LMFDB - Embedded newform 2312.4.a.k.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2312,4,Mod(1,2312)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2312, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("2312.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2312 = 2^{3} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2312.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:	$136.412415933$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 95x^{6} + 756x^{4} - 1780x^{2} + 1152 $ x^8 - 95x^6 + 756x^4 - 1780*x^2 + 1152
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{13} $
Twist minimal:	no (minimal twist has level 136)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.03229$ of defining polynomial
Character	$\chi$	$=$	2312.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.52350 q^{3} -18.2701 q^{5} +13.8757 q^{7} +45.6501 q^{9} +O(q^{10})$	$q-8.52350 q^{3} -18.2701 q^{5} +13.8757 q^{7} +45.6501 q^{9} -60.8512 q^{11} +61.8450 q^{13} +155.725 q^{15} +40.0349 q^{19} -118.270 q^{21} -4.88830 q^{23} +208.795 q^{25} -158.964 q^{27} -113.141 q^{29} -95.1610 q^{31} +518.665 q^{33} -253.510 q^{35} +273.445 q^{37} -527.136 q^{39} +446.056 q^{41} +274.325 q^{43} -834.029 q^{45} -27.6551 q^{47} -150.464 q^{49} -488.605 q^{53} +1111.76 q^{55} -341.237 q^{57} -266.325 q^{59} -502.818 q^{61} +633.428 q^{63} -1129.91 q^{65} -1008.84 q^{67} +41.6654 q^{69} +724.061 q^{71} -188.092 q^{73} -1779.67 q^{75} -844.355 q^{77} +48.3382 q^{79} +122.376 q^{81} -1384.89 q^{83} +964.357 q^{87} -50.3364 q^{89} +858.144 q^{91} +811.105 q^{93} -731.440 q^{95} -1285.79 q^{97} -2777.86 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 132 q^{9}+O(q^{10}) $ 8 * q + 132 * q^9	$ 8 q + 132 q^{9} + 44 q^{13} - 24 q^{15} - 48 q^{19} + 308 q^{21} + 520 q^{25} + 812 q^{33} - 1064 q^{35} - 8 q^{43} + 312 q^{47} + 1124 q^{49} - 472 q^{53} + 1416 q^{55} + 72 q^{59} - 624 q^{67} - 180 q^{69} - 1660 q^{77} + 3156 q^{81} - 2472 q^{83} + 6664 q^{87} + 68 q^{89} + 4036 q^{93}+O(q^{100}) $ 8 * q + 132 * q^9 + 44 * q^13 - 24 * q^15 - 48 * q^19 + 308 * q^21 + 520 * q^25 + 812 * q^33 - 1064 * q^35 - 8 * q^43 + 312 * q^47 + 1124 * q^49 - 472 * q^53 + 1416 * q^55 + 72 * q^59 - 624 * q^67 - 180 * q^69 - 1660 * q^77 + 3156 * q^81 - 2472 * q^83 + 6664 * q^87 + 68 * q^89 + 4036 * q^93

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
$3$	−8.52350	−1.64035	−0.820174	−	0.572114i	$-0.806124\pi$
$3$	−8.52350	−1.64035	−0.820174	+	0.572114i	$0.806124\pi$
$4$	0	0
$5$	−18.2701	−1.63412	−0.817062	−	0.576550i	$-0.804399\pi$
$5$	−18.2701	−1.63412	−0.817062	+	0.576550i	$0.804399\pi$
$6$	0	0
$7$	13.8757	0.749219	0.374609	−	0.927183i	$-0.377777\pi$
$7$	13.8757	0.749219	0.374609	+	0.927183i	$0.377777\pi$
$8$	0	0
$9$	45.6501	1.69074
$10$	0	0
$11$	−60.8512	−1.66794	−0.833969	−	0.551811i	$-0.813937\pi$
$11$	−60.8512	−1.66794	−0.833969	+	0.551811i	$0.813937\pi$
$12$	0	0
$13$	61.8450	1.31944	0.659720	−	0.751512i	$-0.270675\pi$
$13$	61.8450	1.31944	0.659720	+	0.751512i	$0.270675\pi$
$14$	0	0
$15$	155.725	2.68053
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	40.0349	0.483402	0.241701	−	0.970351i	$-0.422295\pi$
$19$	40.0349	0.483402	0.241701	+	0.970351i	$0.422295\pi$
$20$	0	0
$21$	−118.270	−1.22898
$22$	0	0
$23$	−4.88830	−0.0443165	−0.0221583	−	0.999754i	$-0.507054\pi$
$23$	−4.88830	−0.0443165	−0.0221583	+	0.999754i	$0.507054\pi$
$24$	0	0
$25$	208.795	1.67036
$26$	0	0
$27$	−158.964	−1.13306
$28$	0	0
$29$	−113.141	−0.724474	−0.362237	−	0.932086i	$-0.617987\pi$
$29$	−113.141	−0.724474	−0.362237	+	0.932086i	$0.617987\pi$
$30$	0	0
$31$	−95.1610	−0.551336	−0.275668	−	0.961253i	$-0.588899\pi$
$31$	−95.1610	−0.551336	−0.275668	+	0.961253i	$0.588899\pi$
$32$	0	0
$33$	518.665	2.73600
$34$	0	0
$35$	−253.510	−1.22432
$36$	0	0
$37$	273.445	1.21497	0.607487	−	0.794330i	$-0.292178\pi$
$37$	273.445	1.21497	0.607487	+	0.794330i	$0.292178\pi$
$38$	0	0
$39$	−527.136	−2.16434
$40$	0	0
$41$	446.056	1.69908	0.849540	−	0.527525i	$-0.176880\pi$
$41$	446.056	1.69908	0.849540	+	0.527525i	$0.176880\pi$
$42$	0	0
$43$	274.325	0.972888	0.486444	−	0.873712i	$-0.338294\pi$
$43$	274.325	0.972888	0.486444	+	0.873712i	$0.338294\pi$
$44$	0	0
$45$	−834.029	−2.76288
$46$	0	0
$47$	−27.6551	−0.0858280	−0.0429140	−	0.999079i	$-0.513664\pi$
$47$	−27.6551	−0.0858280	−0.0429140	+	0.999079i	$0.513664\pi$
$48$	0	0
$49$	−150.464	−0.438671
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−488.605	−1.26632	−0.633161	−	0.774020i	$-0.718243\pi$
$53$	−488.605	−1.26632	−0.633161	+	0.774020i	$0.718243\pi$
$54$	0	0
$55$	1111.76	2.72562
$56$	0	0
$57$	−341.237	−0.792947
$58$	0	0
$59$	−266.325	−0.587671	−0.293835	−	0.955856i	$-0.594932\pi$
