LMFDB - Embedded newform 1248.2.h.b.623.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1248,2,Mod(623,1248)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1248, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1248.623");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1248 = 2^{5} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1248.h (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:	$9.96533017226$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 16x^{9} + 92x^{6} - 68x^{3} + 27 $ x^12 - 16x^9 + 92x^6 - 68*x^3 + 27
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 312)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			623.7
Root			$-0.691788 - 1.95556i$ of defining polynomial
Character	$\chi$	$=$	1248.623
Dual form			1248.2.h.b.623.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.111731 + 1.72844i) q^{3} -4.04932i q^{5} -2.99062 q^{7} +(-2.97503 - 0.386242i) q^{9} +O(q^{10})$	$q+(-0.111731 + 1.72844i) q^{3} -4.04932i q^{5} -2.99062 q^{7} +(-2.97503 - 0.386242i) q^{9} +3.60555 q^{13} +(6.99902 + 0.452435i) q^{15} +6.14129i q^{17} +(0.334145 - 5.16911i) q^{21} -11.3970 q^{25} +(1.00000 - 5.09902i) q^{27} -7.21110 q^{31} +12.1100i q^{35} -11.6757 q^{37} +(-0.402852 + 6.23199i) q^{39} -12.1236 q^{43} +(-1.56402 + 12.0469i) q^{45} +4.99739i q^{47} +1.94378 q^{49} +(-10.6149 - 0.686173i) q^{51} +(8.89718 + 1.15510i) q^{63} -14.6000i q^{65} +3.10125i q^{71} +(1.27340 - 19.6990i) q^{75} +(8.70163 + 2.29816i) q^{81} +24.8680 q^{85} -10.7828 q^{91} +(0.805704 - 12.4640i) q^{93} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q+O(q^{10}) $ 12 * q	$ 12 q - 60 q^{25} + 12 q^{27} + 84 q^{49} - 60 q^{51} + 84 q^{75}+O(q^{100}) $ 12 * q - 60 * q^25 + 12 * q^27 + 84 * q^49 - 60 * q^51 + 84 * q^75

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1248\mathbb{Z}\right)^\times$.

$n$	$703$	$769$	$833$	$1093$
$\chi(n)$	$-1$	$-1$	$-1$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$	−0.111731	+	1.72844i	−0.0645079	+	0.997917i
$3$	−0.111731	+	1.72844i	−0.0645079	+	0.997917i
$4$		0			0
$5$		−	4.04932i		−	1.81091i	−0.424441	−	0.905455i	$-0.639530\pi$
$5$		−	4.04932i		−	1.81091i	0.424441	−	0.905455i	$-0.360470\pi$
$6$		0			0
$7$	−2.99062			−1.13035			−0.565173	−	0.824972i	$-0.691191\pi$
$7$	−2.99062			−1.13035			−0.565173	+	0.824972i	$0.691191\pi$
$8$		0			0
$9$	−2.97503	−	0.386242i	−0.991677	−	0.128747i
$10$		0			0
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$		0			0
$13$	3.60555			1.00000
$13$	3.60555			1.00000
$14$		0			0
$15$	6.99902	+	0.452435i	1.80714	+	0.116818i
$16$		0			0
$17$			6.14129i			1.48948i	0.667354	+	0.744741i	$0.267427\pi$
$17$			6.14129i			1.48948i	−0.667354	+	0.744741i	$0.732573\pi$
$18$		0			0
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$		0			0
$21$	0.334145	−	5.16911i	0.0729163	−	1.12799i
$22$		0			0
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$		0			0
$25$	−11.3970			−2.27940
$26$		0			0
$27$	1.00000	−	5.09902i	0.192450	−	0.981307i
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$	−7.21110			−1.29515			−0.647576	−	0.762001i	$-0.724217\pi$
$31$	−7.21110			−1.29515			−0.647576	+	0.762001i	$0.724217\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			12.1100i			2.04696i
$36$		0			0
$37$	−11.6757			−1.91948			−0.959738	−	0.280898i	$-0.909368\pi$
$37$	−11.6757			−1.91948			−0.959738	+	0.280898i	$0.909368\pi$
$38$		0			0
$39$	−0.402852	+	6.23199i	−0.0645079	+	0.997917i
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$	−12.1236			−1.84883			−0.924415	−	0.381388i	$-0.875446\pi$
$43$	−12.1236			−1.84883			−0.924415	+	0.381388i	$0.875446\pi$
$44$		0			0
$45$	−1.56402	+	12.0469i	−0.233150	+	1.79584i
$46$		0			0
$47$			4.99739i			0.728944i	0.931214	+	0.364472i	$0.118751\pi$
