LMFDB - Embedded newform 6900.2.a.bc.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6900,2,Mod(1,6900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6900, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6900.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6900.a (trivial)

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:	yes
Analytic conductor:	$55.0967773947$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 13x^{5} + 11x^{4} + 46x^{3} - 32x^{2} - 30x - 2 $ x^7 - x^6 - 13x^5 + 11x^4 + 46x^3 - 32x^2 - 30*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1380)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.04690$ of defining polynomial
Character	$\chi$	$=$	6900.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -0.189781 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -0.189781 q^{7} +1.00000 q^{9} +4.33196 q^{11} +4.67083 q^{13} +0.0397340 q^{17} +7.19716 q^{19} +0.189781 q^{21} +1.00000 q^{23} -1.00000 q^{27} +6.49707 q^{29} +9.17655 q^{31} -4.33196 q^{33} -3.03881 q^{37} -4.67083 q^{39} -8.69330 q^{41} +7.83269 q^{43} -5.03127 q^{47} -6.96398 q^{49} -0.0397340 q^{51} +6.96306 q^{53} -7.19716 q^{57} +4.64368 q^{59} +7.86428 q^{61} -0.189781 q^{63} -2.52865 q^{67} -1.00000 q^{69} -1.70964 q^{71} +6.29127 q^{73} -0.822124 q^{77} -9.82056 q^{79} +1.00000 q^{81} +0.417601 q^{83} -6.49707 q^{87} -3.92081 q^{89} -0.886435 q^{91} -9.17655 q^{93} -0.0584885 q^{97} +4.33196 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 7 q^{3} + q^{7} + 7 q^{9}+O(q^{10}) $ 7 * q - 7 * q^3 + q^7 + 7 * q^9	$ 7 q - 7 q^{3} + q^{7} + 7 q^{9} + 2 q^{13} - q^{17} - 2 q^{19} - q^{21} + 7 q^{23} - 7 q^{27} + 15 q^{29} + 3 q^{31} - 19 q^{37} - 2 q^{39} + 23 q^{41} + 4 q^{43} - 10 q^{47} + 10 q^{49} + q^{51} - 11 q^{53} + 2 q^{57} + 5 q^{59} + 32 q^{61} + q^{63} - 15 q^{67} - 7 q^{69} + 21 q^{71} + 18 q^{73} - 28 q^{77} + 16 q^{79} + 7 q^{81} - 25 q^{83} - 15 q^{87} + 26 q^{89} + 14 q^{91} - 3 q^{93} - 6 q^{97}+O(q^{100}) $ 7 * q - 7 * q^3 + q^7 + 7 * q^9 + 2 * q^13 - q^17 - 2 * q^19 - q^21 + 7 * q^23 - 7 * q^27 + 15 * q^29 + 3 * q^31 - 19 * q^37 - 2 * q^39 + 23 * q^41 + 4 * q^43 - 10 * q^47 + 10 * q^49 + q^51 - 11 * q^53 + 2 * q^57 + 5 * q^59 + 32 * q^61 + q^63 - 15 * q^67 - 7 * q^69 + 21 * q^71 + 18 * q^73 - 28 * q^77 + 16 * q^79 + 7 * q^81 - 25 * q^83 - 15 * q^87 + 26 * q^89 + 14 * q^91 - 3 * q^93 - 6 * q^97

Display 100 coefficients 10 coefficients

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−0.189781	−0.0717305	−0.0358652	−	0.999357i	$-0.511419\pi$
$7$	−0.189781	−0.0717305	−0.0358652	+	0.999357i	$0.511419\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.33196	1.30613	0.653067	−	0.757300i	$-0.273482\pi$
$11$	4.33196	1.30613	0.653067	+	0.757300i	$0.273482\pi$
$12$	0	0
$13$	4.67083	1.29546	0.647728	−	0.761872i	$-0.275719\pi$
$13$	4.67083	1.29546	0.647728	+	0.761872i	$0.275719\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.0397340	0.00963691	0.00481845	−	0.999988i	$-0.498466\pi$
$17$	0.0397340	0.00963691	0.00481845	+	0.999988i	$0.498466\pi$
$18$	0	0
$19$	7.19716	1.65114	0.825571	−	0.564298i	$-0.190853\pi$
$19$	7.19716	1.65114	0.825571	+	0.564298i	$0.190853\pi$
$20$	0	0
$21$	0.189781	0.0414136
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.00000	−0.192450
$28$	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
$33$	−4.33196	−0.754097
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−3.03881	−0.499577	−0.249788	−	0.968300i	$-0.580361\pi$
$37$	−3.03881	−0.499577	−0.249788	+	0.968300i	$0.580361\pi$
$38$	0	0
$39$	−4.67083	−0.747932
$40$	0	0
$41$	−8.69330	−1.35767	−0.678833	−	0.734293i	$-0.737514\pi$
$41$	−8.69330	−1.35767	−0.678833	+	0.734293i	$0.737514\pi$
$42$	0	0
$43$	7.83269	1.19447	0.597237	−	0.802065i	$-0.296265\pi$
$43$	7.83269	1.19447	0.597237	+	0.802065i	$0.296265\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−5.03127	−0.733886	−0.366943	−	0.930243i	$-0.619596\pi$
$47$	−5.03127	−0.733886	−0.366943	+	0.930243i	$0.619596\pi$
$48$	0	0
$49$	−6.96398	−0.994855
$50$	0	0
$51$	−0.0397340	−0.00556387
$52$	0	0
$53$	6.96306	0.956449	0.478225	−	0.878238i	$-0.341280\pi$
