LMFDB - Embedded newform 675.4.b.m.649.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [675,4,Mod(649,675)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(675, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("675.649");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 675 = 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	675.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:	$39.8262892539$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.12559936.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 6x^{3} + 49x^{2} - 42x + 18 $ x^6 - 6x^3 + 49x^2 - 42*x + 18
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 135)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.6
Root			$-2.05655 + 2.05655i$ of defining polynomial
Character	$\chi$	$=$	675.649
Dual form			675.4.b.m.649.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.45876i q^{2} -21.7980 q^{4} +11.8065i q^{7} -75.3201i q^{8} +O(q^{10})$	$q+5.45876i q^{2} -21.7980 q^{4} +11.8065i q^{7} -75.3201i q^{8} +56.2376 q^{11} +34.5961i q^{13} -64.4489 q^{14} +236.770 q^{16} +39.2675i q^{17} +146.561 q^{19} +306.987i q^{22} +23.5777i q^{23} -188.851 q^{26} -257.359i q^{28} +161.003 q^{29} -29.5465 q^{31} +689.908i q^{32} -214.352 q^{34} +217.688i q^{37} +800.039i q^{38} -142.290 q^{41} -468.030i q^{43} -1225.87 q^{44} -128.705 q^{46} +394.318i q^{47} +203.606 q^{49} -754.126i q^{52} +134.780i q^{53} +889.268 q^{56} +878.875i q^{58} -131.195 q^{59} +259.801 q^{61} -161.287i q^{62} -1871.88 q^{64} -445.244i q^{67} -855.954i q^{68} -560.841 q^{71} -88.6681i q^{73} -1188.30 q^{74} -3194.74 q^{76} +663.970i q^{77} -450.342 q^{79} -776.726i q^{82} -284.295i q^{83} +2554.86 q^{86} -4235.82i q^{88} +625.305 q^{89} -408.459 q^{91} -513.948i q^{92} -2152.49 q^{94} +193.261i q^{97} +1111.44i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 34 q^{4}+O(q^{10}) $ 6 * q - 34 * q^4	$ 6 q - 34 q^{4} - 10 q^{11} + 120 q^{14} + 322 q^{16} + 100 q^{19} - 370 q^{26} + 230 q^{29} - 230 q^{31} - 826 q^{34} - 1160 q^{41} - 2830 q^{44} - 570 q^{46} - 1154 q^{49} + 4380 q^{56} + 760 q^{59} - 304 q^{61} - 5874 q^{64} - 80 q^{71} - 5440 q^{74} - 6552 q^{76} - 2026 q^{79} + 3110 q^{86} + 2040 q^{89} - 1264 q^{91} - 7666 q^{94}+O(q^{100}) $ 6 * q - 34 * q^4 - 10 * q^11 + 120 * q^14 + 322 * q^16 + 100 * q^19 - 370 * q^26 + 230 * q^29 - 230 * q^31 - 826 * q^34 - 1160 * q^41 - 2830 * q^44 - 570 * q^46 - 1154 * q^49 + 4380 * q^56 + 760 * q^59 - 304 * q^61 - 5874 * q^64 - 80 * q^71 - 5440 * q^74 - 6552 * q^76 - 2026 * q^79 + 3110 * q^86 + 2040 * q^89 - 1264 * q^91 - 7666 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/675\mathbb{Z}\right)^\times$.

$n$	$326$	$352$
$\chi(n)$	$1$	$-1$

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 $

$2$	5.45876i	1.92996i	0.262320	+	0.964981i	$0.415513\pi$
$2$	5.45876i	1.92996i	−0.262320	+	0.964981i	$0.584487\pi$
$3$	0	0
$3$	0	0
$4$	−21.7980	−2.72475
$5$	0	0
$5$	0	0
$6$	0	0
$7$	11.8065i	0.637492i	0.947840	+	0.318746i	$0.103262\pi$
$7$	11.8065i	0.637492i	−0.947840	+	0.318746i	$0.896738\pi$
$8$	− 75.3201i	− 3.32871i
$9$	0	0
$10$	0	0
$11$	56.2376	1.54148	0.770740	−	0.637150i	$-0.219887\pi$
$11$	56.2376	1.54148	0.770740	+	0.637150i	$0.219887\pi$
$12$	0	0
$13$	34.5961i	0.738094i	0.929411	+	0.369047i	$0.120316\pi$
$13$	34.5961i	0.738094i	−0.929411	+	0.369047i	$0.879684\pi$
$14$	−64.4489	−1.23034
$15$	0	0
$16$	236.770	3.69953
$17$	39.2675i	0.560221i	0.959968	+	0.280111i	$0.0903712\pi$
$17$	39.2675i	0.560221i	−0.959968	+	0.280111i	$0.909629\pi$
$18$	0	0
$19$	146.561	1.76965	0.884825	−	0.465924i	$-0.154278\pi$
$19$	146.561	1.76965	0.884825	+	0.465924i	$0.154278\pi$
$20$	0	0
$21$	0	0
$22$	306.987i	2.97500i
$23$	23.5777i	0.213752i	0.994272	+	0.106876i	$0.0340848\pi$
$23$	23.5777i	0.213752i	−0.994272	+	0.106876i	$0.965915\pi$
$24$	0	0
$25$	0	0
$26$	−188.851	−1.42449
$27$	0	0
$28$	− 257.359i	− 1.73701i
$29$	161.003	1.03095	0.515473	−	0.856906i	$-0.327616\pi$
$29$	161.003	1.03095	0.515473	+	0.856906i	$0.327616\pi$
$30$	0	0
$31$	−29.5465	−0.171184	−0.0855921	−	0.996330i	$-0.527278\pi$
$31$	−29.5465	−0.171184	−0.0855921	+	0.996330i	$0.527278\pi$
$32$	689.908i	3.81124i
$33$	0	0
$34$	−214.352	−1.08121
$35$	0	0
$36$	0	0
$37$	217.688i	0.967233i	0.875280	+	0.483617i	$0.160677\pi$
$37$	217.688i	0.967233i	−0.875280	+	0.483617i	$0.839323\pi$
$38$	800.039i	3.41536i
$39$	0	0
$40$	0	0
$41$	−142.290	−0.541999	−0.270999	−	0.962580i	$-0.587354\pi$
$41$	−142.290	−0.541999	−0.270999	+	0.962580i	$0.587354\pi$
$42$	0	0
$43$	− 468.030i	− 1.65986i	−0.557869	−	0.829929i	$-0.688381\pi$
$43$	− 468.030i	− 1.65986i	0.557869	−	0.829929i	$-0.311619\pi$
$44$	−1225.87	−4.20015
$45$	0	0
$46$	−128.705	−0.412533
$47$	394.318i	1.22377i	0.790946	+	0.611886i	$0.209589\pi$
$47$	394.318i	1.22377i	−0.790946	+	0.611886i	$0.790411\pi$
$48$	0	0
$49$	203.606	0.593604
$50$	0	0
$51$	0	0
$52$	− 754.126i	− 2.01112i
$53$	134.780i	0.349311i	0.984630	+	0.174655i	$0.0558811\pi$
$53$	134.780i	0.349311i	−0.984630	+	0.174655i	$0.944119\pi$
$54$	0	0
