LMFDB - Embedded newform 6384.2.a.cf.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6384,2,Mod(1,6384)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6384, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6384.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6384 = 2^{4} \cdot 3 \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6384.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:	$50.9764966504$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.368464.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 6x^{3} + 6x^{2} + 6x - 4 $ x^5 - 2x^4 - 6x^3 + 6x^2 + 6x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 399)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.09027$ of defining polynomial
Character	$\chi$	$=$	6384.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +0.388134 q^{5} -1.00000 q^{7} +1.00000 q^{9} -6.41859 q^{11} +3.88714 q^{13} +0.388134 q^{15} -4.98727 q^{17} +1.00000 q^{19} -1.00000 q^{21} -3.44207 q^{23} -4.84935 q^{25} +1.00000 q^{27} +0.169801 q^{29} +8.62562 q^{31} -6.41859 q^{33} -0.388134 q^{35} +7.37540 q^{37} +3.88714 q^{39} -8.77968 q^{41} +9.11087 q^{43} +0.388134 q^{45} +4.80672 q^{47} +1.00000 q^{49} -4.98727 q^{51} +8.42933 q^{53} -2.49127 q^{55} +1.00000 q^{57} -2.97652 q^{59} +5.82287 q^{61} -1.00000 q^{63} +1.50873 q^{65} +14.9980 q^{67} -3.44207 q^{69} -4.24879 q^{71} +13.5757 q^{73} -4.84935 q^{75} +6.41859 q^{77} +1.01431 q^{79} +1.00000 q^{81} +4.32147 q^{83} -1.93573 q^{85} +0.169801 q^{87} +13.8955 q^{89} -3.88714 q^{91} +8.62562 q^{93} +0.388134 q^{95} -13.7743 q^{97} -6.41859 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{3} + 4 q^{5} - 5 q^{7} + 5 q^{9} - 8 q^{11} - 6 q^{13} + 4 q^{15} + 12 q^{17} + 5 q^{19} - 5 q^{21} - 12 q^{23} + 15 q^{25} + 5 q^{27} + 4 q^{29} + 8 q^{31} - 8 q^{33} - 4 q^{35} + 2 q^{37} - 6 q^{39}+ \cdots - 8 q^{99}+O(q^{100}) $ 5 * q + 5 * q^3 + 4 * q^5 - 5 * q^7 + 5 * q^9 - 8 * q^11 - 6 * q^13 + 4 * q^15 + 12 * q^17 + 5 * q^19 - 5 * q^21 - 12 * q^23 + 15 * q^25 + 5 * q^27 + 4 * q^29 + 8 * q^31 - 8 * q^33 - 4 * q^35 + 2 * q^37 - 6 * q^39 + 10 * q^41 + 16 * q^43 + 4 * q^45 + 2 * q^47 + 5 * q^49 + 12 * q^51 + 12 * q^55 + 5 * q^57 + 4 * q^59 - 14 * q^61 - 5 * q^63 + 32 * q^65 + 20 * q^67 - 12 * q^69 + 6 * q^71 + 10 * q^73 + 15 * q^75 + 8 * q^77 + 5 * q^81 - 6 * q^83 + 8 * q^85 + 4 * q^87 + 26 * q^89 + 6 * q^91 + 8 * q^93 + 4 * q^95 - 18 * q^97 - 8 * q^99

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
$5$	0.388134	0.173579	0.0867893	−	0.996227i	$-0.472339\pi$
$5$	0.388134	0.173579	0.0867893	+	0.996227i	$0.472339\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−6.41859	−1.93528	−0.967638	−	0.252341i	$-0.918800\pi$
$11$	−6.41859	−1.93528	−0.967638	+	0.252341i	$0.918800\pi$
$12$	0	0
$13$	3.88714	1.07810	0.539049	−	0.842274i	$-0.318784\pi$
$13$	3.88714	1.07810	0.539049	+	0.842274i	$0.318784\pi$
$14$	0	0
$15$	0.388134	0.100216
$16$	0	0
$17$	−4.98727	−1.20959	−0.604795	−	0.796381i	$-0.706745\pi$
$17$	−4.98727	−1.20959	−0.604795	+	0.796381i	$0.706745\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−3.44207	−0.717720	−0.358860	−	0.933391i	$-0.616834\pi$
$23$	−3.44207	−0.717720	−0.358860	+	0.933391i	$0.616834\pi$
$24$	0	0
$25$	−4.84935	−0.969870
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	0.169801	0.0315313	0.0157656	−	0.999876i	$-0.494981\pi$
$29$	0.169801	0.0315313	0.0157656	+	0.999876i	$0.494981\pi$
$30$	0	0
$31$	8.62562	1.54921	0.774604	−	0.632447i	$-0.217949\pi$
$31$	8.62562	1.54921	0.774604	+	0.632447i	$0.217949\pi$
$32$	0	0
$33$	−6.41859	−1.11733
$34$	0	0
$35$	−0.388134	−0.0656066
$36$	0	0
$37$	7.37540	1.21251	0.606254	−	0.795271i	$-0.292671\pi$
$37$	7.37540	1.21251	0.606254	+	0.795271i	$0.292671\pi$
$38$	0	0
$39$	3.88714	0.622440
$40$	0	0
$41$	−8.77968	−1.37116	−0.685578	−	0.727999i	$-0.740450\pi$
$41$	−8.77968	−1.37116	−0.685578	+	0.727999i	$0.740450\pi$
$42$	0	0
$43$	9.11087	1.38939	0.694697	−	0.719302i	$-0.255538\pi$
$43$	9.11087	1.38939	0.694697	+	0.719302i	$0.255538\pi$
$44$	0	0
$45$	0.388134	0.0578596
$46$	0	0
$47$	4.80672	0.701132	0.350566	−	0.936538i	$-0.385989\pi$
$47$	4.80672	0.701132	0.350566	+	0.936538i	$0.385989\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−4.98727	−0.698357
$52$	0	0
$53$	8.42933	1.15786	0.578929	−	0.815378i	$-0.303471\pi$
$53$	8.42933	1.15786	0.578929	+	0.815378i	$0.303471\pi$
$54$	0	0
$55$	−2.49127	−0.335923
$56$	0	0
$57$	1.00000	0.132453
$58$	0	0
$59$	−2.97652	−0.387510	−0.193755	−	0.981050i	$-0.562067\pi$
$59$	−2.97652	−0.387510	−0.193755	+	0.981050i	$0.562067\pi$
$60$	0	0
$61$	5.82287	0.745542	0.372771	−	0.927923i	$-0.378408\pi$
$61$	5.82287	0.745542	0.372771	+	0.927923i	$0.378408\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	1.50873	0.187135
