LMFDB - Embedded newform 315.4.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [315,4,Mod(1,315)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(315, 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("315.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 315 = 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	315.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:	$18.5856016518$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	315.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.43845 q^{2} -5.93087 q^{4} +5.00000 q^{5} -7.00000 q^{7} -20.0388 q^{8} +O(q^{10})$	\(q+1.43845 q^{2} -5.93087 q^{4} +5.00000 q^{5} -7.00000 q^{7} -20.0388 q^{8} +7.19224 q^{10} -7.61553 q^{11} +52.3542 q^{13} -10.0691 q^{14} +18.6222 q^{16} +49.7235 q^{17} +140.600 q^{19} -29.6543 q^{20} -10.9545 q^{22} +23.4470 q^{23} +25.0000 q^{25} +75.3087 q^{26} +41.5161 q^{28} -157.170 q^{29} +127.892 q^{31} +187.098 q^{32} +71.5246 q^{34} -35.0000 q^{35} -115.477 q^{37} +202.246 q^{38} -100.194 q^{40} +188.617 q^{41} +322.186 q^{43} +45.1667 q^{44} +33.7272 q^{46} +76.6477 q^{47} +49.0000 q^{49} +35.9612 q^{50} -310.506 q^{52} +424.172 q^{53} -38.0776 q^{55} +140.272 q^{56} -226.081 q^{58} -107.784 q^{59} +915.511 q^{61} +183.966 q^{62} +120.153 q^{64} +261.771 q^{65} -451.723 q^{67} -294.903 q^{68} -50.3457 q^{70} -907.312 q^{71} +755.956 q^{73} -166.108 q^{74} -833.882 q^{76} +53.3087 q^{77} +22.5834 q^{79} +93.1109 q^{80} +271.316 q^{82} -1112.25 q^{83} +248.617 q^{85} +463.447 q^{86} +152.606 q^{88} -1518.81 q^{89} -366.479 q^{91} -139.061 q^{92} +110.254 q^{94} +703.002 q^{95} -549.987 q^{97} +70.4839 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 7 q^{2} + 17 q^{4} + 10 q^{5} - 14 q^{7} + 63 q^{8}+O(q^{10}) $ 2 * q + 7 * q^2 + 17 * q^4 + 10 * q^5 - 14 * q^7 + 63 * q^8	$ 2 q + 7 q^{2} + 17 q^{4} + 10 q^{5} - 14 q^{7} + 63 q^{8} + 35 q^{10} + 26 q^{11} + 14 q^{13} - 49 q^{14} + 297 q^{16} - 16 q^{17} + 174 q^{19} + 85 q^{20} + 176 q^{22} - 184 q^{23} + 50 q^{25} - 138 q^{26} - 119 q^{28} + 32 q^{29} + 330 q^{31} + 1071 q^{32} - 294 q^{34} - 70 q^{35} - 132 q^{37} + 388 q^{38} + 315 q^{40} - 200 q^{41} + 364 q^{43} + 816 q^{44} - 1120 q^{46} - 292 q^{47} + 98 q^{49} + 175 q^{50} - 1190 q^{52} - 34 q^{53} + 130 q^{55} - 441 q^{56} + 826 q^{58} - 364 q^{59} + 792 q^{61} + 1308 q^{62} + 2809 q^{64} + 70 q^{65} - 788 q^{67} - 1802 q^{68} - 245 q^{70} - 454 q^{71} + 778 q^{73} - 258 q^{74} - 68 q^{76} - 182 q^{77} + 408 q^{79} + 1485 q^{80} - 1890 q^{82} - 1136 q^{83} - 80 q^{85} + 696 q^{86} + 2944 q^{88} - 36 q^{89} - 98 q^{91} - 4896 q^{92} - 1940 q^{94} + 870 q^{95} - 498 q^{97} + 343 q^{98}+O(q^{100}) $ 2 * q + 7 * q^2 + 17 * q^4 + 10 * q^5 - 14 * q^7 + 63 * q^8 + 35 * q^10 + 26 * q^11 + 14 * q^13 - 49 * q^14 + 297 * q^16 - 16 * q^17 + 174 * q^19 + 85 * q^20 + 176 * q^22 - 184 * q^23 + 50 * q^25 - 138 * q^26 - 119 * q^28 + 32 * q^29 + 330 * q^31 + 1071 * q^32 - 294 * q^34 - 70 * q^35 - 132 * q^37 + 388 * q^38 + 315 * q^40 - 200 * q^41 + 364 * q^43 + 816 * q^44 - 1120 * q^46 - 292 * q^47 + 98 * q^49 + 175 * q^50 - 1190 * q^52 - 34 * q^53 + 130 * q^55 - 441 * q^56 + 826 * q^58 - 364 * q^59 + 792 * q^61 + 1308 * q^62 + 2809 * q^64 + 70 * q^65 - 788 * q^67 - 1802 * q^68 - 245 * q^70 - 454 * q^71 + 778 * q^73 - 258 * q^74 - 68 * q^76 - 182 * q^77 + 408 * q^79 + 1485 * q^80 - 1890 * q^82 - 1136 * q^83 - 80 * q^85 + 696 * q^86 + 2944 * q^88 - 36 * q^89 - 98 * q^91 - 4896 * q^92 - 1940 * q^94 + 870 * q^95 - 498 * q^97 + 343 * q^98

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 $

$2$	1.43845	0.508568	0.254284	−	0.967130i	$-0.418160\pi$
$2$	1.43845	0.508568	0.254284	+	0.967130i	$0.418160\pi$
$3$	0	0
$3$	0	0
$4$	−5.93087	−0.741359
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	−20.0388	−0.885599
$9$	0	0
$10$	7.19224	0.227438
$11$	−7.61553	−0.208743	−0.104371	−	0.994538i	$-0.533283\pi$
$11$	−7.61553	−0.208743	−0.104371	+	0.994538i	$0.533283\pi$
$12$	0	0
$13$	52.3542	1.11696	0.558478	−	0.829519i	$-0.311385\pi$
$13$	52.3542	1.11696	0.558478	+	0.829519i	$0.311385\pi$
$14$	−10.0691	−0.192221
$15$	0	0
$16$	18.6222	0.290971
$17$	49.7235	0.709395	0.354697	−	0.934981i	$-0.384584\pi$
$17$	49.7235	0.709395	0.354697	+	0.934981i	$0.384584\pi$
$18$	0	0
$19$	140.600	1.69768	0.848840	−	0.528649i	$-0.177301\pi$
$19$	140.600	1.69768	0.848840	+	0.528649i	$0.177301\pi$
$20$	−29.6543	−0.331546
$21$	0	0
$22$	−10.9545	−0.106160
$23$	23.4470	0.212566	0.106283	−	0.994336i	$-0.466105\pi$
$23$	23.4470	0.212566	0.106283	+	0.994336i	$0.466105\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	75.3087	0.568048
$27$	0	0
$28$	41.5161	0.280207
$29$	−157.170	−1.00641	−0.503204	−	0.864168i	$-0.667845\pi$
$29$	−157.170	−1.00641	−0.503204	+	0.864168i	$0.667845\pi$
$30$	0	0
$31$	127.892	0.740971	0.370485	−	0.928838i	$-0.379191\pi$
$31$	127.892	0.740971	0.370485	+	0.928838i	$0.379191\pi$
$32$	187.098	1.03358
$33$	0	0
$34$	71.5246	0.360776
$35$	−35.0000	−0.169031
$36$	0	0
$37$	−115.477	−0.513090	−0.256545	−	0.966532i	$-0.582584\pi$
$37$	−115.477	−0.513090	−0.256545	+	0.966532i	$0.582584\pi$
$38$	202.246	0.863386
$39$	0	0
$40$	−100.194	−0.396052
$41$	188.617	0.718466	0.359233	−	0.933248i	$-0.383038\pi$
$41$	188.617	0.718466	0.359233	+	0.933248i	$0.383038\pi$
$42$	0	0
$43$	322.186	1.14262	0.571312	−	0.820733i	$-0.306435\pi$