$55$	0	0
$56$	889.268	2.12203
$57$	0	0
$58$	878.875i	1.98969i
$59$	−131.195	−0.289495	−0.144747	−	0.989469i	$-0.546237\pi$
$59$	−131.195	−0.289495	−0.144747	+	0.989469i	$0.546237\pi$
$60$	0	0
$61$	259.801	0.545313	0.272657	−	0.962111i	$-0.412098\pi$
$61$	259.801	0.545313	0.272657	+	0.962111i	$0.412098\pi$
$62$	− 161.287i	− 0.330379i
$63$	0	0
$64$	−1871.88	−3.65602
$65$	0	0
$66$	0	0
$67$	− 445.244i	− 0.811869i	−0.913902	−	0.405935i	$-0.866946\pi$
$67$	− 445.244i	− 0.811869i	0.913902	−	0.405935i	$-0.133054\pi$
$68$	− 855.954i	− 1.52647i
$69$	0	0
$70$	0	0
$71$	−560.841	−0.937459	−0.468729	−	0.883342i	$-0.655288\pi$
$71$	−560.841	−0.937459	−0.468729	+	0.883342i	$0.655288\pi$
$72$	0	0
$73$	− 88.6681i	− 0.142162i	−0.997471	−	0.0710809i	$-0.977355\pi$
$73$	− 88.6681i	− 0.142162i	0.997471	−	0.0710809i	$-0.0226448\pi$
$74$	−1188.30	−1.86672
$75$	0	0
$76$	−3194.74	−4.82186
$77$	663.970i	0.982681i
$78$	0	0
$79$	−450.342	−0.641360	−0.320680	−	0.947188i	$-0.603911\pi$
$79$	−450.342	−0.641360	−0.320680	+	0.947188i	$0.603911\pi$
$80$	0	0
$81$	0	0
$82$	− 776.726i	− 1.04604i
$83$	− 284.295i	− 0.375969i	−0.982172	−	0.187985i	$-0.939804\pi$
$83$	− 284.295i	− 0.375969i	0.982172	−	0.187985i	$-0.0601955\pi$
$84$	0	0
$85$	0	0
$86$	2554.86	3.20346
$87$	0	0
$88$	− 4235.82i	− 5.13114i
$89$	625.305	0.744744	0.372372	−	0.928083i	$-0.378545\pi$
$89$	625.305	0.744744	0.372372	+	0.928083i	$0.378545\pi$
$90$	0	0
$91$	−408.459	−0.470529
$92$	− 513.948i	− 0.582421i
$93$	0	0
$94$	−2152.49	−2.36183
$95$	0	0
$96$	0	0
$97$	193.261i	0.202296i	0.994871	+	0.101148i	$0.0322516\pi$
$97$	193.261i	0.202296i	−0.994871	+	0.101148i	$0.967748\pi$
$98$	1111.44i	1.14563i
$99$	0	0
$100$	0	0
$101$	−1374.86	−1.35449	−0.677245	−	0.735758i	$-0.736826\pi$
$101$	−1374.86	−1.35449	−0.677245	+	0.735758i	$0.736826\pi$
$102$	0	0
$103$	2029.60i	1.94158i	0.239935	+	0.970789i	$0.422874\pi$
$103$	2029.60i	1.94158i	−0.239935	+	0.970789i	$0.577126\pi$
$104$	2605.78	2.45690
$105$	0	0
$106$	−735.732	−0.674156
$107$	− 823.062i	− 0.743630i	−0.928307	−	0.371815i	$-0.878736\pi$
$107$	− 823.062i	− 0.743630i	0.928307	−	0.371815i	$-0.121264\pi$
$108$	0	0
$109$	829.868	0.729238	0.364619	−	0.931157i	$-0.381199\pi$
$109$	829.868	0.729238	0.364619	+	0.931157i	$0.381199\pi$
$110$	0	0
$111$	0	0
$112$	2795.43i	2.35842i
$113$	1503.37i	1.25155i	0.780005	+	0.625773i	$0.215216\pi$
$113$	1503.37i	1.25155i	−0.780005	+	0.625773i	$0.784784\pi$
$114$	0	0
$115$	0	0
$116$	−3509.54	−2.80908
$117$	0	0
$118$	− 716.164i	− 0.558714i
$119$	−463.612	−0.357137
$120$	0	0
$121$	1831.67	1.37616
$122$	1418.19i	1.05243i
$123$	0	0
$124$	644.056	0.466435
$125$	0	0
$126$	0	0
$127$	576.348i	0.402698i	0.979520	+	0.201349i	$0.0645325\pi$
$127$	576.348i	0.402698i	−0.979520	+	0.201349i	$0.935468\pi$
$128$	− 4698.89i	− 3.24474i
$129$	0	0
$130$	0	0
$131$	−2390.04	−1.59403	−0.797017	−	0.603957i	$-0.793590\pi$
$131$	−2390.04	−1.59403	−0.797017	+	0.603957i	$0.793590\pi$
$132$	0	0
$133$	1730.37i	1.12814i
$134$	2430.48	1.56688
$135$	0	0
$136$	2957.63	1.86481
$137$	1002.46i	0.625152i	0.949893	+	0.312576i	$0.101192\pi$
$137$	1002.46i	0.625152i	−0.949893	+	0.312576i	$0.898808\pi$
$138$	0	0
$139$	−131.817	−0.0804356	−0.0402178	−	0.999191i	$-0.512805\pi$
$139$	−131.817	−0.0804356	−0.0402178	+	0.999191i	$0.512805\pi$
$140$	0	0
$141$	0	0
$142$	− 3061.50i	− 1.80926i
$143$	1945.60i	1.13776i
$144$	0	0
$145$	0	0
$146$	484.017	0.274367
$147$	0	0
$148$	− 4745.16i	− 2.63547i
$149$	1019.49	0.560533	0.280267	−	0.959922i	$-0.409577\pi$
$149$	1019.49	0.560533	0.280267	+	0.959922i	$0.409577\pi$
$150$	0	0
$151$	2822.38	1.52107	0.760537	−	0.649295i	$-0.224936\pi$
$151$	2822.38	1.52107	0.760537	+	0.649295i	$0.224936\pi$
$152$	− 11039.0i	− 5.89065i
$153$	0	0
$154$	−3624.45	−1.89654
$155$	0	0
$156$	0	0
$157$	− 476.499i	− 0.242222i	−0.992639	−	0.121111i	$-0.961354\pi$
$157$	− 476.499i	− 0.242222i	0.992639	−	0.121111i	$-0.0386456\pi$
$158$	− 2458.31i	− 1.23780i
$159$	0	0
$160$	0	0
$161$	−278.371	−0.136265
$162$	0	0
$163$	− 2242.26i	− 1.07747i	−0.842476	−	0.538734i	$-0.818903\pi$
$163$	− 2242.26i	− 1.07747i	0.842476	−	0.538734i	$-0.181097\pi$
$164$	3101.64	1.47681
$165$	0	0
$166$	1551.90	0.725607
$167$	95.0390i	0.0440380i	0.999758	+	0.0220190i	$0.00700943\pi$
$167$	95.0390i	0.0440380i	−0.999758	+	0.0220190i	$0.992991\pi$
$168$	0	0
$169$	1000.11	0.455217
$170$	0	0
$171$	0	0
$172$	10202.1i	4.52270i
$173$	2133.76i	0.937727i	0.883271	+	0.468864i	$0.155336\pi$
$173$	2133.76i	0.937727i	−0.883271	+	0.468864i	$0.844664\pi$
$174$	0	0
$175$	0	0
$176$	13315.4	5.70275
$177$	0	0
$178$	3413.39i	1.43733i
$179$	−1704.68	−0.711808	−0.355904	−	0.934523i	$-0.615827\pi$
$179$	−1704.68	−0.711808	−0.355904	+	0.934523i	$0.615827\pi$
$180$	0	0
$181$	−1360.98	−0.558902	−0.279451	−	0.960160i	$-0.590152\pi$
$181$	−1360.98	−0.558902	−0.279451	+	0.960160i	$0.590152\pi$
$182$	− 2229.68i	− 0.908103i
$183$	0	0
$184$	1775.88	0.711518
$185$	0	0
$186$	0	0
$187$	2208.31i	0.863570i
$188$	− 8595.36i	− 3.33447i
$189$	0	0
$190$	0	0
$191$	1096.84	0.415522	0.207761	−	0.978180i	$-0.433382\pi$
