LMFDB - Embedded newform 4140.2.f.c.829.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4140,2,Mod(829,4140)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4140, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))

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

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

chi := DirichletCharacter("4140.829");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4140.f (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:	$33.0580664368$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} + 27x^{12} + 283x^{10} + 1441x^{8} + 3596x^{6} + 3740x^{4} + 772x^{2} + 4 $ x^14 + 27x^12 + 283x^10 + 1441x^8 + 3596x^6 + 3740x^4 + 772x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 1380)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			829.8
Root			$2.04690i$ of defining polynomial
Character	$\chi$	$=$	4140.829
Dual form			4140.2.f.c.829.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.181772 + 2.22867i) q^{5} -0.189781i q^{7} +O(q^{10})$	$q+(0.181772 + 2.22867i) q^{5} -0.189781i q^{7} -4.33196 q^{11} -4.67083i q^{13} -0.0397340i q^{17} -7.19716 q^{19} +1.00000i q^{23} +(-4.93392 + 0.810219i) q^{25} +6.49707 q^{29} +9.17655 q^{31} +(0.422959 - 0.0344969i) q^{35} -3.03881i q^{37} +8.69330 q^{41} -7.83269i q^{43} +5.03127i q^{47} +6.96398 q^{49} +6.96306i q^{53} +(-0.787429 - 9.65450i) q^{55} +4.64368 q^{59} +7.86428 q^{61} +(10.4097 - 0.849027i) q^{65} -2.52865i q^{67} +1.70964 q^{71} -6.29127i q^{73} +0.822124i q^{77} +9.82056 q^{79} +0.417601i q^{83} +(0.0885538 - 0.00722253i) q^{85} -3.92081 q^{89} -0.886435 q^{91} +(-1.30824 - 16.0401i) q^{95} -0.0584885i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q+O(q^{10}) $ 14 * q	$ 14 q + 4 q^{19} - 6 q^{25} + 30 q^{29} + 6 q^{31} + 14 q^{35} - 46 q^{41} - 20 q^{49} - 16 q^{55} + 10 q^{59} + 64 q^{61} + 36 q^{65} - 42 q^{71} - 32 q^{79} - 42 q^{85} + 52 q^{89} + 28 q^{91} + 44 q^{95}+O(q^{100}) $ 14 * q + 4 * q^19 - 6 * q^25 + 30 * q^29 + 6 * q^31 + 14 * q^35 - 46 * q^41 - 20 * q^49 - 16 * q^55 + 10 * q^59 + 64 * q^61 + 36 * q^65 - 42 * q^71 - 32 * q^79 - 42 * q^85 + 52 * q^89 + 28 * q^91 + 44 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4140\mathbb{Z}\right)^\times$.

$n$	$461$	$1657$	$2071$	$3961$
$\chi(n)$	$1$	$-1$	$1$	$1$

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			0
$3$		0			0
$4$		0			0
$5$	0.181772	+	2.22867i	0.0812909	+	0.996690i
$5$	0.181772	+	2.22867i	0.0812909	+	0.996690i
$6$		0			0
$7$		−	0.189781i		−	0.0717305i	−0.999357	−	0.0358652i	$-0.988581\pi$
$7$		−	0.189781i		−	0.0717305i	0.999357	−	0.0358652i	$-0.0114187\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−4.33196			−1.30613			−0.653067	−	0.757300i	$-0.726518\pi$
$11$	−4.33196			−1.30613			−0.653067	+	0.757300i	$0.726518\pi$
$12$		0			0
$13$		−	4.67083i		−	1.29546i	−0.761872	−	0.647728i	$-0.775719\pi$
$13$		−	4.67083i		−	1.29546i	0.761872	−	0.647728i	$-0.224281\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	0.0397340i		−	0.00963691i	−0.999988	−	0.00481845i	$-0.998466\pi$
$17$		−	0.0397340i		−	0.00963691i	0.999988	−	0.00481845i	$-0.00153377\pi$
$18$		0			0
$19$	−7.19716			−1.65114			−0.825571	−	0.564298i	$-0.809147\pi$
$19$	−7.19716			−1.65114			−0.825571	+	0.564298i	$0.809147\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			1.00000i			0.208514i
$23$			1.00000i			0.208514i
$24$		0			0
$25$	−4.93392	+	0.810219i	−0.986784	+	0.162044i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	6.49707			1.20648			0.603238	−	0.797561i	$-0.293877\pi$
$29$	6.49707			1.20648			0.603238	+	0.797561i	$0.293877\pi$
$30$		0			0
$31$	9.17655			1.64816			0.824079	−	0.566475i	$-0.191693\pi$
$31$	9.17655			1.64816			0.824079	+	0.566475i	$0.191693\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	0.422959	−	0.0344969i	0.0714931	−	0.00583104i
$36$		0			0
$37$		−	3.03881i		−	0.499577i	−0.968300	−	0.249788i	$-0.919639\pi$
$37$		−	3.03881i		−	0.499577i	0.968300	−	0.249788i	$-0.0803611\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	8.69330			1.35767			0.678833	−	0.734293i	$-0.262486\pi$
$41$	8.69330			1.35767			0.678833	+	0.734293i	$0.262486\pi$
$42$		0			0
$43$		−	7.83269i		−	1.19447i	−0.802065	−	0.597237i	$-0.796265\pi$
$43$		−	7.83269i		−	1.19447i	0.802065	−	0.597237i	$-0.203735\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			5.03127i			0.733886i	0.930243	+	0.366943i	$0.119596\pi$
$47$			5.03127i			0.733886i	−0.930243	+	0.366943i	$0.880404\pi$
$48$		0			0
$49$	6.96398			0.994855
$50$		0			0
$51$		0			0
$52$		0			0
$53$			6.96306i			0.956449i	0.878238	+	0.478225i	$0.158720\pi$
$53$			6.96306i			0.956449i	−0.878238	+	0.478225i	$0.841280\pi$
$54$		0			0
$55$	−0.787429	−	9.65450i	−0.106177	−	1.30181i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	4.64368			0.604555			0.302278	−	0.953220i	$-0.402253\pi$
$59$	4.64368			0.604555			0.302278	+	0.953220i	$0.402253\pi$
$60$		0			0
$61$	7.86428			1.00692			0.503459	−	0.864019i	$-0.332061\pi$
$61$	7.86428			1.00692			0.503459	+	0.864019i	$0.332061\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	10.4097	−	0.849027i	1.29117	−	0.105309i
$66$		0			0
$67$		−	2.52865i		−	0.308924i	−0.987999	−	0.154462i	$-0.950636\pi$