$53$	6.96306	0.956449	0.478225	+	0.878238i	$0.341280\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−7.19716	−0.953287
$58$	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
$63$	−0.189781	−0.0239102
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−2.52865	−0.308924	−0.154462	−	0.987999i	$-0.549364\pi$
$67$	−2.52865	−0.308924	−0.154462	+	0.987999i	$0.549364\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−1.70964	−0.202897	−0.101448	−	0.994841i	$-0.532348\pi$
$71$	−1.70964	−0.202897	−0.101448	+	0.994841i	$0.532348\pi$
$72$	0	0
$73$	6.29127	0.736337	0.368169	−	0.929759i	$-0.379985\pi$
$73$	6.29127	0.736337	0.368169	+	0.929759i	$0.379985\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−0.822124	−0.0936897
$78$	0	0
$79$	−9.82056	−1.10490	−0.552450	−	0.833546i	$-0.686307\pi$
$79$	−9.82056	−1.10490	−0.552450	+	0.833546i	$0.686307\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	0.417601	0.0458377	0.0229188	−	0.999737i	$-0.492704\pi$
$83$	0.417601	0.0458377	0.0229188	+	0.999737i	$0.492704\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−6.49707	−0.696559
$88$	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
$93$	−9.17655	−0.951564
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−0.0584885	−0.00593861	−0.00296930	−	0.999996i	$-0.500945\pi$
$97$	−0.0584885	−0.00593861	−0.00296930	+	0.999996i	$0.500945\pi$
$98$	0	0
$99$	4.33196	0.435378
$100$	0	0
$101$	−6.61569	−0.658286	−0.329143	−	0.944280i	$-0.606760\pi$
$101$	−6.61569	−0.658286	−0.329143	+	0.944280i	$0.606760\pi$
$102$	0	0
$103$	−16.9236	−1.66753	−0.833766	−	0.552118i	$-0.813820\pi$
$103$	−16.9236	−1.66753	−0.833766	+	0.552118i	$0.813820\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−17.5775	−1.69928	−0.849639	−	0.527364i	$-0.823180\pi$
$107$	−17.5775	−1.69928	−0.849639	+	0.527364i	$0.823180\pi$
$108$	0	0
$109$	−0.243838	−0.0233554	−0.0116777	−	0.999932i	$-0.503717\pi$
$109$	−0.243838	−0.0233554	−0.0116777	+	0.999932i	$0.503717\pi$
$110$	0	0
$111$	3.03881	0.288431
$112$	0	0
$113$	−1.22452	−0.115194	−0.0575968	−	0.998340i	$-0.518344\pi$
$113$	−1.22452	−0.115194	−0.0575968	+	0.998340i	$0.518344\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	4.67083	0.431819
$118$	0	0
$119$	−0.00754075	−0.000691260 0
$120$	0	0
$121$	7.76587	0.705988
$122$	0	0
$123$	8.69330	0.783849
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−20.6764	−1.83474	−0.917368	−	0.398041i	$-0.869690\pi$
$127$	−20.6764	−1.83474	−0.917368	+	0.398041i	$0.869690\pi$
$128$	0	0
$129$	−7.83269	−0.689630
$130$	0	0
$131$	18.3253	1.60109	0.800546	−	0.599272i	$-0.204543\pi$
$131$	18.3253	1.60109	0.800546	+	0.599272i	$0.204543\pi$
$132$	0	0
$133$	−1.36588	−0.118437
$134$	0	0
$135$	0	0
$136$	0	0
$137$	3.46899	0.296376	0.148188	−	0.988959i	$-0.452656\pi$
$137$	3.46899	0.296376	0.148188	+	0.988959i	$0.452656\pi$
$138$	0	0
$139$	6.40878	0.543585	0.271793	−	0.962356i	$-0.412383\pi$
$139$	6.40878	0.543585	0.271793	+	0.962356i	$0.412383\pi$
$140$	0	0
$141$	5.03127	0.423709
$142$	0	0
$143$	20.2339	1.69204
$144$	0	0
$145$	0	0
$146$	0	0
$147$	6.96398	0.574380
$148$	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
$153$	0.0397340	0.00321230
$154$	0	0
$155$	0	0
$156$	0	0
$157$	18.9563	1.51287	0.756437	−	0.654066i	$-0.226938\pi$
$157$	18.9563	1.51287	0.756437	+	0.654066i	$0.226938\pi$
$158$	0	0
$159$	−6.96306	−0.552206
$160$	0	0
$161$	−0.189781	−0.0149568
$162$	0	0
$163$	−13.2785	−1.04005	−0.520026	−	0.854150i	$-0.674078\pi$
$163$	−13.2785	−1.04005	−0.520026	+	0.854150i	$0.674078\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−13.9737	−1.08132	−0.540659	−	0.841242i	$-0.681825\pi$
$167$	−13.9737	−1.08132	−0.540659	+	0.841242i	$0.681825\pi$
$168$	0	0
$169$	8.81667	0.678206
$170$	0	0
$171$	7.19716	0.550381
$172$	0	0
$173$	−12.7917	−0.972534	−0.486267	−	0.873810i	$-0.661642\pi$
$173$	−12.7917	−0.972534	−0.486267	+	0.873810i	$0.661642\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−4.64368	−0.349040
$178$	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
$183$	−7.86428	−0.581344
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0.172126	0.0125871
$188$	0	0
$189$	0.189781	0.0138045
$190$	0	0
