LMFDB - Embedded newform 312.4.a.c.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [312,4,Mod(1,312)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(312, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("312.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 312 = 2^{3} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	312.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:	$18.4085959218$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{113}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 28 $ x^2 - x - 28
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			1.2
Root			$5.81507$ of defining polynomial
Character	$\chi$	$=$	312.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} +13.6301 q^{5} -15.6301 q^{7} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} +13.6301 q^{5} -15.6301 q^{7} +9.00000 q^{9} -50.5206 q^{11} +13.0000 q^{13} -40.8904 q^{15} +2.00000 q^{17} +18.1507 q^{19} +46.8904 q^{21} -64.0000 q^{23} +60.7809 q^{25} -27.0000 q^{27} -103.781 q^{29} +34.4522 q^{31} +151.562 q^{33} -213.041 q^{35} -267.343 q^{37} -39.0000 q^{39} -147.411 q^{41} -166.959 q^{43} +122.671 q^{45} -325.781 q^{47} -98.6985 q^{49} -6.00000 q^{51} -111.041 q^{53} -688.603 q^{55} -54.4522 q^{57} +24.3015 q^{59} -640.987 q^{61} -140.671 q^{63} +177.192 q^{65} -382.589 q^{67} +192.000 q^{69} +510.247 q^{71} -291.781 q^{73} -182.343 q^{75} +789.644 q^{77} -794.685 q^{79} +81.0000 q^{81} +945.507 q^{83} +27.2603 q^{85} +311.343 q^{87} +317.329 q^{89} -203.192 q^{91} -103.357 q^{93} +247.397 q^{95} +1572.88 q^{97} -454.685 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} + 6 q^{5} - 10 q^{7} + 18 q^{9}+O(q^{10}) $ 2 * q - 6 * q^3 + 6 * q^5 - 10 * q^7 + 18 * q^9	$ 2 q - 6 q^{3} + 6 q^{5} - 10 q^{7} + 18 q^{9} - 16 q^{11} + 26 q^{13} - 18 q^{15} + 4 q^{17} - 70 q^{19} + 30 q^{21} - 128 q^{23} - 6 q^{25} - 54 q^{27} - 80 q^{29} - 250 q^{31} + 48 q^{33} - 256 q^{35} - 152 q^{37} - 78 q^{39} - 146 q^{41} - 504 q^{43} + 54 q^{45} - 524 q^{47} - 410 q^{49} - 12 q^{51} - 52 q^{53} - 952 q^{55} + 210 q^{57} - 164 q^{59} - 304 q^{61} - 90 q^{63} + 78 q^{65} - 914 q^{67} + 384 q^{69} - 456 q^{73} + 18 q^{75} + 984 q^{77} - 824 q^{79} + 162 q^{81} + 828 q^{83} + 12 q^{85} + 240 q^{87} + 826 q^{89} - 130 q^{91} + 750 q^{93} + 920 q^{95} + 552 q^{97} - 144 q^{99}+O(q^{100}) $ 2 * q - 6 * q^3 + 6 * q^5 - 10 * q^7 + 18 * q^9 - 16 * q^11 + 26 * q^13 - 18 * q^15 + 4 * q^17 - 70 * q^19 + 30 * q^21 - 128 * q^23 - 6 * q^25 - 54 * q^27 - 80 * q^29 - 250 * q^31 + 48 * q^33 - 256 * q^35 - 152 * q^37 - 78 * q^39 - 146 * q^41 - 504 * q^43 + 54 * q^45 - 524 * q^47 - 410 * q^49 - 12 * q^51 - 52 * q^53 - 952 * q^55 + 210 * q^57 - 164 * q^59 - 304 * q^61 - 90 * q^63 + 78 * q^65 - 914 * q^67 + 384 * q^69 - 456 * q^73 + 18 * q^75 + 984 * q^77 - 824 * q^79 + 162 * q^81 + 828 * q^83 + 12 * q^85 + 240 * q^87 + 826 * q^89 - 130 * q^91 + 750 * q^93 + 920 * q^95 + 552 * q^97 - 144 * 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
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	13.6301	1.21912	0.609559	−	0.792741i	$-0.291347\pi$
$5$	13.6301	1.21912	0.609559	+	0.792741i	$0.291347\pi$
$6$	0	0
$7$	−15.6301	−0.843949	−0.421974	−	0.906608i	$-0.638663\pi$
$7$	−15.6301	−0.843949	−0.421974	+	0.906608i	$0.638663\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−50.5206	−1.38478	−0.692388	−	0.721526i	$-0.743441\pi$
$11$	−50.5206	−1.38478	−0.692388	+	0.721526i	$0.743441\pi$
$12$	0	0
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	0	0
$15$	−40.8904	−0.703858
$16$	0	0
$17$	2.00000	0.0285336	0.0142668	−	0.999898i	$-0.495459\pi$
$17$	2.00000	0.0285336	0.0142668	+	0.999898i	$0.495459\pi$
$18$	0	0
$19$	18.1507	0.219161	0.109581	−	0.993978i	$-0.465049\pi$
$19$	18.1507	0.219161	0.109581	+	0.993978i	$0.465049\pi$
$20$	0	0
$21$	46.8904	0.487254
$22$	0	0
$23$	−64.0000	−0.580214	−0.290107	−	0.956994i	$-0.593691\pi$
$23$	−64.0000	−0.580214	−0.290107	+	0.956994i	$0.593691\pi$
$24$	0	0
$25$	60.7809	0.486247
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	−103.781	−0.664539	−0.332269	−	0.943185i	$-0.607814\pi$
$29$	−103.781	−0.664539	−0.332269	+	0.943185i	$0.607814\pi$
$30$	0	0
$31$	34.4522	0.199606	0.0998032	−	0.995007i	$-0.468179\pi$
$31$	34.4522	0.199606	0.0998032	+	0.995007i	$0.468179\pi$
$32$	0	0
$33$	151.562	0.799500
$34$	0	0
$35$	−213.041	−1.02887
$36$	0	0
$37$	−267.343	−1.18786	−0.593930	−	0.804516i	$-0.702425\pi$
$37$	−267.343	−1.18786	−0.593930	+	0.804516i	$0.702425\pi$
$38$	0	0
$39$	−39.0000	−0.160128
$40$	0	0
$41$	−147.411	−0.561506	−0.280753	−	0.959780i	$-0.590584\pi$
$41$	−147.411	−0.561506	−0.280753	+	0.959780i	$0.590584\pi$
$42$	0	0
$43$	−166.959	−0.592116	−0.296058	−	0.955170i	$-0.595672\pi$
$43$	−166.959	−0.592116	−0.296058	+	0.955170i	$0.595672\pi$
$44$	0	0
$45$	122.671	0.406372
$46$	0	0
$47$	−325.781	−1.01106	−0.505532	−	0.862808i	$-0.668704\pi$