$43$	322.186	1.14262	0.571312	+	0.820733i	$0.306435\pi$
$44$	45.1667	0.154753
$45$	0	0
$46$	33.7272	0.108104
$47$	76.6477	0.237877	0.118938	−	0.992902i	$-0.462051\pi$
$47$	76.6477	0.237877	0.118938	+	0.992902i	$0.462051\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	35.9612	0.101714
$51$	0	0
$52$	−310.506	−0.828065
$53$	424.172	1.09933	0.549666	−	0.835385i	$-0.314755\pi$
$53$	424.172	1.09933	0.549666	+	0.835385i	$0.314755\pi$
$54$	0	0
$55$	−38.0776	−0.0933525
$56$	140.272	0.334725
$57$	0	0
$58$	−226.081	−0.511827
$59$	−107.784	−0.237835	−0.118918	−	0.992904i	$-0.537942\pi$
$59$	−107.784	−0.237835	−0.118918	+	0.992904i	$0.537942\pi$
$60$	0	0
$61$	915.511	1.92163	0.960813	−	0.277197i	$-0.0894053\pi$
$61$	915.511	1.92163	0.960813	+	0.277197i	$0.0894053\pi$
$62$	183.966	0.376834
$63$	0	0
$64$	120.153	0.234673
$65$	261.771	0.499518
$66$	0	0
$67$	−451.723	−0.823684	−0.411842	−	0.911255i	$-0.635114\pi$
$67$	−451.723	−0.823684	−0.411842	+	0.911255i	$0.635114\pi$
$68$	−294.903	−0.525916
$69$	0	0
$70$	−50.3457	−0.0859637
$71$	−907.312	−1.51659	−0.758297	−	0.651909i	$-0.773968\pi$
$71$	−907.312	−1.51659	−0.758297	+	0.651909i	$0.773968\pi$
$72$	0	0
$73$	755.956	1.21203	0.606014	−	0.795454i	$-0.292768\pi$
$73$	755.956	1.21203	0.606014	+	0.795454i	$0.292768\pi$
$74$	−166.108	−0.260941
$75$	0	0
$76$	−833.882	−1.25859
$77$	53.3087	0.0788973
$78$	0	0
$79$	22.5834	0.0321624	0.0160812	−	0.999871i	$-0.494881\pi$
$79$	22.5834	0.0321624	0.0160812	+	0.999871i	$0.494881\pi$
$80$	93.1109	0.130126
$81$	0	0
$82$	271.316	0.365389
$83$	−1112.25	−1.47091	−0.735454	−	0.677575i	$-0.763031\pi$
$83$	−1112.25	−1.47091	−0.735454	+	0.677575i	$0.763031\pi$
$84$	0	0
$85$	248.617	0.317251
$86$	463.447	0.581102
$87$	0	0
$88$	152.606	0.184862
$89$	−1518.81	−1.80892	−0.904458	−	0.426562i	$-0.859725\pi$
$89$	−1518.81	−1.80892	−0.904458	+	0.426562i	$0.859725\pi$
$90$	0	0
$91$	−366.479	−0.422170
$92$	−139.061	−0.157588
$93$	0	0
$94$	110.254	0.120977
$95$	703.002	0.759226
$96$	0	0
$97$	−549.987	−0.575698	−0.287849	−	0.957676i	$-0.592940\pi$
$97$	−549.987	−0.575698	−0.287849	+	0.957676i	$0.592940\pi$
$98$	70.4839	0.0726526
$99$	0	0
$100$	−148.272	−0.148272
$101$	533.299	0.525398	0.262699	−	0.964878i	$-0.415387\pi$
$101$	533.299	0.525398	0.262699	+	0.964878i	$0.415387\pi$
$102$	0	0
$103$	1357.79	1.29890	0.649451	−	0.760404i	$-0.274999\pi$
$103$	1357.79	1.29890	0.649451	+	0.760404i	$0.274999\pi$
$104$	−1049.12	−0.989176
$105$	0	0
$106$	610.149	0.559084
$107$	−913.693	−0.825515	−0.412757	−	0.910841i	$-0.635434\pi$
$107$	−913.693	−0.825515	−0.412757	+	0.910841i	$0.635434\pi$
$108$	0	0
$109$	−160.489	−0.141028	−0.0705139	−	0.997511i	$-0.522464\pi$
$109$	−160.489	−0.141028	−0.0705139	+	0.997511i	$0.522464\pi$
$110$	−54.7727	−0.0474761
$111$	0	0
$112$	−130.355	−0.109977
$113$	1788.48	1.48891	0.744453	−	0.667675i	$-0.232710\pi$
$113$	1788.48	1.48891	0.744453	+	0.667675i	$0.232710\pi$
$114$	0	0
$115$	117.235	0.0950626
$116$	932.157	0.746109
$117$	0	0
$118$	−155.042	−0.120955
$119$	−348.064	−0.268126
$120$	0	0
$121$	−1273.00	−0.956427
$122$	1316.91	0.977277
$123$	0	0
$124$	−758.511	−0.549325
$125$	125.000	0.0894427
$126$	0	0
$127$	−271.015	−0.189360	−0.0946799	−	0.995508i	$-0.530183\pi$
$127$	−271.015	−0.189360	−0.0946799	+	0.995508i	$0.530183\pi$
$128$	−1323.95	−0.914231
$129$	0	0
$130$	376.543	0.254039
$131$	763.151	0.508984	0.254492	−	0.967075i	$-0.418092\pi$
$131$	763.151	0.508984	0.254492	+	0.967075i	$0.418092\pi$
$132$	0	0
$133$	−984.203	−0.641663
$134$	−649.780	−0.418899
$135$	0	0
$136$	−996.400	−0.628240
$137$	240.934	0.150251	0.0751254	−	0.997174i	$-0.476064\pi$
$137$	240.934	0.150251	0.0751254	+	0.997174i	$0.476064\pi$
$138$	0	0
$139$	−103.150	−0.0629427	−0.0314714	−	0.999505i	$-0.510019\pi$
$139$	−103.150	−0.0629427	−0.0314714	+	0.999505i	$0.510019\pi$
$140$	207.580	0.125312
$141$	0	0
$142$	−1305.12	−0.771291
$143$	−398.705	−0.233156
$144$	0	0
$145$	−785.852	−0.450079
$146$	1087.40	0.616398
$147$	0	0
$148$	684.881	0.380384
$149$	3256.71	1.79061	0.895303	−	0.445458i	$-0.146959\pi$
$149$	3256.71	1.79061	0.895303	+	0.445458i	$0.146959\pi$
$150$	0	0
$151$	1471.04	0.792793	0.396396	−	0.918080i	$-0.370261\pi$
$151$	1471.04	0.792793	0.396396	+	0.918080i	$0.370261\pi$
$152$	−2817.47	−1.50346
$153$	0	0
$154$	76.6817	0.0401246
$155$	639.460	0.331372
$156$	0	0
$157$	−1394.22	−0.708730	−0.354365	−	0.935107i	$-0.615303\pi$
$157$	−1394.22	−0.708730	−0.354365	+	0.935107i	$0.615303\pi$
$158$	32.4850	0.0163567
$159$	0	0
$160$	935.488	0.462230
$161$	−164.129	−0.0803426
$162$	0	0
$163$	−3674.27	−1.76559	−0.882794	−	0.469760i	$-0.844341\pi$
$163$	−3674.27	−1.76559	−0.882794	+	0.469760i	$0.844341\pi$
$164$	−1118.67	−0.532641
$165$	0	0
$166$	−1599.91	−0.748056
$167$	−4041.09	−1.87251	−0.936254	−	0.351325i	$-0.885731\pi$
$167$	−4041.09	−1.87251	−0.936254	+	0.351325i	$0.885731\pi$
$168$	0	0
$169$	543.958	0.247591
$170$	357.623	0.161344
$171$	0	0
$172$	−1910.84	−0.847094
$173$	−59.4582	−0.0261302	−0.0130651	−	0.999915i	$-0.504159\pi$
$173$	−59.4582	−0.0261302	−0.0130651	+	0.999915i	$0.504159\pi$
$174$	0	0
$175$	−175.000	−0.0755929
$176$	−141.818	−0.0607381
$177$	0	0
$178$	−2184.73	−0.919957
$179$	2973.32	1.24155	0.620773	−	0.783991i	$-0.286819\pi$
$179$	2973.32	1.24155	0.620773	+	0.783991i	$0.286819\pi$
$180$	0	0
$181$	−676.220	−0.277696	−0.138848	−	0.990314i	$-0.544340\pi$