$191$	−20.3680	−1.47378	−0.736890	−	0.676013i	$-0.763706\pi$
$191$	−20.3680	−1.47378	−0.736890	+	0.676013i	$0.763706\pi$
$192$	0	0
$193$	15.8592	1.14157	0.570786	−	0.821099i	$-0.306639\pi$
$193$	15.8592	1.14157	0.570786	+	0.821099i	$0.306639\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	6.16978	0.439579	0.219789	−	0.975547i	$-0.429463\pi$
$197$	6.16978	0.439579	0.219789	+	0.975547i	$0.429463\pi$
$198$	0	0
$199$	−4.81534	−0.341350	−0.170675	−	0.985327i	$-0.554595\pi$
$199$	−4.81534	−0.341350	−0.170675	+	0.985327i	$0.554595\pi$
$200$	0	0
$201$	2.52865	0.178358
$202$	0	0
$203$	−1.23302	−0.0865411
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1.00000	0.0695048
$208$	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
$213$	1.70964	0.117143
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−1.74154	−0.118223
$218$	0	0
$219$	−6.29127	−0.425125
$220$	0	0
$221$	0.185591	0.0124842
$222$	0	0
$223$	−11.4215	−0.764840	−0.382420	−	0.923989i	$-0.624909\pi$
$223$	−11.4215	−0.764840	−0.382420	+	0.923989i	$0.624909\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	15.7360	1.04443	0.522217	−	0.852813i	$-0.325105\pi$
$227$	15.7360	1.04443	0.522217	+	0.852813i	$0.325105\pi$
$228$	0	0
$229$	3.46335	0.228865	0.114432	−	0.993431i	$-0.463495\pi$
$229$	3.46335	0.228865	0.114432	+	0.993431i	$0.463495\pi$
$230$	0	0
$231$	0.822124	0.0540918
$232$	0	0
$233$	−0.319746	−0.0209473	−0.0104736	−	0.999945i	$-0.503334\pi$
$233$	−0.319746	−0.0209473	−0.0104736	+	0.999945i	$0.503334\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	9.82056	0.637914
$238$	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
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	33.6167	2.13898
$248$	0	0
$249$	−0.417601	−0.0264644
$250$	0	0
$251$	−3.76059	−0.237366	−0.118683	−	0.992932i	$-0.537867\pi$
$251$	−3.76059	−0.237366	−0.118683	+	0.992932i	$0.537867\pi$
$252$	0	0
$253$	4.33196	0.272348
$254$	0	0
$255$	0	0
$256$	0	0
$257$	26.1810	1.63312	0.816561	−	0.577259i	$-0.195878\pi$
$257$	26.1810	1.63312	0.816561	+	0.577259i	$0.195878\pi$
$258$	0	0
$259$	0.576708	0.0358349
$260$	0	0
$261$	6.49707	0.402158
$262$	0	0
$263$	−22.6844	−1.39878	−0.699391	−	0.714739i	$-0.746545\pi$
$263$	−22.6844	−1.39878	−0.699391	+	0.714739i	$0.746545\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	3.92081	0.239950
$268$	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
$273$	0.886435	0.0536495
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−12.5514	−0.754141	−0.377070	−	0.926185i	$-0.623069\pi$
$277$	−12.5514	−0.754141	−0.377070	+	0.926185i	$0.623069\pi$
$278$	0	0
$279$	9.17655	0.549386
$280$	0	0
$281$	21.7956	1.30022	0.650108	−	0.759842i	$-0.274724\pi$
$281$	21.7956	1.30022	0.650108	+	0.759842i	$0.274724\pi$
$282$	0	0
$283$	24.0765	1.43120	0.715601	−	0.698510i	$-0.246153\pi$
$283$	24.0765	1.43120	0.715601	+	0.698510i	$0.246153\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.64982	0.0973860
$288$	0	0
$289$	−16.9984	−0.999907
$290$	0	0
$291$	0.0584885	0.00342866
$292$	0	0
$293$	−27.2212	−1.59028	−0.795139	−	0.606427i	$-0.792602\pi$
$293$	−27.2212	−1.59028	−0.795139	+	0.606427i	$0.792602\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−4.33196	−0.251366
$298$	0	0
$299$	4.67083	0.270121
$300$	0	0
$301$	−1.48650	−0.0856802
$302$	0	0
$303$	6.61569	0.380061
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−6.48076	−0.369877	−0.184938	−	0.982750i	$-0.559209\pi$
$307$	−6.48076	−0.369877	−0.184938	+	0.982750i	$0.559209\pi$
$308$	0	0
$309$	16.9236	0.962750
$310$	0	0
$311$	4.11457	0.233316	0.116658	−	0.993172i	$-0.462782\pi$
$311$	4.11457	0.233316	0.116658	+	0.993172i	$0.462782\pi$
$312$	0	0
$313$	−0.760034	−0.0429597	−0.0214798	−	0.999769i	$-0.506838\pi$
$313$	−0.760034	−0.0429597	−0.0214798	+	0.999769i	$0.506838\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−7.48188	−0.420224	−0.210112	−	0.977677i	$-0.567383\pi$
$317$	−7.48188	−0.420224	−0.210112	+	0.977677i	$0.567383\pi$
$318$	0	0
$319$	28.1450	1.57582
$320$	0	0
$321$	17.5775	0.981079
$322$	0	0
$323$	0.285972	0.0159119
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0.243838	0.0134843
