0
$643$	33.1436	+	33.1436i	1.30706	+	1.30706i	0.923529	+	0.383528i	$0.125291\pi$
$643$	33.1436	+	33.1436i	1.30706	+	1.30706i	0.383528	+	0.923529i	$0.374709\pi$
$644$			37.4700i			1.47652i
$645$		0			0
$646$		0			0
$647$	35.5176	−	35.5176i	1.39634	−	1.39634i	0.586108	−	0.810233i	$-0.300659\pi$
$647$	35.5176	−	35.5176i	1.39634	−	1.39634i	0.810233	−	0.586108i	$-0.199341\pi$
$648$		0			0
$649$		−	78.0000i		−	3.06177i
$650$		0			0
$651$		0			0
$652$	−35.6931	+	35.6931i	−1.39785	+	1.39785i
$653$	−34.2929	−	34.2929i	−1.34198	−	1.34198i	−0.894084	−	0.447899i	$-0.852172\pi$
$653$	−34.2929	−	34.2929i	−1.34198	−	1.34198i	−0.447899	−	0.894084i	$-0.647828\pi$
$654$		0			0
$655$		0			0
$656$	−24.9800			−0.975305
$657$	15.2971	−	15.2971i	0.596795	−	0.596795i
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$		0			0
$663$	7.64853	+	7.64853i	0.297044	+	0.297044i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$			49.9600i			1.93156i
$670$		0			0
$671$	43.7150			1.68760
$672$		0			0
$673$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$673$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$674$		0			0
$675$		0			0
$676$	26.0000			1.00000
$677$	23.2702	−	23.2702i	0.894345	−	0.894345i	−0.100584	−	0.994929i	$-0.532071\pi$
$677$	23.2702	−	23.2702i	0.894345	−	0.894345i	0.994929	+	0.100584i	$0.0320711\pi$
$678$		0			0
$679$		−	13.0000i		−	0.498894i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$683$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	43.7150i		−	1.66541i
$690$		0			0
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$	29.3939	−	29.3939i	1.11739	−	1.11739i
$693$	47.7650	+	47.7650i	1.81444	+	1.81444i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−7.64853	+	7.64853i	−0.289709	+	0.289709i
$698$		0			0
$699$			51.0000i			1.92900i
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$			49.9600i			1.88294i
$705$		0			0
$706$		0			0
$707$		0			0
$708$	−30.5941	−	30.5941i	−1.14980	−	1.14980i
$709$		0			0			−	1.00000i	$-0.5\pi$
$709$		0			0				1.00000i	$0.5\pi$
$710$		0			0
$711$	−33.0000			−1.23760
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−22.9456	+	22.9456i	−0.856919	+	0.856919i
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$			46.0000i			1.70958i
$725$		0			0
$726$		0			0
$727$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$727$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$728$		0			0
$729$		−	27.0000i		−	1.00000i
$730$		0			0
$731$		0			0
$732$	17.1464	−	17.1464i	0.633750	−	0.633750i
$733$	33.1436	+	33.1436i	1.22419	+	1.22419i	0.966129	+	0.258058i	$0.0830827\pi$
$733$	33.1436	+	33.1436i	1.22419	+	1.22419i	0.258058	+	0.966129i	$0.416917\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−63.6867	+	63.6867i	−2.34593	+	2.34593i
$738$		0			0
$739$		0			0			−	1.00000i	$-0.5\pi$
$739$		0			0				1.00000i	$0.5\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$743$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$	15.2971	+	15.2971i	0.559316	+	0.559316i
$749$			68.6950i			2.51006i
$750$		0			0
$751$	−53.0000			−1.93400			−0.966999	−	0.254781i	$-0.917997\pi$
$751$	−53.0000			−1.93400			−0.966999	+	0.254781i	$0.917997\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	37.4700			1.36277
$757$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$757$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$758$		0			0
