LMFDB - Embedded newform 275.4.b.b.199.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [275,4,Mod(199,275)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(275, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

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

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

chi := DirichletCharacter("275.199");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 275 = 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	275.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$16.2255252516$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{17})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9x^{2} + 16 $ x^4 + 9*x^2 + 16
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 55)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.1
Root			$-1.56155i$ of defining polynomial
Character	$\chi$	$=$	275.199
Dual form			275.4.b.b.199.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-5.56155i q^{2} +3.56155i q^{3} -22.9309 q^{4} +19.8078 q^{6} +6.05398i q^{7} +83.0388i q^{8} +14.3153 q^{9} -11.0000 q^{11} -81.6695i q^{12} +4.38447i q^{13} +33.6695 q^{14} +278.378 q^{16} -110.546i q^{17} -79.6155i q^{18} +94.2699 q^{19} -21.5616 q^{21} +61.1771i q^{22} -15.7538i q^{23} -295.747 q^{24} +24.3845 q^{26} +147.147i q^{27} -138.823i q^{28} +256.870 q^{29} -170.702 q^{31} -883.902i q^{32} -39.1771i q^{33} -614.810 q^{34} -328.263 q^{36} -190.853i q^{37} -524.287i q^{38} -15.6155 q^{39} +249.602 q^{41} +119.916i q^{42} -291.602i q^{43} +252.240 q^{44} -87.6155 q^{46} +182.155i q^{47} +991.457i q^{48} +306.349 q^{49} +393.717 q^{51} -100.540i q^{52} +289.902i q^{53} +818.365 q^{54} -502.715 q^{56} +335.747i q^{57} -1428.60i q^{58} -282.725 q^{59} +167.825 q^{61} +949.366i q^{62} +86.6647i q^{63} -2688.85 q^{64} -217.885 q^{66} -176.233i q^{67} +2534.93i q^{68} +56.1080 q^{69} +919.255 q^{71} +1188.73i q^{72} -154.570i q^{73} -1061.44 q^{74} -2161.69 q^{76} -66.5937i q^{77} +86.8466i q^{78} +882.017 q^{79} -137.557 q^{81} -1388.18i q^{82} -277.619i q^{83} +494.425 q^{84} -1621.76 q^{86} +914.857i q^{87} -913.427i q^{88} +977.147 q^{89} -26.5435 q^{91} +361.248i q^{92} -607.963i q^{93} +1013.07 q^{94} +3148.07 q^{96} -1102.94i q^{97} -1703.78i q^{98} -157.469 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 34 q^{4} + 38 q^{6} + 82 q^{9} - 44 q^{11} - 22 q^{14} + 594 q^{16} + 6 q^{19} - 78 q^{21} - 614 q^{24} + 180 q^{26} + 442 q^{29} + 282 q^{31} - 1346 q^{34} - 340 q^{36} + 20 q^{39} - 288 q^{41} + 374 q^{44}+ \cdots - 902 q^{99}+O(q^{100}) $ 4 * q - 34 * q^4 + 38 * q^6 + 82 * q^9 - 44 * q^11 - 22 * q^14 + 594 * q^16 + 6 * q^19 - 78 * q^21 - 614 * q^24 + 180 * q^26 + 442 * q^29 + 282 * q^31 - 1346 * q^34 - 340 * q^36 + 20 * q^39 - 288 * q^41 + 374 * q^44 - 268 * q^46 - 630 * q^49 + 742 * q^51 + 1550 * q^54 - 2250 * q^56 + 172 * q^59 - 310 * q^61 - 5618 * q^64 - 418 * q^66 + 76 * q^69 + 3174 * q^71 - 3182 * q^74 - 5406 * q^76 + 2588 * q^79 + 1132 * q^81 + 782 * q^84 - 2232 * q^86 + 3554 * q^89 + 2780 * q^91 + 1364 * q^94 + 6086 * q^96 - 902 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/275\mathbb{Z}\right)^\times$.

$n$	$101$	$177$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 5.56155i	− 1.96631i	−0.182785	−	0.983153i	$-0.558511\pi$
$2$	− 5.56155i	− 1.96631i	0.182785	−	0.983153i	$-0.441489\pi$
$3$	3.56155i	0.685421i	0.939441	+	0.342711i	$0.111345\pi$
$3$	3.56155i	0.685421i	−0.939441	+	0.342711i	$0.888655\pi$
$4$	−22.9309	−2.86636
$5$	0	0
$5$	0	0
$6$	19.8078	1.34775
$7$	6.05398i	0.326884i	0.986553	+	0.163442i	$0.0522597\pi$
$7$	6.05398i	0.326884i	−0.986553	+	0.163442i	$0.947740\pi$
$8$	83.0388i	3.66983i
$9$	14.3153	0.530198
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	− 81.6695i	− 1.96466i
$13$	4.38447i	0.0935411i	0.998906	+	0.0467705i	$0.0148930\pi$
$13$	4.38447i	0.0935411i	−0.998906	+	0.0467705i	$0.985107\pi$
$14$	33.6695	0.642754
$15$	0	0
$16$	278.378	4.34965
$17$	− 110.546i	− 1.57714i	−0.614943	−	0.788572i	$-0.710821\pi$
$17$	− 110.546i	− 1.57714i	0.614943	−	0.788572i	$-0.289179\pi$
$18$	− 79.6155i	− 1.04253i
$19$	94.2699	1.13826	0.569131	−	0.822247i	$-0.307280\pi$
$19$	94.2699	1.13826	0.569131	+	0.822247i	$0.307280\pi$
$20$	0	0
$21$	−21.5616	−0.224053
$22$	61.1771i	0.592864i
$23$	− 15.7538i	− 0.142821i	−0.997447	−	0.0714107i	$-0.977250\pi$
$23$	− 15.7538i	− 0.142821i	0.997447	−	0.0714107i	$-0.0227501\pi$
$24$	−295.747	−2.51538
$25$	0	0
$26$	24.3845	0.183930
$27$	147.147i	1.04883i
$28$	− 138.823i	− 0.936967i
$29$	256.870	1.64481	0.822407	−	0.568900i	$-0.192631\pi$
$29$	256.870	1.64481	0.822407	+	0.568900i	$0.192631\pi$
$30$	0	0
$31$	−170.702	−0.988998	−0.494499	−	0.869178i	$-0.664648\pi$
$31$	−170.702	−0.988998	−0.494499	+	0.869178i	$0.664648\pi$
$32$	− 883.902i	− 4.88292i
$33$	− 39.1771i	− 0.206662i
$34$	−614.810	−3.10115
$35$	0	0
$36$	−328.263	−1.51974
$37$	− 190.853i	− 0.848002i	−0.905662	−	0.424001i	$-0.860625\pi$
$37$	− 190.853i	− 0.848002i	0.905662	−	0.424001i	$-0.139375\pi$
$38$	− 524.287i	− 2.23817i
$39$	−15.6155	−0.0641150
$40$	0	0
$41$	249.602	0.950764	0.475382	−	0.879780i	$-0.342310\pi$
$41$	249.602	0.950764	0.475382	+	0.879780i	$0.342310\pi$
$42$	119.916i	0.440557i
$43$	− 291.602i	− 1.03416i	−0.855937	−	0.517081i	$-0.827019\pi$
$43$	− 291.602i	− 1.03416i	0.855937	−	0.517081i	$-0.172981\pi$
$44$	252.240	0.864240
$45$	0	0
$46$	−87.6155	−0.280831
$47$	182.155i	0.565321i	0.959220	+	0.282660i	$0.0912169\pi$
$47$	182.155i	0.565321i	−0.959220	+	0.282660i	$0.908783\pi$
$48$	991.457i	2.98134i
$49$	306.349	0.893147
$50$	0	0
$51$	393.717	1.08101