$66$	0	0
$67$	14.9980	1.83230	0.916149	−	0.400837i	$-0.131281\pi$
$67$	14.9980	1.83230	0.916149	+	0.400837i	$0.131281\pi$
$68$	0	0
$69$	−3.44207	−0.414376
$70$	0	0
$71$	−4.24879	−0.504238	−0.252119	−	0.967696i	$-0.581127\pi$
$71$	−4.24879	−0.504238	−0.252119	+	0.967696i	$0.581127\pi$
$72$	0	0
$73$	13.5757	1.58891	0.794455	−	0.607323i	$-0.207757\pi$
$73$	13.5757	1.58891	0.794455	+	0.607323i	$0.207757\pi$
$74$	0	0
$75$	−4.84935	−0.559955
$76$	0	0
$77$	6.41859	0.731466
$78$	0	0
$79$	1.01431	0.114119	0.0570594	−	0.998371i	$-0.481828\pi$
$79$	1.01431	0.114119	0.0570594	+	0.998371i	$0.481828\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	4.32147	0.474343	0.237171	−	0.971468i	$-0.423780\pi$
$83$	4.32147	0.474343	0.237171	+	0.971468i	$0.423780\pi$
$84$	0	0
$85$	−1.93573	−0.209959
$86$	0	0
$87$	0.169801	0.0182046
$88$	0	0
$89$	13.8955	1.47293	0.736463	−	0.676478i	$-0.236495\pi$
$89$	13.8955	1.47293	0.736463	+	0.676478i	$0.236495\pi$
$90$	0	0
$91$	−3.88714	−0.407483
$92$	0	0
$93$	8.62562	0.894435
$94$	0	0
$95$	0.388134	0.0398217
$96$	0	0
$97$	−13.7743	−1.39857	−0.699283	−	0.714845i	$-0.746497\pi$
$97$	−13.7743	−1.39857	−0.699283	+	0.714845i	$0.746497\pi$
$98$	0	0
$99$	−6.41859	−0.645092
$100$	0	0
$101$	8.15009	0.810964	0.405482	−	0.914103i	$-0.367104\pi$
$101$	8.15009	0.810964	0.405482	+	0.914103i	$0.367104\pi$
$102$	0	0
$103$	12.8372	1.26488	0.632442	−	0.774608i	$-0.282053\pi$
$103$	12.8372	1.26488	0.632442	+	0.774608i	$0.282053\pi$
$104$	0	0
$105$	−0.388134	−0.0378780
$106$	0	0
$107$	8.12463	0.785437	0.392719	−	0.919659i	$-0.371535\pi$
$107$	8.12463	0.785437	0.392719	+	0.919659i	$0.371535\pi$
$108$	0	0
$109$	−20.3356	−1.94780	−0.973900	−	0.226978i	$-0.927115\pi$
$109$	−20.3356	−1.94780	−0.973900	+	0.226978i	$0.927115\pi$
$110$	0	0
$111$	7.37540	0.700042
$112$	0	0
$113$	4.16980	0.392262	0.196131	−	0.980578i	$-0.437162\pi$
$113$	4.16980	0.392262	0.196131	+	0.980578i	$0.437162\pi$
$114$	0	0
$115$	−1.33598	−0.124581
$116$	0	0
$117$	3.88714	0.359366
$118$	0	0
$119$	4.98727	0.457182
$120$	0	0
$121$	30.1983	2.74530
$122$	0	0
$123$	−8.77968	−0.791637
$124$	0	0
$125$	−3.82287	−0.341927
$126$	0	0
$127$	6.20025	0.550184	0.275092	−	0.961418i	$-0.411292\pi$
$127$	6.20025	0.550184	0.275092	+	0.961418i	$0.411292\pi$
$128$	0	0
$129$	9.11087	0.802168
$130$	0	0
$131$	−15.5667	−1.36007	−0.680034	−	0.733181i	$-0.738035\pi$
$131$	−15.5667	−1.36007	−0.680034	+	0.733181i	$0.738035\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	0	0
$135$	0.388134	0.0334052
$136$	0	0
$137$	10.7222	0.916058	0.458029	−	0.888937i	$-0.348556\pi$
$137$	10.7222	0.916058	0.458029	+	0.888937i	$0.348556\pi$
$138$	0	0
$139$	−10.0987	−0.856560	−0.428280	−	0.903646i	$-0.640880\pi$
$139$	−10.0987	−0.856560	−0.428280	+	0.903646i	$0.640880\pi$
$140$	0	0
$141$	4.80672	0.404799
$142$	0	0
$143$	−24.9499	−2.08642
$144$	0	0
$145$	0.0659056	0.00547316
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	9.75279	0.798980	0.399490	−	0.916738i	$-0.369187\pi$
$149$	9.75279	0.798980	0.399490	+	0.916738i	$0.369187\pi$
$150$	0	0
$151$	2.20942	0.179800	0.0899002	−	0.995951i	$-0.471345\pi$
$151$	2.20942	0.179800	0.0899002	+	0.995951i	$0.471345\pi$
$152$	0	0
$153$	−4.98727	−0.403197
$154$	0	0
$155$	3.34789	0.268909
$156$	0	0
$157$	−10.3519	−0.826173	−0.413087	−	0.910692i	$-0.635549\pi$
$157$	−10.3519	−0.826173	−0.413087	+	0.910692i	$0.635549\pi$
$158$	0	0
$159$	8.42933	0.668490
$160$	0	0
$161$	3.44207	0.271273
$162$	0	0
$163$	8.52406	0.667656	0.333828	−	0.942634i	$-0.391660\pi$
$163$	8.52406	0.667656	0.333828	+	0.942634i	$0.391660\pi$
$164$	0	0
$165$	−2.49127	−0.193945
$166$	0	0
$167$	−6.99801	−0.541522	−0.270761	−	0.962647i	$-0.587275\pi$
$167$	−6.99801	−0.541522	−0.270761	+	0.962647i	$0.587275\pi$
$168$	0	0
$169$	2.10985	0.162296
$170$	0	0
$171$	1.00000	0.0764719
$172$	0	0
$173$	14.6403	1.11308	0.556542	−	0.830820i	$-0.312128\pi$
$173$	14.6403	1.11308	0.556542	+	0.830820i	$0.312128\pi$
$174$	0	0
$175$	4.84935	0.366577
$176$	0	0
$177$	−2.97652	−0.223729
$178$	0	0
$179$	12.9009	0.964258	0.482129	−	0.876100i	$-0.339864\pi$
$179$	12.9009	0.964258	0.482129	+	0.876100i	$0.339864\pi$
$180$	0	0
$181$	−12.3897	−0.920920	−0.460460	−	0.887680i	$-0.652316\pi$
$181$	−12.3897	−0.920920	−0.460460	+	0.887680i	$0.652316\pi$
$182$	0	0
$183$	5.82287	0.430439
$184$	0	0
$185$	2.86264	0.210466
$186$	0	0
$187$	32.0112	2.34089
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−25.0544	−1.81287	−0.906436	−	0.422343i	$-0.861208\pi$
$191$	−25.0544	−1.81287	−0.906436	+	0.422343i	$0.861208\pi$
$192$	0	0
$193$	8.79939	0.633394	0.316697	−	0.948527i	$-0.397426\pi$
$193$	8.79939	0.633394	0.316697	+	0.948527i	$0.397426\pi$
$194$	0	0
