LMFDB - Embedded newform 3192.2.a.u.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3192,2,Mod(1,3192)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3192, 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("3192.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3192 = 2^{3} \cdot 3 \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3192.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:	$25.4882483252$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.568.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x - 2 $ x^3 - x^2 - 6*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.363328$ of defining polynomial
Character	$\chi$	$=$	3192.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +3.50466 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +3.50466 q^{5} -1.00000 q^{7} +1.00000 q^{9} +0.726656 q^{11} +4.00000 q^{13} -3.50466 q^{15} +3.50466 q^{17} -1.00000 q^{19} +1.00000 q^{21} +0.726656 q^{23} +7.28267 q^{25} -1.00000 q^{27} +2.77801 q^{29} -1.27334 q^{31} -0.726656 q^{33} -3.50466 q^{35} +2.00000 q^{37} -4.00000 q^{39} -3.55602 q^{41} -2.72666 q^{43} +3.50466 q^{45} +0.778008 q^{47} +1.00000 q^{49} -3.50466 q^{51} +0.231321 q^{53} +2.54669 q^{55} +1.00000 q^{57} +12.5653 q^{59} -10.5653 q^{61} -1.00000 q^{63} +14.0187 q^{65} -0.726656 q^{67} -0.726656 q^{69} +11.7873 q^{71} +0.443984 q^{73} -7.28267 q^{75} -0.726656 q^{77} +6.28267 q^{79} +1.00000 q^{81} -7.78734 q^{83} +12.2827 q^{85} -2.77801 q^{87} +7.55602 q^{89} -4.00000 q^{91} +1.27334 q^{93} -3.50466 q^{95} +3.71733 q^{97} +0.726656 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} - 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 - 3 * q^7 + 3 * q^9	$ 3 q - 3 q^{3} - 3 q^{7} + 3 q^{9} - 2 q^{11} + 12 q^{13} - 3 q^{19} + 3 q^{21} - 2 q^{23} + 5 q^{25} - 3 q^{27} + 2 q^{29} - 8 q^{31} + 2 q^{33} + 6 q^{37} - 12 q^{39} + 2 q^{41} - 4 q^{43} - 4 q^{47} + 3 q^{49} - 14 q^{53} + 16 q^{55} + 3 q^{57} + 4 q^{59} + 2 q^{61} - 3 q^{63} + 2 q^{67} + 2 q^{69} + 8 q^{71} + 14 q^{73} - 5 q^{75} + 2 q^{77} + 2 q^{79} + 3 q^{81} + 4 q^{83} + 20 q^{85} - 2 q^{87} + 10 q^{89} - 12 q^{91} + 8 q^{93} + 28 q^{97} - 2 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 - 3 * q^7 + 3 * q^9 - 2 * q^11 + 12 * q^13 - 3 * q^19 + 3 * q^21 - 2 * q^23 + 5 * q^25 - 3 * q^27 + 2 * q^29 - 8 * q^31 + 2 * q^33 + 6 * q^37 - 12 * q^39 + 2 * q^41 - 4 * q^43 - 4 * q^47 + 3 * q^49 - 14 * q^53 + 16 * q^55 + 3 * q^57 + 4 * q^59 + 2 * q^61 - 3 * q^63 + 2 * q^67 + 2 * q^69 + 8 * q^71 + 14 * q^73 - 5 * q^75 + 2 * q^77 + 2 * q^79 + 3 * q^81 + 4 * q^83 + 20 * q^85 - 2 * q^87 + 10 * q^89 - 12 * q^91 + 8 * q^93 + 28 * q^97 - 2 * q^99

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
$5$	3.50466	1.56733	0.783667	−	0.621181i	$-0.213347\pi$
$5$	3.50466	1.56733	0.783667	+	0.621181i	$0.213347\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0.726656	0.219095	0.109548	−	0.993982i	$-0.465060\pi$
$11$	0.726656	0.219095	0.109548	+	0.993982i	$0.465060\pi$
$12$	0	0
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	0	0
$15$	−3.50466	−0.904900
$16$	0	0
$17$	3.50466	0.850006	0.425003	−	0.905192i	$-0.360273\pi$
$17$	3.50466	0.850006	0.425003	+	0.905192i	$0.360273\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	0.726656	0.151518	0.0757592	−	0.997126i	$-0.475862\pi$
$23$	0.726656	0.151518	0.0757592	+	0.997126i	$0.475862\pi$
$24$	0	0
$25$	7.28267	1.45653
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	2.77801	0.515863	0.257932	−	0.966163i	$-0.416959\pi$
$29$	2.77801	0.515863	0.257932	+	0.966163i	$0.416959\pi$
$30$	0	0
$31$	−1.27334	−0.228699	−0.114350	−	0.993441i	$-0.536478\pi$
$31$	−1.27334	−0.228699	−0.114350	+	0.993441i	$0.536478\pi$
$32$	0	0
$33$	−0.726656	−0.126495
$34$	0	0
$35$	−3.50466	−0.592396
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	0	0
$39$	−4.00000	−0.640513
$40$	0	0
$41$	−3.55602	−0.555356	−0.277678	−	0.960674i	$-0.589565\pi$
$41$	−3.55602	−0.555356	−0.277678	+	0.960674i	$0.589565\pi$
$42$	0	0
$43$	−2.72666	−0.415811	−0.207906	−	0.978149i	$-0.566665\pi$
$43$	−2.72666	−0.415811	−0.207906	+	0.978149i	$0.566665\pi$
$44$	0	0
$45$	3.50466	0.522445
$46$	0	0
$47$	0.778008	0.113484	0.0567421	−	0.998389i	$-0.481929\pi$
$47$	0.778008	0.113484	0.0567421	+	0.998389i	$0.481929\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−3.50466	−0.490751
$52$	0	0
$53$	0.231321	0.0317744	0.0158872	−	0.999874i	$-0.494943\pi$
$53$	0.231321	0.0317744	0.0158872	+	0.999874i	$0.494943\pi$
$54$	0	0
$55$	2.54669	0.343395
$56$	0	0
$57$	1.00000	0.132453
$58$	0	0
$59$	12.5653	1.63587	0.817934	−	0.575312i	$-0.195119\pi$
$59$	12.5653	1.63587	0.817934	+	0.575312i	$0.195119\pi$
$60$	0	0
$61$	−10.5653	−1.35275	−0.676377	−	0.736556i	$-0.736451\pi$
