LMFDB - Embedded newform 675.4.b.n.649.4

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			649.4
Root			$1.60096 - 1.60096i$ of defining polynomial
Character	$\chi$	$=$	675.649
Dual form			675.4.b.n.649.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.12612i q^{2} +3.47962 q^{4} +30.7000i q^{7} +24.4070i q^{8} +O(q^{10})$	$q+2.12612i q^{2} +3.47962 q^{4} +30.7000i q^{7} +24.4070i q^{8} +50.1548 q^{11} +15.9592i q^{13} -65.2720 q^{14} -24.0553 q^{16} +105.668i q^{17} +21.3040 q^{19} +106.635i q^{22} -136.137i q^{23} -33.9312 q^{26} +106.824i q^{28} +224.323 q^{29} -225.982 q^{31} +144.112i q^{32} -224.663 q^{34} -416.386i q^{37} +45.2948i q^{38} +76.1411 q^{41} -31.7372i q^{43} +174.519 q^{44} +289.443 q^{46} +60.8026i q^{47} -599.493 q^{49} +55.5320i q^{52} +466.532i q^{53} -749.297 q^{56} +476.938i q^{58} -95.4239 q^{59} -357.174 q^{61} -480.464i q^{62} -498.842 q^{64} +87.8344i q^{67} +367.685i q^{68} +412.693 q^{71} +331.133i q^{73} +885.286 q^{74} +74.1296 q^{76} +1539.75i q^{77} +248.123 q^{79} +161.885i q^{82} +552.505i q^{83} +67.4771 q^{86} +1224.13i q^{88} +291.478 q^{89} -489.949 q^{91} -473.704i q^{92} -129.274 q^{94} +198.606i q^{97} -1274.59i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.12612i	0.751697i	0.926681	+	0.375848i	$0.122649\pi$
$2$	2.12612i	0.751697i	−0.926681	+	0.375848i	$0.877351\pi$
$3$	0	0
$3$	0	0
$4$	3.47962	0.434952
$5$	0	0
$5$	0	0
$6$	0	0
$7$	30.7000i	1.65765i	0.559511	+	0.828823i	$0.310989\pi$
$7$	30.7000i	1.65765i	−0.559511	+	0.828823i	$0.689011\pi$
$8$	24.4070i	1.07865i
$9$	0	0
$10$	0	0
$11$	50.1548	1.37475	0.687375	−	0.726303i	$-0.258763\pi$
$11$	50.1548	1.37475	0.687375	+	0.726303i	$0.258763\pi$
$12$	0	0
$13$	15.9592i	0.340484i	0.985402	+	0.170242i	$0.0544550\pi$
$13$	15.9592i	0.340484i	−0.985402	+	0.170242i	$0.945545\pi$
$14$	−65.2720	−1.24605
$15$	0	0
$16$	−24.0553	−0.375865
$17$	105.668i	1.50755i	0.657134	+	0.753774i	$0.271769\pi$
$17$	105.668i	1.50755i	−0.657134	+	0.753774i	$0.728231\pi$
$18$	0	0
$19$	21.3040	0.257235	0.128618	−	0.991694i	$-0.458946\pi$
$19$	21.3040	0.257235	0.128618	+	0.991694i	$0.458946\pi$
$20$	0	0
$21$	0	0
$22$	106.635i	1.03339i
$23$	− 136.137i	− 1.23420i	−0.786886	−	0.617098i	$-0.788308\pi$
$23$	− 136.137i	− 1.23420i	0.786886	−	0.617098i	$-0.211692\pi$
$24$	0	0
$25$	0	0
$26$	−33.9312	−0.255941
$27$	0	0
$28$	106.824i	0.720997i
$29$	224.323	1.43641	0.718203	−	0.695834i	$-0.244965\pi$
$29$	224.323	1.43641	0.718203	+	0.695834i	$0.244965\pi$
$30$	0	0
$31$	−225.982	−1.30928	−0.654638	−	0.755943i	$-0.727179\pi$
$31$	−225.982	−1.30928	−0.654638	+	0.755943i	$0.727179\pi$
$32$	144.112i	0.796112i
$33$	0	0
$34$	−224.663	−1.13322
$35$	0	0
$36$	0	0
$37$	− 416.386i	− 1.85009i	−0.379854	−	0.925046i	$-0.624026\pi$
$37$	− 416.386i	− 1.85009i	0.379854	−	0.925046i	$-0.375974\pi$
$38$	45.2948i	0.193363i
$39$	0	0
$40$	0	0
$41$	76.1411	0.290030	0.145015	−	0.989429i	$-0.453677\pi$
$41$	76.1411	0.290030	0.145015	+	0.989429i	$0.453677\pi$
$42$	0	0
$43$	− 31.7372i	− 0.112555i	−0.998415	−	0.0562777i	$-0.982077\pi$
$43$	− 31.7372i	− 0.112555i	0.998415	−	0.0562777i	$-0.0179232\pi$
$44$	174.519	0.597950
$45$	0	0
$46$	289.443	0.927741
$47$	60.8026i	0.188701i	0.995539	+	0.0943507i	$0.0300775\pi$
$47$	60.8026i	0.188701i	−0.995539	+	0.0943507i	$0.969922\pi$
$48$	0	0
$49$	−599.493	−1.74779
$50$	0	0
$51$	0	0
$52$	55.5320i	0.148094i
$53$	466.532i	1.20911i	0.796562	+	0.604557i	$0.206650\pi$
$53$	466.532i	1.20911i	−0.796562	+	0.604557i	$0.793350\pi$
$54$	0	0
$55$	0	0
$56$	−749.297	−1.78802
$57$	0	0
$58$	476.938i	1.07974i
$59$	−95.4239	−0.210562	−0.105281	−	0.994443i	$-0.533574\pi$
$59$	−95.4239	−0.210562	−0.105281	+	0.994443i	$0.533574\pi$
$60$	0	0
$61$	−357.174	−0.749696	−0.374848	−	0.927086i	$-0.622305\pi$
$61$	−357.174	−0.749696	−0.374848	+	0.927086i	$0.622305\pi$
$62$	− 480.464i	− 0.984178i
$63$	0	0
$64$	−498.842	−0.974300
$65$	0	0
$66$	0	0
$67$	87.8344i	0.160159i	0.996788	+	0.0800797i	$0.0255175\pi$
$67$	87.8344i	0.160159i	−0.996788	+	0.0800797i	$0.974483\pi$
$68$	367.685i	0.655711i
$69$	0	0
$70$	0	0
$71$	412.693	0.689826	0.344913	−	0.938635i	$-0.387908\pi$
$71$	412.693	0.689826	0.344913	+	0.938635i	$0.387908\pi$
$72$	0	0
$73$	331.133i	0.530906i	0.964124	+	0.265453i	$0.0855216\pi$
$73$	331.133i	0.530906i	−0.964124	+	0.265453i	$0.914478\pi$
$74$	885.286	1.39071
$75$	0	0
$76$	74.1296	0.111885
$77$	1539.75i	2.27885i
$78$	0	0
$79$	248.123	0.353368	0.176684	−	0.984268i	$-0.443463\pi$
$79$	248.123	0.353368	0.176684	+	0.984268i	$0.443463\pi$
$80$	0	0
$81$	0	0
$82$	161.885i	0.218015i
$83$	552.505i	0.730667i	0.930877	+	0.365333i	$0.119045\pi$
$83$	552.505i	0.730667i	−0.930877	+	0.365333i	$0.880955\pi$
$84$	0	0
$85$	0	0
$86$	67.4771	0.0846075
$87$	0	0
$88$	1224.13i	1.48287i