$328$	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
$333$	−3.03881	−0.166526
$334$	0	0
$335$	0	0
$336$	0	0
$337$	22.9098	1.24798	0.623988	−	0.781434i	$-0.285512\pi$
$337$	22.9098	1.24798	0.623988	+	0.781434i	$0.285512\pi$
$338$	0	0
$339$	1.22452	0.0665071
$340$	0	0
$341$	39.7525	2.15272
$342$	0	0
$343$	2.65010	0.143092
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−10.7314	−0.576090	−0.288045	−	0.957617i	$-0.593005\pi$
$347$	−10.7314	−0.576090	−0.288045	+	0.957617i	$0.593005\pi$
$348$	0	0
$349$	−9.67523	−0.517903	−0.258952	−	0.965890i	$-0.583377\pi$
$349$	−9.67523	−0.517903	−0.258952	+	0.965890i	$0.583377\pi$
$350$	0	0
$351$	−4.67083	−0.249311
$352$	0	0
$353$	−29.4875	−1.56946	−0.784730	−	0.619838i	$-0.787198\pi$
$353$	−29.4875	−1.56946	−0.784730	+	0.619838i	$0.787198\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0.00754075	0.000399099 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
$363$	−7.76587	−0.407602
$364$	0	0
$365$	0	0
$366$	0	0
$367$	8.15285	0.425576	0.212788	−	0.977098i	$-0.431746\pi$
$367$	8.15285	0.425576	0.212788	+	0.977098i	$0.431746\pi$
$368$	0	0
$369$	−8.69330	−0.452555
$370$	0	0
$371$	−1.32146	−0.0686066
$372$	0	0
$373$	28.9482	1.49888	0.749442	−	0.662070i	$-0.230322\pi$
$373$	28.9482	1.49888	0.749442	+	0.662070i	$0.230322\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	30.3467	1.56294
$378$	0	0
$379$	2.46596	0.126668	0.0633338	−	0.997992i	$-0.479827\pi$
$379$	2.46596	0.126668	0.0633338	+	0.997992i	$0.479827\pi$
$380$	0	0
$381$	20.6764	1.05928
$382$	0	0
$383$	−19.0579	−0.973813	−0.486906	−	0.873454i	$-0.661875\pi$
$383$	−19.0579	−0.973813	−0.486906	+	0.873454i	$0.661875\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	7.83269	0.398158
$388$	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
$393$	−18.3253	−0.924391
$394$	0	0
$395$	0	0
$396$	0	0
$397$	0.171284	0.00859652	0.00429826	−	0.999991i	$-0.498632\pi$
$397$	0.171284	0.00859652	0.00429826	+	0.999991i	$0.498632\pi$
$398$	0	0
$399$	1.36588	0.0683798
$400$	0	0
$401$	−22.7490	−1.13603	−0.568015	−	0.823018i	$-0.692289\pi$
$401$	−22.7490	−1.13603	−0.568015	+	0.823018i	$0.692289\pi$
$402$	0	0
$403$	42.8621	2.13512
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−13.1640	−0.652515
$408$	0	0
$409$	−11.0110	−0.544460	−0.272230	−	0.962232i	$-0.587761\pi$
$409$	−11.0110	−0.544460	−0.272230	+	0.962232i	$0.587761\pi$
$410$	0	0
$411$	−3.46899	−0.171113
$412$	0	0
$413$	−0.881282	−0.0433651
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−6.40878	−0.313839
$418$	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
$423$	−5.03127	−0.244629
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−1.49249	−0.0722267
$428$	0	0
$429$	−20.2339	−0.976900
$430$	0	0
$431$	27.0949	1.30512	0.652558	−	0.757738i	$-0.273696\pi$
$431$	27.0949	1.30512	0.652558	+	0.757738i	$0.273696\pi$
$432$	0	0
$433$	−35.7515	−1.71811	−0.859053	−	0.511887i	$-0.828947\pi$
$433$	−35.7515	−1.71811	−0.859053	+	0.511887i	$0.828947\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	7.19716	0.344287
$438$	0	0
$439$	−27.8294	−1.32822	−0.664112	−	0.747633i	$-0.731190\pi$
$439$	−27.8294	−1.32822	−0.664112	+	0.747633i	$0.731190\pi$
$440$	0	0
$441$	−6.96398	−0.331618
$442$	0	0
$443$	8.72908	0.414731	0.207366	−	0.978264i	$-0.433511\pi$
$443$	8.72908	0.414731	0.207366	+	0.978264i	$0.433511\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−13.9989	−0.662126
$448$	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
$453$	−14.5758	−0.684831
$454$	0	0
$455$	0	0
$456$	0	0
$457$	27.4809	1.28550	0.642752	−	0.766074i	$-0.277793\pi$
$457$	27.4809	1.28550	0.642752	+	0.766074i	$0.277793\pi$
$458$	0	0
$459$	−0.0397340	−0.00185462
$460$	0	0
$461$	25.3722	1.18170	0.590850	−	0.806782i	$-0.298793\pi$
$461$	25.3722	1.18170	0.590850	+	0.806782i	$0.298793\pi$
$462$	0	0
$463$	32.2448	1.49854	0.749271	−	0.662264i	$-0.230404\pi$
$463$	32.2448	1.49854	0.749271	+	0.662264i	$0.230404\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−0.845923	−0.0391446	−0.0195723	−	0.999808i	$-0.506230\pi$
$467$	−0.845923	−0.0391446	−0.0195723	+	0.999808i	$0.506230\pi$
$468$	0	0
$469$	0.479891	0.0221593
$470$	0	0
$471$	−18.9563	−0.873459