$52$	− 100.540i	− 0.268122i
$53$	289.902i	0.751343i	0.926753	+	0.375671i	$0.122588\pi$
$53$	289.902i	0.751343i	−0.926753	+	0.375671i	$0.877412\pi$
$54$	818.365	2.06232
$55$	0	0
$56$	−502.715	−1.19961
$57$	335.747i	0.780189i
$58$	− 1428.60i	− 3.23421i
$59$	−282.725	−0.623859	−0.311930	−	0.950105i	$-0.600975\pi$
$59$	−282.725	−0.623859	−0.311930	+	0.950105i	$0.600975\pi$
$60$	0	0
$61$	167.825	0.352258	0.176129	−	0.984367i	$-0.443642\pi$
$61$	167.825	0.352258	0.176129	+	0.984367i	$0.443642\pi$
$62$	949.366i	1.94467i
$63$	86.6647i	0.173313i
$64$	−2688.85	−5.25166
$65$	0	0
$66$	−217.885	−0.406361
$67$	− 176.233i	− 0.321347i	−0.987008	−	0.160674i	$-0.948633\pi$
$67$	− 176.233i	− 0.321347i	0.987008	−	0.160674i	$-0.0513667\pi$
$68$	2534.93i	4.52066i
$69$	56.1080	0.0978928
$70$	0	0
$71$	919.255	1.53656	0.768278	−	0.640116i	$-0.221114\pi$
$71$	919.255	1.53656	0.768278	+	0.640116i	$0.221114\pi$
$72$	1188.73i	1.94574i
$73$	− 154.570i	− 0.247823i	−0.992293	−	0.123911i	$-0.960456\pi$
$73$	− 154.570i	− 0.247823i	0.992293	−	0.123911i	$-0.0395438\pi$
$74$	−1061.44	−1.66743
$75$	0	0
$76$	−2161.69	−3.26267
$77$	− 66.5937i	− 0.0985592i
$78$	86.8466i	0.126070i
$79$	882.017	1.25614	0.628068	−	0.778159i	$-0.283846\pi$
$79$	882.017	1.25614	0.628068	+	0.778159i	$0.283846\pi$
$80$	0	0
$81$	−137.557	−0.188692
$82$	− 1388.18i	− 1.86949i
$83$	− 277.619i	− 0.367141i	−0.983007	−	0.183570i	$-0.941234\pi$
$83$	− 277.619i	− 0.367141i	0.983007	−	0.183570i	$-0.0587655\pi$
$84$	494.425	0.642217
$85$	0	0
$86$	−1621.76	−2.03348
$87$	914.857i	1.12739i
$88$	− 913.427i	− 1.10650i
$89$	977.147	1.16379	0.581895	−	0.813264i	$-0.302311\pi$
$89$	977.147	1.16379	0.581895	+	0.813264i	$0.302311\pi$
$90$	0	0
$91$	−26.5435	−0.0305771
$92$	361.248i	0.409377i
$93$	− 607.963i	− 0.677880i
$94$	1013.07	1.11159
$95$	0	0
$96$	3148.07	3.34685
$97$	− 1102.94i	− 1.15451i	−0.816565	−	0.577253i	$-0.804125\pi$
$97$	− 1102.94i	− 1.15451i	0.816565	−	0.577253i	$-0.195875\pi$
$98$	− 1703.78i	− 1.75620i
$99$	−157.469	−0.159861
$100$	0	0
$101$	484.314	0.477139	0.238570	−	0.971125i	$-0.423321\pi$
$101$	484.314	0.477139	0.238570	+	0.971125i	$0.423321\pi$
$102$	− 2189.68i	− 2.12559i
$103$	874.419i	0.836495i	0.908333	+	0.418248i	$0.137356\pi$
$103$	874.419i	0.836495i	−0.908333	+	0.418248i	$0.862644\pi$
$104$	−364.081	−0.343280
$105$	0	0
$106$	1612.31	1.47737
$107$	119.845i	0.108279i	0.998533	+	0.0541394i	$0.0172415\pi$
$107$	119.845i	0.108279i	−0.998533	+	0.0541394i	$0.982758\pi$
$108$	− 3374.20i	− 3.00632i
$109$	−414.621	−0.364344	−0.182172	−	0.983267i	$-0.558313\pi$
$109$	−414.621	−0.364344	−0.182172	+	0.983267i	$0.558313\pi$
$110$	0	0
$111$	679.734	0.581239
$112$	1685.29i	1.42183i
$113$	− 534.453i	− 0.444930i	−0.974941	−	0.222465i	$-0.928590\pi$
$113$	− 534.453i	− 0.444930i	0.974941	−	0.222465i	$-0.0714103\pi$
$114$	1867.28	1.53409
$115$	0	0
$116$	−5890.26	−4.71463
$117$	62.7652i	0.0495953i
$118$	1572.39i	1.22670i
$119$	669.245	0.515543
$120$	0	0
$121$	121.000	0.0909091
$122$	− 933.366i	− 0.692648i
$123$	888.972i	0.651674i
$124$	3914.34	2.83482
$125$	0	0
$126$	481.990	0.340787
$127$	640.121i	0.447256i	0.974675	+	0.223628i	$0.0717902\pi$
$127$	640.121i	0.447256i	−0.974675	+	0.223628i	$0.928210\pi$
$128$	7882.95i	5.44344i
$129$	1038.56	0.708836
$130$	0	0
$131$	1051.05	0.700999	0.350499	−	0.936563i	$-0.386012\pi$
$131$	1051.05	0.700999	0.350499	+	0.936563i	$0.386012\pi$
$132$	898.365i	0.592368i
$133$	570.708i	0.372080i
$134$	−980.129	−0.631867
$135$	0	0
$136$	9179.64	5.78785
$137$	1690.68i	1.05434i	0.849761	+	0.527169i	$0.176746\pi$
$137$	1690.68i	1.05434i	−0.849761	+	0.527169i	$0.823254\pi$
$138$	− 312.047i	− 0.192487i
$139$	−2789.43	−1.70213	−0.851067	−	0.525058i	$-0.824044\pi$
$139$	−2789.43	−1.70213	−0.851067	+	0.525058i	$0.824044\pi$
$140$	0	0
$141$	−648.756	−0.387483
$142$	− 5112.48i	− 3.02134i
$143$	− 48.2292i	− 0.0282037i
$144$	3985.07	2.30618
$145$	0	0
$146$	−859.650	−0.487295
$147$	1091.08i	0.612182i
$148$	4376.43i	2.43068i
$149$	−1090.62	−0.599647	−0.299823	−	0.953995i	$-0.596928\pi$
$149$	−1090.62	−0.599647	−0.299823	+	0.953995i	$0.596928\pi$
$150$	0	0
$151$	623.574	0.336064	0.168032	−	0.985782i	$-0.446259\pi$
$151$	623.574	0.336064	0.168032	+	0.985782i	$0.446259\pi$
$152$	7828.06i	4.17723i
$153$	− 1582.51i	− 0.836198i
$154$	−370.365	−0.193798
$155$	0	0
$156$	358.078	0.183777
$157$	− 2114.96i	− 1.07511i	−0.843228	−	0.537555i	$-0.819348\pi$
$157$	− 2114.96i	− 1.07511i	0.843228	−	0.537555i	$-0.180652\pi$
$158$	− 4905.38i	− 2.46995i
$159$	−1032.50	−0.514986
$160$	0	0
$161$	95.3730	0.0466860
$162$	765.029i	0.371027i
$163$	− 1153.53i	− 0.554301i	−0.960827	−	0.277151i	$-0.910610\pi$
$163$	− 1153.53i	− 0.554301i	0.960827	−	0.277151i	$-0.0893901\pi$
$164$	−5723.60	−2.72523
$165$	0	0
$166$	−1543.99	−0.721911
$167$	− 1100.93i	− 0.510137i	−0.966923	−	0.255068i	$-0.917902\pi$
$167$	− 1100.93i	− 0.510137i	0.966923	−	0.255068i	$-0.0820980\pi$
$168$	− 1790.45i	− 0.822238i
$169$	2177.78	0.991250
$170$	0	0
$171$	1349.51	0.603504
$172$	6686.69i	2.96428i
$173$	2369.25i	1.04122i	0.853795	+	0.520609i	$0.174295\pi$
$173$	2369.25i	1.04122i	−0.853795	+	0.520609i	$0.825705\pi$
$174$	5088.03	2.21679
$175$	0	0
$176$	−3062.16	−1.31147
$177$	− 1006.94i	− 0.427606i
$178$	− 5434.45i	− 2.28837i
$179$	−1226.77	−0.512250	−0.256125	−	0.966644i	$-0.582446\pi$
$179$	−1226.77	−0.512250	−0.256125	+	0.966644i	$0.582446\pi$
$180$	0	0
$181$	−439.606	−0.180528	−0.0902642	−	0.995918i	$-0.528771\pi$
$181$	−439.606	−0.180528	−0.0902642	+	0.995918i	$0.528771\pi$
$182$	147.623i	0.0601239i
$183$	597.717i	0.241445i
$184$	1308.18	0.524131
$185$	0	0