$89$	291.478	0.347153	0.173577	−	0.984820i	$-0.444468\pi$
$89$	291.478	0.347153	0.173577	+	0.984820i	$0.444468\pi$
$90$	0	0
$91$	−489.949	−0.564402
$92$	− 473.704i	− 0.536816i
$93$	0	0
$94$	−129.274	−0.141846
$95$	0	0
$96$	0	0
$97$	198.606i	0.207891i	0.994583	+	0.103946i	$0.0331468\pi$
$97$	198.606i	0.207891i	−0.994583	+	0.103946i	$0.966853\pi$
$98$	− 1274.59i	− 1.31381i
$99$	0	0
$100$	0	0
$101$	816.235	0.804143	0.402071	−	0.915608i	$-0.368290\pi$
$101$	816.235	0.804143	0.402071	+	0.915608i	$0.368290\pi$
$102$	0	0
$103$	1402.37i	1.34155i	0.741660	+	0.670776i	$0.234039\pi$
$103$	1402.37i	1.34155i	−0.741660	+	0.670776i	$0.765961\pi$
$104$	−389.518	−0.367263
$105$	0	0
$106$	−991.902	−0.908887
$107$	− 978.996i	− 0.884515i	−0.896888	−	0.442258i	$-0.854178\pi$
$107$	− 978.996i	− 0.884515i	0.896888	−	0.442258i	$-0.145822\pi$
$108$	0	0
$109$	−2122.96	−1.86553	−0.932766	−	0.360484i	$-0.882612\pi$
$109$	−2122.96	−1.86553	−0.932766	+	0.360484i	$0.882612\pi$
$110$	0	0
$111$	0	0
$112$	− 738.500i	− 0.623051i
$113$	− 1794.80i	− 1.49416i	−0.664732	−	0.747082i	$-0.731454\pi$
$113$	− 1794.80i	− 1.49416i	0.664732	−	0.747082i	$-0.268546\pi$
$114$	0	0
$115$	0	0
$116$	780.559	0.624768
$117$	0	0
$118$	− 202.883i	− 0.158278i
$119$	−3244.02	−2.49898
$120$	0	0
$121$	1184.50	0.889935
$122$	− 759.395i	− 0.563544i
$123$	0	0
$124$	−786.330	−0.569472
$125$	0	0
$126$	0	0
$127$	− 748.894i	− 0.523257i	−0.965169	−	0.261628i	$-0.915741\pi$
$127$	− 748.894i	− 0.523257i	0.965169	−	0.261628i	$-0.0842595\pi$
$128$	92.2971i	0.0637343i
$129$	0	0
$130$	0	0
$131$	−2396.11	−1.59808	−0.799042	−	0.601275i	$-0.794660\pi$
$131$	−2396.11	−1.59808	−0.799042	+	0.601275i	$0.794660\pi$
$132$	0	0
$133$	654.033i	0.426405i
$134$	−186.746	−0.120391
$135$	0	0
$136$	−2579.05	−1.62611
$137$	1004.48i	0.626409i	0.949686	+	0.313205i	$0.101403\pi$
$137$	1004.48i	0.626409i	−0.949686	+	0.313205i	$0.898597\pi$
$138$	0	0
$139$	−2403.02	−1.46634	−0.733172	−	0.680043i	$-0.761961\pi$
$139$	−2403.02	−1.46634	−0.733172	+	0.680043i	$0.761961\pi$
$140$	0	0
$141$	0	0
$142$	877.435i	0.518540i
$143$	800.432i	0.468080i
$144$	0	0
$145$	0	0
$146$	−704.028	−0.399081
$147$	0	0
$148$	− 1448.86i	− 0.804701i
$149$	−509.648	−0.280215	−0.140107	−	0.990136i	$-0.544745\pi$
$149$	−509.648	−0.280215	−0.140107	+	0.990136i	$0.544745\pi$
$150$	0	0
$151$	1443.30	0.777840	0.388920	−	0.921272i	$-0.372848\pi$
$151$	1443.30	0.777840	0.388920	+	0.921272i	$0.372848\pi$
$152$	519.967i	0.277466i
$153$	0	0
$154$	−3273.70	−1.71300
$155$	0	0
$156$	0	0
$157$	− 2155.64i	− 1.09579i	−0.836547	−	0.547895i	$-0.815429\pi$
$157$	− 2155.64i	− 1.09579i	0.836547	−	0.547895i	$-0.184571\pi$
$158$	527.539i	0.265625i
$159$	0	0
$160$	0	0
$161$	4179.41	2.04586
$162$	0	0
$163$	529.909i	0.254636i	0.991862	+	0.127318i	$0.0406369\pi$
$163$	529.909i	0.254636i	−0.991862	+	0.127318i	$0.959363\pi$
$164$	264.942	0.126149
$165$	0	0
$166$	−1174.69	−0.549240
$167$	− 2979.28i	− 1.38050i	−0.723571	−	0.690250i	$-0.757501\pi$
$167$	− 2979.28i	− 1.38050i	0.723571	−	0.690250i	$-0.242499\pi$
$168$	0	0
$169$	1942.30	0.884071
$170$	0	0
$171$	0	0
$172$	− 110.433i	− 0.0489562i
$173$	− 1779.88i	− 0.782205i	−0.920347	−	0.391103i	$-0.872094\pi$
$173$	− 1779.88i	− 0.782205i	0.920347	−	0.391103i	$-0.127906\pi$
$174$	0	0
$175$	0	0
$176$	−1206.49	−0.516720
$177$	0	0
$178$	619.718i	0.260954i
$179$	2836.26	1.18431	0.592157	−	0.805823i	$-0.298276\pi$
$179$	2836.26	1.18431	0.592157	+	0.805823i	$0.298276\pi$
$180$	0	0
$181$	811.890	0.333410	0.166705	−	0.986007i	$-0.446687\pi$
$181$	811.890	0.333410	0.166705	+	0.986007i	$0.446687\pi$
$182$	− 1041.69i	− 0.424259i
$183$	0	0
$184$	3322.70	1.33126
$185$	0	0
$186$	0	0
$187$	5299.77i	2.07250i
$188$	211.570i	0.0820761i
$189$	0	0
$190$	0	0
$191$	1148.46	0.435077	0.217539	−	0.976052i	$-0.430197\pi$
$191$	1148.46	0.435077	0.217539	+	0.976052i	$0.430197\pi$
$192$	0	0
$193$	2150.93i	0.802215i	0.916031	+	0.401107i	$0.131375\pi$
$193$	2150.93i	0.802215i	−0.916031	+	0.401107i	$0.868625\pi$
$194$	−422.261	−0.156271
$195$	0	0
$196$	−2086.00	−0.760206
$197$	− 5057.51i	− 1.82910i	−0.404471	−	0.914551i	$-0.632544\pi$
$197$	− 5057.51i	− 1.82910i	0.404471	−	0.914551i	$-0.367456\pi$
$198$	0	0
$199$	3554.12	1.26605	0.633027	−	0.774130i	$-0.281812\pi$
$199$	3554.12	1.26605	0.633027	+	0.774130i	$0.281812\pi$
$200$	0	0
$201$	0	0
$202$	1735.41i	0.604471i
$203$	6886.73i	2.38105i
$204$	0	0
$205$	0	0
$206$	−2981.61	−1.00844
$207$	0	0
$208$	− 383.905i	− 0.127976i
$209$	1068.50	0.353634
$210$	0	0
$211$	−107.909	−0.0352075	−0.0176038	−	0.999845i	$-0.505604\pi$
$211$	−107.909	−0.0352075	−0.0176038	+	0.999845i	$0.505604\pi$
$212$	1623.35i	0.525907i
$213$	0	0
$214$	2081.46	0.664887
$215$	0	0
$216$	0	0
$217$	− 6937.65i	− 2.17032i
$218$	− 4513.67i	− 1.40231i
$219$	0	0
$220$	0	0