$191$	1096.84	0.415522	0.207761	+	0.978180i	$0.433382\pi$
$192$	0	0
$193$	− 2867.27i	− 1.06938i	−0.845048	−	0.534691i	$-0.820428\pi$
$193$	− 2867.27i	− 1.06938i	0.845048	−	0.534691i	$-0.179572\pi$
$194$	−1054.97	−0.390424
$195$	0	0
$196$	−4438.21	−1.61742
$197$	724.139i	0.261892i	0.991389	+	0.130946i	$0.0418015\pi$
$197$	724.139i	0.261892i	−0.991389	+	0.130946i	$0.958199\pi$
$198$	0	0
$199$	1693.65	0.603315	0.301658	−	0.953416i	$-0.402460\pi$
$199$	1693.65	0.603315	0.301658	+	0.953416i	$0.402460\pi$
$200$	0	0
$201$	0	0
$202$	− 7505.02i	− 2.61411i
$203$	1900.88i	0.657220i
$204$	0	0
$205$	0	0
$206$	−11079.1	−3.74717
$207$	0	0
$208$	8191.30i	2.73060i
$209$	8242.22	2.72788
$210$	0	0
$211$	947.452	0.309124	0.154562	−	0.987983i	$-0.450603\pi$
$211$	947.452	0.309124	0.154562	+	0.987983i	$0.450603\pi$
$212$	− 2937.94i	− 0.951785i
$213$	0	0
$214$	4492.89	1.43518
$215$	0	0
$216$	0	0
$217$	− 348.841i	− 0.109129i
$218$	4530.05i	1.40740i
$219$	0	0
$220$	0	0
$221$	−1358.50	−0.413496
$222$	0	0
$223$	111.866i	0.0335923i	0.999859	+	0.0167961i	$0.00534663\pi$
$223$	111.866i	0.0335923i	−0.999859	+	0.0167961i	$0.994653\pi$
$224$	−8145.41	−2.42964
$225$	0	0
$226$	−8206.51	−2.41544
$227$	1200.70i	0.351072i	0.984473	+	0.175536i	$0.0561658\pi$
$227$	1200.70i	0.351072i	−0.984473	+	0.175536i	$0.943834\pi$
$228$	0	0
$229$	−822.380	−0.237312	−0.118656	−	0.992935i	$-0.537859\pi$
$229$	−822.380	−0.237312	−0.118656	+	0.992935i	$0.537859\pi$
$230$	0	0
$231$	0	0
$232$	− 12126.7i	− 3.43172i
$233$	5329.21i	1.49840i	0.662341	+	0.749202i	$0.269563\pi$
$233$	5329.21i	1.49840i	−0.662341	+	0.749202i	$0.730437\pi$
$234$	0	0
$235$	0	0
$236$	2859.80	0.788802
$237$	0	0
$238$	− 2530.75i	− 0.689260i
$239$	−7085.61	−1.91770	−0.958850	−	0.283914i	$-0.908367\pi$
$239$	−7085.61	−1.91770	−0.958850	+	0.283914i	$0.908367\pi$
$240$	0	0
$241$	−6560.09	−1.75341	−0.876707	−	0.481025i	$-0.840265\pi$
$241$	−6560.09	−1.75341	−0.876707	+	0.481025i	$0.840265\pi$
$242$	9998.63i	2.65593i
$243$	0	0
$244$	−5663.15	−1.48584
$245$	0	0
$246$	0	0
$247$	5070.42i	1.30617i
$248$	2225.45i	0.569822i
$249$	0	0
$250$	0	0
$251$	−714.222	−0.179607	−0.0898033	−	0.995960i	$-0.528624\pi$
$251$	−714.222	−0.179607	−0.0898033	+	0.995960i	$0.528624\pi$
$252$	0	0
$253$	1325.95i	0.329494i
$254$	−3146.14	−0.777191
$255$	0	0
$256$	10675.0	2.60621
$257$	4396.59i	1.06713i	0.845760	+	0.533563i	$0.179147\pi$
$257$	4396.59i	1.06713i	−0.845760	+	0.533563i	$0.820853\pi$
$258$	0	0
$259$	−2570.13	−0.616604
$260$	0	0
$261$	0	0
$262$	− 13046.6i	− 3.07643i
$263$	7550.31i	1.77024i	0.465367	+	0.885118i	$0.345922\pi$
$263$	7550.31i	1.77024i	−0.465367	+	0.885118i	$0.654078\pi$
$264$	0	0
$265$	0	0
$266$	−9445.68	−2.17726
$267$	0	0
$268$	9705.45i	2.21214i
$269$	−5536.86	−1.25497	−0.627487	−	0.778627i	$-0.715917\pi$
$269$	−5536.86	−1.25497	−0.627487	+	0.778627i	$0.715917\pi$
$270$	0	0
$271$	3058.25	0.685518	0.342759	−	0.939423i	$-0.388639\pi$
$271$	3058.25	0.685518	0.342759	+	0.939423i	$0.388639\pi$
$272$	9297.36i	2.07256i
$273$	0	0
$274$	−5472.18	−1.20652
$275$	0	0
$276$	0	0
$277$	4070.19i	0.882865i	0.897294	+	0.441433i	$0.145530\pi$
$277$	4070.19i	0.882865i	−0.897294	+	0.441433i	$0.854470\pi$
$278$	− 719.556i	− 0.155238i
$279$	0	0
$280$	0	0
$281$	−7446.19	−1.58079	−0.790396	−	0.612597i	$-0.790125\pi$
$281$	−7446.19	−1.58079	−0.790396	+	0.612597i	$0.790125\pi$
$282$	0	0
$283$	− 774.651i	− 0.162715i	−0.996685	−	0.0813573i	$-0.974075\pi$
$283$	− 774.651i	− 0.162715i	0.996685	−	0.0813573i	$-0.0259255\pi$
$284$	12225.2	2.55434
$285$	0	0
$286$	−10620.6	−2.19583
$287$	− 1679.95i	− 0.345520i
$288$	0	0
$289$	3371.06	0.686152
$290$	0	0
$291$	0	0
$292$	1932.79i	0.387356i
$293$	− 6749.23i	− 1.34571i	−0.739772	−	0.672857i	$-0.765067\pi$
$293$	− 6749.23i	− 1.34571i	0.739772	−	0.672857i	$-0.234933\pi$
$294$	0	0
$295$	0	0
$296$	16396.3	3.21964
$297$	0	0
$298$	5565.12i	1.08181i
$299$	−815.696	−0.157769
$300$	0	0
$301$	5525.80	1.05815
$302$	15406.7i	2.93561i
$303$	0	0
$304$	34701.2	6.54687
$305$	0	0
$306$	0	0
$307$	2204.39i	0.409808i	0.978782	+	0.204904i	$0.0656883\pi$
$307$	2204.39i	0.409808i	−0.978782	+	0.204904i	$0.934312\pi$
$308$	− 14473.2i	− 2.67756i
$309$	0	0
$310$	0	0
$311$	6032.98	1.10000	0.549998	−	0.835166i	$-0.314628\pi$
$311$	6032.98	1.10000	0.549998	+	0.835166i	$0.314628\pi$
$312$	0	0
$313$	− 5772.96i	− 1.04251i	−0.853400	−	0.521257i	$-0.825463\pi$
$313$	− 5772.96i	− 1.04251i	0.853400	−	0.521257i	$-0.174537\pi$
$314$	2601.09	0.467479
$315$	0	0
$316$	9816.57	1.74755
$317$	− 3302.07i	− 0.585057i	−0.956257	−	0.292528i	$-0.905503\pi$
$317$	− 3302.07i	− 0.585057i	0.956257	−	0.292528i	$-0.0944966\pi$
$318$	0	0
$319$	9054.41	1.58918
$320$	0	0
$321$	0	0
$322$	− 1519.56i	− 0.262986i
$323$	5755.07i	0.991395i
$324$	0	0
$325$	0	0
$326$	12239.9	2.07947
$327$	0	0
$328$	10717.3i	1.80416i
$329$	−4655.53	−0.780144
$330$	0	0
$331$	8053.15	1.33728	0.668642	−	0.743584i	$-0.266876\pi$
$331$	8053.15	1.33728	0.668642	+	0.743584i	$0.266876\pi$
$332$	6197.08i	1.02442i
$333$	0	0
$334$	−518.795	−0.0849916