$47$	−325.781	−1.01106	−0.505532	+	0.862808i	$0.668704\pi$
$48$	0	0
$49$	−98.6985	−0.287751
$50$	0	0
$51$	−6.00000	−0.0164739
$52$	0	0
$53$	−111.041	−0.287786	−0.143893	−	0.989593i	$-0.545962\pi$
$53$	−111.041	−0.287786	−0.143893	+	0.989593i	$0.545962\pi$
$54$	0	0
$55$	−688.603	−1.68820
$56$	0	0
$57$	−54.4522	−0.126533
$58$	0	0
$59$	24.3015	0.0536234	0.0268117	−	0.999641i	$-0.491465\pi$
$59$	24.3015	0.0536234	0.0268117	+	0.999641i	$0.491465\pi$
$60$	0	0
$61$	−640.987	−1.34541	−0.672704	−	0.739911i	$-0.734867\pi$
$61$	−640.987	−1.34541	−0.672704	+	0.739911i	$0.734867\pi$
$62$	0	0
$63$	−140.671	−0.281316
$64$	0	0
$65$	177.192	0.338122
$66$	0	0
$67$	−382.589	−0.697622	−0.348811	−	0.937193i	$-0.613415\pi$
$67$	−382.589	−0.697622	−0.348811	+	0.937193i	$0.613415\pi$
$68$	0	0
$69$	192.000	0.334987
$70$	0	0
$71$	510.247	0.852890	0.426445	−	0.904514i	$-0.359766\pi$
$71$	510.247	0.852890	0.426445	+	0.904514i	$0.359766\pi$
$72$	0	0
$73$	−291.781	−0.467813	−0.233907	−	0.972259i	$-0.575151\pi$
$73$	−291.781	−0.467813	−0.233907	+	0.972259i	$0.575151\pi$
$74$	0	0
$75$	−182.343	−0.280735
$76$	0	0
$77$	789.644	1.16868
$78$	0	0
$79$	−794.685	−1.13176	−0.565880	−	0.824487i	$-0.691464\pi$
$79$	−794.685	−1.13176	−0.565880	+	0.824487i	$0.691464\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	945.507	1.25040	0.625198	−	0.780466i	$-0.285018\pi$
$83$	945.507	1.25040	0.625198	+	0.780466i	$0.285018\pi$
$84$	0	0
$85$	27.2603	0.0347858
$86$	0	0
$87$	311.343	0.383672
$88$	0	0
$89$	317.329	0.377941	0.188971	−	0.981983i	$-0.439485\pi$
$89$	317.329	0.377941	0.188971	+	0.981983i	$0.439485\pi$
$90$	0	0
$91$	−203.192	−0.234069
$92$	0	0
$93$	−103.357	−0.115243
$94$	0	0
$95$	247.397	0.267183
$96$	0	0
$97$	1572.88	1.64641	0.823204	−	0.567746i	$-0.192184\pi$
$97$	1572.88	1.64641	0.823204	+	0.567746i	$0.192184\pi$
$98$	0	0
$99$	−454.685	−0.461592
$100$	0	0
$101$	1820.44	1.79347	0.896735	−	0.442568i	$-0.145932\pi$
$101$	1820.44	1.79347	0.896735	+	0.442568i	$0.145932\pi$
$102$	0	0
$103$	859.015	0.821759	0.410880	−	0.911690i	$-0.365222\pi$
$103$	859.015	0.821759	0.410880	+	0.911690i	$0.365222\pi$
$104$	0	0
$105$	639.123	0.594020
$106$	0	0
$107$	−1126.74	−1.01800	−0.509001	−	0.860766i	$-0.669985\pi$
$107$	−1126.74	−1.01800	−0.509001	+	0.860766i	$0.669985\pi$
$108$	0	0
$109$	−682.659	−0.599879	−0.299940	−	0.953958i	$-0.596967\pi$
$109$	−682.659	−0.599879	−0.299940	+	0.953958i	$0.596967\pi$
$110$	0	0
$111$	802.028	0.685812
$112$	0	0
$113$	771.946	0.642642	0.321321	−	0.946970i	$-0.395873\pi$
$113$	771.946	0.642642	0.321321	+	0.946970i	$0.395873\pi$
$114$	0	0
$115$	−872.329	−0.707349
$116$	0	0
$117$	117.000	0.0924500
$118$	0	0
$119$	−31.2603	−0.0240809
$120$	0	0
$121$	1221.33	0.917603
$122$	0	0
$123$	442.233	0.324185
$124$	0	0
$125$	−875.316	−0.626325
$126$	0	0
$127$	−2182.99	−1.52527	−0.762633	−	0.646832i	$-0.776094\pi$
$127$	−2182.99	−1.52527	−0.762633	+	0.646832i	$0.776094\pi$
$128$	0	0
$129$	500.877	0.341858
$130$	0	0
$131$	881.918	0.588195	0.294097	−	0.955775i	$-0.404981\pi$
$131$	881.918	0.588195	0.294097	+	0.955775i	$0.404981\pi$
$132$	0	0
$133$	−283.699	−0.184961
$134$	0	0
$135$	−368.014	−0.234619
$136$	0	0
$137$	−2764.40	−1.72393	−0.861965	−	0.506967i	$-0.830767\pi$
$137$	−2764.40	−1.72393	−0.861965	+	0.506967i	$0.830767\pi$
$138$	0	0
$139$	1453.40	0.886875	0.443438	−	0.896305i	$-0.353759\pi$
$139$	1453.40	0.886875	0.443438	+	0.896305i	$0.353759\pi$
$140$	0	0
$141$	977.343	0.583738
$142$	0	0
$143$	−656.768	−0.384068
$144$	0	0
$145$	−1414.55	−0.810151
$146$	0	0
$147$	296.096	0.166133
$148$	0	0
$149$	560.123	0.307967	0.153983	−	0.988073i	$-0.450790\pi$
$149$	560.123	0.307967	0.153983	+	0.988073i	$0.450790\pi$
$150$	0	0
$151$	−2962.23	−1.59645	−0.798223	−	0.602363i	$-0.794226\pi$
$151$	−2962.23	−1.59645	−0.798223	+	0.602363i	$0.794226\pi$
$152$	0	0
$153$	18.0000	0.00951120
$154$	0	0
$155$	469.588	0.243344
$156$	0	0
$157$	1837.92	0.934280	0.467140	−	0.884183i	$-0.345284\pi$
$157$	1837.92	0.934280	0.467140	+	0.884183i	$0.345284\pi$
$158$	0	0
$159$	333.123	0.166154
$160$	0	0
$161$	1000.33	0.489671
$162$	0	0
$163$	1793.52	0.861838	0.430919	−	0.902391i	$-0.358190\pi$
$163$	1793.52	0.861838	0.430919	+	0.902391i	$0.358190\pi$
$164$	0	0
$165$	2065.81	0.974685
$166$	0	0
$167$	3919.37	1.81611	0.908054	−	0.418853i	$-0.137568\pi$
$167$	3919.37	1.81611	0.908054	+	0.418853i	$0.137568\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	163.357	0.0730537
$172$	0	0
$173$	3864.60	1.69838	0.849192	−	0.528084i	$-0.177089\pi$
$173$	3864.60	1.69838	0.849192	+	0.528084i	$0.177089\pi$
$174$	0	0
$175$	−950.014	−0.410367
$176$	0	0
$177$	−72.9044	−0.0309595
$178$	0	0
$179$	2829.04	1.18130	0.590650	−	0.806928i	$-0.298871\pi$
$179$	2829.04	1.18130	0.590650	+	0.806928i	$0.298871\pi$
$180$	0	0
$181$	1126.74	0.462707	0.231353	−	0.972870i	$-0.425685\pi$