$61$	−10.5653	−1.35275	−0.676377	+	0.736556i	$0.736451\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	14.0187	1.73880
$66$	0	0
$67$	−0.726656	−0.0887752	−0.0443876	−	0.999014i	$-0.514134\pi$
$67$	−0.726656	−0.0887752	−0.0443876	+	0.999014i	$0.514134\pi$
$68$	0	0
$69$	−0.726656	−0.0874792
$70$	0	0
$71$	11.7873	1.39890	0.699450	−	0.714682i	$-0.253428\pi$
$71$	11.7873	1.39890	0.699450	+	0.714682i	$0.253428\pi$
$72$	0	0
$73$	0.443984	0.0519644	0.0259822	−	0.999662i	$-0.491729\pi$
$73$	0.443984	0.0519644	0.0259822	+	0.999662i	$0.491729\pi$
$74$	0	0
$75$	−7.28267	−0.840931
$76$	0	0
$77$	−0.726656	−0.0828102
$78$	0	0
$79$	6.28267	0.706856	0.353428	−	0.935462i	$-0.385016\pi$
$79$	6.28267	0.706856	0.353428	+	0.935462i	$0.385016\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−7.78734	−0.854771	−0.427386	−	0.904069i	$-0.640565\pi$
$83$	−7.78734	−0.854771	−0.427386	+	0.904069i	$0.640565\pi$
$84$	0	0
$85$	12.2827	1.33224
$86$	0	0
$87$	−2.77801	−0.297834
$88$	0	0
$89$	7.55602	0.800936	0.400468	−	0.916311i	$-0.368848\pi$
$89$	7.55602	0.800936	0.400468	+	0.916311i	$0.368848\pi$
$90$	0	0
$91$	−4.00000	−0.419314
$92$	0	0
$93$	1.27334	0.132040
$94$	0	0
$95$	−3.50466	−0.359571
$96$	0	0
$97$	3.71733	0.377437	0.188719	−	0.982031i	$-0.439567\pi$
$97$	3.71733	0.377437	0.188719	+	0.982031i	$0.439567\pi$
$98$	0	0
$99$	0.726656	0.0730317
$100$	0	0
$101$	−3.50466	−0.348727	−0.174364	−	0.984681i	$-0.555787\pi$
$101$	−3.50466	−0.348727	−0.174364	+	0.984681i	$0.555787\pi$
$102$	0	0
$103$	−12.5653	−1.23810	−0.619050	−	0.785352i	$-0.712482\pi$
$103$	−12.5653	−1.23810	−0.619050	+	0.785352i	$0.712482\pi$
$104$	0	0
$105$	3.50466	0.342020
$106$	0	0
$107$	−6.33402	−0.612333	−0.306167	−	0.951978i	$-0.599046\pi$
$107$	−6.33402	−0.612333	−0.306167	+	0.951978i	$0.599046\pi$
$108$	0	0
$109$	−5.00933	−0.479807	−0.239903	−	0.970797i	$-0.577116\pi$
$109$	−5.00933	−0.479807	−0.239903	+	0.970797i	$0.577116\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	0	0
$113$	8.23132	0.774338	0.387169	−	0.922009i	$-0.373453\pi$
$113$	8.23132	0.774338	0.387169	+	0.922009i	$0.373453\pi$
$114$	0	0
$115$	2.54669	0.237480
$116$	0	0
$117$	4.00000	0.369800
$118$	0	0
$119$	−3.50466	−0.321272
$120$	0	0
$121$	−10.4720	−0.951997
$122$	0	0
$123$	3.55602	0.320635
$124$	0	0
$125$	8.00000	0.715542
$126$	0	0
$127$	6.17997	0.548384	0.274192	−	0.961675i	$-0.411590\pi$
$127$	6.17997	0.548384	0.274192	+	0.961675i	$0.411590\pi$
$128$	0	0
$129$	2.72666	0.240069
$130$	0	0
$131$	−2.79667	−0.244346	−0.122173	−	0.992509i	$-0.538986\pi$
$131$	−2.79667	−0.244346	−0.122173	+	0.992509i	$0.538986\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	−3.50466	−0.301633
$136$	0	0
$137$	−7.45331	−0.636779	−0.318390	−	0.947960i	$-0.603142\pi$
$137$	−7.45331	−0.636779	−0.318390	+	0.947960i	$0.603142\pi$
$138$	0	0
$139$	5.55602	0.471255	0.235628	−	0.971843i	$-0.424285\pi$
$139$	5.55602	0.471255	0.235628	+	0.971843i	$0.424285\pi$
$140$	0	0
$141$	−0.778008	−0.0655201
$142$	0	0
$143$	2.90663	0.243064
$144$	0	0
$145$	9.73599	0.808530
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	−15.0280	−1.23114	−0.615570	−	0.788082i	$-0.711074\pi$
$149$	−15.0280	−1.23114	−0.615570	+	0.788082i	$0.711074\pi$
$150$	0	0
$151$	4.72666	0.384650	0.192325	−	0.981331i	$-0.438397\pi$
$151$	4.72666	0.384650	0.192325	+	0.981331i	$0.438397\pi$
$152$	0	0
$153$	3.50466	0.283335
$154$	0	0
$155$	−4.46264	−0.358448
$156$	0	0
$157$	7.45331	0.594839	0.297420	−	0.954747i	$-0.403874\pi$
$157$	7.45331	0.594839	0.297420	+	0.954747i	$0.403874\pi$
$158$	0	0
$159$	−0.231321	−0.0183449
$160$	0	0
$161$	−0.726656	−0.0572686
$162$	0	0
$163$	16.7453	1.31159	0.655797	−	0.754937i	$-0.272333\pi$
$163$	16.7453	1.31159	0.655797	+	0.754937i	$0.272333\pi$
$164$	0	0
$165$	−2.54669	−0.198259
$166$	0	0
$167$	4.10270	0.317477	0.158738	−	0.987321i	$-0.449257\pi$
$167$	4.10270	0.317477	0.158738	+	0.987321i	$0.449257\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	0	0
$173$	9.00933	0.684967	0.342483	−	0.939524i	$-0.388732\pi$
$173$	9.00933	0.684967	0.342483	+	0.939524i	$0.388732\pi$
$174$	0	0
$175$	−7.28267	−0.550518
$176$	0	0
$177$	−12.5653	−0.944469
$178$	0	0
$179$	−7.68463	−0.574376	−0.287188	−	0.957874i	$-0.592721\pi$
$179$	−7.68463	−0.574376	−0.287188	+	0.957874i	$0.592721\pi$
$180$	0	0
$181$	25.7360	1.91294	0.956470	−	0.291829i	$-0.0942640\pi$
$181$	25.7360	1.91294	0.956470	+	0.291829i	$0.0942640\pi$
$182$	0	0
$183$	10.5653	0.781013
$184$	0	0
$185$	7.00933	0.515336
$186$	0	0
$187$	2.54669	0.186232
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	9.29200	0.672346	0.336173	−	0.941800i	$-0.390867\pi$
$191$	9.29200	0.672346	0.336173	+	0.941800i	$0.390867\pi$
$192$	0	0
$193$	16.0187	1.15305	0.576524	−	0.817080i	$-0.304409\pi$