$221$	−1686.38	−0.513296
$222$	0	0
$223$	1942.03i	0.583174i	0.956544	+	0.291587i	$0.0941832\pi$
$223$	1942.03i	0.583174i	−0.956544	+	0.291587i	$0.905817\pi$
$224$	−4424.24	−1.31967
$225$	0	0
$226$	3815.96	1.12316
$227$	3443.99i	1.00698i	0.864000	+	0.503492i	$0.167952\pi$
$227$	3443.99i	1.00698i	−0.864000	+	0.503492i	$0.832048\pi$
$228$	0	0
$229$	−2069.39	−0.597159	−0.298579	−	0.954385i	$-0.596513\pi$
$229$	−2069.39	−0.597159	−0.298579	+	0.954385i	$0.596513\pi$
$230$	0	0
$231$	0	0
$232$	5475.07i	1.54938i
$233$	2769.24i	0.778622i	0.921106	+	0.389311i	$0.127287\pi$
$233$	2769.24i	0.778622i	−0.921106	+	0.389311i	$0.872713\pi$
$234$	0	0
$235$	0	0
$236$	−332.038	−0.0915842
$237$	0	0
$238$	− 6897.17i	− 1.87848i
$239$	2943.84	0.796742	0.398371	−	0.917224i	$-0.369576\pi$
$239$	2943.84	0.796742	0.398371	+	0.917224i	$0.369576\pi$
$240$	0	0
$241$	4474.54	1.19598	0.597988	−	0.801505i	$-0.295967\pi$
$241$	4474.54	1.19598	0.597988	+	0.801505i	$0.295967\pi$
$242$	2518.40i	0.668961i
$243$	0	0
$244$	−1242.83	−0.326082
$245$	0	0
$246$	0	0
$247$	339.995i	0.0875845i
$248$	− 5515.55i	− 1.41225i
$249$	0	0
$250$	0	0
$251$	−2121.41	−0.533474	−0.266737	−	0.963769i	$-0.585945\pi$
$251$	−2121.41	−0.533474	−0.266737	+	0.963769i	$0.585945\pi$
$252$	0	0
$253$	− 6827.92i	− 1.69671i
$254$	1592.24	0.393330
$255$	0	0
$256$	−4186.97	−1.02221
$257$	4548.24i	1.10393i	0.833866	+	0.551967i	$0.186123\pi$
$257$	4548.24i	1.10393i	−0.833866	+	0.551967i	$0.813877\pi$
$258$	0	0
$259$	12783.1	3.06680
$260$	0	0
$261$	0	0
$262$	− 5094.42i	− 1.20127i
$263$	3499.84i	0.820567i	0.911958	+	0.410284i	$0.134570\pi$
$263$	3499.84i	0.820567i	−0.911958	+	0.410284i	$0.865430\pi$
$264$	0	0
$265$	0	0
$266$	−1390.55	−0.320527
$267$	0	0
$268$	305.630i	0.0696616i
$269$	−2594.25	−0.588008	−0.294004	−	0.955804i	$-0.594988\pi$
$269$	−2594.25	−0.588008	−0.294004	+	0.955804i	$0.594988\pi$
$270$	0	0
$271$	8518.20	1.90939	0.954693	−	0.297593i	$-0.0961838\pi$
$271$	8518.20	1.90939	0.954693	+	0.297593i	$0.0961838\pi$
$272$	− 2541.89i	− 0.566634i
$273$	0	0
$274$	−2135.63	−0.470870
$275$	0	0
$276$	0	0
$277$	− 1890.50i	− 0.410068i	−0.978755	−	0.205034i	$-0.934269\pi$
$277$	− 1890.50i	− 0.410068i	0.978755	−	0.205034i	$-0.0657305\pi$
$278$	− 5109.12i	− 1.10225i
$279$	0	0
$280$	0	0
$281$	3138.68	0.666328	0.333164	−	0.942869i	$-0.391884\pi$
$281$	3138.68	0.666328	0.333164	+	0.942869i	$0.391884\pi$
$282$	0	0
$283$	− 4069.09i	− 0.854708i	−0.904084	−	0.427354i	$-0.859446\pi$
$283$	− 4069.09i	− 0.854708i	0.904084	−	0.427354i	$-0.140554\pi$
$284$	1436.01	0.300041
$285$	0	0
$286$	−1701.81	−0.351854
$287$	2337.54i	0.480768i
$288$	0	0
$289$	−6252.77	−1.27270
$290$	0	0
$291$	0	0
$292$	1152.22i	0.230919i
$293$	− 3637.95i	− 0.725362i	−0.931913	−	0.362681i	$-0.881861\pi$
$293$	− 3637.95i	− 0.725362i	0.931913	−	0.362681i	$-0.118139\pi$
$294$	0	0
$295$	0	0
$296$	10162.7	1.99560
$297$	0	0
$298$	− 1083.57i	− 0.210637i
$299$	2172.64	0.420224
$300$	0	0
$301$	974.334	0.186577
$302$	3068.62i	0.584700i
$303$	0	0
$304$	−512.474	−0.0966856
$305$	0	0
$306$	0	0
$307$	− 6829.07i	− 1.26956i	−0.772692	−	0.634781i	$-0.781090\pi$
$307$	− 6829.07i	− 1.26956i	0.772692	−	0.634781i	$-0.218910\pi$
$308$	5357.75i	0.991189i
$309$	0	0
$310$	0	0
$311$	6601.03	1.20357	0.601785	−	0.798659i	$-0.294457\pi$
$311$	6601.03	1.20357	0.601785	+	0.798659i	$0.294457\pi$
$312$	0	0
$313$	− 2766.59i	− 0.499607i	−0.968297	−	0.249803i	$-0.919634\pi$
$313$	− 2766.59i	− 0.499607i	0.968297	−	0.249803i	$-0.0803660\pi$
$314$	4583.15	0.823701
$315$	0	0
$316$	863.373	0.153698
$317$	4564.41i	0.808716i	0.914601	+	0.404358i	$0.132505\pi$
$317$	4564.41i	0.808716i	−0.914601	+	0.404358i	$0.867495\pi$
$318$	0	0
$319$	11250.9	1.97470
$320$	0	0
$321$	0	0
$322$	8885.92i	1.53787i
$323$	2251.15i	0.387794i
$324$	0	0
$325$	0	0
$326$	−1126.65	−0.191409
$327$	0	0
$328$	1858.38i	0.312841i
$329$	−1866.64	−0.312800
$330$	0	0
$331$	−4665.62	−0.774760	−0.387380	−	0.921920i	$-0.626620\pi$
$331$	−4665.62	−0.774760	−0.387380	+	0.921920i	$0.626620\pi$
$332$	1922.51i	0.317805i
$333$	0	0
$334$	6334.30	1.03772
$335$	0	0
$336$	0	0
$337$	− 3807.06i	− 0.615382i	−0.951486	−	0.307691i	$-0.900444\pi$
$337$	− 3807.06i	− 0.615382i	0.951486	−	0.307691i	$-0.0995564\pi$
$338$	4129.57i	0.664553i
$339$	0	0
$340$	0	0
$341$	−11334.1	−1.79993
$342$	0	0
$343$	− 7874.33i	− 1.23957i
$344$	774.612	0.121408
$345$	0	0
$346$	3784.23	0.587981
$347$	2311.26i	0.357565i	0.983889	+	0.178783i	$0.0572159\pi$
$347$	2311.26i	0.357565i	−0.983889	+	0.178783i	$0.942784\pi$
$348$	0	0
$349$	10873.2	1.66771	0.833854	−	0.551984i	$-0.186129\pi$
$349$	10873.2	1.66771	0.833854	+	0.551984i	$0.186129\pi$
$350$	0	0
$351$	0	0
$352$	7227.89i	1.09445i
$353$	2893.74i	0.436312i	0.975914	+	0.218156i	$0.0700041\pi$
$353$	2893.74i	0.436312i	−0.975914	+	0.218156i	$0.929996\pi$