$181$	1126.74	0.462707	0.231353	+	0.972870i	$0.425685\pi$
$182$	0	0
$183$	1922.96	0.776772
$184$	0	0
$185$	−3643.92	−1.44814
$186$	0	0
$187$	−101.041	−0.0395126
$188$	0	0
$189$	422.014	0.162418
$190$	0	0
$191$	662.136	0.250840	0.125420	−	0.992104i	$-0.459972\pi$
$191$	662.136	0.250840	0.125420	+	0.992104i	$0.459972\pi$
$192$	0	0
$193$	−1290.49	−0.481305	−0.240652	−	0.970611i	$-0.577361\pi$
$193$	−1290.49	−0.481305	−0.240652	+	0.970611i	$0.577361\pi$
$194$	0	0
$195$	−531.576	−0.195215
$196$	0	0
$197$	−1320.01	−0.477396	−0.238698	−	0.971094i	$-0.576721\pi$
$197$	−1320.01	−0.477396	−0.238698	+	0.971094i	$0.576721\pi$
$198$	0	0
$199$	−2711.97	−0.966064	−0.483032	−	0.875603i	$-0.660465\pi$
$199$	−2711.97	−0.966064	−0.483032	+	0.875603i	$0.660465\pi$
$200$	0	0
$201$	1147.77	0.402772
$202$	0	0
$203$	1622.11	0.560837
$204$	0	0
$205$	−2009.23	−0.684541
$206$	0	0
$207$	−576.000	−0.193405
$208$	0	0
$209$	−916.985	−0.303489
$210$	0	0
$211$	1612.06	0.525964	0.262982	−	0.964801i	$-0.415294\pi$
$211$	1612.06	0.525964	0.262982	+	0.964801i	$0.415294\pi$
$212$	0	0
$213$	−1530.74	−0.492416
$214$	0	0
$215$	−2275.67	−0.721859
$216$	0	0
$217$	−538.493	−0.168457
$218$	0	0
$219$	875.343	0.270092
$220$	0	0
$221$	26.0000	0.00791380
$222$	0	0
$223$	−5602.10	−1.68226	−0.841130	−	0.540832i	$-0.818109\pi$
$223$	−5602.10	−1.68226	−0.841130	+	0.540832i	$0.818109\pi$
$224$	0	0
$225$	547.028	0.162082
$226$	0	0
$227$	−1427.23	−0.417308	−0.208654	−	0.977990i	$-0.566908\pi$
$227$	−1427.23	−0.417308	−0.208654	+	0.977990i	$0.566908\pi$
$228$	0	0
$229$	4826.33	1.39272	0.696360	−	0.717692i	$-0.254802\pi$
$229$	4826.33	1.39272	0.696360	+	0.717692i	$0.254802\pi$
$230$	0	0
$231$	−2368.93	−0.674737
$232$	0	0
$233$	−6244.17	−1.75566	−0.877830	−	0.478972i	$-0.841010\pi$
$233$	−6244.17	−1.75566	−0.877830	+	0.478972i	$0.841010\pi$
$234$	0	0
$235$	−4440.44	−1.23261
$236$	0	0
$237$	2384.06	0.653422
$238$	0	0
$239$	6638.69	1.79674	0.898370	−	0.439239i	$-0.144752\pi$
$239$	6638.69	1.79674	0.898370	+	0.439239i	$0.144752\pi$
$240$	0	0
$241$	−4526.19	−1.20978	−0.604891	−	0.796308i	$-0.706784\pi$
$241$	−4526.19	−1.20978	−0.604891	+	0.796308i	$0.706784\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	−1345.28	−0.350802
$246$	0	0
$247$	235.959	0.0607844
$248$	0	0
$249$	−2836.52	−0.721917
$250$	0	0
$251$	2070.36	0.520636	0.260318	−	0.965523i	$-0.416173\pi$
$251$	2070.36	0.520636	0.260318	+	0.965523i	$0.416173\pi$
$252$	0	0
$253$	3233.32	0.803466
$254$	0	0
$255$	−81.7809	−0.0200836
$256$	0	0
$257$	−6154.58	−1.49382	−0.746910	−	0.664925i	$-0.768464\pi$
$257$	−6154.58	−1.49382	−0.746910	+	0.664925i	$0.768464\pi$
$258$	0	0
$259$	4178.60	1.00249
$260$	0	0
$261$	−934.028	−0.221513
$262$	0	0
$263$	4111.51	0.963979	0.481990	−	0.876177i	$-0.339914\pi$
$263$	4111.51	0.963979	0.481990	+	0.876177i	$0.339914\pi$
$264$	0	0
$265$	−1513.51	−0.350845
$266$	0	0
$267$	−951.986	−0.218204
$268$	0	0
$269$	822.851	0.186506	0.0932530	−	0.995642i	$-0.470273\pi$
$269$	822.851	0.186506	0.0932530	+	0.995642i	$0.470273\pi$
$270$	0	0
$271$	1685.41	0.377791	0.188896	−	0.981997i	$-0.439509\pi$
$271$	1685.41	0.377791	0.188896	+	0.981997i	$0.439509\pi$
$272$	0	0
$273$	609.576	0.135140
$274$	0	0
$275$	−3070.69	−0.673343
$276$	0	0
$277$	−185.235	−0.0401794	−0.0200897	−	0.999798i	$-0.506395\pi$
$277$	−185.235	−0.0401794	−0.0200897	+	0.999798i	$0.506395\pi$
$278$	0	0
$279$	310.070	0.0665354
$280$	0	0
$281$	−1008.34	−0.214066	−0.107033	−	0.994255i	$-0.534135\pi$
$281$	−1008.34	−0.214066	−0.107033	+	0.994255i	$0.534135\pi$
$282$	0	0
$283$	−6664.63	−1.39990	−0.699949	−	0.714193i	$-0.746794\pi$
$283$	−6664.63	−1.39990	−0.699949	+	0.714193i	$0.746794\pi$
$284$	0	0
$285$	−742.191	−0.154258
$286$	0	0
$287$	2304.06	0.473882
$288$	0	0
$289$	−4909.00	−0.999186
$290$	0	0
$291$	−4718.63	−0.950554
$292$	0	0
$293$	−7954.02	−1.58593	−0.792967	−	0.609265i	$-0.791465\pi$
$293$	−7954.02	−1.58593	−0.792967	+	0.609265i	$0.791465\pi$
$294$	0	0
$295$	331.232	0.0653732
$296$	0	0
$297$	1364.06	0.266500
$298$	0	0
$299$	−832.000	−0.160922
$300$	0	0
$301$	2609.59	0.499715
$302$	0	0
$303$	−5461.32	−1.03546
$304$	0	0
$305$	−8736.74	−1.64021
$306$	0	0
$307$	8831.69	1.64186	0.820930	−	0.571029i	$-0.193456\pi$
$307$	8831.69	1.64186	0.820930	+	0.571029i	$0.193456\pi$
$308$	0	0
$309$	−2577.04	−0.474443
$310$	0	0
$311$	8043.24	1.46653	0.733264	−	0.679944i	$-0.237996\pi$
$311$	8043.24	1.46653	0.733264	+	0.679944i	$0.237996\pi$
$312$	0	0
$313$	−9939.81	−1.79499	−0.897494	−	0.441026i	$-0.854615\pi$
$313$	−9939.81	−1.79499	−0.897494	+	0.441026i	$0.854615\pi$
$314$	0	0
$315$	−1917.37	−0.342957
$316$	0	0
$317$	6640.48	1.17655	0.588275	−	0.808661i	$-0.299807\pi$
$317$	6640.48	1.17655	0.588275	+	0.808661i	$0.299807\pi$
$318$	0	0
$319$	5243.07	0.920237
$320$	0	0
$321$	3380.22	0.587743
$322$	0	0
$323$	36.3015	0.00625346
$324$	0	0
$325$	790.151	0.134861
$326$	0	0
$327$	2047.98	0.346340