$186$	−3381.22	−1.33292
$187$	1216.01i	0.475527i
$188$	− 4176.98i	− 1.62041i
$189$	−890.823	−0.342846
$190$	0	0
$191$	−4968.96	−1.88241	−0.941207	−	0.337829i	$-0.890307\pi$
$191$	−4968.96	−1.88241	−0.941207	+	0.337829i	$0.890307\pi$
$192$	− 9576.47i	− 3.59960i
$193$	1362.22i	0.508054i	0.967197	+	0.254027i	$0.0817553\pi$
$193$	1362.22i	0.508054i	−0.967197	+	0.254027i	$0.918245\pi$
$194$	−6134.09	−2.27011
$195$	0	0
$196$	−7024.86	−2.56008
$197$	2195.91i	0.794174i	0.917781	+	0.397087i	$0.129979\pi$
$197$	2195.91i	0.794174i	−0.917781	+	0.397087i	$0.870021\pi$
$198$	875.771i	0.314335i
$199$	558.189	0.198839	0.0994194	−	0.995046i	$-0.468301\pi$
$199$	558.189	0.198839	0.0994194	+	0.995046i	$0.468301\pi$
$200$	0	0
$201$	627.663	0.220258
$202$	− 2693.54i	− 0.938202i
$203$	1555.09i	0.537663i
$204$	−9028.27	−3.09856
$205$	0	0
$206$	4863.12	1.64481
$207$	− 225.521i	− 0.0757236i
$208$	1220.54i	0.406871i
$209$	−1036.97	−0.343199
$210$	0	0
$211$	−3002.01	−0.979463	−0.489732	−	0.871873i	$-0.662905\pi$
$211$	−3002.01	−0.979463	−0.489732	+	0.871873i	$0.662905\pi$
$212$	− 6647.71i	− 2.15362i
$213$	3273.97i	1.05319i
$214$	666.523	0.212909
$215$	0	0
$216$	−12218.9	−3.84903
$217$	− 1033.42i	− 0.323287i
$218$	2305.94i	0.716412i
$219$	550.509	0.169863
$220$	0	0
$221$	484.688	0.147528
$222$	− 3780.38i	− 1.14289i
$223$	− 854.595i	− 0.256627i	−0.991734	−	0.128314i	$-0.959044\pi$
$223$	− 854.595i	− 0.256627i	0.991734	−	0.128314i	$-0.0409564\pi$
$224$	5351.12	1.59615
$225$	0	0
$226$	−2972.39	−0.874868
$227$	394.002i	0.115202i	0.998340	+	0.0576010i	$0.0183451\pi$
$227$	394.002i	0.115202i	−0.998340	+	0.0576010i	$0.981655\pi$
$228$	− 7698.97i	− 2.23630i
$229$	491.822	0.141924	0.0709618	−	0.997479i	$-0.477393\pi$
$229$	491.822	0.141924	0.0709618	+	0.997479i	$0.477393\pi$
$230$	0	0
$231$	237.177	0.0675546
$232$	21330.2i	6.03619i
$233$	6884.63i	1.93574i	0.251456	+	0.967869i	$0.419091\pi$
$233$	6884.63i	1.93574i	−0.251456	+	0.967869i	$0.580909\pi$
$234$	349.072	0.0975195
$235$	0	0
$236$	6483.14	1.78820
$237$	3141.35i	0.860982i
$238$	− 3722.04i	− 1.01371i
$239$	−3012.77	−0.815397	−0.407699	−	0.913117i	$-0.633669\pi$
$239$	−3012.77	−0.815397	−0.407699	+	0.913117i	$0.633669\pi$
$240$	0	0
$241$	3106.98	0.830448	0.415224	−	0.909719i	$-0.363703\pi$
$241$	3106.98	0.830448	0.415224	+	0.909719i	$0.363703\pi$
$242$	− 672.948i	− 0.178755i
$243$	3483.05i	0.919496i
$244$	−3848.37	−1.00970
$245$	0	0
$246$	4944.06	1.28139
$247$	413.324i	0.106474i
$248$	− 14174.9i	− 3.62946i
$249$	988.756	0.251646
$250$	0	0
$251$	−834.313	−0.209806	−0.104903	−	0.994482i	$-0.533453\pi$
$251$	−834.313	−0.209806	−0.104903	+	0.994482i	$0.533453\pi$
$252$	− 1987.30i	− 0.496778i
$253$	173.292i	0.0430623i
$254$	3560.07	0.879443
$255$	0	0
$256$	22330.6	5.45182
$257$	− 7536.63i	− 1.82927i	−0.404281	−	0.914635i	$-0.632478\pi$
$257$	− 7536.63i	− 1.82927i	0.404281	−	0.914635i	$-0.367522\pi$
$258$	− 5775.99i	− 1.39379i
$259$	1155.42	0.277198
$260$	0	0
$261$	3677.19	0.872077
$262$	− 5845.48i	− 1.37838i
$263$	6242.10i	1.46351i	0.681565	+	0.731757i	$0.261300\pi$
$263$	6242.10i	1.46351i	−0.681565	+	0.731757i	$0.738700\pi$
$264$	3253.22	0.758416
$265$	0	0
$266$	3174.02	0.731623
$267$	3480.16i	0.797687i
$268$	4041.17i	0.921097i
$269$	−1636.95	−0.371027	−0.185514	−	0.982642i	$-0.559395\pi$
$269$	−1636.95	−0.371027	−0.185514	+	0.982642i	$0.559395\pi$
$270$	0	0
$271$	787.212	0.176457	0.0882283	−	0.996100i	$-0.471880\pi$
$271$	787.212	0.176457	0.0882283	+	0.996100i	$0.471880\pi$
$272$	− 30773.7i	− 6.86003i
$273$	− 94.5360i	− 0.0209582i
$274$	9402.78	2.07315
$275$	0	0
$276$	−1286.60	−0.280596
$277$	1954.96i	0.424052i	0.977264	+	0.212026i	$0.0680061\pi$
$277$	1954.96i	0.424052i	−0.977264	+	0.212026i	$0.931994\pi$
$278$	15513.6i	3.34691i
$279$	−2443.65	−0.524364
$280$	0	0
$281$	−5097.58	−1.08219	−0.541097	−	0.840960i	$-0.681991\pi$
$281$	−5097.58	−1.08219	−0.541097	+	0.840960i	$0.681991\pi$
$282$	3608.09i	0.761910i
$283$	7187.71i	1.50977i	0.655857	+	0.754885i	$0.272307\pi$
$283$	7187.71i	1.50977i	−0.655857	+	0.754885i	$0.727693\pi$
$284$	−21079.3	−4.40432
$285$	0	0
$286$	−268.229	−0.0554571
$287$	1511.09i	0.310789i
$288$	− 12653.4i	− 2.58891i
$289$	−7307.51	−1.48738
$290$	0	0
$291$	3928.20	0.791323
$292$	3544.43i	0.710349i
$293$	− 6388.82i	− 1.27385i	−0.770925	−	0.636926i	$-0.780206\pi$
$293$	− 6388.82i	− 1.27385i	0.770925	−	0.636926i	$-0.219794\pi$
$294$	6068.10	1.20374
$295$	0	0
$296$	15848.2	3.11203
$297$	− 1618.61i	− 0.316234i
$298$	6065.56i	1.17909i
$299$	69.0720	0.0133597
$300$	0	0
$301$	1765.35	0.338051
$302$	− 3468.04i	− 0.660806i
$303$	1724.91i	0.327041i
$304$	26242.6	4.95105
$305$	0	0
$306$	−8801.21	−1.64422
$307$	4882.07i	0.907603i	0.891103	+	0.453802i	$0.149933\pi$
$307$	4882.07i	0.907603i	−0.891103	+	0.453802i	$0.850067\pi$
$308$	1527.05i	0.282506i
$309$	−3114.29	−0.573352
$310$	0	0
$311$	2846.01	0.518914	0.259457	−	0.965755i	$-0.416456\pi$
$311$	2846.01	0.518914	0.259457	+	0.965755i	$0.416456\pi$
$312$	− 1296.70i	− 0.235291i
$313$	− 8009.49i	− 1.44640i	−0.690639	−	0.723200i	$-0.742670\pi$
$313$	− 8009.49i	− 1.44640i	0.690639	−	0.723200i	$-0.257330\pi$
$314$	−11762.5	−2.11400
$315$	0	0
$316$	−20225.4	−3.60053
$317$	4668.89i	0.827227i	0.910453	+	0.413613i	$0.135733\pi$
$317$	4668.89i	0.827227i	−0.910453	+	0.413613i	$0.864267\pi$
$318$	5742.32i	1.01262i
$319$	−2825.57	−0.495930
$320$	0	0
$321$	−426.833	−0.0742165
$322$	− 530.422i	− 0.0917990i
$323$	− 10421.2i	− 1.79520i
$324$	3154.30	0.540860
$325$	0	0
$326$	−6415.39	−1.08993
$327$	− 1476.70i	− 0.249729i
$328$	20726.7i	3.48914i
$329$	−1102.76	−0.184794
$330$	0	0