$59$	−266.325	−0.587671	−0.293835	+	0.955856i	$0.594932\pi$
$60$	0	0
$61$	−502.818	−1.05540	−0.527699	−	0.849432i	$-0.676945\pi$
$61$	−502.818	−1.05540	−0.527699	+	0.849432i	$0.676945\pi$
$62$	0	0
$63$	633.428	1.26674
$64$	0	0
$65$	−1129.91	−2.15613
$66$	0	0
$67$	−1008.84	−1.83955	−0.919776	−	0.392443i	$-0.871630\pi$
$67$	−1008.84	−1.83955	−0.919776	+	0.392443i	$0.871630\pi$
$68$	0	0
$69$	41.6654	0.0726946
$70$	0	0
$71$	724.061	1.21029	0.605143	−	0.796117i	$-0.293116\pi$
$71$	724.061	1.21029	0.605143	+	0.796117i	$0.293116\pi$
$72$	0	0
$73$	−188.092	−0.301569	−0.150785	−	0.988567i	$-0.548180\pi$
$73$	−188.092	−0.301569	−0.150785	+	0.988567i	$0.548180\pi$
$74$	0	0
$75$	−1779.67	−2.73997
$76$	0	0
$77$	−844.355	−1.24965
$78$	0	0
$79$	48.3382	0.0688415	0.0344207	−	0.999407i	$-0.489041\pi$
$79$	48.3382	0.0688415	0.0344207	+	0.999407i	$0.489041\pi$
$80$	0	0
$81$	122.376	0.167868
$82$	0	0
$83$	−1384.89	−1.83146	−0.915729	−	0.401797i	$-0.868386\pi$
$83$	−1384.89	−1.83146	−0.915729	+	0.401797i	$0.868386\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	964.357	1.18839
$88$	0	0
$89$	−50.3364	−0.0599511	−0.0299756	−	0.999551i	$-0.509543\pi$
$89$	−50.3364	−0.0599511	−0.0299756	+	0.999551i	$0.509543\pi$
$90$	0	0
$91$	858.144	0.988549
$92$	0	0
$93$	811.105	0.904384
$94$	0	0
$95$	−731.440	−0.789938
$96$	0	0
$97$	−1285.79	−1.34589	−0.672947	−	0.739691i	$-0.734972\pi$
$97$	−1285.79	−1.34589	−0.672947	+	0.739691i	$0.734972\pi$
$98$	0	0
$99$	−2777.86	−2.82005
$100$	0	0
$101$	−932.025	−0.918217	−0.459109	−	0.888380i	$-0.651831\pi$
$101$	−932.025	−0.918217	−0.459109	+	0.888380i	$0.651831\pi$
$102$	0	0
$103$	−283.119	−0.270840	−0.135420	−	0.990788i	$-0.543238\pi$
$103$	−283.119	−0.270840	−0.135420	+	0.990788i	$0.543238\pi$
$104$	0	0
$105$	2160.80	2.00831
$106$	0	0
$107$	−273.739	−0.247321	−0.123661	−	0.992325i	$-0.539463\pi$
$107$	−273.739	−0.247321	−0.123661	+	0.992325i	$0.539463\pi$
$108$	0	0
$109$	834.584	0.733382	0.366691	−	0.930343i	$-0.380491\pi$
$109$	834.584	0.733382	0.366691	+	0.930343i	$0.380491\pi$
$110$	0	0
$111$	−2330.71	−1.99298
$112$	0	0
$113$	−193.136	−0.160785	−0.0803927	−	0.996763i	$-0.525617\pi$
$113$	−193.136	−0.160785	−0.0803927	+	0.996763i	$0.525617\pi$
$114$	0	0
$115$	89.3095	0.0724187
$116$	0	0
$117$	2823.23	2.23083
$118$	0	0
$119$	0	0
$120$	0	0
$121$	2371.87	1.78202
$122$	0	0
$123$	−3801.96	−2.78708
$124$	0	0
$125$	−1530.94	−1.09545
$126$	0	0
$127$	−1114.62	−0.778793	−0.389396	−	0.921070i	$-0.627316\pi$
$127$	−1114.62	−0.778793	−0.389396	+	0.921070i	$0.627316\pi$
$128$	0	0
$129$	−2338.21	−1.59588
$130$	0	0
$131$	673.762	0.449365	0.224683	−	0.974432i	$-0.427865\pi$
$131$	673.762	0.449365	0.224683	+	0.974432i	$0.427865\pi$
$132$	0	0
$133$	555.513	0.362174
$134$	0	0
$135$	2904.28	1.85156
$136$	0	0
$137$	1213.64	0.756849	0.378425	−	0.925632i	$-0.376466\pi$
$137$	1213.64	0.756849	0.378425	+	0.925632i	$0.376466\pi$
$138$	0	0
$139$	818.573	0.499500	0.249750	−	0.968310i	$-0.419652\pi$
$139$	818.573	0.499500	0.249750	+	0.968310i	$0.419652\pi$
$140$	0	0
$141$	235.718	0.140788
$142$	0	0
$143$	−3763.34	−2.20074
$144$	0	0
$145$	2067.09	1.18388
$146$	0	0
$147$	1282.48	0.719573
$148$	0	0
$149$	2397.27	1.31807	0.659033	−	0.752114i	$-0.270966\pi$
$149$	2397.27	1.31807	0.659033	+	0.752114i	$0.270966\pi$
$150$	0	0
$151$	1197.87	0.645573	0.322787	−	0.946472i	$-0.395380\pi$
$151$	1197.87	0.645573	0.322787	+	0.946472i	$0.395380\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1738.60	0.900952
$156$	0	0
$157$	1926.67	0.979393	0.489696	−	0.871893i	$-0.337108\pi$
$157$	1926.67	0.979393	0.489696	+	0.871893i	$0.337108\pi$
$158$	0	0
$159$	4164.63	2.07721
$160$	0	0
$161$	−67.8287	−0.0332028
$162$	0	0
$163$	−1307.62	−0.628347	−0.314174	−	0.949366i	$-0.601727\pi$
$163$	−1307.62	−0.628347	−0.314174	+	0.949366i	$0.601727\pi$
$164$	0	0
$165$	−9476.04	−4.47096
$166$	0	0
$167$	−2548.82	−1.18104	−0.590520	−	0.807023i	$-0.701077\pi$
$167$	−2548.82	−1.18104	−0.590520	+	0.807023i	$0.701077\pi$
$168$	0	0
$169$	1627.80	0.740921
$170$	0	0
$171$	1827.59	0.817308
$172$	0	0
$173$	−661.688	−0.290793	−0.145397	−	0.989373i	$-0.546446\pi$
$173$	−661.688	−0.290793	−0.145397	+	0.989373i	$0.546446\pi$
$174$	0	0
$175$	2897.18	1.25147
$176$	0	0
$177$	2270.02	0.963985
$178$	0	0
$179$	−2927.34	−1.22235	−0.611173	−	0.791497i	$-0.709302\pi$
$179$	−2927.34	−1.22235	−0.611173	+	0.791497i	$0.709302\pi$
$180$	0	0
$181$	−209.349	−0.0859711	−0.0429856	−	0.999076i	$-0.513687\pi$
$181$	−209.349	−0.0859711	−0.0429856	+	0.999076i	$0.513687\pi$
$182$	0	0
$183$	4285.77	1.73122
$184$	0	0
$185$	−4995.85	−1.98542
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−2205.74	−0.848909
$190$	0	0
$191$	4259.39	1.61361	0.806803	−	0.590820i	$-0.201196\pi$