$328$	0	0
$329$	5092.00	0.853286
$330$	0	0
$331$	−2007.63	−0.333382	−0.166691	−	0.986009i	$-0.553308\pi$
$331$	−2007.63	−0.333382	−0.166691	+	0.986009i	$0.553308\pi$
$332$	0	0
$333$	−2406.08	−0.395954
$334$	0	0
$335$	−5214.74	−0.850483
$336$	0	0
$337$	−4991.43	−0.806826	−0.403413	−	0.915018i	$-0.632176\pi$
$337$	−4991.43	−0.806826	−0.403413	+	0.915018i	$0.632176\pi$
$338$	0	0
$339$	−2315.84	−0.371029
$340$	0	0
$341$	−1740.54	−0.276410
$342$	0	0
$343$	6903.81	1.08680
$344$	0	0
$345$	2616.99	0.408388
$346$	0	0
$347$	−10913.8	−1.68842	−0.844209	−	0.536013i	$-0.819930\pi$
$347$	−10913.8	−1.68842	−0.844209	+	0.536013i	$0.819930\pi$
$348$	0	0
$349$	9148.00	1.40310	0.701549	−	0.712621i	$-0.252492\pi$
$349$	9148.00	1.40310	0.701549	+	0.712621i	$0.252492\pi$
$350$	0	0
$351$	−351.000	−0.0533761
$352$	0	0
$353$	1337.64	0.201686	0.100843	−	0.994902i	$-0.467846\pi$
$353$	1337.64	0.201686	0.100843	+	0.994902i	$0.467846\pi$
$354$	0	0
$355$	6954.74	1.03977
$356$	0	0
$357$	93.7809	0.0139031
$358$	0	0
$359$	7865.89	1.15639	0.578197	−	0.815897i	$-0.303756\pi$
$359$	7865.89	1.15639	0.578197	+	0.815897i	$0.303756\pi$
$360$	0	0
$361$	−6529.55	−0.951968
$362$	0	0
$363$	−3663.99	−0.529778
$364$	0	0
$365$	−3977.02	−0.570319
$366$	0	0
$367$	1935.86	0.275344	0.137672	−	0.990478i	$-0.456038\pi$
$367$	1935.86	0.275344	0.137672	+	0.990478i	$0.456038\pi$
$368$	0	0
$369$	−1326.70	−0.187169
$370$	0	0
$371$	1735.59	0.242877
$372$	0	0
$373$	−5604.36	−0.777970	−0.388985	−	0.921244i	$-0.627174\pi$
$373$	−5604.36	−0.777970	−0.388985	+	0.921244i	$0.627174\pi$
$374$	0	0
$375$	2625.95	0.361609
$376$	0	0
$377$	−1349.15	−0.184310
$378$	0	0
$379$	271.521	0.0367998	0.0183999	−	0.999831i	$-0.494143\pi$
$379$	271.521	0.0367998	0.0183999	+	0.999831i	$0.494143\pi$
$380$	0	0
$381$	6548.96	0.880613
$382$	0	0
$383$	−5763.16	−0.768887	−0.384443	−	0.923149i	$-0.625607\pi$
$383$	−5763.16	−0.768887	−0.384443	+	0.923149i	$0.625607\pi$
$384$	0	0
$385$	10763.0	1.42476
$386$	0	0
$387$	−1502.63	−0.197372
$388$	0	0
$389$	−6596.09	−0.859730	−0.429865	−	0.902893i	$-0.641439\pi$
$389$	−6596.09	−0.859730	−0.429865	+	0.902893i	$0.641439\pi$
$390$	0	0
$391$	−128.000	−0.0165556
$392$	0	0
$393$	−2645.75	−0.339594
$394$	0	0
$395$	−10831.7	−1.37975
$396$	0	0
$397$	7173.42	0.906861	0.453430	−	0.891292i	$-0.350200\pi$
$397$	7173.42	0.906861	0.453430	+	0.891292i	$0.350200\pi$
$398$	0	0
$399$	851.096	0.106787
$400$	0	0
$401$	−3759.83	−0.468222	−0.234111	−	0.972210i	$-0.575218\pi$
$401$	−3759.83	−0.468222	−0.234111	+	0.972210i	$0.575218\pi$
$402$	0	0
$403$	447.878	0.0553608
$404$	0	0
$405$	1104.04	0.135457
$406$	0	0
$407$	13506.3	1.64492
$408$	0	0
$409$	10857.1	1.31259	0.656295	−	0.754505i	$-0.272123\pi$
$409$	10857.1	1.31259	0.656295	+	0.754505i	$0.272123\pi$
$410$	0	0
$411$	8293.20	0.995312
$412$	0	0
$413$	−379.835	−0.0452554
$414$	0	0
$415$	12887.4	1.52438
$416$	0	0
$417$	−4360.20	−0.512038
$418$	0	0
$419$	9802.86	1.14296	0.571481	−	0.820616i	$-0.306369\pi$
$419$	9802.86	1.14296	0.571481	+	0.820616i	$0.306369\pi$
$420$	0	0
$421$	5004.96	0.579398	0.289699	−	0.957118i	$-0.406445\pi$
$421$	5004.96	0.579398	0.289699	+	0.957118i	$0.406445\pi$
$422$	0	0
$423$	−2932.03	−0.337021
$424$	0	0
$425$	121.562	0.0138744
$426$	0	0
$427$	10018.7	1.13546
$428$	0	0
$429$	1970.30	0.221742
$430$	0	0
$431$	−9275.81	−1.03666	−0.518330	−	0.855181i	$-0.673446\pi$
$431$	−9275.81	−1.03666	−0.518330	+	0.855181i	$0.673446\pi$
$432$	0	0
$433$	3160.88	0.350813	0.175406	−	0.984496i	$-0.443876\pi$
$433$	3160.88	0.350813	0.175406	+	0.984496i	$0.443876\pi$
$434$	0	0
$435$	4243.65	0.467741
$436$	0	0
$437$	−1161.65	−0.127160
$438$	0	0
$439$	−14035.4	−1.52591	−0.762954	−	0.646453i	$-0.776252\pi$
$439$	−14035.4	−1.52591	−0.762954	+	0.646453i	$0.776252\pi$
$440$	0	0
$441$	−888.287	−0.0959170
$442$	0	0
$443$	761.046	0.0816217	0.0408108	−	0.999167i	$-0.487006\pi$
$443$	761.046	0.0816217	0.0408108	+	0.999167i	$0.487006\pi$
$444$	0	0
$445$	4325.24	0.460755
$446$	0	0
$447$	−1680.37	−0.177805
$448$	0	0
$449$	8385.22	0.881343	0.440672	−	0.897668i	$-0.354740\pi$
$449$	8385.22	0.881343	0.440672	+	0.897668i	$0.354740\pi$
$450$	0	0
$451$	7447.29	0.777559
$452$	0	0
$453$	8886.70	0.921708
$454$	0	0
$455$	−2769.54	−0.285358
$456$	0	0
$457$	16484.7	1.68736	0.843679	−	0.536848i	$-0.180385\pi$
$457$	16484.7	1.68736	0.843679	+	0.536848i	$0.180385\pi$
$458$	0	0
$459$	−54.0000	−0.00549129
$460$	0	0
$461$	2478.45	0.250396	0.125198	−	0.992132i	$-0.460043\pi$
$461$	2478.45	0.250396	0.125198	+	0.992132i	$0.460043\pi$
$462$	0	0
$463$	−2588.90	−0.259863	−0.129931	−	0.991523i	$-0.541476\pi$
$463$	−2588.90	−0.259863	−0.129931	+	0.991523i	$0.541476\pi$
$464$	0	0
$465$	−1408.77	−0.140494
$466$	0	0
$467$	−13670.7	−1.35461	−0.677306	−	0.735701i	$-0.736853\pi$
$467$	−13670.7	−1.35461	−0.677306	+	0.735701i	$0.736853\pi$
$468$	0	0
$469$	5979.92	0.588757
$470$	0	0
