LMFDB - Embedded newform 95.7.d.a.94.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [95,7,Mod(94,95)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(95, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("95.94");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 95 = 5 \cdot 19 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	95.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:	no
Analytic conductor:	$21.8551379439$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-19}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 5 $ x^2 - x + 5
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			94.1
Root			$0.500000 - 2.17945i$ of defining polynomial
Character	$\chi$	$=$	95.94
Dual form			95.7.d.a.94.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-64.0000 q^{4} +(-27.0000 - 122.049i) q^{5} -313.841i q^{7} -729.000 q^{9} +O(q^{10})$	$q-64.0000 q^{4} +(-27.0000 - 122.049i) q^{5} -313.841i q^{7} -729.000 q^{9} +1062.00 q^{11} +4096.00 q^{16} +1952.79i q^{17} -6859.00 q^{19} +(1728.00 + 7811.15i) q^{20} +12937.2i q^{23} +(-14167.0 + 6590.66i) q^{25} +20085.8i q^{28} +(-38304.0 + 8473.70i) q^{35} +46656.0 q^{36} +70300.3i q^{43} -67968.0 q^{44} +(19683.0 + 88973.8i) q^{45} +193570. i q^{47} +19153.0 q^{49} +(-28674.0 - 129616. i) q^{55} +57062.0 q^{61} +228790. i q^{63} -262144. q^{64} -124978. i q^{68} -676641. i q^{73} +438976. q^{76} -333299. i q^{77} +(-110592. - 499913. i) q^{80} +531441. q^{81} -168916. i q^{83} +(238336. - 52725.2i) q^{85} -827982. i q^{92} +(185193. + 837135. i) q^{95} -774198. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 128 q^{4} - 54 q^{5} - 1458 q^{9}+O(q^{10}) $ 2 * q - 128 * q^4 - 54 * q^5 - 1458 * q^9	$ 2 q - 128 q^{4} - 54 q^{5} - 1458 q^{9} + 2124 q^{11} + 8192 q^{16} - 13718 q^{19} + 3456 q^{20} - 28334 q^{25} - 76608 q^{35} + 93312 q^{36} - 135936 q^{44} + 39366 q^{45} + 38306 q^{49} - 57348 q^{55} + 114124 q^{61} - 524288 q^{64} + 877952 q^{76} - 221184 q^{80} + 1062882 q^{81} + 476672 q^{85} + 370386 q^{95} - 1548396 q^{99}+O(q^{100}) $ 2 * q - 128 * q^4 - 54 * q^5 - 1458 * q^9 + 2124 * q^11 + 8192 * q^16 - 13718 * q^19 + 3456 * q^20 - 28334 * q^25 - 76608 * q^35 + 93312 * q^36 - 135936 * q^44 + 39366 * q^45 + 38306 * q^49 - 57348 * q^55 + 114124 * q^61 - 524288 * q^64 + 877952 * q^76 - 221184 * q^80 + 1062882 * q^81 + 476672 * q^85 + 370386 * q^95 - 1548396 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/95\mathbb{Z}\right)^\times$.