$47$			4.99739i			0.728944i	−0.931214	+	0.364472i	$0.881249\pi$
$48$		0			0
$49$	1.94378			0.277683
$50$		0			0
$51$	−10.6149	−	0.686173i	−1.48638	−	0.0960834i
$52$		0			0
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$		0			0
$63$	8.89718	+	1.15510i	1.12094	+	0.145529i
$64$		0			0
$65$		−	14.6000i		−	1.81091i
$66$		0			0
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			3.10125i			0.368051i	0.982921	+	0.184025i	$0.0589129\pi$
$71$			3.10125i			0.368051i	−0.982921	+	0.184025i	$0.941087\pi$
$72$		0			0
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$	1.27340	−	19.6990i	0.147039	−	2.27465i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$		0			0
$81$	8.70163	+	2.29816i	0.966848	+	0.255351i
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$	24.8680			2.69732
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$	−10.7828			−1.13035
$92$		0			0
$93$	0.805704	−	12.4640i	0.0835476	−	1.29245i
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$	−20.9314	−	1.35306i	−2.04269	−	0.132045i
$106$		0			0
$107$			10.1980i			0.985882i	0.870063	+	0.492941i	$0.164078\pi$
$107$			10.1980i			0.985882i	−0.870063	+	0.492941i	$0.835922\pi$
$108$		0			0
$109$	−0.286752			−0.0274659			−0.0137329	−	0.999906i	$-0.504371\pi$
$109$	−0.286752			−0.0274659			−0.0137329	+	0.999906i	$0.504371\pi$
$110$		0			0
$111$	1.30454	−	20.1808i	0.123821	−	1.91548i
$112$		0			0
$113$			20.3961i			1.91870i	0.282216	+	0.959351i	$0.408930\pi$
$113$			20.3961i			1.91870i	−0.282216	+	0.959351i	$0.591070\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−10.7266	−	1.39261i	−0.991677	−	0.128747i
$118$		0			0
$119$		−	18.3662i		−	1.68363i
$120$		0			0
$121$	−11.0000			−1.00000
$122$		0			0
$123$		0			0
$124$		0			0
$125$			25.9035i			2.31688i
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$		0			0
$129$	1.35458	−	20.9549i	0.119264	−	1.84498i
$130$		0			0
$131$		−	18.8294i		−	1.64513i	−0.568669	−	0.822566i	$-0.692542\pi$
$131$		−	18.8294i		−	1.64513i	0.568669	−	0.822566i	$-0.307458\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−20.6476	−	4.04932i	−1.77706	−	0.348510i
$136$		0			0
$137$		0			0			−	1.00000i	$-0.5\pi$
$137$		0			0				1.00000i	$0.5\pi$
$138$		0			0
$139$	23.5768			1.99976			0.999879	−	0.0155623i	$-0.00495384\pi$
$139$	23.5768			1.99976			0.999879	+	0.0155623i	$0.00495384\pi$
$140$		0			0
$141$	−8.63770	−	0.558363i	−0.727426	−	0.0470227i
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−0.217181	+	3.35972i	−0.0179128	+	0.277105i
$148$		0			0
$149$		−	11.3137i		−	0.926855i	−0.886135	−	0.463428i	$-0.846619\pi$
$149$		−	11.3137i		−	0.926855i	0.886135	−	0.463428i	$-0.153381\pi$
$150$		0			0
$151$	−14.3796			−1.17019			−0.585097	−	0.810964i	$-0.698943\pi$
$151$	−14.3796			−1.17019			−0.585097	+	0.810964i	$0.698943\pi$
$152$		0			0
$153$	2.37202	−	18.2705i	0.191767	−	1.47709i
$154$		0			0
$155$			29.2001i			2.34541i
$156$		0			0
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	14.1421i		−	1.09435i	−0.837018	−	0.547176i	$-0.815703\pi$
$167$		−	14.1421i		−	1.09435i	0.837018	−	0.547176i	$-0.184297\pi$
$168$		0			0
$169$	13.0000			1.00000
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$	34.0840			2.57651
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	14.1945i		−	1.06095i	−0.847702	−	0.530473i	$-0.822014\pi$
$179$		−	14.1945i		−	1.06095i	0.847702	−	0.530473i	$-0.177986\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			47.2787i			3.47600i
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−2.99062	+	15.2492i	−0.217535	+	1.10922i
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$		0			0
$193$		0			0		1.00000			$0$
$193$		0			0		−1.00000			$\pi$
$194$		0			0
$195$	25.2353	+	1.63128i	1.80714	+	0.116818i
$196$		0			0
$197$		−	5.94546i		−	0.423596i	−0.977313	−	0.211798i	$-0.932068\pi$