$471$	−5513.76	−0.539407
$472$	0	0
$473$	8434.86	0.819948
$474$	0	0
$475$	1103.22	0.106566
$476$	0	0
$477$	−999.370	−0.0959288
$478$	0	0
$479$	8034.36	0.766386	0.383193	−	0.923668i	$-0.374824\pi$
$479$	8034.36	0.766386	0.383193	+	0.923668i	$0.374824\pi$
$480$	0	0
$481$	−3475.45	−0.329453
$482$	0	0
$483$	−3000.99	−0.282712
$484$	0	0
$485$	21438.6	2.00716
$486$	0	0
$487$	−7890.43	−0.734188	−0.367094	−	0.930184i	$-0.619647\pi$
$487$	−7890.43	−0.734188	−0.367094	+	0.930184i	$0.619647\pi$
$488$	0	0
$489$	−5380.57	−0.497582
$490$	0	0
$491$	2217.87	0.203852	0.101926	−	0.994792i	$-0.467500\pi$
$491$	2217.87	0.203852	0.101926	+	0.994792i	$0.467500\pi$
$492$	0	0
$493$	−207.562	−0.0189617
$494$	0	0
$495$	−6197.43	−0.562735
$496$	0	0
$497$	−7975.23	−0.719795
$498$	0	0
$499$	−6636.90	−0.595407	−0.297704	−	0.954658i	$-0.596221\pi$
$499$	−6636.90	−0.595407	−0.297704	+	0.954658i	$0.596221\pi$
$500$	0	0
$501$	−11758.1	−1.04853
$502$	0	0
$503$	−9418.80	−0.834917	−0.417459	−	0.908696i	$-0.637079\pi$
$503$	−9418.80	−0.834917	−0.417459	+	0.908696i	$0.637079\pi$
$504$	0	0
$505$	24812.9	2.18645
$506$	0	0
$507$	−507.000	−0.0444116
$508$	0	0
$509$	−11177.6	−0.973353	−0.486676	−	0.873582i	$-0.661791\pi$
$509$	−11177.6	−0.973353	−0.486676	+	0.873582i	$0.661791\pi$
$510$	0	0
$511$	4560.58	0.394810
$512$	0	0
$513$	−490.070	−0.0421776
$514$	0	0
$515$	11708.5	1.00182
$516$	0	0
$517$	16458.6	1.40010
$518$	0	0
$519$	−11593.8	−0.980562
$520$	0	0
$521$	2993.21	0.251699	0.125849	−	0.992049i	$-0.459834\pi$
$521$	2993.21	0.251699	0.125849	+	0.992049i	$0.459834\pi$
$522$	0	0
$523$	−15198.3	−1.27070	−0.635351	−	0.772224i	$-0.719144\pi$
$523$	−15198.3	−1.27070	−0.635351	+	0.772224i	$0.719144\pi$
$524$	0	0
$525$	2850.04	0.236926
$526$	0	0
$527$	68.9044	0.00569549
$528$	0	0
$529$	−8071.00	−0.663352
$530$	0	0
$531$	218.713	0.0178745
$532$	0	0
$533$	−1916.34	−0.155734
$534$	0	0
$535$	−15357.6	−1.24106
$536$	0	0
$537$	−8487.13	−0.682024
$538$	0	0
$539$	4986.31	0.398470
$540$	0	0
$541$	−2043.27	−0.162379	−0.0811894	−	0.996699i	$-0.525872\pi$
$541$	−2043.27	−0.162379	−0.0811894	+	0.996699i	$0.525872\pi$
$542$	0	0
$543$	−3380.22	−0.267144
$544$	0	0
$545$	−9304.74	−0.731323
$546$	0	0
$547$	−4616.69	−0.360869	−0.180434	−	0.983587i	$-0.557750\pi$
$547$	−4616.69	−0.360869	−0.180434	+	0.983587i	$0.557750\pi$
$548$	0	0
$549$	−5768.88	−0.448470
$550$	0	0
$551$	−1883.70	−0.145641
$552$	0	0
$553$	12421.0	0.955148
$554$	0	0
$555$	10931.8	0.836085
$556$	0	0
$557$	18077.9	1.37520	0.687600	−	0.726089i	$-0.258664\pi$
$557$	18077.9	1.37520	0.687600	+	0.726089i	$0.258664\pi$
$558$	0	0
$559$	−2170.46	−0.164223
$560$	0	0
$561$	303.123	0.0228126
$562$	0	0
$563$	13561.8	1.01520	0.507602	−	0.861592i	$-0.330532\pi$
$563$	13561.8	1.01520	0.507602	+	0.861592i	$0.330532\pi$
$564$	0	0
$565$	10521.7	0.783456
$566$	0	0
$567$	−1266.04	−0.0937721
$568$	0	0
$569$	−7280.83	−0.536429	−0.268215	−	0.963359i	$-0.586434\pi$
$569$	−7280.83	−0.536429	−0.268215	+	0.963359i	$0.586434\pi$
$570$	0	0
$571$	−17884.5	−1.31076	−0.655380	−	0.755299i	$-0.727492\pi$
$571$	−17884.5	−1.31076	−0.655380	+	0.755299i	$0.727492\pi$
$572$	0	0
$573$	−1986.41	−0.144823
$574$	0	0
$575$	−3889.98	−0.282127
$576$	0	0
$577$	−12067.2	−0.870645	−0.435323	−	0.900275i	$-0.643366\pi$
$577$	−12067.2	−0.870645	−0.435323	+	0.900275i	$0.643366\pi$
$578$	0	0
$579$	3871.48	0.277881
$580$	0	0
$581$	−14778.4	−1.05527
$582$	0	0
$583$	5609.86	0.398519
$584$	0	0
$585$	1594.73	0.112707
$586$	0	0
$587$	−9202.11	−0.647038	−0.323519	−	0.946222i	$-0.604866\pi$
$587$	−9202.11	−0.647038	−0.323519	+	0.946222i	$0.604866\pi$
$588$	0	0
$589$	625.332	0.0437460
$590$	0	0
$591$	3960.04	0.275625
$592$	0	0
$593$	−16985.1	−1.17621	−0.588106	−	0.808784i	$-0.700126\pi$
$593$	−16985.1	−1.17621	−0.588106	+	0.808784i	$0.700126\pi$
$594$	0	0
$595$	−426.082	−0.0293574
$596$	0	0
$597$	8135.92	0.557757
$598$	0	0
$599$	541.705	0.0369507	0.0184753	−	0.999829i	$-0.494119\pi$
$599$	541.705	0.0369507	0.0184753	+	0.999829i	$0.494119\pi$
$600$	0	0
$601$	−24039.4	−1.63160	−0.815798	−	0.578337i	$-0.803702\pi$
$601$	−24039.4	−1.63160	−0.815798	+	0.578337i	$0.803702\pi$
$602$	0	0
$603$	−3443.30	−0.232541
$604$	0	0
$605$	16646.9	1.11867
$606$	0	0
$607$	22951.8	1.53474	0.767369	−	0.641206i	$-0.221565\pi$
$607$	22951.8	1.53474	0.767369	+	0.641206i	$0.221565\pi$
$608$	0	0
$609$	−4866.33	−0.323799
$610$	0	0
$611$	−4235.15	−0.280419
$612$	0	0
$613$	−19479.1	−1.28345	−0.641725	−	0.766935i	$-0.721781\pi$
$613$	−19479.1	−1.28345	−0.641725	+	0.766935i	$0.721781\pi$
$614$	0	0
$615$	6027.70	0.395220
$616$	0	0
$617$	12586.6	0.821258	0.410629	−	0.911803i	$-0.365309\pi$
$617$	12586.6	0.821258	0.410629	+	0.911803i	$0.365309\pi$
$618$	0	0
$619$	10589.4	0.687598	0.343799	−	0.939043i	$-0.388286\pi$
$619$	10589.4	0.687598	0.343799	+	0.939043i	$0.388286\pi$
$620$	0	0
$621$	1728.00	0.111662
$622$	0	0
$623$	−4959.89	−0.318963
