LMFDB - Embedded newform 2254.4.a.z.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2254,4,Mod(1,2254)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2254, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2254.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2254 = 2 \cdot 7^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2254.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:	$132.990305153$
Analytic rank:	$1$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 200x^{12} + 15521x^{10} - 598294x^{8} + 12167812x^{6} - 125559722x^{4} + 539505876x^{2} - 324615200 $ x^14 - 200x^12 + 15521x^10 - 598294x^8 + 12167812x^6 - 125559722x^4 + 539505876x^2 - 324615200
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2}\cdot 7^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-8.18737$ of defining polynomial
Character	$\chi$	$=$	2254.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.00000 q^{2} -8.18737 q^{3} +4.00000 q^{4} +0.929896 q^{5} +16.3747 q^{6} -8.00000 q^{8} +40.0330 q^{9} -1.85979 q^{10} +24.7919 q^{11} -32.7495 q^{12} +49.9754 q^{13} -7.61340 q^{15} +16.0000 q^{16} +47.0882 q^{17} -80.0660 q^{18} -28.1884 q^{19} +3.71959 q^{20} -49.5838 q^{22} +23.0000 q^{23} +65.4989 q^{24} -124.135 q^{25} -99.9508 q^{26} -106.706 q^{27} -83.2679 q^{29} +15.2268 q^{30} -104.094 q^{31} -32.0000 q^{32} -202.981 q^{33} -94.1763 q^{34} +160.132 q^{36} -131.894 q^{37} +56.3768 q^{38} -409.167 q^{39} -7.43917 q^{40} +214.694 q^{41} -0.790748 q^{43} +99.1676 q^{44} +37.2265 q^{45} -46.0000 q^{46} -45.0818 q^{47} -130.998 q^{48} +248.271 q^{50} -385.528 q^{51} +199.902 q^{52} -643.028 q^{53} +213.412 q^{54} +23.0539 q^{55} +230.789 q^{57} +166.536 q^{58} +368.943 q^{59} -30.4536 q^{60} -82.7460 q^{61} +208.189 q^{62} +64.0000 q^{64} +46.4719 q^{65} +405.961 q^{66} -47.0400 q^{67} +188.353 q^{68} -188.309 q^{69} +321.872 q^{71} -320.264 q^{72} -301.612 q^{73} +263.788 q^{74} +1016.34 q^{75} -112.754 q^{76} +818.334 q^{78} -648.828 q^{79} +14.8783 q^{80} -207.250 q^{81} -429.389 q^{82} +267.912 q^{83} +43.7871 q^{85} +1.58150 q^{86} +681.745 q^{87} -198.335 q^{88} -27.0437 q^{89} -74.4531 q^{90} +92.0000 q^{92} +852.259 q^{93} +90.1635 q^{94} -26.2123 q^{95} +261.996 q^{96} +233.648 q^{97} +992.495 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 28 q^{2} + 56 q^{4} - 112 q^{8} + 22 q^{9} - 92 q^{11} - 268 q^{15} + 224 q^{16} - 44 q^{18} + 184 q^{22} + 322 q^{23} + 130 q^{25} + 196 q^{29} + 536 q^{30} - 448 q^{32} + 88 q^{36} + 628 q^{37}+ \cdots + 1800 q^{99}+O(q^{100}) $ 14 * q - 28 * q^2 + 56 * q^4 - 112 * q^8 + 22 * q^9 - 92 * q^11 - 268 * q^15 + 224 * q^16 - 44 * q^18 + 184 * q^22 + 322 * q^23 + 130 * q^25 + 196 * q^29 + 536 * q^30 - 448 * q^32 + 88 * q^36 + 628 * q^37 + 388 * q^39 + 640 * q^43 - 368 * q^44 - 644 * q^46 - 260 * q^50 + 656 * q^51 - 1260 * q^53 + 380 * q^57 - 392 * q^58 - 1072 * q^60 + 896 * q^64 - 204 * q^65 - 1564 * q^67 - 900 * q^71 - 176 * q^72 - 1256 * q^74 - 776 * q^78 - 404 * q^79 - 4278 * q^81 + 1512 * q^85 - 1280 * q^86 + 736 * q^88 + 1288 * q^92 - 2712 * q^93 - 4752 * q^95 + 1800 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	−8.18737	−1.57566	−0.787830	−	0.615893i	$-0.788795\pi$
$3$	−8.18737	−1.57566	−0.787830	+	0.615893i	$0.788795\pi$
$4$	4.00000	0.500000
$5$	0.929896	0.0831725	0.0415862	−	0.999135i	$-0.486759\pi$
$5$	0.929896	0.0831725	0.0415862	+	0.999135i	$0.486759\pi$
$6$	16.3747	1.11416
$7$	0	0
$7$	0	0
$8$	−8.00000	−0.353553
$9$	40.0330	1.48270
$10$	−1.85979	−0.0588118
$11$	24.7919	0.679549	0.339775	−	0.940507i	$-0.389649\pi$
$11$	24.7919	0.679549	0.339775	+	0.940507i	$0.389649\pi$
$12$	−32.7495	−0.787830
$13$	49.9754	1.06621	0.533103	−	0.846050i	$-0.321026\pi$
$13$	49.9754	1.06621	0.533103	+	0.846050i	$0.321026\pi$
$14$	0	0
$15$	−7.61340	−0.131051
$16$	16.0000	0.250000
$17$	47.0882	0.671797	0.335899	−	0.941898i	$-0.390960\pi$
$17$	47.0882	0.671797	0.335899	+	0.941898i	$0.390960\pi$
$18$	−80.0660	−1.04843
$19$	−28.1884	−0.340361	−0.170181	−	0.985413i	$-0.554435\pi$
$19$	−28.1884	−0.340361	−0.170181	+	0.985413i	$0.554435\pi$
$20$	3.71959	0.0415862
$21$	0	0
$22$	−49.5838	−0.480514
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	65.4989	0.557080
$25$	−124.135	−0.993082
$26$	−99.9508	−0.753922
$27$	−106.706	−0.760577
$28$	0	0
$29$	−83.2679	−0.533188	−0.266594	−	0.963809i	$-0.585898\pi$
$29$	−83.2679	−0.533188	−0.266594	+	0.963809i	$0.585898\pi$
$30$	15.2268	0.0926674
$31$	−104.094	−0.603093	−0.301547	−	0.953451i	$-0.597503\pi$
$31$	−104.094	−0.603093	−0.301547	+	0.953451i	$0.597503\pi$
$32$	−32.0000	−0.176777
$33$	−202.981	−1.07074
$34$	−94.1763	−0.475032
$35$	0	0
$36$	160.132	0.741352
$37$	−131.894	−0.586034	−0.293017	−	0.956107i	$-0.594659\pi$
$37$	−131.894	−0.586034	−0.293017	+	0.956107i	$0.594659\pi$
$38$	56.3768	0.240672
$39$	−409.167	−1.67998
$40$	−7.43917	−0.0294059
$41$	214.694	0.817796	0.408898	−	0.912580i	$-0.365913\pi$
$41$	214.694	0.817796	0.408898	+	0.912580i	$0.365913\pi$
$42$	0	0
$43$	−0.790748	−0.00280437	−0.00140219	−	0.999999i	$-0.500446\pi$
$43$	−0.790748	−0.00280437	−0.00140219	+	0.999999i	$0.500446\pi$
$44$	99.1676	0.339775
$45$	37.2265	0.123320
$46$	−46.0000	−0.147442
$47$	−45.0818	−0.139912	−0.0699558	−	0.997550i	$-0.522286\pi$
$47$	−45.0818	−0.139912	−0.0699558	+	0.997550i	$0.522286\pi$
$48$	−130.998	−0.393915
$49$	0	0
$50$	248.271	0.702215
$51$	−385.528	−1.05852
$52$	199.902	0.533103
$53$	−643.028	−1.66654	−0.833271	−	0.552865i	$-0.813535\pi$
$53$	−643.028	−1.66654	−0.833271	+	0.552865i	$0.813535\pi$