$193$	16.0187	1.15305	0.576524	+	0.817080i	$0.304409\pi$
$194$	0	0
$195$	−14.0187	−1.00390
$196$	0	0
$197$	14.0000	0.997459	0.498729	−	0.866758i	$-0.333800\pi$
$197$	14.0000	0.997459	0.498729	+	0.866758i	$0.333800\pi$
$198$	0	0
$199$	−9.55602	−0.677408	−0.338704	−	0.940893i	$-0.609989\pi$
$199$	−9.55602	−0.677408	−0.338704	+	0.940893i	$0.609989\pi$
$200$	0	0
$201$	0.726656	0.0512544
$202$	0	0
$203$	−2.77801	−0.194978
$204$	0	0
$205$	−12.4626	−0.870429
$206$	0	0
$207$	0.726656	0.0505061
$208$	0	0
$209$	−0.726656	−0.0502639
$210$	0	0
$211$	25.7546	1.77302	0.886511	−	0.462707i	$-0.153122\pi$
$211$	25.7546	1.77302	0.886511	+	0.462707i	$0.153122\pi$
$212$	0	0
$213$	−11.7873	−0.807655
$214$	0	0
$215$	−9.55602	−0.651715
$216$	0	0
$217$	1.27334	0.0864402
$218$	0	0
$219$	−0.443984	−0.0300017
$220$	0	0
$221$	14.0187	0.942997
$222$	0	0
$223$	−11.2920	−0.756168	−0.378084	−	0.925771i	$-0.623417\pi$
$223$	−11.2920	−0.756168	−0.378084	+	0.925771i	$0.623417\pi$
$224$	0	0
$225$	7.28267	0.485511
$226$	0	0
$227$	5.55602	0.368766	0.184383	−	0.982854i	$-0.440971\pi$
$227$	5.55602	0.368766	0.184383	+	0.982854i	$0.440971\pi$
$228$	0	0
$229$	−15.0280	−0.993077	−0.496539	−	0.868015i	$-0.665396\pi$
$229$	−15.0280	−0.993077	−0.496539	+	0.868015i	$0.665396\pi$
$230$	0	0
$231$	0.726656	0.0478105
$232$	0	0
$233$	−6.10270	−0.399801	−0.199901	−	0.979816i	$-0.564062\pi$
$233$	−6.10270	−0.399801	−0.199901	+	0.979816i	$0.564062\pi$
$234$	0	0
$235$	2.72666	0.177867
$236$	0	0
$237$	−6.28267	−0.408103
$238$	0	0
$239$	5.18930	0.335668	0.167834	−	0.985815i	$-0.446323\pi$
$239$	5.18930	0.335668	0.167834	+	0.985815i	$0.446323\pi$
$240$	0	0
$241$	13.7360	0.884813	0.442406	−	0.896815i	$-0.354125\pi$
$241$	13.7360	0.884813	0.442406	+	0.896815i	$0.354125\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	3.50466	0.223905
$246$	0	0
$247$	−4.00000	−0.254514
$248$	0	0
$249$	7.78734	0.493502
$250$	0	0
$251$	10.7967	0.681479	0.340740	−	0.940158i	$-0.389323\pi$
$251$	10.7967	0.681479	0.340740	+	0.940158i	$0.389323\pi$
$252$	0	0
$253$	0.528030	0.0331969
$254$	0	0
$255$	−12.2827	−0.769171
$256$	0	0
$257$	7.45331	0.464925	0.232462	−	0.972605i	$-0.425322\pi$
$257$	7.45331	0.464925	0.232462	+	0.972605i	$0.425322\pi$
$258$	0	0
$259$	−2.00000	−0.124274
$260$	0	0
$261$	2.77801	0.171954
$262$	0	0
$263$	−18.6426	−1.14955	−0.574776	−	0.818311i	$-0.694911\pi$
$263$	−18.6426	−1.14955	−0.574776	+	0.818311i	$0.694911\pi$
$264$	0	0
$265$	0.810702	0.0498010
$266$	0	0
$267$	−7.55602	−0.462421
$268$	0	0
$269$	7.65872	0.466960	0.233480	−	0.972362i	$-0.424989\pi$
$269$	7.65872	0.466960	0.233480	+	0.972362i	$0.424989\pi$
$270$	0	0
$271$	11.1120	0.675008	0.337504	−	0.941324i	$-0.390417\pi$
$271$	11.1120	0.675008	0.337504	+	0.941324i	$0.390417\pi$
$272$	0	0
$273$	4.00000	0.242091
$274$	0	0
$275$	5.29200	0.319120
$276$	0	0
$277$	−4.30133	−0.258442	−0.129221	−	0.991616i	$-0.541248\pi$
$277$	−4.30133	−0.258442	−0.129221	+	0.991616i	$0.541248\pi$
$278$	0	0
$279$	−1.27334	−0.0762331
$280$	0	0
$281$	−0.759350	−0.0452991	−0.0226495	−	0.999743i	$-0.507210\pi$
$281$	−0.759350	−0.0452991	−0.0226495	+	0.999743i	$0.507210\pi$
$282$	0	0
$283$	4.99067	0.296665	0.148332	−	0.988938i	$-0.452609\pi$
$283$	4.99067	0.296665	0.148332	+	0.988938i	$0.452609\pi$
$284$	0	0
$285$	3.50466	0.207598
$286$	0	0
$287$	3.55602	0.209905
$288$	0	0
$289$	−4.71733	−0.277490
$290$	0	0
$291$	−3.71733	−0.217914
$292$	0	0
$293$	−22.1400	−1.29343	−0.646717	−	0.762730i	$-0.723858\pi$
$293$	−22.1400	−1.29343	−0.646717	+	0.762730i	$0.723858\pi$
$294$	0	0
$295$	44.0373	2.56395
$296$	0	0
$297$	−0.726656	−0.0421649
$298$	0	0
$299$	2.90663	0.168095
$300$	0	0
$301$	2.72666	0.157162
$302$	0	0
$303$	3.50466	0.201338
$304$	0	0
$305$	−37.0280	−2.12022
$306$	0	0
$307$	8.17997	0.466856	0.233428	−	0.972374i	$-0.425006\pi$
$307$	8.17997	0.466856	0.233428	+	0.972374i	$0.425006\pi$
$308$	0	0
$309$	12.5653	0.714817
$310$	0	0
$311$	5.76868	0.327112	0.163556	−	0.986534i	$-0.447704\pi$
$311$	5.76868	0.327112	0.163556	+	0.986534i	$0.447704\pi$
$312$	0	0
$313$	13.1120	0.741136	0.370568	−	0.928805i	$-0.379163\pi$
$313$	13.1120	0.741136	0.370568	+	0.928805i	$0.379163\pi$
$314$	0	0
$315$	−3.50466	−0.197465
$316$	0	0
$317$	−31.8060	−1.78640	−0.893201	−	0.449657i	$-0.851546\pi$
$317$	−31.8060	−1.78640	−0.893201	+	0.449657i	$0.851546\pi$
$318$	0	0
$319$	2.01866	0.113023
$320$	0	0
$321$	6.33402	0.353531
$322$	0	0
$323$	−3.50466	−0.195005
$324$	0	0
$325$	29.1307	1.61588
$326$	0	0
$327$	5.00933	0.277017
$328$	0	0
$329$	−0.778008	−0.0428930
$330$	0	0
$331$	−23.8387	−1.31029	−0.655146	−	0.755502i	$-0.727393\pi$
$331$	−23.8387	−1.31029	−0.655146	+	0.755502i	$0.727393\pi$
$332$	0	0
$333$	2.00000	0.109599
$334$	0	0
$335$	−2.54669	−0.139140
$336$	0	0
$337$	32.1214	1.74976	0.874881	−	0.484338i	$-0.160939\pi$