$624$	0	0
$625$	−19528.3	−1.24981
$626$	0	0
$627$	2750.96	0.175219
$628$	0	0
$629$	−534.685	−0.0338940
$630$	0	0
$631$	−11104.0	−0.700547	−0.350273	−	0.936647i	$-0.613911\pi$
$631$	−11104.0	−0.700547	−0.350273	+	0.936647i	$0.613911\pi$
$632$	0	0
$633$	−4836.17	−0.303666
$634$	0	0
$635$	−29754.4	−1.85948
$636$	0	0
$637$	−1283.08	−0.0798077
$638$	0	0
$639$	4592.22	0.284297
$640$	0	0
$641$	15811.9	0.974311	0.487155	−	0.873315i	$-0.338035\pi$
$641$	15811.9	0.974311	0.487155	+	0.873315i	$0.338035\pi$
$642$	0	0
$643$	1406.62	0.0862704	0.0431352	−	0.999069i	$-0.486265\pi$
$643$	1406.62	0.0862704	0.0431352	+	0.999069i	$0.486265\pi$
$644$	0	0
$645$	6827.02	0.416765
$646$	0	0
$647$	−11661.8	−0.708613	−0.354307	−	0.935129i	$-0.615283\pi$
$647$	−11661.8	−0.708613	−0.354307	+	0.935129i	$0.615283\pi$
$648$	0	0
$649$	−1227.72	−0.0742564
$650$	0	0
$651$	1615.48	0.0972590
$652$	0	0
$653$	−5545.04	−0.332303	−0.166152	−	0.986100i	$-0.553134\pi$
$653$	−5545.04	−0.332303	−0.166152	+	0.986100i	$0.553134\pi$
$654$	0	0
$655$	12020.7	0.717078
$656$	0	0
$657$	−2626.03	−0.155938
$658$	0	0
$659$	10950.0	0.647272	0.323636	−	0.946182i	$-0.395095\pi$
$659$	10950.0	0.647272	0.323636	+	0.946182i	$0.395095\pi$
$660$	0	0
$661$	18488.8	1.08795	0.543973	−	0.839103i	$-0.316919\pi$
$661$	18488.8	1.08795	0.543973	+	0.839103i	$0.316919\pi$
$662$	0	0
$663$	−78.0000	−0.00456903
$664$	0	0
$665$	−3866.85	−0.225489
$666$	0	0
$667$	6641.98	0.385575
$668$	0	0
$669$	16806.3	0.971254
$670$	0	0
$671$	32383.0	1.86309
$672$	0	0
$673$	−5865.80	−0.335973	−0.167987	−	0.985789i	$-0.553727\pi$
$673$	−5865.80	−0.335973	−0.167987	+	0.985789i	$0.553727\pi$
$674$	0	0
$675$	−1641.08	−0.0935783
$676$	0	0
$677$	−10258.6	−0.582376	−0.291188	−	0.956666i	$-0.594050\pi$
$677$	−10258.6	−0.582376	−0.291188	+	0.956666i	$0.594050\pi$
$678$	0	0
$679$	−24584.3	−1.38948
$680$	0	0
$681$	4281.70	0.240933
$682$	0	0
$683$	−26215.4	−1.46867	−0.734337	−	0.678785i	$-0.762507\pi$
$683$	−26215.4	−1.46867	−0.734337	+	0.678785i	$0.762507\pi$
$684$	0	0
$685$	−37679.2	−2.10167
$686$	0	0
$687$	−14479.0	−0.804088
$688$	0	0
$689$	−1443.54	−0.0798176
$690$	0	0
$691$	−7290.15	−0.401346	−0.200673	−	0.979658i	$-0.564313\pi$
$691$	−7290.15	−0.401346	−0.200673	+	0.979658i	$0.564313\pi$
$692$	0	0
$693$	7106.80	0.389560
$694$	0	0
$695$	19810.0	1.08120
$696$	0	0
$697$	−294.822	−0.0160218
$698$	0	0
$699$	18732.5	1.01363
$700$	0	0
$701$	15472.1	0.833628	0.416814	−	0.908992i	$-0.363147\pi$
$701$	15472.1	0.833628	0.416814	+	0.908992i	$0.363147\pi$
$702$	0	0
$703$	−4852.46	−0.260333
$704$	0	0
$705$	13321.3	0.711645
$706$	0	0
$707$	−28453.7	−1.51360
$708$	0	0
$709$	17594.7	0.931994	0.465997	−	0.884786i	$-0.345696\pi$
$709$	17594.7	0.931994	0.465997	+	0.884786i	$0.345696\pi$
$710$	0	0
$711$	−7152.17	−0.377253
$712$	0	0
$713$	−2204.94	−0.115814
$714$	0	0
$715$	−8951.84	−0.468223
$716$	0	0
$717$	−19916.1	−1.03735
$718$	0	0
$719$	−12944.2	−0.671400	−0.335700	−	0.941969i	$-0.608973\pi$
$719$	−12944.2	−0.671400	−0.335700	+	0.941969i	$0.608973\pi$
$720$	0	0
$721$	−13426.5	−0.693523
$722$	0	0
$723$	13578.6	0.698469
$724$	0	0
$725$	−6307.89	−0.323130
$726$	0	0
$727$	20231.8	1.03213	0.516063	−	0.856551i	$-0.327397\pi$
$727$	20231.8	1.03213	0.516063	+	0.856551i	$0.327397\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−333.918	−0.0168952
$732$	0	0
$733$	−26581.0	−1.33942	−0.669708	−	0.742624i	$-0.733581\pi$
$733$	−26581.0	−1.33942	−0.669708	+	0.742624i	$0.733581\pi$
$734$	0	0
$735$	4035.83	0.202536
$736$	0	0
$737$	19328.6	0.966050
$738$	0	0
$739$	−1018.30	−0.0506886	−0.0253443	−	0.999679i	$-0.508068\pi$
$739$	−1018.30	−0.0506886	−0.0253443	+	0.999679i	$0.508068\pi$
$740$	0	0
$741$	−707.878	−0.0350939
$742$	0	0
$743$	−31784.2	−1.56938	−0.784691	−	0.619888i	$-0.787178\pi$
$743$	−31784.2	−1.56938	−0.784691	+	0.619888i	$0.787178\pi$
$744$	0	0
$745$	7634.56	0.375448
$746$	0	0
$747$	8509.57	0.416799
$748$	0	0
$749$	17611.1	0.859141
$750$	0	0
$751$	−3549.69	−0.172477	−0.0862385	−	0.996275i	$-0.527485\pi$
$751$	−3549.69	−0.172477	−0.0862385	+	0.996275i	$0.527485\pi$
$752$	0	0
$753$	−6211.07	−0.300589
$754$	0	0
$755$	−40375.7	−1.94625
$756$	0	0
$757$	−18852.3	−0.905151	−0.452576	−	0.891726i	$-0.649495\pi$
$757$	−18852.3	−0.905151	−0.452576	+	0.891726i	$0.649495\pi$
$758$	0	0
$759$	−9699.95	−0.463881
$760$	0	0
$761$	−8131.76	−0.387353	−0.193677	−	0.981065i	$-0.562041\pi$
$761$	−8131.76	−0.387353	−0.193677	+	0.981065i	$0.562041\pi$
$762$	0	0
$763$	10670.1	0.506267
$764$	0	0
$765$	245.343	0.0115953
$766$	0	0
$767$	315.919	0.0148725
$768$	0	0
$769$	−15773.3	−0.739662	−0.369831	−	0.929099i	$-0.620584\pi$
$769$	−15773.3	−0.739662	−0.369831	+	0.929099i	$0.620584\pi$
$770$	0	0
$771$	18463.7	0.862458
$772$	0	0
$773$	23061.4	1.07304	0.536521	−	0.843887i	$-0.319738\pi$
$773$	23061.4	1.07304	0.536521	+	0.843887i	$0.319738\pi$
$774$	0	0
$775$	2094.03	0.0970580