$54$	213.412	0.537809
$55$	23.0539	0.0565198
$56$	0	0
$57$	230.789	0.536294
$58$	166.536	0.377021
$59$	368.943	0.814106	0.407053	−	0.913404i	$-0.366556\pi$
$59$	368.943	0.814106	0.407053	+	0.913404i	$0.366556\pi$
$60$	−30.4536	−0.0655257
$61$	−82.7460	−0.173681	−0.0868405	−	0.996222i	$-0.527677\pi$
$61$	−82.7460	−0.173681	−0.0868405	+	0.996222i	$0.527677\pi$
$62$	208.189	0.426451
$63$	0	0
$64$	64.0000	0.125000
$65$	46.4719	0.0886790
$66$	405.961	0.757126
$67$	−47.0400	−0.0857739	−0.0428869	−	0.999080i	$-0.513656\pi$
$67$	−47.0400	−0.0857739	−0.0428869	+	0.999080i	$0.513656\pi$
$68$	188.353	0.335899
$69$	−188.309	−0.328548
$70$	0	0
$71$	321.872	0.538017	0.269009	−	0.963138i	$-0.413304\pi$
$71$	321.872	0.538017	0.269009	+	0.963138i	$0.413304\pi$
$72$	−320.264	−0.524215
$73$	−301.612	−0.483575	−0.241787	−	0.970329i	$-0.577734\pi$
$73$	−301.612	−0.483575	−0.241787	+	0.970329i	$0.577734\pi$
$74$	263.788	0.414389
$75$	1016.34	1.56476
$76$	−112.754	−0.170181
$77$	0	0
$78$	818.334	1.18792
$79$	−648.828	−0.924036	−0.462018	−	0.886871i	$-0.652874\pi$
$79$	−648.828	−0.924036	−0.462018	+	0.886871i	$0.652874\pi$
$80$	14.8783	0.0207931
$81$	−207.250	−0.284293
$82$	−429.389	−0.578269
$83$	267.912	0.354303	0.177152	−	0.984184i	$-0.443312\pi$
$83$	267.912	0.354303	0.177152	+	0.984184i	$0.443312\pi$
$84$	0	0
$85$	43.7871	0.0558750
$86$	1.58150	0.00198299
$87$	681.745	0.840123
$88$	−198.335	−0.240257
$89$	−27.0437	−0.0322093	−0.0161046	−	0.999870i	$-0.505126\pi$
$89$	−27.0437	−0.0322093	−0.0161046	+	0.999870i	$0.505126\pi$
$90$	−74.4531	−0.0872005
$91$	0	0
$92$	92.0000	0.104257
$93$	852.259	0.950270
$94$	90.1635	0.0989325
$95$	−26.2123	−0.0283087
$96$	261.996	0.278540
$97$	233.648	0.244571	0.122285	−	0.992495i	$-0.460978\pi$
$97$	233.648	0.244571	0.122285	+	0.992495i	$0.460978\pi$
$98$	0	0
$99$	992.495	1.00757
$100$	−496.541	−0.496541
$101$	−1626.81	−1.60271	−0.801353	−	0.598192i	$-0.795886\pi$
$101$	−1626.81	−1.60271	−0.801353	+	0.598192i	$0.795886\pi$
$102$	771.056	0.748490
$103$	884.505	0.846144	0.423072	−	0.906096i	$-0.360952\pi$
$103$	884.505	0.846144	0.423072	+	0.906096i	$0.360952\pi$
$104$	−399.803	−0.376961
$105$	0	0
$106$	1286.06	1.17842
$107$	−86.7017	−0.0783343	−0.0391672	−	0.999233i	$-0.512470\pi$
$107$	−86.7017	−0.0783343	−0.0391672	+	0.999233i	$0.512470\pi$
$108$	−426.824	−0.380288
$109$	−20.1154	−0.0176762	−0.00883812	−	0.999961i	$-0.502813\pi$
$109$	−20.1154	−0.0176762	−0.00883812	+	0.999961i	$0.502813\pi$
$110$	−46.1078	−0.0399655
$111$	1079.87	0.923390
$112$	0	0
$113$	1258.57	1.04775	0.523876	−	0.851794i	$-0.324485\pi$
$113$	1258.57	1.04775	0.523876	+	0.851794i	$0.324485\pi$
$114$	−461.578	−0.379217
$115$	21.3876	0.0173427
$116$	−333.072	−0.266594
$117$	2000.67	1.58087
$118$	−737.886	−0.575660
$119$	0	0
$120$	60.9072	0.0463337
$121$	−716.361	−0.538213
$122$	165.492	0.122811
$123$	−1757.78	−1.28857
$124$	−416.377	−0.301547
$125$	−231.670	−0.165770
$126$	0	0
$127$	1543.27	1.07829	0.539145	−	0.842213i	$-0.318748\pi$
$127$	1543.27	1.07829	0.539145	+	0.842213i	$0.318748\pi$
$128$	−128.000	−0.0883883
$129$	6.47414	0.00441873
$130$	−92.9439	−0.0627055
$131$	2001.30	1.33477	0.667384	−	0.744713i	$-0.267414\pi$
$131$	2001.30	1.33477	0.667384	+	0.744713i	$0.267414\pi$
$132$	−811.922	−0.535369
$133$	0	0
$134$	94.0800	0.0606513
$135$	−99.2255	−0.0632591
$136$	−376.705	−0.237516
$137$	−2170.60	−1.35363	−0.676813	−	0.736155i	$-0.736639\pi$
$137$	−2170.60	−1.35363	−0.676813	+	0.736155i	$0.736639\pi$
$138$	376.619	0.232318
$139$	1546.53	0.943702	0.471851	−	0.881678i	$-0.343586\pi$
$139$	1546.53	0.943702	0.471851	+	0.881678i	$0.343586\pi$
$140$	0	0
$141$	369.101	0.220453
$142$	−643.745	−0.380435
$143$	1238.99	0.724540
$144$	640.528	0.370676
$145$	−77.4305	−0.0443466
$146$	603.223	0.341939
$147$	0	0
$148$	−527.576	−0.293017
$149$	1758.33	0.966767	0.483384	−	0.875409i	$-0.339408\pi$
$149$	1758.33	0.966767	0.483384	+	0.875409i	$0.339408\pi$
$150$	−2032.68	−1.10645
$151$	−1070.02	−0.576670	−0.288335	−	0.957530i	$-0.593102\pi$
$151$	−1070.02	−0.576670	−0.288335	+	0.957530i	$0.593102\pi$
$152$	225.507	0.120336
$153$	1885.08	0.996077
$154$	0	0
$155$	−96.7969	−0.0501607
$156$	−1636.67	−0.839989
$157$	2070.79	1.05266	0.526329	−	0.850281i	$-0.323568\pi$
$157$	2070.79	1.05266	0.526329	+	0.850281i	$0.323568\pi$
$158$	1297.66	0.653392
$159$	5264.71	2.62590
$160$	−29.7567	−0.0147030
$161$	0	0
$162$	414.500	0.201026
$163$	265.619	0.127637	0.0638187	−	0.997962i	$-0.479672\pi$
$163$	265.619	0.127637	0.0638187	+	0.997962i	$0.479672\pi$
$164$	858.778	0.408898
$165$	−188.751	−0.0890560
$166$	−535.824	−0.250530
$167$	−1210.08	−0.560713	−0.280356	−	0.959896i	$-0.590453\pi$
$167$	−1210.08	−0.560713	−0.280356	+	0.959896i	$0.590453\pi$
$168$	0	0
$169$	300.540	0.136796
$170$	−87.5742	−0.0395096
$171$	−1128.47	−0.504655
$172$	−3.16299	−0.00140219
$173$	1516.28	0.666364	0.333182	−	0.942863i	$-0.391878\pi$
$173$	1516.28	0.666364	0.333182	+	0.942863i	$0.391878\pi$
$174$	−1363.49	−0.594057
$175$	0	0
$176$	396.671	0.169887
$177$	−3020.67	−1.28275
$178$	54.0874	0.0227754
$179$	−1605.41	−0.670358	−0.335179	−	0.942154i	$-0.608797\pi$
$179$	−1605.41	−0.670358	−0.335179	+	0.942154i	$0.608797\pi$
$180$	148.906	0.0616601
$181$	1845.59	0.757910	0.378955	−	0.925415i	$-0.376283\pi$
$181$	1845.59	0.757910	0.378955	+	0.925415i	$0.376283\pi$
$182$	0	0
$183$	677.472	0.273662
$184$	−184.000	−0.0737210
$185$	−122.648	−0.0487419
$186$	−1704.52	−0.671942
$187$	1167.41	0.456519
$188$	−180.327	−0.0699558
$189$	0	0