$197$		−	5.94546i		−	0.423596i	0.977313	−	0.211798i	$-0.0679319\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−13.4644			−0.926925			−0.463463	−	0.886117i	$-0.653393\pi$
$211$	−13.4644			−0.926925			−0.463463	+	0.886117i	$0.653393\pi$
$212$		0			0
$213$	−5.36034	−	0.346506i	−0.367284	−	0.0237422i
$214$		0			0
$215$			49.0923i			3.34807i
$216$		0			0
$217$	21.5656			1.46397
$218$		0			0
$219$		0			0
$220$		0			0
$221$			22.1427i			1.48948i
$222$		0			0
$223$	8.39834			0.562395			0.281197	−	0.959650i	$-0.409268\pi$
$223$	8.39834			0.562395			0.281197	+	0.959650i	$0.409268\pi$
$224$		0			0
$225$	33.9064	+	4.40199i	2.26043	+	0.293466i
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$		0			0
$229$	6.26798			0.414200			0.207100	−	0.978320i	$-0.433597\pi$
$229$	6.26798			0.414200			0.207100	+	0.978320i	$0.433597\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	9.96513i		−	0.652837i	−0.945225	−	0.326419i	$-0.894158\pi$
$233$		−	9.96513i		−	0.652837i	0.945225	−	0.326419i	$-0.105842\pi$
$234$		0			0
$235$	20.2360			1.32005
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	19.2985i		−	1.24832i	−0.781297	−	0.624159i	$-0.785442\pi$
$239$		−	19.2985i		−	1.24832i	0.781297	−	0.624159i	$-0.214558\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		0			0
$243$	−4.94449	+	14.7835i	−0.317189	+	0.948362i
$244$		0			0
$245$		−	7.87100i		−	0.502860i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$			10.1980i			0.643695i	0.946792	+	0.321847i	$0.104304\pi$
$251$			10.1980i			0.643695i	−0.946792	+	0.321847i	$0.895696\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	−2.77853	+	42.9830i	−0.173998	+	2.69170i
$256$		0			0
$257$			10.7762i			0.672200i	0.941826	+	0.336100i	$0.109108\pi$
$257$			10.7762i			0.672200i	−0.941826	+	0.336100i	$0.890892\pi$
$258$		0			0
$259$	34.9176			2.16967
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$	−32.3233			−1.96350			−0.981749	−	0.190179i	$-0.939093\pi$
$271$	−32.3233			−1.96350			−0.981749	+	0.190179i	$0.939093\pi$
$272$		0			0
$273$	1.20478	−	18.6375i	0.0729163	−	1.12799i
$274$		0			0
$275$		0			0
$276$		0			0
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$		0			0
$279$	21.4533	+	2.78523i	1.28437	+	0.166747i
$280$		0			0
$281$		0			0			−	1.00000i	$-0.5\pi$
$281$		0			0				1.00000i	$0.5\pi$
$282$		0			0
$283$	−14.0000			−0.832214			−0.416107	−	0.909316i	$-0.636606\pi$
$283$	−14.0000			−0.832214			−0.416107	+	0.909316i	$0.636606\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−20.7154			−1.21856
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	30.2414i		−	1.76672i	−0.468695	−	0.883360i	$-0.655276\pi$
$293$		−	30.2414i		−	1.76672i	0.468695	−	0.883360i	$-0.344724\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	36.2570			2.08982
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$	30.6442			1.73211			0.866055	−	0.499948i	$-0.166648\pi$
$313$	30.6442			1.73211			0.866055	+	0.499948i	$0.166648\pi$
$314$		0			0
$315$	4.67737	−	36.0275i	0.263540	−	2.02992i
$316$		0			0
$317$		−	28.2843i		−	1.58860i	−0.607524	−	0.794301i	$-0.707837\pi$
$317$		−	28.2843i		−	1.58860i	0.607524	−	0.794301i	$-0.292163\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−17.6267	−	1.13944i	−0.983828	−	0.0635972i
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−41.0924			−2.27940
$326$		0			0
$327$	0.0320391	−	0.495635i	0.00177177	−	0.0274087i
$328$		0			0
$329$		−	14.9453i		−	0.823959i
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$		0			0
$333$	34.7356	+	4.50964i	1.90350	+	0.247127i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	27.9626			1.52322			0.761611	−	0.648035i	$-0.224409\pi$
$337$	27.9626			1.52322			0.761611	+	0.648035i	$0.224409\pi$
$338$		0			0
$339$	−35.2535	−	2.27887i	−1.91471	−	0.123771i
$340$		0			0
$341$		0			0
$342$		0			0