$776$	0	0
$777$	−12535.8	−0.578790
$778$	0	0
$779$	−2675.62	−0.123060
$780$	0	0
$781$	−25778.0	−1.18106
$782$	0	0
$783$	2802.08	0.127891
$784$	0	0
$785$	25051.1	1.13900
$786$	0	0
$787$	−38874.7	−1.76078	−0.880390	−	0.474251i	$-0.842719\pi$
$787$	−38874.7	−1.76078	−0.880390	+	0.474251i	$0.842719\pi$
$788$	0	0
$789$	−12334.5	−0.556554
$790$	0	0
$791$	−12065.6	−0.542357
$792$	0	0
$793$	−8332.83	−0.373149
$794$	0	0
$795$	4540.52	0.202561
$796$	0	0
$797$	8956.95	0.398082	0.199041	−	0.979991i	$-0.436217\pi$
$797$	8956.95	0.398082	0.199041	+	0.979991i	$0.436217\pi$
$798$	0	0
$799$	−651.562	−0.0288493
$800$	0	0
$801$	2855.96	0.125980
$802$	0	0
$803$	14740.9	0.647816
$804$	0	0
$805$	13634.6	0.596966
$806$	0	0
$807$	−2468.55	−0.107679
$808$	0	0
$809$	−21956.3	−0.954195	−0.477098	−	0.878850i	$-0.658311\pi$
$809$	−21956.3	−0.954195	−0.477098	+	0.878850i	$0.658311\pi$
$810$	0	0
$811$	14652.2	0.634414	0.317207	−	0.948356i	$-0.397255\pi$
$811$	14652.2	0.634414	0.317207	+	0.948356i	$0.397255\pi$
$812$	0	0
$813$	−5056.23	−0.218118
$814$	0	0
$815$	24446.0	1.05068
$816$	0	0
$817$	−3030.42	−0.129769
$818$	0	0
$819$	−1828.73	−0.0780231
$820$	0	0
$821$	12172.5	0.517444	0.258722	−	0.965952i	$-0.416699\pi$
$821$	12172.5	0.517444	0.258722	+	0.965952i	$0.416699\pi$
$822$	0	0
$823$	−22022.1	−0.932738	−0.466369	−	0.884590i	$-0.654438\pi$
$823$	−22022.1	−0.932738	−0.466369	+	0.884590i	$0.654438\pi$
$824$	0	0
$825$	9212.06	0.388755
$826$	0	0
$827$	7620.60	0.320428	0.160214	−	0.987082i	$-0.448781\pi$
$827$	7620.60	0.320428	0.160214	+	0.987082i	$0.448781\pi$
$828$	0	0
$829$	18727.6	0.784603	0.392302	−	0.919837i	$-0.371679\pi$
$829$	18727.6	0.784603	0.392302	+	0.919837i	$0.371679\pi$
$830$	0	0
$831$	555.705	0.0231976
$832$	0	0
$833$	−197.397	−0.00821057
$834$	0	0
$835$	53421.6	2.21405
$836$	0	0
$837$	−930.209	−0.0384143
$838$	0	0
$839$	6853.37	0.282008	0.141004	−	0.990009i	$-0.454967\pi$
$839$	6853.37	0.282008	0.141004	+	0.990009i	$0.454967\pi$
$840$	0	0
$841$	−13618.5	−0.558388
$842$	0	0
$843$	3025.02	0.123591
$844$	0	0
$845$	2303.49	0.0937783
$846$	0	0
$847$	−19089.6	−0.774410
$848$	0	0
$849$	19993.9	0.808232
$850$	0	0
$851$	17109.9	0.689214
$852$	0	0
$853$	−16744.0	−0.672102	−0.336051	−	0.941844i	$-0.609091\pi$
$853$	−16744.0	−0.672102	−0.336051	+	0.941844i	$0.609091\pi$
$854$	0	0
$855$	2226.57	0.0890611
$856$	0	0
$857$	28271.8	1.12689	0.563446	−	0.826153i	$-0.309475\pi$
$857$	28271.8	1.12689	0.563446	+	0.826153i	$0.309475\pi$
$858$	0	0
$859$	40681.8	1.61589	0.807943	−	0.589260i	$-0.200581\pi$
$859$	40681.8	1.61589	0.807943	+	0.589260i	$0.200581\pi$
$860$	0	0
$861$	−6912.17	−0.273596
$862$	0	0
$863$	16589.8	0.654371	0.327186	−	0.944960i	$-0.393900\pi$
$863$	16589.8	0.654371	0.327186	+	0.944960i	$0.393900\pi$
$864$	0	0
$865$	52675.1	2.07053
$866$	0	0
$867$	14727.0	0.576880
$868$	0	0
$869$	40148.0	1.56723
$870$	0	0
$871$	−4973.66	−0.193486
$872$	0	0
$873$	14155.9	0.548803
$874$	0	0
$875$	13681.3	0.528586
$876$	0	0
$877$	−41833.7	−1.61074	−0.805372	−	0.592769i	$-0.798035\pi$
$877$	−41833.7	−1.61074	−0.805372	+	0.592769i	$0.798035\pi$
$878$	0	0
$879$	23862.0	0.915639
$880$	0	0
$881$	−24676.3	−0.943663	−0.471831	−	0.881689i	$-0.656407\pi$
$881$	−24676.3	−0.943663	−0.471831	+	0.881689i	$0.656407\pi$
$882$	0	0
$883$	30567.6	1.16499	0.582493	−	0.812836i	$-0.302077\pi$
$883$	30567.6	1.16499	0.582493	+	0.812836i	$0.302077\pi$
$884$	0	0
$885$	−993.697	−0.0377432
$886$	0	0
$887$	19020.9	0.720023	0.360012	−	0.932948i	$-0.382773\pi$
$887$	19020.9	0.720023	0.360012	+	0.932948i	$0.382773\pi$
$888$	0	0
$889$	34120.4	1.28725
$890$	0	0
$891$	−4092.17	−0.153864
$892$	0	0
$893$	−5913.16	−0.221586
$894$	0	0
$895$	38560.3	1.44014
$896$	0	0
$897$	2496.00	0.0929086
$898$	0	0
$899$	−3575.48	−0.132646
$900$	0	0
$901$	−222.082	−0.00821158
$902$	0	0
$903$	−7828.77	−0.288511
$904$	0	0
$905$	15357.6	0.564094
$906$	0	0
$907$	−39186.2	−1.43457	−0.717287	−	0.696778i	$-0.754616\pi$
$907$	−39186.2	−1.43457	−0.717287	+	0.696778i	$0.754616\pi$
$908$	0	0
$909$	16384.0	0.597823
$910$	0	0
$911$	−1303.25	−0.0473969	−0.0236985	−	0.999719i	$-0.507544\pi$
$911$	−1303.25	−0.0473969	−0.0236985	+	0.999719i	$0.507544\pi$
$912$	0	0
$913$	−47767.6	−1.73152
$914$	0	0
$915$	26210.2	0.946976
$916$	0	0
$917$	−13784.5	−0.496406
$918$	0	0
$919$	11710.2	0.420330	0.210165	−	0.977666i	$-0.432600\pi$
$919$	11710.2	0.420330	0.210165	+	0.977666i	$0.432600\pi$
$920$	0	0
$921$	−26495.1	−0.947928
$922$	0	0
$923$	6633.21	0.236549
$924$	0	0
$925$	−16249.3	−0.577594
$926$	0	0
$927$	7731.13	0.273920
$928$	0	0
$929$	34967.9	1.23494	0.617470	−	0.786595i	$-0.288158\pi$
$929$	34967.9	1.23494	0.617470	+	0.786595i	$0.288158\pi$
$930$	0	0
$931$	−1791.45	−0.0630638
$932$	0	0
$933$	−24129.7	−0.846700
$934$	0	0
$935$	−1377.21	−0.0481705
$936$	0	0
$937$	−6538.36	−0.227960	−0.113980	−	0.993483i	$-0.536360\pi$
$937$	−6538.36	−0.227960	−0.113980	+	0.993483i	$0.536360\pi$