$195$	1.50873	0.108042
$196$	0	0
$197$	−4.80856	−0.342595	−0.171298	−	0.985219i	$-0.554796\pi$
$197$	−4.80856	−0.342595	−0.171298	+	0.985219i	$0.554796\pi$
$198$	0	0
$199$	−9.61344	−0.681479	−0.340739	−	0.940158i	$-0.610677\pi$
$199$	−9.61344	−0.681479	−0.340739	+	0.940158i	$0.610677\pi$
$200$	0	0
$201$	14.9980	1.05788
$202$	0	0
$203$	−0.169801	−0.0119177
$204$	0	0
$205$	−3.40769	−0.238003
$206$	0	0
$207$	−3.44207	−0.239240
$208$	0	0
$209$	−6.41859	−0.443983
$210$	0	0
$211$	−2.22174	−0.152951	−0.0764756	−	0.997071i	$-0.524367\pi$
$211$	−2.22174	−0.152951	−0.0764756	+	0.997071i	$0.524367\pi$
$212$	0	0
$213$	−4.24879	−0.291122
$214$	0	0
$215$	3.53624	0.241169
$216$	0	0
$217$	−8.62562	−0.585545
$218$	0	0
$219$	13.5757	0.917358
$220$	0	0
$221$	−19.3862	−1.30406
$222$	0	0
$223$	−18.2606	−1.22282	−0.611408	−	0.791315i	$-0.709397\pi$
$223$	−18.2606	−1.22282	−0.611408	+	0.791315i	$0.709397\pi$
$224$	0	0
$225$	−4.84935	−0.323290
$226$	0	0
$227$	5.13736	0.340978	0.170489	−	0.985360i	$-0.445465\pi$
$227$	5.13736	0.340978	0.170489	+	0.985360i	$0.445465\pi$
$228$	0	0
$229$	10.4750	0.692206	0.346103	−	0.938197i	$-0.387505\pi$
$229$	10.4750	0.692206	0.346103	+	0.938197i	$0.387505\pi$
$230$	0	0
$231$	6.41859	0.422312
$232$	0	0
$233$	17.5271	1.14824	0.574118	−	0.818772i	$-0.305345\pi$
$233$	17.5271	1.14824	0.574118	+	0.818772i	$0.305345\pi$
$234$	0	0
$235$	1.86565	0.121702
$236$	0	0
$237$	1.01431	0.0658865
$238$	0	0
$239$	21.7777	1.40868	0.704341	−	0.709862i	$-0.251243\pi$
$239$	21.7777	1.40868	0.704341	+	0.709862i	$0.251243\pi$
$240$	0	0
$241$	10.3611	0.667417	0.333708	−	0.942676i	$-0.391700\pi$
$241$	10.3611	0.667417	0.333708	+	0.942676i	$0.391700\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0.388134	0.0247970
$246$	0	0
$247$	3.88714	0.247333
$248$	0	0
$249$	4.32147	0.273862
$250$	0	0
$251$	−10.9146	−0.688922	−0.344461	−	0.938801i	$-0.611938\pi$
$251$	−10.9146	−0.688922	−0.344461	+	0.938801i	$0.611938\pi$
$252$	0	0
$253$	22.0932	1.38899
$254$	0	0
$255$	−1.93573	−0.121220
$256$	0	0
$257$	−3.28123	−0.204677	−0.102339	−	0.994750i	$-0.532633\pi$
$257$	−3.28123	−0.204677	−0.102339	+	0.994750i	$0.532633\pi$
$258$	0	0
$259$	−7.37540	−0.458285
$260$	0	0
$261$	0.169801	0.0105104
$262$	0	0
$263$	18.0034	1.11014	0.555069	−	0.831804i	$-0.312692\pi$
$263$	18.0034	1.11014	0.555069	+	0.831804i	$0.312692\pi$
$264$	0	0
$265$	3.27171	0.200979
$266$	0	0
$267$	13.8955	0.850394
$268$	0	0
$269$	−0.886737	−0.0540653	−0.0270327	−	0.999635i	$-0.508606\pi$
$269$	−0.886737	−0.0540653	−0.0270327	+	0.999635i	$0.508606\pi$
$270$	0	0
$271$	−7.43917	−0.451898	−0.225949	−	0.974139i	$-0.572548\pi$
$271$	−7.43917	−0.451898	−0.225949	+	0.974139i	$0.572548\pi$
$272$	0	0
$273$	−3.88714	−0.235260
$274$	0	0
$275$	31.1260	1.87697
$276$	0	0
$277$	−7.40189	−0.444736	−0.222368	−	0.974963i	$-0.571379\pi$
$277$	−7.40189	−0.444736	−0.222368	+	0.974963i	$0.571379\pi$
$278$	0	0
$279$	8.62562	0.516402
$280$	0	0
$281$	−1.98187	−0.118228	−0.0591141	−	0.998251i	$-0.518828\pi$
$281$	−1.98187	−0.118228	−0.0591141	+	0.998251i	$0.518828\pi$
$282$	0	0
$283$	−7.83519	−0.465753	−0.232877	−	0.972506i	$-0.574814\pi$
$283$	−7.83519	−0.465753	−0.232877	+	0.972506i	$0.574814\pi$
$284$	0	0
$285$	0.388134	0.0229911
$286$	0	0
$287$	8.77968	0.518248
$288$	0	0
$289$	7.87283	0.463108
$290$	0	0
$291$	−13.7743	−0.807463
$292$	0	0
$293$	3.83178	0.223855	0.111927	−	0.993716i	$-0.464298\pi$
$293$	3.83178	0.223855	0.111927	+	0.993716i	$0.464298\pi$
$294$	0	0
$295$	−1.15529	−0.0672635
$296$	0	0
$297$	−6.41859	−0.372444
$298$	0	0
$299$	−13.3798	−0.773773
$300$	0	0
$301$	−9.11087	−0.525142
$302$	0	0
$303$	8.15009	0.468211
$304$	0	0
$305$	2.26005	0.129410
$306$	0	0
$307$	−1.46279	−0.0834861	−0.0417431	−	0.999128i	$-0.513291\pi$
$307$	−1.46279	−0.0834861	−0.0417431	+	0.999128i	$0.513291\pi$
$308$	0	0
$309$	12.8372	0.730281
$310$	0	0
$311$	12.0213	0.681665	0.340832	−	0.940124i	$-0.389291\pi$
$311$	12.0213	0.681665	0.340832	+	0.940124i	$0.389291\pi$
$312$	0	0
$313$	5.36938	0.303495	0.151748	−	0.988419i	$-0.451510\pi$
$313$	5.36938	0.303495	0.151748	+	0.988419i	$0.451510\pi$
$314$	0	0
$315$	−0.388134	−0.0218689
$316$	0	0
$317$	−12.6674	−0.711471	−0.355735	−	0.934587i	$-0.615770\pi$
$317$	−12.6674	−0.711471	−0.355735	+	0.934587i	$0.615770\pi$
$318$	0	0
$319$	−1.08988	−0.0610218
$320$	0	0
$321$	8.12463	0.453472
$322$	0	0
$323$	−4.98727	−0.277499
$324$	0	0
$325$	−18.8501	−1.04562
$326$	0	0
$327$	−20.3356	−1.12456
$328$	0	0
$329$	−4.80672	−0.265003
$330$	0	0
$331$	−15.2575	−0.838630	−0.419315	−	0.907841i	$-0.637730\pi$
$331$	−15.2575	−0.838630	−0.419315	+	0.907841i	$0.637730\pi$
$332$	0	0
$333$	7.37540	0.404169
$334$	0	0
$335$	5.82123	0.318048
$336$	0	0