$759$		−	56.2050i		−	2.04011i
$760$		0			0
$761$	−24.9800			−0.905524			−0.452762	−	0.891631i	$-0.649561\pi$
$761$	−24.9800			−0.905524			−0.452762	+	0.891631i	$0.649561\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−31.8434	+	31.8434i	−1.14980	+	1.14980i
$768$	19.5959	+	19.5959i	0.707107	+	0.707107i
$769$		0			0			−	1.00000i	$-0.5\pi$
$769$		0			0				1.00000i	$0.5\pi$
$770$		0			0
$771$	−48.0000			−1.72868
$772$	−35.6931	+	35.6931i	−1.28462	+	1.28462i
$773$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$773$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	15.9217	−	15.9217i	0.571187	−	0.571187i
$778$		0			0
$779$		0			0
$780$		0			0
$781$	39.0000			1.39553
$782$		0			0
$783$		0			0
$784$			24.0000i			0.857143i
$785$		0			0
$786$		0			0
$787$	−10.1980	+	10.1980i	−0.363521	+	0.363521i	−0.865107	−	0.501587i	$-0.832750\pi$
$787$	−10.1980	+	10.1980i	−0.363521	+	0.363521i	0.501587	+	0.865107i	$0.332750\pi$
$788$		0			0
$789$			6.00000i			0.213606i
$790$		0			0
$791$	−49.9600			−1.77637
$792$		0			0
$793$	−17.8466	−	17.8466i	−0.633750	−	0.633750i
$794$		0			0
$795$		0			0
$796$	−8.00000			−0.283552
$797$	−1.22474	+	1.22474i	−0.0433827	+	0.0433827i	−0.728465	−	0.685083i	$-0.759766\pi$
$797$	−1.22474	+	1.22474i	−0.0433827	+	0.0433827i	0.685083	+	0.728465i	$0.259766\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−56.2050			−1.98591
$802$		0			0
$803$	−31.8434	−	31.8434i	−1.12373	−	1.12373i
$804$			49.9600i			1.76195i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0			−	1.00000i	$-0.5\pi$
$809$		0			0				1.00000i	$0.5\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$	12.0000			0.420084
$817$		0			0
$818$		0			0
$819$		−	39.0000i		−	1.36277i
$820$		0			0
$821$	−56.2050			−1.96157			−0.980784	−	0.195098i	$-0.937497\pi$
$821$	−56.2050			−1.96157			−0.980784	+	0.195098i	$0.937497\pi$
$822$		0			0
$823$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$823$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$827$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$828$	−22.0454	−	22.0454i	−0.766131	−	0.766131i
$829$		−	14.0000i		−	0.486240i	−0.969996	−	0.243120i	$-0.921829\pi$
$829$		−	14.0000i		−	0.486240i	0.969996	−	0.243120i	$-0.0781709\pi$
$830$		0			0
$831$		0			0
$832$	20.3961	−	20.3961i	0.707107	−	0.707107i
$833$	7.34847	+	7.34847i	0.254609	+	0.254609i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	43.7150i		−	1.50921i	−0.656180	−	0.754604i	$-0.727829\pi$
$839$		−	43.7150i		−	1.50921i	0.656180	−	0.754604i	$-0.272171\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$		0			0
$843$	−30.5941	−	30.5941i	−1.05372	−	1.05372i
$844$			16.0000i			0.550743i
$845$		0			0
$846$		0			0
$847$	71.3863	−	71.3863i	2.45286	−	2.45286i
$848$	−34.2929	−	34.2929i	−1.17762	−	1.17762i
$849$		0			0
$850$		0			0
$851$	−18.7350			−0.642227
$852$	15.2971	−	15.2971i	0.524069	−	0.524069i
$853$	33.1436	+	33.1436i	1.13482	+	1.13482i	0.989366	+	0.145451i	$0.0464632\pi$
$853$	33.1436	+	33.1436i	1.13482	+	1.13482i	0.145451	+	0.989366i	$0.453537\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	11.0227	−	11.0227i	0.376528	−	0.376528i	−0.493320	−	0.869848i	$-0.664217\pi$
$857$	11.0227	−	11.0227i	0.376528	−	0.376528i	0.869848	+	0.493320i	$0.164217\pi$
$858$		0			0
