LMFDB - Embedded newform 684.5.h.f.37.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [684,5,Mod(37,684)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(684, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("684.37");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 684 = 2^{2} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	684.h (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:	$70.7050547493$
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} - 2 x^{13} + 2574 x^{12} - 7948 x^{11} + 5136095 x^{10} - 4313010 x^{9} + 3526383758 x^{8} + \cdots + 13\!\cdots\!00 $ x^14 - 2x^13 + 2574x^12 - 7948x^11 + 5136095x^10 - 4313010x^9 + 3526383758x^8 + 42678039416x^7 + 1723219853217x^6 + 14955695567410x^5 + 301377432088480x^4 + 2343588994038264x^3 + 36239668281491904x^2 + 197066822628281760*x + 1359248846310158400
Coefficient ring:	$\Z[a_1, \ldots, a_{31}]$
Coefficient ring index:	$ 2^{20}\cdot 3^{13} $
Twist minimal:	no (minimal twist has level 228)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			37.4
Root			$17.5385 - 30.3776i$ of defining polynomial
Character	$\chi$	$=$	684.37
Dual form			684.5.h.f.37.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-36.0770 q^{5} +53.4819 q^{7} +O(q^{10})$	$q-36.0770 q^{5} +53.4819 q^{7} -123.750 q^{11} +200.417i q^{13} -247.460 q^{17} +(-70.2082 + 354.107i) q^{19} +865.599 q^{23} +676.551 q^{25} -1120.97i q^{29} -1184.44i q^{31} -1929.47 q^{35} +1150.31i q^{37} +3027.92i q^{41} +1945.49 q^{43} -3412.87 q^{47} +459.317 q^{49} +2263.13i q^{53} +4464.51 q^{55} +328.955i q^{59} -172.620 q^{61} -7230.47i q^{65} -3387.06i q^{67} -2681.09i q^{71} -9323.61 q^{73} -6618.36 q^{77} -12093.6i q^{79} +1128.96 q^{83} +8927.60 q^{85} -5339.41i q^{89} +10718.7i q^{91} +(2532.90 - 12775.1i) q^{95} +2663.91i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 18 q^{5} - 86 q^{7}+O(q^{10}) $ 14 * q - 18 * q^5 - 86 * q^7	$ 14 q - 18 q^{5} - 86 q^{7} - 258 q^{11} - 498 q^{17} + 170 q^{19} + 588 q^{23} + 1560 q^{25} - 534 q^{35} + 1882 q^{43} + 222 q^{47} + 4104 q^{49} + 2702 q^{55} - 2462 q^{61} - 5774 q^{73} + 4578 q^{77} - 17988 q^{83} + 2342 q^{85} + 18270 q^{95}+O(q^{100}) $ 14 * q - 18 * q^5 - 86 * q^7 - 258 * q^11 - 498 * q^17 + 170 * q^19 + 588 * q^23 + 1560 * q^25 - 534 * q^35 + 1882 * q^43 + 222 * q^47 + 4104 * q^49 + 2702 * q^55 - 2462 * q^61 - 5774 * q^73 + 4578 * q^77 - 17988 * q^83 + 2342 * q^85 + 18270 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/684\mathbb{Z}\right)^\times$.

$n$	$325$	$343$	$533$
$\chi(n)$	$-1$	$1$	$1$

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
$5$	−36.0770			−1.44308			−0.721540	−	0.692372i	$-0.756566\pi$
$5$	−36.0770			−1.44308			−0.721540	+	0.692372i	$0.756566\pi$
$6$		0			0
$7$	53.4819			1.09147			0.545734	−	0.837958i	$-0.316251\pi$
$7$	53.4819			1.09147			0.545734	+	0.837958i	$0.316251\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−123.750			−1.02272			−0.511362	−	0.859366i	$-0.670859\pi$
$11$	−123.750			−1.02272			−0.511362	+	0.859366i	$0.670859\pi$
$12$		0			0
$13$			200.417i			1.18590i	0.805238	+	0.592951i	$0.202037\pi$
$13$			200.417i			1.18590i	−0.805238	+	0.592951i	$0.797963\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−247.460			−0.856261			−0.428131	−	0.903717i	$-0.640828\pi$
$17$	−247.460			−0.856261			−0.428131	+	0.903717i	$0.640828\pi$
$18$		0			0
$19$	−70.2082	+	354.107i	−0.194483	+	0.980906i
$19$	−70.2082	+	354.107i	−0.194483	+	0.980906i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	865.599			1.63629			0.818146	−	0.575010i	$-0.195002\pi$
$23$	865.599			1.63629			0.818146	+	0.575010i	$0.195002\pi$
$24$		0			0
$25$	676.551			1.08248
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	1120.97i		−	1.33291i	−0.745547	−	0.666453i	$-0.767812\pi$
$29$		−	1120.97i		−	1.33291i	0.745547	−	0.666453i	$-0.232188\pi$
$30$		0			0
$31$		−	1184.44i		−	1.23250i	−0.787549	−	0.616252i	$-0.788650\pi$
$31$		−	1184.44i		−	1.23250i	0.787549	−	0.616252i	$-0.211350\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−1929.47			−1.57508
$36$		0			0
$37$			1150.31i			0.840255i	0.907465	+	0.420127i	$0.138015\pi$
$37$			1150.31i			0.840255i	−0.907465	+	0.420127i	$0.861985\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			3027.92i			1.80126i	0.434587	+	0.900630i	$0.356894\pi$
$41$			3027.92i			1.80126i	−0.434587	+	0.900630i	$0.643106\pi$
$42$		0			0
$43$	1945.49			1.05219			0.526093	−	0.850427i	$-0.323656\pi$
$43$	1945.49			1.05219			0.526093	+	0.850427i	$0.323656\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−3412.87			−1.54498			−0.772492	−	0.635025i	$-0.780990\pi$
$47$	−3412.87			−1.54498			−0.772492	+	0.635025i	$0.780990\pi$
$48$		0			0
$49$	459.317			0.191302
$50$		0			0
$51$		0			0
$52$		0			0
$53$			2263.13i			0.805672i	0.915272	+	0.402836i	$0.131975\pi$
$53$			2263.13i			0.805672i	−0.915272	+	0.402836i	$0.868025\pi$
$54$		0			0
$55$	4464.51			1.47587
$56$		0			0
$57$		0			0
$58$		0			0
$59$			328.955i			0.0945000i	0.998883	+	0.0472500i	$0.0150457\pi$
$59$			328.955i			0.0945000i	−0.998883	+	0.0472500i	$0.984954\pi$