$335$	0	0
$336$	0	0
$337$	− 3460.33i	− 0.559335i	−0.960097	−	0.279668i	$-0.909776\pi$
$337$	− 3460.33i	− 0.559335i	0.960097	−	0.279668i	$-0.0902242\pi$
$338$	5459.37i	0.878552i
$339$	0	0
$340$	0	0
$341$	−1661.62	−0.263877
$342$	0	0
$343$	6453.51i	1.01591i
$344$	−35252.0	−5.52518
$345$	0	0
$346$	−11647.7	−1.80978
$347$	− 9328.27i	− 1.44314i	−0.692344	−	0.721568i	$-0.743422\pi$
$347$	− 9328.27i	− 1.44314i	0.692344	−	0.721568i	$-0.256578\pi$
$348$	0	0
$349$	−8899.42	−1.36497	−0.682486	−	0.730899i	$-0.739101\pi$
$349$	−8899.42	−1.36497	−0.682486	+	0.730899i	$0.739101\pi$
$350$	0	0
$351$	0	0
$352$	38798.8i	5.87495i
$353$	− 3722.45i	− 0.561264i	−0.959816	−	0.280632i	$-0.909456\pi$
$353$	− 3722.45i	− 0.561264i	0.959816	−	0.280632i	$-0.0905440\pi$
$354$	0	0
$355$	0	0
$356$	−13630.4	−2.02924
$357$	0	0
$358$	− 9305.42i	− 1.37376i
$359$	11029.3	1.62145	0.810727	−	0.585425i	$-0.199072\pi$
$359$	11029.3	1.62145	0.810727	+	0.585425i	$0.199072\pi$
$360$	0	0
$361$	14621.1	2.13166
$362$	− 7429.29i	− 1.07866i
$363$	0	0
$364$	8903.60	1.28208
$365$	0	0
$366$	0	0
$367$	4853.11i	0.690274i	0.938552	+	0.345137i	$0.112167\pi$
$367$	4853.11i	0.690274i	−0.938552	+	0.345137i	$0.887833\pi$
$368$	5582.49i	0.790781i
$369$	0	0
$370$	0	0
$371$	−1591.28	−0.222683
$372$	0	0
$373$	− 12373.8i	− 1.71767i	−0.512254	−	0.858834i	$-0.671189\pi$
$373$	− 12373.8i	− 1.71767i	0.512254	−	0.858834i	$-0.328811\pi$
$374$	−12054.6	−1.66666
$375$	0	0
$376$	29700.1	4.07358
$377$	5570.06i	0.760935i
$378$	0	0
$379$	−11150.6	−1.51127	−0.755634	−	0.654994i	$-0.772671\pi$
$379$	−11150.6	−1.51127	−0.755634	+	0.654994i	$0.772671\pi$
$380$	0	0
$381$	0	0
$382$	5987.39i	0.801941i
$383$	− 2199.59i	− 0.293457i	−0.989177	−	0.146728i	$-0.953126\pi$
$383$	− 2199.59i	− 0.293457i	0.989177	−	0.146728i	$-0.0468743\pi$
$384$	0	0
$385$	0	0
$386$	15651.7	2.06386
$387$	0	0
$388$	− 4212.72i	− 0.551207i
$389$	9218.92	1.20159	0.600794	−	0.799404i	$-0.294851\pi$
$389$	9218.92	1.20159	0.600794	+	0.799404i	$0.294851\pi$
$390$	0	0
$391$	−925.838	−0.119748
$392$	− 15335.6i	− 1.97594i
$393$	0	0
$394$	−3952.90	−0.505442
$395$	0	0
$396$	0	0
$397$	− 1119.36i	− 0.141509i	−0.997494	−	0.0707544i	$-0.977459\pi$
$397$	− 1119.36i	− 0.141509i	0.997494	−	0.0707544i	$-0.0225407\pi$
$398$	9245.23i	1.16438i
$399$	0	0
$400$	0	0
$401$	−12296.9	−1.53137	−0.765683	−	0.643218i	$-0.777599\pi$
$401$	−12296.9	−1.53137	−0.765683	+	0.643218i	$0.777599\pi$
$402$	0	0
$403$	− 1022.19i	− 0.126350i
$404$	29969.2	3.69065
$405$	0	0
$406$	−10376.4	−1.26841
$407$	12242.2i	1.49097i
$408$	0	0
$409$	−2500.22	−0.302269	−0.151134	−	0.988513i	$-0.548293\pi$
$409$	−2500.22	−0.302269	−0.151134	+	0.988513i	$0.548293\pi$
$410$	0	0
$411$	0	0
$412$	− 44241.3i	− 5.29032i
$413$	− 1548.96i	− 0.184551i
$414$	0	0
$415$	0	0
$416$	−23868.1	−2.81305
$417$	0	0
$418$	44992.3i	5.26470i
$419$	−8332.97	−0.971581	−0.485790	−	0.874075i	$-0.661468\pi$
$419$	−8332.97	−0.971581	−0.485790	+	0.874075i	$0.661468\pi$
$420$	0	0
$421$	11374.2	1.31673	0.658367	−	0.752697i	$-0.271248\pi$
$421$	11374.2	1.31673	0.658367	+	0.752697i	$0.271248\pi$
$422$	5171.91i	0.596598i
$423$	0	0
$424$	10151.6	1.16275
$425$	0	0
$426$	0	0
$427$	3067.35i	0.347633i
$428$	17941.1i	2.02621i
$429$	0	0
$430$	0	0
$431$	10030.2	1.12097	0.560484	−	0.828165i	$-0.310615\pi$
$431$	10030.2	1.12097	0.560484	+	0.828165i	$0.310615\pi$
$432$	0	0
$433$	7609.38i	0.844535i	0.906471	+	0.422267i	$0.138766\pi$
$433$	7609.38i	0.844535i	−0.906471	+	0.422267i	$0.861234\pi$
$434$	1904.24	0.210614
$435$	0	0
$436$	−18089.5	−1.98699
$437$	3455.57i	0.378266i
$438$	0	0
$439$	12370.5	1.34490	0.672450	−	0.740143i	$-0.265242\pi$
$439$	12370.5	1.34490	0.672450	+	0.740143i	$0.265242\pi$
$440$	0	0
$441$	0	0
$442$	− 7415.72i	− 0.798032i
$443$	12084.4i	1.29605i	0.761621	+	0.648023i	$0.224404\pi$
$443$	12084.4i	1.29605i	−0.761621	+	0.648023i	$0.775596\pi$
$444$	0	0
$445$	0	0
$446$	−610.647	−0.0648318
$447$	0	0
$448$	− 22100.4i	− 2.33068i
$449$	−625.550	−0.0657495	−0.0328747	−	0.999459i	$-0.510466\pi$
$449$	−625.550	−0.0657495	−0.0328747	+	0.999459i	$0.510466\pi$
$450$	0	0
$451$	−8002.04	−0.835480
$452$	− 32770.4i	− 3.41016i
$453$	0	0
$454$	−6554.33	−0.677555
$455$	0	0
$456$	0	0
$457$	− 1811.22i	− 0.185395i	−0.995694	−	0.0926974i	$-0.970451\pi$
$457$	− 1811.22i	− 0.185395i	0.995694	−	0.0926974i	$-0.0295489\pi$
$458$	− 4489.17i	− 0.458003i
$459$	0	0
$460$	0	0
$461$	11625.0	1.17447	0.587233	−	0.809418i	$-0.300217\pi$
$461$	11625.0	1.17447	0.587233	+	0.809418i	$0.300217\pi$
$462$	0	0
$463$	7291.88i	0.731928i	0.930629	+	0.365964i	$0.119261\pi$
$463$	7291.88i	0.731928i	−0.930629	+	0.365964i	$0.880739\pi$
$464$	38120.6	3.81402
$465$	0	0
$466$	−29090.9	−2.89186
$467$	− 11637.6i	− 1.15316i	−0.817042	−	0.576579i	$-0.804387\pi$
$467$	− 11637.6i	− 1.15316i	0.817042	−	0.576579i	$-0.195613\pi$
$468$	0	0
$469$	5256.78	0.517560
$470$	0	0
$471$	0	0
$472$	9881.66i	0.963644i
$473$	− 26320.9i	− 2.55864i
$474$	0	0
$475$	0	0
$476$	10105.8	0.973109
$477$	0	0
$478$	− 38678.6i	− 3.70109i
$479$	12041.4	1.14862	0.574309	−	0.818639i	$-0.305271\pi$