$354$	0	0
$355$	0	0
$356$	1014.23	0.150995
$357$	0	0
$358$	6030.23i	0.890245i
$359$	9875.47	1.45183	0.725915	−	0.687784i	$-0.241416\pi$
$359$	9875.47	1.45183	0.725915	+	0.687784i	$0.241416\pi$
$360$	0	0
$361$	−6405.14	−0.933830
$362$	1726.17i	0.250624i
$363$	0	0
$364$	−1704.83	−0.245488
$365$	0	0
$366$	0	0
$367$	10722.2i	1.52505i	0.646961	+	0.762523i	$0.276040\pi$
$367$	10722.2i	1.52505i	−0.646961	+	0.762523i	$0.723960\pi$
$368$	3274.82i	0.463891i
$369$	0	0
$370$	0	0
$371$	−14322.5	−2.00428
$372$	0	0
$373$	− 4175.90i	− 0.579678i	−0.957075	−	0.289839i	$-0.906398\pi$
$373$	− 4175.90i	− 0.579678i	0.957075	−	0.289839i	$-0.0936018\pi$
$374$	−11267.9	−1.55789
$375$	0	0
$376$	−1484.01	−0.203543
$377$	3580.03i	0.489074i
$378$	0	0
$379$	−1715.14	−0.232457	−0.116228	−	0.993223i	$-0.537080\pi$
$379$	−1715.14	−0.232457	−0.116228	+	0.993223i	$0.537080\pi$
$380$	0	0
$381$	0	0
$382$	2441.77i	0.327046i
$383$	− 5139.06i	− 0.685624i	−0.939404	−	0.342812i	$-0.888621\pi$
$383$	− 5139.06i	− 0.685624i	0.939404	−	0.342812i	$-0.111379\pi$
$384$	0	0
$385$	0	0
$386$	−4573.14	−0.603022
$387$	0	0
$388$	691.074i	0.0904226i
$389$	−8996.53	−1.17260	−0.586301	−	0.810093i	$-0.699416\pi$
$389$	−8996.53	−1.17260	−0.586301	+	0.810093i	$0.699416\pi$
$390$	0	0
$391$	14385.3	1.86061
$392$	− 14631.8i	− 1.88525i
$393$	0	0
$394$	10752.9	1.37493
$395$	0	0
$396$	0	0
$397$	− 105.823i	− 0.0133781i	−0.999978	−	0.00668905i	$-0.997871\pi$
$397$	− 105.823i	− 0.0133781i	0.999978	−	0.00668905i	$-0.00212921\pi$
$398$	7556.48i	0.951688i
$399$	0	0
$400$	0	0
$401$	14481.8	1.80346	0.901730	−	0.432300i	$-0.142298\pi$
$401$	14481.8	1.80346	0.901730	+	0.432300i	$0.142298\pi$
$402$	0	0
$403$	− 3606.50i	− 0.445788i
$404$	2840.18	0.349763
$405$	0	0
$406$	−14642.0	−1.78983
$407$	− 20883.8i	− 2.54341i
$408$	0	0
$409$	861.065	0.104100	0.0520500	−	0.998644i	$-0.483424\pi$
$409$	861.065	0.104100	0.0520500	+	0.998644i	$0.483424\pi$
$410$	0	0
$411$	0	0
$412$	4879.72i	0.583511i
$413$	− 2929.52i	− 0.349037i
$414$	0	0
$415$	0	0
$416$	−2299.91	−0.271064
$417$	0	0
$418$	2271.75i	0.265825i
$419$	13017.7	1.51780	0.758900	−	0.651207i	$-0.225737\pi$
$419$	13017.7	1.51780	0.758900	+	0.651207i	$0.225737\pi$
$420$	0	0
$421$	14546.6	1.68398	0.841991	−	0.539492i	$-0.181384\pi$
$421$	14546.6	1.68398	0.841991	+	0.539492i	$0.181384\pi$
$422$	− 229.428i	− 0.0264654i
$423$	0	0
$424$	−11386.7	−1.30421
$425$	0	0
$426$	0	0
$427$	− 10965.3i	− 1.24273i
$428$	− 3406.53i	− 0.384722i
$429$	0	0
$430$	0	0
$431$	−3539.94	−0.395622	−0.197811	−	0.980240i	$-0.563383\pi$
$431$	−3539.94	−0.395622	−0.197811	+	0.980240i	$0.563383\pi$
$432$	0	0
$433$	− 669.471i	− 0.0743019i	−0.999310	−	0.0371509i	$-0.988172\pi$
$433$	− 669.471i	− 0.0743019i	0.999310	−	0.0371509i	$-0.0118282\pi$
$434$	14750.3	1.63142
$435$	0	0
$436$	−7387.09	−0.811416
$437$	− 2900.26i	− 0.317478i
$438$	0	0
$439$	−12568.5	−1.36643	−0.683216	−	0.730216i	$-0.739419\pi$
$439$	−12568.5	−1.36643	−0.683216	+	0.730216i	$0.739419\pi$
$440$	0	0
$441$	0	0
$442$	− 3585.45i	− 0.385843i
$443$	− 11060.3i	− 1.18621i	−0.805124	−	0.593106i	$-0.797901\pi$
$443$	− 11060.3i	− 1.18621i	0.805124	−	0.593106i	$-0.202099\pi$
$444$	0	0
$445$	0	0
$446$	−4128.98	−0.438370
$447$	0	0
$448$	− 15314.5i	− 1.61504i
$449$	−18553.9	−1.95014	−0.975072	−	0.221891i	$-0.928777\pi$
$449$	−18553.9	−1.95014	−0.975072	+	0.221891i	$0.928777\pi$
$450$	0	0
$451$	3818.84	0.398719
$452$	− 6245.21i	− 0.649889i
$453$	0	0
$454$	−7322.33	−0.756947
$455$	0	0
$456$	0	0
$457$	− 6802.26i	− 0.696272i	−0.937444	−	0.348136i	$-0.886815\pi$
$457$	− 6802.26i	− 0.696272i	0.937444	−	0.348136i	$-0.113185\pi$
$458$	− 4399.78i	− 0.448882i
$459$	0	0
$460$	0	0
$461$	14894.4	1.50477	0.752386	−	0.658722i	$-0.228903\pi$
$461$	14894.4	1.50477	0.752386	+	0.658722i	$0.228903\pi$
$462$	0	0
$463$	14288.7i	1.43423i	0.696952	+	0.717117i	$0.254539\pi$
$463$	14288.7i	1.43423i	−0.696952	+	0.717117i	$0.745461\pi$
$464$	−5396.17	−0.539895
$465$	0	0
$466$	−5887.74	−0.585288
$467$	13115.1i	1.29956i	0.760122	+	0.649780i	$0.225139\pi$
$467$	13115.1i	1.29956i	−0.760122	+	0.649780i	$0.774861\pi$
$468$	0	0
$469$	−2696.52	−0.265488
$470$	0	0
$471$	0	0
$472$	− 2329.01i	− 0.227122i
$473$	− 1591.77i	− 0.154735i
$474$	0	0
$475$	0	0
$476$	−11287.9	−1.08694
$477$	0	0
$478$	6258.96i	0.598908i
$479$	−7919.79	−0.755458	−0.377729	−	0.925916i	$-0.623295\pi$
$479$	−7919.79	−0.755458	−0.377729	+	0.925916i	$0.623295\pi$
$480$	0	0
$481$	6645.20	0.629927
$482$	9513.40i	0.899011i
$483$	0	0
$484$	4121.62	0.387079
$485$	0	0
$486$	0	0
$487$	− 17003.1i	− 1.58210i	−0.611751	−	0.791051i	$-0.709534\pi$
$487$	− 17003.1i	− 1.58210i	0.611751	−	0.791051i	$-0.290466\pi$
$488$	− 8717.56i	− 0.808659i
$489$	0	0
$490$	0	0
$491$	6391.74	0.587485	0.293743	−	0.955885i	$-0.405099\pi$