$331$	2581.31	0.428645	0.214323	−	0.976763i	$-0.431246\pi$
$331$	2581.31	0.428645	0.214323	+	0.976763i	$0.431246\pi$
$332$	6366.05i	1.05236i
$333$	− 2732.13i	− 0.449609i
$334$	−6122.91	−1.00309
$335$	0	0
$336$	−6002.26	−0.974554
$337$	− 8152.45i	− 1.31778i	−0.752239	−	0.658890i	$-0.771026\pi$
$337$	− 8152.45i	− 1.31778i	0.752239	−	0.658890i	$-0.228974\pi$
$338$	− 12111.8i	− 1.94910i
$339$	1903.48	0.304964
$340$	0	0
$341$	1877.72	0.298194
$342$	− 7505.35i	− 1.18667i
$343$	3931.15i	0.618839i
$344$	24214.3	3.79520
$345$	0	0
$346$	13176.7	2.04735
$347$	− 3426.21i	− 0.530054i	−0.964241	−	0.265027i	$-0.914619\pi$
$347$	− 3426.21i	− 0.530054i	0.964241	−	0.265027i	$-0.0853808\pi$
$348$	− 20978.5i	− 3.23151i
$349$	−1334.33	−0.204656	−0.102328	−	0.994751i	$-0.532629\pi$
$349$	−1334.33	−0.204656	−0.102328	+	0.994751i	$0.532629\pi$
$350$	0	0
$351$	−645.161	−0.0981087
$352$	9722.93i	1.47225i
$353$	− 4406.21i	− 0.664360i	−0.943216	−	0.332180i	$-0.892216\pi$
$353$	− 4406.21i	− 0.664360i	0.943216	−	0.332180i	$-0.107784\pi$
$354$	−5600.16	−0.840805
$355$	0	0
$356$	−22406.8	−3.33584
$357$	2383.55i	0.353364i
$358$	6822.72i	1.00724i
$359$	8623.04	1.26771	0.633853	−	0.773453i	$-0.281472\pi$
$359$	8623.04	1.26771	0.633853	+	0.773453i	$0.281472\pi$
$360$	0	0
$361$	2027.81	0.295642
$362$	2444.89i	0.354974i
$363$	430.948i	0.0623110i
$364$	608.665	0.0876448
$365$	0	0
$366$	3324.23	0.474755
$367$	− 3585.58i	− 0.509989i	−0.966942	−	0.254995i	$-0.917926\pi$
$367$	− 3585.58i	− 0.509989i	0.966942	−	0.254995i	$-0.0820737\pi$
$368$	− 4385.51i	− 0.621224i
$369$	3573.14	0.504093
$370$	0	0
$371$	−1755.06	−0.245602
$372$	13941.1i	1.94305i
$373$	− 9855.90i	− 1.36815i	−0.729413	−	0.684074i	$-0.760207\pi$
$373$	− 9855.90i	− 1.36815i	0.729413	−	0.684074i	$-0.239793\pi$
$374$	6762.91	0.935031
$375$	0	0
$376$	−15126.0	−2.07463
$377$	1126.24i	0.153858i
$378$	4954.36i	0.674139i
$379$	10837.8	1.46887	0.734435	−	0.678679i	$-0.237447\pi$
$379$	10837.8	1.46887	0.734435	+	0.678679i	$0.237447\pi$
$380$	0	0
$381$	−2279.83	−0.306559
$382$	27635.1i	3.70140i
$383$	2025.55i	0.270237i	0.990829	+	0.135119i	$0.0431416\pi$
$383$	2025.55i	0.270237i	−0.990829	+	0.135119i	$0.956858\pi$
$384$	−28075.5	−3.73105
$385$	0	0
$386$	7576.04	0.998990
$387$	− 4174.39i	− 0.548310i
$388$	25291.5i	3.30923i
$389$	978.894	0.127588	0.0637942	−	0.997963i	$-0.479680\pi$
$389$	978.894	0.127588	0.0637942	+	0.997963i	$0.479680\pi$
$390$	0	0
$391$	−1741.52	−0.225250
$392$	25438.9i	3.27770i
$393$	3743.38i	0.480479i
$394$	12212.7	1.56159
$395$	0	0
$396$	3610.90	0.458218
$397$	− 5008.28i	− 0.633144i	−0.948568	−	0.316572i	$-0.897468\pi$
$397$	− 5008.28i	− 0.633144i	0.948568	−	0.316572i	$-0.102532\pi$
$398$	− 3104.39i	− 0.390978i
$399$	−2032.60	−0.255031
$400$	0	0
$401$	−15584.1	−1.94073	−0.970366	−	0.241639i	$-0.922315\pi$
$401$	−15584.1	−1.94073	−0.970366	+	0.241639i	$0.922315\pi$
$402$	− 3490.78i	− 0.433095i
$403$	− 748.437i	− 0.0925119i
$404$	−11105.7	−1.36765
$405$	0	0
$406$	8648.69	1.05721
$407$	2099.39i	0.255682i
$408$	32693.8i	3.96712i
$409$	−15106.6	−1.82634	−0.913171	−	0.407576i	$-0.866374\pi$
$409$	−15106.6	−1.82634	−0.913171	+	0.407576i	$0.866374\pi$
$410$	0	0
$411$	−6021.43	−0.722665
$412$	− 20051.2i	− 2.39770i
$413$	− 1711.61i	− 0.203930i
$414$	−1254.25	−0.148896
$415$	0	0
$416$	3875.45	0.456753
$417$	− 9934.71i	− 1.16668i
$418$	5767.16i	0.674834i
$419$	1518.17	0.177011	0.0885056	−	0.996076i	$-0.471791\pi$
$419$	1518.17	0.177011	0.0885056	+	0.996076i	$0.471791\pi$
$420$	0	0
$421$	4637.05	0.536808	0.268404	−	0.963306i	$-0.413504\pi$
$421$	4637.05	0.536808	0.268404	+	0.963306i	$0.413504\pi$
$422$	16695.8i	1.92592i
$423$	2607.62i	0.299732i
$424$	−24073.2	−2.75730
$425$	0	0
$426$	18208.4	2.07089
$427$	1016.01i	0.115148i
$428$	− 2748.14i	− 0.310366i
$429$	171.771	0.0193314
$430$	0	0
$431$	−11477.9	−1.28276	−0.641380	−	0.767223i	$-0.721638\pi$
$431$	−11477.9	−1.28276	−0.641380	+	0.767223i	$0.721638\pi$
$432$	40962.4i	4.56205i
$433$	− 10204.5i	− 1.13256i	−0.824213	−	0.566280i	$-0.808382\pi$
$433$	− 10204.5i	− 1.13256i	0.824213	−	0.566280i	$-0.191618\pi$
$434$	−5747.44	−0.635682
$435$	0	0
$436$	9507.62	1.04434
$437$	− 1485.11i	− 0.162568i
$438$	− 3061.69i	− 0.334002i
$439$	6919.06	0.752229	0.376115	−	0.926573i	$-0.377260\pi$
$439$	6919.06	0.752229	0.376115	+	0.926573i	$0.377260\pi$
$440$	0	0
$441$	4385.50	0.473545
$442$	− 2695.62i	− 0.290085i
$443$	2912.53i	0.312366i	0.987728	+	0.156183i	$0.0499190\pi$
$443$	2912.53i	0.312366i	−0.987728	+	0.156183i	$0.950081\pi$
$444$	−15586.9	−1.66604
$445$	0	0
$446$	−4752.87	−0.504608
$447$	− 3884.32i	− 0.411011i
$448$	− 16278.2i	− 1.71668i
$449$	1155.53	0.121454	0.0607270	−	0.998154i	$-0.480658\pi$
$449$	1155.53	0.121454	0.0607270	+	0.998154i	$0.480658\pi$
$450$	0	0
$451$	−2745.62	−0.286666
$452$	12255.5i	1.27533i
$453$	2220.89i	0.230346i
$454$	2191.26	0.226522
$455$	0	0
$456$	−27880.0	−2.86316
$457$	2745.62i	0.281039i	0.990078	+	0.140519i	$0.0448772\pi$
$457$	2745.62i	0.281039i	−0.990078	+	0.140519i	$0.955123\pi$
$458$	− 2735.29i	− 0.279065i
$459$	16266.5	1.65416
$460$	0	0
$461$	11224.1	1.13397	0.566984	−	0.823729i	$-0.308110\pi$
$461$	11224.1	1.13397	0.566984	+	0.823729i	$0.308110\pi$
$462$	− 1319.07i	− 0.132833i
$463$	− 15994.8i	− 1.60549i	−0.596325	−	0.802743i	$-0.703373\pi$
$463$	− 15994.8i	− 1.60549i	0.596325	−	0.802743i	$-0.296627\pi$
$464$	71507.0	7.15437
$465$	0	0
$466$	38289.2	3.80625
$467$	6674.60i	0.661378i	0.943740	+	0.330689i	$0.107281\pi$
$467$	6674.60i	0.661378i	−0.943740	+	0.330689i	$0.892719\pi$
$468$	− 1439.26i	− 0.142158i
$469$	1066.91	0.105043
$470$	0	0
$471$	7532.55	0.736904