$n$	$21$	$77$
$\chi(n)$	$-1$	$-1$

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 $

$2$		0			0			−	1.00000i	$-0.5\pi$
$2$		0			0				1.00000i	$0.5\pi$
$3$		0			0			−	1.00000i	$-0.5\pi$
$3$		0			0				1.00000i	$0.5\pi$
$4$	−64.0000			−1.00000
$5$	−27.0000	−	122.049i	−0.216000	−	0.976393i
$5$	−27.0000	−	122.049i	−0.216000	−	0.976393i
$6$		0			0
$7$		−	313.841i		−	0.914988i	−0.889213	−	0.457494i	$-0.848747\pi$
$7$		−	313.841i		−	0.914988i	0.889213	−	0.457494i	$-0.151253\pi$
$8$		0			0
$9$	−729.000			−1.00000
$10$		0			0
$11$	1062.00			0.797896			0.398948	−	0.916973i	$-0.369375\pi$
$11$	1062.00			0.797896			0.398948	+	0.916973i	$0.369375\pi$
$12$		0			0
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$		0			0
$15$		0			0
$16$	4096.00			1.00000
$17$			1952.79i			0.397473i	0.980053	+	0.198737i	$0.0636839\pi$
$17$			1952.79i			0.397473i	−0.980053	+	0.198737i	$0.936316\pi$
$18$		0			0
$19$	−6859.00			−1.00000
$19$	−6859.00			−1.00000
$20$	1728.00	+	7811.15i	0.216000	+	0.976393i
$21$		0			0
$22$		0			0
$23$			12937.2i			1.06330i	0.846963	+	0.531652i	$0.178428\pi$
$23$			12937.2i			1.06330i	−0.846963	+	0.531652i	$0.821572\pi$
$24$		0			0
$25$	−14167.0	+	6590.66i	−0.906688	+	0.421802i
$26$		0			0
$27$		0			0
$28$			20085.8i			0.914988i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−38304.0	+	8473.70i	−0.893388	+	0.197637i
$36$	46656.0			1.00000
$37$		0			0			−	1.00000i	$-0.5\pi$
$37$		0			0				1.00000i	$0.5\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$			70300.3i			0.884203i	0.896965	+	0.442101i	$0.145767\pi$
$43$			70300.3i			0.884203i	−0.896965	+	0.442101i	$0.854233\pi$
$44$	−67968.0			−0.797896
$45$	19683.0	+	88973.8i	0.216000	+	0.976393i
$46$		0			0
$47$			193570.i			1.86442i	0.361914	+	0.932211i	$0.382123\pi$
$47$			193570.i			1.86442i	−0.361914	+	0.932211i	$0.617877\pi$
$48$		0			0
$49$	19153.0			0.162798
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		0			0
$55$	−28674.0	−	129616.i	−0.172346	−	0.779061i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		1.00000			$0$
$59$		0			0		−1.00000			$\pi$
$60$		0			0
$61$	57062.0			0.251395			0.125698	−	0.992069i	$-0.459883\pi$
$61$	57062.0			0.251395			0.125698	+	0.992069i	$0.459883\pi$
$62$		0			0
$63$			228790.i			0.914988i
$64$	−262144.			−1.00000
$65$		0			0
$66$		0			0
$67$		0			0			−	1.00000i	$-0.5\pi$
$67$		0			0				1.00000i	$0.5\pi$
$68$		−	124978.i		−	0.397473i
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$		−	676641.i		−	1.73936i	−0.493616	−	0.869680i	$-0.664325\pi$
$73$		−	676641.i		−	1.73936i	0.493616	−	0.869680i	$-0.335675\pi$
$74$		0			0
$75$		0			0
$76$	438976.			1.00000
$77$		−	333299.i		−	0.730065i
$78$		0			0
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$	−110592.	−	499913.i	−0.216000	−	0.976393i
$81$	531441.			1.00000
$82$		0			0
$83$		−	168916.i		−	0.295418i	−0.989031	−	0.147709i	$-0.952810\pi$
$83$		−	168916.i		−	0.295418i	0.989031	−	0.147709i	$-0.0471899\pi$
$84$		0			0
$85$	238336.	−	52725.2i	0.388090	−	0.0858543i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$		0			0
$92$		−	827982.i		−	1.06330i
$93$		0			0
$94$		0			0
$95$	185193.	+	837135.i	0.216000	+	0.976393i
$96$		0			0
$97$		0			0			−	1.00000i	$-0.5\pi$
$97$		0			0				1.00000i	$0.5\pi$
$98$		0			0
$99$	−774198.			−0.797896
$100$	906688.	−	421802.i	0.906688	−	0.421802i
$101$	−2.06030e6			−1.99970			−0.999852	−	0.0171767i	$-0.994532\pi$
$101$	−2.06030e6			−1.99970			−0.999852	+	0.0171767i	$0.994532\pi$