$491$	6391.74	0.587485	0.293743	+	0.955885i	$0.405099\pi$
$492$	0	0
$493$	23703.8i	2.16545i
$494$	−722.870	−0.0658370
$495$	0	0
$496$	5436.07	0.492111
$497$	12669.7i	1.14349i
$498$	0	0
$499$	2674.12	0.239900	0.119950	−	0.992780i	$-0.461727\pi$
$499$	2674.12	0.239900	0.119950	+	0.992780i	$0.461727\pi$
$500$	0	0
$501$	0	0
$502$	− 4510.36i	− 0.401011i
$503$	− 1263.13i	− 0.111969i	−0.998432	−	0.0559843i	$-0.982170\pi$
$503$	− 1263.13i	− 0.111969i	0.998432	−	0.0559843i	$-0.0178297\pi$
$504$	0	0
$505$	0	0
$506$	14517.0	1.27541
$507$	0	0
$508$	− 2605.86i	− 0.227592i
$509$	−1380.84	−0.120245	−0.0601226	−	0.998191i	$-0.519149\pi$
$509$	−1380.84	−0.120245	−0.0601226	+	0.998191i	$0.519149\pi$
$510$	0	0
$511$	−10165.8	−0.880055
$512$	− 8163.62i	− 0.704657i
$513$	0	0
$514$	−9670.09	−0.829824
$515$	0	0
$516$	0	0
$517$	3049.54i	0.259417i
$518$	27178.3i	2.30530i
$519$	0	0
$520$	0	0
$521$	2689.80	0.226185	0.113092	−	0.993584i	$-0.463924\pi$
$521$	2689.80	0.226185	0.113092	+	0.993584i	$0.463924\pi$
$522$	0	0
$523$	− 7144.18i	− 0.597310i	−0.954361	−	0.298655i	$-0.903462\pi$
$523$	− 7144.18i	− 0.597310i	0.954361	−	0.298655i	$-0.0965380\pi$
$524$	−8337.54	−0.695090
$525$	0	0
$526$	−7441.07	−0.616818
$527$	− 23879.1i	− 1.97379i
$528$	0	0
$529$	−6366.25	−0.523239
$530$	0	0
$531$	0	0
$532$	2275.78i	0.185466i
$533$	1215.15i	0.0987508i
$534$	0	0
$535$	0	0
$536$	−2143.78	−0.172756
$537$	0	0
$538$	− 5515.68i	− 0.442004i
$539$	−30067.4	−2.40278
$540$	0	0
$541$	−3310.57	−0.263091	−0.131546	−	0.991310i	$-0.541994\pi$
$541$	−3310.57	−0.263091	−0.131546	+	0.991310i	$0.541994\pi$
$542$	18110.7i	1.43528i
$543$	0	0
$544$	−15228.0	−1.20018
$545$	0	0
$546$	0	0
$547$	2286.52i	0.178729i	0.995999	+	0.0893643i	$0.0284835\pi$
$547$	2286.52i	0.178729i	−0.995999	+	0.0893643i	$0.971516\pi$
$548$	3495.19i	0.272458i
$549$	0	0
$550$	0	0
$551$	4778.98	0.369494
$552$	0	0
$553$	7617.39i	0.585758i
$554$	4019.42	0.308247
$555$	0	0
$556$	−8361.60	−0.637789
$557$	− 13846.6i	− 1.05332i	−0.850077	−	0.526658i	$-0.823445\pi$
$557$	− 13846.6i	− 1.05332i	0.850077	−	0.526658i	$-0.176555\pi$
$558$	0	0
$559$	506.502	0.0383233
$560$	0	0
$561$	0	0
$562$	6673.21i	0.500876i
$563$	− 16164.4i	− 1.21003i	−0.796213	−	0.605017i	$-0.793166\pi$
$563$	− 16164.4i	− 1.21003i	0.796213	−	0.605017i	$-0.206834\pi$
$564$	0	0
$565$	0	0
$566$	8651.37	0.642481
$567$	0	0
$568$	10072.6i	0.744080i
$569$	496.334	0.0365684	0.0182842	−	0.999833i	$-0.494180\pi$
$569$	496.334	0.0365684	0.0182842	+	0.999833i	$0.494180\pi$
$570$	0	0
$571$	5971.18	0.437629	0.218815	−	0.975766i	$-0.429781\pi$
$571$	5971.18	0.437629	0.218815	+	0.975766i	$0.429781\pi$
$572$	2785.20i	0.203592i
$573$	0	0
$574$	−4969.88	−0.361392
$575$	0	0
$576$	0	0
$577$	9571.82i	0.690607i	0.938491	+	0.345304i	$0.112224\pi$
$577$	9571.82i	0.690607i	−0.938491	+	0.345304i	$0.887776\pi$
$578$	− 13294.1i	− 0.956684i
$579$	0	0
$580$	0	0
$581$	−16961.9	−1.21119
$582$	0	0
$583$	23398.8i	1.66223i
$584$	−8081.97	−0.572662
$585$	0	0
$586$	7734.71	0.545253
$587$	− 5322.18i	− 0.374225i	−0.982339	−	0.187112i	$-0.940087\pi$
$587$	− 5322.18i	− 0.374225i	0.982339	−	0.187112i	$-0.0599128\pi$
$588$	0	0
$589$	−4814.31	−0.336791
$590$	0	0
$591$	0	0
$592$	10016.3i	0.695385i
$593$	− 11066.5i	− 0.766354i	−0.923675	−	0.383177i	$-0.874830\pi$
$593$	− 11066.5i	− 0.766354i	0.923675	−	0.383177i	$-0.125170\pi$
$594$	0	0
$595$	0	0
$596$	−1773.38	−0.121880
$597$	0	0
$598$	4619.29i	0.315881i
$599$	18385.0	1.25407	0.627037	−	0.778989i	$-0.284267\pi$
$599$	18385.0	1.25407	0.627037	+	0.778989i	$0.284267\pi$
$600$	0	0
$601$	−617.830	−0.0419331	−0.0209666	−	0.999780i	$-0.506674\pi$
$601$	−617.830	−0.0419331	−0.0209666	+	0.999780i	$0.506674\pi$
$602$	2071.55i	0.140249i
$603$	0	0
$604$	5022.12	0.338323
$605$	0	0
$606$	0	0
$607$	13839.8i	0.925436i	0.886506	+	0.462718i	$0.153126\pi$
$607$	13839.8i	0.925436i	−0.886506	+	0.462718i	$0.846874\pi$
$608$	3070.15i	0.204788i
$609$	0	0
$610$	0	0
$611$	−970.362	−0.0642498
$612$	0	0
$613$	25655.4i	1.69039i	0.534457	+	0.845196i	$0.320516\pi$
$613$	25655.4i	1.69039i	−0.534457	+	0.845196i	$0.679484\pi$
$614$	14519.4	0.954326
$615$	0	0
$616$	−37580.8	−2.45808
$617$	− 4218.26i	− 0.275236i	−0.990485	−	0.137618i	$-0.956055\pi$
$617$	− 4218.26i	− 0.275236i	0.990485	−	0.137618i	$-0.0439446\pi$
$618$	0	0
$619$	−12182.6	−0.791047	−0.395524	−	0.918456i	$-0.629437\pi$
$619$	−12182.6	−0.791047	−0.395524	+	0.918456i	$0.629437\pi$
$620$	0	0
$621$	0	0
$622$	14034.6i	0.904719i
$623$	8948.40i	0.575458i
$624$	0	0
$625$	0	0
$626$	5882.10	0.375553
$627$	0	0
$628$	− 7500.80i	− 0.476616i
$629$	43998.8	2.78910
$630$	0	0
$631$	16673.2	1.05190	0.525951	−	0.850515i	$-0.323709\pi$
$631$	16673.2	1.05190	0.525951	+	0.850515i	$0.323709\pi$
$632$	6055.95i	0.381159i
$633$	0	0