$190$	52.4246	0.0200173
$191$	151.354	0.0573382	0.0286691	−	0.999589i	$-0.490873\pi$
$191$	151.354	0.0573382	0.0286691	+	0.999589i	$0.490873\pi$
$192$	−523.992	−0.196957
$193$	−402.106	−0.149970	−0.0749851	−	0.997185i	$-0.523891\pi$
$193$	−402.106	−0.149970	−0.0749851	+	0.997185i	$0.523891\pi$
$194$	−467.296	−0.172938
$195$	−380.483	−0.139728
$196$	0	0
$197$	−541.200	−0.195730	−0.0978651	−	0.995200i	$-0.531201\pi$
$197$	−541.200	−0.195730	−0.0978651	+	0.995200i	$0.531201\pi$
$198$	−1984.99	−0.712460
$199$	4526.94	1.61259	0.806297	−	0.591511i	$-0.201468\pi$
$199$	4526.94	1.61259	0.806297	+	0.591511i	$0.201468\pi$
$200$	993.082	0.351108
$201$	385.134	0.135150
$202$	3253.61	1.13328
$203$	0	0
$204$	−1542.11	−0.529262
$205$	199.644	0.0680181
$206$	−1769.01	−0.598314
$207$	920.759	0.309165
$208$	799.606	0.266552
$209$	−698.845	−0.231292
$210$	0	0
$211$	−1026.64	−0.334960	−0.167480	−	0.985875i	$-0.553563\pi$
$211$	−1026.64	−0.334960	−0.167480	+	0.985875i	$0.553563\pi$
$212$	−2572.11	−0.833271
$213$	−2635.29	−0.847732
$214$	173.403	0.0553907
$215$	−0.735314	−0.000233246 0
$216$	853.648	0.268905
$217$	0	0
$218$	40.2309	0.0124990
$219$	2469.40	0.761949
$220$	92.2156	0.0282599
$221$	2353.25	0.716275
$222$	−2159.73	−0.652935
$223$	−1590.16	−0.477512	−0.238756	−	0.971080i	$-0.576740\pi$
$223$	−1590.16	−0.477512	−0.238756	+	0.971080i	$0.576740\pi$
$224$	0	0
$225$	−4969.51	−1.47245
$226$	−2517.13	−0.740873
$227$	−4829.13	−1.41199	−0.705993	−	0.708219i	$-0.749499\pi$
$227$	−4829.13	−1.41199	−0.705993	+	0.708219i	$0.749499\pi$
$228$	923.156	0.268147
$229$	−5916.43	−1.70729	−0.853643	−	0.520859i	$-0.825612\pi$
$229$	−5916.43	−1.70729	−0.853643	+	0.520859i	$0.825612\pi$
$230$	−42.7752	−0.0122631
$231$	0	0
$232$	666.143	0.188511
$233$	3505.15	0.985537	0.492768	−	0.870161i	$-0.335985\pi$
$233$	3505.15	0.985537	0.492768	+	0.870161i	$0.335985\pi$
$234$	−4001.33	−1.11784
$235$	−41.9214	−0.0116368
$236$	1475.77	0.407053
$237$	5312.19	1.45597
$238$	0	0
$239$	−1133.93	−0.306895	−0.153447	−	0.988157i	$-0.549038\pi$
$239$	−1133.93	−0.306895	−0.153447	+	0.988157i	$0.549038\pi$
$240$	−121.814	−0.0327629
$241$	−2264.96	−0.605389	−0.302695	−	0.953088i	$-0.597886\pi$
$241$	−2264.96	−0.605389	−0.302695	+	0.953088i	$0.597886\pi$
$242$	1432.72	0.380574
$243$	4577.89	1.20853
$244$	−330.984	−0.0868405
$245$	0	0
$246$	3515.56	0.911155
$247$	−1408.73	−0.362896
$248$	832.755	0.213226
$249$	−2193.50	−0.558262
$250$	463.340	0.117217
$251$	1409.31	0.354401	0.177200	−	0.984175i	$-0.443296\pi$
$251$	1409.31	0.354401	0.177200	+	0.984175i	$0.443296\pi$
$252$	0	0
$253$	570.214	0.141696
$254$	−3086.53	−0.762466
$255$	−358.501	−0.0880401
$256$	256.000	0.0625000
$257$	2174.35	0.527752	0.263876	−	0.964557i	$-0.414999\pi$
$257$	2174.35	0.527752	0.263876	+	0.964557i	$0.414999\pi$
$258$	−12.9483	−0.00312452
$259$	0	0
$260$	185.888	0.0443395
$261$	−3333.46	−0.790560
$262$	−4002.61	−0.943824
$263$	1107.47	0.259656	0.129828	−	0.991537i	$-0.458558\pi$
$263$	1107.47	0.259656	0.129828	+	0.991537i	$0.458558\pi$
$264$	1623.84	0.378563
$265$	−597.950	−0.138610
$266$	0	0
$267$	221.417	0.0507509
$268$	−188.160	−0.0428869
$269$	−5159.43	−1.16943	−0.584714	−	0.811239i	$-0.698793\pi$
$269$	−5159.43	−1.16943	−0.584714	+	0.811239i	$0.698793\pi$
$270$	198.451	0.0447309
$271$	2325.93	0.521365	0.260683	−	0.965425i	$-0.416052\pi$
$271$	2325.93	0.521365	0.260683	+	0.965425i	$0.416052\pi$
$272$	753.411	0.167949
$273$	0	0
$274$	4341.20	0.957158
$275$	−3077.55	−0.674848
$276$	−753.238	−0.164274
$277$	5121.59	1.11093	0.555463	−	0.831541i	$-0.312541\pi$
$277$	5121.59	1.11093	0.555463	+	0.831541i	$0.312541\pi$
$278$	−3093.05	−0.667298
$279$	−4167.21	−0.894209
$280$	0	0
$281$	−8796.14	−1.86738	−0.933690	−	0.358083i	$-0.883430\pi$
$281$	−8796.14	−1.86738	−0.933690	+	0.358083i	$0.883430\pi$
$282$	−738.202	−0.155884
$283$	−265.327	−0.0557316	−0.0278658	−	0.999612i	$-0.508871\pi$
$283$	−265.327	−0.0557316	−0.0278658	+	0.999612i	$0.508871\pi$
$284$	1287.49	0.269009
$285$	214.610	0.0446049
$286$	−2477.97	−0.512327
$287$	0	0
$288$	−1281.06	−0.262107
$289$	−2695.71	−0.548688
$290$	154.861	0.0313578
$291$	−1912.96	−0.385361
$292$	−1206.45	−0.241787
$293$	1877.36	0.374324	0.187162	−	0.982329i	$-0.440071\pi$
$293$	1877.36	0.374324	0.187162	+	0.982329i	$0.440071\pi$
$294$	0	0
$295$	343.079	0.0677112
$296$	1055.15	0.207194
$297$	−2645.45	−0.516850
$298$	−3516.67	−0.683608
$299$	1149.43	0.222319
$300$	4065.37	0.782380
$301$	0	0
$302$	2140.04	0.407767
$303$	13319.3	2.52532
$304$	−451.015	−0.0850904
$305$	−76.9452	−0.0144455
$306$	−3770.16	−0.704332
$307$	760.529	0.141387	0.0706933	−	0.997498i	$-0.477479\pi$
$307$	760.529	0.141387	0.0706933	+	0.997498i	$0.477479\pi$
$308$	0	0
$309$	−7241.76	−1.33324
$310$	193.594	0.0354690
$311$	−2859.73	−0.521417	−0.260708	−	0.965418i	$-0.583956\pi$
$311$	−2859.73	−0.521417	−0.260708	+	0.965418i	$0.583956\pi$
$312$	3273.34	0.593962
$313$	−5618.47	−1.01462	−0.507308	−	0.861765i	$-0.669359\pi$
$313$	−5618.47	−1.01462	−0.507308	+	0.861765i	$0.669359\pi$
$314$	−4141.59	−0.744342
$315$	0	0
$316$	−2595.31	−0.462018
$317$	−2261.50	−0.400689	−0.200344	−	0.979726i	$-0.564206\pi$
$317$	−2261.50	−0.400689	−0.200344	+	0.979726i	$0.564206\pi$
$318$	−10529.4	−1.85679
$319$	−2064.37	−0.362328
$320$	59.5134	0.0103966
$321$	709.859	0.123428
$322$	0	0
$323$	−1327.34	−0.228654
$324$	−828.999	−0.142147
$325$	−6203.71	−1.05883
$326$	−531.238	−0.0902532
$327$	164.693	0.0278517
$328$	−1717.56	−0.289135
$329$	0	0
$330$	377.502	0.0629721