$181$	−676.220	−0.277696	−0.138848	+	0.990314i	$0.544340\pi$
$182$	−527.161	−0.214702
$183$	0	0
$184$	−469.849	−0.188249
$185$	−577.386	−0.229461
$186$	0	0
$187$	−378.671	−0.148081
$188$	−454.588	−0.176352
$189$	0	0
$190$	1011.23	0.386118
$191$	−16.5589	−0.00627308	−0.00313654	−	0.999995i	$-0.500998\pi$
$191$	−16.5589	−0.00627308	−0.00313654	+	0.999995i	$0.500998\pi$
$192$	0	0
$193$	2694.68	1.00501	0.502506	−	0.864574i	$-0.332411\pi$
$193$	2694.68	1.00501	0.502506	+	0.864574i	$0.332411\pi$
$194$	−791.127	−0.292781
$195$	0	0
$196$	−290.613	−0.105908
$197$	−1027.38	−0.371561	−0.185781	−	0.982591i	$-0.559481\pi$
$197$	−1027.38	−0.371561	−0.185781	+	0.982591i	$0.559481\pi$
$198$	0	0
$199$	2823.77	1.00589	0.502944	−	0.864319i	$-0.332250\pi$
$199$	2823.77	1.00589	0.502944	+	0.864319i	$0.332250\pi$
$200$	−500.971	−0.177120
$201$	0	0
$202$	767.123	0.267201
$203$	1100.19	0.380386
$204$	0	0
$205$	943.087	0.321308
$206$	1953.11	0.660579
$207$	0	0
$208$	974.948	0.325002
$209$	−1070.75	−0.354378
$210$	0	0
$211$	5151.16	1.68067	0.840333	−	0.542071i	$-0.182360\pi$
$211$	5151.16	1.68067	0.840333	+	0.542071i	$0.182360\pi$
$212$	−2515.71	−0.814999
$213$	0	0
$214$	−1314.30	−0.419830
$215$	1610.93	0.510997
$216$	0	0
$217$	−895.244	−0.280061
$218$	−230.855	−0.0717222
$219$	0	0
$220$	225.834	0.0692077
$221$	2603.23	0.792363
$222$	0	0
$223$	−114.496	−0.0343822	−0.0171911	−	0.999852i	$-0.505472\pi$
$223$	−114.496	−0.0343822	−0.0171911	+	0.999852i	$0.505472\pi$
$224$	−1309.68	−0.390656
$225$	0	0
$226$	2572.64	0.757209
$227$	−4744.75	−1.38731	−0.693657	−	0.720306i	$-0.744001\pi$
$227$	−4744.75	−1.38731	−0.693657	+	0.720306i	$0.744001\pi$
$228$	0	0
$229$	−5384.47	−1.55378	−0.776891	−	0.629635i	$-0.783204\pi$
$229$	−5384.47	−1.55378	−0.776891	+	0.629635i	$0.783204\pi$
$230$	168.636	0.0483458
$231$	0	0
$232$	3149.51	0.891274
$233$	1608.91	0.452373	0.226187	−	0.974084i	$-0.427374\pi$
$233$	1608.91	0.452373	0.226187	+	0.974084i	$0.427374\pi$
$234$	0	0
$235$	383.239	0.106382
$236$	639.253	0.176321
$237$	0	0
$238$	−500.672	−0.136360
$239$	3113.11	0.842554	0.421277	−	0.906932i	$-0.361582\pi$
$239$	3113.11	0.842554	0.421277	+	0.906932i	$0.361582\pi$
$240$	0	0
$241$	7136.38	1.90745	0.953724	−	0.300685i	$-0.0972151\pi$
$241$	7136.38	1.90745	0.953724	+	0.300685i	$0.0972151\pi$
$242$	−1831.15	−0.486408
$243$	0	0
$244$	−5429.78	−1.42461
$245$	245.000	0.0638877
$246$	0	0
$247$	7361.01	1.89624
$248$	−2562.81	−0.656203
$249$	0	0
$250$	179.806	0.0454877
$251$	225.504	0.0567079	0.0283539	−	0.999598i	$-0.490973\pi$
$251$	225.504	0.0567079	0.0283539	+	0.999598i	$0.490973\pi$
$252$	0	0
$253$	−178.561	−0.0443717
$254$	−389.841	−0.0963024
$255$	0	0
$256$	−2865.65	−0.699621
$257$	4423.05	1.07355	0.536775	−	0.843725i	$-0.319642\pi$
$257$	4423.05	1.07355	0.536775	+	0.843725i	$0.319642\pi$
$258$	0	0
$259$	808.341	0.193930
$260$	−1552.53	−0.370322
$261$	0	0
$262$	1097.75	0.258853
$263$	−6540.56	−1.53349	−0.766746	−	0.641950i	$-0.778125\pi$
$263$	−6540.56	−1.53349	−0.766746	+	0.641950i	$0.778125\pi$
$264$	0	0
$265$	2120.86	0.491636
$266$	−1415.72	−0.326329
$267$	0	0
$268$	2679.11	0.610645
$269$	−2262.97	−0.512920	−0.256460	−	0.966555i	$-0.582556\pi$
$269$	−2262.97	−0.512920	−0.256460	+	0.966555i	$0.582556\pi$
$270$	0	0
$271$	1615.68	0.362160	0.181080	−	0.983468i	$-0.442041\pi$
$271$	1615.68	0.362160	0.181080	+	0.983468i	$0.442041\pi$
$272$	925.959	0.206414
$273$	0	0
$274$	346.571	0.0764127
$275$	−190.388	−0.0417485
$276$	0	0
$277$	4691.55	1.01765	0.508823	−	0.860871i	$-0.330081\pi$
$277$	4691.55	1.01765	0.508823	+	0.860871i	$0.330081\pi$
$278$	−148.375	−0.0320107
$279$	0	0
$280$	701.359	0.149694
$281$	8119.00	1.72363	0.861813	−	0.507226i	$-0.169329\pi$
$281$	8119.00	1.72363	0.861813	+	0.507226i	$0.169329\pi$
$282$	0	0
$283$	−3633.75	−0.763265	−0.381632	−	0.924314i	$-0.624638\pi$
$283$	−3633.75	−0.763265	−0.381632	+	0.924314i	$0.624638\pi$
$284$	5381.15	1.12434
$285$	0	0
$286$	−573.515	−0.118576
$287$	−1320.32	−0.271554
$288$	0	0
$289$	−2440.58	−0.496759
$290$	−1130.41	−0.228896
$291$	0	0
$292$	−4483.48	−0.898547
$293$	7981.99	1.59151	0.795756	−	0.605618i	$-0.207074\pi$
$293$	7981.99	1.59151	0.795756	+	0.605618i	$0.207074\pi$
$294$	0	0
$295$	−538.920	−0.106363
$296$	2314.03	0.454392
$297$	0	0
$298$	4684.61	0.910645
$299$	1227.55	0.237427
$300$	0	0
$301$	−2255.30	−0.431871
$302$	2116.02	0.403189
$303$	0	0
$304$	2618.28	0.493977
$305$	4577.56	0.859377
$306$	0	0
$307$	−7118.15	−1.32330	−0.661652	−	0.749811i	$-0.730144\pi$
$307$	−7118.15	−1.32330	−0.661652	+	0.749811i	$0.730144\pi$
$308$	−316.167	−0.0584912
$309$	0	0
$310$	919.830	0.168525
$311$	9155.92	1.66940	0.834702	−	0.550703i	$-0.185640\pi$
$311$	9155.92	1.66940	0.834702	+	0.550703i	$0.185640\pi$
$312$	0	0
$313$	−6163.44	−1.11303	−0.556515	−	0.830838i	$-0.687862\pi$
$313$	−6163.44	−1.11303	−0.556515	+	0.830838i	$0.687862\pi$
$314$	−2005.51	−0.360438
$315$	0	0
$316$	−133.939	−0.0238438
$317$	−8658.37	−1.53408	−0.767038	−	0.641601i	$-0.778270\pi$
$317$	−8658.37	−1.53408	−0.767038	+	0.641601i	$0.778270\pi$
$318$	0	0
$319$	1196.94	0.210080
$320$	600.763	0.104949
$321$	0	0
$322$	−236.090	−0.0408597
$323$	6991.14	1.20433
$324$	0	0
$325$	1308.85	0.223391
$326$	−5285.24	−0.897922
$327$	0	0
$328$	−3779.67	−0.636272
$329$	−536.534	−0.0899090
$330$	0	0
$331$	−128.477	−0.0213346	−0.0106673	−	0.999943i	$-0.503396\pi$