$337$	32.1214	1.74976	0.874881	+	0.484338i	$0.160939\pi$
$338$	0	0
$339$	−8.23132	−0.447064
$340$	0	0
$341$	−0.925283	−0.0501069
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−2.54669	−0.137109
$346$	0	0
$347$	−12.8294	−0.688716	−0.344358	−	0.938838i	$-0.611903\pi$
$347$	−12.8294	−0.688716	−0.344358	+	0.938838i	$0.611903\pi$
$348$	0	0
$349$	12.9066	0.690876	0.345438	−	0.938442i	$-0.387730\pi$
$349$	12.9066	0.690876	0.345438	+	0.938442i	$0.387730\pi$
$350$	0	0
$351$	−4.00000	−0.213504
$352$	0	0
$353$	8.53265	0.454147	0.227074	−	0.973878i	$-0.427084\pi$
$353$	8.53265	0.454147	0.227074	+	0.973878i	$0.427084\pi$
$354$	0	0
$355$	41.3107	2.19254
$356$	0	0
$357$	3.50466	0.185487
$358$	0	0
$359$	1.61462	0.0852166	0.0426083	−	0.999092i	$-0.486433\pi$
$359$	1.61462	0.0852166	0.0426083	+	0.999092i	$0.486433\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	10.4720	0.549636
$364$	0	0
$365$	1.55602	0.0814456
$366$	0	0
$367$	12.1027	0.631756	0.315878	−	0.948800i	$-0.397701\pi$
$367$	12.1027	0.631756	0.315878	+	0.948800i	$0.397701\pi$
$368$	0	0
$369$	−3.55602	−0.185119
$370$	0	0
$371$	−0.231321	−0.0120096
$372$	0	0
$373$	−8.01866	−0.415190	−0.207595	−	0.978215i	$-0.566564\pi$
$373$	−8.01866	−0.415190	−0.207595	+	0.978215i	$0.566564\pi$
$374$	0	0
$375$	−8.00000	−0.413118
$376$	0	0
$377$	11.1120	0.572299
$378$	0	0
$379$	22.7453	1.16835	0.584174	−	0.811628i	$-0.301419\pi$
$379$	22.7453	1.16835	0.584174	+	0.811628i	$0.301419\pi$
$380$	0	0
$381$	−6.17997	−0.316609
$382$	0	0
$383$	−9.91595	−0.506682	−0.253341	−	0.967377i	$-0.581529\pi$
$383$	−9.91595	−0.506682	−0.253341	+	0.967377i	$0.581529\pi$
$384$	0	0
$385$	−2.54669	−0.129791
$386$	0	0
$387$	−2.72666	−0.138604
$388$	0	0
$389$	−4.64939	−0.235733	−0.117867	−	0.993029i	$-0.537606\pi$
$389$	−4.64939	−0.235733	−0.117867	+	0.993029i	$0.537606\pi$
$390$	0	0
$391$	2.54669	0.128791
$392$	0	0
$393$	2.79667	0.141073
$394$	0	0
$395$	22.0187	1.10788
$396$	0	0
$397$	1.43466	0.0720033	0.0360016	−	0.999352i	$-0.488538\pi$
$397$	1.43466	0.0720033	0.0360016	+	0.999352i	$0.488538\pi$
$398$	0	0
$399$	−1.00000	−0.0500626
$400$	0	0
$401$	−11.7687	−0.587700	−0.293850	−	0.955852i	$-0.594937\pi$
$401$	−11.7687	−0.587700	−0.293850	+	0.955852i	$0.594937\pi$
$402$	0	0
$403$	−5.09337	−0.253719
$404$	0	0
$405$	3.50466	0.174148
$406$	0	0
$407$	1.45331	0.0720380
$408$	0	0
$409$	−6.01866	−0.297603	−0.148802	−	0.988867i	$-0.547542\pi$
$409$	−6.01866	−0.297603	−0.148802	+	0.988867i	$0.547542\pi$
$410$	0	0
$411$	7.45331	0.367645
$412$	0	0
$413$	−12.5653	−0.618300
$414$	0	0
$415$	−27.2920	−1.33971
$416$	0	0
$417$	−5.55602	−0.272079
$418$	0	0
$419$	−12.7780	−0.624247	−0.312123	−	0.950042i	$-0.601040\pi$
$419$	−12.7780	−0.624247	−0.312123	+	0.950042i	$0.601040\pi$
$420$	0	0
$421$	−6.66805	−0.324981	−0.162490	−	0.986710i	$-0.551953\pi$
$421$	−6.66805	−0.324981	−0.162490	+	0.986710i	$0.551953\pi$
$422$	0	0
$423$	0.778008	0.0378280
$424$	0	0
$425$	25.5233	1.23806
$426$	0	0
$427$	10.5653	0.511293
$428$	0	0
$429$	−2.90663	−0.140333
$430$	0	0
$431$	−30.8994	−1.48837	−0.744185	−	0.667973i	$-0.767162\pi$
$431$	−30.8994	−1.48837	−0.744185	+	0.667973i	$0.767162\pi$
$432$	0	0
$433$	21.2080	1.01919	0.509595	−	0.860415i	$-0.329795\pi$
$433$	21.2080	1.01919	0.509595	+	0.860415i	$0.329795\pi$
$434$	0	0
$435$	−9.73599	−0.466805
$436$	0	0
$437$	−0.726656	−0.0347607
$438$	0	0
$439$	−34.9694	−1.66900	−0.834499	−	0.551010i	$-0.814243\pi$
$439$	−34.9694	−1.66900	−0.834499	+	0.551010i	$0.814243\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	16.8667	0.801360	0.400680	−	0.916218i	$-0.368774\pi$
$443$	16.8667	0.801360	0.400680	+	0.916218i	$0.368774\pi$
$444$	0	0
$445$	26.4813	1.25533
$446$	0	0
$447$	15.0280	0.710799
$448$	0	0
$449$	−36.3713	−1.71647	−0.858235	−	0.513257i	$-0.828439\pi$
$449$	−36.3713	−1.71647	−0.858235	+	0.513257i	$0.828439\pi$
$450$	0	0
$451$	−2.58400	−0.121676
$452$	0	0
$453$	−4.72666	−0.222078
$454$	0	0
$455$	−14.0187	−0.657205
$456$	0	0
$457$	−31.2080	−1.45985	−0.729923	−	0.683529i	$-0.760444\pi$
$457$	−31.2080	−1.45985	−0.729923	+	0.683529i	$0.760444\pi$
$458$	0	0
$459$	−3.50466	−0.163584
$460$	0	0
$461$	−17.5233	−0.816142	−0.408071	−	0.912950i	$-0.633798\pi$
$461$	−17.5233	−0.816142	−0.408071	+	0.912950i	$0.633798\pi$
$462$	0	0
$463$	30.2241	1.40463	0.702316	−	0.711866i	$-0.252150\pi$
$463$	30.2241	1.40463	0.702316	+	0.711866i	$0.252150\pi$
$464$	0	0
$465$	4.46264	0.206950
$466$	0	0
$467$	26.8994	1.24475	0.622377	−	0.782717i	$-0.286167\pi$
$467$	26.8994	1.24475	0.622377	+	0.782717i	$0.286167\pi$
$468$	0	0
$469$	0.726656	0.0335539
$470$	0	0
$471$	−7.45331	−0.343431
$472$	0	0
$473$	−1.98134	−0.0911022
$474$	0	0
$475$	−7.28267	−0.334152
$476$	0	0
$477$	0.231321	0.0105915
$478$	0	0