$343$	15.1212			0.816468
$344$		0			0
$345$		0			0
$346$		0			0
$347$			27.2881i			1.46490i	0.680819	+	0.732452i	$0.261624\pi$
$347$			27.2881i			1.46490i	−0.680819	+	0.732452i	$0.738376\pi$
$348$		0			0
$349$	−29.6194			−1.58549			−0.792745	−	0.609553i	$-0.791349\pi$
$349$	−29.6194			−1.58549			−0.792745	+	0.609553i	$0.791349\pi$
$350$		0			0
$351$	3.60555	−	18.3848i	0.192450	−	0.981307i
$352$		0			0
$353$		0			0			−	1.00000i	$-0.5\pi$
$353$		0			0				1.00000i	$0.5\pi$
$354$		0			0
$355$	12.5580			0.666507
$356$		0			0
$357$	31.7450	+	2.05208i	1.68012	+	0.108608i
$358$		0			0
$359$		−	31.1127i		−	1.64207i	−0.570881	−	0.821033i	$-0.693398\pi$
$359$		−	31.1127i		−	1.64207i	0.570881	−	0.821033i	$-0.306602\pi$
$360$		0			0
$361$	19.0000			1.00000
$362$		0			0
$363$	1.22904	−	19.0129i	0.0645079	−	0.997917i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$	−44.7726	−	2.89422i	−2.31205	−	0.149457i
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			27.3972i			1.39993i	0.714177	+	0.699965i	$0.246801\pi$
$383$			27.3972i			1.39993i	−0.714177	+	0.699965i	$0.753199\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	36.0681	+	4.68263i	1.83344	+	0.238032i
$388$		0			0
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	32.5455	+	2.10383i	1.64171	+	0.106124i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−36.0555			−1.80957			−0.904787	−	0.425864i	$-0.859970\pi$
$397$	−36.0555			−1.80957			−0.904787	+	0.425864i	$0.859970\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$	−26.0000			−1.29515
$404$		0			0
$405$	9.30599	−	35.2357i	0.462418	−	1.75088i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−2.63426	+	40.7512i	−0.129000	+	1.99559i
$418$		0			0
$419$		−	34.9358i		−	1.70673i	−0.521317	−	0.853363i	$-0.674559\pi$
$419$		−	34.9358i		−	1.70673i	0.521317	−	0.853363i	$-0.325441\pi$
$420$		0			0
$421$	−41.0084			−1.99863			−0.999313	−	0.0370678i	$-0.988198\pi$
$421$	−41.0084			−1.99863			−0.999313	+	0.0370678i	$0.988198\pi$
$422$		0			0
$423$	1.93020	−	14.8674i	0.0938495	−	0.722877i
$424$		0			0
$425$		−	69.9922i		−	3.39512i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	23.0908i		−	1.11224i	−0.831100	−	0.556122i	$-0.812289\pi$
$431$		−	23.0908i		−	1.11224i	0.831100	−	0.556122i	$-0.187711\pi$
$432$		0			0
$433$	−40.7566			−1.95864			−0.979319	−	0.202324i	$-0.935151\pi$
$433$	−40.7566			−1.95864			−0.979319	+	0.202324i	$0.935151\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$	−5.78282	−	0.750770i	−0.275372	−	0.0357509i
$442$		0			0
$443$		−	39.5707i		−	1.88006i	−0.341090	−	0.940031i	$-0.610796\pi$
$443$		−	39.5707i		−	1.88006i	0.341090	−	0.940031i	$-0.389204\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	19.5551	+	1.26409i	0.924925	+	0.0597895i
$448$		0			0
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	1.60664	−	24.8543i	0.0754868	−	1.16776i
$454$		0			0
$455$			43.6631i			2.04696i
$456$		0			0
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$		0			0
$459$	31.3146	+	6.14129i	1.46164	+	0.286651i
$460$		0			0
$461$			18.3505i			0.854666i	0.904094	+	0.427333i	$0.140547\pi$
$461$			18.3505i			0.854666i	−0.904094	+	0.427333i	$0.859453\pi$
$462$		0			0
$463$	−7.21110			−0.335128			−0.167564	−	0.985861i	$-0.553590\pi$
$463$	−7.21110			−0.335128			−0.167564	+	0.985861i	$0.553590\pi$
$464$		0			0
$465$	−50.4706	−	3.26255i	−2.34052	−	0.151297i
$466$		0			0
$467$			10.1980i			0.471909i	0.971764	+	0.235954i	$0.0758216\pi$
$467$			10.1980i			0.471909i	−0.971764	+	0.235954i	$0.924178\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$			31.1894i			1.42508i	0.701630	+	0.712541i	$0.252456\pi$
$479$			31.1894i			1.42508i	−0.701630	+	0.712541i	$0.747544\pi$
$480$		0			0
$481$	−42.0974			−1.91948
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	36.0555			1.63383			0.816916	−	0.576757i	$-0.195682\pi$