$67$		−	2.52865i		−	0.308924i	0.987999	−	0.154462i	$-0.0493645\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	1.70964			0.202897			0.101448	−	0.994841i	$-0.467652\pi$
$71$	1.70964			0.202897			0.101448	+	0.994841i	$0.467652\pi$
$72$		0			0
$73$		−	6.29127i		−	0.736337i	−0.929759	−	0.368169i	$-0.879985\pi$
$73$		−	6.29127i		−	0.736337i	0.929759	−	0.368169i	$-0.120015\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			0.822124i			0.0936897i
$78$		0			0
$79$	9.82056			1.10490			0.552450	−	0.833546i	$-0.313693\pi$
$79$	9.82056			1.10490			0.552450	+	0.833546i	$0.313693\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			0.417601i			0.0458377i	0.999737	+	0.0229188i	$0.00729593\pi$
$83$			0.417601i			0.0458377i	−0.999737	+	0.0229188i	$0.992704\pi$
$84$		0			0
$85$	0.0885538	−	0.00722253i	0.00960501	−	0.000783393i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−3.92081			−0.415605			−0.207803	−	0.978171i	$-0.566631\pi$
$89$	−3.92081			−0.415605			−0.207803	+	0.978171i	$0.566631\pi$
$90$		0			0
$91$	−0.886435			−0.0929237
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−1.30824	−	16.0401i	−0.134223	−	1.64568i
$96$		0			0
$97$		−	0.0584885i		−	0.00593861i	−0.999996	−	0.00296930i	$-0.999055\pi$
$97$		−	0.0584885i		−	0.00593861i	0.999996	−	0.00296930i	$-0.000945160\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	6.61569			0.658286			0.329143	−	0.944280i	$-0.393240\pi$
$101$	6.61569			0.658286			0.329143	+	0.944280i	$0.393240\pi$
$102$		0			0
$103$			16.9236i			1.66753i	0.552118	+	0.833766i	$0.313820\pi$
$103$			16.9236i			1.66753i	−0.552118	+	0.833766i	$0.686180\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			17.5775i			1.69928i	0.527364	+	0.849639i	$0.323180\pi$
$107$			17.5775i			1.69928i	−0.527364	+	0.849639i	$0.676820\pi$
$108$		0			0
$109$	0.243838			0.0233554			0.0116777	−	0.999932i	$-0.496283\pi$
$109$	0.243838			0.0233554			0.0116777	+	0.999932i	$0.496283\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	1.22452i		−	0.115194i	−0.998340	−	0.0575968i	$-0.981656\pi$
$113$		−	1.22452i		−	0.115194i	0.998340	−	0.0575968i	$-0.0183438\pi$
$114$		0			0
$115$	−2.22867	+	0.181772i	−0.207824	+	0.0169503i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−0.00754075			−0.000691260
$120$		0			0
$121$	7.76587			0.705988
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−2.70256	−	10.8488i	−0.241724	−	0.970345i
$126$		0			0
$127$		−	20.6764i		−	1.83474i	−0.398041	−	0.917368i	$-0.630310\pi$
$127$		−	20.6764i		−	1.83474i	0.398041	−	0.917368i	$-0.369690\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−18.3253			−1.60109			−0.800546	−	0.599272i	$-0.795457\pi$
$131$	−18.3253			−1.60109			−0.800546	+	0.599272i	$0.795457\pi$
$132$		0			0
$133$			1.36588i			0.118437i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	3.46899i		−	0.296376i	−0.988959	−	0.148188i	$-0.952656\pi$
$137$		−	3.46899i		−	0.296376i	0.988959	−	0.148188i	$-0.0473440\pi$
$138$		0			0
$139$	−6.40878			−0.543585			−0.271793	−	0.962356i	$-0.587617\pi$
$139$	−6.40878			−0.543585			−0.271793	+	0.962356i	$0.587617\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			20.2339i			1.69204i
$144$		0			0
$145$	1.18099	+	14.4798i	0.0980755	+	1.20248i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	13.9989			1.14684			0.573418	−	0.819263i	$-0.305617\pi$
$149$	13.9989			1.14684			0.573418	+	0.819263i	$0.305617\pi$
$150$		0			0
$151$	14.5758			1.18616			0.593081	−	0.805143i	$-0.297911\pi$
$151$	14.5758			1.18616			0.593081	+	0.805143i	$0.297911\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	1.66804	+	20.4515i	0.133980	+	1.64270i
$156$		0			0
$157$			18.9563i			1.51287i	0.654066	+	0.756437i	$0.273062\pi$
$157$			18.9563i			1.51287i	−0.654066	+	0.756437i	$0.726938\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	0.189781			0.0149568
$162$		0			0
$163$			13.2785i			1.04005i	0.854150	+	0.520026i	$0.174078\pi$
$163$			13.2785i			1.04005i	−0.854150	+	0.520026i	$0.825922\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			13.9737i			1.08132i	0.841242	+	0.540659i	$0.181825\pi$
$167$			13.9737i			1.08132i	−0.841242	+	0.540659i	$0.818175\pi$
$168$		0			0
$169$	−8.81667			−0.678206
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	12.7917i		−	0.972534i	−0.873810	−	0.486267i	$-0.838358\pi$
$173$		−	12.7917i		−	0.972534i	0.873810	−	0.486267i	$-0.161642\pi$
$174$		0			0
$175$	0.153764	+	0.936364i	0.0116235	+	0.0707825i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	0.0511629			0.00382410			0.00191205	−	0.999998i	$-0.499391\pi$
$179$	0.0511629			0.00382410			0.00191205	+	0.999998i	$0.499391\pi$
$180$		0			0
$181$	21.7448			1.61628			0.808141	−	0.588989i	$-0.200474\pi$
$181$	21.7448			1.61628			0.808141	+	0.588989i	$0.200474\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	6.77249	−	0.552370i	0.497924	−	0.0406111i
$186$		0			0
$187$			0.172126i			0.0125871i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	20.3680			1.47378			0.736890	−	0.676013i	$-0.236294\pi$
$191$	20.3680			1.47378			0.736890	+	0.676013i	$0.236294\pi$
$192$		0			0
$193$		−	15.8592i		−	1.14157i	−0.821099	−	0.570786i	$-0.806639\pi$