$479$	−7.11929	−0.325289	−0.162644	−	0.986685i	$-0.552002\pi$
$479$	−7.11929	−0.325289	−0.162644	+	0.986685i	$0.552002\pi$
$480$	0	0
$481$	8.00000	0.364769
$482$	0	0
$483$	0.726656	0.0330640
$484$	0	0
$485$	13.0280	0.591570
$486$	0	0
$487$	−16.8294	−0.762611	−0.381306	−	0.924449i	$-0.624525\pi$
$487$	−16.8294	−0.762611	−0.381306	+	0.924449i	$0.624525\pi$
$488$	0	0
$489$	−16.7453	−0.757249
$490$	0	0
$491$	−32.8667	−1.48325	−0.741626	−	0.670813i	$-0.765945\pi$
$491$	−32.8667	−1.48325	−0.741626	+	0.670813i	$0.765945\pi$
$492$	0	0
$493$	9.73599	0.438487
$494$	0	0
$495$	2.54669	0.114465
$496$	0	0
$497$	−11.7873	−0.528734
$498$	0	0
$499$	−1.41600	−0.0633888	−0.0316944	−	0.999498i	$-0.510090\pi$
$499$	−1.41600	−0.0633888	−0.0316944	+	0.999498i	$0.510090\pi$
$500$	0	0
$501$	−4.10270	−0.183295
$502$	0	0
$503$	−4.21266	−0.187833	−0.0939167	−	0.995580i	$-0.529939\pi$
$503$	−4.21266	−0.187833	−0.0939167	+	0.995580i	$0.529939\pi$
$504$	0	0
$505$	−12.2827	−0.546572
$506$	0	0
$507$	−3.00000	−0.133235
$508$	0	0
$509$	24.5840	1.08967	0.544833	−	0.838544i	$-0.316593\pi$
$509$	24.5840	1.08967	0.544833	+	0.838544i	$0.316593\pi$
$510$	0	0
$511$	−0.443984	−0.0196407
$512$	0	0
$513$	1.00000	0.0441511
$514$	0	0
$515$	−44.0373	−1.94052
$516$	0	0
$517$	0.565344	0.0248638
$518$	0	0
$519$	−9.00933	−0.395466
$520$	0	0
$521$	−34.0000	−1.48957	−0.744784	−	0.667306i	$-0.767447\pi$
$521$	−34.0000	−1.48957	−0.744784	+	0.667306i	$0.767447\pi$
$522$	0	0
$523$	41.3107	1.80639	0.903194	−	0.429232i	$-0.141216\pi$
$523$	41.3107	1.80639	0.903194	+	0.429232i	$0.141216\pi$
$524$	0	0
$525$	7.28267	0.317842
$526$	0	0
$527$	−4.46264	−0.194396
$528$	0	0
$529$	−22.4720	−0.977042
$530$	0	0
$531$	12.5653	0.545290
$532$	0	0
$533$	−14.2241	−0.616113
$534$	0	0
$535$	−22.1986	−0.959730
$536$	0	0
$537$	7.68463	0.331616
$538$	0	0
$539$	0.726656	0.0312993
$540$	0	0
$541$	19.1307	0.822493	0.411246	−	0.911524i	$-0.365094\pi$
$541$	19.1307	0.822493	0.411246	+	0.911524i	$0.365094\pi$
$542$	0	0
$543$	−25.7360	−1.10444
$544$	0	0
$545$	−17.5560	−0.752017
$546$	0	0
$547$	−24.5067	−1.04783	−0.523916	−	0.851770i	$-0.675529\pi$
$547$	−24.5067	−1.04783	−0.523916	+	0.851770i	$0.675529\pi$
$548$	0	0
$549$	−10.5653	−0.450918
$550$	0	0
$551$	−2.77801	−0.118347
$552$	0	0
$553$	−6.28267	−0.267166
$554$	0	0
$555$	−7.00933	−0.297529
$556$	0	0
$557$	−3.35061	−0.141970	−0.0709850	−	0.997477i	$-0.522614\pi$
$557$	−3.35061	−0.141970	−0.0709850	+	0.997477i	$0.522614\pi$
$558$	0	0
$559$	−10.9066	−0.461301
$560$	0	0
$561$	−2.54669	−0.107521
$562$	0	0
$563$	−1.02799	−0.0433244	−0.0216622	−	0.999765i	$-0.506896\pi$
$563$	−1.02799	−0.0433244	−0.0216622	+	0.999765i	$0.506896\pi$
$564$	0	0
$565$	28.8480	1.21365
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−41.1566	−1.72537	−0.862687	−	0.505738i	$-0.831220\pi$
$569$	−41.1566	−1.72537	−0.862687	+	0.505738i	$0.831220\pi$
$570$	0	0
$571$	2.01866	0.0844782	0.0422391	−	0.999108i	$-0.486551\pi$
$571$	2.01866	0.0844782	0.0422391	+	0.999108i	$0.486551\pi$
$572$	0	0
$573$	−9.29200	−0.388179
$574$	0	0
$575$	5.29200	0.220692
$576$	0	0
$577$	0.0186574	0.000776718 0	0.000388359	−	1.00000i	$-0.499876\pi$
$577$	0.0186574	0.000776718 0	0.000388359		1.00000i	$0.499876\pi$
$578$	0	0
$579$	−16.0187	−0.665713
$580$	0	0
$581$	7.78734	0.323073
$582$	0	0
$583$	0.168091	0.00696161
$584$	0	0
$585$	14.0187	0.579600
$586$	0	0
$587$	−15.9927	−0.660091	−0.330046	−	0.943965i	$-0.607064\pi$
$587$	−15.9927	−0.660091	−0.330046	+	0.943965i	$0.607064\pi$
$588$	0	0
$589$	1.27334	0.0524672
$590$	0	0
$591$	−14.0000	−0.575883
$592$	0	0
$593$	−39.6447	−1.62801	−0.814006	−	0.580856i	$-0.802718\pi$
$593$	−39.6447	−1.62801	−0.814006	+	0.580856i	$0.802718\pi$
$594$	0	0
$595$	−12.2827	−0.503540
$596$	0	0
$597$	9.55602	0.391102
$598$	0	0
$599$	−5.13795	−0.209931	−0.104965	−	0.994476i	$-0.533473\pi$
$599$	−5.13795	−0.209931	−0.104965	+	0.994476i	$0.533473\pi$
$600$	0	0
$601$	9.73599	0.397139	0.198570	−	0.980087i	$-0.436370\pi$
$601$	9.73599	0.397139	0.198570	+	0.980087i	$0.436370\pi$
$602$	0	0
$603$	−0.726656	−0.0295917
$604$	0	0
$605$	−36.7007	−1.49210
$606$	0	0
$607$	−44.0373	−1.78742	−0.893710	−	0.448646i	$-0.851907\pi$
$607$	−44.0373	−1.78742	−0.893710	+	0.448646i	$0.851907\pi$
$608$	0	0
$609$	2.77801	0.112571
$610$	0	0
$611$	3.11203	0.125899
$612$	0	0
$613$	35.9787	1.45317	0.726583	−	0.687079i	$-0.241107\pi$
$613$	35.9787	1.45317	0.726583	+	0.687079i	$0.241107\pi$
$614$	0	0
$615$	12.4626	0.502542
$616$	0	0
$617$	3.09337	0.124535	0.0622673	−	0.998060i	$-0.480167\pi$
$617$	3.09337	0.124535	0.0622673	+	0.998060i	$0.480167\pi$
$618$	0	0
$619$	−19.5747	−0.786773	−0.393386	−	0.919373i	$-0.628696\pi$
$619$	−19.5747	−0.786773	−0.393386	+	0.919373i	$0.628696\pi$
$620$	0	0
$621$	−0.726656	−0.0291597
$622$	0	0