$487$	36.0555			1.63383			0.816916	+	0.576757i	$0.195682\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			43.3946i			1.95837i	0.202972	+	0.979185i	$0.434940\pi$
$491$			43.3946i			1.95837i	−0.202972	+	0.979185i	$0.565060\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	9.27465i		−	0.416025i
$498$		0			0
$499$		0			0		1.00000			$0$
$499$		0			0		−1.00000			$\pi$
$500$		0			0
$501$	24.4439	+	1.58012i	1.09207	+	0.0705944i
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−1.45250	+	22.4698i	−0.0645079	+	0.997917i
$508$		0			0
$509$			39.5980i			1.75515i	0.479440	+	0.877575i	$0.340840\pi$
$509$			39.5980i			1.75515i	−0.479440	+	0.877575i	$0.659160\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$			22.2477i			0.974690i	0.873210	+	0.487345i	$0.162035\pi$
$521$			22.2477i			0.974690i	−0.873210	+	0.487345i	$0.837965\pi$
$522$		0			0
$523$	34.0000			1.48672			0.743358	−	0.668894i	$-0.233232\pi$
$523$	34.0000			1.48672			0.743358	+	0.668894i	$0.233232\pi$
$524$		0			0
$525$	−3.80824	+	58.9123i	−0.166205	+	2.57114i
$526$		0			0
$527$		−	44.2855i		−	1.92911i
$528$		0			0
$529$	−23.0000			−1.00000
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	41.2951			1.78534
$536$		0			0
$537$	24.5344	+	1.58597i	1.05874	+	0.0684395i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	11.1022			0.477321			0.238661	−	0.971103i	$-0.423292\pi$
$541$	11.1022			0.477321			0.238661	+	0.971103i	$0.423292\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			1.16115i			0.0497382i
$546$		0			0
$547$	20.8953			0.893416			0.446708	−	0.894680i	$-0.352596\pi$
$547$	20.8953			0.893416			0.446708	+	0.894680i	$0.352596\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−81.7185	−	5.28250i	−3.46876	−	0.224229i
$556$		0			0
$557$			46.4387i			1.96767i	0.179080	+	0.983834i	$0.442688\pi$
$557$			46.4387i			1.96767i	−0.179080	+	0.983834i	$0.557312\pi$
$558$		0			0
$559$	−43.7122			−1.84883
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	2.72298i		−	0.114760i	−0.998352	−	0.0573799i	$-0.981725\pi$
$563$		−	2.72298i		−	0.114760i	0.998352	−	0.0573799i	$-0.0182746\pi$
$564$		0			0
$565$	82.5902			3.47460
$566$		0			0
$567$	−26.0232	−	6.87292i	−1.09287	−	0.288636i
$568$		0			0
$569$			47.6239i			1.99650i	0.0591437	+	0.998249i	$0.481163\pi$
$569$			47.6239i			1.99650i	−0.0591437	+	0.998249i	$0.518837\pi$
$570$		0			0
$571$	−46.4832			−1.94526			−0.972631	−	0.232356i	$-0.925357\pi$
$571$	−46.4832			−1.94526			−0.972631	+	0.232356i	$0.925357\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$	−5.63914	+	43.4356i	−0.233150	+	1.79584i
$586$		0			0
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$	10.2764	+	0.664292i	0.422714	+	0.0273253i
$592$		0			0
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$	−74.3708			−3.04890
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	−39.4158			−1.60781			−0.803903	−	0.594761i	$-0.797247\pi$
$601$	−39.4158			−1.60781			−0.803903	+	0.594761i	$0.797247\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$			44.5425i			1.81091i
$606$		0			0
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			18.0183i			0.728944i
$612$		0			0
$613$	−36.0555			−1.45627			−0.728134	−	0.685435i	$-0.759612\pi$
$613$	−36.0555			−1.45627			−0.728134	+	0.685435i	$0.759612\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0			−	1.00000i	$-0.5\pi$
$617$		0			0				1.00000i	$0.5\pi$
$618$		0			0
$619$		0			0		1.00000			$0$
$619$		0			0		−1.00000			$\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	47.9064			1.91626
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	71.7039i		−	2.85902i
$630$		0			0
$631$	44.2857			1.76299			0.881494	−	0.472196i	$-0.156539\pi$
$631$	44.2857			1.76299			0.881494	+	0.472196i	$0.156539\pi$
$632$		0			0