$472$	− 23477.2i	− 2.28946i
$473$	3207.62i	0.311811i
$474$	17470.8	1.69295
$475$	0	0
$476$	−15346.4	−1.47773
$477$	4150.05i	0.398360i
$478$	16755.7i	1.60332i
$479$	11582.0	1.10479	0.552396	−	0.833582i	$-0.313714\pi$
$479$	11582.0	1.10479	0.552396	+	0.833582i	$0.313714\pi$
$480$	0	0
$481$	836.791	0.0793230
$482$	− 17279.6i	− 1.63292i
$483$	339.676i	0.0319996i
$484$	−2774.64	−0.260578
$485$	0	0
$486$	19371.2	1.80801
$487$	− 10618.7i	− 0.988047i	−0.869448	−	0.494024i	$-0.835526\pi$
$487$	− 10618.7i	− 0.988047i	0.869448	−	0.494024i	$-0.164474\pi$
$488$	13936.0i	1.29273i
$489$	4108.34	0.379930
$490$	0	0
$491$	−17948.0	−1.64966	−0.824829	−	0.565382i	$-0.808729\pi$
$491$	−17948.0	−1.64966	−0.824829	+	0.565382i	$0.808729\pi$
$492$	− 20384.9i	− 1.86793i
$493$	− 28396.1i	− 2.59411i
$494$	2298.72	0.209361
$495$	0	0
$496$	−47519.6	−4.30180
$497$	5565.15i	0.502275i
$498$	− 5499.02i	− 0.494813i
$499$	10409.6	0.933865	0.466932	−	0.884293i	$-0.345359\pi$
$499$	10409.6	0.933865	0.466932	+	0.884293i	$0.345359\pi$
$500$	0	0
$501$	3921.04	0.349659
$502$	4640.07i	0.412543i
$503$	− 7319.98i	− 0.648870i	−0.945908	−	0.324435i	$-0.894826\pi$
$503$	− 7319.98i	− 0.648870i	0.945908	−	0.324435i	$-0.105174\pi$
$504$	−7196.54	−0.636030
$505$	0	0
$506$	963.771	0.0846736
$507$	7756.27i	0.679424i
$508$	− 14678.5i	− 1.28200i
$509$	−7619.94	−0.663552	−0.331776	−	0.943358i	$-0.607648\pi$
$509$	−7619.94	−0.663552	−0.331776	+	0.943358i	$0.607648\pi$
$510$	0	0
$511$	935.763	0.0810093
$512$	− 61129.5i	− 5.27650i
$513$	13871.5i	1.19384i
$514$	−41915.4	−3.59690
$515$	0	0
$516$	−23815.0	−2.03178
$517$	− 2003.71i	− 0.170451i
$518$	− 6425.93i	− 0.545057i
$519$	−8438.20	−0.713672
$520$	0	0
$521$	−12413.4	−1.04384	−0.521921	−	0.852994i	$-0.674785\pi$
$521$	−12413.4	−1.04384	−0.521921	+	0.852994i	$0.674785\pi$
$522$	− 20450.9i	− 1.71477i
$523$	2524.30i	0.211051i	0.994417	+	0.105526i	$0.0336525\pi$
$523$	2524.30i	0.211051i	−0.994417	+	0.105526i	$0.966347\pi$
$524$	−24101.5	−2.00931
$525$	0	0
$526$	34715.8	2.87772
$527$	18870.5i	1.55979i
$528$	− 10906.0i	− 0.898909i
$529$	11918.8	0.979602
$530$	0	0
$531$	−4047.31	−0.330769
$532$	− 13086.8i	− 1.06651i
$533$	1094.37i	0.0889355i
$534$	19355.1	1.56850
$535$	0	0
$536$	14634.2	1.17929
$537$	− 4369.19i	− 0.351107i
$538$	9103.96i	0.729553i
$539$	−3369.84	−0.269294
$540$	0	0
$541$	10271.4	0.816269	0.408135	−	0.912922i	$-0.366179\pi$
$541$	10271.4	0.816269	0.408135	+	0.912922i	$0.366179\pi$
$542$	− 4378.12i	− 0.346968i
$543$	− 1565.68i	− 0.123738i
$544$	−97712.2	−7.70106
$545$	0	0
$546$	−525.767	−0.0412102
$547$	7810.11i	0.610487i	0.952274	+	0.305243i	$0.0987378\pi$
$547$	7810.11i	0.610487i	−0.952274	+	0.305243i	$0.901262\pi$
$548$	− 38768.7i	− 3.02211i
$549$	2402.47	0.186767
$550$	0	0
$551$	24215.1	1.87223
$552$	4659.14i	0.359250i
$553$	5339.71i	0.410610i
$554$	10872.6	0.833815
$555$	0	0
$556$	63964.1	4.87892
$557$	− 18348.8i	− 1.39580i	−0.716193	−	0.697902i	$-0.754117\pi$
$557$	− 18348.8i	− 1.39580i	0.716193	−	0.697902i	$-0.245883\pi$
$558$	13590.5i	1.03106i
$559$	1278.52	0.0967365
$560$	0	0
$561$	−4330.89	−0.325936
$562$	28350.5i	2.12792i
$563$	174.680i	0.0130761i	0.999979	+	0.00653807i	$0.00208115\pi$
$563$	174.680i	0.0130761i	−0.999979	+	0.00653807i	$0.997919\pi$
$564$	14876.5	1.11066
$565$	0	0
$566$	39974.8	2.96867
$567$	− 832.765i	− 0.0616805i
$568$	76333.8i	5.63890i
$569$	3208.08	0.236362	0.118181	−	0.992992i	$-0.462294\pi$
$569$	3208.08	0.236362	0.118181	+	0.992992i	$0.462294\pi$
$570$	0	0
$571$	−11660.4	−0.854592	−0.427296	−	0.904112i	$-0.640534\pi$
$571$	−11660.4	−0.854592	−0.427296	+	0.904112i	$0.640534\pi$
$572$	1105.94i	0.0808419i
$573$	− 17697.2i	− 1.29025i
$574$	8403.98	0.611107
$575$	0	0
$576$	−38491.8	−2.78442
$577$	− 12906.9i	− 0.931233i	−0.884987	−	0.465617i	$-0.845833\pi$
$577$	− 12906.9i	− 0.931233i	0.884987	−	0.465617i	$-0.154167\pi$
$578$	40641.1i	2.92465i
$579$	−4851.61	−0.348231
$580$	0	0
$581$	1680.70	0.120012
$582$	− 21846.9i	− 1.55598i
$583$	− 3188.93i	− 0.226538i
$584$	12835.3	0.909468
$585$	0	0
$586$	−35531.8	−2.50478
$587$	27427.0i	1.92850i	0.264986	+	0.964252i	$0.414633\pi$
$587$	27427.0i	1.92850i	−0.264986	+	0.964252i	$0.585367\pi$
$588$	− 25019.4i	− 1.75473i
$589$	−16092.0	−1.12574
$590$	0	0
$591$	−7820.86	−0.544344
$592$	− 53129.3i	− 3.68852i
$593$	− 5332.11i	− 0.369247i	−0.982809	−	0.184623i	$-0.940893\pi$
$593$	− 5332.11i	− 0.369247i	0.982809	−	0.184623i	$-0.0591066\pi$
$594$	−9002.01	−0.621813
$595$	0	0
$596$	25009.0	1.71880
$597$	1988.02i	0.136288i
$598$	− 384.148i	− 0.0262692i
$599$	22329.2	1.52312	0.761558	−	0.648097i	$-0.224435\pi$
$599$	22329.2	1.52312	0.761558	+	0.648097i	$0.224435\pi$
$600$	0	0
$601$	15511.8	1.05281	0.526405	−	0.850234i	$-0.323540\pi$
$601$	15511.8	1.05281	0.526405	+	0.850234i	$0.323540\pi$
$602$	− 9818.10i	− 0.664711i
$603$	− 2522.83i	− 0.170378i
$604$	−14299.1	−0.963281
$605$	0	0
$606$	9593.18	0.643063
$607$	7205.25i	0.481799i	0.970550	+	0.240900i	$0.0774424\pi$
$607$	7205.25i	0.481799i	−0.970550	+	0.240900i	$0.922558\pi$
$608$	− 83325.4i	− 5.55804i
$609$	−5538.52	−0.368526
$610$	0	0
$611$	−798.655	−0.0528807
$612$	36288.3i	2.39684i
$613$	2837.16i	0.186936i	0.995622	+	0.0934682i	$0.0297953\pi$
$613$	2837.16i	0.186936i	−0.995622	+	0.0934682i	$0.970205\pi$
$614$	27151.9	1.78463
$615$	0	0
$616$	5529.86	0.361696
$617$	− 7423.58i	− 0.484379i	−0.970229	−	0.242190i	$-0.922134\pi$
$617$	− 7423.58i	− 0.484379i	0.970229	−	0.242190i	$-0.0778656\pi$
$618$	17320.3i	1.12738i
$619$	−9747.62	−0.632940	−0.316470	−	0.948603i	$-0.602498\pi$
$619$	−9747.62	−0.632940	−0.316470	+	0.948603i	$0.602498\pi$
$620$	0	0