$623$	−7.55602	−0.302725
$624$	0	0
$625$	−8.37605	−0.335042
$626$	0	0
$627$	0.726656	0.0290199
$628$	0	0
$629$	7.00933	0.279480
$630$	0	0
$631$	−23.6519	−0.941569	−0.470784	−	0.882248i	$-0.656029\pi$
$631$	−23.6519	−0.941569	−0.470784	+	0.882248i	$0.656029\pi$
$632$	0	0
$633$	−25.7546	−1.02366
$634$	0	0
$635$	21.6587	0.859500
$636$	0	0
$637$	4.00000	0.158486
$638$	0	0
$639$	11.7873	0.466300
$640$	0	0
$641$	−42.2500	−1.66877	−0.834387	−	0.551179i	$-0.814178\pi$
$641$	−42.2500	−1.66877	−0.834387	+	0.551179i	$0.814178\pi$
$642$	0	0
$643$	−19.9346	−0.786144	−0.393072	−	0.919508i	$-0.628588\pi$
$643$	−19.9346	−0.786144	−0.393072	+	0.919508i	$0.628588\pi$
$644$	0	0
$645$	9.55602	0.376268
$646$	0	0
$647$	7.11929	0.279888	0.139944	−	0.990159i	$-0.455308\pi$
$647$	7.11929	0.279888	0.139944	+	0.990159i	$0.455308\pi$
$648$	0	0
$649$	9.13069	0.358411
$650$	0	0
$651$	−1.27334	−0.0499063
$652$	0	0
$653$	46.6027	1.82370	0.911851	−	0.410520i	$-0.134653\pi$
$653$	46.6027	1.82370	0.911851	+	0.410520i	$0.134653\pi$
$654$	0	0
$655$	−9.80137	−0.382971
$656$	0	0
$657$	0.443984	0.0173215
$658$	0	0
$659$	−45.2780	−1.76378	−0.881890	−	0.471456i	$-0.843729\pi$
$659$	−45.2780	−1.76378	−0.881890	+	0.471456i	$0.843729\pi$
$660$	0	0
$661$	0.205406	0.00798935	0.00399468	−	0.999992i	$-0.498728\pi$
$661$	0.205406	0.00798935	0.00399468	+	0.999992i	$0.498728\pi$
$662$	0	0
$663$	−14.0187	−0.544440
$664$	0	0
$665$	3.50466	0.135905
$666$	0	0
$667$	2.01866	0.0781627
$668$	0	0
$669$	11.2920	0.436574
$670$	0	0
$671$	−7.67738	−0.296382
$672$	0	0
$673$	5.43466	0.209491	0.104745	−	0.994499i	$-0.466597\pi$
$673$	5.43466	0.209491	0.104745	+	0.994499i	$0.466597\pi$
$674$	0	0
$675$	−7.28267	−0.280310
$676$	0	0
$677$	16.5467	0.635941	0.317970	−	0.948101i	$-0.396999\pi$
$677$	16.5467	0.635941	0.317970	+	0.948101i	$0.396999\pi$
$678$	0	0
$679$	−3.71733	−0.142658
$680$	0	0
$681$	−5.55602	−0.212907
$682$	0	0
$683$	−3.32469	−0.127216	−0.0636080	−	0.997975i	$-0.520261\pi$
$683$	−3.32469	−0.127216	−0.0636080	+	0.997975i	$0.520261\pi$
$684$	0	0
$685$	−26.1214	−0.998046
$686$	0	0
$687$	15.0280	0.573353
$688$	0	0
$689$	0.925283	0.0352505
$690$	0	0
$691$	−24.2427	−0.922237	−0.461118	−	0.887339i	$-0.652552\pi$
$691$	−24.2427	−0.922237	−0.461118	+	0.887339i	$0.652552\pi$
$692$	0	0
$693$	−0.726656	−0.0276034
$694$	0	0
$695$	19.4720	0.738614
$696$	0	0
$697$	−12.4626	−0.472056
$698$	0	0
$699$	6.10270	0.230825
$700$	0	0
$701$	−44.0187	−1.66256	−0.831281	−	0.555853i	$-0.812392\pi$
$701$	−44.0187	−1.66256	−0.831281	+	0.555853i	$0.812392\pi$
$702$	0	0
$703$	−2.00000	−0.0754314
$704$	0	0
$705$	−2.72666	−0.102692
$706$	0	0
$707$	3.50466	0.131806
$708$	0	0
$709$	−24.5067	−0.920370	−0.460185	−	0.887823i	$-0.652217\pi$
$709$	−24.5067	−0.920370	−0.460185	+	0.887823i	$0.652217\pi$
$710$	0	0
$711$	6.28267	0.235619
$712$	0	0
$713$	−0.925283	−0.0346521
$714$	0	0
$715$	10.1867	0.380963
$716$	0	0
$717$	−5.18930	−0.193798
$718$	0	0
$719$	−19.7873	−0.737943	−0.368972	−	0.929441i	$-0.620290\pi$
$719$	−19.7873	−0.737943	−0.368972	+	0.929441i	$0.620290\pi$
$720$	0	0
$721$	12.5653	0.467958
$722$	0	0
$723$	−13.7360	−0.510847
$724$	0	0
$725$	20.2313	0.751372
$726$	0	0
$727$	−40.5653	−1.50449	−0.752243	−	0.658886i	$-0.771028\pi$
$727$	−40.5653	−1.50449	−0.752243	+	0.658886i	$0.771028\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−9.55602	−0.353442
$732$	0	0
$733$	25.0093	0.923741	0.461870	−	0.886947i	$-0.347179\pi$
$733$	25.0093	0.923741	0.461870	+	0.886947i	$0.347179\pi$
$734$	0	0
$735$	−3.50466	−0.129271
$736$	0	0
$737$	−0.528030	−0.0194502
$738$	0	0
$739$	−31.6519	−1.16434	−0.582168	−	0.813069i	$-0.697795\pi$
$739$	−31.6519	−1.16434	−0.582168	+	0.813069i	$0.697795\pi$
$740$	0	0
$741$	4.00000	0.146944
$742$	0	0
$743$	0.778008	0.0285423	0.0142712	−	0.999898i	$-0.495457\pi$
$743$	0.778008	0.0285423	0.0142712	+	0.999898i	$0.495457\pi$
$744$	0	0
$745$	−52.6680	−1.92961
$746$	0	0
$747$	−7.78734	−0.284924
$748$	0	0
$749$	6.33402	0.231440
$750$	0	0
$751$	9.08660	0.331575	0.165787	−	0.986162i	$-0.446983\pi$
$751$	9.08660	0.331575	0.165787	+	0.986162i	$0.446983\pi$
$752$	0	0
$753$	−10.7967	−0.393452
$754$	0	0
$755$	16.5653	0.602875
$756$	0	0
$757$	30.3599	1.10345	0.551725	−	0.834026i	$-0.313970\pi$
$757$	30.3599	1.10345	0.551725	+	0.834026i	$0.313970\pi$
$758$	0	0
$759$	−0.528030	−0.0191663
$760$	0	0
$761$	35.5420	1.28840	0.644198	−	0.764859i	$-0.277191\pi$
$761$	35.5420	1.28840	0.644198	+	0.764859i	$0.277191\pi$
$762$	0	0
$763$	5.00933	0.181350
$764$	0	0
$765$	12.2827	0.444081
$766$	0	0
$767$	50.2614	1.81483
$768$	0	0
$769$	2.20541	0.0795290	0.0397645	−	0.999209i	$-0.487339\pi$
$769$	2.20541	0.0795290	0.0397645	+	0.999209i	$0.487339\pi$
$770$	0	0
$771$	−7.45331	−0.268425
$772$	0	0