$331$	−128.477	−0.0213346	−0.0106673	+	0.999943i	$0.503396\pi$
$332$	6596.61	1.09047
$333$	0	0
$334$	−5812.89	−0.952297
$335$	−2258.62	−0.368363
$336$	0	0
$337$	7784.57	1.25832	0.629158	−	0.777277i	$-0.283400\pi$
$337$	7784.57	1.25832	0.629158	+	0.777277i	$0.283400\pi$
$338$	782.455	0.125917
$339$	0	0
$340$	−1474.52	−0.235197
$341$	−973.965	−0.154672
$342$	0	0
$343$	−343.000	−0.0539949
$344$	−6456.22	−1.01191
$345$	0	0
$346$	−85.5274	−0.0132890
$347$	−3740.26	−0.578638	−0.289319	−	0.957233i	$-0.593429\pi$
$347$	−3740.26	−0.578638	−0.289319	+	0.957233i	$0.593429\pi$
$348$	0	0
$349$	5676.86	0.870703	0.435351	−	0.900261i	$-0.356624\pi$
$349$	5676.86	0.870703	0.435351	+	0.900261i	$0.356624\pi$
$350$	−251.728	−0.0384441
$351$	0	0
$352$	−1424.85	−0.215752
$353$	−909.564	−0.137142	−0.0685711	−	0.997646i	$-0.521844\pi$
$353$	−909.564	−0.137142	−0.0685711	+	0.997646i	$0.521844\pi$
$354$	0	0
$355$	−4536.56	−0.678241
$356$	9007.87	1.34106
$357$	0	0
$358$	4276.97	0.631410
$359$	−2678.57	−0.393788	−0.196894	−	0.980425i	$-0.563085\pi$
$359$	−2678.57	−0.393788	−0.196894	+	0.980425i	$0.563085\pi$
$360$	0	0
$361$	12909.5	1.88212
$362$	−972.706	−0.141227
$363$	0	0
$364$	2173.54	0.312979
$365$	3779.78	0.542035
$366$	0	0
$367$	−716.898	−0.101967	−0.0509833	−	0.998700i	$-0.516236\pi$
$367$	−716.898	−0.101967	−0.0509833	+	0.998700i	$0.516236\pi$
$368$	436.633	0.0618508
$369$	0	0
$370$	−830.540	−0.116697
$371$	−2969.21	−0.415508
$372$	0	0
$373$	−2006.15	−0.278484	−0.139242	−	0.990258i	$-0.544467\pi$
$373$	−2006.15	−0.278484	−0.139242	+	0.990258i	$0.544467\pi$
$374$	−544.698	−0.0753092
$375$	0	0
$376$	−1535.93	−0.210664
$377$	−8228.53	−1.12411
$378$	0	0
$379$	7277.53	0.986336	0.493168	−	0.869934i	$-0.335839\pi$
$379$	7277.53	0.986336	0.493168	+	0.869934i	$0.335839\pi$
$380$	−4169.41	−0.562859
$381$	0	0
$382$	−23.8191	−0.00319029
$383$	−5953.94	−0.794339	−0.397170	−	0.917745i	$-0.630008\pi$
$383$	−5953.94	−0.794339	−0.397170	+	0.917745i	$0.630008\pi$
$384$	0	0
$385$	266.543	0.0352839
$386$	3876.16	0.511117
$387$	0	0
$388$	3261.90	0.426799
$389$	−1867.98	−0.243472	−0.121736	−	0.992563i	$-0.538846\pi$
$389$	−1867.98	−0.243472	−0.121736	+	0.992563i	$0.538846\pi$
$390$	0	0
$391$	1165.86	0.150794
$392$	−981.902	−0.126514
$393$	0	0
$394$	−1477.83	−0.188964
$395$	112.917	0.0143834
$396$	0	0
$397$	−2160.07	−0.273075	−0.136538	−	0.990635i	$-0.543597\pi$
$397$	−2160.07	−0.273075	−0.136538	+	0.990635i	$0.543597\pi$
$398$	4061.85	0.511563
$399$	0	0
$400$	465.554	0.0581943
$401$	−1954.81	−0.243438	−0.121719	−	0.992565i	$-0.538841\pi$
$401$	−1954.81	−0.243438	−0.121719	+	0.992565i	$0.538841\pi$
$402$	0	0
$403$	6695.68	0.827632
$404$	−3162.93	−0.389509
$405$	0	0
$406$	1582.57	0.193452
$407$	879.420	0.107104
$408$	0	0
$409$	14895.1	1.80077	0.900384	−	0.435096i	$-0.143286\pi$
$409$	14895.1	1.80077	0.900384	+	0.435096i	$0.143286\pi$
$410$	1356.58	0.163407
$411$	0	0
$412$	−8052.86	−0.962952
$413$	754.489	0.0898934
$414$	0	0
$415$	−5561.25	−0.657810
$416$	9795.34	1.15446
$417$	0	0
$418$	−1540.21	−0.180225
$419$	−12608.9	−1.47013	−0.735067	−	0.677994i	$-0.762849\pi$
$419$	−12608.9	−1.47013	−0.735067	+	0.677994i	$0.762849\pi$
$420$	0	0
$421$	−7862.86	−0.910243	−0.455122	−	0.890429i	$-0.650404\pi$
$421$	−7862.86	−0.910243	−0.455122	+	0.890429i	$0.650404\pi$
$422$	7409.67	0.854732
$423$	0	0
$424$	−8499.91	−0.973567
$425$	1243.09	0.141879
$426$	0	0
$427$	−6408.58	−0.726307
$428$	5419.00	0.612002
$429$	0	0
$430$	2317.23	0.259877
$431$	−14291.0	−1.59715	−0.798575	−	0.601896i	$-0.794412\pi$
$431$	−14291.0	−1.59715	−0.798575	+	0.601896i	$0.794412\pi$
$432$	0	0
$433$	−13759.0	−1.52705	−0.763527	−	0.645776i	$-0.776534\pi$
$433$	−13759.0	−1.52705	−0.763527	+	0.645776i	$0.776534\pi$
$434$	−1287.76	−0.142430
$435$	0	0
$436$	951.838	0.104552
$437$	3296.65	0.360870
$438$	0	0
$439$	−6093.13	−0.662436	−0.331218	−	0.943554i	$-0.607459\pi$
$439$	−6093.13	−0.662436	−0.331218	+	0.943554i	$0.607459\pi$
$440$	763.031	0.0826729
$441$	0	0
$442$	3744.61	0.402970
$443$	−13449.5	−1.44244	−0.721222	−	0.692704i	$-0.756419\pi$
$443$	−13449.5	−1.44244	−0.721222	+	0.692704i	$0.756419\pi$
$444$	0	0
$445$	−7594.05	−0.808972
$446$	−164.697	−0.0174857
$447$	0	0
$448$	−841.068	−0.0886981
$449$	−6893.83	−0.724588	−0.362294	−	0.932064i	$-0.618006\pi$
$449$	−6893.83	−0.724588	−0.362294	+	0.932064i	$0.618006\pi$
$450$	0	0
$451$	−1436.42	−0.149974
$452$	−10607.3	−1.10381
$453$	0	0
$454$	−6825.07	−0.705543
$455$	−1832.40	−0.188800
$456$	0	0
$457$	−11820.1	−1.20990	−0.604948	−	0.796265i	$-0.706806\pi$
$457$	−11820.1	−1.20990	−0.604948	+	0.796265i	$0.706806\pi$
$458$	−7745.28	−0.790203
$459$	0	0
$460$	−695.304	−0.0704755
$461$	−8443.38	−0.853031	−0.426516	−	0.904480i	$-0.640259\pi$
$461$	−8443.38	−0.853031	−0.426516	+	0.904480i	$0.640259\pi$
$462$	0	0
$463$	−1269.74	−0.127451	−0.0637257	−	0.997967i	$-0.520298\pi$
$463$	−1269.74	−0.127451	−0.0637257	+	0.997967i	$0.520298\pi$
$464$	−2926.86	−0.292836
$465$	0	0
$466$	2314.33	0.230063
$467$	16481.8	1.63316	0.816579	−	0.577233i	$-0.195868\pi$
$467$	16481.8	1.63316	0.816579	+	0.577233i	$0.195868\pi$
$468$	0	0
$469$	3162.06	0.311323
$470$	551.268	0.0541024
$471$	0	0
$472$	2159.87	0.210627
$473$	−2453.61	−0.238514
$474$	0	0
$475$	3515.01	0.339536
$476$	2064.32	0.198778
$477$	0	0
$478$	4478.05	0.428496
$479$	1400.21	0.133564	0.0667822	−	0.997768i	$-0.478727\pi$