$331$	4247.57	0.705340	0.352670	−	0.935748i	$-0.385274\pi$
$331$	4247.57	0.705340	0.352670	+	0.935748i	$0.385274\pi$
$332$	1071.65	0.177152
$333$	−5280.12	−0.868915
$334$	2420.17	0.396484
$335$	−43.7423	−0.00713402
$336$	0	0
$337$	5733.71	0.926811	0.463406	−	0.886146i	$-0.346627\pi$
$337$	5733.71	0.926811	0.463406	+	0.886146i	$0.346627\pi$
$338$	−601.080	−0.0967291
$339$	−10304.4	−1.65090
$340$	175.148	0.0279375
$341$	−2580.70	−0.409832
$342$	2256.93	0.356845
$343$	0	0
$344$	6.32598	0.000991495 0
$345$	−175.108	−0.0273261
$346$	−3032.57	−0.471190
$347$	−5198.76	−0.804277	−0.402138	−	0.915579i	$-0.631733\pi$
$347$	−5198.76	−0.804277	−0.402138	+	0.915579i	$0.631733\pi$
$348$	2726.98	0.420062
$349$	−8775.86	−1.34602	−0.673010	−	0.739633i	$-0.734999\pi$
$349$	−8775.86	−1.34602	−0.673010	+	0.739633i	$0.734999\pi$
$350$	0	0
$351$	−5332.67	−0.810932
$352$	−793.341	−0.120128
$353$	−6361.10	−0.959114	−0.479557	−	0.877511i	$-0.659203\pi$
$353$	−6361.10	−0.959114	−0.479557	+	0.877511i	$0.659203\pi$
$354$	6041.34	0.907044
$355$	299.308	0.0447482
$356$	−108.175	−0.0161046
$357$	0	0
$358$	3210.82	0.474015
$359$	−6661.85	−0.979385	−0.489692	−	0.871895i	$-0.662891\pi$
$359$	−6661.85	−0.979385	−0.489692	+	0.871895i	$0.662891\pi$
$360$	−297.812	−0.0436002
$361$	−6064.41	−0.884154
$362$	−3691.18	−0.535924
$363$	5865.11	0.848040
$364$	0	0
$365$	−280.467	−0.0402201
$366$	−1354.94	−0.193508
$367$	−6552.70	−0.932011	−0.466006	−	0.884782i	$-0.654307\pi$
$367$	−6552.70	−0.932011	−0.466006	+	0.884782i	$0.654307\pi$
$368$	368.000	0.0521286
$369$	8594.86	1.21255
$370$	245.296	0.0344657
$371$	0	0
$372$	3409.03	0.475135
$373$	4988.55	0.692486	0.346243	−	0.938145i	$-0.387457\pi$
$373$	4988.55	0.692486	0.346243	+	0.938145i	$0.387457\pi$
$374$	−2334.81	−0.322808
$375$	1896.77	0.261196
$376$	360.654	0.0494663
$377$	−4161.35	−0.568489
$378$	0	0
$379$	3768.99	0.510818	0.255409	−	0.966833i	$-0.417790\pi$
$379$	3768.99	0.510818	0.255409	+	0.966833i	$0.417790\pi$
$380$	−104.849	−0.0141543
$381$	−12635.3	−1.69902
$382$	−302.708	−0.0405443
$383$	−58.3576	−0.00778572	−0.00389286	−	0.999992i	$-0.501239\pi$
$383$	−58.3576	−0.00778572	−0.00389286	+	0.999992i	$0.501239\pi$
$384$	1047.98	0.139270
$385$	0	0
$386$	804.213	0.106045
$387$	−31.6560	−0.00415805
$388$	934.593	0.122285
$389$	−9517.96	−1.24057	−0.620283	−	0.784378i	$-0.712982\pi$
$389$	−9517.96	−1.24057	−0.620283	+	0.784378i	$0.712982\pi$
$390$	760.966	0.0988026
$391$	1083.03	0.140079
$392$	0	0
$393$	−16385.4	−2.10314
$394$	1082.40	0.138402
$395$	−603.343	−0.0768544
$396$	3969.98	0.503785
$397$	3250.14	0.410881	0.205441	−	0.978670i	$-0.434137\pi$
$397$	3250.14	0.410881	0.205441	+	0.978670i	$0.434137\pi$
$398$	−9053.88	−1.14028
$399$	0	0
$400$	−1986.16	−0.248271
$401$	−1602.93	−0.199617	−0.0998086	−	0.995007i	$-0.531823\pi$
$401$	−1602.93	−0.199617	−0.0998086	+	0.995007i	$0.531823\pi$
$402$	−770.268	−0.0955658
$403$	−5202.15	−0.643022
$404$	−6507.22	−0.801353
$405$	−192.721	−0.0236454
$406$	0	0
$407$	−3269.91	−0.398239
$408$	3084.22	0.374245
$409$	8648.78	1.04561	0.522805	−	0.852452i	$-0.324885\pi$
$409$	8648.78	1.04561	0.522805	+	0.852452i	$0.324885\pi$
$410$	−399.287	−0.0480961
$411$	17771.5	2.13285
$412$	3538.02	0.423072
$413$	0	0
$414$	−1841.52	−0.218613
$415$	249.131	0.0294683
$416$	−1599.21	−0.188480
$417$	−12662.0	−1.48695
$418$	1397.69	0.163548
$419$	−920.296	−0.107302	−0.0536509	−	0.998560i	$-0.517086\pi$
$419$	−920.296	−0.107302	−0.0536509	+	0.998560i	$0.517086\pi$
$420$	0	0
$421$	−10677.9	−1.23613	−0.618064	−	0.786128i	$-0.712083\pi$
$421$	−10677.9	−1.23613	−0.618064	+	0.786128i	$0.712083\pi$
$422$	2053.27	0.236853
$423$	−1804.76	−0.207448
$424$	5144.23	0.589212
$425$	−5845.30	−0.667150
$426$	5270.57	0.599437
$427$	0	0
$428$	−346.807	−0.0391672
$429$	−10144.0	−1.14163
$430$	1.47063	0.000164930 0
$431$	−10662.8	−1.19167	−0.595834	−	0.803108i	$-0.703178\pi$
$431$	−10662.8	−1.19167	−0.595834	+	0.803108i	$0.703178\pi$
$432$	−1707.30	−0.190144
$433$	666.389	0.0739599	0.0369799	−	0.999316i	$-0.488226\pi$
$433$	666.389	0.0739599	0.0369799	+	0.999316i	$0.488226\pi$
$434$	0	0
$435$	633.952	0.0698751
$436$	−80.4618	−0.00883812
$437$	−648.334	−0.0709703
$438$	−4938.81	−0.538780
$439$	203.557	0.0221304	0.0110652	−	0.999939i	$-0.496478\pi$
$439$	203.557	0.0221304	0.0110652	+	0.999939i	$0.496478\pi$
$440$	−184.431	−0.0199828
$441$	0	0
$442$	−4706.50	−0.506483
$443$	−15540.0	−1.66665	−0.833327	−	0.552781i	$-0.813567\pi$
$443$	−15540.0	−1.66665	−0.833327	+	0.552781i	$0.813567\pi$
$444$	4319.46	0.461695
$445$	−25.1478	−0.00267892
$446$	3180.33	0.337652
$447$	−14396.1	−1.52330
$448$	0	0
$449$	−3811.42	−0.400606	−0.200303	−	0.979734i	$-0.564193\pi$
$449$	−3811.42	−0.400606	−0.200303	+	0.979734i	$0.564193\pi$
$450$	9939.02	1.04118
$451$	5322.68	0.555733
$452$	5034.27	0.523876
$453$	8760.66	0.908636
$454$	9658.27	0.998425
$455$	0	0
$456$	−1846.31	−0.189609
$457$	−5391.26	−0.551844	−0.275922	−	0.961180i	$-0.588983\pi$
$457$	−5391.26	−0.551844	−0.275922	+	0.961180i	$0.588983\pi$
$458$	11832.9	1.20723
$459$	−5024.59	−0.510954
$460$	85.5505	0.00867133
$461$	1512.84	0.152841	0.0764207	−	0.997076i	$-0.475651\pi$
$461$	1512.84	0.152841	0.0764207	+	0.997076i	$0.475651\pi$
$462$	0	0
$463$	9939.43	0.997678	0.498839	−	0.866695i	$-0.333760\pi$
$463$	9939.43	0.997678	0.498839	+	0.866695i	$0.333760\pi$
$464$	−1332.29	−0.133297
$465$	792.512	0.0790363
$466$	−7010.30	−0.696880
$467$	−8514.31	−0.843673	−0.421836	−	0.906672i	$-0.638614\pi$
$467$	−8514.31	−0.843673	−0.421836	+	0.906672i	$0.638614\pi$