$191$	4259.39	1.61361	0.806803	+	0.590820i	$0.201196\pi$
$192$	0	0
$193$	72.8650	0.0271758	0.0135879	−	0.999908i	$-0.495675\pi$
$193$	72.8650	0.0271758	0.0135879	+	0.999908i	$0.495675\pi$
$194$	0	0
$195$	9630.80	3.53680
$196$	0	0
$197$	−4200.16	−1.51903	−0.759516	−	0.650489i	$-0.774564\pi$
$197$	−4200.16	−1.51903	−0.759516	+	0.650489i	$0.774564\pi$
$198$	0	0
$199$	−2013.74	−0.717337	−0.358669	−	0.933465i	$-0.616769\pi$
$199$	−2013.74	−0.717337	−0.358669	+	0.933465i	$0.616769\pi$
$200$	0	0
$201$	8598.89	3.01751
$202$	0	0
$203$	−1569.91	−0.542790
$204$	0	0
$205$	−8149.47	−2.77651
$206$	0	0
$207$	−223.151	−0.0749279
$208$	0	0
$209$	−2436.17	−0.806284
$210$	0	0
$211$	−1431.34	−0.467001	−0.233500	−	0.972357i	$-0.575018\pi$
$211$	−1431.34	−0.467001	−0.233500	+	0.972357i	$0.575018\pi$
$212$	0	0
$213$	−6171.54	−1.98529
$214$	0	0
$215$	−5011.94	−1.58982
$216$	0	0
$217$	−1320.43	−0.413072
$218$	0	0
$219$	1603.21	0.494679
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−1220.63	−0.366544	−0.183272	−	0.983062i	$-0.558669\pi$
$223$	−1220.63	−0.366544	−0.183272	+	0.983062i	$0.558669\pi$
$224$	0	0
$225$	9531.51	2.82415
$226$	0	0
$227$	4338.39	1.26850	0.634250	−	0.773128i	$-0.281309\pi$
$227$	4338.39	1.26850	0.634250	+	0.773128i	$0.281309\pi$
$228$	0	0
$229$	1887.81	0.544760	0.272380	−	0.962190i	$-0.412189\pi$
$229$	1887.81	0.544760	0.272380	+	0.962190i	$0.412189\pi$
$230$	0	0
$231$	7196.86	2.04986
$232$	0	0
$233$	4991.46	1.40344	0.701720	−	0.712453i	$-0.252416\pi$
$233$	4991.46	1.40344	0.701720	+	0.712453i	$0.252416\pi$
$234$	0	0
$235$	505.261	0.140254
$236$	0	0
$237$	−412.011	−0.112924
$238$	0	0
$239$	−2220.06	−0.600853	−0.300427	−	0.953805i	$-0.597129\pi$
$239$	−2220.06	−0.600853	−0.300427	+	0.953805i	$0.597129\pi$
$240$	0	0
$241$	4989.64	1.33366	0.666828	−	0.745212i	$-0.267652\pi$
$241$	4989.64	1.33366	0.666828	+	0.745212i	$0.267652\pi$
$242$	0	0
$243$	3248.95	0.857697
$244$	0	0
$245$	2748.99	0.716843
$246$	0	0
$247$	2475.96	0.637819
$248$	0	0
$249$	11804.1	3.00423
$250$	0	0
$251$	−1077.08	−0.270855	−0.135427	−	0.990787i	$-0.543241\pi$
$251$	−1077.08	−0.270855	−0.135427	+	0.990787i	$0.543241\pi$
$252$	0	0
$253$	297.459	0.0739173
$254$	0	0
$255$	0	0
$256$	0	0
$257$	6521.35	1.58284	0.791421	−	0.611271i	$-0.209342\pi$
$257$	6521.35	1.58284	0.791421	+	0.611271i	$0.209342\pi$
$258$	0	0
$259$	3794.24	0.910281
$260$	0	0
$261$	−5164.89	−1.22490
$262$	0	0
$263$	−7879.02	−1.84730	−0.923652	−	0.383231i	$-0.874811\pi$
$263$	−7879.02	−1.84730	−0.923652	+	0.383231i	$0.874811\pi$
$264$	0	0
$265$	8926.85	2.06933
$266$	0	0
$267$	429.043	0.0983408
$268$	0	0
$269$	2539.91	0.575691	0.287845	−	0.957677i	$-0.407061\pi$
$269$	2539.91	0.575691	0.287845	+	0.957677i	$0.407061\pi$
$270$	0	0
$271$	−919.402	−0.206087	−0.103044	−	0.994677i	$-0.532858\pi$
$271$	−919.402	−0.206087	−0.103044	+	0.994677i	$0.532858\pi$
$272$	0	0
$273$	−7314.39	−1.62156
$274$	0	0
$275$	−12705.4	−2.78606
$276$	0	0
$277$	6999.15	1.51819	0.759094	−	0.650981i	$-0.225642\pi$
$277$	6999.15	1.51819	0.759094	+	0.650981i	$0.225642\pi$
$278$	0	0
$279$	−4344.11	−0.932168
$280$	0	0
$281$	−1310.63	−0.278242	−0.139121	−	0.990275i	$-0.544428\pi$
$281$	−1310.63	−0.278242	−0.139121	+	0.990275i	$0.544428\pi$
$282$	0	0
$283$	2119.86	0.445275	0.222638	−	0.974901i	$-0.428533\pi$
$283$	2119.86	0.445275	0.222638	+	0.974901i	$0.428533\pi$
$284$	0	0
$285$	6234.42	1.29577
$286$	0	0
$287$	6189.35	1.27298
$288$	0	0
$289$	0	0
$290$	0	0
$291$	10959.4	2.20773
$292$	0	0
$293$	−4192.22	−0.835877	−0.417939	−	0.908475i	$-0.637247\pi$
$293$	−4192.22	−0.835877	−0.417939	+	0.908475i	$0.637247\pi$
$294$	0	0
$295$	4865.78	0.960327
$296$	0	0
$297$	9673.13	1.88987
$298$	0	0
$299$	−302.317	−0.0584730
$300$	0	0
$301$	3806.46	0.728906
$302$	0	0
$303$	7944.11	1.50620
$304$	0	0
$305$	9186.51	1.72465
$306$	0	0
$307$	899.329	0.167190	0.0835952	−	0.996500i	$-0.473360\pi$
$307$	899.329	0.167190	0.0835952	+	0.996500i	$0.473360\pi$
$308$	0	0
$309$	2413.17	0.444273
$310$	0	0
$311$	−8053.67	−1.46843	−0.734215	−	0.678917i	$-0.762450\pi$
$311$	−8053.67	−1.46843	−0.734215	+	0.678917i	$0.762450\pi$
$312$	0	0
$313$	3940.63	0.711622	0.355811	−	0.934558i	$-0.384205\pi$
$313$	3940.63	0.711622	0.355811	+	0.934558i	$0.384205\pi$
$314$	0	0
$315$	−11572.8	−2.07000
$316$	0	0
$317$	−5790.24	−1.02591	−0.512953	−	0.858416i	$-0.671449\pi$
$317$	−5790.24	−1.02591	−0.512953	+	0.858416i	$0.671449\pi$
$318$	0	0
$319$	6884.76	1.20838
$320$	0	0
$321$	2333.22	0.405693
$322$	0	0
$323$	0	0
$324$	0	0
$325$	12912.9	2.20394
$326$	0	0
$327$	−7113.57	−1.20300
$328$	0	0
$329$	−383.735	−0.0643039
$330$	0	0
$331$	6184.41	1.02697	0.513484	−	0.858099i	$-0.328355\pi$
$331$	6184.41	1.02697	0.513484	+	0.858099i	$0.328355\pi$