$60$		0			0
$61$	−172.620			−0.0463907			−0.0231954	−	0.999731i	$-0.507384\pi$
$61$	−172.620			−0.0463907			−0.0231954	+	0.999731i	$0.507384\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	7230.47i		−	1.71135i
$66$		0			0
$67$		−	3387.06i		−	0.754523i	−0.926107	−	0.377262i	$-0.876866\pi$
$67$		−	3387.06i		−	0.754523i	0.926107	−	0.377262i	$-0.123134\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	2681.09i		−	0.531858i	−0.963993	−	0.265929i	$-0.914321\pi$
$71$		−	2681.09i		−	0.531858i	0.963993	−	0.265929i	$-0.0856786\pi$
$72$		0			0
$73$	−9323.61			−1.74960			−0.874799	−	0.484486i	$-0.839007\pi$
$73$	−9323.61			−1.74960			−0.874799	+	0.484486i	$0.839007\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−6618.36			−1.11627
$78$		0			0
$79$		−	12093.6i		−	1.93777i	−0.247503	−	0.968887i	$-0.579610\pi$
$79$		−	12093.6i		−	1.93777i	0.247503	−	0.968887i	$-0.420390\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	1128.96			0.163878			0.0819391	−	0.996637i	$-0.473889\pi$
$83$	1128.96			0.163878			0.0819391	+	0.996637i	$0.473889\pi$
$84$		0			0
$85$	8927.60			1.23565
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	5339.41i		−	0.674082i	−0.941490	−	0.337041i	$-0.890574\pi$
$89$		−	5339.41i		−	0.674082i	0.941490	−	0.337041i	$-0.109426\pi$
$90$		0			0
$91$			10718.7i			1.29437i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	2532.90	−	12775.1i	0.280654	−	1.41553i
$96$		0			0
$97$			2663.91i			0.283124i	0.989929	+	0.141562i	$0.0452124\pi$
$97$			2663.91i			0.283124i	−0.989929	+	0.141562i	$0.954788\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	9704.35			0.951313			0.475657	−	0.879631i	$-0.342210\pi$
$101$	9704.35			0.951313			0.475657	+	0.879631i	$0.342210\pi$
$102$		0			0
$103$		−	16717.6i		−	1.57579i	−0.615809	−	0.787896i	$-0.711171\pi$
$103$		−	16717.6i		−	1.57579i	0.615809	−	0.787896i	$-0.288829\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	19922.5i		−	1.74011i	−0.492957	−	0.870054i	$-0.664084\pi$
$107$		−	19922.5i		−	1.74011i	0.492957	−	0.870054i	$-0.335916\pi$
$108$		0			0
$109$		−	15096.5i		−	1.27064i	−0.772248	−	0.635322i	$-0.780867\pi$
$109$		−	15096.5i		−	1.27064i	0.772248	−	0.635322i	$-0.219133\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	5388.47i		−	0.421996i	−0.977486	−	0.210998i	$-0.932329\pi$
$113$		−	5388.47i		−	0.421996i	0.977486	−	0.210998i	$-0.0676714\pi$
$114$		0			0
$115$	−31228.2			−2.36130
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−13234.6			−0.934582
$120$		0			0
$121$	672.940			0.0459627
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1859.82			−0.119029
$126$		0			0
$127$		−	21665.0i		−	1.34323i	−0.740900	−	0.671616i	$-0.765601\pi$
$127$		−	21665.0i		−	1.34323i	0.740900	−	0.671616i	$-0.234399\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−2636.15			−0.153613			−0.0768065	−	0.997046i	$-0.524472\pi$
$131$	−2636.15			−0.153613			−0.0768065	+	0.997046i	$0.524472\pi$
$132$		0			0
$133$	−3754.87	+	18938.3i	−0.212271	+	1.07063i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	22771.8			1.21326			0.606632	−	0.794983i	$-0.292520\pi$
$137$	22771.8			1.21326			0.606632	+	0.794983i	$0.292520\pi$
$138$		0			0
$139$	11186.0			0.578954			0.289477	−	0.957185i	$-0.406519\pi$
$139$	11186.0			0.578954			0.289477	+	0.957185i	$0.406519\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	24801.6i		−	1.21285i
$144$		0			0
$145$			40441.4i			1.92349i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	4507.58			0.203035			0.101517	−	0.994834i	$-0.467630\pi$
$149$	4507.58			0.203035			0.101517	+	0.994834i	$0.467630\pi$
$150$		0			0
$151$			2197.53i			0.0963786i	0.998838	+	0.0481893i	$0.0153451\pi$
$151$			2197.53i			0.0963786i	−0.998838	+	0.0481893i	$0.984655\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			42731.0i			1.77860i
$156$		0			0
$157$	−40213.8			−1.63146			−0.815728	−	0.578435i	$-0.803664\pi$
$157$	−40213.8			−1.63146			−0.815728	+	0.578435i	$0.803664\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	46293.9			1.78596
$162$		0			0
$163$	−17896.6			−0.673590			−0.336795	−	0.941578i	$-0.609343\pi$
$163$	−17896.6			−0.673590			−0.336795	+	0.941578i	$0.609343\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	37563.9i		−	1.34691i	−0.739229	−	0.673454i	$-0.764810\pi$
$167$		−	37563.9i		−	1.34691i	0.739229	−	0.673454i	$-0.235190\pi$
$168$		0			0
$169$	−11606.2			−0.406364
$170$		0			0
$171$		0			0
$172$		0			0
$173$			4705.16i			0.157211i	0.996906	+	0.0786053i	$0.0250467\pi$
$173$			4705.16i			0.157211i	−0.996906	+	0.0786053i	$0.974953\pi$
$174$		0			0
$175$	36183.3			1.18149
$176$		0			0
$177$		0			0
$178$		0			0
$179$			3805.42i			0.118767i	0.998235	+	0.0593837i	$0.0189135\pi$
$179$			3805.42i			0.118767i	−0.998235	+	0.0593837i	$0.981086\pi$
$180$		0			0
$181$			47436.8i			1.44797i	0.689818	+	0.723983i	$0.257690\pi$
$181$			47436.8i			1.44797i	−0.689818	+	0.723983i	$0.742310\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	41499.7i		−	1.21256i
$186$		0			0
$187$	30623.0			0.875718
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−36508.5			−1.00075			−0.500376	−	0.865808i	$-0.666805\pi$