$468$	8002.66	0.790434
$469$	0	0
$470$	83.8427	0.00822846
$471$	−16954.4	−1.65863
$472$	−2951.54	−0.287830
$473$	−19.6042	−0.00190571
$474$	−10624.4	−1.02952
$475$	3499.18	0.338007
$476$	0	0
$477$	−25742.3	−2.47099
$478$	2267.86	0.217008
$479$	8661.25	0.826185	0.413093	−	0.910689i	$-0.364449\pi$
$479$	8661.25	0.826185	0.413093	+	0.910689i	$0.364449\pi$
$480$	243.629	0.0231668
$481$	−6591.46	−0.624833
$482$	4529.92	0.428075
$483$	0	0
$484$	−2865.44	−0.269106
$485$	217.269	0.0203416
$486$	−9155.78	−0.854557
$487$	20654.5	1.92186	0.960930	−	0.276793i	$-0.0892717\pi$
$487$	20654.5	1.92186	0.960930	+	0.276793i	$0.0892717\pi$
$488$	661.968	0.0614055
$489$	−2174.72	−0.201113
$490$	0	0
$491$	6176.18	0.567673	0.283836	−	0.958873i	$-0.408393\pi$
$491$	6176.18	0.567673	0.283836	+	0.958873i	$0.408393\pi$
$492$	−7031.13	−0.644284
$493$	−3920.93	−0.358195
$494$	2817.46	0.256606
$495$	922.917	0.0838021
$496$	−1665.51	−0.150773
$497$	0	0
$498$	4386.99	0.394751
$499$	−19317.7	−1.73302	−0.866511	−	0.499158i	$-0.833643\pi$
$499$	−19317.7	−1.73302	−0.866511	+	0.499158i	$0.833643\pi$
$500$	−926.680	−0.0828848
$501$	9907.40	0.883493
$502$	−2818.61	−0.250599
$503$	3625.42	0.321370	0.160685	−	0.987006i	$-0.448630\pi$
$503$	3625.42	0.321370	0.160685	+	0.987006i	$0.448630\pi$
$504$	0	0
$505$	−1512.76	−0.133301
$506$	−1140.43	−0.100194
$507$	−2460.63	−0.215543
$508$	6173.06	0.539145
$509$	7028.83	0.612077	0.306039	−	0.952019i	$-0.400996\pi$
$509$	7028.83	0.612077	0.306039	+	0.952019i	$0.400996\pi$
$510$	717.002	0.0622537
$511$	0	0
$512$	−512.000	−0.0441942
$513$	3007.87	0.258871
$514$	−4348.70	−0.373177
$515$	822.498	0.0703759
$516$	25.8966	0.00220937
$517$	−1117.66	−0.0950769
$518$	0	0
$519$	−12414.4	−1.04996
$520$	−371.775	−0.0313528
$521$	16658.2	1.40078	0.700391	−	0.713759i	$-0.253009\pi$
$521$	16658.2	1.40078	0.700391	+	0.713759i	$0.253009\pi$
$522$	6666.93	0.559011
$523$	−12541.8	−1.04859	−0.524297	−	0.851536i	$-0.675672\pi$
$523$	−12541.8	−1.04859	−0.524297	+	0.851536i	$0.675672\pi$
$524$	8005.21	0.667384
$525$	0	0
$526$	−2214.94	−0.183604
$527$	−4901.61	−0.405156
$528$	−3247.69	−0.267685
$529$	529.000	0.0434783
$530$	1195.90	0.0980123
$531$	14769.9	1.20708
$532$	0	0
$533$	10729.4	0.871939
$534$	−442.833	−0.0358863
$535$	−80.6236	−0.00651526
$536$	376.320	0.0303256
$537$	13144.1	1.05626
$538$	10318.9	0.826911
$539$	0	0
$540$	−396.902	−0.0316295
$541$	−14463.9	−1.14945	−0.574726	−	0.818346i	$-0.694891\pi$
$541$	−14463.9	−1.14945	−0.574726	+	0.818346i	$0.694891\pi$
$542$	−4651.85	−0.368661
$543$	−15110.5	−1.19421
$544$	−1506.82	−0.118758
$545$	−18.7053	−0.00147018
$546$	0	0
$547$	754.766	0.0589972	0.0294986	−	0.999565i	$-0.490609\pi$
$547$	754.766	0.0589972	0.0294986	+	0.999565i	$0.490609\pi$
$548$	−8682.40	−0.676813
$549$	−3312.57	−0.257518
$550$	6155.10	0.477190
$551$	2347.19	0.181477
$552$	1506.48	0.116159
$553$	0	0
$554$	−10243.2	−0.785544
$555$	1004.16	0.0768006
$556$	6186.10	0.471851
$557$	13788.9	1.04893	0.524464	−	0.851433i	$-0.324266\pi$
$557$	13788.9	1.04893	0.524464	+	0.851433i	$0.324266\pi$
$558$	8334.42	0.632301
$559$	−39.5179	−0.00299004
$560$	0	0
$561$	−9557.98	−0.719319
$562$	17592.3	1.32044
$563$	21277.1	1.59276	0.796379	−	0.604798i	$-0.206746\pi$
$563$	21277.1	1.59276	0.796379	+	0.604798i	$0.206746\pi$
$564$	1476.40	0.110227
$565$	1170.34	0.0871442
$566$	530.654	0.0394082
$567$	0	0
$568$	−2574.98	−0.190218
$569$	−4588.21	−0.338046	−0.169023	−	0.985612i	$-0.554061\pi$
$569$	−4588.21	−0.338046	−0.169023	+	0.985612i	$0.554061\pi$
$570$	−429.220	−0.0315404
$571$	−13123.8	−0.961845	−0.480922	−	0.876763i	$-0.659698\pi$
$571$	−13123.8	−0.961845	−0.480922	+	0.876763i	$0.659698\pi$
$572$	4955.94	0.362270
$573$	−1239.19	−0.0903455
$574$	0	0
$575$	−2855.11	−0.207072
$576$	2562.11	0.185338
$577$	13006.8	0.938437	0.469219	−	0.883082i	$-0.344536\pi$
$577$	13006.8	0.938437	0.469219	+	0.883082i	$0.344536\pi$
$578$	5391.41	0.387981
$579$	3292.19	0.236302
$580$	−309.722	−0.0221733
$581$	0	0
$582$	3825.93	0.272491
$583$	−15941.9	−1.13250
$584$	2412.89	0.170969
$585$	1860.41	0.131485
$586$	−3754.73	−0.264687
$587$	−4411.66	−0.310202	−0.155101	−	0.987899i	$-0.549570\pi$
$587$	−4411.66	−0.310202	−0.155101	+	0.987899i	$0.549570\pi$
$588$	0	0
$589$	2934.25	0.205270
$590$	−686.157	−0.0478791
$591$	4431.00	0.308404
$592$	−2110.31	−0.146508
$593$	−16590.1	−1.14886	−0.574430	−	0.818554i	$-0.694776\pi$
$593$	−16590.1	−1.14886	−0.574430	+	0.818554i	$0.694776\pi$
$594$	5290.89	0.365468
$595$	0	0
$596$	7033.33	0.483384
$597$	−37063.7	−2.54090
$598$	−2298.87	−0.157204
$599$	−21989.1	−1.49992	−0.749960	−	0.661483i	$-0.769927\pi$
$599$	−21989.1	−1.49992	−0.749960	+	0.661483i	$0.769927\pi$
$600$	−8130.73	−0.553226
$601$	−10366.9	−0.703621	−0.351811	−	0.936071i	$-0.614434\pi$
$601$	−10366.9	−0.703621	−0.351811	+	0.936071i	$0.614434\pi$
$602$	0	0
$603$	−1883.15	−0.127177
$604$	−4280.09	−0.288335
$605$	−666.142	−0.0447645
$606$	−26638.5	−1.78567
$607$	−18204.4	−1.21729	−0.608645	−	0.793442i	$-0.708287\pi$
$607$	−18204.4	−1.21729	−0.608645	+	0.793442i	$0.708287\pi$
$608$	902.030	0.0601680
$609$	0	0
$610$	153.890	0.0102145
$611$	−2252.98	−0.149175
$612$	7540.32	0.498038
$613$	4955.64	0.326520	0.163260	−	0.986583i	$-0.447799\pi$
$613$	4955.64	0.326520	0.163260	+	0.986583i	$0.447799\pi$
$614$	−1521.06	−0.0999754
$615$	−1634.56	−0.107173
$616$	0	0
$617$	6462.50	0.421670	0.210835	−	0.977522i	$-0.432382\pi$
$617$	6462.50	0.421670	0.210835	+	0.977522i	$0.432382\pi$
$618$	14483.5	0.942740