$332$	0	0
$333$	12482.8	2.05421
$334$	0	0
$335$	18431.7	3.00606
$336$	0	0
$337$	−2895.51	−0.468038	−0.234019	−	0.972232i	$-0.575188\pi$
$337$	−2895.51	−0.468038	−0.234019	+	0.972232i	$0.575188\pi$
$338$	0	0
$339$	1646.20	0.263744
$340$	0	0
$341$	5790.66	0.919595
$342$	0	0
$343$	−6847.17	−1.07788
$344$	0	0
$345$	−761.230	−0.118792
$346$	0	0
$347$	1555.79	0.240690	0.120345	−	0.992732i	$-0.461600\pi$
$347$	1555.79	0.240690	0.120345	+	0.992732i	$0.461600\pi$
$348$	0	0
$349$	−1103.98	−0.169325	−0.0846627	−	0.996410i	$-0.526981\pi$
$349$	−1103.98	−0.169325	−0.0846627	+	0.996410i	$0.526981\pi$
$350$	0	0
$351$	−9831.11	−1.49500
$352$	0	0
$353$	9622.07	1.45080	0.725398	−	0.688330i	$-0.241656\pi$
$353$	9622.07	1.45080	0.725398	+	0.688330i	$0.241656\pi$
$354$	0	0
$355$	−13228.6	−1.97776
$356$	0	0
$357$	0	0
$358$	0	0
$359$	3095.00	0.455008	0.227504	−	0.973777i	$-0.426944\pi$
$359$	3095.00	0.455008	0.227504	+	0.973777i	$0.426944\pi$
$360$	0	0
$361$	−5256.21	−0.766323
$362$	0	0
$363$	−20216.6	−2.92313
$364$	0	0
$365$	3436.46	0.492802
$366$	0	0
$367$	1805.83	0.256849	0.128425	−	0.991719i	$-0.459008\pi$
$367$	1805.83	0.256849	0.128425	+	0.991719i	$0.459008\pi$
$368$	0	0
$369$	20362.5	2.87271
$370$	0	0
$371$	−6779.75	−0.948752
$372$	0	0
$373$	6265.65	0.869767	0.434884	−	0.900487i	$-0.356789\pi$
$373$	6265.65	0.869767	0.434884	+	0.900487i	$0.356789\pi$
$374$	0	0
$375$	13049.0	1.79693
$376$	0	0
$377$	−6997.20	−0.955900
$378$	0	0
$379$	−1997.53	−0.270729	−0.135364	−	0.990796i	$-0.543221\pi$
$379$	−1997.53	−0.270729	−0.135364	+	0.990796i	$0.543221\pi$
$380$	0	0
$381$	9500.48	1.27749
$382$	0	0
$383$	−13879.9	−1.85178	−0.925889	−	0.377797i	$-0.876682\pi$
$383$	−13879.9	−1.85178	−0.925889	+	0.377797i	$0.876682\pi$
$384$	0	0
$385$	15426.4	2.04208
$386$	0	0
$387$	12523.0	1.64490
$388$	0	0
$389$	7261.69	0.946483	0.473242	−	0.880933i	$-0.343084\pi$
$389$	7261.69	0.946483	0.473242	+	0.880933i	$0.343084\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−5742.81	−0.737115
$394$	0	0
$395$	−883.143	−0.112496
$396$	0	0
$397$	−7324.04	−0.925902	−0.462951	−	0.886384i	$-0.653209\pi$
$397$	−7324.04	−0.925902	−0.462951	+	0.886384i	$0.653209\pi$
$398$	0	0
$399$	−4734.91	−0.594091
$400$	0	0
$401$	2017.73	0.251274	0.125637	−	0.992076i	$-0.459903\pi$
$401$	2017.73	0.251274	0.125637	+	0.992076i	$0.459903\pi$
$402$	0	0
$403$	−5885.23	−0.727455
$404$	0	0
$405$	−2235.81	−0.274317
$406$	0	0
$407$	−16639.4	−2.02650
$408$	0	0
$409$	−837.375	−0.101236	−0.0506180	−	0.998718i	$-0.516119\pi$
$409$	−837.375	−0.101236	−0.0506180	+	0.998718i	$0.516119\pi$
$410$	0	0
$411$	−10344.5	−1.24150
$412$	0	0
$413$	−3695.45	−0.440294
$414$	0	0
$415$	25301.9	2.99283
$416$	0	0
$417$	−6977.11	−0.819353
$418$	0	0
$419$	−6292.34	−0.733653	−0.366827	−	0.930289i	$-0.619556\pi$
$419$	−6292.34	−0.733653	−0.366827	+	0.930289i	$0.619556\pi$
$420$	0	0
$421$	−3436.25	−0.397797	−0.198899	−	0.980020i	$-0.563736\pi$
$421$	−3436.25	−0.397797	−0.198899	+	0.980020i	$0.563736\pi$
$422$	0	0
$423$	−1262.46	−0.145113
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−6976.96	−0.790724
$428$	0	0
$429$	32076.8	3.60999
$430$	0	0
$431$	1198.92	0.133991	0.0669955	−	0.997753i	$-0.478659\pi$
$431$	1198.92	0.133991	0.0669955	+	0.997753i	$0.478659\pi$
$432$	0	0
$433$	1921.84	0.213297	0.106649	−	0.994297i	$-0.465988\pi$
$433$	1921.84	0.213297	0.106649	+	0.994297i	$0.465988\pi$
$434$	0	0
$435$	−17618.9	−1.94198
$436$	0	0
$437$	−195.702	−0.0214227
$438$	0	0
$439$	17813.5	1.93666	0.968328	−	0.249681i	$-0.0803258\pi$
$439$	17813.5	1.93666	0.968328	+	0.249681i	$0.0803258\pi$
$440$	0	0
$441$	−6868.70	−0.741680
$442$	0	0
$443$	−10707.1	−1.14833	−0.574165	−	0.818739i	$-0.694673\pi$
$443$	−10707.1	−1.14833	−0.574165	+	0.818739i	$0.694673\pi$
$444$	0	0
$445$	919.650	0.0979676
$446$	0	0
$447$	−20433.1	−2.16209
$448$	0	0
$449$	−9542.98	−1.00303	−0.501516	−	0.865149i	$-0.667224\pi$
$449$	−9542.98	−1.00303	−0.501516	+	0.865149i	$0.667224\pi$
$450$	0	0
$451$	−27143.0	−2.83396
$452$	0	0
$453$	−10210.1	−1.05896
$454$	0	0
$455$	−15678.4	−1.61541
$456$	0	0
$457$	17234.9	1.76415	0.882075	−	0.471109i	$-0.156146\pi$
$457$	17234.9	1.76415	0.882075	+	0.471109i	$0.156146\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−11358.1	−1.14751	−0.573755	−	0.819027i	$-0.694514\pi$
$461$	−11358.1	−1.14751	−0.573755	+	0.819027i	$0.694514\pi$
$462$	0	0
$463$	5407.62	0.542793	0.271397	−	0.962468i	$-0.412514\pi$
$463$	5407.62	0.542793	0.271397	+	0.962468i	$0.412514\pi$
$464$	0	0
$465$	−14818.9	−1.47787
$466$	0	0
$467$	5059.61	0.501350	0.250675	−	0.968071i	$-0.419347\pi$
$467$	5059.61	0.501350	0.250675	+	0.968071i	$0.419347\pi$
$468$	0	0
$469$	−13998.5	−1.37823
$470$	0	0
$471$	−16421.9	−1.60655