$479$	12041.4	1.14862	0.574309	+	0.818639i	$0.305271\pi$
$480$	0	0
$481$	−7531.14	−0.713909
$482$	− 35810.0i	− 3.38402i
$483$	0	0
$484$	−39926.7	−3.74969
$485$	0	0
$486$	0	0
$487$	− 7037.81i	− 0.654853i	−0.944877	−	0.327427i	$-0.893819\pi$
$487$	− 7037.81i	− 0.654853i	0.944877	−	0.327427i	$-0.106181\pi$
$488$	− 19568.2i	− 1.81519i
$489$	0	0
$490$	0	0
$491$	−6603.07	−0.606909	−0.303454	−	0.952846i	$-0.598140\pi$
$491$	−6603.07	−0.606909	−0.303454	+	0.952846i	$0.598140\pi$
$492$	0	0
$493$	6322.17i	0.577558i
$494$	−27678.2	−2.52085
$495$	0	0
$496$	−6995.72	−0.633301
$497$	− 6621.58i	− 0.597623i
$498$	0	0
$499$	36.7047	0.00329284	0.00164642	−	0.999999i	$-0.499476\pi$
$499$	36.7047	0.00329284	0.00164642	+	0.999999i	$0.499476\pi$
$500$	0	0
$501$	0	0
$502$	− 3898.76i	− 0.346634i
$503$	− 21242.5i	− 1.88302i	−0.336990	−	0.941508i	$-0.609409\pi$
$503$	− 21242.5i	− 1.88302i	0.336990	−	0.941508i	$-0.390591\pi$
$504$	0	0
$505$	0	0
$506$	−7238.06	−0.635911
$507$	0	0
$508$	− 12563.2i	− 1.09725i
$509$	−11280.4	−0.982309	−0.491155	−	0.871072i	$-0.663425\pi$
$509$	−11280.4	−0.982309	−0.491155	+	0.871072i	$0.663425\pi$
$510$	0	0
$511$	1046.86	0.0906270
$512$	20681.3i	1.78514i
$513$	0	0
$514$	−23999.9	−2.05951
$515$	0	0
$516$	0	0
$517$	22175.5i	1.88642i
$518$	− 14029.7i	− 1.19002i
$519$	0	0
$520$	0	0
$521$	10239.3	0.861023	0.430511	−	0.902585i	$-0.358333\pi$
$521$	10239.3	0.861023	0.430511	+	0.902585i	$0.358333\pi$
$522$	0	0
$523$	− 2822.00i	− 0.235942i	−0.993017	−	0.117971i	$-0.962361\pi$
$523$	− 2822.00i	− 0.235942i	0.993017	−	0.117971i	$-0.0376389\pi$
$524$	52098.1	4.34335
$525$	0	0
$526$	−41215.3	−3.41649
$527$	− 1160.22i	− 0.0959010i
$528$	0	0
$529$	11611.1	0.954310
$530$	0	0
$531$	0	0
$532$	− 37718.7i	− 3.07390i
$533$	− 4922.67i	− 0.400046i
$534$	0	0
$535$	0	0
$536$	−33535.8	−2.70248
$537$	0	0
$538$	− 30224.4i	− 2.42205i
$539$	11450.3	0.915028
$540$	0	0
$541$	−9409.63	−0.747785	−0.373892	−	0.927472i	$-0.621977\pi$
$541$	−9409.63	−0.747785	−0.373892	+	0.927472i	$0.621977\pi$
$542$	16694.2i	1.32302i
$543$	0	0
$544$	−27091.0	−2.13514
$545$	0	0
$546$	0	0
$547$	3836.43i	0.299879i	0.988695	+	0.149940i	$0.0479079\pi$
$547$	3836.43i	0.299879i	−0.988695	+	0.149940i	$0.952092\pi$
$548$	− 21851.6i	− 1.70339i
$549$	0	0
$550$	0	0
$551$	23596.7	1.82441
$552$	0	0
$553$	− 5316.97i	− 0.408862i
$554$	−22218.2	−1.70390
$555$	0	0
$556$	2873.35	0.219167
$557$	6145.92i	0.467524i	0.972294	+	0.233762i	$0.0751036\pi$
$557$	6145.92i	0.467524i	−0.972294	+	0.233762i	$0.924896\pi$
$558$	0	0
$559$	16192.0	1.22513
$560$	0	0
$561$	0	0
$562$	− 40646.9i	− 3.05087i
$563$	− 13247.9i	− 0.991707i	−0.868406	−	0.495854i	$-0.834855\pi$
$563$	− 13247.9i	− 0.991707i	0.868406	−	0.495854i	$-0.165145\pi$
$564$	0	0
$565$	0	0
$566$	4228.63	0.314033
$567$	0	0
$568$	42242.6i	3.12053i
$569$	6544.89	0.482208	0.241104	−	0.970499i	$-0.422491\pi$
$569$	6544.89	0.482208	0.241104	+	0.970499i	$0.422491\pi$
$570$	0	0
$571$	20362.1	1.49234	0.746170	−	0.665755i	$-0.231890\pi$
$571$	20362.1	1.49234	0.746170	+	0.665755i	$0.231890\pi$
$572$	− 42410.2i	− 3.10011i
$573$	0	0
$574$	9170.43	0.666840
$575$	0	0
$576$	0	0
$577$	26247.4i	1.89375i	0.321600	+	0.946876i	$0.395779\pi$
$577$	26247.4i	1.89375i	−0.321600	+	0.946876i	$0.604221\pi$
$578$	18401.8i	1.32425i
$579$	0	0
$580$	0	0
$581$	3356.54	0.239677
$582$	0	0
$583$	7579.71i	0.538455i
$584$	−6678.49	−0.473215
$585$	0	0
$586$	36842.4	2.59718
$587$	− 14098.2i	− 0.991301i	−0.868522	−	0.495650i	$-0.834930\pi$
$587$	− 14098.2i	− 0.991301i	0.868522	−	0.495650i	$-0.165070\pi$
$588$	0	0
$589$	−4330.36	−0.302936
$590$	0	0
$591$	0	0
$592$	51541.9i	3.57831i
$593$	− 3476.71i	− 0.240761i	−0.992728	−	0.120380i	$-0.961589\pi$
$593$	− 3476.71i	− 0.240761i	0.992728	−	0.120380i	$-0.0384115\pi$
$594$	0	0
$595$	0	0
$596$	−22222.8	−1.52732
$597$	0	0
$598$	− 4452.69i	− 0.304488i
$599$	−16179.0	−1.10360	−0.551798	−	0.833978i	$-0.686058\pi$
$599$	−16179.0	−1.10360	−0.551798	+	0.833978i	$0.686058\pi$
$600$	0	0
$601$	8112.16	0.550586	0.275293	−	0.961360i	$-0.411225\pi$
$601$	8112.16	0.550586	0.275293	+	0.961360i	$0.411225\pi$
$602$	30164.0i	2.04218i
$603$	0	0
$604$	−61522.3	−4.14455
$605$	0	0
$606$	0	0
$607$	− 21139.2i	− 1.41353i	−0.707449	−	0.706765i	$-0.750154\pi$
$607$	− 21139.2i	− 1.41353i	0.707449	−	0.706765i	$-0.249846\pi$
$608$	101113.i	6.74456i
$609$	0	0
$610$	0	0
$611$	−13641.9	−0.903258
$612$	0	0
$613$	− 11440.3i	− 0.753785i	−0.926257	−	0.376892i	$-0.876993\pi$
$613$	− 11440.3i	− 0.753785i	0.926257	−	0.376892i	$-0.123007\pi$
$614$	−12033.2	−0.790915
$615$	0	0
$616$	50010.3	3.27106
$617$	21566.2i	1.40717i	0.710612	+	0.703584i	$0.248418\pi$
$617$	21566.2i	1.40717i	−0.710612	+	0.703584i	$0.751582\pi$
$618$	0	0
$619$	15198.3	0.986866	0.493433	−	0.869784i	$-0.335742\pi$
$619$	15198.3	0.986866	0.493433	+	0.869784i	$0.335742\pi$
$620$	0	0
$621$	0	0
$622$	32932.6i	2.12295i
$623$	7382.68i	0.474769i
$624$	0	0
$625$	0	0
$626$	31513.2	2.01201
$627$	0	0
$628$	10386.7i	0.659994i
$629$	−8548.05	−0.541865
$630$	0	0