$193$		−	15.8592i		−	1.14157i	0.821099	−	0.570786i	$-0.193361\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	6.16978i		−	0.439579i	−0.975547	−	0.219789i	$-0.929463\pi$
$197$		−	6.16978i		−	0.439579i	0.975547	−	0.219789i	$-0.0705370\pi$
$198$		0			0
$199$	4.81534			0.341350			0.170675	−	0.985327i	$-0.445405\pi$
$199$	4.81534			0.341350			0.170675	+	0.985327i	$0.445405\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	1.23302i		−	0.0865411i
$204$		0			0
$205$	1.58020	+	19.3745i	0.110366	+	1.35317i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	31.1778			2.15661
$210$		0			0
$211$	−23.9874			−1.65136			−0.825679	−	0.564140i	$-0.809208\pi$
$211$	−23.9874			−1.65136			−0.825679	+	0.564140i	$0.809208\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	17.4565	−	1.42376i	1.19052	−	0.0970999i
$216$		0			0
$217$		−	1.74154i		−	0.118223i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−0.185591			−0.0124842
$222$		0			0
$223$			11.4215i			0.764840i	0.923989	+	0.382420i	$0.124909\pi$
$223$			11.4215i			0.764840i	−0.923989	+	0.382420i	$0.875091\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	15.7360i		−	1.04443i	−0.852813	−	0.522217i	$-0.825105\pi$
$227$		−	15.7360i		−	1.04443i	0.852813	−	0.522217i	$-0.174895\pi$
$228$		0			0
$229$	−3.46335			−0.228865			−0.114432	−	0.993431i	$-0.536505\pi$
$229$	−3.46335			−0.228865			−0.114432	+	0.993431i	$0.536505\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	0.319746i		−	0.0209473i	−0.999945	−	0.0104736i	$-0.996666\pi$
$233$		−	0.319746i		−	0.0209473i	0.999945	−	0.0104736i	$-0.00333392\pi$
$234$		0			0
$235$	−11.2130	+	0.914544i	−0.731457	+	0.0596583i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−2.93697			−0.189977			−0.0949885	−	0.995478i	$-0.530281\pi$
$239$	−2.93697			−0.189977			−0.0949885	+	0.995478i	$0.530281\pi$
$240$		0			0
$241$	−18.6764			−1.20305			−0.601527	−	0.798853i	$-0.705441\pi$
$241$	−18.6764			−1.20305			−0.601527	+	0.798853i	$0.705441\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	1.26586	+	15.5204i	0.0808727	+	0.991562i
$246$		0			0
$247$			33.6167i			2.13898i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	3.76059			0.237366			0.118683	−	0.992932i	$-0.462133\pi$
$251$	3.76059			0.237366			0.118683	+	0.992932i	$0.462133\pi$
$252$		0			0
$253$		−	4.33196i		−	0.272348i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	26.1810i		−	1.63312i	−0.577259	−	0.816561i	$-0.695878\pi$
$257$		−	26.1810i		−	1.63312i	0.577259	−	0.816561i	$-0.304122\pi$
$258$		0			0
$259$	−0.576708			−0.0358349
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	22.6844i		−	1.39878i	−0.714739	−	0.699391i	$-0.753455\pi$
$263$		−	22.6844i		−	1.39878i	0.714739	−	0.699391i	$-0.246545\pi$
$264$		0			0
$265$	−15.5183	+	1.26569i	−0.953284	+	0.0777507i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	0.570220			0.0347669			0.0173835	−	0.999849i	$-0.494466\pi$
$269$	0.570220			0.0347669			0.0173835	+	0.999849i	$0.494466\pi$
$270$		0			0
$271$	21.9506			1.33341			0.666703	−	0.745323i	$-0.267705\pi$
$271$	21.9506			1.33341			0.666703	+	0.745323i	$0.267705\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	21.3735	−	3.50984i	1.28887	−	0.211651i
$276$		0			0
$277$		−	12.5514i		−	0.754141i	−0.926185	−	0.377070i	$-0.876931\pi$
$277$		−	12.5514i		−	0.754141i	0.926185	−	0.377070i	$-0.123069\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−21.7956			−1.30022			−0.650108	−	0.759842i	$-0.725276\pi$
$281$	−21.7956			−1.30022			−0.650108	+	0.759842i	$0.725276\pi$
$282$		0			0
$283$		−	24.0765i		−	1.43120i	−0.698510	−	0.715601i	$-0.746153\pi$
$283$		−	24.0765i		−	1.43120i	0.698510	−	0.715601i	$-0.253847\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	1.64982i		−	0.0973860i
$288$		0			0
$289$	16.9984			0.999907
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	27.2212i		−	1.59028i	−0.606427	−	0.795139i	$-0.707398\pi$
$293$		−	27.2212i		−	1.59028i	0.606427	−	0.795139i	$-0.292602\pi$
$294$		0			0
$295$	0.844091	+	10.3492i	0.0491449	+	0.602555i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	4.67083			0.270121
$300$		0			0
$301$	−1.48650			−0.0856802
$302$		0			0
$303$		0			0
$304$		0			0
$305$	1.42951	+	17.5269i	0.0818533	+	1.00358i
$306$		0			0
$307$		−	6.48076i		−	0.369877i	−0.982750	−	0.184938i	$-0.940791\pi$
$307$		−	6.48076i		−	0.369877i	0.982750	−	0.184938i	$-0.0592085\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−4.11457			−0.233316			−0.116658	−	0.993172i	$-0.537218\pi$
$311$	−4.11457			−0.233316			−0.116658	+	0.993172i	$0.537218\pi$
$312$		0			0
$313$			0.760034i			0.0429597i	0.999769	+	0.0214798i	$0.00683777\pi$
$313$			0.760034i			0.0429597i	−0.999769	+	0.0214798i	$0.993162\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			7.48188i			0.420224i	0.977677	+	0.210112i	$0.0673829\pi$
$317$			7.48188i			0.420224i	−0.977677	+	0.210112i	$0.932617\pi$
$318$		0			0
$319$	−28.1450			−1.57582
$320$		0			0
$321$		0			0
$322$		0			0
$323$			0.285972i			0.0159119i
$324$		0			0
$325$	3.78440	+	23.0455i	0.209921	+	1.27833i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	0.954839			0.0526420
$330$		0			0
$331$	28.5981			1.57189			0.785946	−	0.618295i	$-0.212176\pi$