$859$			41.0000i			1.39890i	0.714681	+	0.699451i	$0.246572\pi$
$859$			41.0000i			1.39890i	−0.714681	+	0.699451i	$0.753428\pi$
$860$		0			0
$861$	39.0000			1.32912
$862$		0			0
$863$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$863$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−17.1464	+	17.1464i	−0.582323	+	0.582323i
$868$		0			0
$869$			68.6950i			2.33032i
$870$		0			0
$871$	52.0000			1.76195
$872$		0			0
$873$	7.64853	+	7.64853i	0.258863	+	0.258863i
$874$		0			0
$875$		0			0
$876$	−24.9800			−0.843996
$877$	−35.6931	+	35.6931i	−1.20527	+	1.20527i	−0.232730	+	0.972541i	$0.574766\pi$
$877$	−35.6931	+	35.6931i	−1.20527	+	1.20527i	−0.972541	+	0.232730i	$0.925234\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$883$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$884$		−	12.4900i		−	0.420084i
$885$		0			0
$886$		0			0
$887$	−13.4722	+	13.4722i	−0.452352	+	0.452352i	−0.896135	−	0.443783i	$-0.853636\pi$
$887$	−13.4722	+	13.4722i	−0.452352	+	0.452352i	0.443783	+	0.896135i	$0.353636\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−56.2050			−1.88294
$892$	40.7922	−	40.7922i	1.36582	−	1.36582i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−22.9456	+	22.9456i	−0.766131	+	0.766131i
$898$		0			0
$899$		0			0
$900$		0			0
$901$	−21.0000			−0.699611
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$907$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		−	59.0000i		−	1.94623i	−0.230319	−	0.973115i	$-0.573977\pi$
$919$		−	59.0000i		−	1.94623i	0.230319	−	0.973115i	$-0.426023\pi$
$920$		0			0
$921$	6.24500			0.205780
$922$		0			0
$923$	−15.9217	−	15.9217i	−0.524069	−	0.524069i
$924$		−	78.0000i		−	2.56601i
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	43.7150i		−	1.43424i	−0.696949	−	0.717121i	$-0.745459\pi$
$929$		−	43.7150i		−	1.43424i	0.696949	−	0.717121i	$-0.254541\pi$
$930$		0			0
$931$		0			0
$932$	41.6413	−	41.6413i	1.36401	−	1.36401i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$937$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−56.2050			−1.83223			−0.916115	−	0.400916i	$-0.868692\pi$
$941$	−56.2050			−1.83223			−0.916115	+	0.400916i	$0.868692\pi$
$942$		0			0
$943$	−22.9456	−	22.9456i	−0.747211	−	0.747211i
$944$			49.9600i			1.62606i
$945$		0			0
$946$		0			0
$947$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$947$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$948$	26.9444	+	26.9444i	0.875113	+	0.875113i
$949$			26.0000i			0.843996i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	33.0681	+	33.0681i	1.07118	+	1.07118i	0.997264	+	0.0739168i	$0.0235499\pi$
$953$	33.0681	+	33.0681i	1.07118	+	1.07118i	0.0739168	+	0.997264i	$0.476450\pi$
$954$		0			0
$955$		0			0
$956$	37.4700			1.21187
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	31.0000			1.00000
$962$		0			0
$963$	−40.4166	−	40.4166i	−1.30241	−	1.30241i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	40.7922	−	40.7922i	1.31179	−	1.31179i	0.391690	−	0.920097i	$-0.371891\pi$
$967$	40.7922	−	40.7922i	1.31179	−	1.31179i	0.920097	−	0.391690i	$-0.128109\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$	−22.0454	+	22.0454i	−0.707107	+	0.707107i
$973$	−48.4407	−	48.4407i	−1.55294	−	1.55294i
$974$		0			0
$975$		0			0
$976$	−28.0000			−0.896258