$619$	24210.9	1.57208	0.786041	−	0.618175i	$-0.212128\pi$
$619$	24210.9	1.57208	0.786041	+	0.618175i	$0.212128\pi$
$620$	−387.188	−0.0250804
$621$	−2454.24	−0.158591
$622$	5719.47	0.368697
$623$	0	0
$624$	−6546.67	−0.419995
$625$	15301.5	0.979295
$626$	11236.9	0.717442
$627$	5721.70	0.364438
$628$	8283.17	0.526329
$629$	−6210.65	−0.393696
$630$	0	0
$631$	26854.7	1.69424	0.847122	−	0.531399i	$-0.178333\pi$
$631$	26854.7	1.69424	0.847122	+	0.531399i	$0.178333\pi$
$632$	5190.62	0.326696
$633$	8405.45	0.527783
$634$	4523.00	0.283330
$635$	1435.08	0.0896840
$636$	21058.8	1.31295
$637$	0	0
$638$	4128.74	0.256204
$639$	12885.5	0.797720
$640$	−119.027	−0.00735148
$641$	−26485.5	−1.63200	−0.816002	−	0.578049i	$-0.803814\pi$
$641$	−26485.5	−1.63200	−0.816002	+	0.578049i	$0.803814\pi$
$642$	−1419.72	−0.0872769
$643$	−4717.54	−0.289334	−0.144667	−	0.989480i	$-0.546211\pi$
$643$	−4717.54	−0.289334	−0.144667	+	0.989480i	$0.546211\pi$
$644$	0	0
$645$	6.02028	0.000367517 0
$646$	2654.68	0.161683
$647$	24950.0	1.51605	0.758026	−	0.652224i	$-0.226164\pi$
$647$	24950.0	1.51605	0.758026	+	0.652224i	$0.226164\pi$
$648$	1658.00	0.100513
$649$	9146.80	0.553225
$650$	12407.4	0.748706
$651$	0	0
$652$	1062.48	0.0638187
$653$	−8336.75	−0.499605	−0.249803	−	0.968297i	$-0.580366\pi$
$653$	−8336.75	−0.499605	−0.249803	+	0.968297i	$0.580366\pi$
$654$	−329.385	−0.0196942
$655$	1861.00	0.111016
$656$	3435.11	0.204449
$657$	−12074.4	−0.716998
$658$	0	0
$659$	2792.93	0.165094	0.0825471	−	0.996587i	$-0.473695\pi$
$659$	2792.93	0.165094	0.0825471	+	0.996587i	$0.473695\pi$
$660$	−755.003	−0.0445280
$661$	−16190.8	−0.952724	−0.476362	−	0.879249i	$-0.658045\pi$
$661$	−16190.8	−0.952724	−0.476362	+	0.879249i	$0.658045\pi$
$662$	−8495.14	−0.498751
$663$	−19266.9	−1.12861
$664$	−2143.30	−0.125265
$665$	0	0
$666$	10560.2	0.614415
$667$	−1915.16	−0.111177
$668$	−4840.33	−0.280356
$669$	13019.3	0.752397
$670$	87.4846	0.00504452
$671$	−2051.43	−0.118025
$672$	0	0
$673$	23556.4	1.34923	0.674616	−	0.738169i	$-0.264309\pi$
$673$	23556.4	1.34923	0.674616	+	0.738169i	$0.264309\pi$
$674$	−11467.4	−0.655354
$675$	13246.0	0.755316
$676$	1202.16	0.0683978
$677$	−10509.5	−0.596621	−0.298311	−	0.954469i	$-0.596423\pi$
$677$	−10509.5	−0.596621	−0.298311	+	0.954469i	$0.596423\pi$
$678$	20608.7	1.16736
$679$	0	0
$680$	−350.297	−0.0197548
$681$	39537.9	2.22481
$682$	5161.39	0.289795
$683$	−16935.1	−0.948762	−0.474381	−	0.880320i	$-0.657328\pi$
$683$	−16935.1	−0.948762	−0.474381	+	0.880320i	$0.657328\pi$
$684$	−4513.87	−0.252328
$685$	−2018.43	−0.112584
$686$	0	0
$687$	48440.0	2.69010
$688$	−12.6520	−0.000701093 0
$689$	−32135.6	−1.77688
$690$	350.217	0.0193225
$691$	7831.98	0.431176	0.215588	−	0.976484i	$-0.430833\pi$
$691$	7831.98	0.431176	0.215588	+	0.976484i	$0.430833\pi$
$692$	6065.14	0.333182
$693$	0	0
$694$	10397.5	0.568710
$695$	1438.11	0.0784900
$696$	−5453.96	−0.297028
$697$	10109.6	0.549393
$698$	17551.7	0.951780
$699$	−28698.0	−1.55287
$700$	0	0
$701$	30651.2	1.65147	0.825735	−	0.564058i	$-0.190761\pi$
$701$	30651.2	1.65147	0.825735	+	0.564058i	$0.190761\pi$
$702$	10665.3	0.573415
$703$	3717.89	0.199463
$704$	1586.68	0.0849437
$705$	343.226	0.0183356
$706$	12722.2	0.678196
$707$	0	0
$708$	−12082.7	−0.641377
$709$	8493.59	0.449906	0.224953	−	0.974370i	$-0.427777\pi$
$709$	8493.59	0.449906	0.224953	+	0.974370i	$0.427777\pi$
$710$	−598.616	−0.0316418
$711$	−25974.5	−1.37007
$712$	216.350	0.0113877
$713$	−2394.17	−0.125754
$714$	0	0
$715$	1152.13	0.0602617
$716$	−6421.65	−0.335179
$717$	9283.91	0.483562
$718$	13323.7	0.692530
$719$	20656.9	1.07145	0.535726	−	0.844392i	$-0.320038\pi$
$719$	20656.9	1.07145	0.535726	+	0.844392i	$0.320038\pi$
$720$	595.625	0.0308300
$721$	0	0
$722$	12128.8	0.625191
$723$	18544.1	0.953888
$724$	7382.37	0.378955
$725$	10336.5	0.529500
$726$	−11730.2	−0.599655
$727$	−3871.26	−0.197493	−0.0987463	−	0.995113i	$-0.531483\pi$
$727$	−3871.26	−0.197493	−0.0987463	+	0.995113i	$0.531483\pi$
$728$	0	0
$729$	−31885.1	−1.61993
$730$	560.935	0.0284399
$731$	−37.2349	−0.00188397
$732$	2709.89	0.136831
$733$	12077.1	0.608562	0.304281	−	0.952582i	$-0.401584\pi$
$733$	12077.1	0.608562	0.304281	+	0.952582i	$0.401584\pi$
$734$	13105.4	0.659031
$735$	0	0
$736$	−736.000	−0.0368605
$737$	−1166.21	−0.0582876
$738$	−17189.7	−0.857402
$739$	−13369.2	−0.665488	−0.332744	−	0.943017i	$-0.607975\pi$
$739$	−13369.2	−0.665488	−0.332744	+	0.943017i	$0.607975\pi$
$740$	−490.591	−0.0243709
$741$	11533.8	0.571800
$742$	0	0
$743$	3177.57	0.156896	0.0784479	−	0.996918i	$-0.475004\pi$
$743$	3177.57	0.156896	0.0784479	+	0.996918i	$0.475004\pi$
$744$	−6818.07	−0.335971
$745$	1635.07	0.0804084
$746$	−9977.10	−0.489662
$747$	10725.3	0.525327
$748$	4669.62	0.228260
$749$	0	0
$750$	−3793.54	−0.184694
$751$	10890.0	0.529135	0.264568	−	0.964367i	$-0.414771\pi$
$751$	10890.0	0.529135	0.264568	+	0.964367i	$0.414771\pi$
$752$	−721.308	−0.0349779
$753$	−11538.5	−0.558415
$754$	8322.69	0.401982
$755$	−995.010	−0.0479631
$756$	0	0
$757$	19313.4	0.927287	0.463644	−	0.886022i	$-0.346542\pi$
$757$	19313.4	0.927287	0.463644	+	0.886022i	$0.346542\pi$
$758$	−7537.98	−0.361203
$759$	−4668.55	−0.223264
$760$	209.698	0.0100086
$761$	−1623.10	−0.0773159	−0.0386579	−	0.999253i	$-0.512308\pi$
$761$	−1623.10	−0.0773159	−0.0386579	+	0.999253i	$0.512308\pi$
$762$	25270.6	1.20139
$763$	0	0
$764$	605.417	0.0286691
$765$	1752.93	0.0828461
$766$	116.715	0.00550534
$767$	18438.1	0.868005
$768$	−2095.97	−0.0984787
$769$	−32253.8	−1.51249	−0.756244	−	0.654290i	$-0.772968\pi$