$331$	28.5981			1.57189			0.785946	+	0.618295i	$0.212176\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	5.63553	−	0.459639i	0.307902	−	0.0251127i
$336$		0			0
$337$			22.9098i			1.24798i	0.781434	+	0.623988i	$0.214488\pi$
$337$			22.9098i			1.24798i	−0.781434	+	0.623988i	$0.785512\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−39.7525			−2.15272
$342$		0			0
$343$		−	2.65010i		−	0.143092i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			10.7314i			0.576090i	0.957617	+	0.288045i	$0.0930053\pi$
$347$			10.7314i			0.576090i	−0.957617	+	0.288045i	$0.906995\pi$
$348$		0			0
$349$	9.67523			0.517903			0.258952	−	0.965890i	$-0.416623\pi$
$349$	9.67523			0.517903			0.258952	+	0.965890i	$0.416623\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	29.4875i		−	1.56946i	−0.619838	−	0.784730i	$-0.712802\pi$
$353$		−	29.4875i		−	1.56946i	0.619838	−	0.784730i	$-0.287198\pi$
$354$		0			0
$355$	0.310765	+	3.81022i	0.0164937	+	0.202225i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	4.53528			0.239363			0.119681	−	0.992812i	$-0.461813\pi$
$359$	4.53528			0.239363			0.119681	+	0.992812i	$0.461813\pi$
$360$		0			0
$361$	32.7991			1.72627
$362$		0			0
$363$		0			0
$364$		0			0
$365$	14.0211	−	1.14358i	0.733900	−	0.0598576i
$366$		0			0
$367$			8.15285i			0.425576i	0.977098	+	0.212788i	$0.0682543\pi$
$367$			8.15285i			0.425576i	−0.977098	+	0.212788i	$0.931746\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	1.32146			0.0686066
$372$		0			0
$373$		−	28.9482i		−	1.49888i	−0.662070	−	0.749442i	$-0.730322\pi$
$373$		−	28.9482i		−	1.49888i	0.662070	−	0.749442i	$-0.269678\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	30.3467i		−	1.56294i
$378$		0			0
$379$	−2.46596			−0.126668			−0.0633338	−	0.997992i	$-0.520173\pi$
$379$	−2.46596			−0.126668			−0.0633338	+	0.997992i	$0.520173\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	19.0579i		−	0.973813i	−0.873454	−	0.486906i	$-0.838125\pi$
$383$		−	19.0579i		−	0.973813i	0.873454	−	0.486906i	$-0.161875\pi$
$384$		0			0
$385$	−1.83224	+	0.149439i	−0.0933796	+	0.00761612i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	37.4975			1.90120			0.950599	−	0.310421i	$-0.100470\pi$
$389$	37.4975			1.90120			0.950599	+	0.310421i	$0.100470\pi$
$390$		0			0
$391$	0.0397340			0.00200943
$392$		0			0
$393$		0			0
$394$		0			0
$395$	1.78510	+	21.8868i	0.0898183	+	1.10124i
$396$		0			0
$397$			0.171284i			0.00859652i	0.999991	+	0.00429826i	$0.00136818\pi$
$397$			0.171284i			0.00859652i	−0.999991	+	0.00429826i	$0.998632\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	22.7490			1.13603			0.568015	−	0.823018i	$-0.307711\pi$
$401$	22.7490			1.13603			0.568015	+	0.823018i	$0.307711\pi$
$402$		0			0
$403$		−	42.8621i		−	2.13512i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			13.1640i			0.652515i
$408$		0			0
$409$	11.0110			0.544460			0.272230	−	0.962232i	$-0.412239\pi$
$409$	11.0110			0.544460			0.272230	+	0.962232i	$0.412239\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	0.881282i		−	0.0433651i
$414$		0			0
$415$	−0.930694	+	0.0759082i	−0.0456860	+	0.00372619i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−30.9587			−1.51243			−0.756215	−	0.654323i	$-0.772954\pi$
$419$	−30.9587			−1.51243			−0.756215	+	0.654323i	$0.772954\pi$
$420$		0			0
$421$	18.0710			0.880727			0.440364	−	0.897819i	$-0.354850\pi$
$421$	18.0710			0.880727			0.440364	+	0.897819i	$0.354850\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	0.0321932	+	0.196044i	0.00156160	+	0.00950954i
$426$		0			0
$427$		−	1.49249i		−	0.0722267i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−27.0949			−1.30512			−0.652558	−	0.757738i	$-0.726304\pi$
$431$	−27.0949			−1.30512			−0.652558	+	0.757738i	$0.726304\pi$
$432$		0			0
$433$			35.7515i			1.71811i	0.511887	+	0.859053i	$0.328947\pi$
$433$			35.7515i			1.71811i	−0.511887	+	0.859053i	$0.671053\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	7.19716i		−	0.344287i
$438$		0			0
$439$	27.8294			1.32822			0.664112	−	0.747633i	$-0.268810\pi$
$439$	27.8294			1.32822			0.664112	+	0.747633i	$0.268810\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			8.72908i			0.414731i	0.978264	+	0.207366i	$0.0664889\pi$
$443$			8.72908i			0.414731i	−0.978264	+	0.207366i	$0.933511\pi$
$444$		0			0
$445$	−0.712694	−	8.73819i	−0.0337849	−	0.414230i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	30.6145			1.44479			0.722394	−	0.691481i	$-0.243042\pi$
$449$	30.6145			1.44479			0.722394	+	0.691481i	$0.243042\pi$
$450$		0			0
$451$	−37.6590			−1.77329
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−0.161129	−	1.97557i	−0.00755385	−	0.0926161i
$456$		0			0
$457$			27.4809i			1.28550i	0.766074	+	0.642752i	$0.222207\pi$
$457$			27.4809i			1.28550i	−0.766074	+	0.642752i	$0.777793\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−25.3722			−1.18170			−0.590850	−	0.806782i	$-0.701207\pi$
$461$	−25.3722			−1.18170			−0.590850	+	0.806782i	$0.701207\pi$
$462$		0			0
$463$		−	32.2448i		−	1.49854i	−0.662264	−	0.749271i	$-0.730404\pi$
$463$		−	32.2448i		−	1.49854i	0.662264	−	0.749271i	$-0.269596\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			0.845923i			0.0391446i	0.999808	+	0.0195723i	$0.00623046\pi$