$191$	−36508.5			−1.00075			−0.500376	+	0.865808i	$0.666805\pi$
$192$		0			0
$193$		−	36804.0i		−	0.988052i	−0.869447	−	0.494026i	$-0.835525\pi$
$193$		−	36804.0i		−	0.988052i	0.869447	−	0.494026i	$-0.164475\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−38154.6			−0.983139			−0.491569	−	0.870838i	$-0.663577\pi$
$197$	−38154.6			−0.983139			−0.491569	+	0.870838i	$0.663577\pi$
$198$		0			0
$199$	25907.9			0.654223			0.327112	−	0.944986i	$-0.393925\pi$
$199$	25907.9			0.654223			0.327112	+	0.944986i	$0.393925\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	59951.9i		−	1.45482i
$204$		0			0
$205$		−	109238.i		−	2.59936i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	8688.23	−	43820.6i	0.198902	−	1.00320i
$210$		0			0
$211$		−	38298.8i		−	0.860241i	−0.902772	−	0.430120i	$-0.858471\pi$
$211$		−	38298.8i		−	0.860241i	0.902772	−	0.430120i	$-0.141529\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−70187.6			−1.51839
$216$		0			0
$217$		−	63346.0i		−	1.34524i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	49595.2i		−	1.01544i
$222$		0			0
$223$			12575.8i			0.252886i	0.991974	+	0.126443i	$0.0403560\pi$
$223$			12575.8i			0.252886i	−0.991974	+	0.126443i	$0.959644\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			81997.6i			1.59129i	0.605762	+	0.795646i	$0.292868\pi$
$227$			81997.6i			1.59129i	−0.605762	+	0.795646i	$0.707132\pi$
$228$		0			0
$229$	−53290.0			−1.01619			−0.508095	−	0.861301i	$-0.669650\pi$
$229$	−53290.0			−1.01619			−0.508095	+	0.861301i	$0.669650\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−26060.8			−0.480039			−0.240020	−	0.970768i	$-0.577154\pi$
$233$	−26060.8			−0.480039			−0.240020	+	0.970768i	$0.577154\pi$
$234$		0			0
$235$	123126.			2.22954
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−46809.6			−0.819481			−0.409741	−	0.912202i	$-0.634381\pi$
$239$	−46809.6			−0.819481			−0.409741	+	0.912202i	$0.634381\pi$
$240$		0			0
$241$			22147.5i			0.381321i	0.981656	+	0.190661i	$0.0610630\pi$
$241$			22147.5i			0.381321i	−0.981656	+	0.190661i	$0.938937\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−16570.8			−0.276065
$246$		0			0
$247$	−70969.2	−	14070.9i	−1.16326	−	0.230637i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	28467.6			0.451860			0.225930	−	0.974144i	$-0.427458\pi$
$251$	28467.6			0.451860			0.225930	+	0.974144i	$0.427458\pi$
$252$		0			0
$253$	−107117.			−1.67347
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	41268.4i		−	0.624815i	−0.949948	−	0.312408i	$-0.898865\pi$
$257$		−	41268.4i		−	0.624815i	0.949948	−	0.312408i	$-0.101135\pi$
$258$		0			0
$259$			61520.7i			0.917111i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	81859.8			1.18348			0.591738	−	0.806131i	$-0.298442\pi$
$263$	81859.8			1.18348			0.591738	+	0.806131i	$0.298442\pi$
$264$		0			0
$265$		−	81647.1i		−	1.16265i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	56995.2i		−	0.787650i	−0.919185	−	0.393825i	$-0.871152\pi$
$269$		−	56995.2i		−	0.787650i	0.919185	−	0.393825i	$-0.128848\pi$
$270$		0			0
$271$	106818.			1.45448			0.727240	−	0.686383i	$-0.240803\pi$
$271$	106818.			1.45448			0.727240	+	0.686383i	$0.240803\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−83722.9			−1.10708
$276$		0			0
$277$	146892.			1.91443			0.957214	−	0.289380i	$-0.0934491\pi$
$277$	146892.			1.91443			0.957214	+	0.289380i	$0.0934491\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	38010.4i		−	0.481382i	−0.970602	−	0.240691i	$-0.922626\pi$
$281$		−	38010.4i		−	0.481382i	0.970602	−	0.240691i	$-0.0773740\pi$
$282$		0			0
$283$	−12866.6			−0.160654			−0.0803268	−	0.996769i	$-0.525596\pi$
$283$	−12866.6			−0.160654			−0.0803268	+	0.996769i	$0.525596\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			161939.i			1.96602i
$288$		0			0
$289$	−22284.8			−0.266816
$290$		0			0
$291$		0			0
$292$		0			0
$293$			27005.9i			0.314574i	0.987553	+	0.157287i	$0.0502748\pi$
$293$			27005.9i			0.314574i	−0.987553	+	0.157287i	$0.949725\pi$
$294$		0			0
$295$		−	11867.7i		−	0.136371i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			173481.i			1.94048i
$300$		0			0
$301$	104049.			1.14843
$302$		0			0
$303$		0			0
$304$		0			0
$305$	6227.61			0.0669456
$306$		0			0
$307$			88088.7i			0.934638i	0.884089	+	0.467319i	$0.154780\pi$
$307$			88088.7i			0.934638i	−0.884089	+	0.467319i	$0.845220\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	108976.			1.12670			0.563352	−	0.826217i	$-0.309511\pi$
$311$	108976.			1.12670			0.563352	+	0.826217i	$0.309511\pi$
$312$		0			0
$313$	34286.2			0.349970			0.174985	−	0.984571i	$-0.444012\pi$
$313$	34286.2			0.349970			0.174985	+	0.984571i	$0.444012\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			2519.32i			0.0250706i	0.999921	+	0.0125353i	$0.00399022\pi$
$317$			2519.32i			0.0250706i	−0.999921	+	0.0125353i	$0.996010\pi$
$318$		0			0
$319$			138720.i			1.36319i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	17373.7	−	87627.2i	0.166528	−	0.839912i
$324$		0			0