$633$	1.50439	−	23.2724i	0.0597940	−	0.924994i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	7.00841			0.277683
$638$		0			0
$639$	1.19783	−	9.22632i	0.0473855	−	0.364988i
$640$		0			0
$641$		−	40.7922i		−	1.61119i	−0.592464	−	0.805597i	$-0.701845\pi$
$641$		−	40.7922i		−	1.61119i	0.592464	−	0.805597i	$-0.298155\pi$
$642$		0			0
$643$		0			0		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$		0			0
$645$	−84.8532	−	5.48513i	−3.34109	−	0.215977i
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−2.40955	+	37.2750i	−0.0944378	+	1.46092i
$652$		0			0
$653$		0			0			−	1.00000i	$-0.5\pi$
$653$		0			0				1.00000i	$0.5\pi$
$654$		0			0
$655$	−76.2463			−2.97919
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	50.9902i		−	1.98630i	−0.116863	−	0.993148i	$-0.537284\pi$
$659$		−	50.9902i		−	1.98630i	0.116863	−	0.993148i	$-0.462716\pi$
$660$		0			0
$661$	7.21110			0.280479			0.140240	−	0.990118i	$-0.455213\pi$
$661$	7.21110			0.280479			0.140240	+	0.990118i	$0.455213\pi$
$662$		0			0
$663$	−38.2725	−	2.47403i	−1.48638	−	0.0960834i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−0.938356	+	14.5161i	−0.0362789	+	0.561223i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	26.6219			1.02620			0.513099	−	0.858330i	$-0.328498\pi$
$673$	26.6219			1.02620			0.513099	+	0.858330i	$0.328498\pi$
$674$		0			0
$675$	−11.3970	+	58.1135i	−0.438670	+	2.23679i
$676$		0			0
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−0.700328	+	10.8339i	−0.0267192	+	0.413337i
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	95.4700i		−	3.62138i
$696$		0			0
$697$		0			0
$698$		0			0
$699$	17.2242	+	1.11341i	0.651478	+	0.0421132i
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	−2.26099	+	34.9768i	−0.0851539	+	1.31730i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	50.4777			1.89573			0.947865	−	0.318671i	$-0.103237\pi$
$709$	50.4777			1.89573			0.947865	+	0.318671i	$0.103237\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$	33.3564	+	2.15624i	1.24572	+	0.0805264i
$718$		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
$724$		0			0
$725$		0			0
$726$		0			0
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	−25.0000	−	10.1980i	−0.925926	−	0.377705i
$730$		0			0
$731$		−	74.4545i		−	2.75380i
$732$		0			0
$733$	−5.12098			−0.189148			−0.0945738	−	0.995518i	$-0.530149\pi$
$733$	−5.12098			−0.189148			−0.0945738	+	0.995518i	$0.530149\pi$
$734$		0			0
$735$	13.6046	+	0.879435i	0.501812	+	0.0324385i
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			53.5892i			1.96600i	0.183611	+	0.982999i	$0.441221\pi$
$743$			53.5892i			1.96600i	−0.183611	+	0.982999i	$0.558779\pi$
$744$		0			0
$745$	−45.8128			−1.67845
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	30.4984i		−	1.11439i
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$		0			0
$753$	−17.6267	−	1.13944i	−0.642354	−	0.0415234i
$754$		0			0
$755$			58.2275i			2.11912i
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0			−	1.00000i	$-0.5\pi$
$761$		0			0				1.00000i	$0.5\pi$
$762$		0			0
$763$	0.857565			0.0310459
$764$		0			0
$765$	−73.9832	−	9.60507i	−2.67487	−	0.347272i
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$	−18.6260	−	1.20403i	−0.670800	−	0.0433622i
$772$		0			0
$773$		−	32.1375i		−	1.15591i	−0.816070	−	0.577953i	$-0.803852\pi$
$773$		−	32.1375i		−	1.15591i	0.816070	−	0.577953i	$-0.196148\pi$
$774$		0			0
$775$	82.1849			2.95217
$776$		0			0
$777$	−3.90138	+	60.3530i	−0.139961	+	2.16515i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$		0			0		1.00000			$0$
$787$		0			0		−1.00000			$\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	60.9968i		−	2.16880i
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$	−30.6904			−1.08575
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	56.0827i		−	1.97176i	−0.167449	−	0.985881i	$-0.553553\pi$