$337$	−19.5379	−1.06430	−0.532148	−	0.846652i	$-0.678615\pi$
$337$	−19.5379	−1.06430	−0.532148	+	0.846652i	$0.678615\pi$
$338$	0	0
$339$	4.16980	0.226473
$340$	0	0
$341$	−55.3643	−2.99814
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−1.33598	−0.0719268
$346$	0	0
$347$	17.7777	0.954356	0.477178	−	0.878807i	$-0.341660\pi$
$347$	17.7777	0.954356	0.477178	+	0.878807i	$0.341660\pi$
$348$	0	0
$349$	13.6987	0.733275	0.366637	−	0.930364i	$-0.380509\pi$
$349$	13.6987	0.733275	0.366637	+	0.930364i	$0.380509\pi$
$350$	0	0
$351$	3.88714	0.207480
$352$	0	0
$353$	8.81013	0.468916	0.234458	−	0.972126i	$-0.424668\pi$
$353$	8.81013	0.468916	0.234458	+	0.972126i	$0.424668\pi$
$354$	0	0
$355$	−1.64910	−0.0875250
$356$	0	0
$357$	4.98727	0.263954
$358$	0	0
$359$	9.44808	0.498651	0.249325	−	0.968420i	$-0.419791\pi$
$359$	9.44808	0.498651	0.249325	+	0.968420i	$0.419791\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	30.1983	1.58500
$364$	0	0
$365$	5.26917	0.275801
$366$	0	0
$367$	1.08439	0.0566044	0.0283022	−	0.999599i	$-0.490990\pi$
$367$	1.08439	0.0566044	0.0283022	+	0.999599i	$0.490990\pi$
$368$	0	0
$369$	−8.77968	−0.457052
$370$	0	0
$371$	−8.42933	−0.437629
$372$	0	0
$373$	−3.64971	−0.188975	−0.0944874	−	0.995526i	$-0.530121\pi$
$373$	−3.64971	−0.188975	−0.0944874	+	0.995526i	$0.530121\pi$
$374$	0	0
$375$	−3.82287	−0.197412
$376$	0	0
$377$	0.660041	0.0339938
$378$	0	0
$379$	−12.8846	−0.661840	−0.330920	−	0.943659i	$-0.607359\pi$
$379$	−12.8846	−0.661840	−0.330920	+	0.943659i	$0.607359\pi$
$380$	0	0
$381$	6.20025	0.317649
$382$	0	0
$383$	17.7242	0.905663	0.452831	−	0.891596i	$-0.350414\pi$
$383$	17.7242	0.905663	0.452831	+	0.891596i	$0.350414\pi$
$384$	0	0
$385$	2.49127	0.126967
$386$	0	0
$387$	9.11087	0.463132
$388$	0	0
$389$	10.4750	0.531102	0.265551	−	0.964097i	$-0.414446\pi$
$389$	10.4750	0.531102	0.265551	+	0.964097i	$0.414446\pi$
$390$	0	0
$391$	17.1665	0.868147
$392$	0	0
$393$	−15.5667	−0.785236
$394$	0	0
$395$	0.393688	0.0198086
$396$	0	0
$397$	−34.3679	−1.72488	−0.862438	−	0.506163i	$-0.831064\pi$
$397$	−34.3679	−1.72488	−0.862438	+	0.506163i	$0.831064\pi$
$398$	0	0
$399$	−1.00000	−0.0500626
$400$	0	0
$401$	−10.7403	−0.536346	−0.268173	−	0.963371i	$-0.586420\pi$
$401$	−10.7403	−0.536346	−0.268173	+	0.963371i	$0.586420\pi$
$402$	0	0
$403$	33.5290	1.67020
$404$	0	0
$405$	0.388134	0.0192865
$406$	0	0
$407$	−47.3397	−2.34654
$408$	0	0
$409$	26.0559	1.28838	0.644191	−	0.764865i	$-0.277194\pi$
$409$	26.0559	1.28838	0.644191	+	0.764865i	$0.277194\pi$
$410$	0	0
$411$	10.7222	0.528886
$412$	0	0
$413$	2.97652	0.146465
$414$	0	0
$415$	1.67731	0.0823358
$416$	0	0
$417$	−10.0987	−0.494535
$418$	0	0
$419$	1.36205	0.0665405	0.0332702	−	0.999446i	$-0.489408\pi$
$419$	1.36205	0.0665405	0.0332702	+	0.999446i	$0.489408\pi$
$420$	0	0
$421$	−1.93675	−0.0943912	−0.0471956	−	0.998886i	$-0.515028\pi$
$421$	−1.93675	−0.0943912	−0.0471956	+	0.998886i	$0.515028\pi$
$422$	0	0
$423$	4.80672	0.233711
$424$	0	0
$425$	24.1850	1.17315
$426$	0	0
$427$	−5.82287	−0.281788
$428$	0	0
$429$	−24.9499	−1.20459
$430$	0	0
$431$	−27.0267	−1.30183	−0.650915	−	0.759151i	$-0.725615\pi$
$431$	−27.0267	−1.30183	−0.650915	+	0.759151i	$0.725615\pi$
$432$	0	0
$433$	−38.2404	−1.83771	−0.918857	−	0.394590i	$-0.870887\pi$
$433$	−38.2404	−1.83771	−0.918857	+	0.394590i	$0.870887\pi$
$434$	0	0
$435$	0.0659056	0.00315993
$436$	0	0
$437$	−3.44207	−0.164656
$438$	0	0
$439$	2.76497	0.131965	0.0659823	−	0.997821i	$-0.478982\pi$
$439$	2.76497	0.131965	0.0659823	+	0.997821i	$0.478982\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−14.0564	−0.667839	−0.333919	−	0.942602i	$-0.608371\pi$
$443$	−14.0564	−0.667839	−0.333919	+	0.942602i	$0.608371\pi$
$444$	0	0
$445$	5.39333	0.255668
$446$	0	0
$447$	9.75279	0.461291
$448$	0	0
$449$	−14.3792	−0.678598	−0.339299	−	0.940679i	$-0.610190\pi$
$449$	−14.3792	−0.678598	−0.339299	+	0.940679i	$0.610190\pi$
$450$	0	0
$451$	56.3531	2.65357
$452$	0	0
$453$	2.20942	0.103808
$454$	0	0
$455$	−1.50873	−0.0707303
$456$	0	0
$457$	27.8024	1.30054	0.650271	−	0.759703i	$-0.274655\pi$
$457$	27.8024	1.30054	0.650271	+	0.759703i	$0.274655\pi$
$458$	0	0
$459$	−4.98727	−0.232786
$460$	0	0
$461$	21.0697	0.981312	0.490656	−	0.871353i	$-0.336757\pi$
$461$	21.0697	0.981312	0.490656	+	0.871353i	$0.336757\pi$
$462$	0	0
$463$	−2.52304	−0.117256	−0.0586278	−	0.998280i	$-0.518673\pi$
$463$	−2.52304	−0.117256	−0.0586278	+	0.998280i	$0.518673\pi$
$464$	0	0
$465$	3.34789	0.155255
$466$	0	0
$467$	−21.2342	−0.982602	−0.491301	−	0.870990i	$-0.663479\pi$
$467$	−21.2342	−0.982602	−0.491301	+	0.870990i	$0.663479\pi$
$468$	0	0
$469$	−14.9980	−0.692544
$470$	0	0
$471$	−10.3519	−0.476991
$472$	0	0
$473$	−58.4789	−2.68886
$474$	0	0
$475$	−4.84935	−0.222504