$773$	−3.87864	−0.139505	−0.0697525	−	0.997564i	$-0.522221\pi$
$773$	−3.87864	−0.139505	−0.0697525	+	0.997564i	$0.522221\pi$
$774$	0	0
$775$	−9.27334	−0.333108
$776$	0	0
$777$	2.00000	0.0717496
$778$	0	0
$779$	3.55602	0.127407
$780$	0	0
$781$	8.56534	0.306492
$782$	0	0
$783$	−2.77801	−0.0992779
$784$	0	0
$785$	26.1214	0.932311
$786$	0	0
$787$	−41.3107	−1.47257	−0.736283	−	0.676674i	$-0.763421\pi$
$787$	−41.3107	−1.47257	−0.736283	+	0.676674i	$0.763421\pi$
$788$	0	0
$789$	18.6426	0.663695
$790$	0	0
$791$	−8.23132	−0.292672
$792$	0	0
$793$	−42.2614	−1.50075
$794$	0	0
$795$	−0.810702	−0.0287526
$796$	0	0
$797$	5.53736	0.196143	0.0980716	−	0.995179i	$-0.468733\pi$
$797$	5.53736	0.196143	0.0980716	+	0.995179i	$0.468733\pi$
$798$	0	0
$799$	2.72666	0.0964622
$800$	0	0
$801$	7.55602	0.266979
$802$	0	0
$803$	0.322624	0.0113852
$804$	0	0
$805$	−2.54669	−0.0897589
$806$	0	0
$807$	−7.65872	−0.269600
$808$	0	0
$809$	−54.5653	−1.91842	−0.959208	−	0.282702i	$-0.908769\pi$
$809$	−54.5653	−1.91842	−0.959208	+	0.282702i	$0.908769\pi$
$810$	0	0
$811$	38.5840	1.35487	0.677434	−	0.735584i	$-0.263092\pi$
$811$	38.5840	1.35487	0.677434	+	0.735584i	$0.263092\pi$
$812$	0	0
$813$	−11.1120	−0.389716
$814$	0	0
$815$	58.6867	2.05571
$816$	0	0
$817$	2.72666	0.0953936
$818$	0	0
$819$	−4.00000	−0.139771
$820$	0	0
$821$	−42.1400	−1.47070	−0.735348	−	0.677689i	$-0.762981\pi$
$821$	−42.1400	−1.47070	−0.735348	+	0.677689i	$0.762981\pi$
$822$	0	0
$823$	−47.5347	−1.65696	−0.828478	−	0.560021i	$-0.810793\pi$
$823$	−47.5347	−1.65696	−0.828478	+	0.560021i	$0.810793\pi$
$824$	0	0
$825$	−5.29200	−0.184244
$826$	0	0
$827$	0.315366	0.0109664	0.00548318	−	0.999985i	$-0.498255\pi$
$827$	0.315366	0.0109664	0.00548318	+	0.999985i	$0.498255\pi$
$828$	0	0
$829$	16.8480	0.585156	0.292578	−	0.956242i	$-0.405487\pi$
$829$	16.8480	0.585156	0.292578	+	0.956242i	$0.405487\pi$
$830$	0	0
$831$	4.30133	0.149211
$832$	0	0
$833$	3.50466	0.121429
$834$	0	0
$835$	14.3786	0.497592
$836$	0	0
$837$	1.27334	0.0440132
$838$	0	0
$839$	47.5093	1.64020	0.820101	−	0.572218i	$-0.193917\pi$
$839$	47.5093	1.64020	0.820101	+	0.572218i	$0.193917\pi$
$840$	0	0
$841$	−21.2827	−0.733885
$842$	0	0
$843$	0.759350	0.0261534
$844$	0	0
$845$	10.5140	0.361692
$846$	0	0
$847$	10.4720	0.359821
$848$	0	0
$849$	−4.99067	−0.171279
$850$	0	0
$851$	1.45331	0.0498189
$852$	0	0
$853$	9.67738	0.331347	0.165674	−	0.986181i	$-0.447020\pi$
$853$	9.67738	0.331347	0.165674	+	0.986181i	$0.447020\pi$
$854$	0	0
$855$	−3.50466	−0.119857
$856$	0	0
$857$	−26.9907	−0.921984	−0.460992	−	0.887404i	$-0.652506\pi$
$857$	−26.9907	−0.921984	−0.460992	+	0.887404i	$0.652506\pi$
$858$	0	0
$859$	38.6213	1.31774	0.658871	−	0.752256i	$-0.271034\pi$
$859$	38.6213	1.31774	0.658871	+	0.752256i	$0.271034\pi$
$860$	0	0
$861$	−3.55602	−0.121189
$862$	0	0
$863$	−0.778008	−0.0264837	−0.0132418	−	0.999912i	$-0.504215\pi$
$863$	−0.778008	−0.0264837	−0.0132418	+	0.999912i	$0.504215\pi$
$864$	0	0
$865$	31.5747	1.07357
$866$	0	0
$867$	4.71733	0.160209
$868$	0	0
$869$	4.56534	0.154869
$870$	0	0
$871$	−2.90663	−0.0984873
$872$	0	0
$873$	3.71733	0.125812
$874$	0	0
$875$	−8.00000	−0.270449
$876$	0	0
$877$	16.3786	0.553066	0.276533	−	0.961004i	$-0.410815\pi$
$877$	16.3786	0.553066	0.276533	+	0.961004i	$0.410815\pi$
$878$	0	0
$879$	22.1400	0.746764
$880$	0	0
$881$	7.60737	0.256299	0.128149	−	0.991755i	$-0.459096\pi$
$881$	7.60737	0.256299	0.128149	+	0.991755i	$0.459096\pi$
$882$	0	0
$883$	18.3413	0.617233	0.308617	−	0.951187i	$-0.400134\pi$
$883$	18.3413	0.617233	0.308617	+	0.951187i	$0.400134\pi$
$884$	0	0
$885$	−44.0373	−1.48030
$886$	0	0
$887$	−4.46264	−0.149841	−0.0749204	−	0.997190i	$-0.523870\pi$
$887$	−4.46264	−0.149841	−0.0749204	+	0.997190i	$0.523870\pi$
$888$	0	0
$889$	−6.17997	−0.207270
$890$	0	0
$891$	0.726656	0.0243439
$892$	0	0
$893$	−0.778008	−0.0260350
$894$	0	0
$895$	−26.9321	−0.900240
$896$	0	0
$897$	−2.90663	−0.0970494
$898$	0	0
$899$	−3.53736	−0.117978
$900$	0	0
$901$	0.810702	0.0270084
$902$	0	0
$903$	−2.72666	−0.0907374
$904$	0	0
$905$	90.1960	2.99822
$906$	0	0
$907$	5.03476	0.167177	0.0835883	−	0.996500i	$-0.473362\pi$
$907$	5.03476	0.167177	0.0835883	+	0.996500i	$0.473362\pi$
$908$	0	0
$909$	−3.50466	−0.116242
$910$	0	0
$911$	21.1379	0.700331	0.350166	−	0.936688i	$-0.386125\pi$
$911$	21.1379	0.700331	0.350166	+	0.936688i	$0.386125\pi$
$912$	0	0
$913$	−5.65872	−0.187276
$914$	0	0
$915$	37.0280	1.22411
$916$	0	0
$917$	2.79667	0.0923540
$918$	0	0
$919$	5.09337	0.168015	0.0840075	−	0.996465i	$-0.473228\pi$
$919$	5.09337	0.168015	0.0840075	+	0.996465i	$0.473228\pi$
$920$	0	0
$921$	−8.17997	−0.269539
$922$	0	0
$923$	47.1493	1.55194
$924$	0	0
$925$	14.5653	0.478906
$926$	0	0
$927$	−12.5653	−0.412700
$928$	0	0
$929$	−43.6447	−1.43194	−0.715968	−	0.698133i	$-0.754014\pi$