$479$	1400.21	0.133564	0.0667822	+	0.997768i	$0.478727\pi$
$480$	0	0
$481$	−6045.72	−0.573100
$482$	10265.3	0.970066
$483$	0	0
$484$	7550.02	0.709055
$485$	−2749.93	−0.257460
$486$	0	0
$487$	−14165.9	−1.31811	−0.659055	−	0.752094i	$-0.729044\pi$
$487$	−14165.9	−1.31811	−0.659055	+	0.752094i	$0.729044\pi$
$488$	−18345.8	−1.70179
$489$	0	0
$490$	352.420	0.0324912
$491$	−4739.28	−0.435603	−0.217801	−	0.975993i	$-0.569888\pi$
$491$	−4739.28	−0.435603	−0.217801	+	0.975993i	$0.569888\pi$
$492$	0	0
$493$	−7815.06	−0.713940
$494$	10588.4	0.964364
$495$	0	0
$496$	2381.63	0.215601
$497$	6351.19	0.573219
$498$	0	0
$499$	−11370.0	−1.02003	−0.510013	−	0.860167i	$-0.670360\pi$
$499$	−11370.0	−1.02003	−0.510013	+	0.860167i	$0.670360\pi$
$500$	−741.359	−0.0663091
$501$	0	0
$502$	324.375	0.0288398
$503$	−9212.48	−0.816629	−0.408314	−	0.912841i	$-0.633883\pi$
$503$	−9212.48	−0.816629	−0.408314	+	0.912841i	$0.633883\pi$
$504$	0	0
$505$	2666.50	0.234965
$506$	−256.851	−0.0225660
$507$	0	0
$508$	1607.36	0.140384
$509$	15938.3	1.38792	0.693960	−	0.720014i	$-0.255864\pi$
$509$	15938.3	1.38792	0.693960	+	0.720014i	$0.255864\pi$
$510$	0	0
$511$	−5291.69	−0.458103
$512$	6469.49	0.558426
$513$	0	0
$514$	6362.33	0.545973
$515$	6788.94	0.580886
$516$	0	0
$517$	−583.713	−0.0496550
$518$	1162.76	0.0986265
$519$	0	0
$520$	−5245.58	−0.442373
$521$	−6442.99	−0.541790	−0.270895	−	0.962609i	$-0.587320\pi$
$521$	−6442.99	−0.541790	−0.270895	+	0.962609i	$0.587320\pi$
$522$	0	0
$523$	986.655	0.0824922	0.0412461	−	0.999149i	$-0.486867\pi$
$523$	986.655	0.0824922	0.0412461	+	0.999149i	$0.486867\pi$
$524$	−4526.15	−0.377339
$525$	0	0
$526$	−9408.26	−0.779885
$527$	6359.24	0.525641
$528$	0	0
$529$	−11617.2	−0.954815
$530$	3050.75	0.250030
$531$	0	0
$532$	5837.18	0.475703
$533$	9874.91	0.802495
$534$	0	0
$535$	−4568.47	−0.369181
$536$	9052.01	0.729454
$537$	0	0
$538$	−3255.16	−0.260855
$539$	−373.161	−0.0298204
$540$	0	0
$541$	−12681.1	−1.00777	−0.503885	−	0.863771i	$-0.668097\pi$
$541$	−12681.1	−1.00777	−0.503885	+	0.863771i	$0.668097\pi$
$542$	2324.06	0.184183
$543$	0	0
$544$	9303.14	0.733215
$545$	−802.443	−0.0630695
$546$	0	0
$547$	−14826.4	−1.15892	−0.579462	−	0.814999i	$-0.696737\pi$
$547$	−14826.4	−1.15892	−0.579462	+	0.814999i	$0.696737\pi$
$548$	−1428.95	−0.111390
$549$	0	0
$550$	−273.863	−0.0212320
$551$	−22098.2	−1.70856
$552$	0	0
$553$	−158.083	−0.0121562
$554$	6748.55	0.517542
$555$	0	0
$556$	611.767	0.0466632
$557$	−1926.46	−0.146547	−0.0732737	−	0.997312i	$-0.523345\pi$
$557$	−1926.46	−0.146547	−0.0732737	+	0.997312i	$0.523345\pi$
$558$	0	0
$559$	16867.8	1.27626
$560$	−651.776	−0.0491832
$561$	0	0
$562$	11678.8	0.876581
$563$	18624.8	1.39422	0.697108	−	0.716966i	$-0.254470\pi$
$563$	18624.8	1.39422	0.697108	+	0.716966i	$0.254470\pi$
$564$	0	0
$565$	8942.41	0.665859
$566$	−5226.96	−0.388172
$567$	0	0
$568$	18181.5	1.34309
$569$	20093.9	1.48045	0.740227	−	0.672357i	$-0.234718\pi$
$569$	20093.9	1.48045	0.740227	+	0.672357i	$0.234718\pi$
$570$	0	0
$571$	4535.25	0.332389	0.166195	−	0.986093i	$-0.446852\pi$
$571$	4535.25	0.332389	0.166195	+	0.986093i	$0.446852\pi$
$572$	2364.66	0.172852
$573$	0	0
$574$	−1899.21	−0.138104
$575$	586.174	0.0425133
$576$	0	0
$577$	10034.6	0.723994	0.361997	−	0.932179i	$-0.382095\pi$
$577$	10034.6	0.723994	0.361997	+	0.932179i	$0.382095\pi$
$578$	−3510.64	−0.252636
$579$	0	0
$580$	4660.79	0.333670
$581$	7785.75	0.555951
$582$	0	0
$583$	−3230.30	−0.229477
$584$	−15148.5	−1.07337
$585$	0	0
$586$	11481.7	0.809391
$587$	11192.6	0.786999	0.393499	−	0.919325i	$-0.371264\pi$
$587$	11192.6	0.786999	0.393499	+	0.919325i	$0.371264\pi$
$588$	0	0
$589$	17981.7	1.25793
$590$	−775.209	−0.0540929
$591$	0	0
$592$	−2150.44	−0.149295
$593$	−20317.6	−1.40699	−0.703493	−	0.710703i	$-0.748377\pi$
$593$	−20317.6	−1.40699	−0.703493	+	0.710703i	$0.748377\pi$
$594$	0	0
$595$	−1740.32	−0.119910
$596$	−19315.1	−1.32748
$597$	0	0
$598$	1765.76	0.120748
$599$	26376.5	1.79919	0.899594	−	0.436727i	$-0.143862\pi$
$599$	26376.5	1.79919	0.899594	+	0.436727i	$0.143862\pi$
$600$	0	0
$601$	9266.13	0.628907	0.314454	−	0.949273i	$-0.398179\pi$
$601$	9266.13	0.628907	0.314454	+	0.949273i	$0.398179\pi$
$602$	−3244.13	−0.219636
$603$	0	0
$604$	−8724.56	−0.587744
$605$	−6365.02	−0.427727
$606$	0	0
$607$	11338.0	0.758149	0.379075	−	0.925366i	$-0.376242\pi$
$607$	11338.0	0.758149	0.379075	+	0.925366i	$0.376242\pi$
$608$	26306.0	1.75469
$609$	0	0
$610$	6584.57	0.437052
$611$	4012.83	0.265698
$612$	0	0
$613$	25712.5	1.69416	0.847078	−	0.531469i	$-0.178360\pi$
$613$	25712.5	1.69416	0.847078	+	0.531469i	$0.178360\pi$
$614$	−10239.1	−0.672990
$615$	0	0
$616$	−1068.24	−0.0698714
$617$	−663.465	−0.0432903	−0.0216451	−	0.999766i	$-0.506890\pi$
$617$	−663.465	−0.0432903	−0.0216451	+	0.999766i	$0.506890\pi$
$618$	0	0
$619$	−12768.7	−0.829108	−0.414554	−	0.910025i	$-0.636062\pi$
$619$	−12768.7	−0.829108	−0.414554	+	0.910025i	$0.636062\pi$
$620$	−3792.56	−0.245666
$621$	0	0
$622$	13170.3	0.849005
$623$	10631.7	0.683706
$624$	0	0
$625$	625.000	0.0400000
$626$	−8865.78	−0.566051
$627$	0	0
$628$	8268.92	0.525423
$629$	−5741.93	−0.363984
$630$	0	0
$631$	−14937.6	−0.942400	−0.471200	−	0.882026i	$-0.656179\pi$
$631$	−14937.6	−0.942400	−0.471200	+	0.882026i	$0.656179\pi$
$632$	−452.544	−0.0284829
$633$	0	0
$634$	−12454.6	−0.780182
$635$	−1355.08	−0.0846843
$636$	0	0
$637$	2565.35	0.159565
$638$	1721.73	0.106840