$467$			0.845923i			0.0391446i	−0.999808	+	0.0195723i	$0.993770\pi$
$468$		0			0
$469$	−0.479891			−0.0221593
$470$		0			0
$471$		0			0
$472$		0			0
$473$			33.9309i			1.56014i
$474$		0			0
$475$	35.5102	−	5.83128i	1.62932	−	0.267557i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−8.01308			−0.366127			−0.183063	−	0.983101i	$-0.558601\pi$
$479$	−8.01308			−0.366127			−0.183063	+	0.983101i	$0.558601\pi$
$480$		0			0
$481$	−14.1938			−0.647180
$482$		0			0
$483$		0			0
$484$		0			0
$485$	0.130351	−	0.0106316i	0.00591895	−	0.000482755i
$486$		0			0
$487$			17.2251i			0.780544i	0.920700	+	0.390272i	$0.127619\pi$
$487$			17.2251i			0.780544i	−0.920700	+	0.390272i	$0.872381\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	9.41331			0.424817			0.212408	−	0.977181i	$-0.431869\pi$
$491$	9.41331			0.424817			0.212408	+	0.977181i	$0.431869\pi$
$492$		0			0
$493$		−	0.258154i		−	0.0116267i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	0.324457i		−	0.0145539i
$498$		0			0
$499$	9.17655			0.410799			0.205399	−	0.978678i	$-0.434151\pi$
$499$	9.17655			0.410799			0.205399	+	0.978678i	$0.434151\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	5.02474i		−	0.224042i	−0.993706	−	0.112021i	$-0.964268\pi$
$503$		−	5.02474i		−	0.224042i	0.993706	−	0.112021i	$-0.0357324\pi$
$504$		0			0
$505$	1.20255	+	14.7442i	0.0535127	+	0.656107i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−14.3326			−0.635282			−0.317641	−	0.948211i	$-0.602891\pi$
$509$	−14.3326			−0.635282			−0.317641	+	0.948211i	$0.602891\pi$
$510$		0			0
$511$	−1.19396			−0.0528178
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−37.7171	+	3.07624i	−1.66201	+	0.135555i
$516$		0			0
$517$		−	21.7952i		−	0.958554i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	25.4727			1.11598			0.557991	−	0.829847i	$-0.311573\pi$
$521$	25.4727			1.11598			0.557991	+	0.829847i	$0.311573\pi$
$522$		0			0
$523$		−	15.7464i		−	0.688543i	−0.938870	−	0.344271i	$-0.888126\pi$
$523$		−	15.7464i		−	0.688543i	0.938870	−	0.344271i	$-0.111874\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	0.364621i		−	0.0158831i
$528$		0			0
$529$	−1.00000			−0.0434783
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	40.6050i		−	1.75880i
$534$		0			0
$535$	−39.1743	+	3.19509i	−1.69365	+	0.138136i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−30.1677			−1.29941
$540$		0			0
$541$	−5.58372			−0.240063			−0.120031	−	0.992770i	$-0.538300\pi$
$541$	−5.58372			−0.240063			−0.120031	+	0.992770i	$0.538300\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	0.0443229	+	0.543434i	0.00189859	+	0.0232781i
$546$		0			0
$547$			13.9544i			0.596648i	0.954465	+	0.298324i	$0.0964276\pi$
$547$			13.9544i			0.596648i	−0.954465	+	0.298324i	$0.903572\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−46.7605			−1.99206
$552$		0			0
$553$		−	1.86376i		−	0.0792550i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			30.6343i			1.29802i	0.760781	+	0.649009i	$0.224816\pi$
$557$			30.6343i			1.29802i	−0.760781	+	0.649009i	$0.775184\pi$
$558$		0			0
$559$	−36.5852			−1.54739
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	40.9163i		−	1.72442i	−0.506553	−	0.862209i	$-0.669080\pi$
$563$		−	40.9163i		−	1.72442i	0.506553	−	0.862209i	$-0.330920\pi$
$564$		0			0
$565$	2.72906	−	0.222584i	0.114812	−	0.00936420i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	17.2131			0.721612			0.360806	−	0.932641i	$-0.382502\pi$
$569$	17.2131			0.721612			0.360806	+	0.932641i	$0.382502\pi$
$570$		0			0
$571$	−14.7070			−0.615469			−0.307734	−	0.951472i	$-0.599571\pi$
$571$	−14.7070			−0.615469			−0.307734	+	0.951472i	$0.599571\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−0.810219	−	4.93392i	−0.0337885	−	0.205759i
$576$		0			0
$577$		−	15.7617i		−	0.656168i	−0.944648	−	0.328084i	$-0.893597\pi$
$577$		−	15.7617i		−	0.656168i	0.944648	−	0.328084i	$-0.106403\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	0.0792528			0.00328796
$582$		0			0
$583$		−	30.1637i		−	1.24925i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			40.4757i			1.67061i	0.549787	+	0.835305i	$0.314709\pi$
$587$			40.4757i			1.67061i	−0.549787	+	0.835305i	$0.685291\pi$
$588$		0			0
$589$	−66.0451			−2.72134
$590$		0			0
$591$		0			0
$592$		0			0
$593$			35.8541i			1.47235i	0.676791	+	0.736175i	$0.263370\pi$
$593$			35.8541i			1.47235i	−0.676791	+	0.736175i	$0.736630\pi$
$594$		0			0
$595$	−0.00137070	−	0.0168058i	−5.61932e−5	−	0.000688972i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−21.0263			−0.859112			−0.429556	−	0.903040i	$-0.641330\pi$
$599$	−21.0263			−0.859112			−0.429556	+	0.903040i	$0.641330\pi$
$600$		0			0
$601$	6.16524			0.251485			0.125743	−	0.992063i	$-0.459869\pi$
$601$	6.16524			0.251485			0.125743	+	0.992063i	$0.459869\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	1.41162	+	17.3075i	0.0573904	+	0.703651i
$606$		0			0
$607$			27.5809i			1.11947i	0.828671	+	0.559736i	$0.189097\pi$
$607$			27.5809i			1.11947i	−0.828671	+	0.559736i	$0.810903\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	23.5002			0.950716
$612$		0			0
$613$		−	21.7384i		−	0.878005i	−0.898486	−	0.439003i	$-0.855332\pi$