$929$	−43.6447	−1.43194	−0.715968	+	0.698133i	$0.754014\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	0	0
$933$	−5.76868	−0.188858
$934$	0	0
$935$	8.92528	0.291888
$936$	0	0
$937$	6.35994	0.207770	0.103885	−	0.994589i	$-0.466873\pi$
$937$	6.35994	0.207770	0.103885	+	0.994589i	$0.466873\pi$
$938$	0	0
$939$	−13.1120	−0.427895
$940$	0	0
$941$	−46.9253	−1.52972	−0.764860	−	0.644196i	$-0.777192\pi$
$941$	−46.9253	−1.52972	−0.764860	+	0.644196i	$0.777192\pi$
$942$	0	0
$943$	−2.58400	−0.0841467
$944$	0	0
$945$	3.50466	0.114007
$946$	0	0
$947$	−16.1613	−0.525172	−0.262586	−	0.964909i	$-0.584575\pi$
$947$	−16.1613	−0.525172	−0.262586	+	0.964909i	$0.584575\pi$
$948$	0	0
$949$	1.77594	0.0576494
$950$	0	0
$951$	31.8060	1.03138
$952$	0	0
$953$	−20.1659	−0.653239	−0.326619	−	0.945156i	$-0.605909\pi$
$953$	−20.1659	−0.653239	−0.326619	+	0.945156i	$0.605909\pi$
$954$	0	0
$955$	32.5653	1.05379
$956$	0	0
$957$	−2.01866	−0.0652539
$958$	0	0
$959$	7.45331	0.240680
$960$	0	0
$961$	−29.3786	−0.947697
$962$	0	0
$963$	−6.33402	−0.204111
$964$	0	0
$965$	56.1400	1.80721
$966$	0	0
$967$	5.80137	0.186560	0.0932798	−	0.995640i	$-0.470265\pi$
$967$	5.80137	0.186560	0.0932798	+	0.995640i	$0.470265\pi$
$968$	0	0
$969$	3.50466	0.112586
$970$	0	0
$971$	18.5840	0.596389	0.298195	−	0.954505i	$-0.403616\pi$
$971$	18.5840	0.596389	0.298195	+	0.954505i	$0.403616\pi$
$972$	0	0
$973$	−5.55602	−0.178118
$974$	0	0
$975$	−29.1307	−0.932929
$976$	0	0
$977$	−4.69396	−0.150173	−0.0750866	−	0.997177i	$-0.523923\pi$
$977$	−4.69396	−0.150173	−0.0750866	+	0.997177i	$0.523923\pi$
$978$	0	0
$979$	5.49063	0.175481
$980$	0	0
$981$	−5.00933	−0.159936
$982$	0	0
$983$	54.9439	1.75244	0.876220	−	0.481912i	$-0.160057\pi$
$983$	54.9439	1.75244	0.876220	+	0.481912i	$0.160057\pi$
$984$	0	0
$985$	49.0653	1.56335
$986$	0	0
$987$	0.778008	0.0247643
$988$	0	0
$989$	−1.98134	−0.0630030
$990$	0	0
$991$	−34.2827	−1.08902	−0.544512	−	0.838753i	$-0.683285\pi$
$991$	−34.2827	−1.08902	−0.544512	+	0.838753i	$0.683285\pi$
$992$	0	0
$993$	23.8387	0.756498
$994$	0	0
$995$	−33.4906	−1.06172
$996$	0	0
$997$	−22.7708	−0.721157	−0.360578	−	0.932729i	$-0.617421\pi$
$997$	−22.7708	−0.721157	−0.360578	+	0.932729i	$0.617421\pi$
$998$	0	0
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3192.2.a.u.1.3	✓	3
3.2	odd	2		9576.2.a.cb.1.1		3
4.3	odd	2		6384.2.a.bw.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3192.2.a.u.1.3	✓	3	1.1	even	1	trivial
6384.2.a.bw.1.3		3	4.3	odd	2
9576.2.a.cb.1.1		3	3.2	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:	\( N \)	\(=\)	\( 3192 = 2^{3} \cdot 3 \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3192.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +3.50466 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +3.50466 q^{5} -1.00000 q^{7} +1.00000 q^{9} +0.726656 q^{11} +4.00000 q^{13} -3.50466 q^{15} +3.50466 q^{17} -1.00000 q^{19} +1.00000 q^{21} +0.726656 q^{23} +7.28267 q^{25} -1.00000 q^{27} +2.77801 q^{29} -1.27334 q^{31} -0.726656 q^{33} -3.50466 q^{35} +2.00000 q^{37} -4.00000 q^{39} -3.55602 q^{41} -2.72666 q^{43} +3.50466 q^{45} +0.778008 q^{47} +1.00000 q^{49} -3.50466 q^{51} +0.231321 q^{53} +2.54669 q^{55} +1.00000 q^{57} +12.5653 q^{59} -10.5653 q^{61} -1.00000 q^{63} +14.0187 q^{65} -0.726656 q^{67} -0.726656 q^{69} +11.7873 q^{71} +0.443984 q^{73} -7.28267 q^{75} -0.726656 q^{77} +6.28267 q^{79} +1.00000 q^{81} -7.78734 q^{83} +12.2827 q^{85} -2.77801 q^{87} +7.55602 q^{89} -4.00000 q^{91} +1.27334 q^{93} -3.50466 q^{95} +3.71733 q^{97} +0.726656 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} - 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 - 3 * q^7 + 3 * q^9	\( 3 q - 3 q^{3} - 3 q^{7} + 3 q^{9} - 2 q^{11} + 12 q^{13} - 3 q^{19} + 3 q^{21} - 2 q^{23} + 5 q^{25} - 3 q^{27} + 2 q^{29} - 8 q^{31} + 2 q^{33} + 6 q^{37} - 12 q^{39} + 2 q^{41} - 4 q^{43} - 4 q^{47} + 3 q^{49} - 14 q^{53} + 16 q^{55} + 3 q^{57} + 4 q^{59} + 2 q^{61} - 3 q^{63} + 2 q^{67} + 2 q^{69} + 8 q^{71} + 14 q^{73} - 5 q^{75} + 2 q^{77} + 2 q^{79} + 3 q^{81} + 4 q^{83} + 20 q^{85} - 2 q^{87} + 10 q^{89} - 12 q^{91} + 8 q^{93} + 28 q^{97} - 2 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 - 3 * q^7 + 3 * q^9 - 2 * q^11 + 12 * q^13 - 3 * q^19 + 3 * q^21 - 2 * q^23 + 5 * q^25 - 3 * q^27 + 2 * q^29 - 8 * q^31 + 2 * q^33 + 6 * q^37 - 12 * q^39 + 2 * q^41 - 4 * q^43 - 4 * q^47 + 3 * q^49 - 14 * q^53 + 16 * q^55 + 3 * q^57 + 4 * q^59 + 2 * q^61 - 3 * q^63 + 2 * q^67 + 2 * q^69 + 8 * q^71 + 14 * q^73 - 5 * q^75 + 2 * q^77 + 2 * q^79 + 3 * q^81 + 4 * q^83 + 20 * q^85 - 2 * q^87 + 10 * q^89 - 12 * q^91 + 8 * q^93 + 28 * q^97 - 2 * q^99

Introduction

L-functions

Modular forms

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