$639$	0	0
$640$	−6619.74	−0.408856
$641$	−10903.0	−0.671829	−0.335914	−	0.941893i	$-0.609045\pi$
$641$	−10903.0	−0.671829	−0.335914	+	0.941893i	$0.609045\pi$
$642$	0	0
$643$	−7623.47	−0.467559	−0.233779	−	0.972290i	$-0.575109\pi$
$643$	−7623.47	−0.467559	−0.233779	+	0.972290i	$0.575109\pi$
$644$	973.426	0.0595627
$645$	0	0
$646$	10056.4	0.612482
$647$	−5384.84	−0.327202	−0.163601	−	0.986527i	$-0.552311\pi$
$647$	−5384.84	−0.327202	−0.163601	+	0.986527i	$0.552311\pi$
$648$	0	0
$649$	820.833	0.0496464
$650$	1882.72	0.113610
$651$	0	0
$652$	21791.6	1.30893
$653$	−297.318	−0.0178177	−0.00890883	−	0.999960i	$-0.502836\pi$
$653$	−297.318	−0.0178177	−0.00890883	+	0.999960i	$0.502836\pi$
$654$	0	0
$655$	3815.76	0.227624
$656$	3512.47	0.209053
$657$	0	0
$658$	−771.776	−0.0457248
$659$	−10324.7	−0.610309	−0.305155	−	0.952303i	$-0.598708\pi$
$659$	−10324.7	−0.610309	−0.305155	+	0.952303i	$0.598708\pi$
$660$	0	0
$661$	4272.98	0.251437	0.125718	−	0.992066i	$-0.459876\pi$
$661$	4272.98	0.251437	0.125718	+	0.992066i	$0.459876\pi$
$662$	−184.808	−0.0108501
$663$	0	0
$664$	22288.2	1.30263
$665$	−4921.01	−0.286960
$666$	0	0
$667$	−3685.17	−0.213928
$668$	23967.2	1.38820
$669$	0	0
$670$	−3248.90	−0.187337
$671$	−6972.10	−0.401125
$672$	0	0
$673$	−23033.4	−1.31927	−0.659637	−	0.751584i	$-0.729290\pi$
$673$	−23033.4	−1.31927	−0.659637	+	0.751584i	$0.729290\pi$
$674$	11197.7	0.639940
$675$	0	0
$676$	−3226.15	−0.183554
$677$	−5113.50	−0.290292	−0.145146	−	0.989410i	$-0.546365\pi$
$677$	−5113.50	−0.290292	−0.145146	+	0.989410i	$0.546365\pi$
$678$	0	0
$679$	3849.91	0.217593
$680$	−4982.00	−0.280957
$681$	0	0
$682$	−1401.00	−0.0786613
$683$	−1341.02	−0.0751286	−0.0375643	−	0.999294i	$-0.511960\pi$
$683$	−1341.02	−0.0751286	−0.0375643	+	0.999294i	$0.511960\pi$
$684$	0	0
$685$	1204.67	0.0671942
$686$	−493.387	−0.0274601
$687$	0	0
$688$	5999.80	0.332471
$689$	22207.2	1.22790
$690$	0	0
$691$	−16809.3	−0.925404	−0.462702	−	0.886514i	$-0.653120\pi$
$691$	−16809.3	−0.925404	−0.462702	+	0.886514i	$0.653120\pi$
$692$	352.639	0.0193718
$693$	0	0
$694$	−5380.16	−0.294277
$695$	−515.748	−0.0281489
$696$	0	0
$697$	9378.71	0.509676
$698$	8165.86	0.442812
$699$	0	0
$700$	1037.90	0.0560415
$701$	−13467.0	−0.725592	−0.362796	−	0.931869i	$-0.618178\pi$
$701$	−13467.0	−0.725592	−0.362796	+	0.931869i	$0.618178\pi$
$702$	0	0
$703$	−16236.1	−0.871064
$704$	−915.025	−0.0489863
$705$	0	0
$706$	−1308.36	−0.0697461
$707$	−3733.09	−0.198582
$708$	0	0
$709$	35514.3	1.88119	0.940597	−	0.339525i	$-0.110266\pi$
$709$	35514.3	1.88119	0.940597	+	0.339525i	$0.110266\pi$
$710$	−6525.61	−0.344932
$711$	0	0
$712$	30435.2	1.60198
$713$	2998.68	0.157506
$714$	0	0
$715$	−1993.52	−0.104271
$716$	−17634.4	−0.920431
$717$	0	0
$718$	−3852.99	−0.200268
$719$	−4993.61	−0.259013	−0.129506	−	0.991579i	$-0.541339\pi$
$719$	−4993.61	−0.259013	−0.129506	+	0.991579i	$0.541339\pi$
$720$	0	0
$721$	−9504.51	−0.490938
$722$	18569.6	0.957186
$723$	0	0
$724$	4010.57	0.205872
$725$	−3929.26	−0.201281
$726$	0	0
$727$	−4223.35	−0.215454	−0.107727	−	0.994181i	$-0.534357\pi$
$727$	−4223.35	−0.215454	−0.107727	+	0.994181i	$0.534357\pi$
$728$	7343.81	0.373873
$729$	0	0
$730$	5437.02	0.275662
$731$	16020.2	0.810572
$732$	0	0
$733$	19030.9	0.958968	0.479484	−	0.877551i	$-0.340824\pi$
$733$	19030.9	0.958968	0.479484	+	0.877551i	$0.340824\pi$
$734$	−1031.22	−0.0518570
$735$	0	0
$736$	4386.87	0.219704
$737$	3440.11	0.171938
$738$	0	0
$739$	−27772.5	−1.38245	−0.691224	−	0.722641i	$-0.742928\pi$
$739$	−27772.5	−1.38245	−0.691224	+	0.722641i	$0.742928\pi$
$740$	3424.40	0.170113
$741$	0	0
$742$	−4271.05	−0.211314
$743$	−26880.7	−1.32726	−0.663631	−	0.748060i	$-0.730985\pi$
$743$	−26880.7	−1.32726	−0.663631	+	0.748060i	$0.730985\pi$
$744$	0	0
$745$	16283.6	0.800783
$746$	−2885.74	−0.141628
$747$	0	0
$748$	2245.85	0.109781
$749$	6395.85	0.312015
$750$	0	0
$751$	−35166.6	−1.70872	−0.854360	−	0.519681i	$-0.826051\pi$
$751$	−35166.6	−1.70872	−0.854360	+	0.519681i	$0.826051\pi$
$752$	1427.35	0.0692154
$753$	0	0
$754$	−11836.3	−0.571688
$755$	7355.21	0.354548
$756$	0	0
$757$	14589.2	0.700467	0.350233	−	0.936662i	$-0.386102\pi$
$757$	14589.2	0.700467	0.350233	+	0.936662i	$0.386102\pi$
$758$	10468.3	0.501619
$759$	0	0
$760$	−14087.3	−0.672370
$761$	782.826	0.0372897	0.0186448	−	0.999826i	$-0.494065\pi$
$761$	782.826	0.0372897	0.0186448	+	0.999826i	$0.494065\pi$
$762$	0	0
$763$	1123.42	0.0533035
$764$	98.2085	0.00465060
$765$	0	0
$766$	−8564.42	−0.403975
$767$	−5642.95	−0.265652
$768$	0	0
$769$	−16548.4	−0.776008	−0.388004	−	0.921658i	$-0.626835\pi$
$769$	−16548.4	−0.776008	−0.388004	+	0.921658i	$0.626835\pi$
$770$	383.409	0.0179443
$771$	0	0
$772$	−15981.8	−0.745075
$773$	−5744.94	−0.267310	−0.133655	−	0.991028i	$-0.542671\pi$
$773$	−5744.94	−0.267310	−0.133655	+	0.991028i	$0.542671\pi$
$774$	0	0
$775$	3197.30	0.148194
$776$	11021.1	0.509837
$777$	0	0
$778$	−2687.00	−0.123822
$779$	26519.7	1.21973
$780$	0	0
$781$	6909.66	0.316578
$782$	1677.03	0.0766888
$783$	0	0
$784$	912.486	0.0415673
$785$	−6971.09	−0.316954
$786$	0	0
$787$	−34744.3	−1.57370	−0.786848	−	0.617146i	$-0.788289\pi$
$787$	−34744.3	−1.57370	−0.786848	+	0.617146i	$0.788289\pi$
$788$	6093.24	0.275460
$789$	0	0
$790$	162.425	0.00731496
$791$	−12519.4	−0.562753
$792$	0	0
$793$	47930.8	2.14637
$794$	−3107.15	−0.138877
$795$	0	0
$796$	−16747.4	−0.745724