$613$		−	21.7384i		−	0.878005i	0.898486	−	0.439003i	$-0.144668\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			25.2023i			1.01461i	0.861767	+	0.507304i	$0.169358\pi$
$617$			25.2023i			1.01461i	−0.861767	+	0.507304i	$0.830642\pi$
$618$		0			0
$619$	−6.45053			−0.259269			−0.129634	−	0.991562i	$-0.541380\pi$
$619$	−6.45053			−0.259269			−0.129634	+	0.991562i	$0.541380\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			0.744096i			0.0298116i
$624$		0			0
$625$	23.6871	−	7.99511i	0.947484	−	0.319804i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−0.120744			−0.00481438
$630$		0			0
$631$	−24.0614			−0.957871			−0.478935	−	0.877850i	$-0.658977\pi$
$631$	−24.0614			−0.957871			−0.478935	+	0.877850i	$0.658977\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	46.0809	−	3.75839i	1.82866	−	0.149147i
$636$		0			0
$637$		−	32.5276i		−	1.28879i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−0.0204978			−0.000809615			−0.000404807	−	1.00000i	$-0.500129\pi$
$641$	−0.0204978			−0.000809615			−0.000404807		1.00000i	$0.500129\pi$
$642$		0			0
$643$		−	46.4037i		−	1.82998i	−0.403471	−	0.914992i	$-0.632196\pi$
$643$		−	46.4037i		−	1.82998i	0.403471	−	0.914992i	$-0.367804\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	16.8386i		−	0.661992i	−0.943632	−	0.330996i	$-0.892615\pi$
$647$		−	16.8386i		−	0.661992i	0.943632	−	0.330996i	$-0.107385\pi$
$648$		0			0
$649$	−20.1162			−0.789631
$650$		0			0
$651$		0			0
$652$		0			0
$653$			40.9087i			1.60088i	0.599413	+	0.800440i	$0.295401\pi$
$653$			40.9087i			1.60088i	−0.599413	+	0.800440i	$0.704599\pi$
$654$		0			0
$655$	−3.33103	−	40.8411i	−0.130154	−	1.59579i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−38.3970			−1.49573			−0.747867	−	0.663849i	$-0.768922\pi$
$659$	−38.3970			−1.49573			−0.747867	+	0.663849i	$0.768922\pi$
$660$		0			0
$661$	−10.8022			−0.420155			−0.210078	−	0.977685i	$-0.567372\pi$
$661$	−10.8022			−0.420155			−0.210078	+	0.977685i	$0.567372\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−3.04410	+	0.248280i	−0.118045	+	0.00962787i
$666$		0			0
$667$			6.49707i			0.251568i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−34.0677			−1.31517
$672$		0			0
$673$			11.3951i			0.439247i	0.975585	+	0.219624i	$0.0704829\pi$
$673$			11.3951i			0.439247i	−0.975585	+	0.219624i	$0.929517\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			3.87400i			0.148890i	0.997225	+	0.0744449i	$0.0237185\pi$
$677$			3.87400i			0.148890i	−0.997225	+	0.0744449i	$0.976282\pi$
$678$		0			0
$679$	−0.0111000			−0.000425979
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	27.9517i		−	1.06954i	−0.844997	−	0.534770i	$-0.820398\pi$
$683$		−	27.9517i		−	1.06954i	0.844997	−	0.534770i	$-0.179602\pi$
$684$		0			0
$685$	7.73122	−	0.630565i	0.295395	−	0.0240927i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	32.5233			1.23904
$690$		0			0
$691$	35.1448			1.33697			0.668485	−	0.743726i	$-0.266943\pi$
$691$	35.1448			1.33697			0.668485	+	0.743726i	$0.266943\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−1.16494	−	14.2830i	−0.0441886	−	0.541786i
$696$		0			0
$697$		−	0.345420i		−	0.0130837i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	7.89656			0.298249			0.149124	−	0.988818i	$-0.452355\pi$
$701$	7.89656			0.298249			0.149124	+	0.988818i	$0.452355\pi$
$702$		0			0
$703$			21.8708i			0.824872i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	1.25553i		−	0.0472191i
$708$		0			0
$709$	16.3686			0.614734			0.307367	−	0.951591i	$-0.400552\pi$
$709$	16.3686			0.614734			0.307367	+	0.951591i	$0.400552\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			9.17655i			0.343665i
$714$		0			0
$715$	−45.0945	+	3.67795i	−1.68644	+	0.137547i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	12.0861			0.450737			0.225368	−	0.974274i	$-0.427641\pi$
$719$	12.0861			0.450737			0.225368	+	0.974274i	$0.427641\pi$
$720$		0			0
$721$	3.21178			0.119613
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−32.0560	+	5.26405i	−1.19053	+	0.195502i
$726$		0			0
$727$			7.89415i			0.292778i	0.989227	+	0.146389i	$0.0467651\pi$
$727$			7.89415i			0.292778i	−0.989227	+	0.146389i	$0.953235\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−0.311224			−0.0115110
$732$		0			0
$733$			17.1444i			0.633242i	0.948552	+	0.316621i	$0.102548\pi$
$733$			17.1444i			0.633242i	−0.948552	+	0.316621i	$0.897452\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			10.9540i			0.403497i
$738$		0			0
$739$	43.1409			1.58696			0.793482	−	0.608593i	$-0.208266\pi$
$739$	43.1409			1.58696			0.793482	+	0.608593i	$0.208266\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	41.0320i		−	1.50532i	−0.658410	−	0.752660i	$-0.728771\pi$
$743$		−	41.0320i		−	1.50532i	0.658410	−	0.752660i	$-0.271229\pi$
$744$		0			0
$745$	2.54461	+	31.1989i	0.0932273	+	1.14304i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	3.33587			0.121890
$750$		0			0
$751$	32.4190			1.18299			0.591494	−	0.806309i	$-0.298538\pi$
$751$	32.4190			1.18299			0.591494	+	0.806309i	$0.298538\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	2.64947	+	32.4846i	0.0964242	+	1.18224i
$756$		0			0
$757$			31.1373i			1.13170i	0.824507	+	0.565852i	$0.191453\pi$