$476$	0	0
$477$	8.42933	0.385953
$478$	0	0
$479$	9.86851	0.450904	0.225452	−	0.974254i	$-0.427614\pi$
$479$	9.86851	0.450904	0.225452	+	0.974254i	$0.427614\pi$
$480$	0	0
$481$	28.6692	1.30720
$482$	0	0
$483$	3.44207	0.156619
$484$	0	0
$485$	−5.34626	−0.242761
$486$	0	0
$487$	−20.7038	−0.938181	−0.469090	−	0.883150i	$-0.655418\pi$
$487$	−20.7038	−0.938181	−0.469090	+	0.883150i	$0.655418\pi$
$488$	0	0
$489$	8.52406	0.385471
$490$	0	0
$491$	32.6041	1.47140	0.735700	−	0.677307i	$-0.236853\pi$
$491$	32.6041	1.47140	0.735700	+	0.677307i	$0.236853\pi$
$492$	0	0
$493$	−0.846844	−0.0381399
$494$	0	0
$495$	−2.49127	−0.111974
$496$	0	0
$497$	4.24879	0.190584
$498$	0	0
$499$	−0.812534	−0.0363740	−0.0181870	−	0.999835i	$-0.505789\pi$
$499$	−0.812534	−0.0363740	−0.0181870	+	0.999835i	$0.505789\pi$
$500$	0	0
$501$	−6.99801	−0.312648
$502$	0	0
$503$	16.6285	0.741427	0.370714	−	0.928747i	$-0.379113\pi$
$503$	16.6285	0.741427	0.370714	+	0.928747i	$0.379113\pi$
$504$	0	0
$505$	3.16333	0.140766
$506$	0	0
$507$	2.10985	0.0937018
$508$	0	0
$509$	33.8641	1.50100	0.750499	−	0.660871i	$-0.229813\pi$
$509$	33.8641	1.50100	0.750499	+	0.660871i	$0.229813\pi$
$510$	0	0
$511$	−13.5757	−0.600552
$512$	0	0
$513$	1.00000	0.0441511
$514$	0	0
$515$	4.98254	0.219557
$516$	0	0
$517$	−30.8524	−1.35689
$518$	0	0
$519$	14.6403	0.642639
$520$	0	0
$521$	28.3645	1.24267	0.621336	−	0.783544i	$-0.286590\pi$
$521$	28.3645	1.24267	0.621336	+	0.783544i	$0.286590\pi$
$522$	0	0
$523$	1.32743	0.0580443	0.0290221	−	0.999579i	$-0.490761\pi$
$523$	1.32743	0.0580443	0.0290221	+	0.999579i	$0.490761\pi$
$524$	0	0
$525$	4.84935	0.211643
$526$	0	0
$527$	−43.0183	−1.87391
$528$	0	0
$529$	−11.1522	−0.484878
$530$	0	0
$531$	−2.97652	−0.129170
$532$	0	0
$533$	−34.1278	−1.47824
$534$	0	0
$535$	3.15344	0.136335
$536$	0	0
$537$	12.9009	0.556715
$538$	0	0
$539$	−6.41859	−0.276468
$540$	0	0
$541$	−19.0875	−0.820637	−0.410319	−	0.911942i	$-0.634583\pi$
$541$	−19.0875	−0.820637	−0.410319	+	0.911942i	$0.634583\pi$
$542$	0	0
$543$	−12.3897	−0.531693
$544$	0	0
$545$	−7.89294	−0.338096
$546$	0	0
$547$	12.7660	0.545834	0.272917	−	0.962038i	$-0.412012\pi$
$547$	12.7660	0.545834	0.272917	+	0.962038i	$0.412012\pi$
$548$	0	0
$549$	5.82287	0.248514
$550$	0	0
$551$	0.169801	0.00723377
$552$	0	0
$553$	−1.01431	−0.0431328
$554$	0	0
$555$	2.86264	0.121512
$556$	0	0
$557$	23.1620	0.981405	0.490703	−	0.871327i	$-0.336740\pi$
$557$	23.1620	0.981405	0.490703	+	0.871327i	$0.336740\pi$
$558$	0	0
$559$	35.4152	1.49790
$560$	0	0
$561$	32.0112	1.35151
$562$	0	0
$563$	9.19938	0.387708	0.193854	−	0.981030i	$-0.437901\pi$
$563$	9.19938	0.387708	0.193854	+	0.981030i	$0.437901\pi$
$564$	0	0
$565$	1.61844	0.0680883
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	42.8405	1.79597	0.897984	−	0.440028i	$-0.145031\pi$
$569$	42.8405	1.79597	0.897984	+	0.440028i	$0.145031\pi$
$570$	0	0
$571$	−10.6358	−0.445095	−0.222547	−	0.974922i	$-0.571437\pi$
$571$	−10.6358	−0.445095	−0.222547	+	0.974922i	$0.571437\pi$
$572$	0	0
$573$	−25.0544	−1.04666
$574$	0	0
$575$	16.6918	0.696096
$576$	0	0
$577$	45.7190	1.90331	0.951653	−	0.307176i	$-0.0993842\pi$
$577$	45.7190	1.90331	0.951653	+	0.307176i	$0.0993842\pi$
$578$	0	0
$579$	8.79939	0.365690
$580$	0	0
$581$	−4.32147	−0.179285
$582$	0	0
$583$	−54.1044	−2.24078
$584$	0	0
$585$	1.50873	0.0623783
$586$	0	0
$587$	−13.8685	−0.572414	−0.286207	−	0.958168i	$-0.592395\pi$
$587$	−13.8685	−0.572414	−0.286207	+	0.958168i	$0.592395\pi$
$588$	0	0
$589$	8.62562	0.355412
$590$	0	0
$591$	−4.80856	−0.197798
$592$	0	0
$593$	−31.7063	−1.30202	−0.651011	−	0.759068i	$-0.725655\pi$
$593$	−31.7063	−1.30202	−0.651011	+	0.759068i	$0.725655\pi$
$594$	0	0
$595$	1.93573	0.0793570
$596$	0	0
$597$	−9.61344	−0.393452
$598$	0	0
$599$	3.91479	0.159954	0.0799771	−	0.996797i	$-0.474515\pi$
$599$	3.91479	0.159954	0.0799771	+	0.996797i	$0.474515\pi$
$600$	0	0
$601$	7.68790	0.313596	0.156798	−	0.987631i	$-0.449883\pi$
$601$	7.68790	0.313596	0.156798	+	0.987631i	$0.449883\pi$
$602$	0	0
$603$	14.9980	0.610766
$604$	0	0
$605$	11.7210	0.476525
$606$	0	0
$607$	−4.11499	−0.167022	−0.0835112	−	0.996507i	$-0.526613\pi$
$607$	−4.11499	−0.167022	−0.0835112	+	0.996507i	$0.526613\pi$
$608$	0	0
$609$	−0.169801	−0.00688069
$610$	0	0
$611$	18.6844	0.755890
$612$	0	0
$613$	14.0997	0.569482	0.284741	−	0.958604i	$-0.408092\pi$
$613$	14.0997	0.569482	0.284741	+	0.958604i	$0.408092\pi$
$614$	0	0
$615$	−3.40769	−0.137411
$616$	0	0
$617$	26.1748	1.05376	0.526879	−	0.849941i	$-0.323362\pi$
$617$	26.1748	1.05376	0.526879	+	0.849941i	$0.323362\pi$
$618$	0	0
$619$	−13.8200	−0.555473	−0.277736	−	0.960657i	$-0.589584\pi$
$619$	−13.8200	−0.555473	−0.277736	+	0.960657i	$0.589584\pi$
$620$	0	0
$621$	−3.44207	−0.138125