$325$			135593.i			1.28372i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−182527.			−1.68630
$330$		0			0
$331$			91819.5i			0.838067i	0.907971	+	0.419033i	$0.137631\pi$
$331$			91819.5i			0.838067i	−0.907971	+	0.419033i	$0.862369\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			122195.i			1.08884i
$336$		0			0
$337$		−	6523.34i		−	0.0574394i	−0.999588	−	0.0287197i	$-0.990857\pi$
$337$		−	6523.34i		−	0.0574394i	0.999588	−	0.0287197i	$-0.00914302\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			146574.i			1.26051i
$342$		0			0
$343$	−103845.			−0.882668
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−35064.9			−0.291215			−0.145608	−	0.989342i	$-0.546514\pi$
$347$	−35064.9			−0.291215			−0.145608	+	0.989342i	$0.546514\pi$
$348$		0			0
$349$	−196315.			−1.61177			−0.805884	−	0.592074i	$-0.798309\pi$
$349$	−196315.			−1.61177			−0.805884	+	0.592074i	$0.798309\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−43972.4			−0.352883			−0.176442	−	0.984311i	$-0.556459\pi$
$353$	−43972.4			−0.352883			−0.176442	+	0.984311i	$0.556459\pi$
$354$		0			0
$355$			96725.9i			0.767513i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	134519.			1.04375			0.521873	−	0.853023i	$-0.325234\pi$
$359$	134519.			1.04375			0.521873	+	0.853023i	$0.325234\pi$
$360$		0			0
$361$	−120463.	−	49722.4i	−0.924353	−	0.381538i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	336368.			2.52481
$366$		0			0
$367$	−163162.			−1.21140			−0.605698	−	0.795695i	$-0.707106\pi$
$367$	−163162.			−1.21140			−0.605698	+	0.795695i	$0.707106\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			121037.i			0.879365i
$372$		0			0
$373$		−	16732.3i		−	0.120265i	−0.998190	−	0.0601323i	$-0.980848\pi$
$373$		−	16732.3i		−	0.120265i	0.998190	−	0.0601323i	$-0.0191523\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	224663.			1.58070
$378$		0			0
$379$			74507.3i			0.518705i	0.965783	+	0.259353i	$0.0835092\pi$
$379$			74507.3i			0.518705i	−0.965783	+	0.259353i	$0.916491\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	90711.2i		−	0.618392i	−0.950998	−	0.309196i	$-0.899940\pi$
$383$		−	90711.2i		−	0.618392i	0.950998	−	0.309196i	$-0.100060\pi$
$384$		0			0
$385$	238771.			1.61087
$386$		0			0
$387$		0			0
$388$		0			0
$389$	50327.5			0.332588			0.166294	−	0.986076i	$-0.446820\pi$
$389$	50327.5			0.332588			0.166294	+	0.986076i	$0.446820\pi$
$390$		0			0
$391$	−214201.			−1.40109
$392$		0			0
$393$		0			0
$394$		0			0
$395$			436303.i			2.79636i
$396$		0			0
$397$	−5586.81			−0.0354473			−0.0177236	−	0.999843i	$-0.505642\pi$
$397$	−5586.81			−0.0354473			−0.0177236	+	0.999843i	$0.505642\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	238219.i		−	1.48145i	−0.671806	−	0.740727i	$-0.734481\pi$
$401$		−	238219.i		−	1.48145i	0.671806	−	0.740727i	$-0.265519\pi$
$402$		0			0
$403$	237382.			1.46163
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	142350.i		−	0.859348i
$408$		0			0
$409$			38177.5i			0.228224i	0.993468	+	0.114112i	$0.0364023\pi$
$409$			38177.5i			0.228224i	−0.993468	+	0.114112i	$0.963598\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			17593.1i			0.103144i
$414$		0			0
$415$	−40729.4			−0.236489
$416$		0			0
$417$		0			0
$418$		0			0
$419$	11321.6			0.0644882			0.0322441	−	0.999480i	$-0.489735\pi$
$419$	11321.6			0.0644882			0.0322441	+	0.999480i	$0.489735\pi$
$420$		0			0
$421$			182111.i			1.02748i	0.857947	+	0.513739i	$0.171740\pi$
$421$			182111.i			1.02748i	−0.857947	+	0.513739i	$0.828260\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−167419.			−0.926888
$426$		0			0
$427$	−9232.04			−0.0506340
$428$		0			0
$429$		0			0
$430$		0			0
$431$			125884.i			0.677668i	0.940846	+	0.338834i	$0.110033\pi$
$431$			125884.i			0.677668i	−0.940846	+	0.338834i	$0.889967\pi$
$432$		0			0
$433$		−	337413.i		−	1.79964i	−0.436258	−	0.899822i	$-0.643696\pi$
$433$		−	337413.i		−	1.79964i	0.436258	−	0.899822i	$-0.356304\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−60772.1	+	306515.i	−0.318230	+	1.60505i
$438$		0			0
$439$		−	79045.7i		−	0.410156i	−0.978746	−	0.205078i	$-0.934255\pi$
$439$		−	79045.7i		−	0.410156i	0.978746	−	0.205078i	$-0.0657448\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	158716.			0.808747			0.404373	−	0.914594i	$-0.367490\pi$
$443$	158716.			0.808747			0.404373	+	0.914594i	$0.367490\pi$
$444$		0			0
$445$			192630.i			0.972755i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			108280.i			0.537099i	0.963266	+	0.268550i	$0.0865443\pi$
$449$			108280.i			0.537099i	−0.963266	+	0.268550i	$0.913456\pi$
$450$		0			0
$451$		−	374703.i		−	1.84219i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	386699.i		−	1.86789i
$456$		0			0
$457$	−192296.			−0.920744			−0.460372	−	0.887726i	$-0.652284\pi$
$457$	−192296.			−0.920744			−0.460372	+	0.887726i	$0.652284\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	155638.			0.732340			0.366170	−	0.930548i	$-0.380669\pi$
$461$	155638.			0.732340			0.366170	+	0.930548i	$0.380669\pi$
$462$		0			0
$463$	244017.			1.13830			0.569152	−	0.822232i	$-0.307272\pi$