$757$			31.1373i			1.13170i	−0.824507	+	0.565852i	$0.808547\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	43.4836			1.57628			0.788139	−	0.615497i	$-0.211045\pi$
$761$	43.4836			1.57628			0.788139	+	0.615497i	$0.211045\pi$
$762$		0			0
$763$		−	0.0462758i		−	0.00167530i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	21.6898i		−	0.783175i
$768$		0			0
$769$	4.35659			0.157103			0.0785514	−	0.996910i	$-0.474971\pi$
$769$	4.35659			0.157103			0.0785514	+	0.996910i	$0.474971\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			52.3629i			1.88336i	0.336508	+	0.941681i	$0.390754\pi$
$773$			52.3629i			1.88336i	−0.336508	+	0.941681i	$0.609246\pi$
$774$		0			0
$775$	−45.2764	+	7.43502i	−1.62637	+	0.267074i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−62.5671			−2.24170
$780$		0			0
$781$	−7.40609			−0.265011
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−42.2472	+	3.44572i	−1.50787	+	0.122983i
$786$		0			0
$787$		−	15.0229i		−	0.535507i	−0.963487	−	0.267754i	$-0.913719\pi$
$787$		−	15.0229i		−	0.535507i	0.963487	−	0.267754i	$-0.0862813\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−0.232392			−0.00826289
$792$		0			0
$793$		−	36.7327i		−	1.30442i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	31.3217i		−	1.10947i	−0.832026	−	0.554737i	$-0.812819\pi$
$797$		−	31.3217i		−	1.10947i	0.832026	−	0.554737i	$-0.187181\pi$
$798$		0			0
$799$	0.199912			0.00707239
$800$		0			0
$801$		0			0
$802$		0			0
$803$			27.2535i			0.961756i
$804$		0			0
$805$	0.0344969	+	0.422959i	0.00121586	+	0.0149073i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−22.0261			−0.774397			−0.387198	−	0.921996i	$-0.626557\pi$
$809$	−22.0261			−0.774397			−0.387198	+	0.921996i	$0.626557\pi$
$810$		0			0
$811$	22.2449			0.781125			0.390563	−	0.920576i	$-0.372281\pi$
$811$	22.2449			0.781125			0.390563	+	0.920576i	$0.372281\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−29.5933	+	2.41366i	−1.03661	+	0.0845468i
$816$		0			0
$817$			56.3731i			1.97225i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−55.8300			−1.94848			−0.974240	−	0.225513i	$-0.927594\pi$
$821$	−55.8300			−1.94848			−0.974240	+	0.225513i	$0.927594\pi$
$822$		0			0
$823$			8.57848i			0.299027i	0.988760	+	0.149514i	$0.0477708\pi$
$823$			8.57848i			0.299027i	−0.988760	+	0.149514i	$0.952229\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			12.4646i			0.433437i	0.976234	+	0.216718i	$0.0695354\pi$
$827$			12.4646i			0.433437i	−0.976234	+	0.216718i	$0.930465\pi$
$828$		0			0
$829$	5.83650			0.202710			0.101355	−	0.994850i	$-0.467682\pi$
$829$	5.83650			0.202710			0.101355	+	0.994850i	$0.467682\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	0.276707i		−	0.00958732i
$834$		0			0
$835$	−31.1428	+	2.54003i	−1.07774	+	0.0879014i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−21.9925			−0.759265			−0.379633	−	0.925137i	$-0.623950\pi$
$839$	−21.9925			−0.759265			−0.379633	+	0.925137i	$0.623950\pi$
$840$		0			0
$841$	13.2119			0.455583
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−1.60262	−	19.6494i	−0.0551320	−	0.675961i
$846$		0			0
$847$		−	1.47381i		−	0.0506409i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	3.03881			0.104169
$852$		0			0
$853$		−	30.7842i		−	1.05403i	−0.849856	−	0.527015i	$-0.823311\pi$
$853$		−	30.7842i		−	1.05403i	0.849856	−	0.527015i	$-0.176689\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			15.0207i			0.513098i	0.966531	+	0.256549i	$0.0825855\pi$
$857$			15.0207i			0.513098i	−0.966531	+	0.256549i	$0.917414\pi$
$858$		0			0
$859$	−0.352932			−0.0120419			−0.00602094	−	0.999982i	$-0.501917\pi$
$859$	−0.352932			−0.0120419			−0.00602094	+	0.999982i	$0.501917\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	8.55180i		−	0.291107i	−0.989350	−	0.145553i	$-0.953504\pi$
$863$		−	8.55180i		−	0.291107i	0.989350	−	0.145553i	$-0.0464962\pi$
$864$		0			0
$865$	28.5084	−	2.32517i	0.969315	−	0.0790582i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−42.5423			−1.44315
$870$		0			0
$871$	−11.8109			−0.400198
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−2.05889	+	0.512894i	−0.0696033	+	0.0173390i
$876$		0			0
$877$		−	46.9633i		−	1.58584i	−0.609327	−	0.792919i	$-0.708560\pi$
$877$		−	46.9633i		−	1.58584i	0.609327	−	0.792919i	$-0.291440\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	47.9481			1.61541			0.807706	−	0.589585i	$-0.200709\pi$
$881$	47.9481			1.61541			0.807706	+	0.589585i	$0.200709\pi$
$882$		0			0
$883$		−	49.6286i		−	1.67014i	−0.550147	−	0.835068i	$-0.685428\pi$
$883$		−	49.6286i		−	1.67014i	0.550147	−	0.835068i	$-0.314572\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			39.1853i			1.31572i	0.753142	+	0.657858i	$0.228537\pi$
$887$			39.1853i			1.31572i	−0.753142	+	0.657858i	$0.771463\pi$
$888$		0			0
$889$	−3.92399			−0.131606
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	36.2108i		−	1.21175i
$894$		0			0
$895$	0.00929999	+	0.114025i	0.000310865	+	0.00381144i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	59.6207			1.98846
$900$		0			0
$901$	0.276670			0.00921721
$902$		0			0
$903$		0			0
$904$		0			0
$905$	3.95261	+	48.4620i	0.131389	+	1.61093i
$906$		0			0