$102$		0			0
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$		0			0
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$		−	1.28549e6i		−	0.914988i
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$	1.57898e6	−	349305.i	1.03820	−	0.229674i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	612864.			0.363683
$120$		0			0
$121$	−643717.			−0.363361
$122$		0			0
$123$		0			0
$124$		0			0
$125$	1.18689e6	+	1.55112e6i	0.607689	+	0.794175i
$126$		0			0
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	1.30228e6			0.579283			0.289642	−	0.957135i	$-0.406464\pi$
$131$	1.30228e6			0.579283			0.289642	+	0.957135i	$0.406464\pi$
$132$		0			0
$133$			2.15263e6i			0.914988i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			4.63738e6i			1.80348i	0.432280	+	0.901739i	$0.357709\pi$
$137$			4.63738e6i			1.80348i	−0.432280	+	0.901739i	$0.642291\pi$
$138$		0			0
$139$	−3.77334e6			−1.40502			−0.702508	−	0.711676i	$-0.747937\pi$
$139$	−3.77334e6			−1.40502			−0.702508	+	0.711676i	$0.747937\pi$
$140$	2.45146e6	−	542317.i	0.893388	−	0.197637i
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−2.98598e6			−1.00000
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−6.24350e6			−1.88742			−0.943711	−	0.330770i	$-0.892692\pi$
$149$	−6.24350e6			−1.88742			−0.943711	+	0.330770i	$0.892692\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$		−	1.42358e6i		−	0.397473i
$154$		0			0
$155$		0			0
$156$		0			0
$157$			7.70448e6i			1.99088i	0.0954122	+	0.995438i	$0.469583\pi$
$157$			7.70448e6i			1.99088i	−0.0954122	+	0.995438i	$0.530417\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	4.06022e6			0.972909
$162$		0			0
$163$			7.51774e6i			1.73590i	0.496652	+	0.867950i	$0.334562\pi$
$163$			7.51774e6i			1.73590i	−0.496652	+	0.867950i	$0.665438\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	−4.82681e6			−1.00000
$170$		0			0
$171$	5.00021e6			1.00000
$172$		−	4.49922e6i		−	0.884203i
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$	2.06842e6	+	4.44618e6i	0.385944	+	0.829608i
$176$	4.34995e6			0.797896
$177$		0			0
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$	−1.25971e6	−	5.69433e6i	−0.216000	−	0.976393i
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$			2.07386e6i			0.317143i
$188$		−	1.23885e7i		−	1.86442i
$189$		0			0
$190$		0			0
$191$	9.37384e6			1.34529			0.672647	−	0.739963i	$-0.265157\pi$
$191$	9.37384e6			1.34529			0.672647	+	0.739963i	$0.265157\pi$
$192$		0			0
$193$		0			0			−	1.00000i	$-0.5\pi$
$193$		0			0				1.00000i	$0.5\pi$
$194$		0			0
$195$		0			0
$196$	−1.22579e6			−0.162798
$197$		−	1.17795e7i		−	1.54073i	−0.637603	−	0.770365i	$-0.720074\pi$
$197$		−	1.17795e7i		−	1.54073i	0.637603	−	0.770365i	$-0.279926\pi$
$198$		0			0
$199$	−1.52712e7			−1.93782			−0.968911	−	0.247410i	$-0.920421\pi$
$199$	−1.52712e7			−1.93782			−0.968911	+	0.247410i	$0.920421\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		−	9.43123e6i		−	1.06330i
$208$		0			0
$209$	−7.28426e6			−0.797896
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	8.58010e6	−	1.89811e6i	0.863330	−	0.190988i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$	1.83514e6	+	8.29544e6i	0.172346	+	0.779061i
$221$		0			0
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$		0			0
$225$	1.03277e7	−	4.80459e6i	0.906688	−	0.421802i
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$		0			0
$229$	−2.66958e6			−0.222298			−0.111149	−	0.993804i	$-0.535453\pi$