$472$	0	0
$473$	−16693.0	−1.62272
$474$	0	0
$475$	8359.09	0.807455
$476$	0	0
$477$	−22304.8	−2.14102
$478$	0	0
$479$	−2151.09	−0.205190	−0.102595	−	0.994723i	$-0.532715\pi$
$479$	−2151.09	−0.205190	−0.102595	+	0.994723i	$0.532715\pi$
$480$	0	0
$481$	16911.2	1.60308
$482$	0	0
$483$	578.138	0.0544641
$484$	0	0
$485$	23491.4	2.19936
$486$	0	0
$487$	−12035.6	−1.11988	−0.559942	−	0.828532i	$-0.689177\pi$
$487$	−12035.6	−1.11988	−0.559942	+	0.828532i	$0.689177\pi$
$488$	0	0
$489$	11145.5	1.03071
$490$	0	0
$491$	−16672.1	−1.53238	−0.766191	−	0.642613i	$-0.777850\pi$
$491$	−16672.1	−1.53238	−0.766191	+	0.642613i	$0.777850\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	50751.7	4.60832
$496$	0	0
$497$	10046.9	0.906769
$498$	0	0
$499$	−6097.31	−0.547000	−0.273500	−	0.961872i	$-0.588181\pi$
$499$	−6097.31	−0.547000	−0.273500	+	0.961872i	$0.588181\pi$
$500$	0	0
$501$	21724.9	1.93732
$502$	0	0
$503$	4307.43	0.381827	0.190913	−	0.981607i	$-0.438855\pi$
$503$	4307.43	0.381827	0.190913	+	0.981607i	$0.438855\pi$
$504$	0	0
$505$	17028.2	1.50048
$506$	0	0
$507$	−13874.6	−1.21537
$508$	0	0
$509$	12433.5	1.08272	0.541362	−	0.840790i	$-0.317909\pi$
$509$	12433.5	1.08272	0.541362	+	0.840790i	$0.317909\pi$
$510$	0	0
$511$	−2609.92	−0.225941
$512$	0	0
$513$	−6364.09	−0.547722
$514$	0	0
$515$	5172.60	0.442587
$516$	0	0
$517$	1682.85	0.143156
$518$	0	0
$519$	5639.90	0.477002
$520$	0	0
$521$	−628.628	−0.0528612	−0.0264306	−	0.999651i	$-0.508414\pi$
$521$	−628.628	−0.0528612	−0.0264306	+	0.999651i	$0.508414\pi$
$522$	0	0
$523$	−10333.5	−0.863963	−0.431981	−	0.901883i	$-0.642185\pi$
$523$	−10333.5	−0.863963	−0.431981	+	0.901883i	$0.642185\pi$
$524$	0	0
$525$	−24694.2	−2.05284
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−12143.1	−0.998036
$530$	0	0
$531$	−12157.8	−0.993600
$532$	0	0
$533$	27586.3	2.24183
$534$	0	0
$535$	5001.24	0.404154
$536$	0	0
$537$	24951.2	2.00507
$538$	0	0
$539$	9155.92	0.731676
$540$	0	0
$541$	−4122.30	−0.327599	−0.163800	−	0.986494i	$-0.552375\pi$
$541$	−4122.30	−0.327599	−0.163800	+	0.986494i	$0.552375\pi$
$542$	0	0
$543$	1784.38	0.141023
$544$	0	0
$545$	−15247.9	−1.19844
$546$	0	0
$547$	21268.5	1.66248	0.831241	−	0.555913i	$-0.187631\pi$
$547$	21268.5	1.66248	0.831241	+	0.555913i	$0.187631\pi$
$548$	0	0
$549$	−22953.7	−1.78441
$550$	0	0
$551$	−4529.58	−0.350212
$552$	0	0
$553$	670.728	0.0515773
$554$	0	0
$555$	42582.1	3.25678
$556$	0	0
$557$	8773.96	0.667441	0.333721	−	0.942672i	$-0.391696\pi$
$557$	8773.96	0.667441	0.333721	+	0.942672i	$0.391696\pi$
$558$	0	0
$559$	16965.6	1.28367
$560$	0	0
$561$	0	0
$562$	0	0
$563$	5507.46	0.412276	0.206138	−	0.978523i	$-0.433910\pi$
$563$	5507.46	0.412276	0.206138	+	0.978523i	$0.433910\pi$
$564$	0	0
$565$	3528.62	0.262743
$566$	0	0
$567$	1698.05	0.125770
$568$	0	0
$569$	13023.4	0.959528	0.479764	−	0.877398i	$-0.340722\pi$
$569$	13023.4	0.959528	0.479764	+	0.877398i	$0.340722\pi$
$570$	0	0
$571$	−13591.0	−0.996089	−0.498044	−	0.867152i	$-0.665948\pi$
$571$	−13591.0	−0.996089	−0.498044	+	0.867152i	$0.665948\pi$
$572$	0	0
$573$	−36304.9	−2.64688
$574$	0	0
$575$	−1020.65	−0.0740246
$576$	0	0
$577$	2171.91	0.156703	0.0783517	−	0.996926i	$-0.475034\pi$
$577$	2171.91	0.156703	0.0783517	+	0.996926i	$0.475034\pi$
$578$	0	0
$579$	−621.065	−0.0445778
$580$	0	0
$581$	−19216.3	−1.37216
$582$	0	0
$583$	29732.2	2.11215
$584$	0	0
$585$	−51580.5	−3.64546
$586$	0	0
$587$	25902.3	1.82130	0.910651	−	0.413176i	$-0.135581\pi$
$587$	25902.3	1.82130	0.910651	+	0.413176i	$0.135581\pi$
$588$	0	0
$589$	−3809.76	−0.266517
$590$	0	0
$591$	35800.1	2.49174
$592$	0	0
$593$	24167.0	1.67356	0.836780	−	0.547539i	$-0.184435\pi$
$593$	24167.0	1.67356	0.836780	+	0.547539i	$0.184435\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	17164.1	1.17668
$598$	0	0
$599$	−14936.3	−1.01883	−0.509415	−	0.860521i	$-0.670138\pi$
$599$	−14936.3	−1.01883	−0.509415	+	0.860521i	$0.670138\pi$
$600$	0	0
$601$	15669.9	1.06354	0.531770	−	0.846889i	$-0.321527\pi$
$601$	15669.9	1.06354	0.531770	+	0.846889i	$0.321527\pi$
$602$	0	0
$603$	−46053.8	−3.11021
$604$	0	0
$605$	−43334.2	−2.91204
$606$	0	0
$607$	−16641.8	−1.11280	−0.556401	−	0.830914i	$-0.687818\pi$
$607$	−16641.8	−1.11280	−0.556401	+	0.830914i	$0.687818\pi$
$608$	0	0
$609$	13381.2	0.890364
$610$	0	0
$611$	−1710.33	−0.113245
$612$	0	0
$613$	17114.6	1.12766	0.563828	−	0.825892i	$-0.309328\pi$
$613$	17114.6	1.12766	0.563828	+	0.825892i	$0.309328\pi$
$614$	0	0
$615$	69462.0	4.55444
$616$	0	0
$617$	21652.6	1.41280	0.706402	−	0.707811i	$-0.250317\pi$
$617$	21652.6	1.41280	0.706402	+	0.707811i	$0.250317\pi$
$618$	0	0
$619$	17883.9	1.16125	0.580625	−	0.814171i	$-0.302808\pi$
$619$	17883.9	1.16125	0.580625	+	0.814171i	$0.302808\pi$