$631$	−4929.66	−0.311009	−0.155504	−	0.987835i	$-0.549700\pi$
$631$	−4929.66	−0.311009	−0.155504	+	0.987835i	$0.549700\pi$
$632$	33919.8i	2.13490i
$633$	0	0
$634$	18025.2	1.12914
$635$	0	0
$636$	0	0
$637$	7043.97i	0.438135i
$638$	49425.8i	3.06706i
$639$	0	0
$640$	0	0
$641$	7535.73	0.464342	0.232171	−	0.972675i	$-0.425417\pi$
$641$	7535.73	0.464342	0.232171	+	0.972675i	$0.425417\pi$
$642$	0	0
$643$	− 15997.2i	− 0.981135i	−0.871403	−	0.490568i	$-0.836789\pi$
$643$	− 15997.2i	− 0.981135i	0.871403	−	0.490568i	$-0.163211\pi$
$644$	6067.93	0.371289
$645$	0	0
$646$	−31415.5	−1.91336
$647$	6020.46i	0.365825i	0.983129	+	0.182913i	$0.0585525\pi$
$647$	6020.46i	0.365825i	−0.983129	+	0.182913i	$0.941447\pi$
$648$	0	0
$649$	−7378.12	−0.446250
$650$	0	0
$651$	0	0
$652$	48876.8i	2.93583i
$653$	− 10948.7i	− 0.656133i	−0.944655	−	0.328067i	$-0.893603\pi$
$653$	− 10948.7i	− 0.656133i	0.944655	−	0.328067i	$-0.106397\pi$
$654$	0	0
$655$	0	0
$656$	−33690.0	−2.00514
$657$	0	0
$658$	− 25413.4i	− 1.50565i
$659$	−12338.3	−0.729334	−0.364667	−	0.931138i	$-0.618817\pi$
$659$	−12338.3	−0.729334	−0.364667	+	0.931138i	$0.618817\pi$
$660$	0	0
$661$	20016.8	1.17786	0.588928	−	0.808186i	$-0.299550\pi$
$661$	20016.8	1.17786	0.588928	+	0.808186i	$0.299550\pi$
$662$	43960.2i	2.58091i
$663$	0	0
$664$	−21413.1	−1.25149
$665$	0	0
$666$	0	0
$667$	3796.08i	0.220367i
$668$	− 2071.66i	− 0.119993i
$669$	0	0
$670$	0	0
$671$	14610.6	0.840589
$672$	0	0
$673$	− 8419.22i	− 0.482225i	−0.970497	−	0.241112i	$-0.922488\pi$
$673$	− 8419.22i	− 0.482225i	0.970497	−	0.241112i	$-0.0775122\pi$
$674$	18889.1	1.07950
$675$	0	0
$676$	−21800.5	−1.24036
$677$	− 25707.1i	− 1.45938i	−0.683776	−	0.729692i	$-0.739663\pi$
$677$	− 25707.1i	− 1.45938i	0.683776	−	0.729692i	$-0.260337\pi$
$678$	0	0
$679$	−2281.74	−0.128962
$680$	0	0
$681$	0	0
$682$	− 9070.41i	− 0.509272i
$683$	− 19624.3i	− 1.09942i	−0.835357	−	0.549708i	$-0.814739\pi$
$683$	− 19624.3i	− 1.09942i	0.835357	−	0.549708i	$-0.185261\pi$
$684$	0	0
$685$	0	0
$686$	−35228.2	−1.96067
$687$	0	0
$688$	− 110815.i	− 6.14069i
$689$	−4662.86	−0.257824
$690$	0	0
$691$	7273.23	0.400415	0.200207	−	0.979754i	$-0.435838\pi$
$691$	7273.23	0.400415	0.200207	+	0.979754i	$0.435838\pi$
$692$	− 46511.8i	− 2.55508i
$693$	0	0
$694$	50920.8	2.78520
$695$	0	0
$696$	0	0
$697$	− 5587.37i	− 0.303639i
$698$	− 48579.8i	− 2.63434i
$699$	0	0
$700$	0	0
$701$	17644.3	0.950664	0.475332	−	0.879807i	$-0.342328\pi$
$701$	17644.3	0.950664	0.475332	+	0.879807i	$0.342328\pi$
$702$	0	0
$703$	31904.5i	1.71166i
$704$	−105270.	−5.63568
$705$	0	0
$706$	20320.0	1.08322
$707$	− 16232.3i	− 0.863476i
$708$	0	0
$709$	−24304.4	−1.28741	−0.643703	−	0.765276i	$-0.722603\pi$
$709$	−24304.4	−1.28741	−0.643703	+	0.765276i	$0.722603\pi$
$710$	0	0
$711$	0	0
$712$	− 47098.1i	− 2.47904i
$713$	− 696.639i	− 0.0365909i
$714$	0	0
$715$	0	0
$716$	37158.6	1.93950
$717$	0	0
$718$	60206.0i	3.12934i
$719$	15170.2	0.786863	0.393431	−	0.919354i	$-0.371288\pi$
$719$	15170.2	0.786863	0.393431	+	0.919354i	$0.371288\pi$
$720$	0	0
$721$	−23962.5	−1.23774
$722$	79812.8i	4.11402i
$723$	0	0
$724$	29666.8	1.52287
$725$	0	0
$726$	0	0
$727$	17487.0i	0.892102i	0.895008	+	0.446051i	$0.147170\pi$
$727$	17487.0i	0.892102i	−0.895008	+	0.446051i	$0.852830\pi$
$728$	30765.2i	1.56625i
$729$	0	0
$730$	0	0
$731$	18378.4	0.929888
$732$	0	0
$733$	18698.0i	0.942194i	0.882082	+	0.471097i	$0.156142\pi$
$733$	18698.0i	0.942194i	−0.882082	+	0.471097i	$0.843858\pi$
$734$	−26492.0	−1.33220
$735$	0	0
$736$	−16266.5	−0.814660
$737$	− 25039.5i	− 1.25148i
$738$	0	0
$739$	35250.3	1.75467	0.877336	−	0.479876i	$-0.159318\pi$
$739$	35250.3	1.75467	0.877336	+	0.479876i	$0.159318\pi$
$740$	0	0
$741$	0	0
$742$	− 8686.43i	− 0.429769i
$743$	− 11133.6i	− 0.549731i	−0.961483	−	0.274866i	$-0.911367\pi$
$743$	− 11133.6i	− 0.549731i	0.961483	−	0.274866i	$-0.0886334\pi$
$744$	0	0
$745$	0	0
$746$	67545.5	3.31503
$747$	0	0
$748$	− 48136.8i	− 2.35301i
$749$	9717.49	0.474058
$750$	0	0
$751$	17197.6	0.835619	0.417809	−	0.908535i	$-0.362798\pi$
$751$	17197.6	0.835619	0.417809	+	0.908535i	$0.362798\pi$
$752$	93362.7i	4.52738i
$753$	0	0
$754$	−30405.6	−1.46858
$755$	0	0
$756$	0	0
$757$	− 804.647i	− 0.0386333i	−0.999813	−	0.0193166i	$-0.993851\pi$
$757$	− 804.647i	− 0.0386333i	0.999813	−	0.0193166i	$-0.00614906\pi$
$758$	− 60868.7i	− 2.91669i
$759$	0	0
$760$	0	0
$761$	−26208.9	−1.24845	−0.624225	−	0.781245i	$-0.714585\pi$
$761$	−26208.9	−1.24845	−0.624225	+	0.781245i	$0.714585\pi$
$762$	0	0
$763$	9797.85i	0.464883i
$764$	−23909.0	−1.13219
$765$	0	0
$766$	12007.0	0.566360
$767$	− 4538.85i	− 0.213674i
$768$	0	0
$769$	−36544.2	−1.71367	−0.856837	−	0.515587i	$-0.827574\pi$
$769$	−36544.2	−1.71367	−0.856837	+	0.515587i	$0.827574\pi$
$770$	0	0
$771$	0	0
$772$	62500.8i	2.91380i
$773$	42387.4i	1.97228i	0.165923	+	0.986139i	$0.446940\pi$
$773$	42387.4i	1.97228i	−0.165923	+	0.986139i	$0.553060\pi$
$774$	0	0
$775$	0	0
$776$	14556.5	0.673385
$777$	0	0
$778$	50323.8i	2.31902i
$779$	−20854.1	−0.959148
$780$	0	0
$781$	−31540.4	−1.44507