$622$	0	0
$623$	−13.8955	−0.556713
$624$	0	0
$625$	22.7630	0.910519
$626$	0	0
$627$	−6.41859	−0.256334
$628$	0	0
$629$	−36.7831	−1.46664
$630$	0	0
$631$	−26.3799	−1.05017	−0.525084	−	0.851050i	$-0.675966\pi$
$631$	−26.3799	−1.05017	−0.525084	+	0.851050i	$0.675966\pi$
$632$	0	0
$633$	−2.22174	−0.0883064
$634$	0	0
$635$	2.40653	0.0955001
$636$	0	0
$637$	3.88714	0.154014
$638$	0	0
$639$	−4.24879	−0.168079
$640$	0	0
$641$	−43.5850	−1.72151	−0.860753	−	0.509024i	$-0.830007\pi$
$641$	−43.5850	−1.72151	−0.860753	+	0.509024i	$0.830007\pi$
$642$	0	0
$643$	37.6911	1.48639	0.743195	−	0.669075i	$-0.233309\pi$
$643$	37.6911	1.48639	0.743195	+	0.669075i	$0.233309\pi$
$644$	0	0
$645$	3.53624	0.139239
$646$	0	0
$647$	41.8752	1.64628	0.823142	−	0.567835i	$-0.192219\pi$
$647$	41.8752	1.64628	0.823142	+	0.567835i	$0.192219\pi$
$648$	0	0
$649$	19.1051	0.749940
$650$	0	0
$651$	−8.62562	−0.338065
$652$	0	0
$653$	20.9665	0.820484	0.410242	−	0.911977i	$-0.365444\pi$
$653$	20.9665	0.820484	0.410242	+	0.911977i	$0.365444\pi$
$654$	0	0
$655$	−6.04196	−0.236079
$656$	0	0
$657$	13.5757	0.529637
$658$	0	0
$659$	−19.0359	−0.741532	−0.370766	−	0.928726i	$-0.620905\pi$
$659$	−19.0359	−0.741532	−0.370766	+	0.928726i	$0.620905\pi$
$660$	0	0
$661$	−18.6957	−0.727178	−0.363589	−	0.931559i	$-0.618449\pi$
$661$	−18.6957	−0.727178	−0.363589	+	0.931559i	$0.618449\pi$
$662$	0	0
$663$	−19.3862	−0.752898
$664$	0	0
$665$	−0.388134	−0.0150512
$666$	0	0
$667$	−0.584467	−0.0226306
$668$	0	0
$669$	−18.2606	−0.705994
$670$	0	0
$671$	−37.3746	−1.44283
$672$	0	0
$673$	42.1036	1.62297	0.811487	−	0.584370i	$-0.198658\pi$
$673$	42.1036	1.62297	0.811487	+	0.584370i	$0.198658\pi$
$674$	0	0
$675$	−4.84935	−0.186652
$676$	0	0
$677$	−41.1045	−1.57977	−0.789887	−	0.613252i	$-0.789861\pi$
$677$	−41.1045	−1.57977	−0.789887	+	0.613252i	$0.789861\pi$
$678$	0	0
$679$	13.7743	0.528608
$680$	0	0
$681$	5.13736	0.196864
$682$	0	0
$683$	2.64451	0.101189	0.0505947	−	0.998719i	$-0.483888\pi$
$683$	2.64451	0.101189	0.0505947	+	0.998719i	$0.483888\pi$
$684$	0	0
$685$	4.16164	0.159008
$686$	0	0
$687$	10.4750	0.399645
$688$	0	0
$689$	32.7660	1.24828
$690$	0	0
$691$	−8.59120	−0.326825	−0.163412	−	0.986558i	$-0.552250\pi$
$691$	−8.59120	−0.326825	−0.163412	+	0.986558i	$0.552250\pi$
$692$	0	0
$693$	6.41859	0.243822
$694$	0	0
$695$	−3.91964	−0.148681
$696$	0	0
$697$	43.7866	1.65854
$698$	0	0
$699$	17.5271	0.662935
$700$	0	0
$701$	24.5073	0.925626	0.462813	−	0.886456i	$-0.346840\pi$
$701$	24.5073	0.925626	0.462813	+	0.886456i	$0.346840\pi$
$702$	0	0
$703$	7.37540	0.278168
$704$	0	0
$705$	1.86565	0.0702645
$706$	0	0
$707$	−8.15009	−0.306516
$708$	0	0
$709$	30.1710	1.13310	0.566548	−	0.824029i	$-0.308278\pi$
$709$	30.1710	1.13310	0.566548	+	0.824029i	$0.308278\pi$
$710$	0	0
$711$	1.01431	0.0380396
$712$	0	0
$713$	−29.6899	−1.11190
$714$	0	0
$715$	−9.68391	−0.362158
$716$	0	0
$717$	21.7777	0.813303
$718$	0	0
$719$	−22.5815	−0.842149	−0.421074	−	0.907026i	$-0.638347\pi$
$719$	−22.5815	−0.842149	−0.421074	+	0.907026i	$0.638347\pi$
$720$	0	0
$721$	−12.8372	−0.478081
$722$	0	0
$723$	10.3611	0.385333
$724$	0	0
$725$	−0.823426	−0.0305813
$726$	0	0
$727$	−16.6904	−0.619013	−0.309507	−	0.950897i	$-0.600164\pi$
$727$	−16.6904	−0.619013	−0.309507	+	0.950897i	$0.600164\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−45.4383	−1.68060
$732$	0	0
$733$	23.0956	0.853055	0.426528	−	0.904475i	$-0.359737\pi$
$733$	23.0956	0.853055	0.426528	+	0.904475i	$0.359737\pi$
$734$	0	0
$735$	0.388134	0.0143165
$736$	0	0
$737$	−96.2660	−3.54601
$738$	0	0
$739$	23.6754	0.870913	0.435456	−	0.900210i	$-0.356587\pi$
$739$	23.6754	0.870913	0.435456	+	0.900210i	$0.356587\pi$
$740$	0	0
$741$	3.88714	0.142798
$742$	0	0
$743$	8.20020	0.300836	0.150418	−	0.988622i	$-0.451938\pi$
$743$	8.20020	0.300836	0.150418	+	0.988622i	$0.451938\pi$
$744$	0	0
$745$	3.78539	0.138686
$746$	0	0
$747$	4.32147	0.158114
$748$	0	0
$749$	−8.12463	−0.296867
$750$	0	0
$751$	53.7070	1.95980	0.979899	−	0.199494i	$-0.0639299\pi$
$751$	53.7070	1.95980	0.979899	+	0.199494i	$0.0639299\pi$
$752$	0	0
$753$	−10.9146	−0.397750
$754$	0	0
$755$	0.857552	0.0312095
$756$	0	0
$757$	48.5136	1.76326	0.881628	−	0.471945i	$-0.156448\pi$
$757$	48.5136	1.76326	0.881628	+	0.471945i	$0.156448\pi$
$758$	0	0
$759$	22.0932	0.801932
$760$	0	0
$761$	0.788644	0.0285883	0.0142942	−	0.999898i	$-0.495450\pi$
$761$	0.788644	0.0285883	0.0142942	+	0.999898i	$0.495450\pi$
$762$	0	0
$763$	20.3356	0.736199
$764$	0	0
$765$	−1.93573	−0.0699863
$766$	0	0
$767$	−11.5702	−0.417774
$768$	0	0
$769$	9.62974	0.347257	0.173629	−	0.984811i	$-0.444451\pi$
$769$	9.62974	0.347257	0.173629	+	0.984811i	$0.444451\pi$
$770$	0	0
$771$	−3.28123	−0.118171
$772$	0	0