$463$	244017.			1.13830			0.569152	+	0.822232i	$0.307272\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−304959.			−1.39832			−0.699162	−	0.714963i	$-0.746443\pi$
$467$	−304959.			−1.39832			−0.699162	+	0.714963i	$0.746443\pi$
$468$		0			0
$469$		−	181146.i		−	0.823538i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−240754.			−1.07610
$474$		0			0
$475$	−47499.5	+	239572.i	−0.210524	+	1.06181i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	246386.			1.07385			0.536927	−	0.843629i	$-0.319585\pi$
$479$	246386.			1.07385			0.536927	+	0.843629i	$0.319585\pi$
$480$		0			0
$481$	−230542.			−0.996460
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	96105.9i		−	0.408570i
$486$		0			0
$487$		−	129143.i		−	0.544519i	−0.962224	−	0.272259i	$-0.912229\pi$
$487$		−	129143.i		−	0.544519i	0.962224	−	0.272259i	$-0.0877709\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	13985.3			0.0580110			0.0290055	−	0.999579i	$-0.490766\pi$
$491$	13985.3			0.0580110			0.0290055	+	0.999579i	$0.490766\pi$
$492$		0			0
$493$			277396.i			1.14132i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	143390.i		−	0.580506i
$498$		0			0
$499$	342933.			1.37724			0.688618	−	0.725125i	$-0.258218\pi$
$499$	342933.			1.37724			0.688618	+	0.725125i	$0.258218\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	120996.			0.478227			0.239114	−	0.970992i	$-0.423143\pi$
$503$	120996.			0.478227			0.239114	+	0.970992i	$0.423143\pi$
$504$		0			0
$505$	−350104.			−1.37282
$506$		0			0
$507$		0			0
$508$		0			0
$509$			224407.i			0.866166i	0.901354	+	0.433083i	$0.142574\pi$
$509$			224407.i			0.866166i	−0.901354	+	0.433083i	$0.857426\pi$
$510$		0			0
$511$	−498645.			−1.90963
$512$		0			0
$513$		0			0
$514$		0			0
$515$			603120.i			2.27399i
$516$		0			0
$517$	422341.			1.58009
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	446907.i		−	1.64642i	−0.567734	−	0.823212i	$-0.692180\pi$
$521$		−	446907.i		−	1.64642i	0.567734	−	0.823212i	$-0.307820\pi$
$522$		0			0
$523$		−	208236.i		−	0.761292i	−0.924721	−	0.380646i	$-0.875702\pi$
$523$		−	208236.i		−	0.761292i	0.924721	−	0.380646i	$-0.124298\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			293100.i			1.05535i
$528$		0			0
$529$	469421.			1.67745
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−606847.			−2.13612
$534$		0			0
$535$			718744.i			2.51112i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−56840.2			−0.195649
$540$		0			0
$541$	525092.			1.79408			0.897038	−	0.441954i	$-0.145715\pi$
$541$	525092.			1.79408			0.897038	+	0.441954i	$0.145715\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			544637.i			1.83364i
$546$		0			0
$547$			273469.i			0.913973i	0.889474	+	0.456986i	$0.151071\pi$
$547$			273469.i			0.913973i	−0.889474	+	0.456986i	$0.848929\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	396945.	+	78701.6i	1.30746	+	0.259227i
$552$		0			0
$553$		−	646792.i		−	2.11502i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−581097.			−1.87300			−0.936501	−	0.350666i	$-0.885955\pi$
$557$	−581097.			−1.87300			−0.936501	+	0.350666i	$0.885955\pi$
$558$		0			0
$559$			389911.i			1.24779i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	425951.i		−	1.34383i	−0.740630	−	0.671913i	$-0.765473\pi$
$563$		−	425951.i		−	1.34383i	0.740630	−	0.671913i	$-0.234527\pi$
$564$		0			0
$565$			194400.i			0.608974i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	25050.3i		−	0.0773729i	−0.999251	−	0.0386865i	$-0.987683\pi$
$569$		−	25050.3i		−	0.0773729i	0.999251	−	0.0386865i	$-0.0123174\pi$
$570$		0			0
$571$	−149988.			−0.460028			−0.230014	−	0.973187i	$-0.573877\pi$
$571$	−149988.			−0.460028			−0.230014	+	0.973187i	$0.573877\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	585622.			1.77126
$576$		0			0
$577$	275186.			0.826561			0.413280	−	0.910604i	$-0.364383\pi$
$577$	275186.			0.826561			0.413280	+	0.910604i	$0.364383\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	60378.8			0.178868
$582$		0			0
$583$		−	280061.i		−	0.823979i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−405498.			−1.17683			−0.588413	−	0.808560i	$-0.700247\pi$
$587$	−405498.			−1.17683			−0.588413	+	0.808560i	$0.700247\pi$
$588$		0			0
$589$	419418.	+	83157.2i	1.20897	+	0.239701i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−585690.			−1.66555			−0.832777	−	0.553609i	$-0.813250\pi$
$593$	−585690.			−1.66555			−0.832777	+	0.553609i	$0.813250\pi$
$594$		0			0
$595$	477465.			1.34868
$596$		0			0
$597$		0			0
$598$		0			0
$599$			630822.i			1.75814i	0.476693	+	0.879070i	$0.341835\pi$
$599$			630822.i			1.75814i	−0.476693	+	0.879070i	$0.658165\pi$
$600$		0			0
$601$		−	9339.76i		−	0.0258575i	−0.999916	−	0.0129288i	$-0.995885\pi$
$601$		−	9339.76i		−	0.0258575i	0.999916	−	0.0129288i	$-0.00411546\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−24277.7			−0.0663279
$606$		0			0
$607$		−	114083.i		−	0.309632i	−0.987943	−	0.154816i	$-0.950522\pi$
$607$		−	114083.i		−	0.309632i	0.987943	−	0.154816i	$-0.0494784\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	683999.i		−	1.83220i
$612$		0			0