$797$	−21748.7	−0.966600	−0.483300	−	0.875455i	$-0.660562\pi$
$797$	−21748.7	−0.966600	−0.483300	+	0.875455i	$0.660562\pi$
$798$	0	0
$799$	3811.19	0.168749
$800$	4677.44	0.206716
$801$	0	0
$802$	−2811.89	−0.123805
$803$	−5757.01	−0.253002
$804$	0	0
$805$	−820.644	−0.0359303
$806$	9631.38	0.420907
$807$	0	0
$808$	−10686.7	−0.465292
$809$	42350.6	1.84050	0.920252	−	0.391325i	$-0.127983\pi$
$809$	42350.6	1.84050	0.920252	+	0.391325i	$0.127983\pi$
$810$	0	0
$811$	−18910.7	−0.818796	−0.409398	−	0.912356i	$-0.634261\pi$
$811$	−18910.7	−0.818796	−0.409398	+	0.912356i	$0.634261\pi$
$812$	−6525.10	−0.282003
$813$	0	0
$814$	1265.00	0.0544696
$815$	−18371.3	−0.789595
$816$	0	0
$817$	45299.4	1.93981
$818$	21425.8	0.915813
$819$	0	0
$820$	−5593.33	−0.238204
$821$	−6593.42	−0.280283	−0.140141	−	0.990132i	$-0.544756\pi$
$821$	−6593.42	−0.280283	−0.140141	+	0.990132i	$0.544756\pi$
$822$	0	0
$823$	26762.4	1.13351	0.566755	−	0.823886i	$-0.308199\pi$
$823$	26762.4	1.13351	0.566755	+	0.823886i	$0.308199\pi$
$824$	−27208.5	−1.15031
$825$	0	0
$826$	1085.29	0.0457169
$827$	24016.7	1.00985	0.504924	−	0.863164i	$-0.331521\pi$
$827$	24016.7	1.00985	0.504924	+	0.863164i	$0.331521\pi$
$828$	0	0
$829$	−28422.3	−1.19077	−0.595383	−	0.803442i	$-0.703000\pi$
$829$	−28422.3	−1.19077	−0.595383	+	0.803442i	$0.703000\pi$
$830$	−7999.56	−0.334541
$831$	0	0
$832$	6290.49	0.262120
$833$	2436.45	0.101342
$834$	0	0
$835$	−20205.4	−0.837411
$836$	6350.46	0.262721
$837$	0	0
$838$	−18137.3	−0.747663
$839$	−6637.09	−0.273108	−0.136554	−	0.990633i	$-0.543603\pi$
$839$	−6637.09	−0.273108	−0.136554	+	0.990633i	$0.543603\pi$
$840$	0	0
$841$	313.546	0.0128560
$842$	−11310.3	−0.462921
$843$	0	0
$844$	−30550.9	−1.24598
$845$	2719.79	0.110726
$846$	0	0
$847$	8911.03	0.361495
$848$	7899.01	0.319874
$849$	0	0
$850$	1788.11	0.0721551
$851$	−2707.59	−0.109066
$852$	0	0
$853$	−1406.88	−0.0564720	−0.0282360	−	0.999601i	$-0.508989\pi$
$853$	−1406.88	−0.0564720	−0.0282360	+	0.999601i	$0.508989\pi$
$854$	−9218.40	−0.369376
$855$	0	0
$856$	18309.3	0.731075
$857$	−27943.4	−1.11380	−0.556901	−	0.830579i	$-0.688010\pi$
$857$	−27943.4	−1.11380	−0.556901	+	0.830579i	$0.688010\pi$
$858$	0	0
$859$	1936.63	0.0769233	0.0384616	−	0.999260i	$-0.487754\pi$
$859$	1936.63	0.0769233	0.0384616	+	0.999260i	$0.487754\pi$
$860$	−9554.20	−0.378832
$861$	0	0
$862$	−20556.8	−0.812259
$863$	−7947.70	−0.313491	−0.156746	−	0.987639i	$-0.550100\pi$
$863$	−7947.70	−0.313491	−0.156746	+	0.987639i	$0.550100\pi$
$864$	0	0
$865$	−297.291	−0.0116858
$866$	−19791.6	−0.776611
$867$	0	0
$868$	5309.58	0.207625
$869$	−171.984	−0.00671365
$870$	0	0
$871$	−23649.6	−0.920019
$872$	3216.00	0.124894
$873$	0	0
$874$	4742.06	0.183527
$875$	−875.000	−0.0338062
$876$	0	0
$877$	38655.0	1.48835	0.744177	−	0.667983i	$-0.232842\pi$
$877$	38655.0	1.48835	0.744177	+	0.667983i	$0.232842\pi$
$878$	−8764.65	−0.336893
$879$	0	0
$880$	−709.088	−0.0271629
$881$	18879.4	0.721978	0.360989	−	0.932570i	$-0.382439\pi$
$881$	18879.4	0.721978	0.360989	+	0.932570i	$0.382439\pi$
$882$	0	0
$883$	−35098.1	−1.33765	−0.668825	−	0.743420i	$-0.733202\pi$
$883$	−35098.1	−1.33765	−0.668825	+	0.743420i	$0.733202\pi$
$884$	−15439.4	−0.587425
$885$	0	0
$886$	−19346.3	−0.733581
$887$	48816.4	1.84791	0.923954	−	0.382504i	$-0.124938\pi$
$887$	48816.4	1.84791	0.923954	+	0.382504i	$0.124938\pi$
$888$	0	0
$889$	1897.11	0.0715713
$890$	−10923.6	−0.411417
$891$	0	0
$892$	679.062	0.0254895
$893$	10776.7	0.403839
$894$	0	0
$895$	14866.6	0.555236
$896$	9267.63	0.345547
$897$	0	0
$898$	−9916.41	−0.368502
$899$	−20100.8	−0.745718
$900$	0	0
$901$	21091.3	0.779860
$902$	−2066.22	−0.0762721
$903$	0	0
$904$	−35839.1	−1.31857
$905$	−3381.10	−0.124190
$906$	0	0
$907$	−6010.83	−0.220051	−0.110026	−	0.993929i	$-0.535093\pi$
$907$	−6010.83	−0.220051	−0.110026	+	0.993929i	$0.535093\pi$
$908$	28140.5	1.02850
$909$	0	0
$910$	−2635.80	−0.0960177
$911$	−25780.1	−0.937576	−0.468788	−	0.883311i	$-0.655309\pi$
$911$	−25780.1	−0.937576	−0.468788	+	0.883311i	$0.655309\pi$
$912$	0	0
$913$	8470.37	0.307041
$914$	−17002.6	−0.615314
$915$	0	0
$916$	31934.6	1.15191
$917$	−5342.06	−0.192378
$918$	0	0
$919$	26731.8	0.959522	0.479761	−	0.877399i	$-0.340723\pi$
$919$	26731.8	0.959522	0.479761	+	0.877399i	$0.340723\pi$
$920$	−2349.25	−0.0841874
$921$	0	0
$922$	−12145.4	−0.433824
$923$	−47501.6	−1.69397
$924$	0	0
$925$	−2886.93	−0.102618
$926$	−1826.46	−0.0648177
$927$	0	0
$928$	−29406.2	−1.04020
$929$	30464.6	1.07590	0.537949	−	0.842977i	$-0.319199\pi$
$929$	30464.6	1.07590	0.537949	+	0.842977i	$0.319199\pi$
$930$	0	0
$931$	6889.42	0.242526
$932$	−9542.22	−0.335371
$933$	0	0
$934$	23708.1	0.830572
$935$	−1893.35	−0.0662238
$936$	0	0
$937$	−28533.4	−0.994819	−0.497409	−	0.867516i	$-0.665715\pi$
$937$	−28533.4	−0.994819	−0.497409	+	0.867516i	$0.665715\pi$
$938$	4548.46	0.158329
$939$	0	0
$940$	−2272.94	−0.0788671
$941$	−34455.8	−1.19365	−0.596827	−	0.802370i	$-0.703572\pi$
$941$	−34455.8	−1.19365	−0.596827	+	0.802370i	$0.703572\pi$
$942$	0	0
$943$	4422.50	0.152722
$944$	−2007.17	−0.0692033
$945$	0	0
$946$	−3529.39	−0.121301
$947$	−2477.68	−0.0850198	−0.0425099	−	0.999096i	$-0.513535\pi$
$947$	−2477.68	−0.0850198	−0.0425099	+	0.999096i	$0.513535\pi$
$948$	0	0
$949$	39577.5	1.35378
$950$	5056.16	0.172677
$951$	0	0
$952$	6974.80	0.237452
$953$	−41690.6	−1.41709	−0.708547	−	0.705663i	$-0.750649\pi$