$907$			39.3323i			1.30601i	0.757355	+	0.653004i	$0.226491\pi$
$907$			39.3323i			1.30601i	−0.757355	+	0.653004i	$0.773509\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	17.8676			0.591980			0.295990	−	0.955191i	$-0.404351\pi$
$911$	17.8676			0.591980			0.295990	+	0.955191i	$0.404351\pi$
$912$		0			0
$913$		−	1.80903i		−	0.0598702i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			3.47780i			0.114847i
$918$		0			0
$919$	20.0858			0.662570			0.331285	−	0.943531i	$-0.392518\pi$
$919$	20.0858			0.662570			0.331285	+	0.943531i	$0.392518\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	7.98544i		−	0.262844i
$924$		0			0
$925$	2.46210	+	14.9932i	0.0809533	+	0.492974i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−56.6085			−1.85727			−0.928633	−	0.371001i	$-0.879015\pi$
$929$	−56.6085			−1.85727			−0.928633	+	0.371001i	$0.879015\pi$
$930$		0			0
$931$	−50.1209			−1.64265
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−0.383612	+	0.0312877i	−0.0125454	+	0.00102322i
$936$		0			0
$937$		−	48.1869i		−	1.57420i	−0.616826	−	0.787099i	$-0.711582\pi$
$937$		−	48.1869i		−	1.57420i	0.616826	−	0.787099i	$-0.288418\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	10.7084			0.349083			0.174542	−	0.984650i	$-0.444156\pi$
$941$	10.7084			0.349083			0.174542	+	0.984650i	$0.444156\pi$
$942$		0			0
$943$			8.69330i			0.283093i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			17.5646i			0.570774i	0.958412	+	0.285387i	$0.0921221\pi$
$947$			17.5646i			0.570774i	−0.958412	+	0.285387i	$0.907878\pi$
$948$		0			0
$949$	−29.3855			−0.953893
$950$		0			0
$951$		0			0
$952$		0			0
$953$			36.3058i			1.17606i	0.808839	+	0.588031i	$0.200097\pi$
$953$			36.3058i			1.17606i	−0.808839	+	0.588031i	$0.799903\pi$
$954$		0			0
$955$	3.70234	+	45.3936i	0.119805	+	1.46890i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−0.658348			−0.0212592
$960$		0			0
$961$	53.2091			1.71642
$962$		0			0
$963$		0			0
$964$		0			0
$965$	35.3449	−	2.88276i	1.13779	−	0.0927994i
$966$		0			0
$967$			30.4808i			0.980195i	0.871668	+	0.490098i	$0.163039\pi$
$967$			30.4808i			0.980195i	−0.871668	+	0.490098i	$0.836961\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−43.2929			−1.38934			−0.694668	−	0.719330i	$-0.744449\pi$
$971$	−43.2929			−1.38934			−0.694668	+	0.719330i	$0.744449\pi$
$972$		0			0
$973$			1.21626i			0.0389916i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	6.01624i		−	0.192477i	−0.995358	−	0.0962383i	$-0.969319\pi$
$977$		−	6.01624i		−	0.192477i	0.995358	−	0.0962383i	$-0.0306811\pi$
$978$		0			0
$979$	16.9848			0.542836
$980$		0			0
$981$		0			0
$982$		0			0
$983$			36.0552i			1.14998i	0.818159	+	0.574992i	$0.194995\pi$
$983$			36.0552i			1.14998i	−0.818159	+	0.574992i	$0.805005\pi$
$984$		0			0
$985$	13.7504	−	1.12149i	0.438124	−	0.0357338i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	7.83269			0.249065
$990$		0			0
$991$	−51.5584			−1.63781			−0.818904	−	0.573931i	$-0.805418\pi$
$991$	−51.5584			−1.63781			−0.818904	+	0.573931i	$0.805418\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	0.875293	+	10.7318i	0.0277487	+	0.340220i
$996$		0			0
$997$		−	18.9389i		−	0.599799i	−0.953971	−	0.299900i	$-0.903047\pi$
$997$		−	18.9389i		−	0.599799i	0.953971	−	0.299900i	$-0.0969532\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4140.2.f.c.829.8		14
3.2	odd	2		1380.2.f.b.829.11	yes	14
5.4	even	2	inner	4140.2.f.c.829.7		14
15.2	even	4		6900.2.a.bd.1.4		7
15.8	even	4		6900.2.a.bc.1.4		7
15.14	odd	2		1380.2.f.b.829.4	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1380.2.f.b.829.4	✓	14	15.14	odd	2
1380.2.f.b.829.11	yes	14	3.2	odd	2
4140.2.f.c.829.7		14	5.4	even	2	inner
4140.2.f.c.829.8		14	1.1	even	1	trivial
6900.2.a.bc.1.4		7	15.8	even	4
6900.2.a.bd.1.4		7	15.2	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4140.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(0.181772 + 2.22867i) q^{5} -0.189781i q^{7} +O(q^{10})\)	\(q+(0.181772 + 2.22867i) q^{5} -0.189781i q^{7} -4.33196 q^{11} -4.67083i q^{13} -0.0397340i q^{17} -7.19716 q^{19} +1.00000i q^{23} +(-4.93392 + 0.810219i) q^{25} +6.49707 q^{29} +9.17655 q^{31} +(0.422959 - 0.0344969i) q^{35} -3.03881i q^{37} +8.69330 q^{41} -7.83269i q^{43} +5.03127i q^{47} +6.96398 q^{49} +6.96306i q^{53} +(-0.787429 - 9.65450i) q^{55} +4.64368 q^{59} +7.86428 q^{61} +(10.4097 - 0.849027i) q^{65} -2.52865i q^{67} +1.70964 q^{71} -6.29127i q^{73} +0.822124i q^{77} +9.82056 q^{79} +0.417601i q^{83} +(0.0885538 - 0.00722253i) q^{85} -3.92081 q^{89} -0.886435 q^{91} +(-1.30824 - 16.0401i) q^{95} -0.0584885i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q+O(q^{10}) \) 14 * q	\( 14 q + 4 q^{19} - 6 q^{25} + 30 q^{29} + 6 q^{31} + 14 q^{35} - 46 q^{41} - 20 q^{49} - 16 q^{55} + 10 q^{59} + 64 q^{61} + 36 q^{65} - 42 q^{71} - 32 q^{79} - 42 q^{85} + 52 q^{89} + 28 q^{91} + 44 q^{95}+O(q^{100}) \) 14 * q + 4 * q^19 - 6 * q^25 + 30 * q^29 + 6 * q^31 + 14 * q^35 - 46 * q^41 - 20 * q^49 - 16 * q^55 + 10 * q^59 + 64 * q^61 + 36 * q^65 - 42 * q^71 - 32 * q^79 - 42 * q^85 + 52 * q^89 + 28 * q^91 + 44 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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