$620$	0	0
$621$	777.062	0.0502132
$622$	0	0
$623$	−698.455	−0.0449165
$624$	0	0
$625$	1871.02	0.119745
$626$	0	0
$627$	20764.7	1.32259
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−18222.4	−1.14964	−0.574820	−	0.818280i	$-0.694928\pi$
$631$	−18222.4	−1.14964	−0.574820	+	0.818280i	$0.694928\pi$
$632$	0	0
$633$	12200.0	0.766044
$634$	0	0
$635$	20364.2	1.27264
$636$	0	0
$637$	−9305.46	−0.578800
$638$	0	0
$639$	33053.4	2.04628
$640$	0	0
$641$	−17105.1	−1.05399	−0.526996	−	0.849868i	$-0.676682\pi$
$641$	−17105.1	−1.05399	−0.526996	+	0.849868i	$0.676682\pi$
$642$	0	0
$643$	−155.386	−0.00953008	−0.00476504	−	0.999989i	$-0.501517\pi$
$643$	−155.386	−0.00953008	−0.00476504	+	0.999989i	$0.501517\pi$
$644$	0	0
$645$	42719.2	2.60786
$646$	0	0
$647$	17013.9	1.03383	0.516914	−	0.856037i	$-0.327081\pi$
$647$	17013.9	1.03383	0.516914	+	0.856037i	$0.327081\pi$
$648$	0	0
$649$	16206.2	0.980199
$650$	0	0
$651$	11254.7	0.677581
$652$	0	0
$653$	−31766.8	−1.90373	−0.951863	−	0.306524i	$-0.900834\pi$
$653$	−31766.8	−1.90373	−0.951863	+	0.306524i	$0.900834\pi$
$654$	0	0
$655$	−12309.7	−0.734318
$656$	0	0
$657$	−8586.43	−0.509876
$658$	0	0
$659$	5545.66	0.327812	0.163906	−	0.986476i	$-0.447591\pi$
$659$	5545.66	0.327812	0.163906	+	0.986476i	$0.447591\pi$
$660$	0	0
$661$	−12808.2	−0.753676	−0.376838	−	0.926279i	$-0.622989\pi$
$661$	−12808.2	−0.753676	−0.376838	+	0.926279i	$0.622989\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−10149.3	−0.591836
$666$	0	0
$667$	553.067	0.0321062
$668$	0	0
$669$	10404.0	0.601260
$670$	0	0
$671$	30597.1	1.76034
$672$	0	0
$673$	13921.9	0.797399	0.398699	−	0.917082i	$-0.369462\pi$
$673$	13921.9	0.797399	0.398699	+	0.917082i	$0.369462\pi$
$674$	0	0
$675$	−33190.9	−1.89262
$676$	0	0
$677$	15570.9	0.883954	0.441977	−	0.897026i	$-0.354277\pi$
$677$	15570.9	0.883954	0.441977	+	0.897026i	$0.354277\pi$
$678$	0	0
$679$	−17841.2	−1.00837
$680$	0	0
$681$	−36978.3	−2.08078
$682$	0	0
$683$	−6547.76	−0.366827	−0.183414	−	0.983036i	$-0.558715\pi$
$683$	−6547.76	−0.366827	−0.183414	+	0.983036i	$0.558715\pi$
$684$	0	0
$685$	−22173.3	−1.23679
$686$	0	0
$687$	−16090.8	−0.893597
$688$	0	0
$689$	−30217.8	−1.67084
$690$	0	0
$691$	12241.5	0.673936	0.336968	−	0.941516i	$-0.390599\pi$
$691$	12241.5	0.673936	0.336968	+	0.941516i	$0.390599\pi$
$692$	0	0
$693$	−38544.8	−2.11284
$694$	0	0
$695$	−14955.4	−0.816244
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−42544.7	−2.30213
$700$	0	0
$701$	−20002.0	−1.07770	−0.538849	−	0.842403i	$-0.681141\pi$
$701$	−20002.0	−1.07770	−0.538849	+	0.842403i	$0.681141\pi$
$702$	0	0
$703$	10947.3	0.587320
$704$	0	0
$705$	−4306.59	−0.230065
$706$	0	0
$707$	−12932.5	−0.687946
$708$	0	0
$709$	−781.933	−0.0414191	−0.0207095	−	0.999786i	$-0.506593\pi$
$709$	−781.933	−0.0414191	−0.0207095	+	0.999786i	$0.506593\pi$
$710$	0	0
$711$	2206.64	0.116393
$712$	0	0
$713$	465.175	0.0244333
$714$	0	0
$715$	68756.5	3.59629
$716$	0	0
$717$	18922.7	0.985609
$718$	0	0
$719$	−3489.63	−0.181003	−0.0905017	−	0.995896i	$-0.528847\pi$
$719$	−3489.63	−0.181003	−0.0905017	+	0.995896i	$0.528847\pi$
$720$	0	0
$721$	−3928.48	−0.202919
$722$	0	0
$723$	−42529.2	−2.18766
$724$	0	0
$725$	−23623.3	−1.21013
$726$	0	0
$727$	−4894.00	−0.249667	−0.124834	−	0.992178i	$-0.539840\pi$
$727$	−4894.00	−0.249667	−0.124834	+	0.992178i	$0.539840\pi$
$728$	0	0
$729$	−30996.6	−1.57479
$730$	0	0
$731$	0	0
$732$	0	0
$733$	5187.08	0.261377	0.130688	−	0.991423i	$-0.458281\pi$
$733$	5187.08	0.261377	0.130688	+	0.991423i	$0.458281\pi$
$734$	0	0
$735$	−23431.0	−1.17587
$736$	0	0
$737$	61389.4	3.06826
$738$	0	0
$739$	−10603.0	−0.527793	−0.263896	−	0.964551i	$-0.585008\pi$
$739$	−10603.0	−0.527793	−0.263896	+	0.964551i	$0.585008\pi$
$740$	0	0
$741$	−21103.8	−1.04625
$742$	0	0
$743$	4784.54	0.236242	0.118121	−	0.992999i	$-0.462313\pi$
$743$	4784.54	0.236242	0.118121	+	0.992999i	$0.462313\pi$
$744$	0	0
$745$	−43798.2	−2.15388
$746$	0	0
$747$	−63220.1	−3.09652
$748$	0	0
$749$	−3798.33	−0.185298
$750$	0	0
$751$	6361.82	0.309116	0.154558	−	0.987984i	$-0.450605\pi$
$751$	6361.82	0.309116	0.154558	+	0.987984i	$0.450605\pi$
$752$	0	0
$753$	9180.47	0.444296
$754$	0	0
$755$	−21885.2	−1.05495
$756$	0	0
$757$	28528.8	1.36975	0.684873	−	0.728662i	$-0.259858\pi$
$757$	28528.8	1.36975	0.684873	+	0.728662i	$0.259858\pi$
$758$	0	0
$759$	−2535.39	−0.121250
$760$	0	0
$761$	−18097.5	−0.862069	−0.431035	−	0.902335i	$-0.641851\pi$
$761$	−18097.5	−0.862069	−0.431035	+	0.902335i	$0.641851\pi$
$762$	0	0
$763$	11580.5	0.549463
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−16470.9	−0.775396
$768$	0	0
$769$	7189.93	0.337159	0.168580	−	0.985688i	$-0.446082\pi$
$769$	7189.93	0.337159	0.168580	+	0.985688i	$0.446082\pi$