$621$	2318.12	0.149795
$622$	− 15828.2i	− 1.02034i
$623$	5915.62i	0.380424i
$624$	−4347.02	−0.278878
$625$	0	0
$626$	−44545.2	−2.84407
$627$	− 3693.22i	− 0.235236i
$628$	48497.9i	3.08165i
$629$	−21098.1	−1.33742
$630$	0	0
$631$	5914.75	0.373157	0.186579	−	0.982440i	$-0.440260\pi$
$631$	5914.75	0.373157	0.186579	+	0.982440i	$0.440260\pi$
$632$	73241.7i	4.60980i
$633$	− 10691.8i	− 0.671345i
$634$	25966.3	1.62658
$635$	0	0
$636$	23676.2	1.47614
$637$	1343.18i	0.0835459i
$638$	15714.6i	0.975150i
$639$	13159.4	0.814679
$640$	0	0
$641$	−25438.0	−1.56746	−0.783728	−	0.621104i	$-0.786684\pi$
$641$	−25438.0	−1.56746	−0.783728	+	0.621104i	$0.786684\pi$
$642$	2373.86i	0.145932i
$643$	− 769.253i	− 0.0471794i	−0.999722	−	0.0235897i	$-0.992490\pi$
$643$	− 769.253i	− 0.0471794i	0.999722	−	0.0235897i	$-0.00750954\pi$
$644$	−2186.99	−0.133819
$645$	0	0
$646$	−57958.0	−3.52992
$647$	25813.2i	1.56850i	0.620445	+	0.784250i	$0.286952\pi$
$647$	25813.2i	1.56850i	−0.620445	+	0.784250i	$0.713048\pi$
$648$	− 11422.6i	− 0.692469i
$649$	3109.98	0.188101
$650$	0	0
$651$	3680.59	0.221588
$652$	26451.3i	1.58883i
$653$	13138.6i	0.787372i	0.919245	+	0.393686i	$0.128800\pi$
$653$	13138.6i	0.787372i	−0.919245	+	0.393686i	$0.871200\pi$
$654$	−8212.72	−0.491044
$655$	0	0
$656$	69483.7	4.13549
$657$	− 2212.72i	− 0.131395i
$658$	6133.08i	0.363362i
$659$	19105.0	1.12933	0.564663	−	0.825322i	$-0.309006\pi$
$659$	19105.0	1.12933	0.564663	+	0.825322i	$0.309006\pi$
$660$	0	0
$661$	−31694.3	−1.86500	−0.932502	−	0.361166i	$-0.882379\pi$
$661$	−31694.3	−1.86500	−0.932502	+	0.361166i	$0.882379\pi$
$662$	− 14356.1i	− 0.842848i
$663$	1726.24i	0.101119i
$664$	23053.2	1.34734
$665$	0	0
$666$	−15194.9	−0.884069
$667$	− 4046.68i	− 0.234915i
$668$	25245.4i	1.46224i
$669$	3043.68	0.175898
$670$	0	0
$671$	−1846.07	−0.106210
$672$	19058.3i	1.09403i
$673$	23110.6i	1.32370i	0.749638	+	0.661848i	$0.230228\pi$
$673$	23110.6i	1.32370i	−0.749638	+	0.661848i	$0.769772\pi$
$674$	−45340.3	−2.59116
$675$	0	0
$676$	−49938.3	−2.84128
$677$	− 17052.4i	− 0.968062i	−0.875051	−	0.484031i	$-0.839172\pi$
$677$	− 17052.4i	− 0.968062i	0.875051	−	0.484031i	$-0.160828\pi$
$678$	− 10586.3i	− 0.599653i
$679$	6677.20	0.377390
$680$	0	0
$681$	−1403.26	−0.0789618
$682$	− 10443.0i	− 0.586341i
$683$	− 28542.8i	− 1.59907i	−0.600623	−	0.799533i	$-0.705081\pi$
$683$	− 28542.8i	− 1.59907i	0.600623	−	0.799533i	$-0.294919\pi$
$684$	−30945.3	−1.72986
$685$	0	0
$686$	21863.3	1.21683
$687$	1751.65i	0.0972774i
$688$	− 81175.6i	− 4.49824i
$689$	−1271.07	−0.0702814
$690$	0	0
$691$	6479.20	0.356701	0.178351	−	0.983967i	$-0.442924\pi$
$691$	6479.20	0.356701	0.178351	+	0.983967i	$0.442924\pi$
$692$	− 54328.9i	− 2.98450i
$693$	− 953.312i	− 0.0522559i
$694$	−19055.1	−1.04225
$695$	0	0
$696$	−75968.6	−4.13733
$697$	− 27592.6i	− 1.49949i
$698$	7420.94i	0.402417i
$699$	−24520.0	−1.32680
$700$	0	0
$701$	21118.6	1.13786	0.568929	−	0.822386i	$-0.307358\pi$
$701$	21118.6	1.13786	0.568929	+	0.822386i	$0.307358\pi$
$702$	3588.10i	0.192912i
$703$	− 17991.7i	− 0.965249i
$704$	29577.3	1.58343
$705$	0	0
$706$	−24505.4	−1.30633
$707$	2932.03i	0.155969i
$708$	23090.0i	1.22567i
$709$	−7072.53	−0.374632	−0.187316	−	0.982300i	$-0.559979\pi$
$709$	−7072.53	−0.374632	−0.187316	+	0.982300i	$0.559979\pi$
$710$	0	0
$711$	12626.4	0.666000
$712$	81141.1i	4.27092i
$713$	2689.20i	0.141250i
$714$	13256.3	0.694822
$715$	0	0
$716$	28130.8	1.46829
$717$	− 10730.1i	− 0.558890i
$718$	− 47957.5i	− 2.49270i
$719$	−22177.9	−1.15034	−0.575170	−	0.818034i	$-0.695064\pi$
$719$	−22177.9	−1.15034	−0.575170	+	0.818034i	$0.695064\pi$
$720$	0	0
$721$	−5293.71	−0.273437
$722$	− 11277.8i	− 0.581323i
$723$	11065.7i	0.569207i
$724$	10080.5	0.517459
$725$	0	0
$726$	2396.74	0.122523
$727$	− 17390.7i	− 0.887186i	−0.896228	−	0.443593i	$-0.853704\pi$
$727$	− 17390.7i	− 0.887186i	0.896228	−	0.443593i	$-0.146296\pi$
$728$	− 2204.14i	− 0.112213i
$729$	−16119.1	−0.818935
$730$	0	0
$731$	−32235.6	−1.63102
$732$	− 13706.2i	− 0.692069i
$733$	21877.0i	1.10238i	0.834379	+	0.551191i	$0.185826\pi$
$733$	21877.0i	1.10238i	−0.834379	+	0.551191i	$0.814174\pi$
$734$	−19941.4	−1.00279
$735$	0	0
$736$	−13924.8	−0.697385
$737$	1938.56i	0.0968899i
$738$	− 19872.2i	− 0.991201i
$739$	−14203.1	−0.706994	−0.353497	−	0.935436i	$-0.615008\pi$
$739$	−14203.1	−0.706994	−0.353497	+	0.935436i	$0.615008\pi$
$740$	0	0
$741$	−1472.07	−0.0729797
$742$	9760.87i	0.482928i
$743$	3933.68i	0.194230i	0.995273	+	0.0971148i	$0.0309614\pi$
$743$	3933.68i	0.194230i	−0.995273	+	0.0971148i	$0.969039\pi$
$744$	50484.5	2.48771
$745$	0	0
$746$	−54814.1	−2.69020
$747$	− 3974.21i	− 0.194657i
$748$	− 27884.2i	− 1.36303i
$749$	−725.537	−0.0353946
$750$	0	0
$751$	22554.3	1.09590	0.547949	−	0.836512i	$-0.315409\pi$
$751$	22554.3	1.09590	0.547949	+	0.836512i	$0.315409\pi$
$752$	50708.0i	2.45895i
$753$	− 2971.45i	− 0.143806i
$754$	6263.65	0.302531
$755$	0	0
$756$	20427.3	0.982719
$757$	11432.0i	0.548883i	0.961604	+	0.274441i	$0.0884929\pi$
$757$	11432.0i	0.548883i	−0.961604	+	0.274441i	$0.911507\pi$
$758$	− 60275.2i	− 2.88825i
$759$	−617.187	−0.0295158
$760$	0	0
$761$	−32660.1	−1.55575	−0.777877	−	0.628416i	$-0.783704\pi$
$761$	−32660.1	−1.55575	−0.777877	+	0.628416i	$0.783704\pi$
$762$	12679.4i	0.602789i
$763$	− 2510.11i	− 0.119098i
$764$	113943.	5.39568
$765$	0	0
$766$	11265.2	0.531369
$767$	− 1239.60i	− 0.0583565i
$768$	79531.8i	3.73679i
$769$	8569.93	0.401872	0.200936	−	0.979604i	$-0.435602\pi$
$769$	8569.93	0.401872	0.200936	+	0.979604i	$0.435602\pi$
$770$	0	0
$771$	26842.1	1.25382
$772$	− 31236.8i	− 1.45627i
$773$	29158.0i	1.35671i	0.734733	+	0.678357i	$0.237307\pi$