$977$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$977$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$978$		0			0
$979$			117.000i			3.73934i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$983$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	47.0000			1.49300			0.746502	−	0.665383i	$-0.231732\pi$
$991$	47.0000			1.49300			0.746502	+	0.665383i	$0.231732\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$997$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$998$		0			0
$999$			18.7350i			0.592749i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	975.2.s.b.857.4	yes	8
3.2	odd	2	inner	975.2.s.b.857.2	yes	8
5.2	odd	4	inner	975.2.s.b.818.4	yes	8
5.3	odd	4	inner	975.2.s.b.818.1	✓	8
5.4	even	2	inner	975.2.s.b.857.1	yes	8
13.12	even	2	inner	975.2.s.b.857.3	yes	8
15.2	even	4	inner	975.2.s.b.818.2	yes	8
15.8	even	4	inner	975.2.s.b.818.3	yes	8
15.14	odd	2	inner	975.2.s.b.857.3	yes	8
39.38	odd	2	inner	975.2.s.b.857.1	yes	8
65.12	odd	4	inner	975.2.s.b.818.3	yes	8
65.38	odd	4	inner	975.2.s.b.818.2	yes	8
65.64	even	2	inner	975.2.s.b.857.2	yes	8
195.38	even	4	inner	975.2.s.b.818.4	yes	8
195.77	even	4	inner	975.2.s.b.818.1	✓	8
195.194	odd	2	CM	975.2.s.b.857.4	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
975.2.s.b.818.1	✓	8	5.3	odd	4	inner
975.2.s.b.818.1	✓	8	195.77	even	4	inner
975.2.s.b.818.2	yes	8	15.2	even	4	inner
975.2.s.b.818.2	yes	8	65.38	odd	4	inner
975.2.s.b.818.3	yes	8	15.8	even	4	inner
975.2.s.b.818.3	yes	8	65.12	odd	4	inner
975.2.s.b.818.4	yes	8	5.2	odd	4	inner
975.2.s.b.818.4	yes	8	195.38	even	4	inner
975.2.s.b.857.1	yes	8	5.4	even	2	inner
975.2.s.b.857.1	yes	8	39.38	odd	2	inner
975.2.s.b.857.2	yes	8	3.2	odd	2	inner
975.2.s.b.857.2	yes	8	65.64	even	2	inner
975.2.s.b.857.3	yes	8	13.12	even	2	inner
975.2.s.b.857.3	yes	8	15.14	odd	2	inner
975.2.s.b.857.4	yes	8	1.1	even	1	trivial
975.2.s.b.857.4	yes	8	195.194	odd	2	CM

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:	\( N \)	\(=\)	\( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	975.s (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.22474 + 1.22474i) q^{3} -2.00000i q^{4} +(2.54951 - 2.54951i) q^{7} +3.00000i q^{9} +6.24500 q^{11} +(2.44949 - 2.44949i) q^{12} +(-2.54951 - 2.54951i) q^{13} -4.00000 q^{16} +(-1.22474 + 1.22474i) q^{17} +6.24500 q^{21} +(-3.67423 - 3.67423i) q^{23} +(-3.67423 + 3.67423i) q^{27} +(-5.09902 - 5.09902i) q^{28} +(7.64853 + 7.64853i) q^{33} +6.00000 q^{36} +(2.54951 - 2.54951i) q^{37} -6.24500i q^{39} +6.24500 q^{41} -12.4900i q^{44} +(-4.89898 - 4.89898i) q^{48} -6.00000i q^{49} -3.00000 q^{51} +(-5.09902 + 5.09902i) q^{52} +(8.57321 + 8.57321i) q^{53} -12.4900i q^{59} +7.00000 q^{61} +(7.64853 + 7.64853i) q^{63} +8.00000i q^{64} +(-10.1980 + 10.1980i) q^{67} +(2.44949 + 2.44949i) q^{68} -9.00000i q^{69} +6.24500 q^{71} +(-5.09902 - 5.09902i) q^{73} +(15.9217 - 15.9217i) q^{77} +11.0000i q^{79} -9.00000 q^{81} -12.4900i q^{84} +18.7350i q^{89} -13.0000 q^{91} +(-7.34847 + 7.34847i) q^{92} +(2.54951 - 2.54951i) q^{97} +18.7350i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 32 q^{16} + 48 q^{36} - 24 q^{51} + 56 q^{61} - 72 q^{81} - 104 q^{91}+O(q^{100}) \) 8 * q - 32 * q^16 + 48 * q^36 - 24 * q^51 + 56 * q^61 - 72 * q^81 - 104 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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