$472$	0	0
$473$	33.9309	1.56014
$474$	0	0
$475$	0	0
$476$	0	0
$477$	6.96306	0.318816
$478$	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
$483$	0.189781	0.00863534
$484$	0	0
$485$	0	0
$486$	0	0
$487$	17.2251	0.780544	0.390272	−	0.920700i	$-0.372381\pi$
$487$	17.2251	0.780544	0.390272	+	0.920700i	$0.372381\pi$
$488$	0	0
$489$	13.2785	0.600474
$490$	0	0
$491$	−9.41331	−0.424817	−0.212408	−	0.977181i	$-0.568131\pi$
$491$	−9.41331	−0.424817	−0.212408	+	0.977181i	$0.568131\pi$
$492$	0	0
$493$	0.258154	0.0116267
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0.324457	0.0145539
$498$	0	0
$499$	−9.17655	−0.410799	−0.205399	−	0.978678i	$-0.565849\pi$
$499$	−9.17655	−0.410799	−0.205399	+	0.978678i	$0.565849\pi$
$500$	0	0
$501$	13.9737	0.624300
$502$	0	0
$503$	−5.02474	−0.224042	−0.112021	−	0.993706i	$-0.535732\pi$
$503$	−5.02474	−0.224042	−0.112021	+	0.993706i	$0.535732\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−8.81667	−0.391562
$508$	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
$513$	−7.19716	−0.317762
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−21.7952	−0.958554
$518$	0	0
$519$	12.7917	0.561493
$520$	0	0
$521$	−25.4727	−1.11598	−0.557991	−	0.829847i	$-0.688427\pi$
$521$	−25.4727	−1.11598	−0.557991	+	0.829847i	$0.688427\pi$
$522$	0	0
$523$	15.7464	0.688543	0.344271	−	0.938870i	$-0.388126\pi$
$523$	15.7464	0.688543	0.344271	+	0.938870i	$0.388126\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0.364621	0.0158831
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	4.64368	0.201518
$532$	0	0
$533$	−40.6050	−1.75880
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−0.0511629	−0.00220784
$538$	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
$543$	−21.7448	−0.933161
$544$	0	0
$545$	0	0
$546$	0	0
$547$	13.9544	0.596648	0.298324	−	0.954465i	$-0.403572\pi$
$547$	13.9544	0.596648	0.298324	+	0.954465i	$0.403572\pi$
$548$	0	0
$549$	7.86428	0.335639
$550$	0	0
$551$	46.7605	1.99206
$552$	0	0
$553$	1.86376	0.0792550
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−30.6343	−1.29802	−0.649009	−	0.760781i	$-0.724816\pi$
$557$	−30.6343	−1.29802	−0.649009	+	0.760781i	$0.724816\pi$
$558$	0	0
$559$	36.5852	1.54739
$560$	0	0
$561$	−0.172126	−0.00726716
$562$	0	0
$563$	−40.9163	−1.72442	−0.862209	−	0.506553i	$-0.830920\pi$
$563$	−40.9163	−1.72442	−0.862209	+	0.506553i	$0.830920\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−0.189781	−0.00797005
$568$	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
$573$	20.3680	0.850887
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−15.7617	−0.656168	−0.328084	−	0.944648i	$-0.606403\pi$
$577$	−15.7617	−0.656168	−0.328084	+	0.944648i	$0.606403\pi$
$578$	0	0
$579$	−15.8592	−0.659086
$580$	0	0
$581$	−0.0792528	−0.00328796
$582$	0	0
$583$	30.1637	1.24925
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−40.4757	−1.67061	−0.835305	−	0.549787i	$-0.814709\pi$
$587$	−40.4757	−1.67061	−0.835305	+	0.549787i	$0.814709\pi$
$588$	0	0
$589$	66.0451	2.72134
$590$	0	0
$591$	−6.16978	−0.253791
$592$	0	0
$593$	35.8541	1.47235	0.736175	−	0.676791i	$-0.236630\pi$
$593$	35.8541	1.47235	0.736175	+	0.676791i	$0.236630\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	4.81534	0.197079
$598$	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
$603$	−2.52865	−0.102975
$604$	0	0
$605$	0	0
$606$	0	0
$607$	27.5809	1.11947	0.559736	−	0.828671i	$-0.310903\pi$
$607$	27.5809	1.11947	0.559736	+	0.828671i	$0.310903\pi$
$608$	0	0
$609$	1.23302	0.0499645
$610$	0	0
$611$	−23.5002	−0.950716
$612$	0	0
$613$	21.7384	0.878005	0.439003	−	0.898486i	$-0.355332\pi$
$613$	21.7384	0.878005	0.439003	+	0.898486i	$0.355332\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−25.2023	−1.01461	−0.507304	−	0.861767i	$-0.669358\pi$
$617$	−25.2023	−1.01461	−0.507304	+	0.861767i	$0.669358\pi$
$618$	0	0
$619$	6.45053	0.259269	0.129634	−	0.991562i	$-0.458620\pi$
$619$	6.45053	0.259269	0.129634	+	0.991562i	$0.458620\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	0.744096	0.0298116
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−31.1778	−1.24512
$628$	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