$773$	29158.0i	1.35671i	−0.734733	+	0.678357i	$0.762693\pi$
$774$	−23216.1	−1.07815
$775$	0	0
$776$	91587.2	4.23684
$777$	4115.09i	0.189998i
$778$	− 5444.17i	− 0.250878i
$779$	23530.0	1.08222
$780$	0	0
$781$	−10111.8	−0.463289
$782$	9685.58i	0.442910i
$783$	37797.6i	1.72513i
$784$	85280.9	3.88488
$785$	0	0
$786$	20819.0	0.944770
$787$	− 8501.30i	− 0.385055i	−0.981292	−	0.192528i	$-0.938332\pi$
$787$	− 8501.30i	− 0.385055i	0.981292	−	0.192528i	$-0.0616685\pi$
$788$	− 50354.2i	− 2.27639i
$789$	−22231.6	−1.00312
$790$	0	0
$791$	3235.56	0.145440
$792$	− 13076.0i	− 0.586662i
$793$	735.823i	0.0329506i
$794$	−27853.8	−1.24496
$795$	0	0
$796$	−12799.7	−0.569944
$797$	37459.5i	1.66485i	0.554139	+	0.832424i	$0.313047\pi$
$797$	37459.5i	1.66485i	−0.554139	+	0.832424i	$0.686953\pi$
$798$	11304.4i	0.501470i
$799$	20136.6	0.891592
$800$	0	0
$801$	13988.2	0.617039
$802$	86671.9i	3.81607i
$803$	1700.27i	0.0747214i
$804$	−14392.9	−0.631339
$805$	0	0
$806$	−4162.47	−0.181907
$807$	− 5830.07i	− 0.254310i
$808$	40216.9i	1.75102i
$809$	−18045.8	−0.784246	−0.392123	−	0.919913i	$-0.628259\pi$
$809$	−18045.8	−0.784246	−0.392123	+	0.919913i	$0.628259\pi$
$810$	0	0
$811$	914.961	0.0396161	0.0198080	−	0.999804i	$-0.493694\pi$
$811$	914.961	0.0396161	0.0198080	+	0.999804i	$0.493694\pi$
$812$	− 35659.5i	− 1.54114i
$813$	2803.70i	0.120947i
$814$	11675.8	0.502750
$815$	0	0
$816$	109602.	4.70201
$817$	− 27489.3i	− 1.17715i
$818$	84016.3i	3.59115i
$819$	−379.979	−0.0162119
$820$	0	0
$821$	15188.9	0.645672	0.322836	−	0.946455i	$-0.395364\pi$
$821$	15188.9	0.645672	0.322836	+	0.946455i	$0.395364\pi$
$822$	33488.5i	1.42098i
$823$	37930.3i	1.60652i	0.595628	+	0.803261i	$0.296903\pi$
$823$	37930.3i	1.60652i	−0.595628	+	0.803261i	$0.703097\pi$
$824$	−72610.7	−3.06980
$825$	0	0
$826$	−9519.22	−0.400988
$827$	29679.9i	1.24797i	0.781437	+	0.623985i	$0.214487\pi$
$827$	29679.9i	1.24797i	−0.781437	+	0.623985i	$0.785513\pi$
$828$	5171.39i	0.217051i
$829$	−29094.2	−1.21892	−0.609458	−	0.792818i	$-0.708613\pi$
$829$	−29094.2	−1.21892	−0.609458	+	0.792818i	$0.708613\pi$
$830$	0	0
$831$	−6962.70	−0.290654
$832$	− 11789.2i	− 0.491245i
$833$	− 33865.8i	− 1.40862i
$834$	−55252.4	−2.29405
$835$	0	0
$836$	23778.6	0.983732
$837$	− 25118.2i	− 1.03729i
$838$	− 8443.41i	− 0.348058i
$839$	−38791.2	−1.59621	−0.798105	−	0.602519i	$-0.794164\pi$
$839$	−38791.2	−1.59621	−0.798105	+	0.602519i	$0.794164\pi$
$840$	0	0
$841$	41593.3	1.70541
$842$	− 25789.2i	− 1.05553i
$843$	− 18155.3i	− 0.741758i
$844$	68838.7	2.80749
$845$	0	0
$846$	14502.4	0.589365
$847$	732.531i	0.0297167i
$848$	80702.4i	3.26808i
$849$	−25599.4	−1.03483
$850$	0	0
$851$	−3006.66	−0.121113
$852$	− 75075.1i	− 3.01881i
$853$	42933.7i	1.72335i	0.507458	+	0.861677i	$0.330585\pi$
$853$	42933.7i	1.72335i	−0.507458	+	0.861677i	$0.669415\pi$
$854$	5650.58	0.226415
$855$	0	0
$856$	−9951.76	−0.397365
$857$	− 2664.36i	− 0.106199i	−0.998589	−	0.0530997i	$-0.983090\pi$
$857$	− 2664.36i	− 0.106199i	0.998589	−	0.0530997i	$-0.0169101\pi$
$858$	− 955.312i	− 0.0380115i
$859$	25002.7	0.993111	0.496556	−	0.868005i	$-0.334598\pi$
$859$	25002.7	0.993111	0.496556	+	0.868005i	$0.334598\pi$
$860$	0	0
$861$	−5381.81	−0.213022
$862$	63834.8i	2.52230i
$863$	− 27509.8i	− 1.08510i	−0.840023	−	0.542551i	$-0.817459\pi$
$863$	− 27509.8i	− 1.08510i	0.840023	−	0.542551i	$-0.182541\pi$
$864$	130063.	5.12135
$865$	0	0
$866$	−56753.1	−2.22696
$867$	− 26026.1i	− 1.01948i
$868$	23697.3i	0.926658i
$869$	−9702.19	−0.378739
$870$	0	0
$871$	772.688	0.0300592
$872$	− 34429.6i	− 1.33708i
$873$	− 15789.0i	− 0.612117i
$874$	−8259.51	−0.319659
$875$	0	0
$876$	−12623.7	−0.486888
$877$	− 25993.8i	− 1.00085i	−0.865779	−	0.500426i	$-0.833177\pi$
$877$	− 25993.8i	− 1.00085i	0.865779	−	0.500426i	$-0.166823\pi$
$878$	− 38480.7i	− 1.47911i
$879$	22754.1	0.873126
$880$	0	0
$881$	−29528.7	−1.12922	−0.564612	−	0.825357i	$-0.690974\pi$
$881$	−29528.7	−1.12922	−0.564612	+	0.825357i	$0.690974\pi$
$882$	− 24390.2i	− 0.931133i
$883$	50497.5i	1.92455i	0.272081	+	0.962274i	$0.412288\pi$
$883$	50497.5i	1.92455i	−0.272081	+	0.962274i	$0.587712\pi$
$884$	−11114.3	−0.422867
$885$	0	0
$886$	16198.2	0.614208
$887$	36471.9i	1.38062i	0.723515	+	0.690309i	$0.242525\pi$
$887$	36471.9i	1.38062i	−0.723515	+	0.690309i	$0.757475\pi$
$888$	56444.3i	2.13305i
$889$	−3875.28	−0.146201
$890$	0	0
$891$	1513.12	0.0568929
$892$	19596.6i	0.735586i
$893$	17171.8i	0.643484i
$894$	−21602.8	−0.808173
$895$	0	0
$896$	−47723.2	−1.77937
$897$	246.004i	0.00915700i
$898$	− 6426.54i	− 0.238816i
$899$	−43848.2	−1.62672
$900$	0	0
$901$	32047.7	1.18498
$902$	15269.9i	0.563673i
$903$	6287.40i	0.231707i
$904$	44380.3	1.63282
$905$	0	0
$906$	12351.6	0.452930
$907$	15130.2i	0.553902i	0.960884	+	0.276951i	$0.0893240\pi$
$907$	15130.2i	0.553902i	−0.960884	+	0.276951i	$0.910676\pi$
$908$	− 9034.81i	− 0.330210i
$909$	6933.12	0.252978
$910$	0	0
$911$	13937.0	0.506864	0.253432	−	0.967353i	$-0.418441\pi$
$911$	13937.0	0.506864	0.253432	+	0.967353i	$0.418441\pi$
$912$	93464.6i	3.39355i
$913$	3053.81i	0.110697i
$914$	15269.9	0.552608
$915$	0	0
$916$	−11277.9	−0.406804
$917$	6363.04i	0.229145i
$918$	− 90467.3i	− 3.25258i
$919$	40897.5	1.46799	0.733996	−	0.679153i	$-0.237653\pi$
$919$	40897.5	1.46799	0.733996	+	0.679153i	$0.237653\pi$
$920$	0	0
$921$	−17387.7	−0.622091
$922$	− 62423.5i	− 2.22973i
$923$	4030.45i	0.143731i
$924$	−5438.68	−0.193636
$925$	0	0
$926$	−88955.7	−3.15688
$927$	12517.6i	0.443508i
$928$	− 227048.i	− 8.03149i
$929$	12154.1	0.429239	0.214620	−	0.976698i	$-0.431149\pi$
$929$	12154.1	0.429239	0.214620	+	0.976698i	$0.431149\pi$