$782$	− 5053.92i	− 0.231110i
$783$	0	0
$784$	48207.8	2.19606
$785$	0	0
$786$	0	0
$787$	− 26849.4i	− 1.21611i	−0.793896	−	0.608054i	$-0.791950\pi$
$787$	− 26849.4i	− 1.21611i	0.793896	−	0.608054i	$-0.208050\pi$
$788$	− 15784.8i	− 0.713592i
$789$	0	0
$790$	0	0
$791$	−17749.5	−0.797851
$792$	0	0
$793$	8988.09i	0.402492i
$794$	6110.31	0.273107
$795$	0	0
$796$	−36918.3	−1.64389
$797$	− 31307.0i	− 1.39141i	−0.718328	−	0.695704i	$-0.755092\pi$
$797$	− 31307.0i	− 1.39141i	0.718328	−	0.695704i	$-0.244908\pi$
$798$	0	0
$799$	−15483.9	−0.685583
$800$	0	0
$801$	0	0
$802$	− 67125.7i	− 2.95548i
$803$	− 4986.48i	− 0.219140i
$804$	0	0
$805$	0	0
$806$	5579.90	0.243851
$807$	0	0
$808$	103554.i	4.50870i
$809$	−10011.9	−0.435106	−0.217553	−	0.976049i	$-0.569807\pi$
$809$	−10011.9	−0.435106	−0.217553	+	0.976049i	$0.569807\pi$
$810$	0	0
$811$	4603.68	0.199331	0.0996653	−	0.995021i	$-0.468223\pi$
$811$	4603.68	0.199331	0.0996653	+	0.995021i	$0.468223\pi$
$812$	− 41435.5i	− 1.79076i
$813$	0	0
$814$	−66827.4	−2.87752
$815$	0	0
$816$	0	0
$817$	− 68594.8i	− 2.93737i
$818$	− 13648.1i	− 0.583367i
$819$	0	0
$820$	0	0
$821$	35429.0	1.50607	0.753033	−	0.657983i	$-0.228590\pi$
$821$	35429.0	1.50607	0.753033	+	0.657983i	$0.228590\pi$
$822$	0	0
$823$	− 28297.6i	− 1.19853i	−0.800550	−	0.599265i	$-0.795459\pi$
$823$	− 28297.6i	− 1.19853i	0.800550	−	0.599265i	$-0.204541\pi$
$824$	152870.	6.46295
$825$	0	0
$826$	8455.40	0.356176
$827$	41059.9i	1.72647i	0.504800	+	0.863236i	$0.331566\pi$
$827$	41059.9i	1.72647i	−0.504800	+	0.863236i	$0.668434\pi$
$828$	0	0
$829$	−9030.68	−0.378345	−0.189173	−	0.981944i	$-0.560581\pi$
$829$	−9030.68	−0.378345	−0.189173	+	0.981944i	$0.560581\pi$
$830$	0	0
$831$	0	0
$832$	− 64759.8i	− 2.69849i
$833$	7995.10i	0.332550i
$834$	0	0
$835$	0	0
$836$	−179664.	−7.43280
$837$	0	0
$838$	− 45487.7i	− 1.87511i
$839$	5227.81	0.215118	0.107559	−	0.994199i	$-0.465697\pi$
$839$	5227.81	0.215118	0.107559	+	0.994199i	$0.465697\pi$
$840$	0	0
$841$	1532.87	0.0628508
$842$	62089.0i	2.54125i
$843$	0	0
$844$	−20652.6	−0.842288
$845$	0	0
$846$	0	0
$847$	21625.6i	0.877290i
$848$	31911.8i	1.29228i
$849$	0	0
$850$	0	0
$851$	−5132.58	−0.206748
$852$	0	0
$853$	− 30002.9i	− 1.20432i	−0.798377	−	0.602158i	$-0.794308\pi$
$853$	− 30002.9i	− 1.20432i	0.798377	−	0.602158i	$-0.205692\pi$
$854$	−16743.9	−0.670918
$855$	0	0
$856$	−61993.1	−2.47533
$857$	15671.7i	0.624662i	0.949973	+	0.312331i	$0.101110\pi$
$857$	15671.7i	0.624662i	−0.949973	+	0.312331i	$0.898890\pi$
$858$	0	0
$859$	31306.8	1.24351	0.621755	−	0.783212i	$-0.286420\pi$
$859$	31306.8	1.24351	0.621755	+	0.783212i	$0.286420\pi$
$860$	0	0
$861$	0	0
$862$	54752.3i	2.16342i
$863$	− 13212.3i	− 0.521150i	−0.965454	−	0.260575i	$-0.916088\pi$
$863$	− 13212.3i	− 0.521150i	0.965454	−	0.260575i	$-0.0839121\pi$
$864$	0	0
$865$	0	0
$866$	−41537.8	−1.62992
$867$	0	0
$868$	7604.05i	0.297348i
$869$	−25326.2	−0.988643
$870$	0	0
$871$	15403.7	0.599236
$872$	− 62505.7i	− 2.42742i
$873$	0	0
$874$	−18863.1	−0.730039
$875$	0	0
$876$	0	0
$877$	− 15440.6i	− 0.594519i	−0.954797	−	0.297260i	$-0.903927\pi$
$877$	− 15440.6i	− 0.594519i	0.954797	−	0.297260i	$-0.0960727\pi$
$878$	67527.5i	2.59561i
$879$	0	0
$880$	0	0
$881$	27563.1	1.05406	0.527029	−	0.849847i	$-0.323306\pi$
$881$	27563.1	1.05406	0.527029	+	0.849847i	$0.323306\pi$
$882$	0	0
$883$	19897.7i	0.758337i	0.925328	+	0.379169i	$0.123790\pi$
$883$	19897.7i	0.758337i	−0.925328	+	0.379169i	$0.876210\pi$
$884$	29612.6	1.12667
$885$	0	0
$886$	−65965.9	−2.50132
$887$	40061.6i	1.51650i	0.651963	+	0.758251i	$0.273946\pi$
$887$	40061.6i	1.51650i	−0.651963	+	0.758251i	$0.726054\pi$
$888$	0	0
$889$	−6804.66	−0.256716
$890$	0	0
$891$	0	0
$892$	− 2438.45i	− 0.0915306i
$893$	57791.6i	2.16565i
$894$	0	0
$895$	0	0
$896$	55477.5	2.06850
$897$	0	0
$898$	− 3414.72i	− 0.126894i
$899$	−4757.07	−0.176482
$900$	0	0
$901$	−5292.47	−0.195691
$902$	− 43681.2i	− 1.61244i
$903$	0	0
$904$	113234.	4.16603
$905$	0	0
$906$	0	0
$907$	− 27839.6i	− 1.01918i	−0.860417	−	0.509591i	$-0.829797\pi$
$907$	− 27839.6i	− 1.01918i	0.860417	−	0.509591i	$-0.170203\pi$
$908$	− 26172.9i	− 0.956584i
$909$	0	0
$910$	0	0
$911$	−22251.0	−0.809232	−0.404616	−	0.914487i	$-0.632595\pi$
$911$	−22251.0	−0.809232	−0.404616	+	0.914487i	$0.632595\pi$
$912$	0	0
$913$	− 15988.1i	− 0.579549i
$914$	9887.02	0.357805
$915$	0	0
$916$	17926.3	0.646616
$917$	− 28218.0i	− 1.01618i
$918$	0	0
$919$	−29480.5	−1.05819	−0.529093	−	0.848564i	$-0.677468\pi$
$919$	−29480.5	−1.05819	−0.529093	+	0.848564i	$0.677468\pi$
$920$	0	0
$921$	0	0
$922$	63457.9i	2.26668i
$923$	− 19402.9i	− 0.691933i
$924$	0	0
$925$	0	0
$926$	−39804.6	−1.41259
$927$	0	0
$928$	111077.i	3.92919i
$929$	42351.8	1.49571	0.747857	−	0.663860i	$-0.231083\pi$
$929$	42351.8	1.49571	0.747857	+	0.663860i	$0.231083\pi$
$930$	0	0
$931$	29840.7	1.05047
$932$	− 116166.i	− 4.08278i
$933$	0	0
$934$	63526.9	2.22555
$935$	0	0
$936$	0	0
$937$	35930.6i	1.25272i	0.779532	+	0.626362i	$0.215457\pi$
$937$	35930.6i	1.25272i	−0.779532	+	0.626362i	$0.784543\pi$
$938$	28695.5i	0.998872i