$633$	23.9874	0.953412
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−32.5276	−1.28879
$638$	0	0
$639$	−1.70964	−0.0676323
$640$	0	0
$641$	0.0204978	0.000809615 0	0.000404807	−	1.00000i	$-0.499871\pi$
$641$	0.0204978	0.000809615 0	0.000404807		1.00000i	$0.499871\pi$
$642$	0	0
$643$	46.4037	1.82998	0.914992	−	0.403471i	$-0.132196\pi$
$643$	46.4037	1.82998	0.914992	+	0.403471i	$0.132196\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	16.8386	0.661992	0.330996	−	0.943632i	$-0.392615\pi$
$647$	16.8386	0.661992	0.330996	+	0.943632i	$0.392615\pi$
$648$	0	0
$649$	20.1162	0.789631
$650$	0	0
$651$	1.74154	0.0682562
$652$	0	0
$653$	40.9087	1.60088	0.800440	−	0.599413i	$-0.204599\pi$
$653$	40.9087	1.60088	0.800440	+	0.599413i	$0.204599\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	6.29127	0.245446
$658$	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
$663$	−0.185591	−0.00720775
$664$	0	0
$665$	0	0
$666$	0	0
$667$	6.49707	0.251568
$668$	0	0
$669$	11.4215	0.441581
$670$	0	0
$671$	34.0677	1.31517
$672$	0	0
$673$	−11.3951	−0.439247	−0.219624	−	0.975585i	$-0.570483\pi$
$673$	−11.3951	−0.439247	−0.219624	+	0.975585i	$0.570483\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−3.87400	−0.148890	−0.0744449	−	0.997225i	$-0.523718\pi$
$677$	−3.87400	−0.148890	−0.0744449	+	0.997225i	$0.523718\pi$
$678$	0	0
$679$	0.0111000	0.000425979 0
$680$	0	0
$681$	−15.7360	−0.603004
$682$	0	0
$683$	−27.9517	−1.06954	−0.534770	−	0.844997i	$-0.679602\pi$
$683$	−27.9517	−1.06954	−0.534770	+	0.844997i	$0.679602\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−3.46335	−0.132135
$688$	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
$693$	−0.822124	−0.0312299
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−0.345420	−0.0130837
$698$	0	0
$699$	0.319746	0.0120939
$700$	0	0
$701$	−7.89656	−0.298249	−0.149124	−	0.988818i	$-0.547645\pi$
$701$	−7.89656	−0.298249	−0.149124	+	0.988818i	$0.547645\pi$
$702$	0	0
$703$	−21.8708	−0.824872
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.25553	0.0472191
$708$	0	0
$709$	−16.3686	−0.614734	−0.307367	−	0.951591i	$-0.599448\pi$
$709$	−16.3686	−0.614734	−0.307367	+	0.951591i	$0.599448\pi$
$710$	0	0
$711$	−9.82056	−0.368300
$712$	0	0
$713$	9.17655	0.343665
$714$	0	0
$715$	0	0
$716$	0	0
$717$	2.93697	0.109683
$718$	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
$723$	18.6764	0.694583
$724$	0	0
$725$	0	0
$726$	0	0
$727$	7.89415	0.292778	0.146389	−	0.989227i	$-0.453235\pi$
$727$	7.89415	0.292778	0.146389	+	0.989227i	$0.453235\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	0.311224	0.0115110
$732$	0	0
$733$	−17.1444	−0.633242	−0.316621	−	0.948552i	$-0.602548\pi$
$733$	−17.1444	−0.633242	−0.316621	+	0.948552i	$0.602548\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−10.9540	−0.403497
$738$	0	0
$739$	−43.1409	−1.58696	−0.793482	−	0.608593i	$-0.791734\pi$
$739$	−43.1409	−1.58696	−0.793482	+	0.608593i	$0.791734\pi$
$740$	0	0
$741$	−33.6167	−1.23494
$742$	0	0
$743$	−41.0320	−1.50532	−0.752660	−	0.658410i	$-0.771229\pi$
$743$	−41.0320	−1.50532	−0.752660	+	0.658410i	$0.771229\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0.417601	0.0152792
$748$	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
$753$	3.76059	0.137043
$754$	0	0
$755$	0	0
$756$	0	0
$757$	31.1373	1.13170	0.565852	−	0.824507i	$-0.308547\pi$
$757$	31.1373	1.13170	0.565852	+	0.824507i	$0.308547\pi$
$758$	0	0
$759$	−4.33196	−0.157240
$760$	0	0
$761$	−43.4836	−1.57628	−0.788139	−	0.615497i	$-0.788955\pi$
$761$	−43.4836	−1.57628	−0.788139	+	0.615497i	$0.788955\pi$
$762$	0	0
$763$	0.0462758	0.00167530
$764$	0	0
$765$	0	0
$766$	0	0
$767$	21.6898	0.783175
$768$	0	0
$769$	−4.35659	−0.157103	−0.0785514	−	0.996910i	$-0.525029\pi$
$769$	−4.35659	−0.157103	−0.0785514	+	0.996910i	$0.525029\pi$
$770$	0	0
$771$	−26.1810	−0.942884
$772$	0	0
$773$	52.3629	1.88336	0.941681	−	0.336508i	$-0.109246\pi$
$773$	52.3629	1.88336	0.941681	+	0.336508i	$0.109246\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−0.576708	−0.0206893
$778$	0	0
$779$	−62.5671	−2.24170
$780$	0	0
$781$	−7.40609	−0.265011
$782$	0	0