$229$	−2.66958e6			−0.222298			−0.111149	+	0.993804i	$0.535453\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	4.94104e6i		−	0.390616i	−0.980742	−	0.195308i	$-0.937429\pi$
$233$		−	4.94104e6i		−	0.390616i	0.980742	−	0.195308i	$-0.0625707\pi$
$234$		0			0
$235$	2.36251e7	−	5.22639e6i	1.82041	−	0.402715i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−1.53322e7			−1.12308			−0.561541	−	0.827449i	$-0.689791\pi$
$239$	−1.53322e7			−1.12308			−0.561541	+	0.827449i	$0.689791\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		0			0
$243$		0			0
$244$	−3.65197e6			−0.251395
$245$	−517131.	−	2.33761e6i	−0.0351643	−	0.158955i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−5.08340e6			−0.321464			−0.160732	−	0.986998i	$-0.551386\pi$
$251$	−5.08340e6			−0.321464			−0.160732	+	0.986998i	$0.551386\pi$
$252$		−	1.46426e7i		−	0.914988i
$253$			1.37393e7i			0.848406i
$254$		0			0
$255$		0			0
$256$	1.67772e7			1.00000
$257$		0			0			−	1.00000i	$-0.5\pi$
$257$		0			0				1.00000i	$0.5\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	3.18370e7i		−	1.75011i	−0.484024	−	0.875054i	$-0.660825\pi$
$263$		−	3.18370e7i		−	1.75011i	0.484024	−	0.875054i	$-0.339175\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$	−2.84226e7			−1.42809			−0.714045	−	0.700100i	$-0.753139\pi$
$271$	−2.84226e7			−1.42809			−0.714045	+	0.700100i	$0.753139\pi$
$272$			7.99861e6i			0.397473i
$273$		0			0
$274$		0			0
$275$	−1.50454e7	+	6.99928e6i	−0.723443	+	0.336554i
$276$		0			0
$277$			3.01347e7i			1.41784i	0.705289	+	0.708920i	$0.250817\pi$
$277$			3.01347e7i			1.41784i	−0.705289	+	0.708920i	$0.749183\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$			3.07828e7i			1.35815i	0.734068	+	0.679076i	$0.237619\pi$
$283$			3.07828e7i			1.35815i	−0.734068	+	0.679076i	$0.762381\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	2.03242e7			0.842015
$290$		0			0
$291$		0			0
$292$			4.33050e7i			1.73936i
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	2.20631e7			0.809035
$302$		0			0
$303$		0			0
$304$	−2.80945e7			−1.00000
$305$	−1.54067e6	−	6.96437e6i	−0.0543014	−	0.245461i
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$			2.13311e7i			0.730065i
$309$		0			0
$310$		0			0
$311$	−4.42879e7			−1.47233			−0.736164	−	0.676804i	$-0.763365\pi$
$311$	−4.42879e7			−1.47233			−0.736164	+	0.676804i	$0.763365\pi$
$312$		0			0
$313$			5.23342e7i			1.70668i	0.521353	+	0.853341i	$0.325427\pi$
$313$			5.23342e7i			1.70668i	−0.521353	+	0.853341i	$0.674573\pi$
$314$		0			0
$315$	2.79236e7	−	6.17733e6i	0.893388	−	0.197637i
$316$		0			0
$317$		0			0			−	1.00000i	$-0.5\pi$
$317$		0			0				1.00000i	$0.5\pi$
$318$		0			0
$319$		0			0
$320$	7.07789e6	+	3.19945e7i	0.216000	+	0.976393i
$321$		0			0
$322$		0			0
$323$		−	1.33942e7i		−	0.397473i
$324$	−3.40122e7			−1.00000
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	6.07501e7			1.70592
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$			1.08106e7i			0.295418i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$		0			0			−	1.00000i	$-0.5\pi$
$337$		0			0				1.00000i	$0.5\pi$
$338$		0			0
$339$		0			0
$340$	−1.52535e7	+	3.37442e6i	−0.388090	+	0.0858543i
$341$		0			0
$342$		0			0
$343$		−	4.29340e7i		−	1.06395i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	5.40634e7i		−	1.29394i	−0.762515	−	0.646971i	$-0.776035\pi$
$347$		−	5.40634e7i		−	1.29394i	0.762515	−	0.646971i	$-0.223965\pi$
$348$		0			0
$349$	−4.62042e7			−1.08694			−0.543469	−	0.839429i	$-0.682890\pi$