$769$	−32253.8	−1.51249	−0.756244	+	0.654290i	$0.772968\pi$
$770$	0	0
$771$	−17802.2	−0.831557
$772$	−1608.43	−0.0749851
$773$	19470.2	0.905946	0.452973	−	0.891524i	$-0.350363\pi$
$773$	19470.2	0.905946	0.452973	+	0.891524i	$0.350363\pi$
$774$	63.3120	0.00294019
$775$	12921.8	0.598921
$776$	−1869.19	−0.0864689
$777$	0	0
$778$	19035.9	0.877212
$779$	−6051.90	−0.278346
$780$	−1521.93	−0.0698640
$781$	7979.83	0.365609
$782$	−2166.06	−0.0990511
$783$	8885.18	0.405531
$784$	0	0
$785$	1925.62	0.0875522
$786$	32770.8	1.48715
$787$	9749.79	0.441604	0.220802	−	0.975319i	$-0.429132\pi$
$787$	9749.79	0.441604	0.220802	+	0.975319i	$0.429132\pi$
$788$	−2164.80	−0.0978651
$789$	−9067.25	−0.409129
$790$	1206.69	0.0543442
$791$	0	0
$792$	−7939.96	−0.356230
$793$	−4135.27	−0.185180
$794$	−6500.28	−0.290537
$795$	4895.63	0.218403
$796$	18107.8	0.806297
$797$	−21513.8	−0.956158	−0.478079	−	0.878317i	$-0.658667\pi$
$797$	−21513.8	−0.956158	−0.478079	+	0.878317i	$0.658667\pi$
$798$	0	0
$799$	−2122.82	−0.0939923
$800$	3972.33	0.175554
$801$	−1082.64	−0.0477568
$802$	3205.86	0.141151
$803$	−7477.53	−0.328613
$804$	1540.54	0.0675752
$805$	0	0
$806$	10404.3	0.454685
$807$	42242.2	1.84262
$808$	13014.4	0.566642
$809$	18997.9	0.825624	0.412812	−	0.910816i	$-0.364547\pi$
$809$	18997.9	0.825624	0.412812	+	0.910816i	$0.364547\pi$
$810$	385.442	0.0167198
$811$	−1464.38	−0.0634048	−0.0317024	−	0.999497i	$-0.510093\pi$
$811$	−1464.38	−0.0634048	−0.0317024	+	0.999497i	$0.510093\pi$
$812$	0	0
$813$	−19043.2	−0.821494
$814$	6539.81	0.281597
$815$	246.998	0.0106159
$816$	−6168.45	−0.264631
$817$	22.2899	0.000954500 0
$818$	−17297.6	−0.739358
$819$	0	0
$820$	798.574	0.0340090
$821$	−40491.2	−1.72126	−0.860628	−	0.509234i	$-0.829929\pi$
$821$	−40491.2	−1.72126	−0.860628	+	0.509234i	$0.829929\pi$
$822$	−35543.0	−1.50816
$823$	−8810.58	−0.373168	−0.186584	−	0.982439i	$-0.559742\pi$
$823$	−8810.58	−0.373168	−0.186584	+	0.982439i	$0.559742\pi$
$824$	−7076.04	−0.299157
$825$	25197.0	1.06333
$826$	0	0
$827$	17019.8	0.715644	0.357822	−	0.933790i	$-0.383519\pi$
$827$	17019.8	0.715644	0.357822	+	0.933790i	$0.383519\pi$
$828$	3683.04	0.154583
$829$	−27122.9	−1.13633	−0.568166	−	0.822914i	$-0.692347\pi$
$829$	−27122.9	−1.13633	−0.568166	+	0.822914i	$0.692347\pi$
$830$	−498.261	−0.0208372
$831$	−41932.4	−1.75044
$832$	3198.43	0.133276
$833$	0	0
$834$	25324.0	1.05143
$835$	−1125.25	−0.0466359
$836$	−2795.38	−0.115646
$837$	11107.5	0.458699
$838$	1840.59	0.0758738
$839$	2294.57	0.0944188	0.0472094	−	0.998885i	$-0.484967\pi$
$839$	2294.57	0.0944188	0.0472094	+	0.998885i	$0.484967\pi$
$840$	0	0
$841$	−17455.5	−0.715710
$842$	21355.8	0.874074
$843$	72017.2	2.94235
$844$	−4106.55	−0.167480
$845$	279.471	0.0113776
$846$	3609.52	0.146688
$847$	0	0
$848$	−10288.5	−0.416635
$849$	2172.33	0.0878141
$850$	11690.6	0.471746
$851$	−3033.56	−0.122197
$852$	−10541.1	−0.423866
$853$	−24642.9	−0.989165	−0.494582	−	0.869131i	$-0.664679\pi$
$853$	−24642.9	−0.989165	−0.494582	+	0.869131i	$0.664679\pi$
$854$	0	0
$855$	−1049.36	−0.0419734
$856$	693.614	0.0276954
$857$	−24067.9	−0.959327	−0.479663	−	0.877453i	$-0.659241\pi$
$857$	−24067.9	−0.959327	−0.479663	+	0.877453i	$0.659241\pi$
$858$	20288.1	0.807253
$859$	−16504.8	−0.655571	−0.327785	−	0.944752i	$-0.606302\pi$
$859$	−16504.8	−0.655571	−0.327785	+	0.944752i	$0.606302\pi$
$860$	−2.94125	−0.000116623 0
$861$	0	0
$862$	21325.6	0.842637
$863$	−15498.1	−0.611309	−0.305655	−	0.952142i	$-0.598875\pi$
$863$	−15498.1	−0.611309	−0.305655	+	0.952142i	$0.598875\pi$
$864$	3414.59	0.134452
$865$	1409.99	0.0554231
$866$	−1332.78	−0.0522975
$867$	22070.7	0.864546
$868$	0	0
$869$	−16085.7	−0.627928
$870$	−1267.90	−0.0494092
$871$	−2350.84	−0.0914526
$872$	160.924	0.00624949
$873$	9353.64	0.362626
$874$	1296.67	0.0501836
$875$	0	0
$876$	9877.62	0.380975
$877$	43574.1	1.67776	0.838878	−	0.544319i	$-0.183212\pi$
$877$	43574.1	1.67776	0.838878	+	0.544319i	$0.183212\pi$
$878$	−407.115	−0.0156486
$879$	−15370.7	−0.589807
$880$	368.863	0.0141299
$881$	−31820.6	−1.21687	−0.608435	−	0.793604i	$-0.708202\pi$
$881$	−31820.6	−1.21687	−0.608435	+	0.793604i	$0.708202\pi$
$882$	0	0
$883$	8773.52	0.334374	0.167187	−	0.985925i	$-0.446532\pi$
$883$	8773.52	0.334374	0.167187	+	0.985925i	$0.446532\pi$
$884$	9413.00	0.358137
$885$	−2808.91	−0.106690
$886$	31080.0	1.17850
$887$	−46161.4	−1.74740	−0.873702	−	0.486461i	$-0.838287\pi$
$887$	−46161.4	−1.74740	−0.873702	+	0.486461i	$0.838287\pi$
$888$	−8638.92	−0.326468
$889$	0	0
$890$	50.2957	0.00189429
$891$	−5138.12	−0.193191
$892$	−6360.65	−0.238756
$893$	1270.78	0.0476205
$894$	28792.2	1.07713
$895$	−1492.87	−0.0557553
$896$	0	0
$897$	−9410.84	−0.350300
$898$	7622.84	0.283271
$899$	8667.72	0.321562
$900$	−19878.0	−0.736223
$901$	−30279.0	−1.11958
$902$	−10645.4	−0.392962
$903$	0	0
$904$	−10068.5	−0.370437
$905$	1716.21	0.0630373
$906$	−17521.3	−0.642502
$907$	52766.7	1.93174	0.965870	−	0.259027i	$-0.0834019\pi$
$907$	52766.7	1.93174	0.965870	+	0.259027i	$0.0834019\pi$
$908$	−19316.5	−0.705993
$909$	−65125.9	−2.37634
$910$	0	0
$911$	−4410.66	−0.160408	−0.0802040	−	0.996778i	$-0.525557\pi$
$911$	−4410.66	−0.160408	−0.0802040	+	0.996778i	$0.525557\pi$
$912$	3692.62	0.134073
$913$	6642.05	0.240767
$914$	10782.5	0.390212
$915$	629.979	0.0227612
$916$	−23665.7	−0.853643
$917$	0	0
$918$	10049.2	0.361299
$919$	−30972.6	−1.11174	−0.555872	−	0.831268i	$-0.687616\pi$
$919$	−30972.6	−1.11174	−0.555872	+	0.831268i	$0.687616\pi$
$920$	−171.101	−0.00613155
$921$	−6226.73	−0.222777
$922$	−3025.68	−0.108075