$773$	−35.2618	−1.26828	−0.634139	−	0.773219i	$-0.718645\pi$
$773$	−35.2618	−1.26828	−0.634139	+	0.773219i	$0.718645\pi$
$774$	0	0
$775$	−41.8287	−1.50253
$776$	0	0
$777$	−7.37540	−0.264591
$778$	0	0
$779$	−8.77968	−0.314565
$780$	0	0
$781$	27.2712	0.975841
$782$	0	0
$783$	0.169801	0.00606820
$784$	0	0
$785$	−4.01793	−0.143406
$786$	0	0
$787$	40.3469	1.43821	0.719106	−	0.694900i	$-0.244551\pi$
$787$	40.3469	1.43821	0.719106	+	0.694900i	$0.244551\pi$
$788$	0	0
$789$	18.0034	0.640938
$790$	0	0
$791$	−4.16980	−0.148261
$792$	0	0
$793$	22.6343	0.803767
$794$	0	0
$795$	3.27171	0.116036
$796$	0	0
$797$	5.29192	0.187449	0.0937247	−	0.995598i	$-0.470123\pi$
$797$	5.29192	0.187449	0.0937247	+	0.995598i	$0.470123\pi$
$798$	0	0
$799$	−23.9724	−0.848083
$800$	0	0
$801$	13.8955	0.490975
$802$	0	0
$803$	−87.1365	−3.07498
$804$	0	0
$805$	1.33598	0.0470872
$806$	0	0
$807$	−0.886737	−0.0312146
$808$	0	0
$809$	−54.6544	−1.92155	−0.960774	−	0.277333i	$-0.910549\pi$
$809$	−54.6544	−1.92155	−0.960774	+	0.277333i	$0.910549\pi$
$810$	0	0
$811$	−34.1414	−1.19887	−0.599433	−	0.800425i	$-0.704607\pi$
$811$	−34.1414	−1.19887	−0.599433	+	0.800425i	$0.704607\pi$
$812$	0	0
$813$	−7.43917	−0.260903
$814$	0	0
$815$	3.30847	0.115891
$816$	0	0
$817$	9.11087	0.318749
$818$	0	0
$819$	−3.88714	−0.135828
$820$	0	0
$821$	−47.8638	−1.67046	−0.835229	−	0.549902i	$-0.814665\pi$
$821$	−47.8638	−1.67046	−0.835229	+	0.549902i	$0.814665\pi$
$822$	0	0
$823$	8.87160	0.309244	0.154622	−	0.987974i	$-0.450584\pi$
$823$	8.87160	0.309244	0.154622	+	0.987974i	$0.450584\pi$
$824$	0	0
$825$	31.1260	1.08367
$826$	0	0
$827$	−22.8022	−0.792910	−0.396455	−	0.918054i	$-0.629760\pi$
$827$	−22.8022	−0.792910	−0.396455	+	0.918054i	$0.629760\pi$
$828$	0	0
$829$	−17.9774	−0.624381	−0.312190	−	0.950020i	$-0.601063\pi$
$829$	−17.9774	−0.624381	−0.312190	+	0.950020i	$0.601063\pi$
$830$	0	0
$831$	−7.40189	−0.256769
$832$	0	0
$833$	−4.98727	−0.172799
$834$	0	0
$835$	−2.71616	−0.0939967
$836$	0	0
$837$	8.62562	0.298145
$838$	0	0
$839$	41.0481	1.41714	0.708569	−	0.705641i	$-0.249341\pi$
$839$	41.0481	1.41714	0.708569	+	0.705641i	$0.249341\pi$
$840$	0	0
$841$	−28.9712	−0.999006
$842$	0	0
$843$	−1.98187	−0.0682591
$844$	0	0
$845$	0.818905	0.0281712
$846$	0	0
$847$	−30.1983	−1.03762
$848$	0	0
$849$	−7.83519	−0.268903
$850$	0	0
$851$	−25.3866	−0.870242
$852$	0	0
$853$	10.2504	0.350966	0.175483	−	0.984482i	$-0.443851\pi$
$853$	10.2504	0.350966	0.175483	+	0.984482i	$0.443851\pi$
$854$	0	0
$855$	0.388134	0.0132739
$856$	0	0
$857$	35.3195	1.20649	0.603246	−	0.797555i	$-0.293874\pi$
$857$	35.3195	1.20649	0.603246	+	0.797555i	$0.293874\pi$
$858$	0	0
$859$	29.0481	0.991109	0.495555	−	0.868577i	$-0.334965\pi$
$859$	29.0481	0.991109	0.495555	+	0.868577i	$0.334965\pi$
$860$	0	0
$861$	8.77968	0.299211
$862$	0	0
$863$	−37.0768	−1.26211	−0.631054	−	0.775739i	$-0.717378\pi$
$863$	−37.0768	−1.26211	−0.631054	+	0.775739i	$0.717378\pi$
$864$	0	0
$865$	5.68241	0.193208
$866$	0	0
$867$	7.87283	0.267375
$868$	0	0
$869$	−6.51043	−0.220851
$870$	0	0
$871$	58.2994	1.97540
$872$	0	0
$873$	−13.7743	−0.466189
$874$	0	0
$875$	3.82287	0.129236
$876$	0	0
$877$	−40.8400	−1.37907	−0.689535	−	0.724253i	$-0.742185\pi$
$877$	−40.8400	−1.37907	−0.689535	+	0.724253i	$0.742185\pi$
$878$	0	0
$879$	3.83178	0.129243
$880$	0	0
$881$	−20.8316	−0.701835	−0.350918	−	0.936406i	$-0.614130\pi$
$881$	−20.8316	−0.701835	−0.350918	+	0.936406i	$0.614130\pi$
$882$	0	0
$883$	−17.8077	−0.599276	−0.299638	−	0.954053i	$-0.596866\pi$
$883$	−17.8077	−0.599276	−0.299638	+	0.954053i	$0.596866\pi$
$884$	0	0
$885$	−1.15529	−0.0388346
$886$	0	0
$887$	12.0913	0.405985	0.202993	−	0.979180i	$-0.434933\pi$
$887$	12.0913	0.405985	0.202993	+	0.979180i	$0.434933\pi$
$888$	0	0
$889$	−6.20025	−0.207950
$890$	0	0
$891$	−6.41859	−0.215031
$892$	0	0
$893$	4.80672	0.160851
$894$	0	0
$895$	5.00727	0.167375
$896$	0	0
$897$	−13.3798	−0.446738
$898$	0	0
$899$	1.46464	0.0488485
$900$	0	0
$901$	−42.0393	−1.40053
$902$	0	0
$903$	−9.11087	−0.303191
$904$	0	0
$905$	−4.80886	−0.159852
$906$	0	0
$907$	48.9499	1.62536	0.812678	−	0.582713i	$-0.198009\pi$
$907$	48.9499	1.62536	0.812678	+	0.582713i	$0.198009\pi$
$908$	0	0
$909$	8.15009	0.270321
$910$	0	0
$911$	−27.4903	−0.910796	−0.455398	−	0.890288i	$-0.650503\pi$
$911$	−27.4903	−0.910796	−0.455398	+	0.890288i	$0.650503\pi$
$912$	0	0
$913$	−27.7377	−0.917985
$914$	0	0
$915$	2.26005	0.0747150
$916$	0	0
$917$	15.5667	0.514057
$918$	0	0
$919$	−26.3177	−0.868141	−0.434070	−	0.900879i	$-0.642923\pi$
$919$	−26.3177	−0.868141	−0.434070	+	0.900879i	$0.642923\pi$
$920$	0	0
$921$	−1.46279	−0.0482007
$922$	0	0
$923$	−16.5156	−0.543618
$924$	0	0
$925$	−35.7659	−1.17598
$926$	0	0
$927$	12.8372	0.421628
$928$	0	0