$770$	0	0
$771$	−55584.7	−2.59641
$772$	0	0
$773$	5316.50	0.247375	0.123688	−	0.992321i	$-0.460528\pi$
$773$	5316.50	0.247375	0.123688	+	0.992321i	$0.460528\pi$
$774$	0	0
$775$	−19869.2	−0.920931
$776$	0	0
$777$	−32340.2	−1.49318
$778$	0	0
$779$	17857.8	0.821338
$780$	0	0
$781$	−44060.0	−2.01868
$782$	0	0
$783$	17985.3	0.820871
$784$	0	0
$785$	−35200.3	−1.60045
$786$	0	0
$787$	8498.14	0.384912	0.192456	−	0.981306i	$-0.438355\pi$
$787$	8498.14	0.384912	0.192456	+	0.981306i	$0.438355\pi$
$788$	0	0
$789$	67156.8	3.03022
$790$	0	0
$791$	−2679.91	−0.120463
$792$	0	0
$793$	−31096.8	−1.39253
$794$	0	0
$795$	−76088.0	−3.39442
$796$	0	0
$797$	19009.1	0.844841	0.422420	−	0.906400i	$-0.361181\pi$
$797$	19009.1	0.844841	0.422420	+	0.906400i	$0.361181\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−2297.86	−0.101362
$802$	0	0
$803$	11445.7	0.502999
$804$	0	0
$805$	1239.23	0.0542575
$806$	0	0
$807$	−21648.9	−0.944334
$808$	0	0
$809$	−16963.6	−0.737215	−0.368608	−	0.929585i	$-0.620165\pi$
$809$	−16963.6	−0.737215	−0.368608	+	0.929585i	$0.620165\pi$
$810$	0	0
$811$	−24943.1	−1.07999	−0.539994	−	0.841669i	$-0.681574\pi$
$811$	−24943.1	−1.07999	−0.539994	+	0.841669i	$0.681574\pi$
$812$	0	0
$813$	7836.52	0.338055
$814$	0	0
$815$	23890.3	1.02680
$816$	0	0
$817$	10982.6	0.470296
$818$	0	0
$819$	39174.3	1.67138
$820$	0	0
$821$	−24317.6	−1.03373	−0.516863	−	0.856068i	$-0.672900\pi$
$821$	−24317.6	−1.03373	−0.516863	+	0.856068i	$0.672900\pi$
$822$	0	0
$823$	4969.93	0.210499	0.105250	−	0.994446i	$-0.466436\pi$
$823$	4969.93	0.210499	0.105250	+	0.994446i	$0.466436\pi$
$824$	0	0
$825$	108295.	4.57011
$826$	0	0
$827$	13711.0	0.576516	0.288258	−	0.957553i	$-0.406924\pi$
$827$	13711.0	0.576516	0.288258	+	0.957553i	$0.406924\pi$
$828$	0	0
$829$	35645.2	1.49338	0.746689	−	0.665173i	$-0.231642\pi$
$829$	35645.2	1.49338	0.746689	+	0.665173i	$0.231642\pi$
$830$	0	0
$831$	−59657.3	−2.49036
$832$	0	0
$833$	0	0
$834$	0	0
$835$	46567.1	1.92996
$836$	0	0
$837$	15127.1	0.624696
$838$	0	0
$839$	39704.8	1.63380	0.816902	−	0.576777i	$-0.195690\pi$
$839$	39704.8	1.63380	0.816902	+	0.576777i	$0.195690\pi$
$840$	0	0
$841$	−11588.1	−0.475137
$842$	0	0
$843$	11171.2	0.456413
$844$	0	0
$845$	−29740.1	−1.21076
$846$	0	0
$847$	32911.4	1.33512
$848$	0	0
$849$	−18068.7	−0.730406
$850$	0	0
$851$	−1336.68	−0.0538434
$852$	0	0
$853$	4378.77	0.175763	0.0878817	−	0.996131i	$-0.471990\pi$
$853$	4378.77	0.175763	0.0878817	+	0.996131i	$0.471990\pi$
$854$	0	0
$855$	−33390.3	−1.33558
$856$	0	0
$857$	−14706.1	−0.586173	−0.293086	−	0.956086i	$-0.594682\pi$
$857$	−14706.1	−0.586173	−0.293086	+	0.956086i	$0.594682\pi$
$858$	0	0
$859$	22056.6	0.876092	0.438046	−	0.898953i	$-0.355671\pi$
$859$	22056.6	0.876092	0.438046	+	0.898953i	$0.355671\pi$
$860$	0	0
$861$	−52755.0	−2.08813
$862$	0	0
$863$	13451.2	0.530571	0.265286	−	0.964170i	$-0.414534\pi$
$863$	13451.2	0.530571	0.265286	+	0.964170i	$0.414534\pi$
$864$	0	0
$865$	12089.1	0.475192
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−2941.44	−0.114823
$870$	0	0
$871$	−62392.0	−2.42718
$872$	0	0
$873$	−58696.2	−2.27556
$874$	0	0
$875$	−21242.9	−0.820734
$876$	0	0
$877$	−36821.0	−1.41774	−0.708870	−	0.705339i	$-0.750795\pi$
$877$	−36821.0	−1.41774	−0.708870	+	0.705339i	$0.750795\pi$
$878$	0	0
$879$	35732.4	1.37113
$880$	0	0
$881$	−20929.4	−0.800372	−0.400186	−	0.916434i	$-0.631055\pi$
$881$	−20929.4	−0.800372	−0.400186	+	0.916434i	$0.631055\pi$
$882$	0	0
$883$	14362.9	0.547394	0.273697	−	0.961816i	$-0.411753\pi$
$883$	14362.9	0.547394	0.273697	+	0.961816i	$0.411753\pi$
$884$	0	0
$885$	−41473.4	−1.57527
$886$	0	0
$887$	−15832.4	−0.599323	−0.299662	−	0.954046i	$-0.596874\pi$
$887$	−15832.4	−0.599323	−0.299662	+	0.954046i	$0.596874\pi$
$888$	0	0
$889$	−15466.2	−0.583486
$890$	0	0
$891$	−7446.71	−0.279993
$892$	0	0
$893$	−1107.17	−0.0414894
$894$	0	0
$895$	53482.7	1.99746
$896$	0	0
$897$	2576.80	0.0959161
$898$	0	0
$899$	10766.6	0.399429
$900$	0	0
$901$	0	0
$902$	0	0
$903$	−32444.4	−1.19566
$904$	0	0
$905$	3824.82	0.140487
$906$	0	0
$907$	−34058.7	−1.24686	−0.623429	−	0.781880i	$-0.714261\pi$
$907$	−34058.7	−1.24686	−0.623429	+	0.781880i	$0.714261\pi$
$908$	0	0
$909$	−42547.0	−1.55247
$910$	0	0
$911$	19317.1	0.702531	0.351265	−	0.936276i	$-0.385752\pi$
$911$	19317.1	0.702531	0.351265	+	0.936276i	$0.385752\pi$
$912$	0	0
$913$	84271.9	3.05476
$914$	0	0
$915$	−78301.2	−2.82903
$916$	0	0
$917$	9348.93	0.336673
$918$	0	0
$919$	−19221.2	−0.689934	−0.344967	−	0.938615i	$-0.612110\pi$
$919$	−19221.2	−0.689934	−0.344967	+	0.938615i	$0.612110\pi$
$920$	0	0
$921$	−7665.43	−0.274250
$922$	0	0
$923$	44779.6	1.59690
$924$	0	0
$925$	57093.9	2.02945
$926$	0	0
$927$	−12924.4	−0.457921
$928$	0	0