$634$	−9704.48	−0.607909
$635$	0	0
$636$	0	0
$637$	− 9567.44i	− 0.595096i
$638$	23920.7i	1.48437i
$639$	0	0
$640$	0	0
$641$	−26270.8	−1.61878	−0.809388	−	0.587275i	$-0.800201\pi$
$641$	−26270.8	−1.61878	−0.809388	+	0.587275i	$0.800201\pi$
$642$	0	0
$643$	− 11556.3i	− 0.708767i	−0.935100	−	0.354384i	$-0.884691\pi$
$643$	− 11556.3i	− 0.708767i	0.935100	−	0.354384i	$-0.115309\pi$
$644$	14542.7	0.889851
$645$	0	0
$646$	−4786.22	−0.291504
$647$	25395.2i	1.54310i	0.636166	+	0.771552i	$0.280519\pi$
$647$	25395.2i	1.54310i	−0.636166	+	0.771552i	$0.719481\pi$
$648$	0	0
$649$	−4785.96	−0.289469
$650$	0	0
$651$	0	0
$652$	1843.88i	0.110754i
$653$	− 6182.84i	− 0.370526i	−0.982689	−	0.185263i	$-0.940686\pi$
$653$	− 6182.84i	− 0.370526i	0.982689	−	0.185263i	$-0.0593137\pi$
$654$	0	0
$655$	0	0
$656$	−1831.60	−0.109012
$657$	0	0
$658$	− 3968.70i	− 0.235131i
$659$	−8885.32	−0.525224	−0.262612	−	0.964901i	$-0.584584\pi$
$659$	−8885.32	−0.525224	−0.262612	+	0.964901i	$0.584584\pi$
$660$	0	0
$661$	−7171.60	−0.422001	−0.211001	−	0.977486i	$-0.567672\pi$
$661$	−7171.60	−0.422001	−0.211001	+	0.977486i	$0.567672\pi$
$662$	− 9919.66i	− 0.582385i
$663$	0	0
$664$	−13485.0	−0.788133
$665$	0	0
$666$	0	0
$667$	− 30538.7i	− 1.77281i
$668$	− 10366.7i	− 0.600451i
$669$	0	0
$670$	0	0
$671$	−17914.0	−1.03064
$672$	0	0
$673$	4250.52i	0.243455i	0.992564	+	0.121728i	$0.0388435\pi$
$673$	4250.52i	0.243455i	−0.992564	+	0.121728i	$0.961157\pi$
$674$	8094.27	0.462581
$675$	0	0
$676$	6758.47	0.384528
$677$	− 2233.65i	− 0.126804i	−0.997988	−	0.0634019i	$-0.979805\pi$
$677$	− 2233.65i	− 0.126804i	0.997988	−	0.0634019i	$-0.0201950\pi$
$678$	0	0
$679$	−6097.23	−0.344610
$680$	0	0
$681$	0	0
$682$	− 24097.6i	− 1.35300i
$683$	− 6071.62i	− 0.340153i	−0.985431	−	0.170076i	$-0.945599\pi$
$683$	− 6071.62i	− 0.340153i	0.985431	−	0.170076i	$-0.0544014\pi$
$684$	0	0
$685$	0	0
$686$	16741.8	0.931784
$687$	0	0
$688$	763.450i	0.0423056i
$689$	−7445.49	−0.411684
$690$	0	0
$691$	−765.784	−0.0421589	−0.0210794	−	0.999778i	$-0.506710\pi$
$691$	−765.784	−0.0421589	−0.0210794	+	0.999778i	$0.506710\pi$
$692$	− 6193.29i	− 0.340222i
$693$	0	0
$694$	−4914.02	−0.268780
$695$	0	0
$696$	0	0
$697$	8045.70i	0.437235i
$698$	23117.8i	1.25361i
$699$	0	0
$700$	0	0
$701$	−23564.3	−1.26963	−0.634814	−	0.772665i	$-0.718923\pi$
$701$	−23564.3	−1.26963	−0.634814	+	0.772665i	$0.718923\pi$
$702$	0	0
$703$	− 8870.67i	− 0.475909i
$704$	−25019.3	−1.33942
$705$	0	0
$706$	−6152.43	−0.327974
$707$	25058.4i	1.33298i
$708$	0	0
$709$	−24172.3	−1.28041	−0.640204	−	0.768205i	$-0.721150\pi$
$709$	−24172.3	−1.28041	−0.640204	+	0.768205i	$0.721150\pi$
$710$	0	0
$711$	0	0
$712$	7114.13i	0.374457i
$713$	30764.5i	1.61590i
$714$	0	0
$715$	0	0
$716$	9869.11	0.515120
$717$	0	0
$718$	20996.4i	1.09134i
$719$	12829.0	0.665424	0.332712	−	0.943028i	$-0.392036\pi$
$719$	12829.0	0.665424	0.332712	+	0.943028i	$0.392036\pi$
$720$	0	0
$721$	−43052.9	−2.22382
$722$	− 13618.1i	− 0.701957i
$723$	0	0
$724$	2825.06	0.145018
$725$	0	0
$726$	0	0
$727$	− 24724.7i	− 1.26133i	−0.776054	−	0.630666i	$-0.782782\pi$
$727$	− 24724.7i	− 1.26133i	0.776054	−	0.630666i	$-0.217218\pi$
$728$	− 11958.2i	− 0.608792i
$729$	0	0
$730$	0	0
$731$	3353.62	0.169683
$732$	0	0
$733$	20172.2i	1.01648i	0.861216	+	0.508239i	$0.169703\pi$
$733$	20172.2i	1.01648i	−0.861216	+	0.508239i	$0.830297\pi$
$734$	−22796.6	−1.14637
$735$	0	0
$736$	19618.9	0.982559
$737$	4405.32i	0.220179i
$738$	0	0
$739$	16452.8	0.818979	0.409489	−	0.912315i	$-0.365707\pi$
$739$	16452.8	0.818979	0.409489	+	0.912315i	$0.365707\pi$
$740$	0	0
$741$	0	0
$742$	− 30451.4i	− 1.50661i
$743$	20864.6i	1.03021i	0.857126	+	0.515107i	$0.172248\pi$
$743$	20864.6i	1.03021i	−0.857126	+	0.515107i	$0.827752\pi$
$744$	0	0
$745$	0	0
$746$	8878.47	0.435742
$747$	0	0
$748$	18441.2i	0.901438i
$749$	30055.2	1.46621
$750$	0	0
$751$	15525.7	0.754381	0.377191	−	0.926136i	$-0.376890\pi$
$751$	15525.7	0.754381	0.377191	+	0.926136i	$0.376890\pi$
$752$	− 1462.63i	− 0.0709262i
$753$	0	0
$754$	−7611.56	−0.367635
$755$	0	0
$756$	0	0
$757$	30105.7i	1.44546i	0.691132	+	0.722729i	$0.257112\pi$
$757$	30105.7i	1.44546i	−0.691132	+	0.722729i	$0.742888\pi$
$758$	− 3646.60i	− 0.174737i
$759$	0	0
$760$	0	0
$761$	21739.9	1.03557	0.517786	−	0.855510i	$-0.326756\pi$
$761$	21739.9	1.03557	0.517786	+	0.855510i	$0.326756\pi$
$762$	0	0
$763$	− 65175.0i	− 3.09239i
$764$	3996.21	0.189238
$765$	0	0
$766$	10926.3	0.515381
$767$	− 1522.89i	− 0.0716929i
$768$	0	0
$769$	−1942.22	−0.0910772	−0.0455386	−	0.998963i	$-0.514500\pi$
$769$	−1942.22	−0.0910772	−0.0455386	+	0.998963i	$0.514500\pi$
$770$	0	0
$771$	0	0
$772$	7484.42i	0.348925i
$773$	7921.02i	0.368563i	0.982873	+	0.184282i	$0.0589958\pi$
$773$	7921.02i	0.368563i	−0.982873	+	0.184282i	$0.941004\pi$
$774$	0	0