$953$	−41690.6	−1.41709	−0.708547	+	0.705663i	$0.750649\pi$
$954$	0	0
$955$	−82.7943	−0.00280541
$956$	−18463.5	−0.624635
$957$	0	0
$958$	2014.13	0.0679266
$959$	−1686.54	−0.0567895
$960$	0	0
$961$	−13434.6	−0.450962
$962$	−8696.44	−0.291460
$963$	0	0
$964$	−42325.0	−1.41410
$965$	13473.4	0.449455
$966$	0	0
$967$	−20641.3	−0.686430	−0.343215	−	0.939257i	$-0.611516\pi$
$967$	−20641.3	−0.686430	−0.343215	+	0.939257i	$0.611516\pi$
$968$	25509.5	0.847010
$969$	0	0
$970$	−3955.63	−0.130936
$971$	−6626.17	−0.218995	−0.109497	−	0.993987i	$-0.534924\pi$
$971$	−6626.17	−0.218995	−0.109497	+	0.993987i	$0.534924\pi$
$972$	0	0
$973$	722.048	0.0237901
$974$	−20377.0	−0.670349
$975$	0	0
$976$	17048.8	0.559138
$977$	−41961.0	−1.37405	−0.687027	−	0.726632i	$-0.741085\pi$
$977$	−41961.0	−1.37405	−0.687027	+	0.726632i	$0.741085\pi$
$978$	0	0
$979$	11566.5	0.377598
$980$	−1453.06	−0.0473637
$981$	0	0
$982$	−6817.21	−0.221533
$983$	16781.7	0.544510	0.272255	−	0.962225i	$-0.412231\pi$
$983$	16781.7	0.544510	0.272255	+	0.962225i	$0.412231\pi$
$984$	0	0
$985$	−5136.88	−0.166167
$986$	−11241.6	−0.363087
$987$	0	0
$988$	−43657.2	−1.40579
$989$	7554.27	0.242884
$990$	0	0
$991$	50319.8	1.61298	0.806489	−	0.591250i	$-0.201365\pi$
$991$	50319.8	1.61298	0.806489	+	0.591250i	$0.201365\pi$
$992$	23928.3	0.765851
$993$	0	0
$994$	9135.85	0.291521
$995$	14118.9	0.449847
$996$	0	0
$997$	12949.5	0.411348	0.205674	−	0.978621i	$-0.434061\pi$
$997$	12949.5	0.411348	0.205674	+	0.978621i	$0.434061\pi$
$998$	−16355.2	−0.518753
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.4.a.m.1.1		2
3.2	odd	2		105.4.a.c.1.2	✓	2
5.4	even	2		1575.4.a.m.1.2		2
7.6	odd	2		2205.4.a.bh.1.1		2
12.11	even	2		1680.4.a.bk.1.1		2
15.2	even	4		525.4.d.i.274.2		4
15.8	even	4		525.4.d.i.274.3		4
15.14	odd	2		525.4.a.p.1.1		2
21.20	even	2		735.4.a.k.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.a.c.1.2	✓	2	3.2	odd	2
315.4.a.m.1.1		2	1.1	even	1	trivial
525.4.a.p.1.1		2	15.14	odd	2
525.4.d.i.274.2		4	15.2	even	4
525.4.d.i.274.3		4	15.8	even	4
735.4.a.k.1.2		2	21.20	even	2
1575.4.a.m.1.2		2	5.4	even	2
1680.4.a.bk.1.1		2	12.11	even	2
2205.4.a.bh.1.1		2	7.6	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:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	315.a (trivial)

\(f(q)\)	\(=\)	\(q+1.43845 q^{2} -5.93087 q^{4} +5.00000 q^{5} -7.00000 q^{7} -20.0388 q^{8} +O(q^{10})\)	\(q+1.43845 q^{2} -5.93087 q^{4} +5.00000 q^{5} -7.00000 q^{7} -20.0388 q^{8} +7.19224 q^{10} -7.61553 q^{11} +52.3542 q^{13} -10.0691 q^{14} +18.6222 q^{16} +49.7235 q^{17} +140.600 q^{19} -29.6543 q^{20} -10.9545 q^{22} +23.4470 q^{23} +25.0000 q^{25} +75.3087 q^{26} +41.5161 q^{28} -157.170 q^{29} +127.892 q^{31} +187.098 q^{32} +71.5246 q^{34} -35.0000 q^{35} -115.477 q^{37} +202.246 q^{38} -100.194 q^{40} +188.617 q^{41} +322.186 q^{43} +45.1667 q^{44} +33.7272 q^{46} +76.6477 q^{47} +49.0000 q^{49} +35.9612 q^{50} -310.506 q^{52} +424.172 q^{53} -38.0776 q^{55} +140.272 q^{56} -226.081 q^{58} -107.784 q^{59} +915.511 q^{61} +183.966 q^{62} +120.153 q^{64} +261.771 q^{65} -451.723 q^{67} -294.903 q^{68} -50.3457 q^{70} -907.312 q^{71} +755.956 q^{73} -166.108 q^{74} -833.882 q^{76} +53.3087 q^{77} +22.5834 q^{79} +93.1109 q^{80} +271.316 q^{82} -1112.25 q^{83} +248.617 q^{85} +463.447 q^{86} +152.606 q^{88} -1518.81 q^{89} -366.479 q^{91} -139.061 q^{92} +110.254 q^{94} +703.002 q^{95} -549.987 q^{97} +70.4839 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 7 q^{2} + 17 q^{4} + 10 q^{5} - 14 q^{7} + 63 q^{8}+O(q^{10}) \) 2 * q + 7 * q^2 + 17 * q^4 + 10 * q^5 - 14 * q^7 + 63 * q^8	\( 2 q + 7 q^{2} + 17 q^{4} + 10 q^{5} - 14 q^{7} + 63 q^{8} + 35 q^{10} + 26 q^{11} + 14 q^{13} - 49 q^{14} + 297 q^{16} - 16 q^{17} + 174 q^{19} + 85 q^{20} + 176 q^{22} - 184 q^{23} + 50 q^{25} - 138 q^{26} - 119 q^{28} + 32 q^{29} + 330 q^{31} + 1071 q^{32} - 294 q^{34} - 70 q^{35} - 132 q^{37} + 388 q^{38} + 315 q^{40} - 200 q^{41} + 364 q^{43} + 816 q^{44} - 1120 q^{46} - 292 q^{47} + 98 q^{49} + 175 q^{50} - 1190 q^{52} - 34 q^{53} + 130 q^{55} - 441 q^{56} + 826 q^{58} - 364 q^{59} + 792 q^{61} + 1308 q^{62} + 2809 q^{64} + 70 q^{65} - 788 q^{67} - 1802 q^{68} - 245 q^{70} - 454 q^{71} + 778 q^{73} - 258 q^{74} - 68 q^{76} - 182 q^{77} + 408 q^{79} + 1485 q^{80} - 1890 q^{82} - 1136 q^{83} - 80 q^{85} + 696 q^{86} + 2944 q^{88} - 36 q^{89} - 98 q^{91} - 4896 q^{92} - 1940 q^{94} + 870 q^{95} - 498 q^{97} + 343 q^{98}+O(q^{100}) \) 2 * q + 7 * q^2 + 17 * q^4 + 10 * q^5 - 14 * q^7 + 63 * q^8 + 35 * q^10 + 26 * q^11 + 14 * q^13 - 49 * q^14 + 297 * q^16 - 16 * q^17 + 174 * q^19 + 85 * q^20 + 176 * q^22 - 184 * q^23 + 50 * q^25 - 138 * q^26 - 119 * q^28 + 32 * q^29 + 330 * q^31 + 1071 * q^32 - 294 * q^34 - 70 * q^35 - 132 * q^37 + 388 * q^38 + 315 * q^40 - 200 * q^41 + 364 * q^43 + 816 * q^44 - 1120 * q^46 - 292 * q^47 + 98 * q^49 + 175 * q^50 - 1190 * q^52 - 34 * q^53 + 130 * q^55 - 441 * q^56 + 826 * q^58 - 364 * q^59 + 792 * q^61 + 1308 * q^62 + 2809 * q^64 + 70 * q^65 - 788 * q^67 - 1802 * q^68 - 245 * q^70 - 454 * q^71 + 778 * q^73 - 258 * q^74 - 68 * q^76 - 182 * q^77 + 408 * q^79 + 1485 * q^80 - 1890 * q^82 - 1136 * q^83 - 80 * q^85 + 696 * q^86 + 2944 * q^88 - 36 * q^89 - 98 * q^91 - 4896 * q^92 - 1940 * q^94 + 870 * q^95 - 498 * q^97 + 343 * q^98

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