$783$	−6.49707	−0.232186
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−15.0229	−0.535507	−0.267754	−	0.963487i	$-0.586281\pi$
$787$	−15.0229	−0.535507	−0.267754	+	0.963487i	$0.586281\pi$
$788$	0	0
$789$	22.6844	0.807587
$790$	0	0
$791$	0.232392	0.00826289
$792$	0	0
$793$	36.7327	1.30442
$794$	0	0
$795$	0	0
$796$	0	0
$797$	31.3217	1.10947	0.554737	−	0.832026i	$-0.312819\pi$
$797$	31.3217	1.10947	0.554737	+	0.832026i	$0.312819\pi$
$798$	0	0
$799$	−0.199912	−0.00707239
$800$	0	0
$801$	−3.92081	−0.138535
$802$	0	0
$803$	27.2535	0.961756
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−0.570220	−0.0200727
$808$	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
$813$	−21.9506	−0.769843
$814$	0	0
$815$	0	0
$816$	0	0
$817$	56.3731	1.97225
$818$	0	0
$819$	−0.886435	−0.0309746
$820$	0	0
$821$	55.8300	1.94848	0.974240	−	0.225513i	$-0.0724058\pi$
$821$	55.8300	1.94848	0.974240	+	0.225513i	$0.0724058\pi$
$822$	0	0
$823$	−8.57848	−0.299027	−0.149514	−	0.988760i	$-0.547771\pi$
$823$	−8.57848	−0.299027	−0.149514	+	0.988760i	$0.547771\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−12.4646	−0.433437	−0.216718	−	0.976234i	$-0.569535\pi$
$827$	−12.4646	−0.433437	−0.216718	+	0.976234i	$0.569535\pi$
$828$	0	0
$829$	−5.83650	−0.202710	−0.101355	−	0.994850i	$-0.532318\pi$
$829$	−5.83650	−0.202710	−0.101355	+	0.994850i	$0.532318\pi$
$830$	0	0
$831$	12.5514	0.435403
$832$	0	0
$833$	−0.276707	−0.00958732
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−9.17655	−0.317188
$838$	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
$843$	−21.7956	−0.750680
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−1.47381	−0.0506409
$848$	0	0
$849$	−24.0765	−0.826304
$850$	0	0
$851$	−3.03881	−0.104169
$852$	0	0
$853$	30.7842	1.05403	0.527015	−	0.849856i	$-0.323311\pi$
$853$	30.7842	1.05403	0.527015	+	0.849856i	$0.323311\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−15.0207	−0.513098	−0.256549	−	0.966531i	$-0.582586\pi$
$857$	−15.0207	−0.513098	−0.256549	+	0.966531i	$0.582586\pi$
$858$	0	0
$859$	0.352932	0.0120419	0.00602094	−	0.999982i	$-0.498083\pi$
$859$	0.352932	0.0120419	0.00602094	+	0.999982i	$0.498083\pi$
$860$	0	0
$861$	−1.64982	−0.0562259
$862$	0	0
$863$	−8.55180	−0.291107	−0.145553	−	0.989350i	$-0.546496\pi$
$863$	−8.55180	−0.291107	−0.145553	+	0.989350i	$0.546496\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	16.9984	0.577297
$868$	0	0
$869$	−42.5423	−1.44315
$870$	0	0
$871$	−11.8109	−0.400198
$872$	0	0
$873$	−0.0584885	−0.00197954
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−46.9633	−1.58584	−0.792919	−	0.609327i	$-0.791440\pi$
$877$	−46.9633	−1.58584	−0.792919	+	0.609327i	$0.791440\pi$
$878$	0	0
$879$	27.2212	0.918147
$880$	0	0
$881$	−47.9481	−1.61541	−0.807706	−	0.589585i	$-0.799291\pi$
$881$	−47.9481	−1.61541	−0.807706	+	0.589585i	$0.799291\pi$
$882$	0	0
$883$	49.6286	1.67014	0.835068	−	0.550147i	$-0.185428\pi$
$883$	49.6286	1.67014	0.835068	+	0.550147i	$0.185428\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−39.1853	−1.31572	−0.657858	−	0.753142i	$-0.728537\pi$
$887$	−39.1853	−1.31572	−0.657858	+	0.753142i	$0.728537\pi$
$888$	0	0
$889$	3.92399	0.131606
$890$	0	0
$891$	4.33196	0.145126
$892$	0	0
$893$	−36.2108	−1.21175
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−4.67083	−0.155955
$898$	0	0
$899$	59.6207	1.98846
$900$	0	0
$901$	0.276670	0.00921721
$902$	0	0
$903$	1.48650	0.0494675
$904$	0	0
$905$	0	0
$906$	0	0
$907$	39.3323	1.30601	0.653004	−	0.757355i	$-0.273509\pi$
$907$	39.3323	1.30601	0.653004	+	0.757355i	$0.273509\pi$
$908$	0	0
$909$	−6.61569	−0.219429
$910$	0	0
$911$	−17.8676	−0.591980	−0.295990	−	0.955191i	$-0.595649\pi$
$911$	−17.8676	−0.591980	−0.295990	+	0.955191i	$0.595649\pi$
$912$	0	0
$913$	1.80903	0.0598702
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−3.47780	−0.114847
$918$	0	0
$919$	−20.0858	−0.662570	−0.331285	−	0.943531i	$-0.607482\pi$
$919$	−20.0858	−0.662570	−0.331285	+	0.943531i	$0.607482\pi$
$920$	0	0
$921$	6.48076	0.213548
$922$	0	0
$923$	−7.98544	−0.262844
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−16.9236	−0.555844
$928$	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