$775$	0	0
$776$	−4847.40	−0.224241
$777$	0	0
$778$	− 19127.7i	− 0.881441i
$779$	1622.11	0.0746060
$780$	0	0
$781$	20698.5	0.948338
$782$	30585.0i	1.39861i
$783$	0	0
$784$	14421.0	0.656933
$785$	0	0
$786$	0	0
$787$	− 1361.40i	− 0.0616630i	−0.999525	−	0.0308315i	$-0.990184\pi$
$787$	− 1361.40i	− 0.0616630i	0.999525	−	0.0308315i	$-0.00981553\pi$
$788$	− 17598.2i	− 0.795571i
$789$	0	0
$790$	0	0
$791$	55100.4	2.47679
$792$	0	0
$793$	− 5700.22i	− 0.255260i
$794$	224.992	0.0100563
$795$	0	0
$796$	12367.0	0.550672
$797$	27917.5i	1.24076i	0.784300	+	0.620381i	$0.213022\pi$
$797$	27917.5i	1.24076i	−0.784300	+	0.620381i	$0.786978\pi$
$798$	0	0
$799$	−6424.90	−0.284476
$800$	0	0
$801$	0	0
$802$	30790.1i	1.35565i
$803$	16607.9i	0.729863i
$804$	0	0
$805$	0	0
$806$	7667.84	0.335097
$807$	0	0
$808$	19921.9i	0.867388i
$809$	−5275.87	−0.229283	−0.114641	−	0.993407i	$-0.536572\pi$
$809$	−5275.87	−0.229283	−0.114641	+	0.993407i	$0.536572\pi$
$810$	0	0
$811$	−23769.1	−1.02916	−0.514579	−	0.857443i	$-0.672052\pi$
$811$	−23769.1	−1.02916	−0.514579	+	0.857443i	$0.672052\pi$
$812$	23963.2i	1.03564i
$813$	0	0
$814$	44401.4	1.91188
$815$	0	0
$816$	0	0
$817$	− 676.129i	− 0.0289532i
$818$	1830.73i	0.0782516i
$819$	0	0
$820$	0	0
$821$	−34209.3	−1.45422	−0.727108	−	0.686523i	$-0.759136\pi$
$821$	−34209.3	−1.45422	−0.727108	+	0.686523i	$0.759136\pi$
$822$	0	0
$823$	− 1240.13i	− 0.0525252i	−0.999655	−	0.0262626i	$-0.991639\pi$
$823$	− 1240.13i	− 0.0525252i	0.999655	−	0.0262626i	$-0.00836061\pi$
$824$	−34227.8	−1.44706
$825$	0	0
$826$	6228.50	0.262370
$827$	− 26971.0i	− 1.13407i	−0.823694	−	0.567034i	$-0.808091\pi$
$827$	− 26971.0i	− 1.13407i	0.823694	−	0.567034i	$-0.191909\pi$
$828$	0	0
$829$	26743.1	1.12042	0.560208	−	0.828352i	$-0.310721\pi$
$829$	26743.1	1.12042	0.560208	+	0.828352i	$0.310721\pi$
$830$	0	0
$831$	0	0
$832$	− 7961.13i	− 0.331734i
$833$	− 63347.3i	− 2.63488i
$834$	0	0
$835$	0	0
$836$	3717.96	0.153814
$837$	0	0
$838$	27677.3i	1.14093i
$839$	5273.14	0.216983	0.108492	−	0.994097i	$-0.465398\pi$
$839$	5273.14	0.216983	0.108492	+	0.994097i	$0.465398\pi$
$840$	0	0
$841$	25931.9	1.06326
$842$	30927.7i	1.26584i
$843$	0	0
$844$	−375.483	−0.0153136
$845$	0	0
$846$	0	0
$847$	36364.3i	1.47520i
$848$	− 11222.6i	− 0.454464i
$849$	0	0
$850$	0	0
$851$	−56685.5	−2.28338
$852$	0	0
$853$	− 5043.46i	− 0.202444i	−0.994864	−	0.101222i	$-0.967725\pi$
$853$	− 5043.46i	− 0.202444i	0.994864	−	0.101222i	$-0.0322752\pi$
$854$	23313.5	0.934157
$855$	0	0
$856$	23894.4	0.954081
$857$	− 14478.5i	− 0.577103i	−0.957464	−	0.288552i	$-0.906826\pi$
$857$	− 14478.5i	− 0.577103i	0.957464	−	0.288552i	$-0.0931737\pi$
$858$	0	0
$859$	7873.88	0.312751	0.156376	−	0.987698i	$-0.450019\pi$
$859$	7873.88	0.312751	0.156376	+	0.987698i	$0.450019\pi$
$860$	0	0
$861$	0	0
$862$	− 7526.35i	− 0.297388i
$863$	− 24166.0i	− 0.953209i	−0.879118	−	0.476604i	$-0.841867\pi$
$863$	− 24166.0i	− 0.953209i	0.879118	−	0.476604i	$-0.158133\pi$
$864$	0	0
$865$	0	0
$866$	1423.38	0.0558525
$867$	0	0
$868$	− 24140.4i	− 0.943983i
$869$	12444.6	0.485792
$870$	0	0
$871$	−1401.77	−0.0545317
$872$	− 51815.2i	− 2.01225i
$873$	0	0
$874$	6166.29	0.238647
$875$	0	0
$876$	0	0
$877$	6411.82i	0.246878i	0.992352	+	0.123439i	$0.0393923\pi$
$877$	6411.82i	0.246878i	−0.992352	+	0.123439i	$0.960608\pi$
$878$	− 26722.2i	− 1.02714i
$879$	0	0
$880$	0	0
$881$	22908.5	0.876059	0.438029	−	0.898961i	$-0.355677\pi$
$881$	22908.5	0.876059	0.438029	+	0.898961i	$0.355677\pi$
$882$	0	0
$883$	− 39198.4i	− 1.49392i	−0.664869	−	0.746960i	$-0.731513\pi$
$883$	− 39198.4i	− 1.49392i	0.664869	−	0.746960i	$-0.268487\pi$
$884$	−5867.97	−0.223259
$885$	0	0
$886$	23515.6	0.891672
$887$	5422.19i	0.205253i	0.994720	+	0.102626i	$0.0327246\pi$
$887$	5422.19i	0.205253i	−0.994720	+	0.102626i	$0.967275\pi$
$888$	0	0
$889$	22991.1	0.867375
$890$	0	0
$891$	0	0
$892$	6757.51i	0.253653i
$893$	1295.34i	0.0485406i
$894$	0	0
$895$	0	0
$896$	−2833.53	−0.105649
$897$	0	0
$898$	− 39447.9i	− 1.46592i
$899$	−50693.0	−1.88065
$900$	0	0
$901$	−49297.6	−1.82280
$902$	8119.32i	0.299716i
$903$	0	0
$904$	43805.7	1.61168
$905$	0	0
$906$	0	0
$907$	− 30727.0i	− 1.12489i	−0.826836	−	0.562444i	$-0.809861\pi$
$907$	− 30727.0i	− 1.12489i	0.826836	−	0.562444i	$-0.190139\pi$
$908$	11983.8i	0.437990i
$909$	0	0
$910$	0	0
$911$	4101.34	0.149159	0.0745793	−	0.997215i	$-0.476239\pi$
$911$	4101.34	0.149159	0.0745793	+	0.997215i	$0.476239\pi$
$912$	0	0
$913$	27710.8i	1.00448i
$914$	14462.4	0.523385
$915$	0	0
$916$	−7200.69	−0.259735
$917$	− 73560.7i	− 2.64906i
$918$	0	0
$919$	50926.3	1.82797	0.913984	−	0.405750i	$-0.132990\pi$
$919$	50926.3	1.82797	0.913984	+	0.405750i	$0.132990\pi$
$920$	0	0
$921$	0	0
$922$	31667.2i	1.13113i
$923$	6586.26i	0.234875i
$924$	0	0
$925$	0	0
$926$	−30379.4	−1.07811