$929$	34753.9	1.22738	0.613692	−	0.789546i	$-0.289684\pi$
$929$	34753.9	1.22738	0.613692	+	0.789546i	$0.289684\pi$
$930$	0	0
$931$	−6023.81	−0.212054
$932$	0	0
$933$	68645.5	2.40874
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−649.069	−0.0226298	−0.0113149	−	0.999936i	$-0.503602\pi$
$937$	−649.069	−0.0226298	−0.0113149	+	0.999936i	$0.503602\pi$
$938$	0	0
$939$	−33588.0	−1.16731
$940$	0	0
$941$	13716.6	0.475183	0.237591	−	0.971365i	$-0.423642\pi$
$941$	13716.6	0.475183	0.237591	+	0.971365i	$0.423642\pi$
$942$	0	0
$943$	−2180.46	−0.0752973
$944$	0	0
$945$	40299.0	1.38722
$946$	0	0
$947$	43131.8	1.48004	0.740019	−	0.672586i	$-0.234816\pi$
$947$	43131.8	1.48004	0.740019	+	0.672586i	$0.234816\pi$
$948$	0	0
$949$	−11632.6	−0.397903
$950$	0	0
$951$	49353.1	1.68284
$952$	0	0
$953$	−4974.39	−0.169083	−0.0845415	−	0.996420i	$-0.526943\pi$
$953$	−4974.39	−0.169083	−0.0845415	+	0.996420i	$0.526943\pi$
$954$	0	0
$955$	−77819.4	−2.63683
$956$	0	0
$957$	−58682.3	−1.98216
$958$	0	0
$959$	16840.2	0.567046
$960$	0	0
$961$	−20735.4	−0.696028
$962$	0	0
$963$	−12496.2	−0.418157
$964$	0	0
$965$	−1331.25	−0.0444087
$966$	0	0
$967$	83.3000	0.00277016	0.00138508	−	0.999999i	$-0.499559\pi$
$967$	83.3000	0.00277016	0.00138508	+	0.999999i	$0.499559\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1577.72	0.0521438	0.0260719	−	0.999660i	$-0.491700\pi$
$971$	1577.72	0.0521438	0.0260719	+	0.999660i	$0.491700\pi$
$972$	0	0
$973$	11358.3	0.374235
$974$	0	0
$975$	−110063.	−3.61523
$976$	0	0
$977$	−30161.6	−0.987672	−0.493836	−	0.869555i	$-0.664406\pi$
$977$	−30161.6	−0.987672	−0.493836	+	0.869555i	$0.664406\pi$
$978$	0	0
$979$	3063.03	0.0999948
$980$	0	0
$981$	38098.8	1.23996
$982$	0	0
$983$	3072.86	0.0997040	0.0498520	−	0.998757i	$-0.484125\pi$
$983$	3072.86	0.0997040	0.0498520	+	0.998757i	$0.484125\pi$
$984$	0	0
$985$	76737.3	2.48229
$986$	0	0
$987$	3270.77	0.105481
$988$	0	0
$989$	−1340.98	−0.0431150
$990$	0	0
$991$	22582.4	0.723869	0.361934	−	0.932204i	$-0.382116\pi$
$991$	22582.4	0.723869	0.361934	+	0.932204i	$0.382116\pi$
$992$	0	0
$993$	−52712.8	−1.68458
$994$	0	0
$995$	36791.1	1.17222
$996$	0	0
$997$	−17226.0	−0.547195	−0.273597	−	0.961844i	$-0.588214\pi$
$997$	−17226.0	−0.547195	−0.273597	+	0.961844i	$0.588214\pi$
$998$	0	0
$999$	−43467.8	−1.37664

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2312.4.a.k.1.2		8
17.4	even	4		136.4.b.b.33.7	yes	8
17.13	even	4		136.4.b.b.33.2	✓	8
17.16	even	2	inner	2312.4.a.k.1.7		8
51.38	odd	4		1224.4.c.e.577.1		8
51.47	odd	4		1224.4.c.e.577.8		8
68.47	odd	4		272.4.b.f.33.7		8
68.55	odd	4		272.4.b.f.33.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.4.b.b.33.2	✓	8	17.13	even	4
136.4.b.b.33.7	yes	8	17.4	even	4
272.4.b.f.33.2		8	68.55	odd	4
272.4.b.f.33.7		8	68.47	odd	4
1224.4.c.e.577.1		8	51.38	odd	4
1224.4.c.e.577.8		8	51.47	odd	4
2312.4.a.k.1.2		8	1.1	even	1	trivial
2312.4.a.k.1.7		8	17.16	even	2	inner

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:	\( N \)	\(=\)	\( 2312 = 2^{3} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2312.a (trivial)

\(f(q)\)	\(=\)	\(q-8.52350 q^{3} -18.2701 q^{5} +13.8757 q^{7} +45.6501 q^{9} +O(q^{10})\)	\(q-8.52350 q^{3} -18.2701 q^{5} +13.8757 q^{7} +45.6501 q^{9} -60.8512 q^{11} +61.8450 q^{13} +155.725 q^{15} +40.0349 q^{19} -118.270 q^{21} -4.88830 q^{23} +208.795 q^{25} -158.964 q^{27} -113.141 q^{29} -95.1610 q^{31} +518.665 q^{33} -253.510 q^{35} +273.445 q^{37} -527.136 q^{39} +446.056 q^{41} +274.325 q^{43} -834.029 q^{45} -27.6551 q^{47} -150.464 q^{49} -488.605 q^{53} +1111.76 q^{55} -341.237 q^{57} -266.325 q^{59} -502.818 q^{61} +633.428 q^{63} -1129.91 q^{65} -1008.84 q^{67} +41.6654 q^{69} +724.061 q^{71} -188.092 q^{73} -1779.67 q^{75} -844.355 q^{77} +48.3382 q^{79} +122.376 q^{81} -1384.89 q^{83} +964.357 q^{87} -50.3364 q^{89} +858.144 q^{91} +811.105 q^{93} -731.440 q^{95} -1285.79 q^{97} -2777.86 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 132 q^{9}+O(q^{10}) \) 8 * q + 132 * q^9	\( 8 q + 132 q^{9} + 44 q^{13} - 24 q^{15} - 48 q^{19} + 308 q^{21} + 520 q^{25} + 812 q^{33} - 1064 q^{35} - 8 q^{43} + 312 q^{47} + 1124 q^{49} - 472 q^{53} + 1416 q^{55} + 72 q^{59} - 624 q^{67} - 180 q^{69} - 1660 q^{77} + 3156 q^{81} - 2472 q^{83} + 6664 q^{87} + 68 q^{89} + 4036 q^{93}+O(q^{100}) \) 8 * q + 132 * q^9 + 44 * q^13 - 24 * q^15 - 48 * q^19 + 308 * q^21 + 520 * q^25 + 812 * q^33 - 1064 * q^35 - 8 * q^43 + 312 * q^47 + 1124 * q^49 - 472 * q^53 + 1416 * q^55 + 72 * q^59 - 624 * q^67 - 180 * q^69 - 1660 * q^77 + 3156 * q^81 - 2472 * q^83 + 6664 * q^87 + 68 * q^89 + 4036 * q^93

Introduction

L-functions

Modular forms

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