$938$	0	0
$939$	29819.4	1.03634
$940$	0	0
$941$	−26190.2	−0.907309	−0.453654	−	0.891178i	$-0.649880\pi$
$941$	−26190.2	−0.907309	−0.453654	+	0.891178i	$0.649880\pi$
$942$	0	0
$943$	9434.31	0.325793
$944$	0	0
$945$	5752.11	0.198007
$946$	0	0
$947$	44316.2	1.52068	0.760340	−	0.649526i	$-0.225032\pi$
$947$	44316.2	1.52068	0.760340	+	0.649526i	$0.225032\pi$
$948$	0	0
$949$	−3793.15	−0.129748
$950$	0	0
$951$	−19921.4	−0.679282
$952$	0	0
$953$	−10052.0	−0.341676	−0.170838	−	0.985299i	$-0.554647\pi$
$953$	−10052.0	−0.341676	−0.170838	+	0.985299i	$0.554647\pi$
$954$	0	0
$955$	9025.00	0.305803
$956$	0	0
$957$	−15729.2	−0.531299
$958$	0	0
$959$	43208.0	1.45491
$960$	0	0
$961$	−28604.0	−0.960157
$962$	0	0
$963$	−10140.7	−0.339334
$964$	0	0
$965$	−17589.6	−0.586767
$966$	0	0
$967$	−10809.8	−0.359482	−0.179741	−	0.983714i	$-0.557526\pi$
$967$	−10809.8	−0.359482	−0.179741	+	0.983714i	$0.557526\pi$
$968$	0	0
$969$	−108.904	−0.00361044
$970$	0	0
$971$	16280.1	0.538056	0.269028	−	0.963132i	$-0.413298\pi$
$971$	16280.1	0.538056	0.269028	+	0.963132i	$0.413298\pi$
$972$	0	0
$973$	−22716.8	−0.748477
$974$	0	0
$975$	−2370.45	−0.0778618
$976$	0	0
$977$	56327.0	1.84448	0.922242	−	0.386613i	$-0.126355\pi$
$977$	56327.0	1.84448	0.922242	+	0.386613i	$0.126355\pi$
$978$	0	0
$979$	−16031.6	−0.523364
$980$	0	0
$981$	−6143.93	−0.199960
$982$	0	0
$983$	32733.6	1.06210	0.531048	−	0.847342i	$-0.321798\pi$
$983$	32733.6	1.06210	0.531048	+	0.847342i	$0.321798\pi$
$984$	0	0
$985$	−17992.0	−0.582002
$986$	0	0
$987$	−15276.0	−0.492645
$988$	0	0
$989$	10685.4	0.343554
$990$	0	0
$991$	−2430.76	−0.0779168	−0.0389584	−	0.999241i	$-0.512404\pi$
$991$	−2430.76	−0.0779168	−0.0389584	+	0.999241i	$0.512404\pi$
$992$	0	0
$993$	6022.90	0.192478
$994$	0	0
$995$	−36964.6	−1.17775
$996$	0	0
$997$	−38619.5	−1.22677	−0.613387	−	0.789783i	$-0.710193\pi$
$997$	−38619.5	−1.22677	−0.613387	+	0.789783i	$0.710193\pi$
$998$	0	0
$999$	7218.25	0.228604

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	312.4.a.c.1.2	✓	2
3.2	odd	2		936.4.a.d.1.1		2
4.3	odd	2		624.4.a.q.1.2		2
8.3	odd	2		2496.4.a.v.1.1		2
8.5	even	2		2496.4.a.be.1.1		2
12.11	even	2		1872.4.a.x.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
312.4.a.c.1.2	✓	2	1.1	even	1	trivial
624.4.a.q.1.2		2	4.3	odd	2
936.4.a.d.1.1		2	3.2	odd	2
1872.4.a.x.1.1		2	12.11	even	2
2496.4.a.v.1.1		2	8.3	odd	2
2496.4.a.be.1.1		2	8.5	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 312 = 2^{3} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	312.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} +13.6301 q^{5} -15.6301 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} +13.6301 q^{5} -15.6301 q^{7} +9.00000 q^{9} -50.5206 q^{11} +13.0000 q^{13} -40.8904 q^{15} +2.00000 q^{17} +18.1507 q^{19} +46.8904 q^{21} -64.0000 q^{23} +60.7809 q^{25} -27.0000 q^{27} -103.781 q^{29} +34.4522 q^{31} +151.562 q^{33} -213.041 q^{35} -267.343 q^{37} -39.0000 q^{39} -147.411 q^{41} -166.959 q^{43} +122.671 q^{45} -325.781 q^{47} -98.6985 q^{49} -6.00000 q^{51} -111.041 q^{53} -688.603 q^{55} -54.4522 q^{57} +24.3015 q^{59} -640.987 q^{61} -140.671 q^{63} +177.192 q^{65} -382.589 q^{67} +192.000 q^{69} +510.247 q^{71} -291.781 q^{73} -182.343 q^{75} +789.644 q^{77} -794.685 q^{79} +81.0000 q^{81} +945.507 q^{83} +27.2603 q^{85} +311.343 q^{87} +317.329 q^{89} -203.192 q^{91} -103.357 q^{93} +247.397 q^{95} +1572.88 q^{97} -454.685 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} + 6 q^{5} - 10 q^{7} + 18 q^{9}+O(q^{10}) \) 2 * q - 6 * q^3 + 6 * q^5 - 10 * q^7 + 18 * q^9	\( 2 q - 6 q^{3} + 6 q^{5} - 10 q^{7} + 18 q^{9} - 16 q^{11} + 26 q^{13} - 18 q^{15} + 4 q^{17} - 70 q^{19} + 30 q^{21} - 128 q^{23} - 6 q^{25} - 54 q^{27} - 80 q^{29} - 250 q^{31} + 48 q^{33} - 256 q^{35} - 152 q^{37} - 78 q^{39} - 146 q^{41} - 504 q^{43} + 54 q^{45} - 524 q^{47} - 410 q^{49} - 12 q^{51} - 52 q^{53} - 952 q^{55} + 210 q^{57} - 164 q^{59} - 304 q^{61} - 90 q^{63} + 78 q^{65} - 914 q^{67} + 384 q^{69} - 456 q^{73} + 18 q^{75} + 984 q^{77} - 824 q^{79} + 162 q^{81} + 828 q^{83} + 12 q^{85} + 240 q^{87} + 826 q^{89} - 130 q^{91} + 750 q^{93} + 920 q^{95} + 552 q^{97} - 144 q^{99}+O(q^{100}) \) 2 * q - 6 * q^3 + 6 * q^5 - 10 * q^7 + 18 * q^9 - 16 * q^11 + 26 * q^13 - 18 * q^15 + 4 * q^17 - 70 * q^19 + 30 * q^21 - 128 * q^23 - 6 * q^25 - 54 * q^27 - 80 * q^29 - 250 * q^31 + 48 * q^33 - 256 * q^35 - 152 * q^37 - 78 * q^39 - 146 * q^41 - 504 * q^43 + 54 * q^45 - 524 * q^47 - 410 * q^49 - 12 * q^51 - 52 * q^53 - 952 * q^55 + 210 * q^57 - 164 * q^59 - 304 * q^61 - 90 * q^63 + 78 * q^65 - 914 * q^67 + 384 * q^69 - 456 * q^73 + 18 * q^75 + 984 * q^77 - 824 * q^79 + 162 * q^81 + 828 * q^83 + 12 * q^85 + 240 * q^87 + 826 * q^89 - 130 * q^91 + 750 * q^93 + 920 * q^95 + 552 * q^97 - 144 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more