$349$	−4.62042e7			−1.08694			−0.543469	+	0.839429i	$0.682890\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	6.61463e7i		−	1.50377i	−0.659295	−	0.751885i	$-0.729145\pi$
$353$		−	6.61463e7i		−	1.50377i	0.659295	−	0.751885i	$-0.270855\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	7.96053e7			1.72052			0.860258	−	0.509858i	$-0.170302\pi$
$359$	7.96053e7			1.72052			0.860258	+	0.509858i	$0.170302\pi$
$360$		0			0
$361$	4.70459e7			1.00000
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−8.25834e7	+	1.82693e7i	−1.69830	+	0.375702i
$366$		0			0
$367$		−	9.68013e7i		−	1.95832i	−0.203095	−	0.979159i	$-0.565100\pi$
$367$		−	9.68013e7i		−	1.95832i	0.203095	−	0.979159i	$-0.434900\pi$
$368$			5.29908e7i			1.06330i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$	−1.18524e7	−	5.35767e7i	−0.216000	−	0.976393i
$381$		0			0
$382$		0			0
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$		0			0
$385$	−4.06788e7	+	8.99907e6i	−0.712831	+	0.157694i
$386$		0			0
$387$		−	5.12489e7i		−	0.884203i
$388$		0			0
$389$	6.58748e7			1.11910			0.559552	−	0.828795i	$-0.310973\pi$
$389$	6.58748e7			1.11910			0.559552	+	0.828795i	$0.310973\pi$
$390$		0			0
$391$	−2.52636e7			−0.422635
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$	4.95487e7			0.797896
$397$			1.09135e8i			1.74418i	0.489346	+	0.872090i	$0.337236\pi$
$397$			1.09135e8i			1.74418i	−0.489346	+	0.872090i	$0.662764\pi$
$398$		0			0
$399$		0			0
$400$	−5.80280e7	+	2.69953e7i	−0.906688	+	0.421802i
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$		0			0
$404$	1.31859e8			1.99970
$405$	−1.43489e7	−	6.48619e7i	−0.216000	−	0.976393i
$406$		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
$415$	−2.06161e7	+	4.56073e6i	−0.288444	+	0.0638102i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	4.11183e7			0.558976			0.279488	−	0.960149i	$-0.409835\pi$
$419$	4.11183e7			0.558976			0.279488	+	0.960149i	$0.409835\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$		−	1.41113e8i		−	1.86442i
$424$		0			0
$425$	−1.28701e7	−	2.76651e7i	−0.167655	−	0.360384i
$426$		0			0
$427$		−	1.79084e7i		−	0.230024i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		0			0
$433$		0			0			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	8.87363e7i		−	1.06330i
$438$		0			0
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$	−1.39625e7			−0.162798
$442$		0			0
$443$		−	1.71860e8i		−	1.97681i	−0.151846	−	0.988404i	$-0.548522\pi$
$443$		−	1.71860e8i		−	1.97681i	0.151846	−	0.988404i	$-0.451478\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$			8.22715e7i			0.914988i
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$			1.21229e8i			1.27015i	0.772449	+	0.635077i	$0.219032\pi$
$457$			1.21229e8i			1.27015i	−0.772449	+	0.635077i	$0.780968\pi$
$458$		0			0
$459$		0			0
$460$	−1.01054e8	+	2.23555e7i	−1.03820	+	0.229674i
$461$	−1.95676e8			−1.99726			−0.998631	−	0.0523154i	$-0.983340\pi$
$461$	−1.95676e8			−1.99726			−0.998631	+	0.0523154i	$0.983340\pi$
$462$		0			0
$463$			1.90683e8i			1.92118i	0.277961	+	0.960592i	$0.410341\pi$
$463$			1.90683e8i			1.92118i	−0.277961	+	0.960592i	$0.589659\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			1.16046e8i			1.13941i	0.821849	+	0.569706i	$0.192943\pi$
$467$			1.16046e8i			1.13941i	−0.821849	+	0.569706i	$0.807057\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$			7.46589e7i			0.705502i
$474$		0			0
$475$	9.71715e7	−	4.52053e7i	0.906688	−	0.421802i
$476$	−3.92233e7			−0.363683