$613$	238807.			0.635516			0.317758	−	0.948172i	$-0.397070\pi$
$613$	238807.			0.635516			0.317758	+	0.948172i	$0.397070\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	357356.			0.938708			0.469354	−	0.883010i	$-0.344487\pi$
$617$	357356.			0.938708			0.469354	+	0.883010i	$0.344487\pi$
$618$		0			0
$619$	−441626.			−1.15259			−0.576293	−	0.817243i	$-0.695501\pi$
$619$	−441626.			−1.15259			−0.576293	+	0.817243i	$0.695501\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	285562.i		−	0.735739i
$624$		0			0
$625$	−355748.			−0.910714
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	284655.i		−	0.719478i
$630$		0			0
$631$	−692824.			−1.74006			−0.870029	−	0.493000i	$-0.835900\pi$
$631$	−692824.			−1.74006			−0.870029	+	0.493000i	$0.835900\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			781608.i			1.93839i
$636$		0			0
$637$			92055.1i			0.226866i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	393624.i		−	0.958001i	−0.877815	−	0.479001i	$-0.840999\pi$
$641$		−	393624.i		−	0.958001i	0.877815	−	0.479001i	$-0.159001\pi$
$642$		0			0
$643$	106391.			0.257325			0.128663	−	0.991688i	$-0.458932\pi$
$643$	106391.			0.257325			0.128663	+	0.991688i	$0.458932\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−300617.			−0.718133			−0.359066	−	0.933312i	$-0.616905\pi$
$647$	−300617.			−0.718133			−0.359066	+	0.933312i	$0.616905\pi$
$648$		0			0
$649$		−	40708.0i		−	0.0966474i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	283673.			0.665259			0.332630	−	0.943058i	$-0.392064\pi$
$653$	283673.			0.665259			0.332630	+	0.943058i	$0.392064\pi$
$654$		0			0
$655$	95104.6			0.221676
$656$		0			0
$657$		0			0
$658$		0			0
$659$			5134.49i			0.0118230i	0.999983	+	0.00591149i	$0.00188170\pi$
$659$			5134.49i			0.0118230i	−0.999983	+	0.00591149i	$0.998118\pi$
$660$		0			0
$661$		−	617871.i		−	1.41415i	−0.707140	−	0.707074i	$-0.750015\pi$
$661$		−	617871.i		−	1.41415i	0.707140	−	0.707074i	$-0.249985\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	135465.	−	683238.i	0.306325	−	1.54500i
$666$		0			0
$667$		−	970314.i		−	2.18102i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	21361.6			0.0474449
$672$		0			0
$673$			720383.i			1.59050i	0.606283	+	0.795249i	$0.292660\pi$
$673$			720383.i			1.59050i	−0.606283	+	0.795249i	$0.707340\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	772887.i		−	1.68631i	−0.537667	−	0.843157i	$-0.680694\pi$
$677$		−	772887.i		−	1.68631i	0.537667	−	0.843157i	$-0.319306\pi$
$678$		0			0
$679$			142471.i			0.309020i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	398829.i		−	0.854959i	−0.904025	−	0.427480i	$-0.859402\pi$
$683$		−	398829.i		−	0.854959i	0.904025	−	0.427480i	$-0.140598\pi$
$684$		0			0
$685$	−821537.			−1.75084
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−453571.			−0.955448
$690$		0			0
$691$	140719.			0.294711			0.147355	−	0.989084i	$-0.452924\pi$
$691$	140719.			0.294711			0.147355	+	0.989084i	$0.452924\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−403556.			−0.835477
$696$		0			0
$697$		−	749287.i		−	1.54235i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	525885.			1.07017			0.535087	−	0.844797i	$-0.320279\pi$
$701$	525885.			1.07017			0.535087	+	0.844797i	$0.320279\pi$
$702$		0			0
$703$	−407332.	−	80761.1i	−0.824211	−	0.163415i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	519007.			1.03833
$708$		0			0
$709$	216268.			0.430230			0.215115	−	0.976589i	$-0.430987\pi$
$709$	216268.			0.430230			0.215115	+	0.976589i	$0.430987\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	1.02525e6i		−	2.01674i
$714$		0			0
$715$			894766.i			1.75024i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	628920.			1.21657			0.608285	−	0.793718i	$-0.291858\pi$
$719$	628920.			1.21657			0.608285	+	0.793718i	$0.291858\pi$
$720$		0			0
$721$		−	894088.i		−	1.71993i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	758397.i		−	1.44285i
$726$		0			0
$727$	315233.			0.596434			0.298217	−	0.954498i	$-0.403608\pi$
$727$	315233.			0.596434			0.298217	+	0.954498i	$0.403608\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−481431.			−0.900947
$732$		0			0
$733$	−856030.			−1.59324			−0.796620	−	0.604481i	$-0.793381\pi$
$733$	−856030.			−1.59324			−0.796620	+	0.604481i	$0.793381\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			419146.i			0.771668i
$738$		0			0
$739$	−453021.			−0.829525			−0.414762	−	0.909930i	$-0.636135\pi$
$739$	−453021.			−0.829525			−0.414762	+	0.909930i	$0.636135\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			643391.i			1.16546i	0.812666	+	0.582730i	$0.198016\pi$
$743$			643391.i			1.16546i	−0.812666	+	0.582730i	$0.801984\pi$
$744$		0			0
$745$	−162620.			−0.292996
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	1.06549e6i		−	1.89927i
$750$		0			0
$751$			282773.i			0.501370i	0.968069	+	0.250685i	$0.0806558\pi$
$751$			282773.i			0.501370i	−0.968069	+	0.250685i	$0.919344\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	79280.3i		−	0.139082i
$756$		0			0
$757$	194797.			0.339931			0.169965	−	0.985450i	$-0.445634\pi$