$933$	−4.11457	−0.134705
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−48.1869	−1.57420	−0.787099	−	0.616826i	$-0.788418\pi$
$937$	−48.1869	−1.57420	−0.787099	+	0.616826i	$0.788418\pi$
$938$	0	0
$939$	0.760034	0.0248028
$940$	0	0
$941$	−10.7084	−0.349083	−0.174542	−	0.984650i	$-0.555844\pi$
$941$	−10.7084	−0.349083	−0.174542	+	0.984650i	$0.555844\pi$
$942$	0	0
$943$	−8.69330	−0.283093
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−17.5646	−0.570774	−0.285387	−	0.958412i	$-0.592122\pi$
$947$	−17.5646	−0.570774	−0.285387	+	0.958412i	$0.592122\pi$
$948$	0	0
$949$	29.3855	0.953893
$950$	0	0
$951$	7.48188	0.242617
$952$	0	0
$953$	36.3058	1.17606	0.588031	−	0.808839i	$-0.299903\pi$
$953$	36.3058	1.17606	0.588031	+	0.808839i	$0.299903\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−28.1450	−0.909800
$958$	0	0
$959$	−0.658348	−0.0212592
$960$	0	0
$961$	53.2091	1.71642
$962$	0	0
$963$	−17.5775	−0.566426
$964$	0	0
$965$	0	0
$966$	0	0
$967$	30.4808	0.980195	0.490098	−	0.871668i	$-0.336961\pi$
$967$	30.4808	0.980195	0.490098	+	0.871668i	$0.336961\pi$
$968$	0	0
$969$	−0.285972	−0.00918674
$970$	0	0
$971$	43.2929	1.38934	0.694668	−	0.719330i	$-0.255551\pi$
$971$	43.2929	1.38934	0.694668	+	0.719330i	$0.255551\pi$
$972$	0	0
$973$	−1.21626	−0.0389916
$974$	0	0
$975$	0	0
$976$	0	0
$977$	6.01624	0.192477	0.0962383	−	0.995358i	$-0.469319\pi$
$977$	6.01624	0.192477	0.0962383	+	0.995358i	$0.469319\pi$
$978$	0	0
$979$	−16.9848	−0.542836
$980$	0	0
$981$	−0.243838	−0.00778515
$982$	0	0
$983$	36.0552	1.14998	0.574992	−	0.818159i	$-0.305005\pi$
$983$	36.0552	1.14998	0.574992	+	0.818159i	$0.305005\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−0.954839	−0.0303929
$988$	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
$993$	−28.5981	−0.907532
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−18.9389	−0.599799	−0.299900	−	0.953971i	$-0.596953\pi$
$997$	−18.9389	−0.599799	−0.299900	+	0.953971i	$0.596953\pi$
$998$	0	0
$999$	3.03881	0.0961436

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1380.2.f.b.829.4	✓	14	5.3	odd	4
1380.2.f.b.829.11	yes	14	5.2	odd	4
4140.2.f.c.829.7		14	15.8	even	4
4140.2.f.c.829.8		14	15.2	even	4
6900.2.a.bc.1.4		7	1.1	even	1	trivial
6900.2.a.bd.1.4		7	5.4	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:	\( N \)	\(=\)	\( 6900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6900.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -0.189781 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -0.189781 q^{7} +1.00000 q^{9} +4.33196 q^{11} +4.67083 q^{13} +0.0397340 q^{17} +7.19716 q^{19} +0.189781 q^{21} +1.00000 q^{23} -1.00000 q^{27} +6.49707 q^{29} +9.17655 q^{31} -4.33196 q^{33} -3.03881 q^{37} -4.67083 q^{39} -8.69330 q^{41} +7.83269 q^{43} -5.03127 q^{47} -6.96398 q^{49} -0.0397340 q^{51} +6.96306 q^{53} -7.19716 q^{57} +4.64368 q^{59} +7.86428 q^{61} -0.189781 q^{63} -2.52865 q^{67} -1.00000 q^{69} -1.70964 q^{71} +6.29127 q^{73} -0.822124 q^{77} -9.82056 q^{79} +1.00000 q^{81} +0.417601 q^{83} -6.49707 q^{87} -3.92081 q^{89} -0.886435 q^{91} -9.17655 q^{93} -0.0584885 q^{97} +4.33196 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 7 q^{3} + q^{7} + 7 q^{9}+O(q^{10}) \) 7 * q - 7 * q^3 + q^7 + 7 * q^9	\( 7 q - 7 q^{3} + q^{7} + 7 q^{9} + 2 q^{13} - q^{17} - 2 q^{19} - q^{21} + 7 q^{23} - 7 q^{27} + 15 q^{29} + 3 q^{31} - 19 q^{37} - 2 q^{39} + 23 q^{41} + 4 q^{43} - 10 q^{47} + 10 q^{49} + q^{51} - 11 q^{53} + 2 q^{57} + 5 q^{59} + 32 q^{61} + q^{63} - 15 q^{67} - 7 q^{69} + 21 q^{71} + 18 q^{73} - 28 q^{77} + 16 q^{79} + 7 q^{81} - 25 q^{83} - 15 q^{87} + 26 q^{89} + 14 q^{91} - 3 q^{93} - 6 q^{97}+O(q^{100}) \) 7 * q - 7 * q^3 + q^7 + 7 * q^9 + 2 * q^13 - q^17 - 2 * q^19 - q^21 + 7 * q^23 - 7 * q^27 + 15 * q^29 + 3 * q^31 - 19 * q^37 - 2 * q^39 + 23 * q^41 + 4 * q^43 - 10 * q^47 + 10 * q^49 + q^51 - 11 * q^53 + 2 * q^57 + 5 * q^59 + 32 * q^61 + q^63 - 15 * q^67 - 7 * q^69 + 21 * q^71 + 18 * q^73 - 28 * q^77 + 16 * q^79 + 7 * q^81 - 25 * q^83 - 15 * q^87 + 26 * q^89 + 14 * q^91 - 3 * q^93 - 6 * q^97

Introduction

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