$923$	16085.7	0.573637
$924$	0	0
$925$	16372.7	0.581980
$926$	−19878.9	−0.705465
$927$	35409.4	1.25458
$928$	2664.57	0.0942553
$929$	35284.1	1.24611	0.623054	−	0.782179i	$-0.285892\pi$
$929$	35284.1	1.24611	0.623054	+	0.782179i	$0.285892\pi$
$930$	−1585.02	−0.0558871
$931$	0	0
$932$	14020.6	0.492768
$933$	23413.7	0.821576
$934$	17028.6	0.596567
$935$	1085.57	0.0379698
$936$	−16005.3	−0.558921
$937$	−36839.6	−1.28441	−0.642207	−	0.766531i	$-0.721981\pi$
$937$	−36839.6	−1.28441	−0.642207	+	0.766531i	$0.721981\pi$
$938$	0	0
$939$	46000.5	1.59869
$940$	−167.685	−0.00581840
$941$	1752.33	0.0607058	0.0303529	−	0.999539i	$-0.490337\pi$
$941$	1752.33	0.0607058	0.0303529	+	0.999539i	$0.490337\pi$
$942$	33908.7	1.17283
$943$	4937.97	0.170522
$944$	5903.09	0.203527
$945$	0	0
$946$	39.2083	0.00134754
$947$	−46405.9	−1.59239	−0.796193	−	0.605043i	$-0.793156\pi$
$947$	−46405.9	−1.59239	−0.796193	+	0.605043i	$0.793156\pi$
$948$	21248.8	0.727983
$949$	−15073.2	−0.515590
$950$	−6998.36	−0.239007
$951$	18515.7	0.631349
$952$	0	0
$953$	−54948.0	−1.86772	−0.933861	−	0.357636i	$-0.883583\pi$
$953$	−54948.0	−1.86772	−0.933861	+	0.357636i	$0.883583\pi$
$954$	51484.7	1.74725
$955$	140.744	0.00476896
$956$	−4535.72	−0.153447
$957$	16901.8	0.570905
$958$	−17322.5	−0.584201
$959$	0	0
$960$	−487.258	−0.0163814
$961$	−18955.4	−0.636279
$962$	13182.9	0.441824
$963$	−3470.93	−0.116147
$964$	−9059.84	−0.302695
$965$	−373.917	−0.0124734
$966$	0	0
$967$	−22916.6	−0.762096	−0.381048	−	0.924555i	$-0.624437\pi$
$967$	−22916.6	−0.762096	−0.381048	+	0.924555i	$0.624437\pi$
$968$	5730.89	0.190287
$969$	10867.4	0.360281
$970$	−434.537	−0.0143837
$971$	−27574.4	−0.911334	−0.455667	−	0.890150i	$-0.650599\pi$
$971$	−27574.4	−0.911334	−0.455667	+	0.890150i	$0.650599\pi$
$972$	18311.6	0.604263
$973$	0	0
$974$	−41309.0	−1.35896
$975$	50792.1	1.66836
$976$	−1323.94	−0.0434203
$977$	−1829.39	−0.0599051	−0.0299525	−	0.999551i	$-0.509536\pi$
$977$	−1829.39	−0.0599051	−0.0299525	+	0.999551i	$0.509536\pi$
$978$	4349.44	0.142208
$979$	−670.465	−0.0218878
$980$	0	0
$981$	−805.282	−0.0262086
$982$	−12352.4	−0.401405
$983$	34523.6	1.12018	0.560088	−	0.828433i	$-0.310767\pi$
$983$	34523.6	1.12018	0.560088	+	0.828433i	$0.310767\pi$
$984$	14062.3	0.455578
$985$	−503.259	−0.0162794
$986$	7841.86	0.253282
$987$	0	0
$988$	−5634.91	−0.181448
$989$	−18.1872	−0.000584752 0
$990$	−1845.83	−0.0592570
$991$	−11285.5	−0.361751	−0.180875	−	0.983506i	$-0.557893\pi$
$991$	−11285.5	−0.361751	−0.180875	+	0.983506i	$0.557893\pi$
$992$	3331.02	0.106613
$993$	−34776.4	−1.11138
$994$	0	0
$995$	4209.59	0.134123
$996$	−8773.98	−0.279131
$997$	−40931.1	−1.30020	−0.650100	−	0.759849i	$-0.725273\pi$
$997$	−40931.1	−1.30020	−0.650100	+	0.759849i	$0.725273\pi$
$998$	38635.4	1.22543
$999$	14073.9	0.445724

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2254.4.a.z.1.1	✓	14
7.6	odd	2	inner	2254.4.a.z.1.14	yes	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2254.4.a.z.1.1	✓	14	1.1	even	1	trivial
2254.4.a.z.1.14	yes	14	7.6	odd	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:	\( N \)	\(=\)	\( 2254 = 2 \cdot 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2254.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} -8.18737 q^{3} +4.00000 q^{4} +0.929896 q^{5} +16.3747 q^{6} -8.00000 q^{8} +40.0330 q^{9} -1.85979 q^{10} +24.7919 q^{11} -32.7495 q^{12} +49.9754 q^{13} -7.61340 q^{15} +16.0000 q^{16} +47.0882 q^{17} -80.0660 q^{18} -28.1884 q^{19} +3.71959 q^{20} -49.5838 q^{22} +23.0000 q^{23} +65.4989 q^{24} -124.135 q^{25} -99.9508 q^{26} -106.706 q^{27} -83.2679 q^{29} +15.2268 q^{30} -104.094 q^{31} -32.0000 q^{32} -202.981 q^{33} -94.1763 q^{34} +160.132 q^{36} -131.894 q^{37} +56.3768 q^{38} -409.167 q^{39} -7.43917 q^{40} +214.694 q^{41} -0.790748 q^{43} +99.1676 q^{44} +37.2265 q^{45} -46.0000 q^{46} -45.0818 q^{47} -130.998 q^{48} +248.271 q^{50} -385.528 q^{51} +199.902 q^{52} -643.028 q^{53} +213.412 q^{54} +23.0539 q^{55} +230.789 q^{57} +166.536 q^{58} +368.943 q^{59} -30.4536 q^{60} -82.7460 q^{61} +208.189 q^{62} +64.0000 q^{64} +46.4719 q^{65} +405.961 q^{66} -47.0400 q^{67} +188.353 q^{68} -188.309 q^{69} +321.872 q^{71} -320.264 q^{72} -301.612 q^{73} +263.788 q^{74} +1016.34 q^{75} -112.754 q^{76} +818.334 q^{78} -648.828 q^{79} +14.8783 q^{80} -207.250 q^{81} -429.389 q^{82} +267.912 q^{83} +43.7871 q^{85} +1.58150 q^{86} +681.745 q^{87} -198.335 q^{88} -27.0437 q^{89} -74.4531 q^{90} +92.0000 q^{92} +852.259 q^{93} +90.1635 q^{94} -26.2123 q^{95} +261.996 q^{96} +233.648 q^{97} +992.495 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 28 q^{2} + 56 q^{4} - 112 q^{8} + 22 q^{9} - 92 q^{11} - 268 q^{15} + 224 q^{16} - 44 q^{18} + 184 q^{22} + 322 q^{23} + 130 q^{25} + 196 q^{29} + 536 q^{30} - 448 q^{32} + 88 q^{36} + 628 q^{37}+ \cdots + 1800 q^{99}+O(q^{100}) \) 14 * q - 28 * q^2 + 56 * q^4 - 112 * q^8 + 22 * q^9 - 92 * q^11 - 268 * q^15 + 224 * q^16 - 44 * q^18 + 184 * q^22 + 322 * q^23 + 130 * q^25 + 196 * q^29 + 536 * q^30 - 448 * q^32 + 88 * q^36 + 628 * q^37 + 388 * q^39 + 640 * q^43 - 368 * q^44 - 644 * q^46 - 260 * q^50 + 656 * q^51 - 1260 * q^53 + 380 * q^57 - 392 * q^58 - 1072 * q^60 + 896 * q^64 - 204 * q^65 - 1564 * q^67 - 900 * q^71 - 176 * q^72 - 1256 * q^74 - 776 * q^78 - 404 * q^79 - 4278 * q^81 + 1512 * q^85 - 1280 * q^86 + 736 * q^88 + 1288 * q^92 - 2712 * q^93 - 4752 * q^95 + 1800 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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