$757$	194797.			0.339931			0.169965	+	0.985450i	$0.445634\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−41877.9			−0.0723129			−0.0361565	−	0.999346i	$-0.511511\pi$
$761$	−41877.9			−0.0723129			−0.0361565	+	0.999346i	$0.511511\pi$
$762$		0			0
$763$		−	807391.i		−	1.38687i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−65928.2			−0.112068
$768$		0			0
$769$	668908.			1.13113			0.565566	−	0.824703i	$-0.308658\pi$
$769$	668908.			1.13113			0.565566	+	0.824703i	$0.308658\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			784341.i			1.31264i	0.754482	+	0.656320i	$0.227888\pi$
$773$			784341.i			1.31264i	−0.754482	+	0.656320i	$0.772112\pi$
$774$		0			0
$775$		−	801333.i		−	1.33416i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−1.07221e6	−	212585.i	−1.76687	−	0.350313i
$780$		0			0
$781$			331784.i			0.543943i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	1.45079e6			2.35432
$786$		0			0
$787$		−	1.06921e6i		−	1.72628i	−0.504963	−	0.863141i	$-0.668494\pi$
$787$		−	1.06921e6i		−	1.72628i	0.504963	−	0.863141i	$-0.331506\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	288186.i		−	0.460595i
$792$		0			0
$793$		−	34596.0i		−	0.0550149i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	218300.i		−	0.343666i	−0.985126	−	0.171833i	$-0.945031\pi$
$797$		−	218300.i		−	0.343666i	0.985126	−	0.171833i	$-0.0549690\pi$
$798$		0			0
$799$	844547.			1.32291
$800$		0			0
$801$		0			0
$802$		0			0
$803$	1.15379e6			1.78935
$804$		0			0
$805$	−1.67015e6			−2.57729
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−1347.17			−0.00205838			−0.00102919	−	0.999999i	$-0.500328\pi$
$809$	−1347.17			−0.00205838			−0.00102919	+	0.999999i	$0.500328\pi$
$810$		0			0
$811$			766781.i			1.16582i	0.812539	+	0.582908i	$0.198085\pi$
$811$			766781.i			1.16582i	−0.812539	+	0.582908i	$0.801915\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	645656.			0.972045
$816$		0			0
$817$	−136590.	+	688913.i	−0.204632	+	1.03210i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−1.06169e6			−1.57510			−0.787552	−	0.616248i	$-0.788652\pi$
$821$	−1.06169e6			−1.57510			−0.787552	+	0.616248i	$0.788652\pi$
$822$		0			0
$823$	−582030.			−0.859302			−0.429651	−	0.902995i	$-0.641363\pi$
$823$	−582030.			−0.859302			−0.429651	+	0.902995i	$0.641363\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	542914.i		−	0.793816i	−0.917858	−	0.396908i	$-0.870083\pi$
$827$		−	542914.i		−	0.793816i	0.917858	−	0.396908i	$-0.129917\pi$
$828$		0			0
$829$		−	23610.7i		−	0.0343558i	−0.999852	−	0.0171779i	$-0.994532\pi$
$829$		−	23610.7i		−	0.0343558i	0.999852	−	0.0171779i	$-0.00546817\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−113662.			−0.163805
$834$		0			0
$835$			1.35519e6i			1.94370i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	465771.i		−	0.661681i	−0.943687	−	0.330840i	$-0.892668\pi$
$839$		−	465771.i		−	0.661681i	0.943687	−	0.330840i	$-0.107332\pi$
$840$		0			0
$841$	−549302.			−0.776639
$842$		0			0
$843$		0			0
$844$		0			0
$845$	418716.			0.586416
$846$		0			0
$847$	35990.1			0.0501668
$848$		0			0
$849$		0			0
$850$		0			0
$851$			995706.i			1.37490i
$852$		0			0
$853$	671549.			0.922953			0.461476	−	0.887153i	$-0.347320\pi$
$853$	671549.			0.922953			0.461476	+	0.887153i	$0.347320\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			355473.i			0.483999i	0.970276	+	0.241999i	$0.0778032\pi$
$857$			355473.i			0.483999i	−0.970276	+	0.241999i	$0.922197\pi$
$858$		0			0
$859$	334322.			0.453084			0.226542	−	0.974001i	$-0.427258\pi$
$859$	334322.			0.453084			0.226542	+	0.974001i	$0.427258\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			339010.i			0.455188i	0.973756	+	0.227594i	$0.0730859\pi$
$863$			339010.i			0.455188i	−0.973756	+	0.227594i	$0.926914\pi$
$864$		0			0
$865$		−	169748.i		−	0.226868i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			1.49658e6i			1.98181i
$870$		0			0
$871$	678825.			0.894791
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−99466.9			−0.129916
$876$		0			0
$877$			882336.i			1.14719i	0.819140	+	0.573594i	$0.194451\pi$
$877$			882336.i			1.14719i	−0.819140	+	0.573594i	$0.805549\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−314197.			−0.404809			−0.202404	−	0.979302i	$-0.564876\pi$
$881$	−314197.			−0.404809			−0.202404	+	0.979302i	$0.564876\pi$
$882$		0			0
$883$	639126.			0.819719			0.409859	−	0.912149i	$-0.365578\pi$
$883$	639126.			0.819719			0.409859	+	0.912149i	$0.365578\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	807641.i		−	1.02653i	−0.858230	−	0.513265i	$-0.828436\pi$
$887$		−	807641.i		−	1.02653i	0.858230	−	0.513265i	$-0.171564\pi$
$888$		0			0
$889$		−	1.15868e6i		−	1.46609i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	239611.	−	1.20852e6i	0.300472	−	1.51548i
$894$		0			0
$895$		−	137288.i		−	0.171391i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−1.32772e6			−1.64281
$900$		0			0
$901$		−	560034.i		−	0.689866i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	1.71138e6i		−	2.08953i
$906$		0			0
$907$		−	1.57926e6i		−	1.91972i	−0.280479	−	0.959860i	$-0.590493\pi$