$939$	0	0
$940$	0	0
$941$	21564.0	0.747041	0.373521	−	0.927622i	$-0.378151\pi$
$941$	21564.0	0.747041	0.373521	+	0.927622i	$0.378151\pi$
$942$	0	0
$943$	− 3354.87i	− 0.115853i
$944$	−31063.1	−1.07099
$945$	0	0
$946$	143679.	4.93807
$947$	− 16632.4i	− 0.570729i	−0.958419	−	0.285364i	$-0.907885\pi$
$947$	− 16632.4i	− 0.570729i	0.958419	−	0.285364i	$-0.0921146\pi$
$948$	0	0
$949$	3067.57	0.104929
$950$	0	0
$951$	0	0
$952$	34919.3i	1.18880i
$953$	− 39557.8i	− 1.34460i	−0.740279	−	0.672299i	$-0.765307\pi$
$953$	− 39557.8i	− 1.34460i	0.740279	−	0.672299i	$-0.234693\pi$
$954$	0	0
$955$	0	0
$956$	154452.	5.22526
$957$	0	0
$958$	65731.4i	2.21679i
$959$	−11835.5	−0.398529
$960$	0	0
$961$	−28918.0	−0.970696
$962$	− 41110.6i	− 1.37782i
$963$	0	0
$964$	142997.	4.77762
$965$	0	0
$966$	0	0
$967$	10666.2i	0.354707i	0.984147	+	0.177354i	$0.0567536\pi$
$967$	10666.2i	0.354707i	−0.984147	+	0.177354i	$0.943246\pi$
$968$	− 137961.i	− 4.58083i
$969$	0	0
$970$	0	0
$971$	−31821.8	−1.05171	−0.525855	−	0.850574i	$-0.676254\pi$
$971$	−31821.8	−1.05171	−0.525855	+	0.850574i	$0.676254\pi$
$972$	0	0
$973$	− 1556.30i	− 0.0512771i
$974$	38417.7	1.26384
$975$	0	0
$976$	61513.0	2.01740
$977$	11126.5i	0.364347i	0.983266	+	0.182173i	$0.0583132\pi$
$977$	11126.5i	0.364347i	−0.983266	+	0.182173i	$0.941687\pi$
$978$	0	0
$979$	35165.7	1.14801
$980$	0	0
$981$	0	0
$982$	− 36044.5i	− 1.17131i
$983$	991.225i	0.0321619i	0.999871	+	0.0160810i	$0.00511895\pi$
$983$	991.225i	0.0321619i	−0.999871	+	0.0160810i	$0.994881\pi$
$984$	0	0
$985$	0	0
$986$	−34511.2	−1.11467
$987$	0	0
$988$	− 110525.i	− 3.55898i
$989$	11035.1	0.354798
$990$	0	0
$991$	−48714.9	−1.56153	−0.780767	−	0.624822i	$-0.785171\pi$
$991$	−48714.9	−1.56153	−0.780767	+	0.624822i	$0.785171\pi$
$992$	− 20384.4i	− 0.652424i
$993$	0	0
$994$	36145.6	1.15339
$995$	0	0
$996$	0	0
$997$	42207.2i	1.34074i	0.742028	+	0.670369i	$0.233864\pi$
$997$	42207.2i	1.34074i	−0.742028	+	0.670369i	$0.766136\pi$
$998$	200.362i	0.00635505i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	675.4.b.m.649.6		6
3.2	odd	2		675.4.b.n.649.1		6
5.2	odd	4		135.4.a.e.1.1	✓	3
5.3	odd	4		675.4.a.s.1.3		3
5.4	even	2	inner	675.4.b.m.649.1		6
15.2	even	4		135.4.a.h.1.3	yes	3
15.8	even	4		675.4.a.p.1.1		3
15.14	odd	2		675.4.b.n.649.6		6
20.7	even	4		2160.4.a.bi.1.2		3
45.2	even	12		405.4.e.q.271.1		6
45.7	odd	12		405.4.e.v.271.3		6
45.22	odd	12		405.4.e.v.136.3		6
45.32	even	12		405.4.e.q.136.1		6
60.47	odd	4		2160.4.a.bq.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.4.a.e.1.1	✓	3	5.2	odd	4
135.4.a.h.1.3	yes	3	15.2	even	4
405.4.e.q.136.1		6	45.32	even	12
405.4.e.q.271.1		6	45.2	even	12
405.4.e.v.136.3		6	45.22	odd	12
405.4.e.v.271.3		6	45.7	odd	12
675.4.a.p.1.1		3	15.8	even	4
675.4.a.s.1.3		3	5.3	odd	4
675.4.b.m.649.1		6	5.4	even	2	inner
675.4.b.m.649.6		6	1.1	even	1	trivial
675.4.b.n.649.1		6	3.2	odd	2
675.4.b.n.649.6		6	15.14	odd	2
2160.4.a.bi.1.2		3	20.7	even	4
2160.4.a.bq.1.2		3	60.47	odd	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:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	675.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+5.45876i q^{2} -21.7980 q^{4} +11.8065i q^{7} -75.3201i q^{8} +O(q^{10})\)	\(q+5.45876i q^{2} -21.7980 q^{4} +11.8065i q^{7} -75.3201i q^{8} +56.2376 q^{11} +34.5961i q^{13} -64.4489 q^{14} +236.770 q^{16} +39.2675i q^{17} +146.561 q^{19} +306.987i q^{22} +23.5777i q^{23} -188.851 q^{26} -257.359i q^{28} +161.003 q^{29} -29.5465 q^{31} +689.908i q^{32} -214.352 q^{34} +217.688i q^{37} +800.039i q^{38} -142.290 q^{41} -468.030i q^{43} -1225.87 q^{44} -128.705 q^{46} +394.318i q^{47} +203.606 q^{49} -754.126i q^{52} +134.780i q^{53} +889.268 q^{56} +878.875i q^{58} -131.195 q^{59} +259.801 q^{61} -161.287i q^{62} -1871.88 q^{64} -445.244i q^{67} -855.954i q^{68} -560.841 q^{71} -88.6681i q^{73} -1188.30 q^{74} -3194.74 q^{76} +663.970i q^{77} -450.342 q^{79} -776.726i q^{82} -284.295i q^{83} +2554.86 q^{86} -4235.82i q^{88} +625.305 q^{89} -408.459 q^{91} -513.948i q^{92} -2152.49 q^{94} +193.261i q^{97} +1111.44i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 34 q^{4}+O(q^{10}) \) 6 * q - 34 * q^4	\( 6 q - 34 q^{4} - 10 q^{11} + 120 q^{14} + 322 q^{16} + 100 q^{19} - 370 q^{26} + 230 q^{29} - 230 q^{31} - 826 q^{34} - 1160 q^{41} - 2830 q^{44} - 570 q^{46} - 1154 q^{49} + 4380 q^{56} + 760 q^{59} - 304 q^{61} - 5874 q^{64} - 80 q^{71} - 5440 q^{74} - 6552 q^{76} - 2026 q^{79} + 3110 q^{86} + 2040 q^{89} - 1264 q^{91} - 7666 q^{94}+O(q^{100}) \) 6 * q - 34 * q^4 - 10 * q^11 + 120 * q^14 + 322 * q^16 + 100 * q^19 - 370 * q^26 + 230 * q^29 - 230 * q^31 - 826 * q^34 - 1160 * q^41 - 2830 * q^44 - 570 * q^46 - 1154 * q^49 + 4380 * q^56 + 760 * q^59 - 304 * q^61 - 5874 * q^64 - 80 * q^71 - 5440 * q^74 - 6552 * q^76 - 2026 * q^79 + 3110 * q^86 + 2040 * q^89 - 1264 * q^91 - 7666 * q^94

Introduction

L-functions

Modular forms

Varieties

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Database

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