$477$		0			0
$478$		0			0
$479$	1.87497e8			1.70603			0.853015	−	0.521886i	$-0.174771\pi$
$479$	1.87497e8			1.70603			0.853015	+	0.521886i	$0.174771\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	4.11979e7			0.363361
$485$		0			0
$486$		0			0
$487$		0			0			−	1.00000i	$-0.5\pi$
$487$		0			0				1.00000i	$0.5\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−1.09684e8			−0.926610			−0.463305	−	0.886199i	$-0.653337\pi$
$491$	−1.09684e8			−0.926610			−0.463305	+	0.886199i	$0.653337\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	2.09033e7	+	9.44902e7i	0.172346	+	0.779061i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	2.47627e8			1.99295			0.996475	−	0.0838961i	$-0.0267364\pi$
$499$	2.47627e8			1.99295			0.996475	+	0.0838961i	$0.0267364\pi$
$500$	−7.59612e7	−	9.92719e7i	−0.607689	−	0.794175i
$501$		0			0
$502$		0			0
$503$			2.34711e8i			1.84429i	0.386843	+	0.922146i	$0.373566\pi$
$503$			2.34711e8i			1.84429i	−0.386843	+	0.922146i	$0.626434\pi$
$504$		0			0
$505$	5.56280e7	+	2.51458e8i	0.431936	+	1.95250i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		1.00000			$0$
$509$		0			0		−1.00000			$\pi$
$510$		0			0
$511$	−2.12357e8			−1.59149
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$			2.05571e8i			1.48762i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$		0			0			−	1.00000i	$-0.5\pi$
$523$		0			0				1.00000i	$0.5\pi$
$524$	−8.33460e7			−0.579283
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−1.93356e7			−0.130614
$530$		0			0
$531$		0			0
$532$		−	1.37769e8i		−	0.914988i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	2.03405e7			0.129896
$540$		0			0
$541$	−3.05822e8			−1.93142			−0.965709	−	0.259626i	$-0.916401\pi$
$541$	−3.05822e8			−1.93142			−0.965709	+	0.259626i	$0.916401\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0			−	1.00000i	$-0.5\pi$
$547$		0			0				1.00000i	$0.5\pi$
$548$		−	2.96792e8i		−	1.80348i
$549$	−4.15982e7			−0.251395
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	2.41494e8			1.40502
$557$		−	1.82082e8i		−	1.05366i	−0.849970	−	0.526831i	$-0.823380\pi$
$557$		−	1.82082e8i		−	1.05366i	0.849970	−	0.526831i	$-0.176620\pi$
$558$		0			0
$559$		0			0
$560$	−1.56893e8	+	3.47083e7i	−0.893388	+	0.197637i
$561$		0			0
$562$		0			0
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		−	1.66788e8i		−	0.914988i
$568$		0			0
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$	3.51908e8			1.89026			0.945130	−	0.326696i	$-0.105935\pi$
$571$	3.51908e8			1.89026			0.945130	+	0.326696i	$0.105935\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−8.52647e7	−	1.83281e8i	−0.448503	−	0.964084i
$576$	1.91103e8			1.00000
$577$			1.40693e8i			0.732394i	0.930537	+	0.366197i	$0.119340\pi$
$577$			1.40693e8i			0.732394i	−0.930537	+	0.366197i	$0.880660\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−5.30127e7			−0.270304
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	3.09139e8i		−	1.52841i	−0.644975	−	0.764204i	$-0.723132\pi$
$587$		−	3.09139e8i		−	1.52841i	0.644975	−	0.764204i	$-0.276868\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$			4.15855e8i			1.99424i	0.0758206	+	0.997121i	$0.475842\pi$
$593$			4.15855e8i			1.99424i	−0.0758206	+	0.997121i	$0.524158\pi$
$594$		0			0
$595$	−1.65473e7	−	7.47995e7i	−0.0785556	−	0.355098i
$596$	3.99584e8			1.88742
$597$		0			0
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	1.73804e7	+	7.85651e7i	0.0784861	+	0.354784i
$606$		0			0