$907$		−	1.57926e6i		−	1.91972i	0.280479	−	0.959860i	$-0.409507\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			779982.i			0.939827i	0.882712	+	0.469914i	$0.155715\pi$
$911$			779982.i			0.939827i	−0.882712	+	0.469914i	$0.844285\pi$
$912$		0			0
$913$	−139708.			−0.167602
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−140987.			−0.167664
$918$		0			0
$919$	−1.37039e6			−1.62261			−0.811304	−	0.584624i	$-0.801242\pi$
$919$	−1.37039e6			−1.62261			−0.811304	+	0.584624i	$0.801242\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	537338.			0.630731
$924$		0			0
$925$			778243.i			0.909561i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−1.19573e6			−1.38548			−0.692741	−	0.721186i	$-0.743597\pi$
$929$	−1.19573e6			−1.38548			−0.692741	+	0.721186i	$0.743597\pi$
$930$		0			0
$931$	−32247.8	+	162647.i	−0.0372050	+	0.187650i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−1.10479e6			−1.26373
$936$		0			0
$937$	425588.			0.484742			0.242371	−	0.970184i	$-0.422075\pi$
$937$	425588.			0.484742			0.242371	+	0.970184i	$0.422075\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	701178.i		−	0.791861i	−0.918281	−	0.395930i	$-0.870422\pi$
$941$		−	701178.i		−	0.791861i	0.918281	−	0.395930i	$-0.129578\pi$
$942$		0			0
$943$			2.62096e6i			2.94739i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−1.10181e6			−1.22859			−0.614297	−	0.789075i	$-0.710560\pi$
$947$	−1.10181e6			−1.22859			−0.614297	+	0.789075i	$0.710560\pi$
$948$		0			0
$949$		−	1.86861e6i		−	2.07485i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			1.17507e6i			1.29383i	0.762562	+	0.646915i	$0.223941\pi$
$953$			1.17507e6i			1.29383i	−0.762562	+	0.646915i	$0.776059\pi$
$954$		0			0
$955$	1.31712e6			1.44417
$956$		0			0
$957$		0			0
$958$		0			0
$959$	1.21788e6			1.32424
$960$		0			0
$961$	−479371.			−0.519068
$962$		0			0
$963$		0			0
$964$		0			0
$965$			1.32778e6i			1.42584i
$966$		0			0
$967$	1.01498e6			1.08544			0.542721	−	0.839913i	$-0.317394\pi$
$967$	1.01498e6			1.08544			0.542721	+	0.839913i	$0.317394\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	206763.i		−	0.219298i	−0.993970	−	0.109649i	$-0.965027\pi$
$971$		−	206763.i		−	0.219298i	0.993970	−	0.109649i	$-0.0349726\pi$
$972$		0			0
$973$	598247.			0.631909
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	777874.i		−	0.814930i	−0.913221	−	0.407465i	$-0.866413\pi$
$977$		−	777874.i		−	0.814930i	0.913221	−	0.407465i	$-0.133587\pi$
$978$		0			0
$979$			660749.i			0.689399i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			1.16284e6i			1.20341i	0.798718	+	0.601705i	$0.205512\pi$
$983$			1.16284e6i			1.20341i	−0.798718	+	0.601705i	$0.794488\pi$
$984$		0			0
$985$	1.37651e6			1.41875
$986$		0			0
$987$		0			0
$988$		0			0
$989$	1.68402e6			1.72169
$990$		0			0
$991$			1.08199e6i			1.10174i	0.834593	+	0.550868i	$0.185703\pi$
$991$			1.08199e6i			1.10174i	−0.834593	+	0.550868i	$0.814297\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−934680.			−0.944097
$996$		0			0
$997$	194370.			0.195542			0.0977710	−	0.995209i	$-0.468829\pi$
$997$	194370.			0.195542			0.0977710	+	0.995209i	$0.468829\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	684.5.h.f.37.4		14
3.2	odd	2		228.5.h.a.37.6	✓	14
12.11	even	2		912.5.o.c.721.13		14
19.18	odd	2	inner	684.5.h.f.37.3		14
57.56	even	2		228.5.h.a.37.13	yes	14
228.227	odd	2		912.5.o.c.721.6		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
228.5.h.a.37.6	✓	14	3.2	odd	2
228.5.h.a.37.13	yes	14	57.56	even	2
684.5.h.f.37.3		14	19.18	odd	2	inner
684.5.h.f.37.4		14	1.1	even	1	trivial
912.5.o.c.721.6		14	228.227	odd	2
912.5.o.c.721.13		14	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	684.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-36.0770 q^{5} +53.4819 q^{7} +O(q^{10})\)	\(q-36.0770 q^{5} +53.4819 q^{7} -123.750 q^{11} +200.417i q^{13} -247.460 q^{17} +(-70.2082 + 354.107i) q^{19} +865.599 q^{23} +676.551 q^{25} -1120.97i q^{29} -1184.44i q^{31} -1929.47 q^{35} +1150.31i q^{37} +3027.92i q^{41} +1945.49 q^{43} -3412.87 q^{47} +459.317 q^{49} +2263.13i q^{53} +4464.51 q^{55} +328.955i q^{59} -172.620 q^{61} -7230.47i q^{65} -3387.06i q^{67} -2681.09i q^{71} -9323.61 q^{73} -6618.36 q^{77} -12093.6i q^{79} +1128.96 q^{83} +8927.60 q^{85} -5339.41i q^{89} +10718.7i q^{91} +(2532.90 - 12775.1i) q^{95} +2663.91i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 18 q^{5} - 86 q^{7}+O(q^{10}) \) 14 * q - 18 * q^5 - 86 * q^7	\( 14 q - 18 q^{5} - 86 q^{7} - 258 q^{11} - 498 q^{17} + 170 q^{19} + 588 q^{23} + 1560 q^{25} - 534 q^{35} + 1882 q^{43} + 222 q^{47} + 4104 q^{49} + 2702 q^{55} - 2462 q^{61} - 5774 q^{73} + 4578 q^{77} - 17988 q^{83} + 2342 q^{85} + 18270 q^{95}+O(q^{100}) \) 14 * q - 18 * q^5 - 86 * q^7 - 258 * q^11 - 498 * q^17 + 170 * q^19 + 588 * q^23 + 1560 * q^25 - 534 * q^35 + 1882 * q^43 + 222 * q^47 + 4104 * q^49 + 2702 * q^55 - 2462 * q^61 - 5774 * q^73 + 4578 * q^77 - 17988 * q^83 + 2342 * q^85 + 18270 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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