$809$		−	56.0827i		−	1.97176i	0.167449	−	0.985881i	$-0.446447\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$	3.61151	−	55.8689i	0.126661	−	1.95941i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$	32.0792	+	4.16477i	1.12094	+	0.145529i
$820$		0			0
$821$		−	7.84160i		−	0.273674i	−0.990594	−	0.136837i	$-0.956306\pi$
$821$		−	7.84160i		−	0.273674i	0.990594	−	0.136837i	$-0.0436936\pi$
$822$		0			0
$823$		0			0		1.00000			$0$
$823$		0			0		−1.00000			$\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			11.9373i			0.413604i
$834$		0			0
$835$	−57.2660			−1.98177
$836$		0			0
$837$	−7.21110	+	36.7696i	−0.249252	+	1.27094i
$838$		0			0
$839$			53.7401i			1.85531i	0.373432	+	0.927657i	$0.378181\pi$
$839$			53.7401i			1.85531i	−0.373432	+	0.927657i	$0.621819\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	52.6412i		−	1.81091i
$846$		0			0
$847$	32.8968			1.13035
$848$		0			0
$849$	1.56423	−	24.1982i	0.0536844	−	0.830480i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−52.3973			−1.79405			−0.897025	−	0.441980i	$-0.854276\pi$
$853$	−52.3973			−1.79405			−0.897025	+	0.441980i	$0.854276\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	40.7922i		−	1.39343i	−0.717346	−	0.696717i	$-0.754643\pi$
$857$		−	40.7922i		−	1.39343i	0.717346	−	0.696717i	$-0.245357\pi$
$858$		0			0
$859$	−50.0000			−1.70598			−0.852989	−	0.521929i	$-0.825213\pi$
$859$	−50.0000			−1.70598			−0.852989	+	0.521929i	$0.825213\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			55.4854i			1.88874i	0.328880	+	0.944372i	$0.393329\pi$
$863$			55.4854i			1.88874i	−0.328880	+	0.944372i	$0.606671\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	2.31456	−	35.8055i	0.0786065	−	1.21602i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		−	77.4673i		−	2.61887i
$876$		0			0
$877$	40.4349			1.36539			0.682694	−	0.730704i	$-0.260808\pi$
$877$	40.4349			1.36539			0.682694	+	0.730704i	$0.260808\pi$
$878$		0			0
$879$	52.2705	+	3.37890i	1.76304	+	0.113967i
$880$		0			0
$881$		−	39.9762i		−	1.34683i	−0.739263	−	0.673417i	$-0.764826\pi$
$881$		−	39.9762i		−	1.34683i	0.739263	−	0.673417i	$-0.235174\pi$
$882$		0			0
$883$	59.2772			1.99484			0.997418	−	0.0718139i	$-0.0228788\pi$
$883$	59.2772			1.99484			0.997418	+	0.0718139i	$0.0228788\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$	−57.4781			−1.92128
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$	−4.05103	+	62.6682i	−0.134810	+	2.08547i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	57.9364			1.92375			0.961874	−	0.273495i	$-0.0881796\pi$
$907$	57.9364			1.92375			0.961874	+	0.273495i	$0.0881796\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$			56.3115i			1.85957i
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			11.1817i			0.368051i
$924$		0			0
$925$	133.068			4.37525
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0			−	1.00000i	$-0.5\pi$
$929$		0			0				1.00000i	$0.5\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	2.00000			0.0653372			0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000			0.0653372			0.0326686	+	0.999466i	$0.489599\pi$
$938$		0			0
$939$	−3.42391	+	52.9667i	−0.111735	+	1.72850i
$940$		0			0
$941$			48.3348i			1.57567i	0.615887	+	0.787835i	$0.288798\pi$
$941$			48.3348i			1.57567i	−0.615887	+	0.787835i	$0.711202\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$	61.7489	+	12.1100i	2.00869	+	0.393937i
$946$		0			0
$947$		0			0			−	1.00000i	$-0.5\pi$
$947$		0			0				1.00000i	$0.5\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	48.8878	+	3.16023i	1.58529	+	0.102478i
$952$		0			0
$953$			52.2588i			1.69283i	0.532524	+	0.846415i	$0.321244\pi$
$953$			52.2588i			1.69283i	−0.532524	+	0.846415i	$0.678756\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	21.0000			0.677419