$607$		0			0			−	1.00000i	$-0.5\pi$
$607$		0			0				1.00000i	$0.5\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$			9.11092e7i			0.397473i
$613$		−	3.43599e8i		−	1.49166i	−0.666134	−	0.745832i	$-0.732052\pi$
$613$		−	3.43599e8i		−	1.49166i	0.666134	−	0.745832i	$-0.267948\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			4.69696e8i			1.99968i	0.0177783	+	0.999842i	$0.494341\pi$
$617$			4.69696e8i			1.99968i	−0.0177783	+	0.999842i	$0.505659\pi$
$618$		0			0
$619$	−4.70840e8			−1.98519			−0.992594	−	0.121480i	$-0.961236\pi$
$619$	−4.70840e8			−1.98519			−0.992594	+	0.121480i	$0.961236\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	1.57267e8	−	1.86740e8i	0.644166	−	0.764886i
$626$		0			0
$627$		0			0
$628$		−	4.93086e8i		−	1.99088i
$629$		0			0
$630$		0			0
$631$	−1.23521e8			−0.491647			−0.245824	−	0.969315i	$-0.579058\pi$
$631$	−1.23521e8			−0.491647			−0.245824	+	0.969315i	$0.579058\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$		0			0
$643$		−	5.31686e8i		−	1.99996i	−0.00603535	−	0.999982i	$-0.501921\pi$
$643$		−	5.31686e8i		−	1.99996i	0.00603535	−	0.999982i	$-0.498079\pi$
$644$	−2.59854e8			−0.972909
$645$		0			0
$646$		0			0
$647$		−	4.82072e8i		−	1.77991i	−0.456046	−	0.889956i	$-0.650735\pi$
$647$		−	4.82072e8i		−	1.77991i	0.456046	−	0.889956i	$-0.349265\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		−	4.81135e8i		−	1.73590i
$653$			3.57517e8i			1.28398i	0.766714	+	0.641988i	$0.221890\pi$
$653$			3.57517e8i			1.28398i	−0.766714	+	0.641988i	$0.778110\pi$
$654$		0			0
$655$	−3.51616e7	−	1.58942e8i	−0.125125	−	0.565609i
$656$		0			0
$657$			4.93271e8i			1.73936i
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	2.62727e8	−	5.81211e7i	0.893388	−	0.197637i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$	6.05998e7			0.200588
$672$		0			0
$673$		0			0			−	1.00000i	$-0.5\pi$
$673$		0			0				1.00000i	$0.5\pi$
$674$		0			0
$675$		0			0
$676$	3.08916e8			1.00000
$677$		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	$0.5\pi$
$684$	−3.20014e8			−1.00000
$685$	5.65988e8	−	1.25209e8i	1.76090	−	0.389551i
$686$		0			0
$687$		0			0
$688$			2.87950e8i			0.884203i
$689$		0			0
$690$		0			0
$691$	−2.21024e8			−0.669892			−0.334946	−	0.942237i	$-0.608718\pi$
$691$	−2.21024e8			−0.669892			−0.334946	+	0.942237i	$0.608718\pi$
$692$		0			0
$693$			2.42975e8i			0.730065i
$694$		0			0
$695$	1.01880e8	+	4.60533e8i	0.303483	+	1.37185i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−1.32379e8	−	2.84556e8i	−0.385944	−	0.829608i
$701$	2.94922e8			0.856156			0.428078	−	0.903742i	$-0.359191\pi$
$701$	2.94922e8			0.856156			0.428078	+	0.903742i	$0.359191\pi$
$702$		0			0
$703$		0			0
$704$	−2.78397e8			−0.797896
$705$		0			0
$706$		0			0
$707$			6.46605e8i			1.82971i
$708$		0			0
$709$	−3.01929e8			−0.847161			−0.423580	−	0.905859i	$-0.639227\pi$
$709$	−3.01929e8			−0.847161			−0.423580	+	0.905859i	$0.639227\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	6.00444e8			1.61542			0.807711	−	0.589579i	$-0.200706\pi$
$719$	6.00444e8			1.61542			0.807711	+	0.589579i	$0.200706\pi$
$720$	8.06216e7	+	3.64437e8i	0.216000	+	0.976393i
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	7.56680e8i		−	1.96929i	−0.174579	−	0.984643i	$-0.555856\pi$
$727$		−	7.56680e8i		−	1.96929i	0.174579	−	0.984643i	$-0.444144\pi$
$728$		0			0
$729$	−3.87420e8			−1.00000
$730$		0			0
$731$	−1.37282e8			−0.351447
$732$		0			0
$733$			7.87665e8i			2.00000i	0.00166557	+	0.999999i	$0.499470\pi$