$927$	0	0
$928$	32327.6i	1.14354i
$929$	−22040.8	−0.778402	−0.389201	−	0.921153i	$-0.627249\pi$
$929$	−22040.8	−0.778402	−0.389201	+	0.921153i	$0.627249\pi$
$930$	0	0
$931$	−12771.6	−0.449593
$932$	9635.89i	0.338663i
$933$	0	0
$934$	−27884.3	−0.976875
$935$	0	0
$936$	0	0
$937$	28111.2i	0.980097i	0.871695	+	0.490049i	$0.163021\pi$
$937$	28111.2i	0.980097i	−0.871695	+	0.490049i	$0.836979\pi$
$938$	− 5733.12i	− 0.199566i
$939$	0	0
$940$	0	0
$941$	52811.3	1.82954	0.914770	−	0.403974i	$-0.132371\pi$
$941$	52811.3	1.82954	0.914770	+	0.403974i	$0.132371\pi$
$942$	0	0
$943$	− 10365.6i	− 0.357954i
$944$	2295.45	0.0791427
$945$	0	0
$946$	3384.30	0.116314
$947$	− 1826.23i	− 0.0626657i	−0.999509	−	0.0313329i	$-0.990025\pi$
$947$	− 1826.23i	− 0.0626657i	0.999509	−	0.0313329i	$-0.00997519\pi$
$948$	0	0
$949$	−5284.63	−0.180765
$950$	0	0
$951$	0	0
$952$	− 79176.9i	− 2.69552i
$953$	41164.2i	1.39920i	0.714534	+	0.699601i	$0.246639\pi$
$953$	41164.2i	1.39920i	−0.714534	+	0.699601i	$0.753361\pi$
$954$	0	0
$955$	0	0
$956$	10243.4	0.346545
$957$	0	0
$958$	− 16838.4i	− 0.567875i
$959$	−30837.4	−1.03837
$960$	0	0
$961$	21276.8	0.714202
$962$	14128.5i	0.473514i
$963$	0	0
$964$	15569.7	0.520192
$965$	0	0
$966$	0	0
$967$	− 7088.78i	− 0.235739i	−0.993029	−	0.117870i	$-0.962394\pi$
$967$	− 7088.78i	− 0.235739i	0.993029	−	0.117870i	$-0.0376065\pi$
$968$	28910.2i	0.959927i
$969$	0	0
$970$	0	0
$971$	2355.90	0.0778625	0.0389312	−	0.999242i	$-0.487605\pi$
$971$	2355.90	0.0778625	0.0389312	+	0.999242i	$0.487605\pi$
$972$	0	0
$973$	− 73773.0i	− 2.43068i
$974$	36150.6	1.18926
$975$	0	0
$976$	8591.95	0.281784
$977$	− 16795.1i	− 0.549972i	−0.961448	−	0.274986i	$-0.911327\pi$
$977$	− 16795.1i	− 0.549972i	0.961448	−	0.274986i	$-0.0886732\pi$
$978$	0	0
$979$	14619.0	0.477249
$980$	0	0
$981$	0	0
$982$	13589.6i	0.441611i
$983$	14733.8i	0.478063i	0.971012	+	0.239032i	$0.0768300\pi$
$983$	14733.8i	0.478063i	−0.971012	+	0.239032i	$0.923170\pi$
$984$	0	0
$985$	0	0
$986$	−50397.2	−1.62776
$987$	0	0
$988$	1183.05i	0.0380950i
$989$	−4320.61	−0.138915
$990$	0	0
$991$	36809.2	1.17990	0.589950	−	0.807440i	$-0.299147\pi$
$991$	36809.2	1.17990	0.589950	+	0.807440i	$0.299147\pi$
$992$	− 32566.6i	− 1.04233i
$993$	0	0
$994$	−26937.3	−0.859556
$995$	0	0
$996$	0	0
$997$	− 21210.0i	− 0.673749i	−0.941550	−	0.336875i	$-0.890630\pi$
$997$	− 21210.0i	− 0.673749i	0.941550	−	0.336875i	$-0.109370\pi$
$998$	5685.49i	0.180332i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	675.4.b.n.649.4		6
3.2	odd	2		675.4.b.m.649.3		6
5.2	odd	4		675.4.a.p.1.2		3
5.3	odd	4		135.4.a.h.1.2	yes	3
5.4	even	2	inner	675.4.b.n.649.3		6
15.2	even	4		675.4.a.s.1.2		3
15.8	even	4		135.4.a.e.1.2	✓	3
15.14	odd	2		675.4.b.m.649.4		6
20.3	even	4		2160.4.a.bq.1.1		3
45.13	odd	12		405.4.e.q.136.2		6
45.23	even	12		405.4.e.v.136.2		6
45.38	even	12		405.4.e.v.271.2		6
45.43	odd	12		405.4.e.q.271.2		6
60.23	odd	4		2160.4.a.bi.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.4.a.e.1.2	✓	3	15.8	even	4
135.4.a.h.1.2	yes	3	5.3	odd	4
405.4.e.q.136.2		6	45.13	odd	12
405.4.e.q.271.2		6	45.43	odd	12
405.4.e.v.136.2		6	45.23	even	12
405.4.e.v.271.2		6	45.38	even	12
675.4.a.p.1.2		3	5.2	odd	4
675.4.a.s.1.2		3	15.2	even	4
675.4.b.m.649.3		6	3.2	odd	2
675.4.b.m.649.4		6	15.14	odd	2
675.4.b.n.649.3		6	5.4	even	2	inner
675.4.b.n.649.4		6	1.1	even	1	trivial
2160.4.a.bi.1.1		3	60.23	odd	4
2160.4.a.bq.1.1		3	20.3	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:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	675.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.12612i q^{2} +3.47962 q^{4} +30.7000i q^{7} +24.4070i q^{8} +O(q^{10})\)	\(q+2.12612i q^{2} +3.47962 q^{4} +30.7000i q^{7} +24.4070i q^{8} +50.1548 q^{11} +15.9592i q^{13} -65.2720 q^{14} -24.0553 q^{16} +105.668i q^{17} +21.3040 q^{19} +106.635i q^{22} -136.137i q^{23} -33.9312 q^{26} +106.824i q^{28} +224.323 q^{29} -225.982 q^{31} +144.112i q^{32} -224.663 q^{34} -416.386i q^{37} +45.2948i q^{38} +76.1411 q^{41} -31.7372i q^{43} +174.519 q^{44} +289.443 q^{46} +60.8026i q^{47} -599.493 q^{49} +55.5320i q^{52} +466.532i q^{53} -749.297 q^{56} +476.938i q^{58} -95.4239 q^{59} -357.174 q^{61} -480.464i q^{62} -498.842 q^{64} +87.8344i q^{67} +367.685i q^{68} +412.693 q^{71} +331.133i q^{73} +885.286 q^{74} +74.1296 q^{76} +1539.75i q^{77} +248.123 q^{79} +161.885i q^{82} +552.505i q^{83} +67.4771 q^{86} +1224.13i q^{88} +291.478 q^{89} -489.949 q^{91} -473.704i q^{92} -129.274 q^{94} +198.606i q^{97} -1274.59i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 34 q^{4}+O(q^{10}) \) 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

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Database

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