$962$		0			0
$963$	3.93891	−	30.3395i	0.126929	−	0.977677i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−61.6559			−1.98272			−0.991360	−	0.131166i	$-0.958128\pi$
$967$	−61.6559			−1.98272			−0.991360	+	0.131166i	$0.958128\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	60.3120i		−	1.93551i	−0.251902	−	0.967753i	$-0.581056\pi$
$971$		−	60.3120i		−	1.93551i	0.251902	−	0.967753i	$-0.418944\pi$
$972$		0			0
$973$	−70.5091			−2.26042
$974$		0			0
$975$	4.59130	−	71.0259i	0.147039	−	2.27465i
$976$		0			0
$977$		0			0			−	1.00000i	$-0.5\pi$
$977$		0			0				1.00000i	$0.5\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	0.853097	+	0.110756i	0.0272373	+	0.00353615i
$982$		0			0
$983$			25.5010i			0.813357i	0.913571	+	0.406678i	$0.133313\pi$
$983$			25.5010i			0.813357i	−0.913571	+	0.406678i	$0.866687\pi$
$984$		0			0
$985$	−24.0751			−0.767095
$986$		0			0
$987$	25.8321	+	1.66985i	0.822243	+	0.0531519i
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		1.00000			$0$
$997$		0			0		−1.00000			$\pi$
$998$		0			0
$999$	−11.6757	+	59.5347i	−0.369403	+	1.88359i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1248.2.h.b.623.7		12
3.2	odd	2	inner	1248.2.h.b.623.6		12
4.3	odd	2		312.2.h.b.155.3	✓	12
8.3	odd	2	inner	1248.2.h.b.623.8		12
8.5	even	2		312.2.h.b.155.9	yes	12
12.11	even	2		312.2.h.b.155.10	yes	12
13.12	even	2	inner	1248.2.h.b.623.8		12
24.5	odd	2		312.2.h.b.155.4	yes	12
24.11	even	2	inner	1248.2.h.b.623.5		12
39.38	odd	2	inner	1248.2.h.b.623.5		12
52.51	odd	2		312.2.h.b.155.9	yes	12
104.51	odd	2	CM	1248.2.h.b.623.7		12
104.77	even	2		312.2.h.b.155.3	✓	12
156.155	even	2		312.2.h.b.155.4	yes	12
312.77	odd	2		312.2.h.b.155.10	yes	12
312.155	even	2	inner	1248.2.h.b.623.6		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.2.h.b.155.3	✓	12	4.3	odd	2
312.2.h.b.155.3	✓	12	104.77	even	2
312.2.h.b.155.4	yes	12	24.5	odd	2
312.2.h.b.155.4	yes	12	156.155	even	2
312.2.h.b.155.9	yes	12	8.5	even	2
312.2.h.b.155.9	yes	12	52.51	odd	2
312.2.h.b.155.10	yes	12	12.11	even	2
312.2.h.b.155.10	yes	12	312.77	odd	2
1248.2.h.b.623.5		12	24.11	even	2	inner
1248.2.h.b.623.5		12	39.38	odd	2	inner
1248.2.h.b.623.6		12	3.2	odd	2	inner
1248.2.h.b.623.6		12	312.155	even	2	inner
1248.2.h.b.623.7		12	1.1	even	1	trivial
1248.2.h.b.623.7		12	104.51	odd	2	CM
1248.2.h.b.623.8		12	8.3	odd	2	inner
1248.2.h.b.623.8		12	13.12	even	2	inner

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:	\( N \)	\(=\)	\( 1248 = 2^{5} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1248.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.111731 + 1.72844i) q^{3} -4.04932i q^{5} -2.99062 q^{7} +(-2.97503 - 0.386242i) q^{9} +O(q^{10})\)	\(q+(-0.111731 + 1.72844i) q^{3} -4.04932i q^{5} -2.99062 q^{7} +(-2.97503 - 0.386242i) q^{9} +3.60555 q^{13} +(6.99902 + 0.452435i) q^{15} +6.14129i q^{17} +(0.334145 - 5.16911i) q^{21} -11.3970 q^{25} +(1.00000 - 5.09902i) q^{27} -7.21110 q^{31} +12.1100i q^{35} -11.6757 q^{37} +(-0.402852 + 6.23199i) q^{39} -12.1236 q^{43} +(-1.56402 + 12.0469i) q^{45} +4.99739i q^{47} +1.94378 q^{49} +(-10.6149 - 0.686173i) q^{51} +(8.89718 + 1.15510i) q^{63} -14.6000i q^{65} +3.10125i q^{71} +(1.27340 - 19.6990i) q^{75} +(8.70163 + 2.29816i) q^{81} +24.8680 q^{85} -10.7828 q^{91} +(0.805704 - 12.4640i) q^{93} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q+O(q^{10}) \) 12 * q	\( 12 q - 60 q^{25} + 12 q^{27} + 84 q^{49} - 60 q^{51} + 84 q^{75}+O(q^{100}) \) 12 * q - 60 * q^25 + 12 * q^27 + 84 * q^49 - 60 * q^51 + 84 * q^75

Introduction

L-functions

Modular forms

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Representations

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Database

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