$929$	27.5260	0.903099	0.451550	−	0.892246i	$-0.350871\pi$
$929$	27.5260	0.903099	0.451550	+	0.892246i	$0.350871\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	12.0213	0.393559
$934$	0	0
$935$	12.4246	0.406329
$936$	0	0
$937$	−18.9196	−0.618075	−0.309038	−	0.951050i	$-0.600007\pi$
$937$	−18.9196	−0.618075	−0.309038	+	0.951050i	$0.600007\pi$
$938$	0	0
$939$	5.36938	0.175223
$940$	0	0
$941$	33.7139	1.09904	0.549521	−	0.835480i	$-0.314810\pi$
$941$	33.7139	1.09904	0.549521	+	0.835480i	$0.314810\pi$
$942$	0	0
$943$	30.2202	0.984106
$944$	0	0
$945$	−0.388134	−0.0126260
$946$	0	0
$947$	0.655109	0.0212882	0.0106441	−	0.999943i	$-0.496612\pi$
$947$	0.655109	0.0212882	0.0106441	+	0.999943i	$0.496612\pi$
$948$	0	0
$949$	52.7705	1.71300
$950$	0	0
$951$	−12.6674	−0.410768
$952$	0	0
$953$	33.6974	1.09157	0.545783	−	0.837927i	$-0.316232\pi$
$953$	33.6974	1.09157	0.545783	+	0.837927i	$0.316232\pi$
$954$	0	0
$955$	−9.72446	−0.314676
$956$	0	0
$957$	−1.08988	−0.0352309
$958$	0	0
$959$	−10.7222	−0.346237
$960$	0	0
$961$	43.4013	1.40004
$962$	0	0
$963$	8.12463	0.261812
$964$	0	0
$965$	3.41534	0.109944
$966$	0	0
$967$	31.1237	1.00087	0.500436	−	0.865774i	$-0.333173\pi$
$967$	31.1237	1.00087	0.500436	+	0.865774i	$0.333173\pi$
$968$	0	0
$969$	−4.98727	−0.160214
$970$	0	0
$971$	−0.890148	−0.0285662	−0.0142831	−	0.999898i	$-0.504547\pi$
$971$	−0.890148	−0.0285662	−0.0142831	+	0.999898i	$0.504547\pi$
$972$	0	0
$973$	10.0987	0.323749
$974$	0	0
$975$	−18.8501	−0.603687
$976$	0	0
$977$	−28.1686	−0.901192	−0.450596	−	0.892728i	$-0.648789\pi$
$977$	−28.1686	−0.901192	−0.450596	+	0.892728i	$0.648789\pi$
$978$	0	0
$979$	−89.1898	−2.85052
$980$	0	0
$981$	−20.3356	−0.649267
$982$	0	0
$983$	32.9894	1.05220	0.526099	−	0.850424i	$-0.323654\pi$
$983$	32.9894	1.05220	0.526099	+	0.850424i	$0.323654\pi$
$984$	0	0
$985$	−1.86636	−0.0594673
$986$	0	0
$987$	−4.80672	−0.153000
$988$	0	0
$989$	−31.3602	−0.997197
$990$	0	0
$991$	−32.9379	−1.04631	−0.523154	−	0.852238i	$-0.675245\pi$
$991$	−32.9379	−1.04631	−0.523154	+	0.852238i	$0.675245\pi$
$992$	0	0
$993$	−15.2575	−0.484183
$994$	0	0
$995$	−3.73130	−0.118290
$996$	0	0
$997$	−36.5637	−1.15798	−0.578992	−	0.815334i	$-0.696554\pi$
$997$	−36.5637	−1.15798	−0.578992	+	0.815334i	$0.696554\pi$
$998$	0	0
$999$	7.37540	0.233347

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6384.2.a.cf.1.3		5
4.3	odd	2		399.2.a.g.1.4	✓	5
12.11	even	2		1197.2.a.o.1.2		5
20.19	odd	2		9975.2.a.bp.1.2		5
28.27	even	2		2793.2.a.bg.1.4		5
76.75	even	2		7581.2.a.w.1.2		5
84.83	odd	2		8379.2.a.cb.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
399.2.a.g.1.4	✓	5	4.3	odd	2
1197.2.a.o.1.2		5	12.11	even	2
2793.2.a.bg.1.4		5	28.27	even	2
6384.2.a.cf.1.3		5	1.1	even	1	trivial
7581.2.a.w.1.2		5	76.75	even	2
8379.2.a.cb.1.2		5	84.83	odd	2
9975.2.a.bp.1.2		5	20.19	odd	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:	\( N \)	\(=\)	\( 6384 = 2^{4} \cdot 3 \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6384.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +0.388134 q^{5} -1.00000 q^{7} +1.00000 q^{9} -6.41859 q^{11} +3.88714 q^{13} +0.388134 q^{15} -4.98727 q^{17} +1.00000 q^{19} -1.00000 q^{21} -3.44207 q^{23} -4.84935 q^{25} +1.00000 q^{27} +0.169801 q^{29} +8.62562 q^{31} -6.41859 q^{33} -0.388134 q^{35} +7.37540 q^{37} +3.88714 q^{39} -8.77968 q^{41} +9.11087 q^{43} +0.388134 q^{45} +4.80672 q^{47} +1.00000 q^{49} -4.98727 q^{51} +8.42933 q^{53} -2.49127 q^{55} +1.00000 q^{57} -2.97652 q^{59} +5.82287 q^{61} -1.00000 q^{63} +1.50873 q^{65} +14.9980 q^{67} -3.44207 q^{69} -4.24879 q^{71} +13.5757 q^{73} -4.84935 q^{75} +6.41859 q^{77} +1.01431 q^{79} +1.00000 q^{81} +4.32147 q^{83} -1.93573 q^{85} +0.169801 q^{87} +13.8955 q^{89} -3.88714 q^{91} +8.62562 q^{93} +0.388134 q^{95} -13.7743 q^{97} -6.41859 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{3} + 4 q^{5} - 5 q^{7} + 5 q^{9} - 8 q^{11} - 6 q^{13} + 4 q^{15} + 12 q^{17} + 5 q^{19} - 5 q^{21} - 12 q^{23} + 15 q^{25} + 5 q^{27} + 4 q^{29} + 8 q^{31} - 8 q^{33} - 4 q^{35} + 2 q^{37} - 6 q^{39}+ \cdots - 8 q^{99}+O(q^{100}) \) 5 * q + 5 * q^3 + 4 * q^5 - 5 * q^7 + 5 * q^9 - 8 * q^11 - 6 * q^13 + 4 * q^15 + 12 * q^17 + 5 * q^19 - 5 * q^21 - 12 * q^23 + 15 * q^25 + 5 * q^27 + 4 * q^29 + 8 * q^31 - 8 * q^33 - 4 * q^35 + 2 * q^37 - 6 * q^39 + 10 * q^41 + 16 * q^43 + 4 * q^45 + 2 * q^47 + 5 * q^49 + 12 * q^51 + 12 * q^55 + 5 * q^57 + 4 * q^59 - 14 * q^61 - 5 * q^63 + 32 * q^65 + 20 * q^67 - 12 * q^69 + 6 * q^71 + 10 * q^73 + 15 * q^75 + 8 * q^77 + 5 * q^81 - 6 * q^83 + 8 * q^85 + 4 * q^87 + 26 * q^89 + 6 * q^91 + 8 * q^93 + 4 * q^95 - 18 * q^97 - 8 * q^99

Introduction

L-functions

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