$930$	0	0
$931$	28879.5	1.01664
$932$	− 157870.i	− 5.54852i
$933$	10136.2i	0.355675i
$934$	37121.1	1.30047
$935$	0	0
$936$	−5211.95	−0.182006
$937$	− 15754.1i	− 0.549267i	−0.961549	−	0.274634i	$-0.911443\pi$
$937$	− 15754.1i	− 0.549267i	0.961549	−	0.274634i	$-0.0885566\pi$
$938$	− 5933.67i	− 0.206547i
$939$	28526.2	0.991393
$940$	0	0
$941$	−4217.53	−0.146108	−0.0730539	−	0.997328i	$-0.523275\pi$
$941$	−4217.53	−0.146108	−0.0730539	+	0.997328i	$0.523275\pi$
$942$	− 41892.7i	− 1.44898i
$943$	− 3932.18i	− 0.135789i
$944$	−78704.5	−2.71357
$945$	0	0
$946$	17839.4	0.613116
$947$	49839.0i	1.71019i	0.518471	+	0.855095i	$0.326502\pi$
$947$	49839.0i	1.71019i	−0.518471	+	0.855095i	$0.673498\pi$
$948$	− 72033.9i	− 2.46788i
$949$	677.708	0.0231816
$950$	0	0
$951$	−16628.5	−0.566999
$952$	55573.3i	1.89196i
$953$	12845.7i	0.436635i	0.975878	+	0.218318i	$0.0700569\pi$
$953$	12845.7i	0.436635i	−0.975878	+	0.218318i	$0.929943\pi$
$954$	23080.7	0.783298
$955$	0	0
$956$	69085.4	2.33722
$957$	− 10063.4i	− 0.339921i
$958$	− 64413.9i	− 2.17236i
$959$	−10235.3	−0.344646
$960$	0	0
$961$	−651.937	−0.0218837
$962$	− 4653.86i	− 0.155973i
$963$	1715.62i	0.0574092i
$964$	−71245.7	−2.38036
$965$	0	0
$966$	1889.13	0.0629210
$967$	− 38829.6i	− 1.29129i	−0.763638	−	0.645645i	$-0.776589\pi$
$967$	− 38829.6i	− 1.29129i	0.763638	−	0.645645i	$-0.223411\pi$
$968$	10047.7i	0.333621i
$969$	37115.6	1.23047
$970$	0	0
$971$	−11438.8	−0.378053	−0.189026	−	0.981972i	$-0.560533\pi$
$971$	−11438.8	−0.378053	−0.189026	+	0.981972i	$0.560533\pi$
$972$	− 79869.3i	− 2.63561i
$973$	− 16887.1i	− 0.556400i
$974$	−59056.4	−1.94280
$975$	0	0
$976$	46718.7	1.53220
$977$	− 12084.1i	− 0.395706i	−0.980232	−	0.197853i	$-0.936603\pi$
$977$	− 12084.1i	− 0.395706i	0.980232	−	0.197853i	$-0.0633969\pi$
$978$	− 22848.8i	− 0.747058i
$979$	−10748.6	−0.350896
$980$	0	0
$981$	−5935.44	−0.193174
$982$	99818.8i	3.24373i
$983$	23502.2i	0.762569i	0.924458	+	0.381284i	$0.124518\pi$
$983$	23502.2i	0.762569i	−0.924458	+	0.381284i	$0.875482\pi$
$984$	−73819.1	−2.39153
$985$	0	0
$986$	−157926.	−5.10081
$987$	− 3927.55i	− 0.126662i
$988$	− 9477.87i	− 0.305194i
$989$	−4593.84	−0.147700
$990$	0	0
$991$	−18664.1	−0.598268	−0.299134	−	0.954211i	$-0.596698\pi$
$991$	−18664.1	−0.598268	−0.299134	+	0.954211i	$0.596698\pi$
$992$	150884.i	4.82919i
$993$	9193.47i	0.293803i
$994$	30950.9	0.987627
$995$	0	0
$996$	−22673.0	−0.721308
$997$	24528.4i	0.779158i	0.920993	+	0.389579i	$0.127380\pi$
$997$	24528.4i	0.779158i	−0.920993	+	0.389579i	$0.872620\pi$
$998$	− 57893.7i	− 1.83626i
$999$	28083.4	0.889410

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	275.4.b.b.199.1		4
5.2	odd	4		275.4.a.c.1.2		2
5.3	odd	4		55.4.a.b.1.1	✓	2
5.4	even	2	inner	275.4.b.b.199.4		4
15.2	even	4		2475.4.a.l.1.1		2
15.8	even	4		495.4.a.e.1.2		2
20.3	even	4		880.4.a.r.1.2		2
55.43	even	4		605.4.a.g.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.4.a.b.1.1	✓	2	5.3	odd	4
275.4.a.c.1.2		2	5.2	odd	4
275.4.b.b.199.1		4	1.1	even	1	trivial
275.4.b.b.199.4		4	5.4	even	2	inner
495.4.a.e.1.2		2	15.8	even	4
605.4.a.g.1.2		2	55.43	even	4
880.4.a.r.1.2		2	20.3	even	4
2475.4.a.l.1.1		2	15.2	even	4

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:	\( N \)	\(=\)	\( 275 = 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	275.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-5.56155i q^{2} +3.56155i q^{3} -22.9309 q^{4} +19.8078 q^{6} +6.05398i q^{7} +83.0388i q^{8} +14.3153 q^{9} -11.0000 q^{11} -81.6695i q^{12} +4.38447i q^{13} +33.6695 q^{14} +278.378 q^{16} -110.546i q^{17} -79.6155i q^{18} +94.2699 q^{19} -21.5616 q^{21} +61.1771i q^{22} -15.7538i q^{23} -295.747 q^{24} +24.3845 q^{26} +147.147i q^{27} -138.823i q^{28} +256.870 q^{29} -170.702 q^{31} -883.902i q^{32} -39.1771i q^{33} -614.810 q^{34} -328.263 q^{36} -190.853i q^{37} -524.287i q^{38} -15.6155 q^{39} +249.602 q^{41} +119.916i q^{42} -291.602i q^{43} +252.240 q^{44} -87.6155 q^{46} +182.155i q^{47} +991.457i q^{48} +306.349 q^{49} +393.717 q^{51} -100.540i q^{52} +289.902i q^{53} +818.365 q^{54} -502.715 q^{56} +335.747i q^{57} -1428.60i q^{58} -282.725 q^{59} +167.825 q^{61} +949.366i q^{62} +86.6647i q^{63} -2688.85 q^{64} -217.885 q^{66} -176.233i q^{67} +2534.93i q^{68} +56.1080 q^{69} +919.255 q^{71} +1188.73i q^{72} -154.570i q^{73} -1061.44 q^{74} -2161.69 q^{76} -66.5937i q^{77} +86.8466i q^{78} +882.017 q^{79} -137.557 q^{81} -1388.18i q^{82} -277.619i q^{83} +494.425 q^{84} -1621.76 q^{86} +914.857i q^{87} -913.427i q^{88} +977.147 q^{89} -26.5435 q^{91} +361.248i q^{92} -607.963i q^{93} +1013.07 q^{94} +3148.07 q^{96} -1102.94i q^{97} -1703.78i q^{98} -157.469 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 34 q^{4} + 38 q^{6} + 82 q^{9} - 44 q^{11} - 22 q^{14} + 594 q^{16} + 6 q^{19} - 78 q^{21} - 614 q^{24} + 180 q^{26} + 442 q^{29} + 282 q^{31} - 1346 q^{34} - 340 q^{36} + 20 q^{39} - 288 q^{41} + 374 q^{44}+ \cdots - 902 q^{99}+O(q^{100}) \) 4 * q - 34 * q^4 + 38 * q^6 + 82 * q^9 - 44 * q^11 - 22 * q^14 + 594 * q^16 + 6 * q^19 - 78 * q^21 - 614 * q^24 + 180 * q^26 + 442 * q^29 + 282 * q^31 - 1346 * q^34 - 340 * q^36 + 20 * q^39 - 288 * q^41 + 374 * q^44 - 268 * q^46 - 630 * q^49 + 742 * q^51 + 1550 * q^54 - 2250 * q^56 + 172 * q^59 - 310 * q^61 - 5618 * q^64 - 418 * q^66 + 76 * q^69 + 3174 * q^71 - 3182 * q^74 - 5406 * q^76 + 2588 * q^79 + 1132 * q^81 + 782 * q^84 - 2232 * q^86 + 3554 * q^89 + 2780 * q^91 + 1364 * q^94 + 6086 * q^96 - 902 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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