$733$			7.87665e8i			2.00000i	−0.00166557	+	0.999999i	$0.500530\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−7.32517e8			−1.81503			−0.907517	−	0.420016i	$-0.862024\pi$
$739$	−7.32517e8			−1.81503			−0.907517	+	0.420016i	$0.862024\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0			−	1.00000i	$-0.5\pi$
$743$		0			0				1.00000i	$0.5\pi$
$744$		0			0
$745$	1.68574e8	+	7.62014e8i	0.407683	+	1.84287i
$746$		0			0
$747$			1.23140e8i			0.295418i
$748$		−	1.32727e8i		−	0.317143i
$749$		0			0
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$			7.92863e8i			1.86442i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	5.58953e7i		−	0.128851i	−0.997923	−	0.0644255i	$-0.979479\pi$
$757$		−	5.58953e7i		−	0.128851i	0.997923	−	0.0644255i	$-0.0205215\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−7.84034e8			−1.77902			−0.889510	−	0.456915i	$-0.848954\pi$
$761$	−7.84034e8			−1.77902			−0.889510	+	0.456915i	$0.848954\pi$
$762$		0			0
$763$		0			0
$764$	−5.99926e8			−1.34529
$765$	−1.73747e8	+	3.84367e7i	−0.388090	+	0.0858543i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	6.84693e8			1.50563			0.752813	−	0.658235i	$-0.228697\pi$
$769$	6.84693e8			1.50563			0.752813	+	0.658235i	$0.228697\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	7.84507e7			0.162798
$785$	9.40325e8	−	2.08021e8i	1.94388	−	0.430029i
$786$		0			0
$787$		0			0			−	1.00000i	$-0.5\pi$
$787$		0			0				1.00000i	$0.5\pi$
$788$			7.53885e8i			1.54073i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$	9.77357e8			1.93782
$797$		0			0

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:	\( N \)	\(=\)	\( 95 = 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	95.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-64.0000 q^{4} +(-27.0000 - 122.049i) q^{5} -313.841i q^{7} -729.000 q^{9} +O(q^{10})\)	\(q-64.0000 q^{4} +(-27.0000 - 122.049i) q^{5} -313.841i q^{7} -729.000 q^{9} +1062.00 q^{11} +4096.00 q^{16} +1952.79i q^{17} -6859.00 q^{19} +(1728.00 + 7811.15i) q^{20} +12937.2i q^{23} +(-14167.0 + 6590.66i) q^{25} +20085.8i q^{28} +(-38304.0 + 8473.70i) q^{35} +46656.0 q^{36} +70300.3i q^{43} -67968.0 q^{44} +(19683.0 + 88973.8i) q^{45} +193570. i q^{47} +19153.0 q^{49} +(-28674.0 - 129616. i) q^{55} +57062.0 q^{61} +228790. i q^{63} -262144. q^{64} -124978. i q^{68} -676641. i q^{73} +438976. q^{76} -333299. i q^{77} +(-110592. - 499913. i) q^{80} +531441. q^{81} -168916. i q^{83} +(238336. - 52725.2i) q^{85} -827982. i q^{92} +(185193. + 837135. i) q^{95} -774198. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 128 q^{4} - 54 q^{5} - 1458 q^{9}+O(q^{10}) \) 2 * q - 128 * q^4 - 54 * q^5 - 1458 * q^9	\( 2 q - 128 q^{4} - 54 q^{5} - 1458 q^{9} + 2124 q^{11} + 8192 q^{16} - 13718 q^{19} + 3456 q^{20} - 28334 q^{25} - 76608 q^{35} + 93312 q^{36} - 135936 q^{44} + 39366 q^{45} + 38306 q^{49} - 57348 q^{55} + 114124 q^{61} - 524288 q^{64} + 877952 q^{76} - 221184 q^{80} + 1062882 q^{81} + 476672 q^{85} + 370386 q^{95} - 1548396 q^{99}+O(q^{100}) \) 2 * q - 128 * q^4 - 54 * q^5 - 1458 * q^9 + 2124 * q^11 + 8192 * q^16 - 13718 * q^19 + 3456 * q^20 - 28334 * q^25 - 76608 * q^35 + 93312 * q^36 - 135936 * q^44 + 39366 * q^45 + 38306 * q^49 - 57348 * q^55 + 114124 * q^61 - 524288 * q^64 + 877952 * q^76 - 221184 * q^80 + 1062882 * q^81 + 476672 * q^85 + 370386 * q^95 - 1548396 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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